Efficient Multi-Carrier Communication on the Digital Subscriber Loop

Efﬁcient Multi-Carrier Communication on the Digital Subscriber Loop

Donnacha Daly

Department of Electrical and Electronic Engineering Faculty of Engineering and Architecture University College Dublin National University of Ireland

A thesis presented to the National University of Ireland Faculty of Engineering and Architecture in fulﬁllment of the requirements for the degree of Doctor of Philosophy

May 2003

Research Supervisors: Dr. C. Heneghan Prof. A. D. Fagan Head of Department: Prof. T. Brazil Abstract

This thesis explores three distinct philosophies for improving the efficiency of multi-carrier communication on the digital subscriber loop. The first topic discussed is impulse response shortening for discrete multitone transceivers. The minimum mean-squared error impulse response shortener is reformulated to allow near-optimal rate performance. It is demonstrated that the best existing eigen-filter designed channel shortener is a particular case of the proposed reformulation. An adaptive time-domain LMS algorithm is provided as an alternative to eigen-decomposition. The next part of the thesis examines bit- and power- loading algorithms for multitone systems. The problem of rate-optimal loading has already been solved. It is shown, however, that the rate-optimal solution does not give best value for complexity, and that near optimal schemes can perform very well at a fraction of the computational cost. The final section of the thesis is a brief exposition of the use of wavelet packets to achieve multi-carrier communication. It is proposed that the non-uniform spectral decomposition afforded by wavelet packet modulation allows reduced inter-symbol interference effects in a dispersive channel.

Associated Publications

D. Daly , C. Heneghan and A.D. Fagan, “Minimum mean-squared error impulse response shortening for discrete multitone transceivers”, accepted for publication, IEEE Trans. Signal Process..

D. Daly , C. Heneghan and A.D. Fagan, “Power- and bit- loading algorithms for multitone communications”, accepted for IEEE Int. Symp. Image, Signal Process., Analysis, Rome, Italy, Sep. 2003.

D. Daly , C. Heneghan and A.D. Fagan, “Optimal wavelet packet modulation”, in Proc. Irish Signals, Syst. Conf., Cork, pp. 47-52, June 2002.

D. Daly , C. Heneghan, A.D. Fagan and M.Vetterli, “Optimal wavelet packet modulation under ﬁnite complexity constraint”, in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., Orlando FLA, Vol III, pp. 2789-2792, May 2002.

D. Daly , C. Heneghan, and A.D. Fagan, “A minimum mean-squared error interpretation of residual ISI channel shortening for discrete multitone transceivers” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., Salt Lake City, UT, May 2001. Acknowledgment

I would primarily like to thank my family, that is my father Peter, my mother Marian and my brother Peter, who have been so supportive of me through the years. I would also like to express gratitude to those who have guided my thought both towards stimulating subjects and away from inevitible dead ends, over the course of my research. In particular, my supervisors Conor and Tony have, in their wisdom, steered me through a well focussed postgraduate career. I am grateful to Prof. Martin Vetterli of the Swiss Federal Institute of Technology, under whom it was a true inspiration to work. As I am notoriously pedantic (and vocal) when it comes to chewing over the intricacies of a particular problem, my gratitude is further extended to those who have had the patience to pick bones with me, whether it was at the whiteboard at two o’clock in the morning, or over a pint in Captain Cook’s in Lausanne. In no particular order, my eager co-conversants were Brian Clerkin, David Naughton, Ger Baldwin, Finbarr O’Regan, Mark Flannagan, Mark Herro, Damien Piguet, Julius Kusuma and Andrea Ridolﬁ. Finally, it remains to mention those in Mas- sana and Advanced Communication Networks with whom I had the pleasure of cooperating, much to the beneﬁt of my signal processing knowhow. Again in no order, I would like to thank Brian Murray, Philip Curran, Ed Lalor, Ciaran McElroy, Alan Harnedy, Carl Murray, Albert Molina, Stephan Horvath and Antony Jamin. This work was sponsered by Science Foundation Ireland.

For Dad

I wandered out in the world for years You just stayed in your room I saw the crescent You saw the whole of the moon

—Mike Scott, 1985

ii Contents

1 Introduction 1

1.1 Evolution of the Digital Subscriber Loop ...... 2

1.1.1 Voice Band Modems ...... 2

1.1.2 ISDN ...... 4

1.1.3 xDSL ...... 4

1.2 Multitone Communications — Literature Survey ...... 6

1.2.1 The Filterbank Transceiver ...... 6

1.2.2 Block Transforms — DMT ...... 8

1.2.3 Lapped Transforms — DWMT ...... 9

1.2.4 Over-Interpolated Filterbanks ...... 11

1.2.5 Near-Perfect Reconstruction ...... 12

1.2.6 Nonuniform Filterbanks ...... 12

1.2.7 Equalization ...... 15

1.2.8 Echo Cancellation ...... 16

1.2.9 Timing and Synchronization ...... 17

1.2.10 Peak-to-Average Power Ratio ...... 19

1.2.11 Bitstream Operations: Coding, Scrambling, and Interleaving ...... 20

1.3 Thesis Outline ...... 21

iii CONTENTS

2 DMT Communication on the Digital Subscriber Loop 22

2.1 DSL Channel Modeling ...... 22

2.1.1 The ADSL Test Loops ...... 23

2.1.2 The VDSL Test Loops ...... 25

2.1.3 Other DSL Elements ...... 25

2.1.4 Primary Line Constants: R, L, G, C ...... 26

2.1.5 Secondary Line Constants: Z0, γp ...... 28

2.1.6 ABCD Matrices ...... 31

2.1.7 Loop Transfer Function ...... 34

2.1.8 DAC, Anti-Aliasing Filters and Splitter ...... 34

2.1.9 Loop Impulse Response ...... 36

2.1.10 Noise, Crosstalk and Radio Frequency Interference ...... 36

2.2 Discrete Multitone Modulation ...... 42

2.2.1 DMT by Fast Fourier Transform ...... 42

2.2.2 Guard Band Insertion ...... 44

2.2.3 Cyclic Preﬁx vs. Guard Band ...... 46

2.2.4 Channel Estimation ...... 49

2.3 Chapter Summary ...... 49

3 Time Domain Equalization 50

3.1 Channel Shortening: State of The Art ...... 50

3.1.1 Minimum Mean Squared Error Impulse Response Shortening ...... 52

3.1.2 The Maximum Shortening Signal to Noise Ratio TEQ ...... 55

3.1.3 Maximum Bit Rate Impulse Response Shortening ...... 57

3.1.4 The Minimum Intersymbol Interference TEQ ...... 60

3.1.5 Summary of Existing Channel Shortening Algorithms ...... 61

iv CONTENTS

3.2 The DIR-optimized MMSE TEQ ...... 63

3.2.1 Special Case I: The min-ISI TEQ ...... 64

3.2.2 Special Case II: The MSSNR TEQ ...... 65

3.2.3 Special Case III: High SNR ...... 66

3.2.4 Adaptive LMS Implementation ...... 66

3.3 Simulation Results ...... 69

3.3.1 Simulation Setup ...... 69

3.3.2 Figures of Merit ...... 69

3.3.3 Performance Results ...... 70

3.3.4 Adaptive Implementation ...... 74

3.4 Chapter Summary ...... 78

4 Power- and Bit-Loading 79

4.1 Water-Pouring ...... 80

4.1.1 Power-Loading ...... 81

4.1.2 Bit-Loading ...... 83

4.2 Modulation Schemes ...... 85

4.2.1 Pulse Amplitude Modulation ...... 85

4.2.2 Quadrature Amplitude Modulation ...... 88

4.2.3 A Note on the Modulation Gap ...... 92

4.3 Loading Algorithms ...... 95

4.3.1 Optimal Loading ...... 95

4.3.2 Near-Optimal Loading ...... 100

4.3.3 Default Loading ...... 102

4.3.4 Constant Power Bit-Loading ...... 103

4.4 Simulation Results ...... 104

4.4.1 Willink’s Test Channel ...... 105

4.4.2 Loading for the DSL channels ...... 111

4.5 Chapter Summary ...... 116

v CONTENTS

5 Wavelet Packet Modulation 117

5.1 Introduction ...... 118

5.1.1 The Two-Channel Transmultiplexer ...... 118

5.1.2 Polyphase Analysis ...... 119

5.1.3 Perfect Reconstruction ...... 124

5.1.4 Worked Example ...... 126

5.1.5 Wavelet Packet Trees ...... 126

5.1.6 Equivalent Branch Filters ...... 129

5.1.7 Wavelet Packet Modulation: State of the Art ...... 132

5.2 Optimal Wavelet Packet Modulation ...... 134

5.2.1 Signal to Interference Ratio ...... 134

5.2.2 Tree Structuring ...... 135

5.2.3 Simulation Results ...... 137

5.2.4 Noise Effects ...... 141

5.3 Chapter Summary ...... 144

6 Conclusion 145

6.1 Summary of Novel Results ...... 145

6.1.1 Channel Shortening ...... 145

6.1.2 Bit-Loading ...... 146

6.1.3 Wavelet Packet Modulation ...... 146

6.2 Future Research ...... 147

6.2.1 Channel Shortening ...... 147

6.2.2 Bit-loading ...... 147

6.2.3 Wavelet Packet Modulation ...... 148

6.3 Concluding Remarks ...... 148

vi CONTENTS

A DSL Test Loop Responses 149

B Solution to the Generalized Eigen-Problem 153

List of Figures 156

List of Tables 162

Bibliography 164

vii Chapter 1

Introduction

Over the last 50 years there has been an ever-increasing demand for the means to reliably commu- nicate digital-data at high speeds. Table 1.1 lists some examples of contemporary residential and commercial applications which require a digital link [1]. This demand has been paralleled by a massive research effort in all facets of communications engineering, giving rise to a large variety of enabling signal processing techniques and implementation technologies. Due to the extensive proliferation of the Public Switched Telephone Network (PSTN) in the developed world, it has always been the focus of much of this research. The popularity of the PSTN as a digital transmission medium can also be attributed to the growth in the use of Personal Computers (PCs) in the home leading to the development of cheap dial-up voice-band modems. Once the restriction to voice-band frequencies on the PSTN was removed in the 1980s, Integrated Digital Services Network (ISDN) and subsequently the Digital Subscriber Loop (DSL) were born and broad-band digital communication over copper became a reality.

This thesis is concerned with the engineering challenges presented by the DSL to reliable digital communications, and in particular how multi-carrier techniques can be used to overcome these obstacles. This chapter gives a brief history of DSL, from the ﬁrst voice-band modems in the 1950s with rates of only hundreds of bits per second (bps) up to the emerging Very high-speed DSL (VDSL) standard with unidirectional rates of over 50 Mbps. The review presented is far from exhaustive, and the reader is referred to [1–7] for a more comprehensive chronicle. Adhering to the divide and conquer paradigm, the malign effects of the dispersive DSL channel can be more easily ameliorated by a subdivision of the allocated transmission spectrum, a process known as multi-carrier or multitone communication. The principles of multitone communication are presented, together with a literature review of the topic and the chapter concludes with an outline of the thesis.

1 CHAPTER 1. INTRODUCTION

TABLE 1.1: RESIDENTIAL AND COMMERCIAL APPLICATIONS REQUIRING DIGITAL INTER-CONNECTIVITY [1]

Residential Applications Commercial Applications Downstream Upstream De- Downstream Upstream De- Application Demand mand Application Demand mand Rate (kbps) Rate (kbps) Rate (kbps) Rate (kbps) Voice Voice 16–64 16–64 16–64 16–64 Telephony Telephony Internet 14–3000 14–384 Facsimile 9–128 9–128 E-mail 9–128 9–64 Internet 28–3000 14–384 Remote HDTV 12000–24000 0 128–6000 64–1500 Access Pay-per-view 1500–6000 0 Video Phone 384–1500 384–1500 Video Video phone 128–1500 128–1500 384–3000 384–3000 Conference

1.1 Evolution of the Digital Subscriber Loop

1.1.1 Voice Band Modems

The PSTN was essentially developed as an analogue communications network used for carry- ing voice signals at frequencies below 4 kHz. In fact the voice-band filters were only designed to pass frequencies in the approximate range 300 Hz–3.3 kHz, since this was deemed to allow sufficiently undistorted male speech for intelligible conversation (the female voice occupies a marginally higher spectral band and, accordingly, there is slightly more distortion of a woman’s voice than a man’s during a telephone conversation). Before the 1950s, low speed telegraphy allowed digital transmission at rates below 100 bps, but it was not until the 1950s, after the invention of the transistor, the advent of the digital computer, the characterization of telephone transmission characteristics and the development of Shannon’s theories of communication [8], that the first digital voice-band modems appeared. Modem is an abbreviation for modulate-demodulate, referring to the signaling functions at the transmitter and receiver of the communication link.

In 1962 the Bell 103 modem used Frequency Shift Keying (FSK) modulation to get 300 bps transmission on the PSTN. In 1964, the European telecommunications standardization body Comite´ Consultatif Internationale de Tel´ egraphie´ et Tel´ ephonie´ (CCITT), now the International Telecom- munications Union (ITU), produced the V.21 standard for the same data rate. These standards marked the beginning of the rush to achieve the maximum throughput over the telephone network. The Shannon capacity for reliable data transmission over a band-limited channel is C =

2 CHAPTER 1. INTRODUCTION

TABLE 1.2: VOICE-BAND DIGITAL MODEM STANDARDS. THE V-SERIES MODEMS ARE CCITT/ITU-T STANDARDS. THE bis SUFFIX ON SOME STANDARDS STANDS FOR second. THIS TABLE IS COMPILED FROM SOURCES [1–7].

Year Standard Transmission Rate Comment

1962 Bell 103 0.3 kbps Full duplex asynchronous FSK, 2- wire PSTN 1962 Bell 201 1.2 kbps FSK, Fixed equalizer 1962 Bell 202 2.4 kbps 4-phase PSK 1964 V.21 0.3 kbps Full duplex asynchronous FSK, 2- wire PSTN 1964 V.22 1.2 kbps Full duplex asynchronous PSK, 2- wire PSTN 1964 V.23 1.2/0.6 kbps Full duplex asymmetric FSK, 4-wire leased 1967 Milgo 4400/48 4.8 kbps 8-phase PSK, Manual Equalizer 1971 Codex 9600 C 9.6 kbps 16-QAM, Adaptive equalizer 1973 VA 3400 1.2 kbps Full duplex PSK 1980 Paradyne MP14400 14.4 kbps 64-QAM rectangular constellation 1981 Codex/ESE SP14.4 14.4 kbps 64-QAM hexagonal Full duplex, Synchronous/asynchronous, Trellis 1988 V.32 9.6 kbps coding, Echo Cancellation, 2-wire PSTN Full duplex, 1991 V.32 bis 14.4 kbps Synchronous/asynchronous, 2-wire PSTN 1993 V.34 28.8 kbps Full duplex, Asynchronous, Band- width optimization, Constellation shaping, Channel Dependent Pre- coding, 2-wire PSTN 1995 V.34 bis 36.6 kbps Full duplex, 10 b/Hz approaching capacity 1996 V.90 56 kbps Not strictly voice-band. Full duplex, Asymmetric Pulse Code Mod- ulation

3 CHAPTER 1. INTRODUCTION

B log2(1 + SNR) bps, where B is the channel bandwidth and SNR is the receiver Signal to Noise Ratio. During the 1960s sequential decoding of binary convolutional codes allowed communication within 3dB of the Shannon limit, and combined with the shrinking size and power consumption of digital Integrated Circuits (ICs), digital communication began to gain real commercial and military value. In the 1970s, techniques such as echo cancellation [9, 10], adaptive equalization [11,12], and timing-recovery/synchronization [13] allowed further gains (for example echo cancellation allows full duplex communication). A signiﬁcant milestone of voice-band communications was the perfection of trellis codes and Viterbi decoding which give signiﬁcant error rate improvement over uncoded multilevel modulation [14, 15]. Some of the voice-band modem standards from the last few decades are listed in Table 1.2.

1.1.2 ISDN

In order to achieve higher rates, the restriction to voice-band frequencies was removed for ISDN, ﬁrst standardized by CCITT in 1976. The standard evolved into Basic rate ISDN, which has a bandwidth of about 80kHz, supporting a transmission rate of 144 kbps [16–18]. ISDN allows full duplex communication using echo cancellation for ranges up to 18,000 feet. The ISDN is an integrated network in which the same digital switches and digital paths are used to establish connections for different services. However, standardization discrepancies have made ISDN installation difﬁcult, and incompatibility with the Plain Old Telephone Service (POTS) has precluded deployment in many areas. DSL evolved as a technology to overcome these problems, and further increase transmission rates.

1.1.3 xDSL

While it became clear that evolution of the DSL from the PSTN was the way forward in terms of improved digital services, there existed many opinions as to the exact nature of the technology and algorithms involved. All sorts of considerations had to be accounted for, such as different user demands, federal regulations on radio emissions (egress) and range variations across different server areas. As a result, various ﬂavours of DSL sprung up to meet these different criteria, the entire family of which is encompassed by the generic expression xDSL. While there existed multiple DSL proposals, they all shared the common objectives:

utilization of existing PSTN infrastructure. • utilization of existing network equipment. •

4 CHAPTER 1. INTRODUCTION

TABLE 1.3: XDSL FLAVOURS AND RATES

Standard Meaning Rate Comments

IDSL ISDN DSL 144 kbps Symmetric, cannot share twisted pair with POTS. SDSL Single line DSL 768 kbps Symmetric, unstandardized. HDSL High bit rate DSL 1.544 Mbps Symmetric, over two separate twisted pairs, cannot co-exist with POTS. HDSL2 2.048 Mbps Same as HDSL, over single twisted pair. Undergoing standardization. CDSL Consumer DSL 1 Mbps downstream, Asymmetric, over single wire 128 kbps upstream pair. MDSL Medium bit rate DSL 1 Mbps downstream, Asymmetric, slower version of 128 kbps upstream ADSL. ADSL Asymmetric DSL 1.5–8 Mbps down- Single twisted pair, can coexist stream, 16–640 kbps with POTS. upstream RADSL Rate adaptive DSL 1.5–8 Mbps down- Optimally selects rate for a stream, 16–640 kbps given twisted pair. upstream VDSL Very high speed DSL 13–52 Mbps down- Under development, draft stream, 1.5–6 Mbps standards exist. Designed upstream for shorter range than ADSL ( 1km). '

Adhering to these basics allowed for rapid design, development and deployment over the installed base of copper. Table 1.3 lists existing and emerging DSL standards. It should be noted that ADSL is the most widely implemented standard, and boasts an extensive availability of documen- tation and service trial data. Furthermore, both ADSL and VDSL are designed to coexist with the POTS, by use of a splitter which is basically a ﬁlter used to separate the voice-band and broadband signals. For these reasons, this thesis is focused on multitone signal processing methods for communication in ADSL and VDSL environments, although the theory within is readily extended to other wireline applications. Multitone ADSL has been standardized by the American National Standards Institute (ANSI) [19] the International Telecommunications Union (ITU) [20] and the European Telecommunications Standards Institute (ETSI) [21]. These similar standards all propose a 1.1 MHz bandwidth with unidirectional rates up to 7 Mbps downstream, depending on the loop. While the VDSL standards have not been ﬁnalized, both ANSI and ETSI have draft standards based on a multi-carrier Physical Layer (PHY) [22,23]. ETSI specify a bandwidth

5 CHAPTER 1. INTRODUCTION

SYNTHESIS BANK CHANNEL ANALYSIS BANK x n Xˆ X ( ) 0 0 ↑M G0(z) F0(z) ↓M

X Xˆ1 1 ↑M G1(z) F1(z) ↓M H(z)

XˆN−1 ↑M GN−1(z) FN− z M XN−1 1( ) ↓ u(n)

FIGURE 1.1: N -CHANNEL FILTERBANK TRANSCEIVER: THE SYNTHESIS BANK TRANSMITS AND THE ANALYSIS BANK RE- CEIVES. ALSO KNOWN AS A TRANSMULTIPLEXER. of 30 MHz [24, pp. 18], whereas ANSI deﬁne 20 MHz as the highest usable frequency [2, pp. 293]. Unidirectional downstream rates of up to 52 Mbps are catered for on certain loops [25]. In this thesis, all simulations are based on the ANSI standards. Extension to ETSI compliance is straightforward.

1.2 Multitone Communications — Literature Survey

The DSL presents significant impairments to broadband communications signals, such as attenuation, dispersion, echo and crosstalk. This is due to the telegraphic effects of twisted pair on an electromagnetic signal. The nature of the twisted pair environment is examined in more detail in Chapter 2. For now, it is asserted that the dispersive nature of the channel gives rise to Inter- symbol Interference (ISI) leading to imperfect detection at the receiver. This can be equivalently attributed to the non-flat spectral characteristics of the transmission medium. The ISI effects of a dispersive communications channel can be mitigated by dividing the allocated spectrum into N approximately nondispersive subchannels, a process known as multitone [26] or multi-carrier [27] modulation. This can be achieved by means of an N-band filterbank transceiver, the most familiar embodiments of which are the Discrete Fourier Transform (DFT) based Discrete Multitone (DMT) [28–32] and the more spectrally efficient Discrete Wavelet Multitone (DWMT) [33–41].

1.2.1 The Filterbank Transceiver

General multitone communication can be realized by an N-channel ﬁlterbank transceiver as shown in Fig 1.1 [42–45]. This analysis is restricted to Finite Impulse Response (FIR) implementation only. Subchannel rates and transmit powers are determined by the spectral properties of the channel H(z), and the power budget at the transmitter [46–58]. This is called power- and bit-loading, and a signiﬁcant part of this thesis is concerned with the optimization thereof.

CHAPTER 1. INTRODUCTION

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FIGURE 1.2: SPECTRAL INTERPRETATION OF FILTERBANK BASED MULTITONE MODULATION: (A) BASEBAND SPEC- TRUM OF ith-SUBCHANNEL DATA; (B) accordion EFFECT OF UPSAMPLING; (C) ith SUBCHANNEL FILTER RESPONSE; (D) MODULATED SUBCHANNEL DATA.

The subchannel symbols Xi(z) are formed by ﬁrst parsing the binary transmit data a(n) into time slots ai(n) and subsequently mapping these lower rate data streams onto their relevant constellation symbols Xi(z), which correspond to distinct frequency bins. For this reason the ﬁlterbank transceiver is also referred to as a transmultiplexer, since a time-division multiplexed (TDM) signal is transformed to its frequency division multiplexed (FDM) counterpart. Ideally the subchannels will be orthogonal and consequently this scheme is sometimes called Orthogonal Frequency Division Multiplexing (OFDM). The terms DMT and OFDM are used interchangeably throughout the literature, although OFDM more commonly refers to multitone communication in a wireless environment, whereas DMT usually refers to guided-wave engineering, where bit-loading and channel-shortening can be performed. The other difference between DMT and OFDM is that in a wireless environment, the synthesized signal x(n) is further modulated to a carrier-band of choice, whereas it is transmitted in its baseband state over twisted-pair. In the wireless case, by using complex baseband equivalent channel- and noise- models, c[n] and v[n] respectively in Figure 1.1, the analysis remains unchanged. For more on OFDM for wireless environments, see [59–61].

The synthesis ﬁlterbank modulates the N independent data symbols Xi(z) onto frequency isolated signals si(n), which are added to generate a common transmit signal x(n). Upsampling on each subchannel by M has an accordion effect on the subchannel spectrum, creating multiple copies of the original spectrum across the band, as shown in Fig’s 1.2.a, 1.2.b. The orthogonal synthesis

ﬁlters Gi(z) are chosen to isolate a particular copy of the original spectrum in the relevant band so it may be recovered at the receiver by its analysis bank counterparts Fi(z) as in Fig’s 1.2.c,

7 CHAPTER 1. INTRODUCTION

1.2.d. This requires interpolation at a rate M N. For M = N the system is said to be ≥ maximally decimated, (or minimally interpolated), yielding maximum spectral efﬁciency. For M > N the system is over-interpolated and redundancy is inherent in the transmitted symbols (the subchannels are not packed as tightly as they could be). If M < N, data is irretrievably lost due to uncancellable subchannel aliasing.

The synthesis and analysis ﬁlters are of length γM where γ is the genus of the ﬁlterbank, and indicates the degree of overlap of the transmit pulses. If γ = 1 we have a block transform such as DMT, whereas if γ > 1 we have lapped transform , such as DWMT, with its associated improvement in spectral isolation.

In an ideal channel, zero ISI (Xˆ = X , i) is achieved only if the ﬁlters satisfy i i ∀

g (n)f ∗(n + kM) = Mδ(i j)δ(k) , k Z, (1.1) i j − ∈ n X where δ(n) is the Kronecker delta function. This is also known as Perfect Reconstruction (PR). It is usual [36] to consider ﬁlters with f (n) = g ( n) which simpliﬁes the constraint to i i∗ −

g (n)g∗(n + kM) = Mδ(i j)δ(k) , k Z, (1.2) i j − ∈ n X i.e. the filterbanks must be paraunitary. This means that the subchannel filters are not only orthogonal to each-other (unitary filterbank), but to time-shifted versions of themselves, where the shift is in multiples of M. Although this theory of PR-filterbanks was developed for source coding applications, where the analysis bank precedes the synthesis bank, the system remains PR when the same filterbanks are mapped to the transmultiplexer configuration used in communications (Fig. 1.1) [62].

1.2.2 Block Transforms — DMT

When γ = 1, the filterbank has no memory and data blocks can be (de)modulated individually, thus implementing a block transform. This is a popular choice, since for a maximally decimated orthogonal filterbank, the ith subchannel filter is then constrained to be a modulated version of the in j 2π rectangular pulse of length M: g (n) = W , 0 n M 1, where W = e M is the primitive i M ≤ ≤ − M th th M root of unity [38, 43]. Thus gi(n) is the i row of the Inverse-DFT (IDFT) matrix. In the absence of a channel, the analysis matrix is then simply the DFT matrix, for perfect reconstruction. Note that if frequencies were indexed highest to lowest, the synthesis bank would comprise of DFT filters and not IDFT filters, but this is moot, since we would still have perfect reconstruction [39]. Convention dictates an IDFT transmitter and DFT receiver, a system widely referred to as DMT, or Block-DMT, which can be implemented efficiently by parallelizable butterfly algorithms

8 CHAPTER 1. INTRODUCTION

[63–65]. In fact, the number of Multiply-Accumulate (MAC) operations required per transmitted information bit is lower for Block-DMT than that for decision-feedback equalized single carrier modulation over all distances of interest in VDSL [66].

The main disadvantage of Block-DMT is that the rectangular low-pass prototype results in sinc- shaped subband spectral responses, the first sidelobes of which are only 13dB down, [33]. A dispersive channel will thus introduce Inter-Carrier Interference (ICI) at significant levels. To mitigate this we can increase filterbank genus γ, and design subchannels with greater spectral isolation.

1.2.3 Lapped Transforms — DWMT

If γ > 1, each output block of M samples from the synthesis bank is dependent on the preceding γ 1 blocks of N input subsymbols X (z). Similarly, demodulation of a particular block of − i receive samples cannot be undertaken until γ 1 preceding blocks are available for processing. − We call this a lapped transform, and much work has been done on the particular cases γ = 2, the Lapped Orthogonal Transform (LOT) [67] and γ = 2l, l Z, the Extended Lapped Transform ∈ (ELT) [68, 69]. Introducing transmitter memory allows us to choose non-rectangular transmit windows, while still maintaining subchannel orthogonality. The longer pulses are agreeable to tighter spectral containment (see Fig 1.3) in accordance with the Heisenberg uncertainty principle, which puts a lower bound on the subdivision area of our time/frequency-plane tiling at a given sampling frequency [42]. Note, however, that increasing γ beyond a certain point does not give value for added complexity, since ICI will no longer be the dominant impairment, and also system delay is increasing [36].

General ELT design is intensive, even prohibitive [44] and relies on Householder factorizations of the filterbank into a cascade of paraunitary lattice structures, which are optimized to minimize spectral leakage between subchannels. The design procedure is greatly accelerated if we restrain the subchannel filters to be modulated versions of a low pass, linear-phase prototype p0(n), since instead of designing N filters, we now only design one. A popular choice of the modulation func- 1 π tions are the cosines at frequencies ωi = (i + 2 ) N , which was originally propsed for prototypes of length 2lM, l Z [70] and later extended to arbitrary length filters [71]. The advantage of ∈ using cosine modulation is that the filterbank can be implemented efficiently by the use of the Discrete Cosine Transform (DCT) [72–76]. These schemes have also been generalized to complex modulation [77]. Modulated filterbanks implementing lapped transforms with applications to communications are generally referred to as DWMT, to distinguish from DMT which uses a rectangular prototype.

Prototype ﬁlter design is not trivial if perfect reconstruction is desired. It is seen that if the 2N- polyphases of p0(n) are implemented by two-channel lossless lattices, and p0(n) is constrained

9 CHAPTER 1. INTRODUCTION

Block−DMT Transceiver Frequency Response: Number of Complex Subchannels is 16 0

−5

−10

−15

−20

Magnitude Squared (dB) −25

−30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

DWMT Transceiver Frequency Response: Number of Real Subchannels is 32. Overlap is 4 0

−5

−10

−15

−20

Magnitude Squared (dB) −25

−30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized Frequency (× π rad / sample)

FIGURE 1.3: BLOCK- AND LAPPED-TRANSFORM FILTER- BANK SPECTRAL RESPONSES: BETTER STOPBAND ATTENU- ATION IS SEEN FOR THE DWMT FILTERBANK (ALTHOUGH THERE ARE TWICE AS MANY SUBCHANNELS IN THE DWMT CASE, SPECTRAL EFFICIENCY IS EQUIVALENT, SINCE THESE SUBCHANNELS ARE CONFINED TO REAL CO-EFFICIENTS).

10 CHAPTER 1. INTRODUCTION

to be linear phase, filterbank paraunitariness is guaranteed, and the problem is to optimize the π lattice coefficients to produce a prototype with good stopband attenuation above w0 = 2N . This nonlinear optimization can produce good results, Rizos, Proakis and Nguyen having demonstrated stopband attenuations ranging from 23dB to 58dB for γ = 2, 4, 6, 8, 10 [36]. Another comparison example is made by Renfors et al. [40] for the case of single tone radio frequency interference. Although the common perception is that switching off the relevent subchannel in block-DMT will still allow near-capacity transmission it is shown that this is not the case. If the tonal interference is not centered on the band of interest, there is significant leakage into other channels, due to the high side-lobes of block-DMT. This is where the good frequency localization of lapped transforms offers a significant advantage.

1.2.4 Over-Interpolated Filterbanks

Perfect Reconstruction breaks down once a dispersive channel is encountered. Traditionally, com- bative measures against ISI have been categorized in two distinct ﬁelds. Channel equalization attempts to invert degrading channel effects, whereas channel coding deliberately introduces redundancies in the transmit data that may be interpreted at the receiver to reveal the nature of the channel distortion. This information is then used to recover the original data. A modern approach is to combine these philosophies, a technique known as vector coding [78]. It has been shown that general vector coding can be implemented by means of an over-interpolated ﬁlterbank transceiver [79]. Improved performance is seen when the subchannels are isolated in frequency which completes the transition from vector coding to over-interpolated multitone modulation. Both block- [80–82] and lapped-transforms [83] with over-interpolation have been developed for ISI-free communication.

The redundancy may take the form of zero-padding [84], cyclic-prefix [85–87], or general precoding samples in the time-domain [88], or that of redundant tones in the frequency domain [89, 90], depending on the modulation in question. The amount of redundancy required is, of course, channel dependent. As an example, consider the cyclic prefix used in DMT (explained in Chapter 2), which is a special case of over-interpolation. The cyclic prefix converts linear convolution in the channel to pseudo-periodic convolution, which is equivalent to easily recoverable scalar multiplication in the discrete frequency domain. This only works if the cyclic prefix length is greater than the channel length p. Thus the required redundancy is L > p. More generally the minimum redundancy for ISI-free filterbank transceivers of interpolation ratio N is equal to the maximum number of congruous zeros of the channel H(z) with respect to N [91]. Examination of this involved prerequisite is beyond the scope of this thesis; suffice to say that in the mean, shorter channels require less redundancy. We emphasise the fact that these schemes cancel all ISI perfectly. The only distortion which occurs is when the shortened channel is not short enough. For

11 CHAPTER 1. INTRODUCTION

this reason a channel shortening equalizer is commonly employed in a ﬁlterbank receiver, generally referred to as a Time-Domain Equalizer (TEQ). This thesis is particularly concerned with the Minimum Mean-Squared Error (MMSE) design of the TEQ. For more on oversampled ﬁlterbanks see [92, 93].

1.2.5 Near-Perfect Reconstruction

As an alternative to seeking zero-ISI communication, contemporary schemes attempt to achieve nearly zero-ISI, acknowledging the fact that subchannel orthogonality will be lost in any case due to channel effects. Furthermore, ﬁlterbank stopband attenuation can be greatly improved if the PR-conditions are relaxed [94], a paradigm which is called Near Perfect Reconstruction (NPR). Although NPR-transmultiplexers are not considered in detail in this thesis, they are included in this literature survey for completeness.

The disadvantage of PR-transmultiplexing is that full channel equalization is difficult; there are bound to be channels that cannot be equalized by a FIR filter [37]. A solution is to incorporate channel knowledge into the design of our synthesis and analysis filters, and minimize overall channel ISI and ICI while still maintaining perfect reconstruction [41, 95]. However, solution of the resulting non-linear optimization is often impractical [96] and also filterbanks used as equalizers can result in seriously degraded performance on subchannels which may even be some distance from channel nulls [94] . As a compromise we can design the transmultiplexer for zero crosstalk [97] (as opposed to zero distortion in the PR case). The main adavantage of this scheme is that ICI introduced in the non-ideal transmission channel tends to be localized, originating largely from the direct neighbours of the disturbed subchannel. Thus, in the absence of any other crosstalk, the amount of work that must be done by the channel equalizer is reduced. Cherubini et al. [98] present a scheme they call Filtered Multitone, in which efficient realization of highly frequency- localized subchannels is made possible by relaxing the PR condition and possibly oversampling. They claim to get better performance than either critical rate Block-DMT or DWMT on an ISI channel, based on the fact that PR conditions are destroyed anyway in this case. Of course, the simplest NPR solution for subchannel spectral isolation is to impose a smooth time-domain window on the transmit pulse, such as a Hanning window [94]. This has proved a popular low-cost solution in such diverse applications as IEEE 802.11a [99] broadband radio-LAN and the Eu- ropean standard for VDSL [22], although it is far from optimal. For contemporary results on NPR-Transmultiplexers, consult [100–113] and references therein.

1.2.6 Nonuniform Filterbanks

While the transmultiplexer in Fig. 1.1 shows N sub-channels, each with a sampling rate M, it is also possible to consider an N-band system with a different sampling rate Mi on each band.

CHAPTER 1. INTRODUCTION

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FIGURE 1.4: UNIFORM vs. NONUNIFORM SUBBAND DECOM- POSITION. THE UNIFORM CASE HAS THE SAME SAMPLING RATIO M ON EACH SUBCHANNEL. THE NONUNIFORM CASE HAS A DIFFERENT SAMPLING RATIO Mi ON EACH SUBCHAN- NEL.

The N subchannel spectra will each be “accordioned” by different amounts, and the design of the filterbank coefficients will have to account for this added freedom. Essentially, the different subchannels are signalled at different rates and accordingly, there will be a nonuniform N-band spectral decomposition, as illustrated by Fig. 1.4. Initially, this approach was developed for audio subband-coding applications, due to the nonuniform frequency sensitivities of the human auditory system (the Bark scale). However, an emerging philosophy is application of nonuniform filterbanks to spectrally shaped multitone communications. It is argued that this added freedom can be used to shape the ISI and ICI introduced by a channel with non-flat spectral characteristics. The problem is that design of nonuniform filterbanks is nontrivial, and is still an area of active research.

It was ﬁrst shown in [114], that if each subband of the ﬁlterbank had a rational sampling factor

Mi, a direct approach could be taken to nonuniform PR filterbank design. The first stipulation ni 1 on the sampling ratio Mi = , mi, ni Z, is that = 1 (critical sampling), which mi ∈ i Mi preserves the system sampling density, i.e. the overall sampleP processing rate does not decrease (inefficient use of bandwidth) or increase (irretrievable loss of data). Secondly, the set of sampling rates must belong to a compatible set [115]. However, this technique is not straightforward and simplifications have been sought.

One of the simplest and most popular approaches is a technique called recombination which only requires the design of uniform PR-structures [116, 117]. In the subband coding approach, where analysis precedes synthesis, an analysis PR-filterbank is constructed with the finest spectral resolution required in the design. Subsequently, those subchannels which have too fine a spectral resolution are recombined using smaller PR synthesis banks. At the synthesis bank, a reverse structure is used, whereby small analysis banks decombine broad subchannels and a large synthesis bank recovers the original signal. Since only PR-structures have been used, the entire structure is PR, and the design procedure is relatively straightforward. This indirect technique maps easily

CHAPTER 1. INTRODUCTION

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FIGURE 1.5: A 3-CHANNEL NONUNIFORM SYNTHESIS FIL- TERBANK, BASED ON THE RECOMBINATION TECHNIQUE. to the transmultiplexer configuration, as shown for the synthesis bank in Fig. 1.5. A disadvantage of recombination structures has been shown to be that an equivalent N-band multirate structure based on N subchannel filters, does not exist. This makes it difficult to specify spectral properties jω jω of Gi e and Fi e in the design, e.g. stopband attenuation [114]. Furthermore, the system is only Linear Time Invariant (LTI) if the number of subchannels in the outer filterbank, and the number of subchannels in the recombination filterbank are co-prime [118,119]; otherwise the system is Linear Periodically Time Varying (LPTV), which is undesirable for communications applications. A way to simplify the design procedure is to consider NPR structures, and also cosine modulated filterbanks in which only a number of prototypes need be designed. [107, 112, 120, 121].

Another simple approach to nonuniform filterbank design takes the opposite stance to the recombination philosophy; instead of designing the most complex filterbank and simplifying as necessary, tree structured filterbanks begin with the simplest filterbank structure, and iterate as necessary e.g. Fig. 1.6. This is by far the simplest approach to nonuniform filterbank design, but the cost is that there is little control over the stopband properties of the resulting subchannels, and the frequency resolution can only occurr in multiples of the number of subbands in the root tree. This technique is known as a hierarchical lapped transform [38]. In the particular case where a two- channel filterbank called a Quadrature Mirror Filter (QMF) constitutes the root tree, the scheme is referred to as Wavelet Packet Modulation (WPM). In this thesis it is shown how WPM schemes may be optimized to reduce ISI and ICI in the dispersive channel. For a recent review and the open research problems in nonuniform filterbanks, see [122].

CHAPTER 1. INTRODUCTION

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FIGURE 1.6: A 6 CHANNEL NONUNIFORM SYNTHESIS BANK BASE ON THE HEIRERARCHICAL LAPPED TRANSFORM. IN THIS CASE THE ROOT TREE IS A 2-CHANNEL TRANSMULTI- PLEXER, KNOWN AS A QUADRATURE MIRROR FILTER AND THE RESULTING STRUCTURE IS CALLED A WAVELET PACKET TREE.

1.2.7 Equalization

The type of equalization required depends not only on the channel, but also on whether the transmitter has memory. Overlapped transforms introduce a priori ISI which must be accounted for at the receiver, whereas block-DMT allows for much simpler channel equalization at the expense of some redundancy.

In Block-DMT there is no a priori ISI, so any ISI is entirely caused by the dispersive channel [123]. ICI however is caused not only by the channel, but by the finite length of the transmultiplexer filters which causes frequency delocalization. These effects can both be overcome by the special trick of cyclically extending the transmit DMT-frames, by the length of the channel memory L. The increased time between transmitted DMT-frames, although incurring a bit-rate N reduction factor of N+L , ensures that successive frames are independent, completely negating the effects of channel ISI (and we know there is no a priori ISI). The fact that the extension is cyclical disguises linear channel convolution as pseudo-circular for the duration of the transmitted frame, which corresponds exactly to the requirement of a pseudocirculant filterbank for crosstalk-free transmission [42]. Thus the sole channel impairment is amplitude distortion which is easily accounted for. The key issue here is the requirement that the cyclic prefix is longer than the channel memory. This is often restrictive, especially at high sampling rates, where channel memory can extend into the many hundreds of samples. To overcome this a Time-Domain Equalizer (TEQ) is employed to optimally squeeze the channel energy into a time-frame of less than L samples. MMSE-TEQ design is the major contribution of this thesis.

15 CHAPTER 1. INTRODUCTION

Equalization for DWMT is still largely an open research area [40] and it is certainly a more difficult problem than channel-shortening for block-DMT [124]. There are two approaches to be found in the literature: both pre-detection equalization and post-detection combining. The focus of a pre- detection equalizer for DWMT is not to shorten the channel, since a priori ISI in the transmitted signal renders guard-band insertion ineffectual anyway. Rather, emphasis is placed on forcing a linear phase channel, since the group delay must be the same over all the subchannels for correct polyphase decomposition [125, 126]. This is equivalent to matched filter detection, where the TEQ is matched to the channel, rather than to the transmit pulse [127]. Amplitude equalization can then be achieved in the frequency domain after the analysis bank, in a post-detection combiner. (Linear phase constraints are not as vital in block-DMT since non-linear phase effects are easily compensated for by rotation of distorted constellation points after cyclic-prefix equalization [36]).

Post-detection combination is generally implemented by means of a Multi-Input Multi-Output (MIMO) filter which negates ISI between DWMT-frames, and ICI between the subchannels. Pre- vious work [128,129] has focussed on MMSE solution for the MIMO filter co-efficients, although it is well known that receivers based on maximum likelihood detection via the Viterbi algorithm are superior to those based on linear- or decision-feedback- equalizers. To some extent then, there is an argument [97, 130] for a channel-shortening pre-detection TEQ, (probably constrained to match the channel for linear phase purposes) which would reduce system complexity when combined with a Viterbi decoder (the computational complexity of which grows exponentially with channel memory).

It has been found that the system matrix which defines combined synthesis/channel/analysis distortion on DWMT-symbols is sparse, with its most significant entries near the diagonal. This makes sense, since the ISI will be defined by the transmultiplexer genus γ, while the ICI should be frequency localized if the transmultiplexer is well designed. The MIMO equalizer that must combat this distortion can take advantage of this sparse structure. Neurohr [128] presents an adaptive solution for the MIMO post-detection combiner which does just that. For equalization of overinterpolated DWMT see [131].

1.2.8 Echo Cancellation

Full Duplex (FDX) Communication is possible for multitone communication i.e. up- and downstream signals can use the same part of the spectrum simultaneously [132], as can be done for single-tone communication. To achieve this, a hybrid coupler is used at the Analogue Front End (AFE) [133]. This is a bridge circuit, which uses line matching to isolate the up- and downstream signal components, as in Fig. 1.7. Echo is the primary source of distortion in the case of a poorly designed hybrid, whereby the transmit signal is not fully decoupled from the received signal, resulting in performance degradation. However, since the transmit signal is known, this information can be used to cancel echo on the line [134–136].

CHAPTER 1. INTRODUCTION

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FIGURE 1.7: HYBRID EQUIVALENT CIRCUIT: A SIM- PLE ELECTRICAL BRIDGE IN WHICH ONE BRANCH IS THE TWISTED PAIR LOOP COUPLED THROUGH A LINE TRANS- FORMER. IF THE IMPEDENCE RATIO ON THE RIGHT AND BOT- TOM OF THE BRIDGE MATCHES THAT ON THE LEFT AND TOP, THERE WILL BE NO ECHO [133].

Data-driven Echo-Cancellers, use gradient algorithms to optimize the canceller, based on the expected statistics of the received signal, together with knowledge of the actual transmit data [134, 135, 137–139]. A Simple Time-Domain Canceller is presented in [140] and [141]. More complicated, Frequency Domain cancellation is outlined in [142], [143], and [132]. Frequency domain cancellation is desirable as it allows optimization of SNR for each of the tones separately, which is an underlying advantage of multitone systems over single-tone. Another simpliﬁcation allowed by multitone echo cancellation, is that complexity can be reduced by only cancelling echo between adjacent or near-adjacent bins. The resulting loss in performance can be small enough to render this trade-off worthwhile, since echo interference on disturbed tones will be localized around the disturbing tones. A combination of time and frequency methods is used in [141]. Implementation issues (VLSI, ﬁxed point) are considered in [144–147].

1.2.9 Timing and Synchronization

The model we have been using for the DSL channel is a discrete-time equivalent sampled channel impulse response h(n), subject to sampled receiver noise u(n). Of course, the validity of this model is highly dependent on how we convert the DMT-signal x(n) to the actual continuous-time line signal x(t) and, more signiﬁcantly, how we sample the received signal y(t) at the ADC of the receiver. Assuming a simple zero-order hold DAC and some low-pass ﬁltering, the received signal will correspond to a series of superimposed pulse-shapes which must be sampled at the phase which corresponds to the maximum-SNR estimate of the transmit sample at the receiver, or possilbly a Maximum-Likelihood (ML) estimate of the best sampling instant, if data- or decision-

CHAPTER 1. INTRODUCTION

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FIGURE 1.8: A) SAMPLE TIMING: INCORRECT SAMPLING PHASE WILL REDUCE SNR, AS SHOWN DOTTED. B) MUL- TITONE SYMBOL TIMING: INCORRECT SYMBOL ALIGNMENT WILL REDUCE IN BLOCK PROCESSING OF A DIFFERENT SET OF SAMPLES, AS SHOWN DOTTED. aided sampling is employed [148]. This is greatly facilitated by matched filtering at the receiver before sampling, which will not only improve the sampled SNR, but will go some way towards linearizing channel phase, which is crucial for filterbank based reception. Timing errors at this point will result in annihilation of subchannel orthogonality [149], and ICI will become a dominant communication impairment. Furthermore, equalizer design based on imperfectly synchronized data, will be mismatched to the channel, which can have a severe effect on rate performance [150]. To negate the effect of timing errors, it is thus necessary to employ some form of timing recovery, either in analogue form, based on a Voltage Controlled crystal Oscillator (VCO) tuned by a Phase Locked Loop (PLL), or more commonly in multi-carrier schemes, all-digital timing error correction [151] based on interpolation of the sampled signal, and resampling at a better estimate of the correct phase. The timing error-estimate is not always perfect and is in fact a noisy estimate, which varies about some mean value. This effect is called timing-jitter, and one method to overcome this problem is to filter the timing-error estimate to remove noise artefacts [152]. Another synchronization method is to embed timing information in the multitone data in the form of pilot tones [153], which are generally prescribed in multitone communication standards for the purposes of both timing and equalization.

As well as sample synchronization at the ADC, it is also important to achieve multitone-symbol- synchronization in the digital receiver (the two are distinguished in Fig. 1.8). More concisely, the window of N-samples chosen as the received multitone symbol must be chosen correctly from the continuous stream of samples output by the ADC for block processing. Failure to align this window correctly corresponds to a time-shift forward or backward by the number of misaligned samples ∆, or, in the frequency domain, the received symbols are scaled by the complex exponen- tial ejωi∆ on each subchannel. This can be visualized as a rotation of the transmit constellation points by an appropriate angle, and so multitone-symbol synchronism is more easily retained by frequency domain methods which employ a rotor for constellation orientation [154]. It is also possible to use correlation filters to detect the cyclic prefix in DMT-symbols, since the inherrent pseudo-periodicity will cause the receive signal to have a peak in its auto-correlation sequence at a delay equal to the cyclic prefix length. This is useful in the detection of symbol transition boundaries.

18 CHAPTER 1. INTRODUCTION

Another synchronization issue arises in the multiuser case, where Near End Cross-Talk (NEXT) occurs between neighbouring users on a bundled DSL cable. In the case that the disturbers are synchronized with the disturbed line, the NEXT so introduced will remain orthogonal between subcarriers on different lines. As described above, orthogonality is removed by imperfect synchronization, and the unfortunate problem of non-orthogonal NEXT will arise [155]. It has been proposed to use a method known as Zipper-DMT, which uses interleaved up and downstream signals in conjunction with tone grouping and pulse shaping (reduces subband sidelobes), in order to mitigate the effects of non-orthogonal NEXT [156]. Other methods attempt to achieve user-wide synchronization by exploitation of cyclic redundancy in DMT schemes [155, 157], although this is not possible with DWMT, as no cyclic preﬁx is used. Timing-recovery and synchronization for ﬁlterbank schemes are studied in detail in [150, 158–160].

As an aside, wireless multitone modems such as OFDM exhibit a signiﬁcant sensitivity to the phase noise of the oscillator used for frequency down-conversion at the portable receiver, a phenomenon often modeled as Wiener phase noise [161]. Furthermore, there is the problem of carrier frequency offset at the down-conversion stage, which can cause severe breakdown of subchannel orthogonality [162]. The synchronizability of OFDM signals is discussed in [163]. For more on frequency offset estimation, and carrier phase jitter see [164–176]. These problems do not arise in multi-carrier DSL, which is a baseband scheme.

In this thesis, it is assumed that receiver synchronism has been achieved by any of the above methods, and we retain a discrete-time equivalent channel model.

1.2.10 Peak-to-Average Power Ratio

A major problem with multitone signals, which does not arise in single-tone systems, occurs when the individual subband signals, which are basically just modulated waveforms at different frequencies, superimpose constructively at the synthesis bank output. What may happen, is that the alignment of several waveform peaks from the different subchannels can cause an amplitude spike in the multitone signal [177–180]. From a theoretical point of view, this is of no concern to the system designer. However, from a practical point of view the signal will have a higher Peak-to-Average Power Ratio (PAPR) and both the Digital to Analog Converter (DAC) and power amplifier will require a higher dynamic range if they are to accurately represent all possible signalling levels (in single-tone communication, the signalling levels are taken from a prescribed finite alphabet, and the required dynamic range be specified explicitly). In the event that the DAC and transmit amplifier have insufficient dynamic range, the multitone symbol will be distorted, resulting in significant degradation of rate performance [181–183]. There are various methods for overcoming the effects of high PAPR signals in multitone, which can be categorized under three broadly distinct headings: coding, clipping and signal correction [184].

19 CHAPTER 1. INTRODUCTION

Coding schemes, such as Golay- and Reed-Muller-codes can be used to reduce multitone PAPR at the cost of bandwidth efficiency [185–191]. Nonlinear block codes embed the desired sequences in a larger set of sequences, and subsequently only a subset of the generated sequences are transmitted, specifically those with low peak powers. These schemes are popular because there is no out-of-band signal generated. Another solution is to limit, or clip, the multitone signal above a certain amplitude. When clipping is undertaken, the nonlinearity introduces out-of-band harmonics, which may conflict with regulatory stipulations and, of more concern, introduce clipping noise as the dominant signal impairment. This may be overcome by combining PAPR reduction techniques with out-of-band filtering. It is also possible to reduce PAPR by signal correction. For example pilot tones may be introduced (again at the loss of spectral efficiency) with such phase and amplitude as to reduce offending signal peaks. Other approaches include decision-aided reconstruction [192], envelope elimination and restoration [184], partial transmit sequencing [193], OFDM necklaces [194] and adaptive data predistortion [195]. In this thesis it is assumed that the high-PAPR nature of multitone signals is appropriately accounted for by any of the above techniques.

1.2.11 Bitstream Operations: Coding, Scrambling, and Interleaving

As well as coding for PAPR reduction, it is also customary to introduce error control codes in a multitone system. This is called Forward Error Correction (FEC). As an example, the ETSI proposal for VDSL communication recommends a variable rate Reed-Solomon (RS) coder [22, pp. 65]. Even though error control codes increase the net bitrate, they also decrease the probability of bit-error in the receiver, resulting in a net coding gain γc. Block-codes, such as Hamming- or RS- codes, pad blocks of data with parity check bits, which do not alter the transmit sequence, rather, they only extend it [196]. Convolutional (trellis) codes can be used in applications where message bits arrive serially, and cannot be block processed. A convolutional encoder can be viewed as a sort of Infinite Impulse Response (IIR) filter, with the exception that internal operations are modulo-2 arithmetic and the output sequence is necessarily longer than the input sequence (since there are redundancies). There are two fundamental advantages of trellis codes over block-codes; they can be designed much more flexibly (just use a different shift register structure) and there exists an optimal decoding procedure: the Viterbi algorithm. A good tutorial on trellis coding can be found in [14, 15]. Coding for OFDM is presented in [197], where both an inner convolutional code and an outer RS-code are employed. Turbo-codes, which have been show to give near Shannon limit performance, have also been considered for ADSL in [198]. A standard binary turbo encoder consists of two recursive convolutional codes separated by an interleaver. While conventional RS 6 codes for ADSL can give a coding gain γc = 5 dB at a Bit Error Rate (BER) of 10− , turbo codes can achieve γc = 6 dB.

A scrambler randomizes bits that are passed to the modulator, so that different bit-patterns will be equally probable. A descrambler removes the randomness. An even distribution of ones and

20 CHAPTER 1. INTRODUCTION

zeros makes transmitted power output more stable. The ETSI VDSL draft proposes a 23 bit randomization proces.

An interleaver systematically separates contiguous bits. This improves robustness to impulse (bursty) noise, in the channel. The FEC can only handle so many bits in error per block; by spreading out the effects of a noise burst over time with an interleaver, the FEC will not be irretrievably swamped. Each interleaved bit will be separated by a number of bits from its originally adjacent bits. De-interleaving is the reverse process. Interleaving is operated on FEC blocks. The simplest method is called block interleaving. In the interleaving algorithm, the bits are written into a buffer in rows but read out from columns. Each bit is then separated from its adjacent bit by the number of bits that equals the number of rows. In a de-interleaving algorithm, the bits are written into a buffer in columns but read out from rows. This simply reverses the interleaving process.

The focus of this thesis is modulation and equalization, and so bitstream operations are not developed or discussed further. For performance calculations, the coding gain γc will be used as a measure of some theoretical code. However, coding has not been simulated for the multi-carrier system.

1.3 Thesis Outline

Although many facets of multi-carrier communications have been mentioned in the above introduction, it is only proposed to examine three paradigms in detail in this thesis. In fact each of these three different problems is presented with its own literature survey in its own chapter, so that the chapters are somewhat self contained.

In Section 1.2.4 and Section 1.2.7 the need for a channel shortening equalizer was explained. MMSE channel shortening is developed in Chapter 3, where it is seen that we can achieve near- optimal rate performance by appropriate spectral shaping in the TEQ. Chapter 4 deals with the problem of power- and bit-loading in the multitone transmitter, as alluded to in Section 1.2.1. It is purported that near-optimal rate performance can be had at a fraction of the computational cost of the optimal loading scheme. In Chapter 5 the particular case of nonuniform transmultiplexing as implemented by wavelet packets is advanced (introduced above in Section 1.2.6). It is seen that a non-uniform subband decomposition can reduce combined ISI/ICI in a dispersive channel.

Before undertaking any of this however, it is necessary to present a valid discrete-time equivalent sampled channel impulse response h(n) which will accurately represent the DSL communications channel. Chapter 2 begins by developing this model from the first principles of transmission-line theory. The DSL noise environment is also characterized. Chapter 2 also presents DMT and cyclic prefix equalization in detail, as a particular case of the over-interpolated filterbank transceiver. DMT is the multi-carrier scheme that is employed in testing the theories of Chapters 3 and 4.

21 Chapter 2

DMT Communication on the Digital Subscriber Loop

In this chapter, a discrete-time equivalent channel model for the DSL is developed from first principles, with particular emphasis on the test loops prescribed by the ADSL and VDSL standards. The model incorporates the ISI effects of the transmission line, the coloured noise model for crosstalk, and the narrow-band model for radio frequency interference. It is shown how the malign nature of these channels, from a communications point of view, necessitates advanced modulation and equalization techniques to achieve throughput with an acceptable error rate. DMT communication is introduced as an effective solution to the problem of communicating in these harsh environments. The fast Fourier transform, together with cyclic prefix equalization, are presented as efficient methods to implement and recover the subband decomposition at the transmitter and receiver respectively.

2.1 DSL Channel Modeling

The subscriber loop is a spider-web of Unshielded Twisted Pair (UTP), spanning much of North America, Europe and the developed world. It is estimated that there are upwards of half a billion users of the DSL, more than 20% of whom are located in America. Around 70% of users are residential, whilst the rest are businesses and institutions [2]. With so many users of the DSL, it proves economical to lay many loops in parallel when installing new cable. It is common to have between 50 and 150 twisted pairs in the same binder or bundle between Central Ofﬁces (CO). The reason a twisted pair is used, in conjunction with differential signaling, is that electromagnetic interference, i.e. crosstalk, between pairs in a binder, will affect each wire in a pair almost identically, and since only the voltage difference is received, crosstalk effects are reduced [199].

22 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

Furthermore, pair twisting allows opposing fields in a wire pair to cancel each other; the tighter the twist, the more effective the cancellation. In fact, if the UTP is represented by a double (bifilar) 3p helix with pitch p, then at a radial distance of 2 the maximum magnetic fields from the cable are 50 dB below those from an equivalent untwisted pair [200].

In the USA, the DSL wire gauges lie in the range AWG#19–AWG#26 (AWG = American Wire Gauge). Lower gauge represents a larger diameter. The most frequently used gauges in the DSL are AWG#24 (0.0201in) and AWG#26 (0.01594in). In Europe the diameter of the wire is expressed in hundredths of a centimeter (4, 6, 8, 10). New wire-types are also to be introduced for VDSL transmission, such as drop wire (e.g. DW10) and Flat Pair (FP) [25]. A cost-saving feature of DSL installation is the use of bridge-taps, whereby lengths of unterminated cable are installed off the main loop in anticipation of projected consumer demand (Fig. 2.1). These bridge-taps cause reﬂections and distortion of the line-signal and, since excavation and removal of bridge-taps is unfeasible, DSL modems must contend with these impairments. In order to facilitate the development of broad-band communication on the DSL, the ANSI and ESTI standardization bodies speciﬁed test loops which would hopefully be representative of a large cross-section of the DSL links encountered in practice. This section describes the standard ADSL and VDSL test-loops, and develops appropriate discrete-time models thereof.

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FIGURE 2.1: BRIDGE-TAP IN A DSL LOOP.

2.1.1 The ADSL Test Loops

A Carrier Serving Area (CSA) is a plant administration area subsection of the main loop plant. It consists of AWG#24 and AWG#26 wire surrounding a remote CO. Lengths in the CSA are limited to 12 kft for AWG#24, including bridge-taps, and to 9 kft for AWG#26. The total length of bridge taps on a line must not be greater than 2.5 kft and the longest length for one bridge tap is 2 kft. Approximately 75–80% of all local loops in the US would fall into the CSA deﬁnition [202]. The CSA was primarily used for HDSL development [203–205]. However, ANSI working group T1.6 produced a new subset of lines to simulate the much higher capacity of ADSL. These generic lines are referred to as ADSL CSA test-loops. CSA test loops 1–8 are shown in Fig. 2.2 [201].

CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

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FIGURE 2.2: CONFIGURATION OF THE EIGHT STAN- DARD CSA LOOPS [201]. NUMBERS REPRESENT LENGTH/THICKNESS IN FEET/GAUGE. VERTICAL LINES REPRESENT BRIDGE-TAPS.

300m 500m VDSL-1 AWG #26 AWG #24

500m 76.2m AWG #24 DW-10 VDSL-3 91m 45.7m AWG #24 AWG #26

300m 45.7m 45.7m VDSL-4 AWG #26 AWG #26 AWG #26

500m 701m 30.4m 26.2m 15.2m VDSL-7 AWG #26 AWG #24 AWG #24 AWG #24 AWG #24

15.2m AWG = American Wire Gauge AWG #24 DW = Drop Wire

FIGURE 2.3: FOUR OF THE ANSI VDSL TEST LOOPS [25]. VERTICAL LINES REPRESENT BRIDGE TAPS.

24 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

TABLE 2.1: VDSL TEST LOOP DESCRIPTIONS [25]

Loop Description

VDSL-0 Null loop VDSL-1 Range stress limit, underground cable VDSL-2 Flat-wire vertical drop, horizontal aerial cable in other section VDSL-3 Reinforced wire, vertical drop, horizontal aerial cable in other section VDSL-4 Bridge-tap, horizontal aerial section VDSL-5 Short loop test with bridge taps and various crosstalk VDSL-6 Medium loop test with bridge taps and various crosstalk VDSL-7 Long loop test with bridge taps and various crosstalk

2.1.2 The VDSL Test Loops

ANSI’s draft VDSL proposes to use portions of the spectrum up to 20 MHz. Since attenuation becomes severe at these frequencies, the channel lengths over which transmission remains feasible are considerably reduced, compared to ADSL. For this reason, it is proposed that VDSL will not have the reach achievable by ADSL, but rather will be used as a high speed solution to the last leg in Fibre To The Curb (FTTC) server networks. Accordingly, a different set of test loops has been speciﬁed, to represent the environments most commonly encountered in VDSL transmission [25]. The various loops are described in Table 2.1. For our purposes it is considered sufﬁcient to model only four of the loops, since

the null loop (VDSL-0) is trivial • VDSL-2 is identical to VDSL-3, but of higher grade • VDSL-7 incorporates VDSL-5 and VDSL-6, since it is a worst case scenario. •

Fig 2.3 illustrates the 4 test loop conﬁgurations of interest.

2.1.3 Other DSL Elements

As well as the actual line and bridge-taps, there are a number of other elements which will have to be included in the discrete-time channel model. The Digital to Analog Converter (DAC) will have some low-pass ﬁltering effect, as will anti-aliasing ﬁlters at the transmitter and the receiver. The splitter, which is used to isolate the broad-band signal from the POTS signal will have a high-pass

CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

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FIGURE 2.4: TRANSMISSION LINE SCHEMATIC. BLOWN-UP SECTION SHOWS INCREMENTAL SECTION OF LINE AT DIS- TANCE x. nature, blocking out low-pass signals in the voice-band. Loading coils are also a common feature of the DSL, whereby inductive elements are included in the circuit to ﬂatten the pass-band at voice-band frequencies, where ampliﬁers are not feasible. However, due to the severe distortions introduced at higher-frequencies, broad-band communication is not usually possible for circuits containing loading coils [206], and they will not be included in our model.

Since the test loops described in this section are comprised of cascaded two-port sections, it proves beneﬁcial to consider their ABCD matrices, since the overall ABCD matrix of any loop is given by the product of the ABCD matrices of its cascaded subsections. In order to determine the ABCD matrices for twisted pair and bridged taps, it is necessary to recall the ﬁrst principles of transmission line theory [3].

2.1.4 Primary Line Constants: R, L, G, C

The transmission line model is shown in Fig 2.4, for a line of length l. An incremental section of twisted pair at distance x from the origin is modelled as having incremental-resistance R∂x, -inductance L∂x, -conductance G∂x and -capacitance C∂x, also shown in Fig. 2.4. R, L, G and C are known as the primary line constants and they have frequency dependent characteristics determined partly from theory and partly from empirical methods, i.e. curve ﬁtting. More detail can be found in the ANSI report on parametric modelling of twisted pair cabling [207]. The frequency dependence is speciﬁed by the following equations:

1 R(f) = 1 1 (2.1) 4 4 2 + 4 4 2 √roc+acf √ros+asf

26 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

TABLE 2.2: LINE CONSTANTS FOUND BY CURVE FITTING, FOR THE WIRE-TYPES ENCOUNTERED IN THE ADSL AND VDSL TEST LOOPS [25].

Wire-type Line Constant AWG#24 AWG#26 DW10 FP roc Ω/km 174.55888 286.17578 180.93 41.6 r Ω/km os ∞ ∞ ∞ ∞ 4 4 2 ac Ω /km Hz 0.053073481 0.14769620 0.0497223 0,001218 4 4 2 as Ω /km Hz 0 0 0 0 −4 −4 −4 −4 l0 H/km 6.1729593 10 6.7536888 10 7.2887 10 10 10 × × × × −6 −6 −6 −6 l∞ H/km 478.97099 10 488.95186 10 543.43 10 911 10 × × × × b 1.1529766 0.92930728 0.75577086 1.195 fm Hz 553760.63 806338.63 718888 174877 −9 −9 −9 −9 c∞ F/km 50 10 49 10 63.8 10 22.8 10 × × × × −9 −9 c0 F/km 0 0 51 10 31.78 10 × × ce 0 0 -0.11584622 -0.1109 −12 −8 −8 −8 g0 f/Hz.km 0.23487476 10 4.3 10 8.9 10 5.3 10 × × × × ge 1.38 0.70 0.856 0.88

27 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

AWG−24 AWG−26 1800 DW10 FP

1600

1400

1200

1000 ohms/km

800

600

400

200

0 2 4 6 10 10 10 10 frequency (Hz)

FIGURE 2.5: INCREMENTAL RESISTANCE R vs. FREQUENCY IN THE RANGE 0–100 MHZ, FOR WIRE-TYPES AWG#24, AWG#26, DW10 AND FLAT-PAIR.

b l + l f 0 fm L(f) = ∞ (2.2) h bi 1 + f fm

ce C(f) = c + ch0f −i (2.3) ∞ ge G(f) = g0f . (2.4)

The thirteen line constants roc, r , ac, as, l0, l , b, , fm, c , c0, ce, g0 and ge are fundamental prop- ∞ ∞ ∞ erties of a particular wire-type, e.g. AWG#26 has different line constants to AWG#24. These line constants have been tabulated in Table 2.2 for the wire-types found in the ADSL and VDSL test loops described in the last section. Using these values in (2.1)–(2.4), the frequency characteristics of the primary constants were simulated in MATLAB and have been plotted in Fig’s 2.5–2.8. These ﬁgures compare well with published data [2].

2.1.5 Secondary Line Constants: Z0, γp

Applying Kirchoff’s current law to the incremental line section of Fig. 2.4, we get

∂ i(x + ∂x, t) = i(x, t) v(x + ∂x, t)G∂x C∂x v(x + ∂x, t) (2.5) − − ∂t i(x + dx, t) i(x, t) ∂ − = Gv(x + ∂x, t) C v(x + ∂x, t). (2.6) ⇒ ∂x − − ∂t

28 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

−4 x 10 L

9.5

8.5

7.5 henrys/km 7

6.5

5.5 AWG−24 AWG−26 DW10 5 FP

0 2 4 6 10 10 10 10 frequency (Hz)

FIGURE 2.6: INCREMENTAL INDUCTANCE L vs. FREQUENCY IN THE RANGE 0–100 MHZ, FOR WIRE-TYPES AWG#24, AWG#26, DW10 AND FLAT-PAIR.

−2 10

−4 10

−6 10 siemens/km −8 10

−10 10

AWG−24 AWG−26 −12 10 DW10 FP

0 2 4 6 10 10 10 10 frequency (Hz)

FIGURE 2.7: INCREMENTAL CONDUCTANCE G vs. FRE- QUENCY IN THE RANGE 0–100 MHZ, FOR WIRE-TYPES AWG#24, AWG#26, DW10 AND FLAT-PAIR.

29 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

−7 x 10 C 1

0.9

0.8

0.7

0.6

0.5 farads/km 0.4

0.3

0.2

AWG−24 0.1 AWG−26 DW10 FP 0 0 2 4 6 10 10 10 10 frequency (Hz)

FIGURE 2.8: INCREMENTAL CAPACITANCE C vs. FREQUENCY IN THE RANGE 0–100 MHZ, FOR WIRE-TYPES AWG#24, AWG#26, DW10 AND FLAT-PAIR.

As ∂x 0, we obtain the differential equation → ∂i(x, t) ∂ = Gv(x, t) C v(x, t). (2.7) ∂x − − ∂t A similar application of Kirchoff’s voltage law yields

∂v(x, t) ∂ = Ri(x, t) L i(x, t). (2.8) ∂x − − ∂t Equations (2.7) and (2.8) are called the Telegrapher equations and they have the coupled frequency domain equivalents

dI(x, jω) = (G + jωC)V (x, jω) (2.9) dx − dV (x, jω) = (R + jωL)I(x, jω) (2.10) dx −

(I(x, jω) and V (x, jω) are the Fourier transforms of i(x, t) and v(x, t), respectively). Decoupling can be achieved by noting that

d2V (x, jω) = (R + jωL)(G + jωC)V (x, jω) (2.11) dx2 or if we deﬁne the propagation constant γp = (R + jωL)(G + jωC), p d2V (x, jω) = γ2V (x, jω) (2.12) dx2 p

30 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

Re[gamma] 12

6 Np/km

2 AWG−24 AWG−26 DW10 FP

1 2 3 4 5 6 7 8 9 10 6 frequency (Hz) x 10

FIGURE 2.9: REAL PART OF PROPAGATION CONSTANT γp vs. FREQUENCY IN THE RANGE 0–100 MHZ, FOR WIRE- TYPES AWG#24, AWG#26, DW10 AND FLAT-PAIR. which has the solution + γpx γpx V (x, jω) = V e− + V −e . (2.13)

+ V and V − are constants that can be determined from boundary conditions. The current can be determined in a similar manner, giving

1 + γpx γpx I(x, jω) = V e− V −e , (2.14) Z0 − where we deﬁne the characteristic impedance

R + jωL Z0 = . (2.15) sG + jωC

γp and Z0 are the secondary line constants, and are shown in Fig’s 2.9–2.12 over a range of frequencies, for each of the wire-types of interest.

2.1.6 ABCD Matrices

The ABCD matrix for the two-port transmission line model in Fig. 2.4 is deﬁned as

V A B V 1 = 2 (2.16) "I1 # "C D# "I2 #

31 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

Im[gamma]

300

250

200

150 Rad/km

100

50 AWG−24 AWG−26 DW10 FP

1 2 3 4 5 6 7 8 9 10 6 frequency (Hz) x 10

FIGURE 2.10: IMAGINARY PART OF PROPAGATION CON- STANT γp vs. FREQUENCY IN THE RANGE 0–100 MHZ, FOR WIRE-TYPES AWG#24, AWG#26, DW10 AND FLAT-PAIR.

|Z0|

AWG−24 AWG−26 85 DW10 FP

65 ohms

40 0 2 4 6 10 10 10 10 frequency (Hz)

FIGURE 2.11: MAGNITUDE OF CHARACTERISTIC IMPE- DENCE Z0 vs. FREQUENCY IN THE RANGE 0–100 MHZ, FOR WIRE-TYPES AWG#24, AWG#26, DW10 AND FLAT-PAIR.

32 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

arg(Z0)

0 AWG−24 AWG−26 DW10 FP −0.1

−0.2

−0.3

Rad −0.4

−0.5

−0.6

−0.7

0 2 4 6 10 10 10 10 frequency (Hz)

FIGURE 2.12: PHASE OF CHARACTERISTIC IMPEDENCE Z0 vs. FREQUENCY IN THE RANGE 0–100 MHZ, FOR WIRE- TYPES AWG#24, AWG#26, DW10 AND FLAT-PAIR.

It is possible to deﬁne the voltages and currents at points x = 0 and x = l, using (2.13) and − (2.14):

+ γpl γpl V1 = V e + V −e− (2.17) + V2 = V + V − (2.18)

1 + γpl γpl I1 = V e V −e− (2.19) Z0 − h i 1 + I2 = V V − . (2.20) Z0 − Substituting into (2.16) and simplifying gives

V cosh(γpl) Z0 sinh(γpl) V 1 = 2 (2.21) I 1 sinh(γ l) cosh(γ l) I " 1 # " Z0 p p # " 2 #

Thus, we have an expression for the ABCD matrix of a length of twisted pair.

For bridge taps, the equations are the same, but the connections are different. A bridge tap can be considered as a two-port network with only a shunt impedence, since the far end is unconnected to the circuit. Thus the effective ABCD matrix is given by

A B 1 0 1 0 = C = . (2.22) C D bridge 1 1 tanh(γ l) 1 " # " Abridge # " Z0 p #

Since the overall DSL test loops are conﬁgured as a cascade of sections, each of which is either a

33 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

length of twisted pair, or a bridge-tap, it is now possible to determine the overall test-loop ABCD matrix as A B A B = (2.23) "C D# "C D# tot Yi i A B where is the ABCD matrix of the ith subsection of the test-loop. C D " #i

2.1.7 Loop Transfer Function

Returning to Fig. 2.4, it is seen that

V2 = ZloadI2 (2.24)

V1 = AV2 + BI 2 (2.25)

I1 = CV2 + DI2 (2.26) which have the trivial solution

V2 Zload = = Hl(jω). (2.27) V1 B + AZload

Hl(jω) is the unsampled, continuous-frequency transfer function of the DSL test-loop. The load impedance Zload is designed to be a real constant, 135 Ω in ETSI loops [24] and 100 Ω in ANSI loops [25]. As asserted previously, the ANSI test-loops will be the main focus of this thesis.

2.1.8 DAC, Anti-Aliasing Filters and Splitter

DAC Splitter AA

P (jω) Hs(jω) Ha(jω) Hl(jω) Channel

Ha(jω) Hs(jω) AA Splitter ADC

FIGURE 2.13: THE ANALOGUE CHANNEL, COMPRISING OF THE DAC, THE SPLITTERS, THE AA-FILTERS, THE DSL AND THE ADC.

It is necessary to include the ﬁltering effects of the transmitter and the receiver in our overall channel model, in addition to the effect of the DSL itself. Fig 2.13 illustrates the overall analogue channel H(jω) which is comprised of the DAC, P (jω), splitters, Hs(jω), anti-aliasing ﬁlters,

Ha(jω), the DSL, Hl(jω) and the ADC. As described in Chapter 1, the ADC is assumed to

34 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

have achieved synchronism by appropriate timing recovery mechanisms, and, furthermore, to have inﬁnite amplitude resolution, which allows us ignore ADC effects.

It is also assumed that the DAC pulse response is rectangular. That is to say, that if the discrete- time DMT signal x(n) is considered to be a stream of scaled Dirac-pulses in continuous time at the DAC input, then the DAC output x(t) is a stream of scaled rectangular pulses p(t), each of length T , where T = 1 is the sample period, and f is the DAC sampling frequency: fs s

x(t) = x(n)δ(t nT ). (2.28) − n X The concept is illustrated in Fig. 2.14.a. Since the DAC pulse is rectangular in the time domain, its frequency-domain counterpart P (jω) must have a sinc shaped response

ωT P (jω) = sinc . (2.29) 2

P (jω) a) DAC

Ca(jω)

FIGURE 2.14: a) FILTERING EFFECT OF THE DAC. THE DISCRETE-TIME MULTITONE SIGNAL IS REPRESENTED AS A CONTINUOUS-TIME STREAM OF DIRAC-PULSES AT THE IN- PUT TO THE DAC. b) LOW- PASS ANTI-ALIAS FILTERING REMOVES SIGNAL COMPONENTS ABOVE THE PRESCRIBED BANDWIDTH.

Before transmission, an anti-aliasing filter Ha(jω) is used to supress out-of-band signal components, as illustrated in Fig. 2.14.b. The same filter is used at the receiver, before the ADC. Since the anti-aliasing filters are analogue components with a very high cut-off frequency, it is assumed in this thesis that they are well designed, and have a brick-wall response over the range of frequencies of interest.

Of more interest is the non-ideal splitter filter Hs(jω), which has the much tougher job of isolating signals above the voice-band. This high-pass filter has a low cut-off frequency, a large pass-band and must have a narrow transition band. For the purposes of this thesis, we use a fifth order Chebyshev high-pass filter with cut-off frequency 5.4 kHz and pass-band ripple of 0.5 dB, as described in [208].

35 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

2.1.9 Loop Impulse Response

The ANSI document Simulation of VDSL Test Loops [209] gives a nice overview of how the effective discrete-time impulse response may be derived from the analogue system description. From the previous section it is clear that the overall analogue channel is

H(jω) = P (jω)Hs(jω)Ha(jω)Hl(jω)Ha(jω)Hs(jω). (2.30)

The filtering effects of each of these components are defined above and it remains to find the sampled equivalent impulse response h(n) by Inverse Discrete-Time Fourier Transform (IDTFT)

ωs 1 2 j ω h(n) = H(jω)e ωs dω, (2.31) 2π ωs Z− 2 where ωs = 2πfs. Using (2.27) and (2.29–2.31), it is now possible to compute the frequency response and sampled impulse response of each of the ADSL and VDSL test-loops of interest. This has been done using MATLAB . Fig. 2.15 shows the magnitude squared frequency responses of the four VDSL test loops, while the impulse response of the test-loop VDSL-1 is shown in Fig. 2.16. For response plots of all of the ADSL and VDSL test loops, see Appendix A. It is of interest that the sampled VDSL impulse responses are not considerably longer in discrete-duration than the sampled ADSL responses, despite the higher sampling frequency. More speciﬁcally, ADSL allows for 256 tones of bandwidth 4.3125 kHz, implying a sampling frequency

f ADSL = 2 256 4.3125 kHz = 2.208 MHz (2.32) s × × whereas VDSL allows for 4096 tones of the same bandwidth, yielding

f VDSL = 2 4096 4.3125 kHz = 35.328 MHz. (2.33) s × × However, since the VDSL channels are shorter, system transients decay over much the same discrete-duration as in the ADSL loops, namely a couple of hundred samples.

Also of note is the similarity of all the ADSL pulse responses. This is due to the splitter, which has a similar high-pass effect on all channels. The effect is not as noticeable on the VDSL loops, since the range of frequencies is much higher.

2.1.10 Noise, Crosstalk and Radio Frequency Interference

ISI in the channel is, of course, not the only obstacle to DSL communications. The signal is also corrupted by thermal background noise, coloured crosstalk from other lines, and tonal Radio

36 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

−20

−40

−60

−80

−100 Magnitude Squared (dB)

−120

−140 VDSL1 VDSL3 VDSL4 VDSL7 −160 0 2 4 6 8 10 12 14 16 Frequency (MHz)

FIGURE 2.15: MAGNITUDE SQUARED FREQUENCY RESPONSES FOR THE VDSL TEST LOOPS.

−3 x 10 Test Loop VDSL1: Impulse Response 20

Amplitude (V) 5

−5 0 100 200 300 400 500 600 700 800 900 1000 Sample

FIGURE 2.16: SAMPLED IMPULSE RESPONSE FOR VDSL TEST LOOP NUMBER 1. THE SAMPLING RATE IS fs = 35.328 MHZ

37 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

Frequency Interferenc (RFI) 1. In order to represent these non-idealities in the system model, we use spectral factorization of the analogue noise Power Spectral Density (PSD), of the system to generate a noise modeling ﬁlter Hu(jω), which will ﬁlter a white-noise signal, to produce noise sequence u(n), with the desired spectral characteristics. Since the different noise processes are independent, the overall noise PSD, (f), is the sum of the PSD’s of each of the processes. Su Crosstalk is by far the dominant contribution to capacity limiting noise for DSL systems [210]. Crosstalk is the unwanted coupling of signals from one pair to another along a multipair cable. When crosstalk is measured at the same end of the cable as the crosstalk signal source, it is called Near-End Crosstalk (NEXT). When crosstalk is measured at the end of the cable opposite to the location of the crosstalk disturber signal, the measurement is termed Far End Crosstalk (FEXT). Since FEXT is attenuated over the length of the DSL connection, its effects are far less severe than that of NEXT, which is a more local phenomenon (although it is still a distributed effect). The analogue PSD of crosstalk in a binder depends on the number of disturbers, and their nature. For example, any given binder may be transporting up- or downstream- ISDN, ADSL, or other DSL services, on any or all of its twisted pairs. Intuitively, crosstalk results from incremental capacitative unbalance between pairs along the cable, and so the mean crosstalk power increases with frequency (capacitative admittance increases with frequency). In fact, single interferer NEXT 3 coupling gain follows an f 2 characteristic [211]. Crosstalk modeling is beyond the scope of this thesis, but the reader is referred to [212–216] for more information on the topic. The analogue PSD’s for crosstalk within a binder are reproduced from [25] in Table 2.3.

While ADSL only uses a bandwidth up to 1 or 2 MHz, VDSL, with its higher sampling rates, may occupy a bandwidth well into the amateur radio frequency range. The internationally recognized amateur radio frequency bands are shown in Table 2.4. It is undesirable either for VDSL users to interfere with the amateur radio users, by unwanted electromagnetic leakage (egress), or for RFI to corrupt the VDSL signal (ingress). RFI tends to be narrowband compared to the broadband signal used in multi-carrier communications. An advantage of multitone communication in the tonal RFI environment is that relevant tones can be nulled, solving both ingress and egress problems simultaneously. The analogue PSD for RFI can be modelled as a sum of scaled Kronecker delta functions at the appropriate frequencies

(f) = A δ(f f ). (2.34) SRFI I − I XI A truer model can be had by using time-varying amplitudes and frequencies of the different sources (I) of interference , although stationary RFI is used in this thesis.

As well as crosstalk and RFI, background noise is included to complete the channel model. This is taken to be Added White Gaussian Noise (AWGN) with constant PSD, N0/2. This yields the 1As mentioned in the introduction, noise effects of the ADC, timing jitter, imperfect echo-cancellation, and DAC clipping are assumed to be apppropriately minimized. Furthermore, impulse noise is beyond the scope of this thesis

38 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

TABLE 2.3: CROSSTALK POWER SPECTRAL DENSITIES IN A DSL BINDER [25]. Nd IS THE NUMBER OF DISTURBERS. d(f) IS THE PSD OF THE DISTURBING SIGNAL. l IS THE S LENGTH OF THE LOOP IN FEET.

Crosstalk Crosstalk PSD

3 Nd 6 −13 NEXT NEXT(f) = d(f) 10 f 2 S S 49 2 Nd 6 −13 2 FEXT FEXT(f) = d(f) H(f) 9 10 lf S S | | 49 × Crosstalk Crosstalk PSD Constants Disturber

2 2 f 1 ISDN ISDN(f) = KISDN sinc 4 f0 = 80 kHz, KISDN = S f0 f0 1+ f 2 f0 5 Vp 9 R , Vp = 2.5V, R = 135Ω

2 2 f 1 HDSL HDSL(f) = KHDSL sinc 4 f0 = 392 kHz, f3dB = S f0 f0 1+ f 2 f3dB 5 Vp 196 kHz, KISDN = 9 R , Vp = 2.7V, R = 135Ω

2 f ADSL,us(f) = KADSL,us(f)sinc f0 Upstream S f = 276 kHz −38dBm/Hz 28kHz≤f≤138kHz 0 KADSL,us(f) = f−138000 ADSL −38−24 43125 dBm/Hz n 2 f f0 = 2.208 MHz, ADSL ds(f) = KADSL dssinc lpf (f) hpf (f) S , , f0 S S K = 110.4 mW, Downstream 1 1 ADSL,ds lpf (f) = 8 , hpf(f)= f 8 ADSL 1+ f 1+ 3dB,h f3dB,l = 1.104 MHz, S f3dB,l S f f3dB,h = 20 kHz

TABLE 2.4: THE INTERNATIONALLY RECOGNIZED AMATEUR RADIO BANDS [25, 217].

1.81–2.0 MHz 3.5–3.8 MHz 7.0–7.1 MHz 10.1–10.15 MHz 14–14.35 MHz 18.068–18.168 MHz 21–21.45 MHz 28–29.7 MHz

39 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

overall analogue noise PSD

N (f) = (f) + (f) + (f) + 0 . (2.35) Su SNEXT SFEXT SRF I 2

In order to generate the discrete-time sampled equivalent noise signal u(n), spectral factorization is required in order to generate a filter with response H (jω) 2 = (f). This can be done easily | u | Su using the Yule-Walker equations in MATLAB . Filtering a random, white, gaussian, unit-variance signal µ(n) with the resulting filter Hu(jω) will produce a noise signal u(n) with the appropriate PSD, (f), provided a sufficient number of coefficients has been used in the noise modeling filter Su Hu(jω). A typical noisy channel might

have a background noise PSD N0 = 140 dBm/Hz • 2 − co-exist in a 50-pair binder with 5 FEXT and 5 NEXT interferers from each of ISDN, HDSL • and ADSL

be subject to ten simultaneous sources of RFI whose powers range from 60 to 10 dBm. • − −

Fig. 2.17 shows the PSD (f) of such a noise signal. For reference, note that a typical DSL Su transmitter may signal at a transmit power level of 20 dBm. ∼

40 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

isdn next hdsl next adsl (up) next −50 adsl (down) next isdn fext hdsl fext adsl (up) fext adsl (down) fext −100 rfi awgn total

−150 Magnitude (dBm)

−200

−250

−300 0 2 4 6 8 10 12 14 16 Frequency (MHz)

FIGURE 2.17: TYPICAL NOISE PSD IN A VDSL SYSTEM. IT IS DIFFICULT TO DISTINGUISH THE NOISE TYPES, BE- CAUSE THEY ARE SUPERIMPOSED. NOTICE, HOWEVER, THE 10 RFI SPIKES ACROSS THE BAND, THE FLAT AWGN N0 = 140 dBm/Hz, AND THE VARIOUS CROSSTALK DISTURBING − SIGNALS IN THE ADSL AND HDSL BANDS.

41 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

a) b) a1

Ω0

◦ 0 90 a1 + jb1 b1 a2 + jb2 a2 2Ω0 . . ◦ 90 I aN−1 + jbN−1 b2 F x(n) Σ x(t) 0 F x(t) − P/S DAC aN−1 jbN−1 T . aN−2 − jbN−2 . . aN−1 . . − (N 1)Ω0 a1 − jb1 ◦ 90 bN−1

FIGURE 2.18: a) ANALOGUE FDM IMPLEMENTED BY A BANK OF OSCILLATORS. b) DMT IMPLEMENTED BY IFFT. THE TWO SYSTEMS ARE IDENTICAL.

2.2 Discrete Multitone Modulation

The previous section provided a discrete-time equivalent sampled channel h(n) and noise model u(n). This section gives an explicit description of the signals on the channel in a block-DMT system. The first Frequency Division Multiplexed (FDM) schemes were implemented using analogue components by the U.S. military and, commercially, in the Collins Kineplex system [218], in the 1960’s. Literally, a bank of oscillators at different frequencies has each of its oscillators modulated independently by the subchannel data, in continuous time. Chapter 1 described how general discrete-time multitone communication could be implemented by a multi-rate filterbank as opposed to a bank of oscillators. In this section, it is shown that the particular case of Fourier- transform based block-DMT is the direct digital equivalent of the analogue system. Furthermore, it is seen that a cyclic prefix used in conjunction with the DMT signal can achieve ISI-free communication.

2.2.1 DMT by Fast Fourier Transform

Fig. 2.18.a illustrates analogue FDM implemented by a bank of oscillators. Binary data is mapped to a real multilevel symbol stream, X(n), which is then parsed into N low-rate symbol streams 2

2This analysis assumes N is even, which is universally true of all standards which employ DMT.

42 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

Xi(n). Xi(n) is further parsed into two streams ai(n) and bi(n), which are used to modulate the in-phase and quadrature components of the ith subchannel carrier. In effect, these are N parallel

QAM signals, modulated on frequencies at multiples of a fundamental carrier frequency Ω0 (in this section only, analogue angular frequency is denoted by Ω to distinguish it from digital angular frequency, which is denoted ω). Note that the diagram only shows N 1 channels, since the DC − signal X0 is not generally communicated. The line signal x(t) can be written

N 1 − x(t) = [a (t) cos(iΩ t) b (t) sin(iΩ t)] (2.36) i 0 − i 0 Xi=0

(summing from i = 0 is allowed by deﬁning the DC signal X0 = 0).

As mentioned in Chapter 1, the IFFT is used as the modulator in block-DMT. By analysing the signals in Fig 2.18.b, it can be seen why this is so. Given real binary data, rather than parsing onto N real multilevel symbol streams, the data is parsed onto N complex symbol streams

Xi(n) = ai(n) + jbi(n). This removes the need to reparse Xi(n) into its in-phase and quadrature components (this cannot be done in the analogue system, since it cannot handle complex numbers, only real signals; clearly, complex numbers are not a problem for the digital system). In order to ensure a real signal at the output of the IFFT, Hermitian symmetry is enforced on the complex vector of symbols Xi(n). This can be achieved by conjugately mirroring the symbols Xi(n) and applying a 2N-point IFFT to the concatenation of the original symbols and their conjugate image

∆ Xi(n) = XN∗ i(n) , N i 2N 1. (2.37) − ≤ ≤ − The further precaution is taken of transmitting zero signal on the DC and Nyquist tones, which are unavailable in a real communications system. Applying the IFFT to the complex subchannel symbols yields

2N 1 1 − j 2πin x(n) = X (n)e 2N (2.38) 2N i Xi=0 N 1 (2.37) 1 − j 2πin j 2πin = X (n)e 2N + X∗(n)e− 2N (2.39) 2N i i Xi=0 N 1 1 − j 2πin = Re X (n)e 2N (2.40) N i Xi=0 n o N 1 1 − 2πin 2πin = Re (a (n) + jb (n)) cos + j sin (2.41) N i i 2N 2N Xi=0 N 1 1 − 2πin 2πin = a (n) cos b (n) sin (2.42) N i 2N − i 2N Xi=0

43 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

At this point, Shannon’s sampling theorem is recalled, which states that a signal band-limited to 1 B is completely determined by samples spaced at intervals of 2B . A particular case of this theorem states that the output of an ideal DAC stimulated by a sampled sinusoid is the corresponding continuous-time sinusoid, provided that the sampling rate is at least twice the sinusoidal frequency. The signal x(n), which is a sum of sampled sinusoids, is band-limited to ω = π because we de- 2π(N 1) ﬁned it so (summation of (2.42) contains maximum frequency 2N− ). Therefore, the sampling theorem, together with the system linearity, dictate that the DAC output x(t) must be a sum of continuous-time sinusoids at the corresponding analogue frequencies (if the sampling frequency is fs then the analogue frequency is Ω = fsω). Deﬁning

2πin 2πi ω = = ω n , where ω = , (2.43) i 2N 0 0 2N

(2.42) becomes N 1 1 − x(n) = [a (n) cos (iω n) b (n) sin (iω n)] (2.44) N i 0 − i 0 Xi=0 which produces N 1 1 − x(t) = [a (n) cos (iΩ t) b (n) sin (iΩ t)] (2.45) N i 0 − i 0 Xi=0 at the DAC output, according to the sampling theorem discussion above (this assumes an ideal DAC, which gives perfect reconstruction of sampled band-limited signals). The IDFT modulator output (2.45) is now seen to be equivalent (within a scale factor) to the analogue FDM signal of (2.36). It is trivial to show that in the absence of a channel, the subchannel-symbols can be perfectly recovered by the inverse operator, namely an FFT demodulator. It remains to see how this might be affected by the discrete-time channel h(n).

2.2.2 Guard Band Insertion

The channel output y(n) is given by linear convolution

p 1 − y(n) = x(n) ? h(n) = h(k)x(n k), (2.46) − Xk=0 where p is the duration of the channel in samples (see Chapter 1). The analysis in this section ignores channel noise u(n) and focuses on the ISI structure. Block-DMT demodulates in blocks of 2N samples, so it is useful to write the nth sample of the lth block as [219]

p 1 − y (n) = h(k)x (n k) , 0 n 2N 1. (2.47) l l − ≤ ≤ − Xk=0

44 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

a) b)

xl(n) yl(n)

c) xl−1(n) xl(n) xl+1(n)

yl−1(n) yl(n) yl+1(n)

xl−1(n) xl(n) xl+1(n)

yl−1(n) yl(n) yl+1(n)

FIGURE 2.19: DMT SIGNALS: a) THE lth TRANSMIT BLOCK. b) THE lth RECEIVE BLOCK. c) CONSECUTIVE TRANSMIT BLOCKS. d) CONSECUTIVE RECEIVE BLOCKS, WITH IBI. e) CONSECUTIVE TRANSMIT BLOCKS WITH GUARD BAND. f) CONSECUTIVE RECEIVE BLOCKS WITH GUARD BAND. IBI IS REDUCED.

Clearly the dispersive channel causes block spreading, as illustrated in (Fig’s 2.19.a, 2.19.b). The summation of (2.47) includes terms x (n k) with n k < 0. Assuming the lth transmit block l − − starts at x (0), the contribution x (n k) for n k < 0 to y (n) must be from the tails of the l l − − l previously transmitted blocks xl m(n) i.e. a dispersive channel will cause block-overlap. This − is sometimes called Inter-Block Interference (IBI), to distinguish it from ICI. Assuming a well designed system, the number of block samples 2N will exceed the channel dispersion duration p and blocks will not interefere with other blocks further away than their nearest neighbours, as illustrated in Fig’s 2.19.c, 2.19.d. That is to say, the IBI contribution from blocks xl m reduces to − an IBI contribution only from the preceding transmit block xl 1. This expands the expression for − the received block sample:

yl(n) = yl l(n) + yl l 1(n), (2.48) | | − where

th th yl l(n) is the component of the n sample of the l received block yl, contributed by the • | samples of the transmit block xl,

th th yl l 1(n) is the component of the n sample of the l received block yl, contributed by the • | − samples of the previous transmit block xl 1. −

45 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

Rewriting (2.48) using (2.47) gives

n p 1 − yl(n) = h(k)xl(n k) + h(n)xl 1(2N + k 1) , 0 n 2N 1 (2.49) − − − ≤ ≤ − Xk=0 k=Xn+1 Desired IBI | {z } | {z } It is clear from (2.49) that the samples of block xl 1 contribute to unwanted IBI only over the − ﬁrst p 1 samples of the receive block y . To eliminate IBI it is thus sufﬁcient to insert a guard- − l band of length L between transmit-blocks, which is longer than the channel length p, as shown in Fig’s 2.19.e, 2.19.f. (2.49) must be altered to account for the guard time. Instead of negative n k indices in xl(n k) representing samples from the previous transmit block xl 1, they are − − − now zero samples from the guard interval. (2.49) reduces to an IBI-free expression

n y (n) = h(k)x (n k) + 0 , 0 n 2N 1 (2.50) l l − ≤ ≤ − Xk=0 IBI Desired |{z} This technique comes at the cost| of a reduction{z }in spectral efﬁciency, since every 2N information bearing samples must be augmented by L redundant samples. If IBI mitigation must come at the cost of spectral efﬁciency, it is seen that there exists a better solution than the guard-band, which also allows compensation for ICI.

2.2.3 Cyclic Preﬁx vs. Guard Band

Guard time guarantees IBI-free transmission, but ICI is still a problem, since within a block there is inter-sample interference, which removes subband orthogonality

n y (n) = h(k)x (n k) , 0 n 2N 1. (2.51) l l − ≤ ≤ − Xk=0 Linear convolution is equivalent to multiplication of -transforms Z

Y (z) = H(z)X(z). (2.52)

If we could convert to the -domain, the transmit data X(z) could easily be recovered after Z channel estimation Y (z) Xˆ(z) = , (2.53) Hˆ (z) where the caret denotes estimated values. However, the demodulator at the receiver is not a Z- transform, but a DFT, corresponding to the reverse operation of the IFFT modulator at the transmitter. As explained in [220, pp. 542], multiplication of DFT’s corresponds to circular convolution,

46 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

L 2N

CP xl

xl−1 CP xl CP xl+1 n

FIGURE 2.20: CYCLIC PREFIX INSERTION. THE LAST L SAMPLES OF EACH DMT BLOCK ARE REPRODUCED AT THE BEGINNING OF THE BLOCK. not linear convolution. Thus to implement simple frequency domain correction by division, the convolution in the channel must be made appear circular, even though it is physically linear. To achieve this, an element of periodicity is introduced in the transmit signal, since the channel is beyond our control. Cyclic extension of transmit DMT symbol xl(n) by L samples gives a quasi- CP periodic signal which we call xl (n), where CP means Cyclic Preﬁx. Furthermore, since a guard time of length L has been stipulated for IBI-free DMT transmission, it makes sense to insert the CP in place of the guard-band: in either case these samples are discarded at the receiver. The CP CP process is shown in Fig. 2.20. It is next shown that linear convolution of xl (n) with the channel h(n) is equivalent to circular convolution of the original sequence xl(n) with h(n).

Proof That Cyclic Extension Emulates Circular Convolution

CP The cyclically preﬁxed symbol xl (n) is written in terms of the original DMT symbol as

xCP(0) x (2N L) l l − xCP(1) x (2N L + 1)  l   l −  . .  .   .       xCP(L 1)   x (2N 1)   l −  =  l −  (2.54)  CP     x (L)   xl(0)   l     CP     xl (L + 1)   xl(1)   .   .   .   .   .   .      xCP(2N + L 1)  x (2N 1)   l −   l −      CP Implementing linear convolution of the cyclically extended transmit signal xl (n) with the chan- CP nel h(n) gives the received samples yl (n):

p 1 − yCP(n) = h(n)xCP(n k) , 0 n < 2N + L, (2.55) l l − ≤ Xk=0

47 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

S/P, QAM I Xi(l) xl(n) CP and F BITS and conjugate F . . P/S mirroring . T .

CP xl (n) u(n) Channel h(n)

CP yl (n)

de-CP yl(n) F Yi(l) F Xˆi(l) P/S and F E and . T . . de-QAM S/P . . Q .

FIGURE 2.21: DMT SYSTEM which, from (2.54), is equivalent to

n p 1 − yCP(n) = h(n)x (n k) + h(n)x (n k + 2N L) , 0 n < 2N + L. (2.56) l l − l − − ≤ Xk=0 k=Xn+1 CP The ﬁrst L samples of yl (n) are discarded at the receiver to get yl(n)

n p 1 − y (n) = h(n)x (n k) + h(n)x (n k + 2N) , 0 n < 2N (2.57) l l − l − ≤ Xk=0 k=Xn+1 (note a shift in the n index of the second term, by L samples to the right, due to this truncation). Comparison with (2.49) shows that replacing the guard-time with a cyclic-preﬁx has not exacer- bated IBI in any way. It remains to show that this replacement implements circular convolution.

Using modulo notation, where [n]2N is the residue of n modulo 2N, we can rewrite (2.57) as

n p 1 − y (n) = h(n)x (n k) + h(n)x ([n k] ) , 0 n < 2N (2.58) l l − l − 2N ≤ Xk=0 k=Xn+1 and finally noting that for 0 k n, we get n k = [n k] , the received DMT block reduces ≤ ≤ − − 2N to p 1 − y (n) = h(n)x ([n k] ) , 0 n < 2N. (2.59) l l − 2N ≤ Xk=0 This is the definition of circular convolution [220, pp. 542]. Thus, (2.59) is equivalent to mul- th th tiplication of DFT’s. Defining the i discrete-Fourier co-efficient of the l block by Xi(l) we get

Yi(l) = HiXi(l) (2.60)

(note that we have assumed a stationary channel, which obviates the need for an l index on Hi).

48 CHAPTER 2. DMT COMMUNICATION ON THE DIGITAL SUBSCRIBER LOOP

th Clearly, Xi(l) and Yi(l) are the i transmit and receive subchannel symbol streams. Note that the subchannel signals are completely decoupled and can be recovered independently. Each received subchannel symbol has simply been scaled by the ith Fourier coefﬁcient of the discrete-channel h(n). Thus, after some channel estimation, the recovered subchannel data is

Yi(l) Xˆi(l) = , i. (2.61) Hˆi ∀

This is called Frequency-domain EQualization (FEQ). The entire DMT system is summarized in Fig. 2.21.

2.2.4 Channel Estimation

Perfect recovery of subchannel data requires channel estimation. This is usually done by using a training sequence during the initialisation of the modem, in order to gain channel information. For example, the ADSL standard [20] speciﬁes a pseudo-random binary training sequence of period 512 samples. A simple least-squares, unbiased, frequency-domain, cross-correlation channel estimator uses Np periods of this training sequence to get [221]

Np 1 1 − Y (l) Hˆ = i , i. (2.62) i N training ∀ p Xi Xl=0 training Xi is independent of l, since the same DMT-block is transmitted each time. This is the channel estimation procedure that is used in later chapters of this thesis (a popular emerging scheme uses channel information from the cyclic preﬁx to provide channel tracking in decision directed mode, with very good results, but this has not been simulated [222].)

2.3 Chapter Summary

This chapter developed a discrete-time equivalent sampled channel model h(n) and sampled noise model u(n) for the ADSL and VDSL test loops. It was shown that block-DMT is the direct digital equivalent of analogue FDM. It was shown how the dispersive channel h(n) causes the ISI on the DMT-block, and how this can be eliminated by a sufficiently long cyclic prefix together with a one tap equalizer on each channel. A simple channel estimation algorithm was given for setting the FEQ. The next chapter discusses what happens in the event of insufficient cyclic prefix length. ISI is not completely eliminated, and residual-ISI remains. This residual-ISI can be reduced by designing a time-domain equalizer, which will shorten the effective channel length, and thus the required cyclic prefix length.

49 Chapter 3

Time Domain Equalization

A short neck denotes a good mind . . . You see, the messages go quicker to the brain, because they’ve shorter to go.

—Muriel Spark, 1960

In this chapter, methods of impulse response shortening for DMT are reviewed and compared in terms of complexity and performance. There is a particular emphasis on design methods for the Time Domain Equalizer (TEQ) which do not rely on nonlinear optimization, but which can be formulated as an eigen-problem and solved either directly, by eigen-decomposition or adaptively, by gradient based update algorithms. The original Minimum Mean-Squared Error (MMSE) solution is of the latter form, but has been discredited as yielding suboptimal rate performance. The main contribution of this chapter is a reformulation of the MMSE algorithm to yield near-optimal rate performance. It is demonstrated that various existing eigen-solutions for the TEQ are in fact particular cases of the reformulated MMSE-TEQ. An adaptive time-domain LMS algorithm is provided which solves for our rate-optimal TEQ.

3.1 Channel Shortening: State of The Art

Fig. 3.1.a shows a communication link with discrete-time equivalent Channel Impulse Response (CIR) h(n), sampled receiver noise u(n), and TEQ with coefﬁcients w(n). The FIR ﬁlters h(n) and w(n) are of length p and q respectively. We can represent the system as a Shortened Channel Impulse Response (SCIR) c(n) = h(n) w(n), subject to equalized noise v(n) = u(n) w(n) ∗ ∗ as in Fig. 3.1.b. The object is to specify w(n) which will “best” shorten the channel to a desired length, L.

CHAPTER 3. TIME DOMAIN EQUALIZATION

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FIGURE 3.1: a) DISCRETE-TIME EQUIVALENT COMMUNICA- TION CHANNEL INCLUDING SAMPLED RECEIVER NOISE AND TEQ b) EQUALIZED CHANNEL AND NOISE c) DESIRED IM- PULSE RESPONSE.

By Least-Squares (LS) modelling the -transform H(z) with q 1 poles and L 1 zeros, and Z − − subsequently choosing the zeros of W (z) as the poles of H(z), the equalized channel should be an L-tap FIR [223]. Unfortunately, LS shortening is generally recognised to give poor results (e.g. [224]), since insufficient TEQ length results in errors which are spread over the entire SCIR duration. Better solutions such as the Maximum Shortening SNR (MSSNR) and Maximum Composite Shortening SNR (MCSSNR) TEQ’s [223,225–227], maximize the proportion of SCIR energy within a target window, which is seen to yield reductions in ISI. However, system noise is not addressed in these schemes, and we shall show that these solutions are in fact zero-forcing (ZF) in philosophy. Noise problems are addressed in [228] which Minimizes the Delay Spread (MDS) of c(n). This MDS-TEQ does no better in terms of reducing ISI alone but can show improvements in reducing combined noise and ISI terms. MMSE impulse reponse shortening was originally proposed by Falconer and Magee [229] for complexity reduction in maximum- likelihood receivers, but has since been extensively relied upon by designers as a suboptimal, but cost effective solution for the TEQ in multitone systems [230–234] (for more on channel shortening for the Viterbi algorithm see [97, 235–237]; for other uses of MMSE channel shortening see [238, 239]). In this approach, the TEQ is chosen to minimize the mean squared-error between the equalized signal, and the signal seen by communication over a virtual channel with a Desired Impulse Response (DIR) b(n), as illustrated by Fig. 3.1.c. The advantage of this method is that the quadratic formulation of the error in terms of the TEQ coefficients allows for efficient solution of the channel-shortener by an adaptive gradient based search [240–244]. The disadvantage is that no attempt is made to optimize the bit-rate directly and as a result poorer performance is generally observed than in other methods.

None of the above solutions directly address bitrate optimization of a multitone system. It is claimed that the optimal TEQ in terms of capacity is that which will Maximize the Geometric- average Signal to Noise Ratio (MGSNR) across the subchannels at the demodulator [245–247]. The GSNR is deﬁned as 1 N 1 N − SNRgeo = SNRi , (3.1) " # Yi=0

51 CHAPTER 3. TIME DOMAIN EQUALIZATION

th where SNRi is the i subchannel SNR. The solution is analytically intractable due to the nonlinear dependence of GSNR on w(n), but by computationally exorbitant nonlinear optimization it is possible to extricate a viable TEQ. The final result is highly dependent on initial conditions, and local minima in the design function may obfuscate iterative attempts to seek the global optimum [248]. Furthermore, a number of inappropriate approximations made in the derivation actually result in worse rate performance than in other “suboptimal” designs [208,249,250]. To overcome this last problem, the SNR can be redefined in terms of the signal, noise and ISI paths of the equalized channel, resulting in a TEQ which directly Maximizes the Bit-Rate (MBR-TEQ) [208]. However, the rate is again expressed as a nonlinear function of w(n) and the prescribed optimization is not considered a practical solution to the equalization problem, although it does serve as a benchmark for achievable performance, coming close to the matched filter bound. As a compromise, the same authors propose the minimum-ISI TEQ which minimizes the distortion term in the MBR formulation by a frequency weighting of the noise and ISI at the receiver. Since this distortion is quadratic in w(n), no nonlinear optimization is required, and the TEQ is found directly by eigen-analysis of system signal correlation matrices. Other rate-boosting eigenfilter solutions rely on a similar frequency shaping of ISI and noise to maximize throughput [251–256], with the pos- sibility of fractional spacing for further gains [257, 258]. It is also seen that subchannel-selective dual- and poly-path TEQ designs can give measurable rate improvement [224, 259].

Four of the above algorithms are now presented in detail:

1. Minimum Mean-Squared Error (MMSE)

2. Maximum Shortening Signal to Noise Ratio (MSSNR)

3. Maximum Bit Rate (MBR)

4. Minimum Inter Symbol Interference (min-ISI).

These four are of particular interest because they include the best existing non-linear optimization (MBR), the best existing quadratic solution (min-ISI), and the two seminal TEQ algorithms which deﬁned the paradigm (MMSE and MSSNR).

3.1.1 Minimum Mean Squared Error Impulse Response Shortening

The philosophy of MMSE-Impulse response shortening is to minimize the mean-squared value J of an error signal e(n), deﬁned as the difference between communication over the equalized channel c(n) and communication over a shorter desired channel b(n) with target length L, as in Fig’s 3.1.b, 3.1.c. More explicitly, we deﬁne the MSE

J = E e2(n) = E (d(n) z(n))2 (3.2) − h i

52 CHAPTER 3. TIME DOMAIN EQUALIZATION

L 1 q 1 2 − − = E b(l)x(n ∆ l) w(l)y(n l) . (3.3)  − − − − !  Xl=0 Xl=0   A synchronization delay ∆ is incorporated in the DIR channel to avoid dormant taps in the TEQ as shown in Fig. 3.1.c i.e. we do not waste any equalization potential of the TEQ upon ﬂat channel delay. Choosing ∆ is a highly nonlinear optimization problem [260, 261], which requires exhaustive search over a large range of delays, for that which gives the lowest MSE (or, possibly, highest rate) at the demodulator. It should be noted that, for fairness of comparison the optimum synchronization delay ∆ has been found for each of the impulse response shortening algorithms tested in this thesis.

Assuming wide sense stationarity of the signal x(n) and the noise u(n), we can rewrite (3.3) using auto-correlation matrices and vector representation. First, we deﬁne the correlation sequence notation r (k) =∆ E[α(n)β(n k)]. Then we get αβ −

J = bT R˜ b bT R∆ w wT R∆ b + wT R w, (3.4) xx − xy − yx yy where the L 1 and q 1 column vectors b and w hold the DIR and TEQ taps × × T b = b(0) b(1) . . . b(L 1) (3.5) − h iT w = w(0) w(1) . . . w(q 1) . (3.6) − h i The L L and q q input and output auto-correlation matrices R˜ and R , and the L q × × xx yy × ∆ input-output cross-correlation matrix Rxy have entries

˜ Rxx = rxx(m n) 0 m < L, 0 n < L (3.7) m,n − ≤ ≤ h i R∆ = r (m n + ∆) 0 m < L, 0 n < q (3.8) xy m,n xy − ≤ ≤ [Ryy] = ryy(m n) 0 m < q , 0 n < q (3.9) m,n − ≤ ≤

∆ ∆ ∆ T and Ryx = Rxy . TEQ-optimized MMSE

In the original MMSE solution by Falconer and Magee, (3.4) is minimized by partial differentiation with respect to the TEQ coefﬁcients w(n)

∂J = 2R∆ b + 2R w = 0 (3.10) ∂w − yx yy

1 ∆ w = R− R b, (3.11) ⇒ opt yy yx

53 CHAPTER 3. TIME DOMAIN EQUALIZATION

and so we term this the TEQ-optimized MMSE algorithm. Alternatively [231], the same solution can be obtained by application of the orthogonality principle

E [e(n)y∗(n)] = 0. (3.12)

In either case, back-substitution of (3.11) into the MSE expression of (3.4) will yield a quadratic form in b, T ˜ ∆ 1 ∆ ∆ T J = b Rxx RxyRy−y Ryx b = b Rx yb, (3.13) − | h i where the symmetric L L matrix Rx y is appropriately deﬁned. In minimizing this, a Unit × | Energy Constraint (UEC) bT b = 1 is applied to the DIR coefﬁcients to avoid the trivial null TEQ solution. The minimum eigenvalue λmin of the matrix Rx y equals the minimum power of the | error signal e(n) under this constraint, and so the solution for b is taken as the unit magnitude eigenvector corresponding to this eigenvalue. The MMSE solution wopt is thus determined by (3.11).

Regarding the constraint applied in this minimization, Moon and Zeng [262] claim that a Unit Tap Constraint (UTC), whereby the first tap of the DIR is constrained to unity, has a tendency to whiten the noise samples at the receiver, offering an advantage over other constraints at the input to a Viterbi decoder (this result was developed specifically for magnetic recording channels). However, a more recent work by Al-Dahir and Cioffi [260] claims that the UEC gives better MSE performance than other design constraints, by “optimally squeezing” the CIR energy into our desired window. In fact, not only does the UEC give lower MSE than the UTC but there is a higher DMT-throughput in all cases (the most significant benchmark) [263]. As we are more concerned with channel shortening and DMT transmission rate than Viterbi detection, the discussion is confined to UEC systems for all the TEQ algorithms considered.

General MMSE Shortening

A more general MMSE solution was proposed by Al-Dhahir and Cioffi in [260] and Nafie and Gatherer in [242]. Rather than differentiating with respect to the TEQ coefficients and then optimizing for the DIR coefficients as in the TEQ-optimized MMSE algorithm described previously, both sets of coefficients can be optimized simultaneously by defining a new coefficient vector g as the concatenation of the TEQ coefficient vector and the DIR coefficient vector:

b g = . (3.14) "w#

The MSE J can be rewritten from (3.4) in terms of g as

J = gT g (3.15) R

CHAPTER 3. TIME DOMAIN EQUALIZATION

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FIGURE 3.2: SHORTENED CHANNEL IMPULSE RESPONSE AND TARGET WINDOW. IMPERFECT SHORTENING CAUSES TAILS OUTSIDE THE WINDOW WHICH CONTRIBUTE TO ISI. where ∆ ∆ R˜ R = xx − yx . (3.16) ∆ R " Rxy Ryy # − The optimal solutions for b and w are thus extracted directly from the eigen-solution

T gopt = arg min g g g R (3.17) n o s.t. gT g =! 1.

The UEC gT g =! 1 precludes a null solution for the TEQ as before 1. This general MMSE algorithm affords a higher degree of freedom to the coefficients b and w than the TEQ-optimized MMSE solution. As can be seen from (3.11) the TEQ-optimized solution constrains b and w to maintain a specific relationship with one another. This is not enforced in the general MMSE solution and therefore this approach gives a lower MSE than can be achieved with the former, as verified later in our simulations. However, it is emphasised that this does not necessarily make it a better TEQ, since lower MSE does not translate directly to a higher communication rate in the multi-carrier system. More comment on this point is provided in subsequent sections.

3.1.2 The Maximum Shortening Signal to Noise Ratio TEQ

It is obvious that good channel shortening will ensure that most of the energy of c(n) will fall within a window of L taps, as illustrated in Fig. 3.2. Thus, it is intuitive to optimize the TEQ by minimizing a cost function which is a measure of the energy of c(n) outside the desired window. This basic approach to channel shortening was pioneered by Melsa, Younce and Rohrs [223] and can be formulated as a quadratic minimization in w(n).

Applying vector notation to Fig’s. 3.1.a, 3.1.b, we can express the (p + q 1) 1 SCIR tap vector − × as c = Hw, (3.18)

1In this thesis the notation α =! β is taken to mean “α is constrained equal to β”.

55 CHAPTER 3. TIME DOMAIN EQUALIZATION

where the (p + q 1) q transposed channel convolution matrix H has entries − × m = 0, 1, . . . , p + q 1 [H]m,n = h(m n) , − . (3.19) − ( n = 0, 1, . . . , q 1 − An expression for the SCIR energy follows as

T T T εc = c c = w H Hw. (3.20)

It is desirable that most of this energy will fall within a window of L taps. Referring to Fig. 3.2, this window is positioned with synchronisation delay ∆ to avoid having dormant taps in the TEQ, analogously to the offset in (3.3). As described in [223], by partitioning the vector c into the L 1 × vector c (containing the samples of c from within the desired window) and the (p L) 1 win − × vector cwall (containing the samples of c from outside the window) we can express the window energy εwin, and the residual energy εwall as

T T T εwin = cwincwin = w HwinHwinw (3.21)

T T T εwall = cwallcwall = w HwallHwallw (3.22) where the L q and (p + q L 1) q matrices H and H are extracted from H as × − − × win wall

h∆ h∆ 1 h∆ q+1 − · · · − h∆+1 h∆ h∆ q+2 H =  · · · −  , (3.23) win . . .. .  . . . .     h h h   ∆+L 1 ∆+L 2 ∆+L q   − − · · · −  h 0 0 0 · · ·  h1 h0 0  . . ·.· · .  . . .. .       h∆ 1 h∆ 2 h∆ q   − − · · · −   h∆+L h∆+L 1 h∆+L q+1   − · · · −  Hwall =  . . . .  . (3.24)  . . .. .       hp 1 hp 2 hp q   − − · · · −   0 h h   M 1 p q 1   . .− ·.· · −. −   . . .. .       0 0 hp 1   · · · −    The Shortening SNR is deﬁned as the ratio εwin (i.e. energy outside- over energy inside- the win- εwall dow) and the MSSNR solution is the vector w which maximizes this ratio (this is also sometimes

56 CHAPTER 3. TIME DOMAIN EQUALIZATION

called the minimum Residual-ISI (RISI) solution where the RISI = 1/SSNR). Melsa, Younce and

Rohrs [223] proposed to minimize εwall such that εwin = 1. Yin and Yue and , independently,

Schur, Speidel and Angerbauer [225,226] proposed to maximize εwin such that εwall = 1. Miloso- vic, Pessoa and Evans [227] proposed to maximize εwin such that εc = 1. Again, these UECs preclude a null solution for w. By noting that εc = εwin + εwall, it is obvious that these solutions are in fact identical. However, there are subtleties of complexity and implementation which are not so obvious. For example, Melsa’s solution encounters a singularity when q > L and Milosovic’s solution is numerically better conditioned than the others. These implementation issues are not considered important to our analysis, and we formalize the MSSNR solution as that provided by Melsa. Using (3.21) and (3.22)

T T wopt = arg min w HwallHwallw w (3.25) T n T o s.t. w HwinHwinw = 1.

This is a generalized eigen-problem, expressed in terms of quadratic forms in w. The well-known solution is given in Appendix B.

3.1.3 Maximum Bit Rate Impulse Response Shortening

The bit rate of a multitone system is maximized by increasing the number of bits per DMT block, transmitted at a target error probability, by appropriate choice of subchannel constellations. The total number of bits per block is Rb = i bi (i.e. the sum over the number of bits bi transmitted on each sub-carrier). A good approximationP to the number of bits which can be transmitted at a ﬁxed error probability for a given modulation is given by a modiﬁcation of the Shannon capacity

1 SNR b = log 1 + i bits per sample per dimension (3.26) i 2 2 Γ where the SNR gap to capacity Γ is given in decibels by

Γ = γ γ + γ . (3.27) m − c d

γm is the modulation gap, γc is the coding gain described in the previous chapter and γd is a design margin [30]. The concept of a capacity gap is explained in greater detail in the next chapter. The

MBR TEQ maximizes the total bit-rate Rb, by ﬁrst redeﬁning the subchannel signal to noise ratios

SNRi to incorporate an ISI term, secondly by expressing Rb as a function of the TEQ co-efﬁcients w and, ﬁnally, performing a nonlinear optimization of Rb with respect to w. The analysis proceeds as follows.

th Calling the i subchannel Signal to (Interference plus Noise) Ratio SINRi, the overall achievable

57 CHAPTER 3. TIME DOMAIN EQUALIZATION

a) Channel u(n) TEQ x(n) z(n) = zsig(n) + zisi(n) + v(n) h(n) w(n)

b) x(n) zsig(n) h(n) w(n)

Signal Path hsig(n) φ(n)

c) x(n) zisi(n) h(n) w(n)

ISI Path hisi(n) ψ(n) = 1 − φ(n)

d) u(n) v(n) w(n)

Noise Path

FIGURE 3.3: a) ACTUAL DISCRETE-TIME SYSTEM (CHAN- NEL, NOISE AND TEQ). b) EQUIVALENT SIGNAL PATH. c) EQUIVALENT ISI PATH. d) EQUIVALENT NOISE PATH. bit-rate is N 1 1 − SINR R = log 1 + i bits per symbol. (3.28) b 2 2 Γ Xi=0 It is necessary to express the SINR as a function of w. Knowing that the cyclic prefix (or some other redundancy) can eliminate all ISI if the channel is sufficiently shortened, it seems reasonable that the ISI measure should reflect the shortcomings of the TEQ in achieving this goal. More specifically, we can define a signal path, a noise path and an ISI path in the system, as shown in Fig. 3.3. In the ideal case, the channel is shortened to length L, and there is no ISI. Thus, the signal path hsig(n) would be an impulse response of length L plus, maybe, some system synchronization delay ∆. This can be isolated from the actual SCIR c(n) by multiplication by a rectangular window φ(n) (similar to the window in Fig. 3.2) as illustrated in Fig. 3.3.b:

hsig(n) = φ(n)c(n), (3.29) where 1, ∆ n ∆ + L 1 φ(n) = ≤ ≤ − (3.30) 0, otherwise.  By extension, the ISI path must be formed by the tails left after this windowing. This can be

58 CHAPTER 3. TIME DOMAIN EQUALIZATION

speciﬁed by multiplication of c(n) by ψ(n) = 1 φ(n): −

hisi(n) = ψ(n)c(n), (3.31) as shown in Fig. 3.3.c. The noise path hnoise(n) is, of course, just the TEQ w(n), as described in Section 3.1, padded to appropriate length. In order to convert to matrix notation, it is beneﬁcial to deﬁne column vectors

T φ = φ(0) φ(1) . . . φ(N 1) (3.32) − h i T ψ = ψ(0) ψ(1) . . . ψ(N 1) , (3.33) − h i and the α β identity- and zero-matrices as Iα β and 0α β, respectively. Using (3.18), we can × × × then deﬁne three N-length column vectors to describe the paths of interest more succinctly:

hsig = ΦHw (3.34) hisi = ΨHw (3.35) hnoise = Θw, (3.36) with Φ = diag(φ), Ψ = diag(ψ) and the q N zero-padding matrix →

Iq q Θ = × . (3.37) "0(N q) q# − × The function diag forms a diagonal matrix from its vector input. Since we are interested in the SINR on each subchannel, we take Fourier transforms of the above paths, in order to ﬁnd their spectral responses. If the DFT vector is

H j 2πi j 2π2i j 2π(N−1)i qi = 1 e N e N . . . e N , (3.38) h i then the ith spectral co-efﬁcient of the paths described in Fig. 3.3 can be written simply from (3.34)–(3.36) as

sig H Hi = qi ΦHw (3.39) isi H Hi = qi ΨHw (3.40) noise H Hi = qi Θw. (3.41)

There is a slight discrepancy here, in that the channel matrix H has been truncated (or possibly, to our good fortune, zero-padded) from height p + q 1 to height N, for dimension matching. − This should not affect the analysis signiﬁcantly, since, in N-band multitone communications, it is assumed that a channel model of order N is sufﬁcient to represent the physical channel.

59 CHAPTER 3. TIME DOMAIN EQUALIZATION

However, note that this thesis later presents an alternative to the MBR-TEQ which does not make this approximation. The SINR on a particular subchannel is now given by

2 ( 2πi ) Hsig Sx N i SINRi = (3.42) 2πi noise 2 2πi isi 2 u( ) H + x( ) H S N i S N i

wT HT ΦT q ( 2πi )qH ΦHw = iSx N i , (3.43) wT HT ΨT q ( 2πi )qH ΨH + ΘT q ( 2πi )qH Θ w iSx N i iSu N i h i where x(ω) 2πi is the sampled power spectral density of a signal x(n). While the expression S |ω= N of (3.43) appears complicated, it should be realized that it is of the form

T w Aiw SINRi = T (3.44) w Biw which from (3.28) allows us to write

N 1 − 1 wT A w w i opt = arg min log2 1 + T . (3.45) w Γ w Biw Xi=0 This theory was developed by Arslan et al [208, 249, 250], who propose to minimize this nonlinear function of w using the Broyden-Fletcher-Goldfarb-Shannon quasi-Newton algorithm in the MATLAB optimization toolbox. This is the best existing rate-maximizing solution available in the literature, although even its authors recognize that it is not a practical solution. Instead, they propose a near-optimal solution that can be formulated as a generalized eigen-problem identical to that described in Appendix B. It is called the min-ISI TEQ.

3.1.4 The Minimum Intersymbol Interference TEQ

In order to avoid nonlinear-optimization, it is possible to minimize the distortion power on each subchannel, which is the denominator of (3.43), by minimizing the sum of the distortion powers over all the subchannels. This is possible because power terms are always non-negative. In other words it is proposed to reformulate the optimization as

N 1 N 1 T T T − 2πi H T T − 2πi H wopt = arg min w H Ψ qi x( )q ΨHw + w Θ qi u( )q Θw w S N i S N i i=0 i=0 P s.t. wT HT ΦT ΦHw = 1. P (3.46) The UEC on the signal path is, of course, only to prevent the null solution, since now we have a generalized eigen-problem, which would otherwise have its minimum at the origin. The solution

60 CHAPTER 3. TIME DOMAIN EQUALIZATION

is given in Appendix B. Although this does not a priori guarantee the same solution as (3.45), it is reported that the performance is virtually identical in the ADSL testbed.

Interpreting the above result, this TEQ approaches optimal rate-performance by minimizing a frequency weighted system distortion. The proposed frequency weighting is directly related to the transmitted power spectrum and the received noise spectrum. It is argued that this weighting will have the desirable effect of placing ISI into the spectral regions of low SNR, in effect maximizing rate by indirect application of the water-pouring theorem [264].

3.1.5 Summary of Existing Channel Shortening Algorithms

It is useful to present the methods reviewed so far in this chapter in a concise summary, so they may be compared more easily. This is done in Table 3.1, which is an extended version of Table 2.1 from [250]. It can be seen that there are four main categories of TEQ. There is very little conceptual difference between the algorithms within each category, and we confine our discussion to a few specific examples (denoted by a dagger). In the next section it is shown that under certain assumptions all three eigen-filter categories (i.e. not the MBR solutions) are particular cases of an MMSE formulation, namely the DIR-optimized MMSE TEQ.

61 CHAPTER 3. TIME DOMAIN EQUALIZATION

TABLE 3.1: SUMMARY OF EXISTING CHANNEL SHORTEN- ING ALGORITHMS (MODIFIED FROM [250]). THOSE MARKED WITH A DAGGER ARE SIMULATED IN THIS THESIS.

Advantages Disdvantages 1. Adaptive or iterative 1. Deep notches in frequency 2. Off-line (initialization) 2. SCIR-DIR difference 3. Attempts to maximize channel capacity 3. Slow or uncertain convergence 4. Direct ISI minimization 4. Requires eigen-decomposition 5. Frequency weighting 5. Requires nonlinear optimization 6. Optimize subchannels 6. Numerical instabilities possible

Advantages Disadvantages

1 2 3 4 5 6 1 2 3 4 5 6 MMSE Algorithms Chow et al. [230] √ √ √ √ Chow et al. [231] √ √ √ √ √

Bladel et al. [232]† √ √ √ √ √

Naife et al. [242]† √ √ √ √ Lashkarin et al. [243] √ √ √ √ √ Boroujeny et al. [248] √ √ √ √ MSSNR Algorithms

Melsa et al. [223]† √ √ √ √ Yin et al. [225] √ √ √ √ Schur et al. [226] √ √ √ √ Milosovic et al. [227] √ √ √ √ MBR Algorithms Al-Dhahir et al. [246] √ √ √ √ √ √ √ Henkel et al. [247] √ √ √ √ √ √ √ Evans et al. [208] √ √ √ √ √ √ √ √ √ min-ISI Algorithms

Evans et al. [208]† √ √ √ √ √ √ √ √ Van Kerckhove et al. [251] √ √ √ √ √ √ Wang et al. [252] √ √ √ √ √ √ Van Acker et al. [256] √ √ √ √ √ Wang et al. [254] √ √ √ √ Frenger et al. [255] √ √ √ √

62 CHAPTER 3. TIME DOMAIN EQUALIZATION

3.2 The DIR-optimized MMSE TEQ

By deﬁning the tall (p + q 1) L windowing matrix Γ with entries given by the Kronecker delta − × function, 0 m < p + q 1 [Γ]m,n = δ(m n + ∆) ≤ − (3.47) − 0 n < L  ≤ p 1  and using y(n) = − h(l)x(n l) + u(n) from Fig. 3.1, we can rewrite the system correlation l=0 − matrices of (3.7)–(3.9)P in a manner similar to that used in [265]

˜ T Rxx = Γ RxxΓ (3.48) ∆ T Rxy = Γ RxxH (3.49) T Ryy = H RxxH + Ruu. (3.50)

R is the (p + q 1)-dimensional auto-correlation matrix of x(n). The dimension p + q 1 xx − − relates to the fact that each sample of z(n) at the equalizer output, depends on p + q 1 samples − of x(n) at the transmitter. R is the q q receiver-noise auto-correlation matrix with entries uu ×

[R ] = r (m n) 0 m < q, 0 n < q. (3.51) uu m,n uu − ≤ ≤

In all of the above, we have assumed that the noise u(n) and the signal x(n) are uncorrelated wide-sense stationary stochastic processes. Substituting (3.48)–(3.51) in (3.4) we obtain

T T T T T T J =b Γ RxxΓb b Γ RxxHw w H RxxΓb − − (3.52) T T T + w H RxxHw + w Ruuw.

We now propose an alternative approach to Falconer and Magee, whereby we minimize the MSE J of (3.52) by partial differentiation with respect to the DIR coefﬁcients b instead of the TEQ coefﬁcients w, a process we call DIR-optimized MMSE channel shortening.

∂J = 2ΓT R Γb 2ΓT R Hw = 0 (3.53) ∂b xx − xx b = ΓT Hw. (3.54) ⇒ opt As in (3.12), we can arrive at this result by the route of the orthogonality principle, but this time the inputs of estimation to which the error signal must be orthogonal, are taken to be the signal samples at the input to the DIR, i.e.

E [e(n)x∗(n ∆)] = 0. (3.55) −

63 CHAPTER 3. TIME DOMAIN EQUALIZATION

This is justifiable by the fact that we are performing an error minimization with respect to not one, but two sets of filter coefficients b and w, and hence have two (equally valid) contending inputs of estimation, x(n ∆) and y(n). Regardless of how it is derived, (3.54) used in (3.52), yields − the quadratic form in w,

J = wT HT ΦT R Φ ΦT R R Φ + R Hw + wT R w (3.56) xx − xx − xx xx uu T T T T = w H Ψ RxxΨHw + w Ruuw, (3.57) where Φ = ΓΓT and Ψ = I Φ. We minimize the MSE J subject to the same UEC as before, −

bT b = wT HT ΓΓT Hw = 1. (3.58)

In order to obtain a solution of the same format as in the literature [208], we note that ΓΓT = ΦT Φ and so wT HT ΨT R ΨHw w = arg min xx opt w T ( +w Ruuw ) (3.59) s.t. wT HT ΦT ΦHw = 1. This is again a generalized eigen-problem whose solution is in Appendix B. In the next section we will show that this solution is in fact close to rate-optimal.

3.2.1 Special Case I: The min-ISI TEQ

As explained in Section 3.1.3, it is reasonable to assume that a channel model of order N is sufﬁcient to represent the physical channel. In the particular case of the shortened channel, we T postulate that truncation of the SCIR c = c(0) c(1) . . . from length p+q 1 to length N − should thus have no appreciable affect onhour analysis. Furthermore,i under the same approximation, the auto-correlation sequence rxx(n) and the sampled power spectral density x(ω) 2πi S |ω= N of the signal x(n) are related by the DFT

N 1 − 2πni 1 2 j r (n) = ( πi )e N . (3.60) xx N Sx N Xi=0 Replacing p + q 1 by N, the N-dimensional auto-correlation matrix of x(n) has entries − N 1 − 2πmi 2πni 1 j 2πi j [R ] = r (m n) = e N ( )e− N . (3.61) xx m,n xx − N Sx N Xi=0

64 CHAPTER 3. TIME DOMAIN EQUALIZATION

Using the DFT vector qi from (3.38) we can write the overall input-signal correlation matrix

N 1 1 − R = q ( 2πi )qH , (3.62) xx N iSx N i Xi=0 which is real, due to the symmetry of x(ω) about π for real signals (this symmetry is enforced in N S 2 DMT in which only 2 complex tones are then available for use) . A similar expression relates the noise correlation matrix to its sampled spectral density

N 1 1 − R = q ( 2πi )qH . (3.63) uu N iSu N i Xi=0 Substituting (3.62) and (3.63) into (3.59), and scaling by N, which does not affect the solution yields

T T T N 1 2πi H w H Ψ i=0− qi x( N )qi ΨHw wopt = arg min S w T T N 1 2πi H  +w Θ P − q ( )q Θw  (3.64)  i=0 iSu N i  T T T s.t.w H Φ ΦHwP = 1,  where Θ is a q N zero padding matrix for dimension matching. Equation (3.64) is exactly the → near-optimal min-ISI solution of (3.46).

We conclude that the min-ISI TEQ is a special case of the DIR-optimized MMSE algorithm. While the min-ISI algorithm gives near-optimal rate performance, the DIR-optimized MMSE solution is nearer-optimal (i.e. gives higher rate), since no subband approximation has been made. In the case where the channel has been under-represented by the restriction to N-subchannels, the under-representation error manifests itself in the deviation of the min-ISI solution from the MMSE solution. However, the min-ISI TEQ is generally a good solution, since the N-band approximation will hold in well designed multitone systems.

3.2.2 Special Case II: The MSSNR TEQ

We consider the scenario where transmitted data is uncorrelated, or more speciﬁcally, rxx(n) = 2 σxδ(n), as is frequently the case in practice. We will assume, without loss of generality, that signal 2 power σx = 1 (in the exception, relevant signals will be scaled by the constant σx). As a further restriction, suppose we disregard noise in the optimization, i.e. the error signal is purely a measure

2The analysis in Chapter 2 assumed 2N complex channels, of which N could be used; this analysis has reverted to N channels to maintain generality for non-block DMT such as N-band over-interpolated transmultiplexers.

65 CHAPTER 3. TIME DOMAIN EQUALIZATION

of ISI. Then the optimization of (3.59) degenerates to

T T T wopt = arg min w H Ψ ΨHw w (3.65) n o s.t. wT HT ΦT ΦHw = 1.

Using (3.23) and (3.24), (3.65) can be expressed as

T T wopt = arg min w HwallHwallw w (3.66) T n T o s.t. w HwinHwinw = 1. which is exactly the MSSNR solution in (3.25), as we ﬁrst reported in [266]. The suboptimality of this solution under general conditions, is now well explained in terms of the omissions from the MMSE formulation. In fact, since the MSSNR-TEQ ignores system noise, we suggest that the algorithm should be referred to as a Zero-Forcing (ZF) TEQ.

3.2.3 Special Case III: High SNR

Returning brieﬂy to the TEQ-optimized MMSE channel shortener of Section 3.1.1, if we use the expanded MSE expression of (3.52) instead of that in (3.4) we get

∂J = 2bT ΓT R H + 2wT HT R H + 2wT R = 0 (3.67) ∂w − xx xx uu

In high SNR channels, the last term diminishes in signiﬁcance and we get ΓT Hw b which ' is the DIR-optimized solution of (3.54). Furthermore, if the DMT-signal x(n) is spectrally white (a valid approximation for high SNR systems in which most of the tones are in use), the solution degenerates to the MSSNR solution as described in Section 3.2.2. For this reason, we see that for high SNR systems, the MSSNR and TEQ-optimized MMSE solutions converge, as we will show in our simulations.

3.2.4 Adaptive LMS Implementation

The TEQ designs discussed so far depend on eigen-decomposition and possible inversion of quadratic forms deﬁning various cost-functions. Although efﬁcient solutions exist [267, 268], these processes tend to be numerically intensive and may result in computational errors if the matrices involved are ill-conditioned. For this reason it is common to use adaptive LMS algorithms to minimize the quadratic functions, via the method of steepest descent [12]. We now provide a time-domain LMS solution for the DIR-optimized TEQ which we have introduced in this chapter.

66 CHAPTER 3. TIME DOMAIN EQUALIZATION

The steepest descent algorithm for coefﬁcient update is given by [12]

µ w = w + ( J) (3.68) n+1 n 2 −∇ where J is the quadratic cost function in question and µ is a small step parameter. Using (3.4) and (3.54), the cost function for the DIR-optimized TEQ is

T ∆ 1 ∆ J = w R R R˜ − R w (3.69) yy − yx xx xy ∆ T h i = w Ry xw (3.70) | where Ry x is appropriately deﬁned. We have not used the simpliﬁed cost function of (3.59) | since H and Ruu are not necessarily known at the receiver. Rather, we have a known training ˜ sequence xt(n) and corresponding receiver signal yt(n) which allow us estimate Ryy and Rxx. The cost-function gradient is

J = 2Ry xw (3.71) ∇ | which we can substitute in (3.68) to give the TEQ update

wn+1 = wn µRy xwn. (3.72) − |

Since Ry x is not known at the receiver, the LMS algorithm is used to replace the steepest descent | algorithm by approximating the correlation values by their instantaneous estimates. We deﬁne the current TEQ input vector as

T yn = y(n) y(n 1) . . . y(n q + 1) (3.73) − − h i and the current DIR-input vector as

T xn = x(n ∆) x(n ∆ 1) . . . x(n ∆ L + 1) . (3.74) − − − − − h i These can be used to replace the relevant correlation matrices by the instantaneous estimates

˜ˆ ∆ T Rxx(n) = xnxn (3.75) ˆ ∆ ∆ T Rxy(n) = xnyn (3.76) ˆ ∆ ∆ T Ryx(n) = ynxn (3.77) ˆ ∆ T Ryy(n) = ynyn . (3.78)

The above equations and (3.70) will give

ˆ ˆ ˆ ∆ ˜ˆ 1 ˆ ∆ Ry x(n) = Ryy(n) Ryx(n)Rxx− (n)Rxy(n) (3.79) | −

67 CHAPTER 3. TIME DOMAIN EQUALIZATION

which can be used in (3.72) to provide the time domain LMS TEQ-coefﬁcient update

ˆ wn+1 = wn µRy x(n)wn. (3.80) − | ˆ Note that the uninvertible, rank one matrix R˜ xx(n) from (3.75) must be replaced by the known correlataion matrix of the training sequence in (3.79), which in the case of a white input sequence ˜ˆ 2 is simpliﬁed to Rxx(n) = σxI.

It remains to apply a unit energy constraint, requisite for non-trivial solution of the TEQ coefficient vector. Although we stipulated the constraint wT HT ΦT ΦHw = 1 for the DIR-optimized solution, it is better to apply a UEC on the TEQ coefficients w directly since, not only will this be simpler to apply but, more importatntly, it will guarantee that the solution vector will point along the shallowest direction of the convex cost function (in this case the MSE J(w)). In the case of a two-tap TEQ, the constraint wT AT Aw = 1 defines an ellipse in 2-dimensional space for some A, whereas the constraint wT w = 1 describes a circle (both curves are centered on the origin). Using the circular constraint, the eigen-solution will lie along the trough of the performance sur- face. However, this is not guaranteed using an elliptical constraint, especially if the major axes of the cost-function contours and the elliptical constraint contours point in roughly the same direction. Since this rarely happens in practise, the DIR-UEC is seen to work well, but we argue that it is more sensible to apply the UEC directly to the parameters over which we are optimizing, as the circular constraint never displays this pathology. For a q-tap TEQ the argument is the same, except the constraint is either an ellipsoid or spheroid in q-space as opposed to an ellipse or circle in 2-space. For this reason we use the more appropriate constraint wT w = 1, without loss of rate-optimality. In order to ensure the LMS update equation of (3.80) does not deviate from this constraint, the TEQ coefficient vector is normalized after each update. Thus, the overall update can be seen as a cost-function minimizing update followed by a projection back onto the constraint space. The algorithm is summarized as

ˆ w˜ n+1 = wn µRy x(n)wn (3.81) − | w˜ w n+1 n+1 = T . (3.82) w˜ n+1w˜ n+1

An implementation of this algorithm is provided in the next section.

68 CHAPTER 3. TIME DOMAIN EQUALIZATION

3.3 Simulation Results

3.3.1 Simulation Setup

For comparison purposes, we use the same simulation environment as used in [208], namely DMT modulation for the eight standard ADSL test conﬁgurations described in Chapter 2. We consider an N = 512 tone system, each tone having the same bandwidth Bi = 4.3125 kHz, corresponding to transmitter signaling at a frequency 2.208 MHz. The transmitter power budget is

Pb = 23 dBm, and the added white Gaussian noise has spectral density -140 dBm/Hz. Near End Crosstalk (NEXT) noise is modeled as eight upstream ADSL disturbers. Synchronization delay ∆ is optimized by exhaustive search over a range of reasonable delays. We assume a modulation SNR gap γm = 9.8dB, coding gain γc = 4.2dB and design margin γd = 6dB giving an overall SNR gap Γ = γ γ + γ dB. Simulation is facilitated by the DMT-TEQ toolbox for m − c d MATLABTM developed at the University of Texas [263]. As well as our DIR-optimized results, we attempted to reproduce the results of [208] for the TEQ-optimized MMSE, the general-MMSE, the MSSNR and the min-ISI TEQ solutions, in order to demonstrate relative performances. The nonlinearly-optimized MBR TEQ is considered superfluous to the analysis, since the min-ISI solution is reported to give virtually identical results at a fraction of the computational cost and, furthermore, the MBR solution can never be used in practice. As an alternative measure of achievable performance the matched filter bound is defined as a limiting system performance in the next section.

3.3.2 Figures of Merit

In order to determine the performance of the TEQ designs, it is necessary to define some figures of merit. Some of these have been defined previously, but they are grouped here for clarity. Using T h = h(0) h(1) . . . h(N 1) , we define: − h i

MFBi Subchannel Matched Filter Bound (Maximum SNR)

2πi H 2 x( ) q h MFB = S N i i ( 2πi ) u N S

RMFB Maximum bit-rate assuming SNR gap Γ

N 1 − MFB R = B log 1 + i bps. MFB i 2 Γ Xi=0

69 CHAPTER 3. TIME DOMAIN EQUALIZATION

MSE Mean Squared Error

MSE = bT R˜ b bT R∆ w wT R∆ b + wT R w xx − xy − yx yy

SNR Conventional Signal to Noise Ratio

T T w H RxxHw SNR = T w Ruuw

SSNR Shortening Signal to Noise Ratio

wT HT H w SSNR = win win T T w HwallHwallw

SINRi Subchannel Signal to (Interference + Noise) Ratio

T T T 2πi H w H Φ qi x( N )qi ΦHw SINRi = S , wT HT ΨT q ( 2πi )qH ΨH + ΘT q ( 2πi )qH Θ w iSx N i iSu N i h i SINR Overall Signal to (Interference + Noise) Ratio

wT HT ΦT R ΦHw SINR = xx T T T T w H Ψ RxxΨHw + w Ruuw

Rb Achievable bit-rate assuming SNR gap Γ

N 1 − SINR R = B log 1 + i bps b i 2 Γ Xi=0 ∆ Synchronization delay chosen to minimize the relevant cost function.

These measures are all deﬁned at the output of the TEQ. The MFB is achieved when the TEQ is perfectly shortening, and there is thus no ISI path.

3.3.3 Performance Results

The bit-rate results relative to the Matched Filter Bound (MFB) for all methods on the eight test loops are listed in Table 3.2 for a q = 17 tap TEQ, attempting to shorten the channel to length L = 32 (compare to Table I in [208]). It is seen that the DIR-optimized MMSE solution gives near-optimal rate-performance. To probe further, we consider in more detail CSA-loop #1 whose frequency response (including the splitter) and noise Power Spectral Density (PSD) are shown in

70 CHAPTER 3. TIME DOMAIN EQUALIZATION

TABLE 3.2: ACHIEVABLE BIT RATES FOR THE EIGHT CSA- LOOPS EQUALIZED WITH THE TEQ-OPTIMIZED MMSE, GEN-MMSE, MSSNR, MIN-ISI AND DIR-OPTIMIZED MMSE ALGORITHMS, AS A PERCENTAGE OF THE MATCHED FILTER BOUND RMFB. THE NUMBER OF TEQ TAPS IS q = 17. IT IS ASSUMED THAT THE CHANNEL IMPULSE RESPONSE IS KNOWN AT THE RECEIVER.

Bit Rate as a percentage of MFB Bit Rate

Loop TEQ-opt. general min DIR-opt. RMFB MMSE MMSE MSSNR ISI MMSE (Mbps)

1 62% 92% 62% 98% 99% 8.45 2 75% 90% 75% 97% 97% 9.68 3 82% 91% 82% 99% 99% 8.11 4 61% 92% 61% 98% 99% 8.05 5 72% 85% 72% 98% 99% 8.53 6 80% 93% 80% 99% 99% 7.77 7 74% 77% 74% 99% 99% 7.75 8 71% 56% 71% 99% 99% 6.90

Fig. 3.4. Fig. 3.5 shows the TEQ frequency responses obtained by the different algorithms, in attempting to shorten this test loop. It is seen that the min-ISI and DIR-optimized solutions give similar spectral shaping, as expected. The general MMSE solution gets closer to these solutions than the TEQ-optimized or MSSNR solutions thanks to its extra design freedom. It is notable that better rate performance is achieved when the disturbing crosstalk band is suppressed by the equalizer. Table 3.3 gives the various ﬁgures of merit, also for shortening of CSA-loop #1. As is to be expected, the MSSNR solution gives the maximum SSNR, the general-MMSE solution gives the lowest MSE and the DIR-optimized TEQ gives the best rate performance. These results were consistent across all the loops tested.

It has been reported that the min-ISI solution can perform well for the ADSL test-loops with a relatively small number of taps. For example, [208, Table II] provides the performance for the same simulation environment equalized with a q = 3 tap TEQ. It is seen that the min-ISI TEQ gets close to the MFB. In order to show that the DIR-optimized TEQ will also have this property, we recreate the previous simulations for q = 3 and the results are given in Table 3.4. Once again, the DIR-optimized solution is seen to give better performance, even at this reduced number of taps.

In generating the results of Tables 3.2-3.4, we assumed that the channel impulse response was known, and used (3.48)–(3.50) to generate the relevant correlation matrices. In an actual commu-

71 CHAPTER 3. TIME DOMAIN EQUALIZATION

−20 Channel Frequency Response Noise Spectral Density −40

−60

−80

−100 Magnitude Squared (dB)

−120

−140 0 0.2 0.4 0.6 0.8 1 Normalized Frequency

FIGURE 3.4: CSA-LOOP #1 CHANNEL FREQUENCY RE- SPONSE (INCLUDING THE SPLITTER) AND NOISE POWER SPECTRAL DENSITY.

−20

−40

−60

−80 Magnitude Squared (dB) TEQ−optimized MMSE MSSNR −100 min−ISI DIR−optimized MMSE general MMSE −120 0 0.2 0.4 0.6 0.8 1 Normalized Frequency

FIGURE 3.5: TEQ FREQUENCY RESPONSES FOR CSA-LOOP #1 EQUALIZED WITH THE TEQ-OPTIMIZED MMSE, GEN- MMSE, MSSNR, MIN-ISI, AND DIR-OPTIMIZED MMSE ALGORITHMS. THE NUMBER OF TEQ TAPS IS q = 17. IT IS ASSUMED THAT THE CHANNEL IMPULSE RESPONSE IS KNOWN AT THE RECEIVER.

72 CHAPTER 3. TIME DOMAIN EQUALIZATION

TABLE 3.3: FIGURES OF MERIT FOR CSA-LOOP #1 EQUAL- IZED WITH THE TEQ-OPTIMIZED MMSE, GEN-MMSE, MSSNR, MIN-ISI AND DIR-OPTIMIZED MMSE ALGO- RITHMS. THE NUMBER OF TEQ TAPS IS q = 17. IT IS AS- SUMED THAT THE CHANNEL IMPULSE RESPONSE IS KNOWN AT THE RECEIVER.

Loop TEQ-opt. general min DIR-opt. CSA-#1 MMSE MMSE MSSNR ISI MMSE

MSE 9.9 10−3 5.5 10−9 — — 9.1 10−9 × × × SNR (dB) 42.7 23.7 42.7 40.2 40.4 SSNR (dB) 33.0 11.9 33.0 28.0 28.1 SINR (dB) 40.8 23.8 40.8 40.8 40.9

Rb (Mbps) 5.2 7.2 5.2 8.3 8.4 Optimum ∆ 26 26 26 26 26

TABLE 3.4: ACHIEVABLE BIT RATES FOR THE EIGHT CSA- LOOPS EQUALIZED WITH THE TEQ-OPTIMIZED MMSE, GEN-MMSE, MSSNR, MIN-ISI AND DIR-OPTIMIZED MMSE ALGORITHMS, AS A PERCENTAGE OF THE MATCHED FILTER BOUND RMFB. THE NUMBER OF TEQ TAPS IS q = 3. IT IS ASSUMED THAT THE CHANNEL IMPULSE RESPONSE IS KNOWN AT THE RECEIVER.

Bit Rate as a percentage of MFB Bit Rate

Loop TEQ-opt. general min DIR-opt. RMFB MMSE MMSE MSSNR ISI MMSE (Mbps)

1 95.1% 95.4% 95.1% 96.5% 97.7% 8.45 2 96.0% 95.2% 96.0% 96.8% 97.0% 9.68 3 75.7% 96.6% 75.7% 97.9% 98.8% 8.11 4 95.5% 96.2% 95.5% 97.2% 98.1% 8.05 5 95.1% 93.9% 95.1% 96.9% 96.9% 8.53 6 94.5% 98.5% 94.5% 99.1% 99.5% 7.77 7 93.2% 93.2% 93.2% 96.0% 96.7% 7.75 8 94.1% 97.8% 94.1% 97.9% 98.6% 6.90

73 CHAPTER 3. TIME DOMAIN EQUALIZATION

TABLE 3.5: ACHIEVABLE BIT RATES FOR THE EIGHT CSA- LOOPS EQUALIZED WITH THE TEQ-OPTIMIZED MMSE, GEN-MMSE, MSSNR, MIN-ISI AND DIR-OPTIMIZED MMSE ALGORITHMS, AS A PERCENTAGE OF THE MATCHED FILTER BOUND RMFB. THE CHANNEL IMPULSE RESPONSE IS ESTIMATED USING A TRAINING SEQUENCE.

Bit Rate as a percentage of MFB Bit Rate

Loop TEQ-opt. general min DIR-opt. RMFB MMSE MMSE MSSNR ISI MMSE (Mbps)

1 31% 38% 31% 37% 56% 8.45 2 62% 32% 63% 28% 79% 9.68 3 67% 66% 37% 50% 73% 8.11 4 67% 40% 68% 37% 57% 8.05 5 63% 39% 63% 34% 53% 8.53 6 30% 76% 38% 53% 84% 7.77 7 25% 53% 25% 38% 66% 7.75 8 55% 68% 44% 75% 85% 6.90 nications environment a periodic training sequence is used to estimate the required correlation matrices (we used the exact values in Table 3.2 for fairness of comparison with the results of [208]). In order to demonstrate more realistic results, we used 400 cycles of the pseudo-random binary training sequence of period 512 samples, specified for ADSL, together with the least-squares, unbiased, frequency-domain, cross-correlation channel estimator detailed in Chapter 2. It is seen in Table 3.5 that there is a significant degradation in performance as compared to the case of perfect channel estimation, as is to be expected. However, a significant result is that the DIR-optimized TEQ is far less sensitive to imperfect channel estimation than is the min-ISI solution. This can be explained by the fact that our MMSE algorithm does not approximate the overall channel by N narrow-band subchannels, which could be an entirely invalid assumption in the likely event of imperfect channel estimation.

3.3.4 Adaptive Implementation

In order to demonstrate the adaptive algorithm introduced in Section 3.2.4, we attempted to shorten the ADSL loops using a training sequence of 400 DMT symbols in conjuction with the time- domain LMS coefﬁcient updates provided in (3.81) and (3.82). The same simulation environment was used as in the last section and we used a 3-tap TEQ in order that the tap-vector updates could

74 CHAPTER 3. TIME DOMAIN EQUALIZATION

10 MSE 5

0 0 0.5 1 1.5 2 5 x 10 1

0.5

TEQ Vector −0.5

−1 0 0.5 1 1.5 2 5 Update n x 10

FIGURE 3.6: A THREE TAP TEQ IS TRAINED USING THE PRO- POSED LMS ALGORITHM TO SHORTEN ADSL TEST LOOP #1. THE TOP CHART SHOWS MSE AND THE BOTTOM CHART SHOWS THE CONVERGENCE OF THE TEQ COEFFICIENT VEC- TOR.

4 be easily viewed. The step size used was µ = 10− . Fig. 3.6 shows the MSE along with the tap- weight updates for shortening of ADSL loop #1. It can be seen that the coefﬁcients converge in a manner that minimizes the cost function. The eigen-solution for the 3-tap DIR-optimized TEQ is found from (3.59) as T wopt = 0.5181 1.0000 0.4820 (3.83) − h i whereas the LMS solution converges to the values

T wlms = 0.5346 1.0000 0.5102 (3.84) − h i which is seen to be quite close to optimal. In order to emphasise the proximity of these solutions, the original and shortened channel are shown in Fig. 3.7 for the eigen-solution and the LMS

75 CHAPTER 3. TIME DOMAIN EQUALIZATION

solution respectively. It can be seen that the LMS solution gives shortening very similar to the eigen-solution. The performance of the LMS solution for each of the 8 test loops is given in Table 3.6. For comparative purposes, results for the eigen-solution are also presented, both for the case where the channel is estimated at the receiver and the case where the channel is known at the receiver. The number of TEQ taps is q = 3. It is seen that the LMS gives good performance relative to the optimal eigen-solution in all cases and actually outperforms the channel estimated solution in most cases (test loop number 8 is an exception). The reason that the LMS solution does so well as compared to the channel estimated case is that the channel estimation algorithm used in this example is far from optimal. In particular, the channel Fourier coefﬁcient estimates obtained from (2.62) do not involve the use of the cyclic preﬁx, and will thus be distorted estimates. More involved channel estimation procedures are beyond the scope of this thesis and the reader is referred to [221] for more detail.

TABLE 3.6: ACHIEVABLE BIT RATES AS A PERCENTAGE OF THE MATCHED FILTER BOUND RMFB FOR THE EIGHT CSA- LOOPS EQUALIZED WITH THE DIR-OPTIMIZED MMSE AL- GORITHM. THE TEQ IS IMPLEMENTED BY TIME-DOMAIN LMS ALGORITHM AND BY EIGEN-SOLUTION. IN THE CASE OF THE EIGEN-SOLUTION, RESULTS ARE PRESENTED FOR BOTH THE CASE WHERE THE CHANNEL IS ESTIMATED AT THE RECEIVER AND THE CASE WHERE THE CHANNEL IS KNOWN AT THE RECEIVER. THE NUMBER OF TEQ TAPS IS q = 3.

Bit Rate as a percentage of MFB Bit Rate

Loop LMS eigen-solution eigen-solution RMFB solution (channel estimated) (channel known) (Mbps)

1 80.3% 60.1% 97.7% 8.45 2 73.9% 59.5% 97.0% 9.68 3 80.5% 62.3% 98.8% 8.11 4 74.8% 54.2% 98.1% 8.05 5 74.9% 60.3% 96.9% 8.53 6 74.5% 61.8% 99.5% 7.77 7 77.5% 54.7% 96.7% 7.75 8 81.8% 96.9% 98.6% 6.90

76 CHAPTER 3. TIME DOMAIN EQUALIZATION

CIR and shortened−CIR

1 shortened CIR CIR 0.8

0.6

0.4

0.2

magnitude −0.2

−0.4

−0.6

−0.8

−1 0 100 200 300 400 500 sample CIR and shortened−CIR

1 shortened CIR CIR 0.8

0.6

0.4

0.2

magnitude −0.2

−0.4

−0.6

−0.8

−1 0 100 200 300 400 500 sample

FIGURE 3.7: TOP: ORIGINAL AND SHORTENED CHANNEL IMPULSE RESPONSE FOR ADSL LOOP #1, USING THE DIR- OPTIMIZED MMSE EIGEN-SOLUTION FOR THE TEQ. THE NUMBER OF TEQ TAPS IS q = 3 AND THE CHANNEL IM- PULSE RESPONSE IS ASSUMED KNOWN AT THE RECEIVER. BOTTOM: SAME PLOTS FOR THE TEQ OBTAINED BY TIME- DOMAIN LMS.

77 CHAPTER 3. TIME DOMAIN EQUALIZATION

3.4 Chapter Summary

This Chapter presented a review of channel shortening methods for DMT. It was shown that the MMSE-TEQ, can be formulated to give near-optimal rate performance in a multitone system, by optimization with respect to the DIR tap weights. We have demonstrated this both theoretically, by reduction to the near-optimal min-ISI solution, and numerically, by simulation in the ADSL testbed. We have also shown that the MSSNR solution is a degenerate case of the DIR-optimized MMSE formulation. Furthermore, the DIR-optimized MMSE TEQ is seen to be more robust than other methods under imperfect channel estimation. We presented a new time-domain LMS algorithm for update of the TEQ-coefﬁcient vector in an adaptive system. This provides the DIR- optimized solution without the need for eigen-decomposition or matrix inversion.

In the next chapter it is seen that rate can be maximized not only by equalization, but also by choice of modulation. This is achieved by power- and bit-loading of a multitone symbol at the transmitter.

78 Chapter 4

Power- and Bit-Loading

Adam Smith said, the best result comes from everyone in the group doing what’s best for himself, right? That’s what he said, right? Incomplete! Incomplete! Because the best result would come from everyone in the group doing what’s best for himself and the group.

— Russel Crowe as John Nash, A Beautiful Mind, 2002

In the previous chapters, it has been shown that ISI-free multitone communication is possible in the harsh DSL environment if appropriate redundancy is included in the transmit signal. It has been shown how the amount of redundancy required can be reduced by use of a channel shortening filter. While the above techniques reduce equalization complexity at the receiver, they introduce the problem of fixing the power and communication rate on each subchannel at the transmitter. We can attempt to maximize the bit-rate subject to a fixed power budget, or minimize the transmit power subject to a fixed bit-rate. These are called the bit-rate- and margin-maximization paradigms respectively [47] and present essentially the same engineering challenge. We tackle bit-rate maximization in this chapter. For results on margin maximization, see [269].

Various attempts at solving this problem [49, 53, 58] stem from the seminal work by Kalet [26], which shows how an intelligent allocation of bits and power across the subchannels should approximate the water-pouring solution for transmission over independent parallel Gaussian channels. The loading is chosen to give the same maximum allowed probability of error on any subchannel. Subsequent work [52] has demonstrated that bit-rate maximization is achieved by a probability of error distribution which is non-uniform across the tones. However, this relies on a computationally exorbitant nonlinear optimization which can be implementationally impractical. The best existing near-optimal solution [55, 57] reverts to uniform subchannel error probabilities, which allows for speedier convergence.

To the best of the author’s knowledge, there has been no comparison made between optimal and near-optimal loading algorithms in the literature and we attempt to do so in this chapter. We

79 CHAPTER 4. POWER- AND BIT-LOADING

will also consider two alternative near-optimal constrained optimizations. In the first, we impose uniform power distribution and in the second we impose both uniform power and error distributions across the tones. It is seen that near-optimal loading offers better value for complexity than optimal loading, since almost the same data-rates are achieved, with significantly less training. Before examining the various loading algorithms, however, the concept of water-pouring is explained, which is the optimization of bandwidth efficiency by appropriate spectral shaping of the system SNR. It is shown how multi-carrier systems are particularly suited to the implementation of water-pouring, both directly, by power-loading and, indirectly, by bit-loading.

4.1 Water-Pouring

The DSL channel model has been presented as the linear time-invariant FIR channel h(n) subject to sampled receiver noise u(n). The well known result for the capacity of a spectrally shaped discrete-time Gaussian-noise channel is given as [264, 270]

π jω 2 1 x(ω) H e C = log2 1 + S dω bits per sample (4.1) 2π 0 u(ω) ! Z S ( (ω) and (ω) were deﬁned in Chapter 3 as the power spectral densities of the transmit signal Sx Su x(n) and the receiver noise u(n), respectively). It is possible to maximize the channel capacity, since the transmit spectral characteristics (ω) are under our control. More formally, we wish Sx to maximize the capacity with respect to the transmit spectrum, subject to a system constraint on the total power budget Pb at the transmitter and subject to the fundamental constraint that the PSD must be positive for all frequencies:

π jω 2 opt 1 x(ω) H e x (ω) = max log2 1 + S dω S Sx(ω) (2π 0 u(ω) ! ) Z S (4.2) 1 π s.t. (ω) 0 and (ω) dω = P . Sx ≥ π Sx b Z0 The ubiquitous method of Lagrange multipliers allows conversion of a linearly constrained optimization problem into an easily solved unconstrained optimization problem. In the case of a linear inequality deﬁning the constraint we must extend the method to incorporate the Kuhn-Tucker conditions [271]. The result is rather simple; optimization occurs either at the Lagrange solution, in which case the constraint is inactive, or at the constrained optimum, which corresponds to evaluation of the optimization function at the bound of the inequality. Simply speaking, we can proceed with Lagrange multipliers and, if the constraint becomes active, we ignore the Lagrangian solution. We adopt this method in solving the linearly constrained capacity optimizaion problem deﬁned in (4.2). In order to maximize the capacity (a positive quantity), we can minimize its negative. The

80 CHAPTER 4. POWER- AND BIT-LOADING

Lagrange cost functional for minimization is deﬁned as

1 π J = C + λ (ω) dω P b . (4.3) − π Sx − Z0 The second term is a measure of how far the solution deviates from within the system constraints. λ is the Lagrange multiplier, which is a positive constant. To minimize, we differentiate.

dJ dC λ = + =! 0 (4.4) d (ω) −d (ω) π Sx Sx dC λ = . (4.5) ⇒ d (ω) π Sx Now,

dC 1 1 = (4.6) u(ω) d x(ω) 2ln2 π (ω) + S S x H(ejω) 2 S | | (4.5) u(ω) 1 x(ω) + S = . (4.7) ⇒ S H (ejω) 2 2ln2 λ | | Finally, to incorporate the Kuhn-Tucker conditions, we choose

opt 1 u(ω) x (ω) = max S 2 , 0 (4.8) S 2ln2 λ − H (ejω) ! | | 1 1 = max , 0 , (4.9) 2ln2 λ − ρ(ω) where the Gain to Noise Ratio (GNR) is deﬁned as

H ejω 2 ρ(ω) = (4.10) u(ω) S and λ takes the smallest value allowed by the power budget. It is evident that we ensure (ω) 0. Sx ≥ The spectrum allocation which maximizes capacity can be understood by a water-filling analogy [272,273]. We allocate power exactly in the way a receptacle fashioned in the shape of the inverse of the GNR would store water as it were filled. The concept is illustrated in Figure 4.1. This is an intuitive result; power is spent more wisely in communicating over bands of higher GNR. In the next section, it is shown that the spectral flexibility of multi-carrier systems afford us an efficient and simple method of implementing this spectral power shaping.

4.1.1 Power-Loading

Multi-carrier systems propitiously convert a dispersive communications environment into N parallel, approximately non-dispersive, subchannels with added Gaussian noise. In fact, as we saw

81 CHAPTER 4. POWER- AND BIT-LOADING

Su(ω) 2 |H(ejω )|

1 2ln2 λ

Sx(ω)

FIGURE 4.1: WATER-POURING FOR MAXIMUM CAPACITY

th in Chapter 2, if the channel h(n) has i Fourier coefﬁcient Hi, then each subchannel can be 1 modelled by a noisy channel of complex gain Hi, bandwidth Bi and added uniform noise power spectral density 2πi Ni . The subchannel-noise variance is 2 Ni . The capacity of each Su N = 2 σui = 2 subchannel is thus determined independently in terms of the subchannel power Pi by the Shannon capacity for such a bandlimited Gaussian channel

2 1 Pi Hi Ci log2 1 + | | (4.11) ' 2 BiNi ! 1 = log (1 + SNR ) bits per sample per dimension, (4.12) 2 2 i where the received subchannel signal to noise ratio SNRi is appropriately deﬁned. It is also convenient to deﬁne the subchannel GNR

SNRi ρi = . (4.13) Pi

The above approximation is generally valid for systems with a large number of subchannels, where the subbands can be well modeled as flat AWGN channels. By repeating the Kuhn-Tucker analysis of the previous section, it is now shown that the multitone system can theoretically approach the water-filling capacity, as the number of subchannels approaches infinity.

Subchannel orthogonality ensures that the total system capacity is additive over the capacities of the individual subchannels: C = i Ci. Furthermore, the total transmit power equals the sum of the powers over all the subchannels: Pb = i Pi. Deﬁning the vector of subchannel powers as T P P = P0 P1 . . . PN 1 , the optimumPpower allocation must be found as − h i

Popt = max C P (4.14) s.t. P = P and P 0 i. i b i ≥ ∀ Xi 1 Uniform multitone systems have a common bandwidth Bi for each subchannel, and we could drop the subscript i. However, we maintain non-uniform generality so that the analysis applies to the Wavelet Packet Modulation scheme presented in the next chapter.

82 CHAPTER 4. POWER- AND BIT-LOADING

The Lagrange cost functional for minimization is

J = C + λ Pi Pb (4.15) − − ! Xi

= Ci + λ Pi Pb (4.16) − − ! Xi Xi 2 (4.12) 1 Pi Hi = log2 1 + | | + λ Pi Pb . (4.17) − 2 BiNi ! − ! Xi Xi Differentiating,

dJ =! 0 (4.18) dPi dC i = λ i. (4.19) ⇒ dPi ∀

Multi-carrier power loading is thus seen to achieve optimality when all subchannels operate at the same slope on their Capacity-Power curves. Finishing the differentiation,

dC 1 ρ i = i (4.20) dPi 2ln2 (1 + Piρi) (4.19) = λ (4.21) 1 1 Pi + = . (4.22) ⇒ ρi 2ln2 λ

Applying the Kuhn-Tucker conditions on this Lagrangian solution,

1 1 P opt max , 0 (4.23) i ' 2ln2 λ − ρ i which, in the limiting case of an inﬁnite number of subchannels, corresponds exactly to the water pouring solution of (4.9). The capacity at optimal power loading can then be found by back- substitution into (4.12):

1 ρ Copt max log i , 0 . (4.24) i ' 2 2 2ln2 λ

4.1.2 Bit-Loading

The trouble with the preceding analysis is that its usefuleness is somewhat limited, since the Shan- non capacity cannot be achieved with conventional modulation schemes. Thus, even if the power spectrum is optimally shaped (which is not always possible, given the granularity of available power levels) it remains to optimize the actual multitone data rate Rb. As stipulated in Chapter 3,

83 CHAPTER 4. POWER- AND BIT-LOADING

the total number of bits which are modulated onto a multitone symbol equals the sum over the bits on each of the sub-channels, i.e. Rb = i bi where bi is the number of bits modulated onto each tone in a single multitone symboling intervP al. The power-loading solution must be reﬁned from (4.14) to give

Popt = max Rb = max bi P P Xi (4.25) s.t. P = P and P 0 i. i b i ≥ ∀ Xi It is clear that the Kuhn-Tucker solution to this optimization problem is of the identical form to that presented for (4.14) in the previous section, with Ci replaced by bi. However, we do not immediately have a closed form expression for bi as a function of Pi as we did for Ci in (4.12). In fact, for a given subchannel SNR, it is impossible to stipulate the number of zero-error bits that can be modulated on a given tone; rather, we are constrained to specification of the number of bits bi we can transmit at a prescribed probability of error pi, for the given transmit power Pi, and particular modulation scheme (e.g. PAM, QAM, PSK etc. ). Thus, not only can the subchannel powers be optimized in order to maximize the overall rate Rb, but, jointly, the subchannel rates can also be optimized, for a target error probability. This is called bit-loading. Defining the subchannel T bit-allocation vector b = b0 b1 . . . bN 1 , the optimal power-loading solution is refined − from (4.25) to give h i

Popt, bopt = max Rb { } P,b { } (4.26) s.t. P = P and P 0 i. i b i ≥ ∀ Xi

The argument is incomplete however. Each subchannel is constrained to operate at an error rate pi such that the overall system probability of error pe is less than a target probability of error pmax, and this constraint should be included in the optimization. The total probability of error is given by b p p = i i i p . (4.27) e b ≤ max P i i Optimal loading solution is given by P

Popt, bopt = max Rb { } P,b { } s.t. P = P and P 0 i i b i ≥ ∀ (4.28) Xi b p s.t. i i i = p and p 0 i. R max i ≥ ∀ P b

84 CHAPTER 4. POWER- AND BIT-LOADING

−3 −1 1 3 −7 −5 −3 −1 1 3 5 7 00 01 11 10 000 001 011 010 110 111 101 100

bi = 2 bi = 3

FIGURE 4.2: PULSE AMPLITUDE MODULATION AND GRAY CODING FOR 2 AND 3 BIT CONSTELLATIONS.

4.2 Modulation Schemes

Before beginning the discussion on optimal loading, it is instructive to examine the more common modulation schemes used in multi-carrier systems, in order to gain a fundamental understanding of the interraction between bit-allocation, power-loading and error rate for a given modulation. The methods discussed here are Pulse Amplitude Modulation (PAM) which is generally used for real coefficient filterbanks, such as DWMT and WPM, and Quadrature Amplitude Modulation (QAM), which is used for complex coefficient filterbanks, such as FFT-based block-DMT.

4.2.1 Pulse Amplitude Modulation

In Pulse Amplitude Modulation (PAM) multitone systems, the transmitted subchannel symbols Xi are taken from the real ﬁnite alphabet

X 1, 3 . . . 2bi 1 , (4.29) i ∈ − n o where bi is the number of bits per symbol on tone i. These symbols are deﬁned by a Gray coding of the input binary data as illustrated in Fig. 4.2 for bi = 2 and bi = 3. It is assumed that each signalling level is equiprobable, which can be enforced by the use of a scrambler on the binary data (see Chapter 1). It is possible to ﬁnd a closed form expression for the PAM symbol error rate in terms of the bit-allocation, and power-allocation as follows.

Suppose the desired energy of the constellation on subchannel i is . The constellation points X Ei i must be scaled accordingly. The discrete values of Xi in (4.29) can be rewritten

X = 2l 1 2bi , l 1, 2, . . . , 2bi (4.30) i − − ∈ n o The root mean square (rms) value of Xi is thus given by

2 E X2 = E (2l 1 2bi ) , l 1, 2, . . . , 2bi (4.31) i − − ∈ q r h i n o 2bi 2 2 = v (2l 1)2 (4.32) u bi u2 − u Xl=1 t

85 CHAPTER 4. POWER- AND BIT-LOADING

22bi 1 = − . (4.33) r 3

i The transmit signaling levels for energy i are given by scaling Xi by E 2 E E[Xi ] q 3 3 3 ¯ i i bi i Xi 2b E , 3 2b E . . . 2 1 2b E . (4.34) ∈ ( 2 i 1 2 i 1 − 2 i 1) r − r − r −

The received signalling levels Yi in a noise free channel correspond to the transmit levels X¯i scaled by the constant scalar channel gain Hi:

2 2 2 3 i Hi 3 i Hi bi 3 i Hi Yi E | | , 3 E | | . . . 2 1 E | | . (4.35) ∈ s 22bi 1 s 22bi 1 − s 22bi 1   − − − 

Subchannel AWGN ui(n) is added at the receiver. A symbol error occurs ifthe added noise u (n) is greater than half the distance between received signaling levels. The probability of this | i | happening is given by 2 pam 3 i Hi q = u ui(n) E | | . (4.36) i P i | | ≥ s 22bi 1  −   This formula needs a slight modiﬁcation, since at either of the two extreme levels

2 bi 3 i Hi Yi = 2 1 E | | (4.37) − s 22bi 1 − an error can occur in one direction only (see Fig. 4.2). The modiﬁed PAM symbol error probability is thus given by

bi 2 pam 2 1 3 i Hi q = − u ui(n) E | | . (4.38) i 2bi P i | | ≥ s 22bi 1  −   The Probability Distribution Function (PDF) of a zero-mean Gaussian random variable x of vari- 2 ance σx from random variable space X is deﬁned as

1 x2 2σ2 X (x) = e− x . (4.39) 2 P 2πσx p Thus the probability that x is greater than a certain threshold x is given by | | 0

( x x ) = (x x ) + (x x ) (4.40) PX | | ≥ 0 PX ≤ − 0 PX ≥ 0 as illustrated in Fig. 4.3. Using (4.39),

86 CHAPTER 4. POWER- AND BIT-LOADING

PX (x)

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FIGURE 4.3: GAUSSIAN PROBABILITY DISTRIBUTION FUNC- TION, AND GRAPHICAL REPRESENTATION OF TAIL ERROR PROBABILITIES .

1 x2 x0 1 x2 ∞ 2σ2 2σ2 X ( x x0) = e− x dx + e− x dx (4.41) 2 2 P | | ≥ x0 2πσ 2πσ Z x Z−∞ x x2 ∞ 1 2 = 2 p e− 2σx dx. p (4.42) 2πσ2 Zx0 x By deﬁning the complementary error functionp erfc as

∆ 2 ∞ x2 erfc(x) = e− dx, (4.43) √π Zx (4.43) into (4.42) gives

x0 X ( x x0) = erfc . (4.44) 2 P | | ≥ 2σx ! This can be used in (4.38) to give the theoretical probabilityp of symbol error for coherently detected pulse amplitude modulation on subchannel i as

2bi 1 3 H 2 qpam = − erfc Ei | i| . (4.45) i bi 2bi 2 2 s2 1 2σu  − i   Since the subchannel-noise variance is 2 Ni , σui = 2

bi 2 pam 2 1 3 i Hi qi = b− erfc 2b E | | (4.46) 2 i s2 i 1 Ni  −   2bi 1 3SNR = − erfc i (4.47) 2bi 22bi 1 r − !

This formula has been used in MATLAB to replicate the theoretical error-rate curves for PAM provided by Proakis [274, pp. 276] and the result is shown in Fig. 4.4. Furthermore, a discrete-time baseband PAM communication signal was simulated over an AWGN channel and the resulting symbol error rate was measured over a range of SNRs. These measured results are plotted on the same graph and it can be seen that they match the theory well.

87 CHAPTER 4. POWER- AND BIT-LOADING

Probability of Symbol Error for PAM −1 10

b = 1 b = 2 b = 3 b = 4 −2 10

−3 10

−4 10

⊕⊗ Measured PAM Symbol Error Probability: q

−5 Theoretical 10

−6 10 −6 −2 2 6 10 14 18 22 SNR per bit (dB)

FIGURE 4.4: PULSE AMPLITUDE MODULATION SYMBOL ERROR PROBABILITY CURVES.

4.2.2 Quadrature Amplitude Modulation

The analysis for the error rate performance of Quadrature Amplitude Modulation (QAM) is similar to above, except that the transmit constellation points are taken from a ﬁnite complex alphabet i.e. QAM is the extension of PAM to two orthogonal signaling dimensions. In Chapter 2 we wrote

Xi = ai + jbi as the complex subchannel symbol for block-DMT. In order not to confuse this bi with the subchannel bitloading bi, we redeﬁne Xi = αi +jβi. For square QAM, which is the form used in the ADSL and VDSL standards, the real in-phase and quadrature componenets of Xi are chosen from the ﬁnite PAM alphabet described in the previous section

bi α , β 1, 3 . . . (2 2 1) . (4.48) { i i} ∈ − n o Obviously, the number of bits along each dimension is half the total number of bits bi. For even values of bi the constellation is thus square as illustrated in Fig. 4.5 for bi = 2, 4. However, there is a problem for tones loaded with an odd number of bits, since the standards do not allow for a fractional number of bits in either dimension. To overcome this, the constellation is chosen to

ﬁt the nearest available square, with a truncation of extremities, as shown in Fig. 4.6 for bi = 5.

88 CHAPTER 4. POWER- AND BIT-LOADING

βi 3 1000 1001 0001 0000

1010 1011 0011 0010 βi 1

01 1 00 αi −3 −1 1 3 −1 αi 1101 1111 0111 0110 −1 1 11 −1 10 1100 1110 0101 0100 −3

bi = 2 bi = 4

FIGURE 4.5: QUADRATURE AMPLITUDE MODULATION CONSTELLATION FOR bi = 2, 4.

More generally, the numerical identity

2 2 n n 3 n 5 n−3 n−5 2 9 2 − 4 2 − 3 2 2 4 2 2 , n 5 (4.49) ≡ × − × ≡ × − ≥ allows us write the number of constellation points 2bi as a square minus 4 smaller squares. By choosing the 4 corners of the outer square for truncation, we get cross-shaped constellations, e.g. Fig. 4.6.

The special examples bi = 1 and bi = 3 are chosen somewhat differently, since it is more difﬁcult to approximate a square with this reduced number of bits. The ANSI VDSL standard [23] uses

BPSK for bi = 1 and a St. Brigid’s Cross shape for bi = 3 as shown in Fig. 4.7, whereas the

ETSI VDSL standard [22] uses the staggered QAM of Fig. 4.8 for bi = 3 and (somewhat inefﬁ- ciently) does not allow transmission for bi = 1. Once again, Gray coding is applied to ensure that neighbouring constellation points differ by exactly one bit. However, it is noted that cross-shaped constellations for an odd-number of bits cannot be perfectly Gray coded due to their irregular shape [196]; the code used is chosen to minimize the number of symbols which differ by more than one bit. The theory for QAM symbol error rates is developed for square-QAM constellations. In practice, the result is also a tight upper bound on the cross-constellation performance.

The QAM receiver chooses, from the set of possible QAM symbols, the one that is the shortest

Euclidean distance from the received noise-corrupted symbol. For even values of bi, the optimal decision can be made by an independent PAM decision along each dimension. If the QAM symbol qam error probability is qi , then the probability of a correct QAM decision equals the probability of a correct PAM decision along each dimension:

1 qqam = (1 qpam)(1 qpam). (4.50) − i − i − i

bi i It is important to realise that only 2 bits lie along each dimension, communicated with energy E2 pam and this must be accounted for in application of the formula for qi from (4.47).

89 CHAPTER 4. POWER- AND BIT-LOADING

αi 5 10111 10011 11011 11111

10010 00111 00011 01011 01111 11010 3

10110 00110 00010 01010 01110 11110 1

βi −5 −3 −1 1 3 5 −1 10100 00100 00000 01000 01100 11100

10000 00101 00001 01001 01101 11000 −3

10101 10001 11001 11101 −5

bi = 5

FIGURE 4.6: QUADRATURE AMPLITUDE MODULATION CONSTELLATION FOR bi = 5.

αi αi 3 101

0 100 010 000 1 1

βi βi −1 1 −3 −1 1 3 −1 −1 1 011 001 111

110 −3

bi = 1 bi = 3

FIGURE 4.7: ANSI QUADRATURE AMPLITUDE MODULATION CONSTELLATION FOR bi = 1, 3.

αi 3 010 101

100 000 1

βi −3 −1 1 3 −1 011 111

110 001 −3

bi = 3

FIGURE 4.8: ETSI QUADRATURE AMPLITUDE MODULATION CONSTELLATION FOR bi = 3.

90 CHAPTER 4. POWER- AND BIT-LOADING

Probability of Symbol Error for QAM −1 10

−2 10

b = 2 b = 4 b = 6 −3 10

−4 10

QAM Symbol Error Probability: q ⊗⊕ Measured −5 Theoretical 10

−6 10 −6 −4 −2 0 2 4 6 8 10 12 14 16 18 20 22 SNR per bit (dB)

FIGURE 4.9: QUADRATURE AMPLITUDE MODULATION SYMBOL ERROR PROBABILITY CURVES.

We get

qqam = 1 (1 qpam)2 (4.51) i − − i = 2qpam (qpam)2 (4.52) i − i 2qpam for small probability of error (4.53) ≈ i bi 2 2 1 3 SNR = 2 − erfc i . (4.54) bi (2bi 1) 2 2 2 s − ! This provides a tight upper bound for the probability of QAM symbol error [274]. Although the above derivation was for even values of bi it is seen that the resulting error rate expression also gives a tight upper bound for odd values [26]. (4.54) has been used to generate the error-rate curves for QAM in MATLAB and the result is shown in Fig. 4.9. Furthermore, a discrete-time baseband QAM communication signal was simulated over a complex AWGN channel and the resulting symbol error rate was measured over a range of SNRs. These measured results are plotted on the same graph and it can be seen that they match the theory well.

91 CHAPTER 4. POWER- AND BIT-LOADING

4.2.3 A Note on the Modulation Gap

In this section, we give a brief exposition of the theory behind the modulation gap [275, 276]. In the previous chapter it was asserted that the number of bits which could be loaded on a complex subchannel for a given error probability and capacity gap Γ was

SNR b = log 1 + i , (4.55) i 2 Γ or equivalently 1 SNR ˜b = log 1 + i per dimension. (4.56) i 2 2 Γ Γ was deﬁned in Chapter 2 as Γ = γ γ + γ dB but, since there is no design margin or coding m − c d gain in the following analysis, we consider Γ = γm as the modulation gap.

Considering QAM ﬁrst, if the symbol error probability of (4.54) is approximated by the upper bound 3SNR qqam 2erfc i (4.57) i ≈ 2 (2bi 1) s − ! then we can express the symbol error probability per dimension as q˜i = qi/2:

3SNR q˜qam erfc i . (4.58) i  ˜  ≈ v2 22bi 1 u u −  t  ˜bi only occurs once in this formula and we can explicitly isolate

1 SNRi ˜bi log 1 + bits per dimension. (4.59) ≈ 2 2 2 1 qam 2 3 erfc− (q˜i ) !

This is of the same form as (4.56) which gives the the QAM modulation gap for a particular probability of symbol error per dimension by inspection as [30, 208]

2 1 qam 2 Γ erfc− (q˜ ) . (4.60) qam ≈ 3 i 7 For example, the oft quoted result that Γqam = 9.8 dB for q˜i = 10− (e.g. [30]) is derived from this expression.

In repeating the analysis for PAM it is conventional, for comparison purposes, to consider the same signal to noise ratio per dimension as used in the analysis for QAM above. Using the formula for

PAM symbol error probability from (4.47) we get the error probability per dimension as q˜i = qi,

92 CHAPTER 4. POWER- AND BIT-LOADING

SNR Gap to Capacity 10

6 Modulation Gap dB

3 −7 −6 −5 −4 −3 −2 10 10 10 10 10 10 Symbol Error Probability

FIGURE 4.10: MODULATION GAP Γ VS SYMBOL ERROR PROBABILITY PER DIMENSION. since PAM has only one signalling dimension:

pam 3SNRi q˜i erfc ˜ , (4.61) ≈ s2(22bi 1)! − which can be rearranged to give

1 SNRi ˜bi log 1 + . (4.62) ≈ 2 2 2 1 pam 2 3 erfc− (q˜i ) !

By inspection, the modulation gap for PAM is given as

2 1 pam 2 Γ erfc− (q˜ ) , (4.63) pam ≈ 3 i which is the same expression as that for QAM. Thus, bi-bit PAM requires the same SNR per dimension as 2bi-bit QAM for the same probability of error per dimension [274, p. 285]. What this means is that the bandwidth efﬁciency per dimension of PAM is identical to that of QAM [196, p. 432].

Fig. 4.10 shows the value of the modulation gap for different values of the symbol error probability per dimension. It is seen that the gap is higher at lower error rates, since extra power is required to achieve these low error rates. An important question is, how valid is the gap approximation and under what conditions is it not applicable? In order to answer this question, we compared the value of bi(qi) obtained by the gap approximation using (4.56) with the actual correct value of bi(qi) by numerical inversion of the implicit equation for qi(bi) given for PAM in (4.47) and for QAM in (4.54). The results are shown in Fig. 4.11. It is evident that the approximation gap holds well for higher SNRs and constellation sizes, while the degradation in its usefulness at lower values is still relatively minor. It should be noted that constellations with less than 1-bit are seldom permitted in multitone modulation schemes. Also shown on the same graphs is the channel

93 CHAPTER 4. POWER- AND BIT-LOADING

Bits per PAM−Symbol vs. SNR for different Symbol Error Rates Bits per PAM−Symbol vs SNR for different Symbol Error Rates 8 8

7 7

6 6

5 5

4 4 Bits/symbol

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Capacity Capacity 0 0 10 15 20 25 30 35 40 10 15 20 25 30 35 40 SNR (dB) SNR (dB) Bits per QAM−Symbol vs. SNR for different Symbol Error Rates Bits per QAM−Symbol vs SNR for different Symbol Error Rate 14 14

12 12

10 10

8 8 Γ ≈ 9 dB Bits/symbol 6 Bits/symbol 6

q = 10−3 q = 10−3 4 q = 10−4 4 q = 10−4 q = 10−5 q = 10−5 q = 10−6 q = 10−6 2 2

Capacity Capacity 0 0 10 15 20 25 30 35 40 10 15 20 25 30 35 40 SNR (dB) SNR (dB)

FIGURE 4.11: THE TOP TWO GRAPHS SHOW THE MAXI- MUM PAM CONSTELLATION SIZE POSSIBLE FOR A GIVEN SNR AND TARGET ERROR RATE. THE FIRST GRAPH USES THE APPROXIMATION GAP TO DETERMINE bi. THE SECOND USES EXACT NUMERICAL INVERSION OF qi VIA THE MAT- LAB FUNCTION fminbnd. IT IS SEEN THAT THE GRAPHS ARE ALMOST IDENTICAL AND THE APPROXIMATION GAP HOLDS VERY WELL. THE BOTTOM GRAPHS SHOW THE SAME CURVES FOR QAM. AGAIN THE APPROXIMATION IS SEEN TO BE VERY CLOSE. FOR ILLUSTRATIVE PURPOSES, THE “GAP” TO CA- PACITY Γ IS ILLUSTRATED ON THE THIRD GRAPH FOR q˜i = 10−6.

94 CHAPTER 4. POWER- AND BIT-LOADING

1 capacity Ci = 2 log2(1 + SNRi) bits/dimension. This illustrates very well why the capacity “gap” is so called; regardless of the SNR or the constellation size, the SNR gap to capacity is constant for 1 SNRi a ﬁxed symbol error rate. The rate-approximation formula bi = 2 log2(1 + Γ ) bits/dimension is thus extremely useful for speeding up the PAM and QAM loading algorithms presented in the next section.

4.3 Loading Algorithms

Using the results for the theoretical error-rate performance of PAM and QAM signaling from the previous sections, it will now be shown how the multitone system may be optimally loaded to maximize rate Rb for a prescribed maximum error rate pmax.

4.3.1 Optimal Loading

The optimal loading problem is deﬁned in (4.28) and is reiterated here as

Popt, bopt = max Rb { } P,b { } s.t. P = P and P 0 i i b i ≥ ∀ (4.64) Xi b p s.t. i i i = p and p 0 i. R max i ≥ ∀ P b The solution to this optimization was developed by Willink et al. [52] as follows. Deﬁne a La- grangian cost functional for optimization within the system constraints:

J = Rb + λ1 Pi Pb + λ2 (pe pmax) . (4.65) − − ! − Xi Locally optimum b and P are achieved when the gradient J = 0. Differentiation yields ∇ N 1 ∂Rb − ∂Pj ∂pe + λ1 + λ2 = 0 , i (4.66) − ∂bi ∂bi ∂bi ∀ Xj=1 N 1 ∂Rb − ∂Pj ∂pe + λ1 + λ2 = 0 , i. (4.67) − ∂Pi ∂Pi ∂Pi ∀ Xj=1 Simpliﬁcations are possible using subchannel orthogonality:

∂p 1 e = , i (4.68) ∂bi λ1 ∀

95 CHAPTER 4. POWER- AND BIT-LOADING

∂p λ e = 1 , i. (4.69) ∂Pi −λ2 ∀

Gray coding ensures that neighbouring symbols only differ by one bit, which means that, for a small error probability, the probability of more than one bit-error resulting from a symbol detection error is negligible [277, 278]. Thus, we can relate the subchannel symbol error rate qi to the subchannel bit error rate by qi = bipi which gives the overall bit error probability as

q p = i i . (4.70) e R P b We can perform the differentiations of (4.68) and (4.69) using (4.70) to get

N 1 N 1 ∂pe 1 − ∂qj 1 − = 2 qj (4.71) ∂bi Rb ∂bi − Rb Xj=1 Xj=1 N 1 1 − ∂qj = pe , (4.72) Rb  ∂bi −  Xj=1 N 1  ∂p 1 − ∂q e = j (4.73) ∂Pi Rb ∂Pi Xj=1 which, together with the optimality conditions of (4.68) and (4.69), yield

N 1 − ∂qj = ξ1 , i (4.74) ∂bi ∀ Xj=1 N 1 − ∂qj = ξ2 , i (4.75) ∂Pi ∀ Xj=1 by appropriate deﬁnition of constants ξ1 and ξ2.

Having designed for minimal ICI we can assume independent subchannels and so

∂qi ∂qi = ξ1 and = ξ2. (4.76) ∂bi ∂Pi

This result is analogous to Water-Pouring over two variables, which is intuitively pleasing. Maxi- mum bit-rate occurs when all subchannels operate at constant slopes on their error-rate/power and error-rate/bitload curves respectively. If two subchannels operate at different slopes, it proves ben- eﬁcial to remove bits or power from the poor channel and give it to the good, all the time ensuring we never exceed a maximum tolerable error rate pmax, or power budget Pb.

The analysis can be furthered by substitution of the symbol error probability qi for the appropriate modulation scheme in (4.76). Since the derivation for QAM is provided in [52], we proceed with

96 CHAPTER 4. POWER- AND BIT-LOADING

pulse amplitude modulation. We simplify the PAM symbol error probability from (4.47) to

qpam erfc(w ), (4.77) i ' i where 2 3SNRi wi = (4.78) 22bi 1 − 2bi 1 and we have approximated b− 1. This last approximation is good for large b and we will 2 i ≈ i describe a lower threshold on bi for the analysis to hold.

A particular case of the Leibniz integral rule [279, p.11] for integrals to inﬁnity states that

∂ ∂a(z) ∞ f(x) dx = f (a(z)) (4.79) ∂z − ∂z Za(z) Using this in evaluating the partial derivatives of the complementary error funcion (4.43), it is straightforward to obtain the following differentials from (4.77):

pam ∂qi 2ln2 w2 = wie− i (4.80) ∂bi √π pam ∂qi 1 w2 = − wie− i . (4.81) ∂Pi Pi√π

These can be used in (4.76) to give the optimal loading conditions as

w2 wie− i = µ1 (4.82) wi w2 e− i = µ2 (4.83) Pi with µi constant. We achieve locally optimal power and bit-loading independently on each subchannel by iterative solution of the above nonlinear, channel dependent, simultaneous equations in bi and Pi.

The cost function J achieves a strictly global minimum when the Hessian 2J is positive semi- ∇ deﬁnite, corresponding to a convex cost function. The Hessian of J is positive semi-deﬁnite if

2 pam 2 pam 2 pam ∆ ∂ qi ∂ qi ∂ qi = 2 2 0 , i. (4.84) D ∂bi ∂Pi − ∂bi∂Pi ≥ ∀

These partial second derivatives can be computed from (4.80) and (4.81) as

2 pam 2 ∂ qi (2ln2) 2 w2 1 = w e− i w (4.85) ∂b2 √π i i − 2w i i 2 pam 2 ∂ qi 1 wi w2 1 = e− i w (4.86) ∂P 2 √π P i − 2w i i i

97 CHAPTER 4. POWER- AND BIT-LOADING

2 pam 2 pam 2 ∂ qi ∂ qi 1 wi w2 1 = = 2ln2 e− i w (4.87) ∂b ∂P ∂P ∂b − √π P i − 2w i i i i i i which, when substituted in (4.84), give = 0, indicating that the solution will return a global D 2bi 1 minimum. In the case that b is ’‘small”, the approximation b− 1 does not hold, and the i 2 i ≈ minimum is not necessarily global. It can be shown [52] that the crossover point

2 1 wi + 2 bi log , (4.88) ≥ 2 w2 1 i − 2 ! is sufﬁcient for positive-semi-deﬁniteness of the Hessian 2J. More intuitively, the rate function ∇ Rb is generally concave with respect to transmit power and by using its negative in the cost functional J, we will obtain the singular optimal operating point on a convex hull again corresponding to a global minimum, provided most of the tones are in use. As a result Rb will be a maximum under the given constraints, as desired.

Unrestricted power- and bit-loading range and granularity have been assumed in the above optimization. Due to implementational complexity constraints, we are often restricted to ﬁnite integer bitloading b b b , b Z or peak-power constraint P P for all used tones [280]. min ≤ i ≤ max i ∈ i ≤ max It is reported that the deviation from optimality imposed by these restrictions is within the bounds of quantization noise seen at the DAC/ADC of the digital system, and as such can be ignored. However, ﬁne gain-scaling across the tones, whereby power is scaled to readjust the subchannel error-rates, can largely restore performance where this does not hold.

The solution to the nonlinear optimization of (4.82) and (4.83) can be achieved by convex simplex nonlinear programming techniques [281, ch. 10],[282]. An appropriate algorithm was given by Willink et al. [52], and is detailed explicitly in Table 4.1. This involved procedure can be summarized in words as follows:

1. Pick a feasible initial solution such as ﬁxed error probability and power on all tones. For

a ﬁxed power, determine the change in overall bit-error probability pe which would result from a small increase or decrease of the rate on any subchannel.

2. Increase the rate on the subchannel with the least corresponding error-rate degradation and decrease the rate on the subchannel with the best error-rate improvement (provided that there is a net error rate improvement). The overall data rate and power have not changed after this bit-swapping, but there is now less overall probability of error. Repeat this step until there is no further possible reduction of error probability.

3. Since we are operating below the target error rate, increase the number of bits on the subchannel which gives the lowest increase in error probability. Repeat until the error probability matches the target value. We have increased the rate at no cost in power or error- performance.

98 CHAPTER 4. POWER- AND BIT-LOADING

TABLE 4.1: WILLINK’S OPTIMAL LOADING ALGORITHM

− ∆ Define ∆p = ∆b[qi P ] b,i |{b0 ... bi ... bN−1, } ∆b[qi P ] − |{b0 ... bi−∆b ... bN−1, } + ∆ ∆p = ∆b[qi P ] b,i |{b0 ... bi+∆b ... bN−1, } ∆b[qi P ] − |{b0 ... bi ... bN−1, } − ∆ ∆p = ∆P [qi b ] P,i |{ ,P0 ... Pi−∆P ... PN−1} ∆P [qi b ] − |{ ,P0 ... Pi ... PN−1} + ∆ ∆p = ∆P [qi b ] P,i |{ ,P0 ... Pi ... PN−1} ∆P [qi b ] − |{ ,P0 ... Pi+∆p ... PN−1} Step 1 Initialize with feasible, suboptimal solution. e.g. Pi = P /N, pi = pe , i b ∀ Step 2 REPEAT + + Calculate ∆pb,I = mini ∆pb,i − − Calculate ∆pb,J = maxi ∆pb,i If ∆p+ < ∆p− transfer ∆b bits | b,I | | b,J | from subchannel J to subchannel I UNTIL ∆p+ ∆p− | b,I | ≥ | b,J | Step 3 REPEAT + + Find ∆pb,I = mini ∆pb,i + If ∆pb,I pmax pe increment bI by ∆b. + ≤ − UNTIL ∆p > pmax pe b,I − Step 4 REPEAT + + Calculate ∆pP,I = maxi ∆pP,i − − Calculate ∆pP,J = mini ∆pP,i If ∆p+ > ∆p− transfer power ∆P | P,I | | P,J | from subchannel J to subchannel I UNTIL ∆p+ ∆p− | b,I | ≤ | b,J | Step 5 Repeat step 3 Step 6 REPEAT from step 2 UNTIL no further change in b or P

99 CHAPTER 4. POWER- AND BIT-LOADING

4. For a ﬁxed rate, determine the change in overall bit-error probability pe which would result from a small increase or decrease of the power on any subchannel. Increase the power on the subchannel with the biggest error-rate improvement and decrease the power on the subchannel with the least error-rate degradation (provided that there is a net error rate improvement). The data rate and power have not changed after this power-swapping, but there is now less overall probability of error. Repeat this step until there is no further possible reduction of error probability.

5. Repeat step 3.

6. Repeat optimization from step 2, until no further increase in bit-rate is possible.

4.3.2 Near-Optimal Loading

The previous optimization requires a search over a very large parameter space. By removing a vector of free variables from the system, namely the subchannel error probability vector p =∆ T [p0, p1 . . . pN 1] , there is a signiﬁcant reduction in convergence time, at the cost of a reduction − in the overall bit-rate R . Imposing the constraint p = p , i, ensures p = p , and it can b i max ∀ e max be removed from the Lagrange functional of (4.65). Our cost functional is now

J = Rb + λ Pi Pb . (4.89) − − ! Xi If we imagine that the power required for a given error probability is a function of the bit-rate, minimization of the functional in (4.89) occurs for

∂J = 0 , i (4.90) ∂bi ∀ which, by subchannel orthogonality, gives

∂P 1 i = , i. (4.91) ∂bi λ ∀

1 Maximum bit-rate occurs when all subchannels operate at the same slope λ on their power/bitload curves. If two subchannels operate at different slopes, it proves beneﬁcial to remove bits from the poor channel and give them to the good, which also affects the power loading for a ﬁxed error probability. This is known as bit-tightness [47]. The advantage of this approach is that Pi(bi) can usually be obtained explicitly. For example, from (4.47) for PAM,

2bi bi 2 2 1 1 2 bipe Pi(bi) = − erfc− . (4.92) 3ρ 2bi 1 i −

100 CHAPTER 4. POWER- AND BIT-LOADING

TABLE 4.2: CAMPELLO’S NEAR-OPTIMAL LOADING

− ∆ Define ∆P = ∆b[Pi ] b,i |{b0 ... bi ... bN−1} ∆b[Pi ] − |{b0 ... bi−∆b ... bN−1} + ∆ ∆P = ∆b[Pi ] b,i |{b0 ... bi+∆b ... bN−1} ∆b[Pi ] − |{b0 ... bi ... bN−1} Step 1 Initialize with feasible, suboptimal solution. e.g. Pi = P /N, pi = pe , i b ∀ Step 2 REPEAT + + Calculate ∆Pb,I = mini ∆Pb,i − − Calculate ∆Pb,J = maxi ∆Pb,i If ∆P + < ∆P − transfer ∆b bits | b,I | | b,J | from subchannel J to subchannel I UNTIL ∆P + ∆P − | b,I | ≥ | b,J | Step 3 REPEAT + + Find ∆Pb,I = mini ∆Pb,i + If ∆Pb,I Pb i Pi increment bI by ∆b. + ≤ − UNTIL ∆P > P Pi b,I b − Pi Step 4 REPEAT from step 2PUNTIL no further change in b or P

This solution, provided by Campello [47,55], lends itself nicely to a convex simplex optimization similar to Willink’s. The algorithm is presented in Table 4.2.

An identical solution, provided by Krongold et al. [54, 57], is obtained by differentiating with respect to the subchannel power instead of the bitloading. From (4.89)

∂R ∂b (P ) b = i i = λ , i. (4.93) ∂Pi ∂Pi ∀

The reason the result is identical is that the cost function is convex, so whether all subchannels operate at the same slopes on their power/rate curves or their rate/power curves is irrelevant. What is not provided in [57] is the following simpliﬁcation. In Section 4.2.3 we saw that bi(Pi) 1 ' 2 log2 (1 + ρiPi/Γ) bits per dimension, for a ﬁxed error probability. This is easily differentiated to provide 1 Γ P opt max , 0 (4.94) i ' 2ln2 λ − ρ i (we have included the Kuhn-Tucker conditions). Back-substitution gives:

ρ bopt max log i , 0 . (4.95) i ' 2 2Γln2 λ All that must be determined is λ which is chosen so that i Pi = Pb. A fast bisection algorithm is provided in [57] to solve for the optimal slope λ and thisPis listed in Table 4.3.

101 CHAPTER 4. POWER- AND BIT-LOADING

TABLE 4.3: KRONGOLD’S NEAR-OPTIMAL LOADING. A BI- SECTION ALGORITHM IS USED TO FIND λ.

Step 1 Using (4.94) pick λl λu ≤ such that Pi(λl) Pb Pi(λu) i ≤ ≤ i i Pi(λu)−Pi(λl) Step 2 λnew bP(λu)−b (λ ) P ← P i i i l Step 3 if PiP(λnew) > P then λu λnew i b ← else λl λnew P ← Step 4 REPEAT from 2 UNTIL i Pi(λnew) = Pb P

4.3.3 Default Loading

By removing the vector of free variables p, corresponding to the subchannel error rates, from the optimization, the previous near-optimal algorithms should benefit from a faster convergence time. This concept can be extended, by removing the power-loading free variable vector P to provide the simplest loading algorithm, which we will refer to as default loading. Power and error probability are fixed across all the tones. Since there are no free variables, bi can be determined explicitly from (4.56). For more efficiency, this can be done offline and the values provided in lookup- tables. The default loading algorithm provides a good starting point for all the other algorithms, since it is almost instantaneous. Only minimial optimization is required insofar as power must be transferred from unused tones to the used tones. The algorithm is summarized as follows:

1. Load power Pi = Pb/Nu on all used tones. Nu is the number of used tones and initially N = N. The probability of error is chosen p = p , i. From (4.56) determine b u i max ∀ i on each tone.

2. Determine the set of tones I which have b < b , i I, where b is the minimum i min ∈ min allowed number of bits per subchannel use (usually bmin = 1 although for some QAM

schemes bmin = 2).

3. Turn off the tone i I with minimum b . and redistribute the saved power among the ∈ i remaining tones. The number of used tones has reduced by one: N new N old 1 u ← u − 4. Repeat from 1 until all tones have b b or b = 0. i ≥ min i

The bit rate Rb = i bi is logarithmically dependent on power as we saw in Fig. 4.11. Fur- thermore, it is also approximatelyP logarithmically dependent on error rate, as illustrated for QAM signalling in Fig. 4.12. However, the bit-rate is linearly dependent on bit-loading. What this means is that the overall data rate is changed far more quickly by a change of subchannel rate, than by a change of subchannel power or by a change in subchannel error-rate. For example, from

102 CHAPTER 4. POWER- AND BIT-LOADING

Bits/Symbol vs Symbol Error Rate for Fixed SNR 10

9 SNR = 30dB

8 SNR = 25dB 7 SNR = 20dB 6

5 SNR = 15dB

Bits/Symbol 4 SNR = 10dB 3

2 SNR = 5dB

1 SNR = 0dB 0 −6 −5 −4 −3 −2 −1 10 10 10 10 10 10 Symbol Error Rate

FIGURE 4.12: THE NUMBER OF POSSIBLE QAM BITS PER SYMBOL AS A FUNCTION OF THE SYMBOL ERROR RATE FOR DIFFERENT SNRS.

Fig. 4.11.c it is seen that the SNR has to increase by about 3dB for an increase of 1 bit per QAM 6 symbol at a symbol error probability of 10− per dimension. This corresponds to a doubling in power on that subchannel. Similarly from Fig 4.12, the allowable error rate would have to be increased from 1 bit per million to 1 bit per thousand at an SNR of 30 dB, just to allow an extra bit per symbol. Since rate appears far more sensitive to bit-loading than either power- or error- loading, we expect the default scheme, which only focuses on bit-loading, to approach optimal performance. This will be seen to be the case in our simulations.

4.3.4 Constant Power Bit-Loading

All the loading algorithms presented above form a subset of the set of possible loading algorithms, based on different constraints on the free variable vectors b, P and p. For completeness, Table 4.4 lists all possible loading combinations. Numbers 3, 6 and 8 have been discussed in detail in this chapter. Of the others, only number 7 is of interest since it provides useful rate optimization and has not been studied previously in the literature. In this instance the vector of free variables we remove from the optimization is P. We impose P = P /N , i and optimize the bit and error i b ∀ distributions over the tones (in the case of unused tones, Pi = 0 and the extra power is divided

103 CHAPTER 4. POWER- AND BIT-LOADING

TABLE 4.4: DIFFERENT LOADING ALGORITHMS BASED ON OPTIMIZATION OVER THE BITLOADING b, THE POWER- LOADING P AND THE EROOR-RATE LOADING p.

1. Fix b, P, p No loading Not feasible. The error rate cannot be speciﬁed independently of the power- and bit-loading. 2. Fix b, p Load P This is margin maximization, since Rb is ﬁxed. We are only interested in rate maximization in this thesis. 3. Fix P, p Load b Default loading. 4. Fix P, b Load p Not feasible (see number 1). 5. Fix b Load p, P Margin maximization again (see 2). 6. Fix p Load b, P Solutions by Kalet [26], Krongold et al. [57] and Campello [55]. 7. Fix P Load b, p This has not been studied before, to the best of the author’s knowledge. 8. Fix Nothing Load b, P, p Willink’s global optimization [52]. between the used tones). The Lagrange functional of (4.65) becomes

J = R + λ (p p ) . (4.96) − b e − max

Setting J = 0 yields ∂pe = λ similar to the previous methods. The solution to this algorithm ∇ ∂bi is achieved by exactly the same method as Willink’s algorithm, but omitting steps 4 and 5 since power optimization is not required. Alternatively, a fast bisection algorithm similar to that provided by Krongold et al. can be used to ﬁnd optimal λ, since all tones operate at the same slope on their error-rate/bit-rate curves. However, explicit formulae for bi(λ) and hence qi(λ) do not exist in this case and they must be provided by lookup table.

It is not expected that this new solution will perform signiﬁcantly better than that provided by the default loading, since at the error probabilities of interest in multitone communication (10 7 − → 5 10− ), it can be seen that the bitloading varies only minimally (cf. Fig. 4.12) with allowable error rate. However, we provide this solution for completeness on the topic.

4.4 Simulation Results

In this section we examine the loading algorithms previously introduced when applied to various test channels. The ﬁrst test channel is that provided by Willink [52], which is a very simple

104 CHAPTER 4. POWER- AND BIT-LOADING

−5

−10

−15 Channel Magnitude Squared (dB)

−20 0 0.2 0.4 0.6 0.8 1 Normalized Frequency

FIGURE 4.13: WILLINK’S TEST CHANNEL. channel, but is useful for comparative purposes. Subsequently we apply the loading algorithms to the DSL test channels, to see how they perform. The ﬁgures of merit used to compare the different solutions are

1. a measure of time to completion of the loading, i.e. algorithmic complexity

2. the achievable bit-rate upon completion of the loading.

4.4.1 Willink’s Test Channel

The simple test channel provided by Willink et al. is bandlimited to 1 Hz. The subchannel attenuations range up to 18 dB. The channel characteristic is assumed to be piecewise constant over each of N=256 subchannels. Although no explicit transfer function is provided, it is found by inspection that the channel model

jω 6ω 6(π ω) 20 log H e = 5 5 cos(ω) 8 e− + e− − dB , ω [0, π) (4.97) 10 − − − ∈ h i is a good approximation. This test channel is illustrated in Fig. 4.13. The following algorithms have each been used to load this test channel:

1. Willink’s optimal loading (Table 4.4 no. 8).

105 CHAPTER 4. POWER- AND BIT-LOADING

2. Campellos near-optimal loading (Table 4.4 no. 6).

3. New near-optimal loading (Table 4.4 no. 7).

4. Default loading (Table 4.4 no. 3).

5 The target bit error probability is pmax = 10− . The noise spectral density is white and is chosen as

-40 dB/Hz. The power budget is Pb = 30 dB. For fairness of comparison, we use the same starting point for each of the algorithms tested, namely that obtained by default loading. Furthermore, a generic greedy algorithm is used to implement each of the loading schemes. This allows us compare convergence rates in terms of the number of iterations, or simply CPU-time elapsed, since the complexity of each iteration is identical. We use elapsed CPU-time as our figure of merit. The figure is given for a 2.2 GHz Intel Pentium processor running MATLAB , with 512 Mb RAM. We have only included results for PAM-loading schemes. In fact, the results are identical for each dimension of a QAM system since the spectral efficiency of QAM per dimension is equivalent to that of PAM.

The results of the loading are provided in Fig’s 4.14–4.17. The purpose of each algorithm can be seen very clearly. Willink’s solution has a non-uniform distribution of bits, power, and error across the tones (compares well to her published data). Campello’s solution constrains the error distribution to uniform. The new loading algorithm imposes uniform power. The default loading scheme only optimizes the bit-loading. The relative performances of the different schemes are given in Table 4.5. It is seen that all algorithms complete relatively quickly, since the channel is

TABLE 4.5: RELATIVE PERFORMANCE OF FOUR LOADING ALGORITHMS WHEN USED TO LOAD WILLINK’S TEST CHAN- NEL.

Bitrate Rb Algorithm (per multitone symbol) CPU Time

1. Willink 1.2648 kb 6.5 sec 2. Campello 1.2646 kb 3.5 sec 3. New 1.2648 kb 5.1 sec 4. Default 1.2645 kb 0.5 sec quite simple. The default scheme is considerably quicker than any other method. More interest- ingly, the achieved loading is seen to be almost identical for all schemes. The optimal scheme has not previously been compared to near-optimal schemes, and this initial result would suggest that optimal loading is an “overkill” solution; even the default solution achieves approximately the same bit-rate as Willink’s optimal loading. This was predicted in the previous section, where we

106 CHAPTER 4. POWER- AND BIT-LOADING

40 9 8 35 7 6 5 30 4 3 25 2 Bitloading

Bits per Subchannel 1 Capacity: 1/2 log(1+SNR) 20 0 0 0.5 1 0 64 128 192 256

Channel Gain to Noise Ratio (dB) Normalized Frequency Subchannel Index −3 x 10 −4 0 −4.5 −2 −4 −5 −6 −8 (Error Probability) −5.5 10 log

Subchannel Power (dB) −10 −6 0 64 128 192 256 0 64 128 192 256 Subchannel Index Subchannel Index

FIGURE 4.14: WILLINK’S OPTIMAL LOADING OF HER TEST CHANNEL; RESULTS IN NON-UNIFORM POWER, NON- UNIFORM BIT-RATES AND NON-UNIFORM ERROR-RATES ACROSS THE TONES.

107 CHAPTER 4. POWER- AND BIT-LOADING

40 9 8 35 7 6 5 30 4 3 25 2 Bitloading

Bits per Subchannel 1 Capacity: 1/2 log(1+SNR) 20 0 0 0.5 1 0 64 128 192 256

Channel Gain to Noise Ratio (dB) Normalized Frequency Subchannel Index

−4 0 −4.5 −0.02

−0.04 −5 −0.06

(Error Probability) −5.5 −0.08 10 log Subchannel Power (dB) −6 0 64 128 192 256 0 64 128 192 256 Subchannel Index Subchannel Index

FIGURE 4.15: CAMPELLO’S NEAR-OPTIMAL LOADING OF WILLINK’S TEST CHANNEL; RESULTS IN NON-UNIFORM POWER, NON-UNIFORM BIT-RATES AND UNIFORM ERROR- RATES ACROSS THE TONES.

108 CHAPTER 4. POWER- AND BIT-LOADING

40 9 8 35 7 6 5 30 4 3 25 2 Bitloading

Bits per Subchannel 1 Capacity: 1/2 log(1+SNR) 20 0 0 0.5 1 0 64 128 192 256

Channel Gain to Noise Ratio (dB) Normalized Frequency Subchannel Index

1 −4

0.5 −4.5

0 −5

−0.5 (Error Probability) −5.5 10 log Subchannel Power (dB) −1 −6 0 64 128 192 256 0 64 128 192 256 Subchannel Index Subchannel Index

FIGURE 4.16: NEW LOADING ALGORITHM APPLIED TO WILLINK’S TEST CHANNEL; RESULTS IN UNIFORM POWER, NON-UNIFORM BIT-RATES AND NON-UNIFORM ERROR- RATES ACROSS THE TONES.

109 CHAPTER 4. POWER- AND BIT-LOADING

40 9 8 35 7 6 5 30 4 3 25 2 Bitloading

Bits per Subchannel 1 Capacity: 1/2 log(1+SNR) 20 0 0 0.5 1 0 64 128 192 256

Channel Gain to Noise Ratio (dB) Normalized Frequency Subchannel Index

1 −4

0.5 −4.5

0 −5

−0.5 (Error Probability) −5.5 10 log Subchannel Power (dB) −1 −6 0 64 128 192 256 0 64 128 192 256 Subchannel Index Subchannel Index

FIGURE 4.17: DEFAULT LOADING OF WILLINK’S TEST CHANNEL; RESULTS IN UNIFORM POWER, NON-UNIFORM BIT-RATES AND UNIFORM ERROR-RATES ACROSS THE TONES.

110 CHAPTER 4. POWER- AND BIT-LOADING

saw that the overall bit-rate is far more sensitive to changes in subchannel rates, than to changes in sub-channel power or subchannel error-rates. To see if this characteristic manifests itself in a more severe channel, we present results for loading of the DSL test channels.

4.4.2 Loading for the DSL channels

A simulation setup similar to the previous chapter is used to test the four loading algorithms of interest. The eight standard DSL carrier-serving-area (CSA) are our test channels. We consider a

256 tone system, each tone having the same bandwidth Bi = 4.3125kHz. The transmitter power budget is Pb = 20dBm, and the added white gaussian noise has spectral density -120 dBm/Hz. 5 The target bit error probability is pmax = 10− . Again, we use the same starting point for each of the algorithms tested, namely that obtained by default loading. The same generic greedy algorithm is used to implement each of the loading schemes.

TABLE 4.6: COMPARISON OF 4 LOADING ALGORITHMS FOR THE 8 ADSL TEST CHANNELS DEFINED IN CHAPTER 2

Willink’s Loading CSA-1 CSA-2 CSA-3 CSA-4 CSA-5 CSA-6 CSA-7 CSA-8

Rb (kb/symbol) 0.9456 1.0791 0.8986 0.8808 0.9542 0.8543 0.8569 0.7911 CPU-time (sec) 1,793 55.3 438 166 717 264 406 1,582

Campello’s Loading CSA-1 CSA-2 CSA-3 CSA-4 CSA-5 CSA-6 CSA-7 CSA-8

Rb (kb/symbol) 0.9451 1.0775 0.8977 0.8789 0.9533 0.8533 0.8566 0.7910 CPU-time (sec) 5.0 2.9 5.6 4.5 5.0 4.2 5.2 6.1

New Loading CSA-1 CSA-2 CSA-3 CSA-4 CSA-5 CSA-6 CSA-7 CSA-8

Rb (kb/symbol) 0.9417 1.0790 0.8915 0.8792 0.9475 0.8505 0.8507 0.7741 CPU-time (sec) 28.2 19.4 32.6 25.7 28.7 32.7 34.6 31.5

Default Loading CSA-1 CSA-2 CSA-3 CSA-4 CSA-5 CSA-6 CSA-7 CSA-8

Rb (kb/symbol) 0.9385 1.0774 0.8885 0.8769 0.9450 0.8474 0.8475 0.7686 CPU-time (sec) 0.55 0.62 0.57 0.58 0.57 0.56 0.57 0.56

The performance of each of the four loading algorithms is presented in Table 4.6. In the usual case that bitloading must take integer values, some rounding and subsequent gain scaling to restore error performance, must be undertaken. However, since this nonlinearity will bias the comparison somewhat, we only consider inﬁnite granularity loading 2. An immediate observation is that there

2The algorithms achieve identical rates when integer loading is enforced

111 CHAPTER 4. POWER- AND BIT-LOADING

100 12 11 90 10 9 80 8 7 70 6 5 60 4

Bits per Subchannel 3 50 2

Channel Gain to Noise Ratio (dB) Bitloading 1 Capacity: 1/2 log(1+SNR) 40 0 0 0.2 0.4 0.6 0.8 1 0 64 128 192 256 Normalized Frequency Subchannel Index

−4

−34.08

−34.09 −4.5

−34.1 −5 −34.11 (Error Probability)

−34.12 10

log −5.5 Subchannel Power (dB) −34.13

−34.14 −6 0 64 128 192 256 0 64 128 192 256 Subchannel Index Subchannel Index

FIGURE 4.18: WILLINK’S OPTIMAL LOADING OF ADSL TEST LOOP NO. 2. is very little to be gained by implementing a nonlinear search for the optimal loading, on any of the test channels considered. In fact default loading achieves over 99% of the optimal rate on all occasions without the need for nonlinear programming. As a precaution, we considered alternative starting points for the nonlinear algorithms. We achieved the same solutions as before, but the algorithms took longer to converge, indicating that the default loading solution is in fact close to optimum. The new algorithm presented is also seen to provide near optimum rates. but takes longer to converge than Campello’s algorithm. However, we conclude that the new algorithm is still not a particularly worthwhile solution (at least for the channels considered) when compared to the default solution and we include it more for completeness than for innovation.

For illustration purposes, we include the loading plots for CSA-Loop#2 for each of the algorithms tested. The channel GNR, bitloading, power-loading, and the resulting error probability distributions are shown in Fig.’s 4.18–4.21. Once again, the proximity of solutions from the various algorithms is strongly evidenced by these graphs.

112 CHAPTER 4. POWER- AND BIT-LOADING

100 12 11 90 10 9 80 8 7 70 6 5 60 4

Bits per Subchannel 3 50 2

Channel Gain to Noise Ratio (dB) Bitloading 1 Capacity: 1/2 log(1+SNR) 40 0 0 0.2 0.4 0.6 0.8 1 0 64 128 192 256 Normalized Frequency Subchannel Index

−34 −4

−34.1 −4.5 −34.2

−34.3 −5 −34.4 (Error Probability)

−34.5 10

log −5.5 Subchannel Power (dB) −34.6

−34.7 −6 0 64 128 192 256 0 64 128 192 256 Subchannel Index Subchannel Index

FIGURE 4.19: CAMPELLO’S NEAR-OPTIMAL LOADING OF ADSL TEST LOOP NO. 2

113 CHAPTER 4. POWER- AND BIT-LOADING

100 12 11 90 10 9 80 8 7 70 6 5 60 4

Bits per Subchannel 3 50 2

Channel Gain to Noise Ratio (dB) Bitloading 1 Capacity: 1/2 log(1+SNR) 40 0 0 0.2 0.4 0.6 0.8 1 0 64 128 192 256 Normalized Frequency Subchannel Index

−34 −4

−34.05 −4.5

−34.1 −5 (Error Probability) 10

−34.15 log −5.5 Subchannel Power (dB)

−34.2 −6 0 64 128 192 256 0 64 128 192 256 Subchannel Index Subchannel Index

FIGURE 4.20: NEW LOADING ALGORITHM APPLIED TO ADSL TEST LOOP NO. 2.

114 CHAPTER 4. POWER- AND BIT-LOADING

100 12 11 90 10 9 80 8 7 70 6 5 60 4

Bits per Subchannel 3 50 2

Channel Gain to Noise Ratio (dB) Bitloading 1 Capacity: 1/2 log(1+SNR) 40 0 0 0.2 0.4 0.6 0.8 1 0 64 128 192 256 Normalized Frequency Subchannel Index

−34 −4

−34.05 −4.5

−34.1 −5 (Error Probability) 10

−34.15 log −5.5 Subchannel Power (dB)

−34.2 −6 0 64 128 192 256 0 64 128 192 256 Subchannel Index Subchannel Index

FIGURE 4.21: DEFAULT LOADING OF ADSL TEST LOOP NO. 2

115 CHAPTER 4. POWER- AND BIT-LOADING

4.5 Chapter Summary

This chapter reviewed some information theoretic aspects of PAM and QAM modulation, speciﬁ- cally

Capacity of ISI + AWGN channel • Water-Pouring • Multi-tone water-pouring • PAM and QAM error rates • The modulation gap • Multitone loading •

We have presented and compared existing optimal and near optimal loading algorithms for general multitone transmitters. This comparison is necessary since, despite the wealth of literature on various loading schemes, there is a paucity of information on their merits relative to each other. Furthermore we have presented alternative near-optimal solutions and included them in the comparison. It is seen that the nonlinear optimization required for true- or near- optimality is somewhat superﬂuous; near-optimal loading can be achieved by direct application of an explicit loading function which requires no iteration, a scheme we have termed default loading. It is recommended that default loading is the best solution to the loading problem in terms of value for complexity, at least for the types of dispersive channel encountered in the DSL environment.

116 Chapter 5

Wavelet Packet Modulation

The reasonable man adapts himself to the world The unreasonable one persists in trying to adapt the world to himself Therefore all progress depends on the unreasonable man

— George Bernard Shaw, 1903

Multi-carrier schemes divide the broadband dispersive channel into parallel, independent, AWGN subchannels of equal bandwidth and interspacing. However, subchannel decomposition cannot be achieved without some spectral leakage. Thus, there is a tradeoff between reducing ISI (more subchannels) and reducing ICI (fewer subchannels). Conventional multicarrier schemes such as DMT and DWMT do not capitalize on this fact, and blindly decompose the spectrum using a uniform spectral distribution. Subject to a complexity constraint, such as a ﬁnite allowable number of subchannels, or maximum frequency resolution, a uniform subchannel distribution may not be optimal, particularly if it results in wasting resources on subdividing parts of the spectrum which are already approximately nondispersive, or if a large number of subchannels operate in adverse conditions, such as deep fades or channel nulls. By intelligent nonuniform allocation of subchannels we may achieve a scenario whereby the combined ISI and ICI is minimized for a given communications channel. The concept is illustrated in Fig. 5.1, where the nonuniform subchannel allocation gives a better approximation to N independent AWGN channels.

One means of achieving a nonuniform subchannel distribution is by wavelet packet modulation, which can be implemented by an iterated two-channel transmultiplexer. In this chapter we present the concept of wavelet packet modulation from first principles, together with a review of the history- and state-of-the-art. Optimal tree structuring is identified as an open problem for multitone communications. We present a training based tree selection algorithm which will achieve minimum ISI/ICI for a given communication channel. The algorithm requires no ad hoc tuning and converges within a fixed known time for a maximum tree depth.

117 CHAPTER 5. WAVELET PACKET MODULATION

a) SNR(ω)

ω b) SNR(ω)

FIGURE 5.1: a) UNIFORM SUBCHANNEL ALLOCATION b) NONUNIFORM SUBCHANNEL ALLOCATION

5.1 Introduction

5.1.1 The Two-Channel Transmultiplexer

The synthesis bank of a two-channel transmultiplexer modulates independent data signals x0(n) and x1(n) onto the one signal x(n) as demonstrated in Fig. 5.2. Upsampling xi(n) by 2 cre- jω 2 ates a copy of its spectrum Xi e within the band to produce the signal xi↑ (n) with spectrum 2 jω Xi↑ e , as illustrated in Fig’s 5.3.a and 5.3.b for i = 0. The job of the synthesis-bank ﬁlters jω 2 jω gi(n)is to select the appropriate image Si e from Xi↑ e for subchannel i, as illustrated by Fig’s 5.3.c and 5.3.d. The transmitted signal is given by the sum of the modulated signals x(n) = s0(n) + s1(n). Filtering by fi(n) at the analysis bank selects the appropriate subband

s0 r0 ˆ x0 ↑ 2 g0 f0 ↓ 2 x0 x SYNTHESIS ANALYSIS ˆ x1 ↑ 2 g1 f1 ↓ 2 x1 s1 r1

FIGURE 5.2: A TWO-CHANNEL TRANSMULTIPLEXER signal ri(n) for subchannel i and the detected data xî(n) is then attained by downsampling. We design the filters gi(n) and fi(n) such that our estimates xî(n) are good — ideally they are exact, which is the condition for Perfect Reconstruction (PR):

xˆ (n ∆ ) = x (n) , i (5.1) i − s i ∀

(∆s is a system delay). A two channel transmultiplexer which gives perfect reconstruction is commonly referred to as a Quadrature Mirror Filter- (QMF) bank. g (n), f (n) and g (n), f (n) { 0 0 } { 1 1 } form orthogonal low- and high-pass quadrature mirror pairs, respectively. For example an orthogonal transmultiplexer can be deﬁned by

f (n) = g ( n) i = 0, 1 (5.2) i i −

118 CHAPTER 5. WAVELET PACKET MODULATION

X ejω 0

(a)

−ω − π π ω 2 2 X↑2 ejω 0

(b)

−ω ω G ejω 0

(c)

−ω ω S ejω 0

(d)

−ω ω −π − π 0 π π 2 2

FIGURE 5.3: SPECTRAL INTERPRETATION OF TWO- CHANNEL TRANSMULTIPLEXER: a) BASEBAND SPECTRUM OF 0th-SUBCHANNEL DATA. b) EFFECT OF UPSAMPLING BY 2. c) FILTER RESPONSE. d) MODULATED SUBCHANNEL DATA.

g (n) = ( 1)ng (K 1 n), (5.3) 1 − 0 − − where K is the filter order. Appropriate choice of the fundamental filter g0(n) will then specify the system e.g. Haar wavelets (K = 2) form the simplest PR system, while longer wavelets (e.g. Daubechie’s) result in lapped transforms with better spectral containment at the expense of time domain pulse spreading [39, 62]. In this section, the PR conditions for a QMF-bank are derived from first principles.

When we upsample a signal by two we shift all signal elements by one index and introduce zeros in the gaps produced. Similarly, when a signal is downsampled by two, half of the samples are discarded, and the discrete-time indexing is changed on the remaining half. It is instructive to track signal operations upon each of these signal components or polyphases as they are more commonly known. We work through some polyphase identities essential to the exposition and which ultimately enhance our understanding of the modulation scheme.

5.1.2 Polyphase Analysis

First we consider the case of upsampling followed by ﬁltering, which is found in the synthesis bank of the transmultiplexer. In this study it is assumed that the upsampler is a zero-phase process, and that the downsampler is at a phase of one. For a sampling ratio of two, what this means is that the upsampler shifts the signal to even-numbered sampling instants n = 0, 2, 4 . . . and the { } downsampler retains samples at odd-valued sampling instants n = 1, 3, 5 . . . . It is perfectly { } feasible to perform the following analysis with both zero-phase, or both unit-phase up- and downsamplers. However, the calculations are less instructive than the tidy solution afforded by this

119 CHAPTER 5. WAVELET PACKET MODULATION

x(n) x↑2(n) s(n) X(z) S(z) ↑ 2 g(n) Geven(z) ↑ 2

⇔ z−1

Godd(z) ↑ 2 a) b)

FIGURE 5.4: a) UPSAMPLING FOLLOWED BY FILTERING. b) POLYPHASE IMPLEMENTATION. assumption. Note that it is not usually necessary to make this assumption in the polyphase theory for the filterbank configuration (analysis bank comes first) since, in that case, the phase selection elements can be made to cancel at their connections. Assuming causal FIR filter g(n) in Fig. 5.4.a, signal s(n) is given by

2 s(0) g(0) x↑ (0) 2  s(1)  g(1) g(0)  x↑ (1) 2  s(2)  = g(2) g(1) g(0)  x↑ (2) (5.4)      2   s(3)  g(3) g(2) g(1) g(0)  x↑ (3)        .   ..   .   .   .  .              g(0) x(0) g(1) g(0)   0  = g(2) g(1) g(0)  x(1) (5.5)     g(3) g(2) g(1) g(0)   0       ..   .   .  .          g(0) x(0) g(1)  x(1) = g(2) g(0)  x(2) (5.6)     g(3) g(1)  x(3)      ..   .   .  .          Consider the odd and even phases of s(n):

s(0) g(0) x(0)

∆ s(2) g(2) g(0)  x(1) s (n) = = (5.7) even s(4) g(4) g(2) g(0) x(2)        .   .   .   .   ..  .             

120 CHAPTER 5. WAVELET PACKET MODULATION

s(1) g(1) x(0)

∆ s(3) g(3) g(1)  x(1) s (n) = = (5.8) odd s(5) g(5) g(3) g(1) x(2)        .   .   .   .   ..  .              It is clear that only the odd components of the ﬁlter g(n) affect sodd(n), and similarly for the even components. Furthermore, the matrix multiplications of (5.7) and (5.8) are exact convo- ∆ T ∆ lutions of the input signal with ﬁlters geven(n) = g(0) g(2) g(4) . . . and godd(n) = T h i g(1) g(3) g(5) . . . . Since time-domain convolution is equivalent to multiplication of h -transforms, it is convenienti to proceed in the -domain. (5.7) and (5.8) become Z Z

Seven(z) = Geven(z)X(z) (5.9)

Sodd(z) = Godd(z)X(z) (5.10)

The system output s(n) is found by interleaving signals sodd(n) and seven(n) which is easily shown to be implementable in the -domain as Z

2 1 2 S(z) = Seven(z ) + z− Sodd(z ). (5.11)

Substitution of (5.9) and (5.10) into (5.11) yields

2 2 1 2 2 S(z) = Geven(z )X(z ) + z− Godd(z )X(z ) (5.12) which gives the polyphase form of Fig. 5.4.b. Comparing this to Fig. 5.4.a, the advantage of the polyphase structure is that the ﬁltering operations precede the upsampling, so we now have two ﬁlters running at half the original sampling rate.

A similar study can be made of a ﬁlter followed by a downsampler, which occurs in the ﬁlterbank receiver. Recall that the downsamplers considered here are unit phase (i.e. odd phase). Referring to Fig. 5.5.a, the output signal xˆ(n), in terms of the even and odd phases of the input signal s(n)

s(n) r(n) xˆ(n) S(z) Xˆ(z) f(n) ↓ 2 ↓ 2 Feven(z)

⇔ z−1

↓ 2 Fodd(z) a) b)

FIGURE 5.5: a) FILTERING FOLLOWED BY DOWNSAMPLING. b) POLYPHASE IMPLEMENTATION.

121 CHAPTER 5. WAVELET PACKET MODULATION

is given by

xˆ(0) r(1) xˆ(1) r(3) = (note the sampling phase) (5.13) xˆ(2) r(5)      .   .   .   .          f(1) f(0) s(0) f(3) f(2) f(1) f(0)  s(1) = (5.14) f(5) f(4) f(3) f(2) f(1) f(0) s(2)      .   .   ..  .          f(1) s(0) f(0) s(1) f(3) f(1) s(2) f(2) f(0) s(3) =     +     f(5) f(3) f(1) s(4) f(4) f(2) f(0) s(5)          .   .   .   .   ..   .   ..   .                  ∆ = fodd(n) ? seven(n) + feven(n) ? sodd(n), (5.15) where the even and odd subscripts retain the meanings given them in (5.8) and (5.7). The direct convolutions in (5.15) are more succinctly represented by -domain multiplications: Z

Xˆ(z) = Fodd(z)Seven(z) + Feven(z)Sodd(z). (5.16)

This leads to the polyphase form of Fig. 5.5.b. Comparing to Fig. 5.5.a, it is once again seen that polyphase implementation is computationally more efﬁcient, since no ﬁltering is done until the signal has been downsampled.

The polyphase decomposition of upsampling and ﬁltering described previously can be applied to each of the branches of the two-channel synthesis bank, as shown in Fig’s 5.6.a and 5.6.b. Referring to (5.12), we get

2 2 1 2 2 S0(z) = G0,even(z )X0(z ) + z− G0,odd(z )X0(z ), (5.17) 2 2 1 2 2 S1(z) = G1,even(z )X1(z ) + z− G1,odd(z )X1(z ). (5.18)

Now, X(z) = S0(z) + S1(z), which can be written in matrix form:

X (z2) X(z) = 1 G (z2) 0 (5.19) 1 z− p 2 "X1(z )# h i

122 CHAPTER 5. WAVELET PACKET MODULATION

S z X z 0( ) 0( ) ↑ 2 G0(z) X(z)

X (z) 1 ↑ 2 G1(z) S (z) 1 a)

G0,even(z) ↑ 2 X0(z) S0(z)

−1 G0,odd(z) ↑ 2 z X(z)

G1,even(z) ↑ 2 X1(z) S1(z)

−1 G1,odd(z) ↑ 2 z b)

X z 0( ) ↑ 2 X(z) Gp(z) X z 1( ) ↑ 2 z−1 c)

FIGURE 5.6: POLYPHASE IMPLEMENTATION OF TWO-CHANNEL SYNTHESIS BANK

where the two-input, two-output polyphase ﬁlter transfer matrix Gp(z) is given by

∆ G0,even(z) G1,even(z) Gp(z) = . (5.20) " G0,odd(z) G1,odd(z) #

This polyphase form results in the implementation illustrated in Fig. 5.6.c.

Applying the polyphase interpretation of ﬁltering followed by downsampling of (5.16) to each of the branches of the analysis bank in Fig. 5.7.a yields the polyphase structure in Fig. 5.7.b. We can write,

Xˆ (z) F (z) F (z) X (z) 0 = 0,odd 0,even even (5.21) ˆ "X1(z)# "F1,odd(z) F1,even(z)# "Xodd(z)#

∆ Xeven(z) = Fp(z) (5.22) "Xodd(z)# where the two-input, two-output analysis bank polyphase transfer matrix Fp(z) is appropriately deﬁned. The resulting polyphase form is shown in Fig. 5.7.c.

123 CHAPTER 5. WAVELET PACKET MODULATION

Xˆ z F0(z) ↓ 2 0( ) X(z)

Xˆ z F1(z) ↓ 2 1( ) a)

↓ 2 F0,even Xˆ0(z)

−1 z ↓ 2 F0,odd X(z)

↓ 2 F1,even Xˆ1(z)

−1 z ↓ 2 F1,odd b)

X(z) Xˆ0(z) ↓ 2 Fp(z) Xˆ1(z) z−1 ↓ 2 c)

FIGURE 5.7: POLYPHASE IMPLEMENTATION OF TWO-CHANNEL ANALYSIS BANK

5.1.3 Perfect Reconstruction

Connecting the polyphase synthesis and analysis banks of Fig’s 5.6.c and 5.7.c we get the polyphase implementation of the two-channel transmultiplexer which is shown in Fig. 5.8. The phase selection operations are highlighted by the dashed box. Some simpliﬁcation is possible here; consider the signals a, b, c, d and x

x(0) a(0) 0 a(0)  x(1)   0   b(0)   b(0)   x(2)  =  a(1)  +  0  =  a(1)  . (5.23)          x(3)   0   b(1)   b(1)           .   .   .   .   .   .   .   .                  Also, recalling that the downsamplers are odd-phase,

124 CHAPTER 5. WAVELET PACKET MODULATION

X z Xˆ z X z a ( ) c 0( ) 0( ) ↑ 2 ↓ 2 Gp(z) Fp(z) Xˆ z X z 1( ) 1( ) ↑ 2 z−1 z−1 ↓ 2 b d

FIGURE 5.8: POLYPHASE IMPLEMENTATION OF TWO-CHANNEL TRANSMULTIPLEXER

c(0) x(1) b(0)  c(1)   x(3)   b(1)   c(2)  =  x(5)  =  b(2)  (5.24)        c(3)   x(7)   b(3)         .   .   .   .   .   .              and d(0) x(0) a(0)  d(1)   x(2)   a(1)   d(2)  =  x(4)  =  a(2)  . (5.25)        d(3)   x(6)   a(3)         .   .   .   .   .   .              Since c = a and d = b, it is apparent that the phase manipulations enclosed within the dashed box of Fig. 5.8 cancel each other out. This was made possible by the choice of odd-phase downsampling and even-phase upsampling. The transmultiplexer operation is thus simply described by the following matrix equation:

Xˆ (z) X (z) 0 = F (z)G (z) 0 . (5.26) ˆ p p " X1(z) # " X1(z) #

Perfect reconstruction is achieved when the output is the same as the input, subject to some scale factor ζs and system delay ∆s. In terms of the polyphase matrices, the PR-conditions on the transmultiplexer co-efﬁcients are given in closed form as

∆s Fp(z)Gp(z) = ζsz− I. (5.27)

It is possible to satisfy (5.27) with causal, FIR, polyphase ﬁlters [42–44] although treatment of this extensive topic is omitted for brevity. Instead, the simplest PR-QMF, namely the Haar ﬁlterbank, is presented as an illustrative example.

125 CHAPTER 5. WAVELET PACKET MODULATION

1.5

|{G0, F0} ejω | |{G1, F1} ejω | 1

0.5 Magnitude

0 0 0.2 0.4 0.6 0.8 1 Normalized Frequency

FIGURE 5.9: THE HAAR FILTERBANK FREQUENCY RESPONSE

5.1.4 Worked Example

Consider the well known Haar ﬁlterbank,

1 1 g0(n) = 1 1 g1(n) = 1 1 √2 √2 − (5.28) 1 h i 1 h i f0(n) = 1 1 f1(n) = 1 1 √2 √2 − h i h i The frequency response is shown in Fig. 5.9. In the -domain Z 1 1 1 1 G0(z) = 1 + z− G1(z) = 1 + z− √2 √2 − (5.29) F (z) = 1 1 + z 1 F (z) = 1 1 z 1 0 √2 − 1 √2 − − Using the polyphase forms deﬁned in (5.20) and (5.22) we write,

1 1 1 1 G 1 F 1 p(z) = √ − p(z) = √ (5.30) 2 "1 1 # 2 " 1 1# − and so, 1 1 1 1 1 Fp(z)Gp(z) = − = I (5.31) 2 " 1 1 # " 1 1 # − which satisﬁes the PR criterion of (5.27).

5.1.5 Wavelet Packet Trees

Assume we have a two-channel PR transmultiplexer, the synthesis and analysis banks of which we will represent by the schematics of Fig. 5.10 for simplicity. Now, suppose we attach another synthesis bank (identical to the one in the current transmultiplexer) onto one of the branches of the existing synthesis bank, and perform a similar operation at the corresponding branch of the analysis bank, as illustrated in Fig. 5.11.a. A branch is deﬁned as one of the arms of the two-channel

126 CHAPTER 5. WAVELET PACKET MODULATION

s0 r0 ˆ x0 ↑ 2 g0 f0 ↓ 2 x0 x0 xˆ0 x x SYNTHESIS ANALYSIS ⇔ ˆ ˆ x1 ↑ 2 g1 f1 ↓ 2 x1 x1 x1 s1 r1 SYNTHESIS ANALYSIS

FIGURE 5.10: TWO CHANNEL TRANSMULTIPLEXER AND SCHEMATICS FOR THE SYNTHESIS AND ANALYSIS BANKS transmultiplexer, while a node is the point of branch interconnection. An important question is, does the given structure give perfect reconstruction? The answer is yes [43]. Since the original tree is PR, any subband can be divided into two orthogonal subchannels. By extension, we have a PR three-channel transmultiplexer. This is called a Wavelet Packet Tree (WPT) since the branch ﬁlters are governed by the same two-scale equations found in wavelet analysis.

x0 xˆ0 x0 xˆ0 x1 xˆ1 x2 xˆ2 x1 x xˆ1 x x3 xˆ3 x4 xˆ4 x2 xˆ2 x5 xˆ5

FIGURE 5.11: a) ITERATED TWO CHANNEL TRANSMULTI- PLEXER, SPLIT ON SUBCHANNEL 0. b) ARBITRARY WAVELET PACKET TREE.

We can generate any arbitrary PR-WPT for the synthesis bank by a similar successive splitting of subbands, each iteration yielding an extra subchannel (communicating at half the rate) squeezed into the same bandwidth as the original. Providing the analysis bank is similarly iterated, the resulting scheme will be PR, by extrapolation of the above argument. In general, we must split m = N 1 nodes to get an arbitrary N-channel WPT as illustrated in Fig. 5.11.b, where five node- − splits (the root node is counted as a split) produce six sub-channels. We might ask the question, how many WPT arrangements are there for a finite number of node splits? For example, the four- channel WPT is defined by three node splits. There are five possible arrangements of these three node splits, as illustrated in Fig. 5.12. Tree enumeration is a combinatorics problem related to the generation of rooted binary trees (also called rooted trivalent trees). A tree is an undirected graph that is connected and has no loops or cycles. A binary tree has only two branches at each node, and a rooted tree has a specified first node (the root) [283]. This is just a formal definition for the structures we have been considering so far. The number bm of rooted ordered binary trees on m

FIGURE 5.12: POSSIBLE ARRANGEMENTS OF A 4-BAND WPT.

127 CHAPTER 5. WAVELET PACKET MODULATION

2 x0 n0 1 x1 n0 2 x0 x2 n1 0 x3 n0 2 x x x4 n2 1 x1 x5 n1 2 x6 n3 x2 x7 x3

FIGURE 5.13: θ AND AN EXAMPLE OF τ Θ FOR D = 3 D ∈ D vertices is given by [284–286] 1 2m b = . (5.32) m m + 1 m The integers which make up this sequence are called the Catalan numbers1 the ﬁrst few of which are given in Table 5.1.

TABLE 5.1: THE CATALAN NUMBERS

m 1 2 3 4 5 6 7 8 9 10 11 12 . . .

bm 1 2 5 14 42 132 429 1,430 4,862 16,796 5,8786 208,012 . . .

Limiting the WPT to a predefined number of subchannels is a useful constraint in applications which are delay insensitive, such as subband coding for data storage. In these cases, the Catalan numbers give a good indication of the spectral diversity afforded by a fixed number of subchannels. For example, a ten-channel system could yield any of 4,862 distinct spectral decompositions. However, for multi-carrier communication applications, it turns out that a more useful limit on the WPT is the maximum depth of iteration D on any subchannel. The reason for this is that this will fix the maximum system delay for the transmultiplexer. Furthermore, it will also specify the maximum sampling rate needed in the system, which is required for setting the system clock at implementation. We define θD as the complete WPT of depth D and ΘD as the set of admissable j th subtrees τ of θD, as shown in Fig. 5.13. Node ni is defined as the i point of branch interconnection (out of possible 2j) at tree depth or scale j, 0 j D. A node-split decomposes the signal ≤ ≤ space j spanned by the wavelet packet at node nj into subspaces j+1 and j+1 whose direct Wi i W2i+1 W2i sum is the original space: j = j+1 j+1. (5.33) Wi W2i+1 ⊕ W2i In words, these orthogonal subspaces span the original space completely and without redundancy.

Thus any arbitrary tree generated in the above manner will give rise to a basis for l2(Z), the space of square summable real sequences [289]. The cardinality γD of the set of WPT’s ΘD is given

1Named after E.C. Catalan (1814–1894), they arise in a host of problems in combinatorics. For example, the number of ways a polygon with m + 2 sides can be cut into m triangles [287], or the number of ways m sets of parentheses can be nested [288] are both given by the mth Catalan number.

128 CHAPTER 5. WAVELET PACKET MODULATION

2 recursively by γD = γD 1 + 1, γ0 = 1 [284, 290]. This number grows very quickly with D, as − listed in Table 5.2. For example, if the maximum allowed tree depth is D = 6, corresponding

TABLE 5.2: CARDINALITY OF THE SET OF TREES ΘD OF MAXIMUM DEPTH D.

D 0 1 2 3 4 5 6 7 . . .

γD 1 2 5 26 677 458,330 210,066,388,901 44,127,887,745,906,175,987,802 . . . to a full tree with 64 subchannels, there are over 210 billion possible subtrees as contenders for the optimal WPT. Clearly, it is impractical to perform an exhaustive search over all the available trees in order to determine that which will minimize combined ISI and ICI for a given channel. However, before developing a more useful approach to tree selection, it is instructive to introduce the concept of equivalent branch ﬁlters.

5.1.6 Equivalent Branch Filters

th It is convenient to define the i subchannel in terms of its own sampler Mi and equivalent filter Ti(z), rather than by a sequence of shared samplers and filters, as illustrated for a general synthesis bank in Fig. 5.14. To do this, we us the Noble identities [44] given in Fig. 5.15, which allow movement of a sampler back and forth between the input and output of a filter. j Suppose that the sequence of synthesis filters (starting from the root) leading up to node ni is Gi0(z), Gi1(z) . . . Gij(z) where Gil(z) G (z), G (z) and G (z) are the -transforms of ∈ { 0 1 } k Z the root filters gk(n). Then by a simple repeated application of the relevant Noble identity, the equivalent branch filter representation is [289]

j il 2l TiS (z) = G (z ) (5.34) l=1 Yj Mi = 2 (5.35) where the S superscript denotes synthesis. For the analysis bank we would get TiA(z) instead of il il TiS (z) by replacing G and Gk with F and Fk respectively and the upsampler by a downsampler of the same sampling ratio Mi.

Note that increasing branch-ﬁlter index i does not necessarily correspond to increasing frequency. D In the special case of the full tree θD the frequency ordering is given by the 2 -bit Gray code [291, Gray 292] e.g. i = 0 1 2 3 00 01 11 10 so the frequency ordering is 0 1 3 2 . { } → { } { } In general, it is not trivial to specify the branch-ﬁlter frequency ordering. However, it is not usually necessary, since the same tree structure is used at the synthesis and analysis banks, and they can

129 CHAPTER 5. WAVELET PACKET MODULATION

X0(z) X1 z X0 z S ( ) ( ) ↑M0 T0 (z)

X z S 1( ) ↑M1 T1 (z) X(z) ⇔ X(z)

XN−1 z S ( ) ↑MN−1 TN−1(z) XN−1(z)

FIGURE 5.14: EQUIVALENT BRANCH FILTER REPRESENTATION. be used without knowledge of the frequency-ordering. In fact, frequency reordering is seen to be counter-productive, since it results in an associated undesirable time re-ordering [292].

↓ 2 H(z) ⇔ H(z2) ↓ 2

H(z) ↑ 2 ⇔ ↑ 2 H(z2)

FIGURE 5.15: THE NOBLE IDENTITIES.

To illustrate the concept of equivalent branch filters we consider the example of the three-channel WPT shown in Fig. 5.16.a based on iterated Haar filterbanks from (5.28). We use the relevant Noble-identity of Fig. 5.15 to move the internal expanders to the fore, as shown in Fig. 5.16.b. We S define the equivalent synthesis branch filters Ti (z):

S 1 1 T (z) = G0(z) = (1 + z− ) (5.36) 0 √2 S 2 1 2 1 T (z) = G (z )G (z) = (1 + z− )( 1 + z− ) (5.37) 1 0 1 2 − S 2 1 2 1 T (z) = G (z )G (z) = ( 1 + z− )( 1 + z− ), (5.38) 2 1 1 2 − − which allows us to draw the equivalent 3-channel system of Fig. 5.16.c. The equivalent branch jω S ﬁlter responses are given by Ti e (we drop the superscript, since the analysis bank has the same magnitude response). F or example,

jω 1 2jω jω T e = (1 + e− )( 1 + e− ) 1 2 − j ω j ω jω jω e 2 e− 2 1 jω e + e− j ω = 2e− ( 2j)e− 2 − 2 2 − 2j

ω = 2 sin cos (ω) (5.39) 2

130 CHAPTER 5. WAVELET PACKET MODULATION

x0 ↑2 G0(z) x x1 ↑2 G0(z)

↑2 G1(z) x2 ↑2 G1(z) a)

x0 x0 S ↑2 G0(z) ↑2 T0 (z)

x1 2 x x1 S x ↑4 G0(z ) G1(z) ↑4 T1 (z)

x2 2 x2 S ↑4 G1(z ) G1(z) ↑4 T2 (z) b) c)

FIGURE 5.16: SAMPLE WAVELET PACKET TREE, WITH 3 SUBCHANNELS AND EQUIVALENT BRANCH FILTER REPRE- SENTATION.

Similarly,

ω T ejω = √2 cos (5.40) 0 2 ω T ejω = 2 sin sin (ω) . (5.41) 2 2 We plot these responses for clarity in Figure 5.17. This is a good example of the unordered frequency indexing in WPM, since T1 selects a higher band than T2. It is also clear that the Haar ﬁlterbank has poor spectral isolation, which will be one of the reasons for developing an optimal tree structure in the next section. Equivalent branch ﬁlters are useful for examination of subband structuring once the optimal WPT has been selected.

2.5

T1 ejω 2

T2 ejω jω 1.5 T0 e

1 Magnitude

0.5

0 0 0.2 0.4 0.6 0.8 1 Normalized Frequency

FIGURE 5.17: SUBCHANNEL MAGNITUDE FREQUENCY RE- SPONSES FOR THE SAMPLE 3-CHANNEL WPT.

131 CHAPTER 5. WAVELET PACKET MODULATION

From the previous analysis, it is clear that wavelet packet modulation can be used to give perfect- reconstruction, nonuniform multi-carrier transmultiplexing. The transmit and receive ﬁlterbanks are wavelet packet trees formed by the iteration of a PR two-channel transmultiplexer. Increased frequency resolution is achieved by increasing the depth of iteration. We now present a review of the history- and state-of-the-art of wavelet packet modulation.

5.1.7 Wavelet Packet Modulation: State of the Art

f f f

a) t b) t c) t

FIGURE 5.18: TIME-FREQUENCY PLANE TILINGS FOR DIF- FERENT WAVELET PACKET TREE CONFIGURATIONS. a) THE FULL-TREE (OR REGULAR-TREE) CORRESPONDS TO THE SHORT TIME FOURIER TRANSFORM (GABOR TRANSFORM). B) THE LOGARITHMIC TREE ONLY SPLITS THE HIGH-PASS CHANNEL AT EACH STAGE; KNOWN AS THE DISCRETE WAVELET TRANSFORM. C) AN ARBITRARY WAVELET PACKET TILING.

Wavelet Packets were originally developed in the early 1990s for multi-resolution analysis and subband coding applications. In these schemes, a signal is analysed by the analysis bank (hence the name), possibly quantized (with variable bit-allocation across the subbands) and lossily recon- structed at the synthesis bank. The WPT decomposes the signal of interest into different time-scale or time-frequency components as illustrated in Fig. 5.18 (for a thorough overview of wavelet theory and its applications see [289, 293, 294]). The most common decompositions have been the full (or regular) tree, which has the same bandwidth on all subchannels, corresponding to a Short Time Fourier Transform (STFT)2 and the logarithmic tree, which has a node split on all high-pass channels up to the maximum depth, also known as the Discrete Wavelet Transform (DWT). The question arose [295] as to what was the optimal time-frequency tiling for a given signal, and subsequently, how to best allocate bits to the subbands of the optimal tree. This was ﬁrst addressed by Coifman and Wickerhauser [296], who used an heuristic approach to maximize the rate, and has since been solved more formally using rate-distortion theory and binary tree-search to minimize an entropy based cost-function, which jointly provides best basis selection and optimal bit-allocation [297–300].

The idea of swapping the analysis and synthesis banks to provide a transmultiplexer conﬁguration for communications became popular in the mid-1990s. The majority of the research on WPTs

2Also called the Gabor transform.

132 CHAPTER 5. WAVELET PACKET MODULATION

f f f

t t t

FIGURE 5.19: EFFECT OF IMPULSE NOISE AND A JAMMER (SHOWN BY ARROWS) ON a) REGULAR MULTITONE b) SIN- GLE TONE AND c) INTELLIGENTLY STRUCTURED WAVELET PACKET MODULATION. FEWER SYMBOLS ARE hit BY INTER- FERENCE, BY OPTIMAL TIME-FREQUENCY TILING.

for communications has been for unstructured trees, i.e. the full tree θD. We will refer to this as unstructured-WPM. In [301, 302] Haar and Daubechies wavelets were seen to give reduced ISI and ICI over FFT-based OFDM in a two-path wireless channel. Unstructured WPM has been shown to perform better than OFDM over an impulsive noise channel [303, 304] and over a nonlinear channel [305–307]. Wavelet based schemes are also seen to have better synchronization properties than regular multitone, [307–310]. A low-stopband, low delay scheme was presented in [311]. Different wavelets for the fundamental QMF have been considered, with limited success, in [307,312–314]. Equalization of unstructured-WPM was considered in [304,315] and trellis coded WPM was presented in [307, 316–318].

There has only been limited research into structured trees for multitone communications applications, which seems to be the natural communications equivalent to the optimal tree structuring used in subband coding. The main focus has been on structured-WPM for improved performance in an impulsive noise channel subject to a frequency jammer [292,319–321]. The intuitive understanding is that fewer symbols will get hit by interference if an intelligent time-frequency tiling is used, as illustrated in Fig. 5.19. However, this is a somewhat academic application, since time- localized noise statistics are rarely available for receiver design in a real communication environment, and will change faster than any adaptive-tree algorithm will be able to track (adaptive trees, which maintain orthogonality during adaptation, were developed for subband coding applications in [322–327] but have not yet been extended to communications applications).

The only scheme which has addressed structured trees for improvements in stationary noise channels was developed by Akansu et al. [123,328–330]. In this scheme, the WPT is chosen to mimic variations in the channel magnitude response. A measure is made of the flatness of a particular subband. If the subband is not “flat” then it is decomposed into two flatter bands, using a node split. If it is “flat”, and of the same order as its neighbour, then the two subands are combined by pruning the relevant node. If either of these spectral unevenness tests fail, then the WPT is no longer altered along that branch. The problem with this scheme is that it does not provide any attempt to minimize ISI and ICI directly, even though it shows good interference reduction for the channels on which it was tested. Furthermore, there are a couple of threshold constants which

133 CHAPTER 5. WAVELET PACKET MODULATION

must be deﬁned in order to allow a measure of channel ﬂatness, and choice of these constants is somewhat ad hoc and may even vary from channel to channel.

For these reasons, we present an entirely new tree-structuring algorithm [331], which directly minimizes the combined ISI/ICI at the receiver for any channel, by use of a training sequence, and appropriate tree horticulture. There are no ad hoc parameters and the algorithm is self terminating, for a given maximum tree-depth.

5.2 Optimal Wavelet Packet Modulation

We seek the tree τ Θ which will outperform all others in terms of minimizing the combined opt ∈ D ISI/ICI for a given channel. Theoretically, using ideal brickwall subchannels, there is no ICI and performance will asymptotically mimic waterfilling as the number of subchannels is increased. In this case optimality is achieved by the complete WPT θD. However, a finite complexity budget constrains us to nonideal filters, which result in interband energy leakages. Variable partitioning of the channel spectrum via appropriate WPT structuring can ameliorate the negative effects of ICI thus introduced. More specifically, by merging subbands over regions of relatively constant magnitude, there may be less aliasing than if we performed subchannel decomposition blindly. Also, by spreading information over multiple frequency bins, it may be possible to utilize portions of the spectrum which would otherwise have lain dormant. In this section, we define an interference measure which can be used for aiding tree selection. We present an efficient, training-based, tree- growing algorithm for minimization of this interference measure. Simulation results are presented for a simple test channel, and also for the more realistic case of the VDSL testbed. ISI/ICI is the dominant impairment in DSL communications and the goal of this chapter is to demonstrate the interference mitigation capability of structured-WPM. For these reasons we only consider noise-free dispersive channels in this chapter.

5.2.1 Signal to Interference Ratio

In an ideal channel, the transmitted constellation points are received undistorted on all subchannels as Xˆ (n) = X (n) , i, n. In the presence of a dispersive channel, the constellation points will i i ∀ be distorted at the receiver, both by ISI from neighbouring symbols X (n k) , n = k, and by i − 6 ICI from neighbouring subchannels Xi l(n) , i = l. This will result in a shift in the received − 6 constellation points from their desired location, as illustrated in Fig 5.20 for the case of binary PAM signalling.

The squared Euclidean distance from the received constellation point to the transmitted constellation point is a measure of instantaneous subchannel interference. The expected value of this norm

134 CHAPTER 5. WAVELET PACKET MODULATION

Xˆ − X i i | {z } ˆ Xi Xi

1 0 -1

FIGURE 5.20: TRANSMITTED BINARY PAM SYMBOL Xi AND INTERFERENCE CORRUPTED RECEIVER ESTIMATE Xˆi. is the subchannel interference power, which we deﬁne

2 σ2 = E Xˆ (n) X (n) . (5.42) Ii i − i

Given a subchannel transmit signal variance of 2 , we define the subchannel Signal to Interfernce σXi Ratio (SIR) as σ2 SIR = Xi . (5.43) i σ2 Ii (in this case a “subchannel” is one of the N time-frequency tiles given by a particular wavelet packet decomposition; this is explained in more detail in the next section). We use 5.43 to define the average subchannel SIR as N 1 1 − SIR = SIR (5.44) av N i Xi=0 The optimal tree will maximize this measure at the receiver. By reducing interference at the receiver, the equalizer should give better performance for a fixed complexity.

5.2.2 Tree Structuring

In our proposed algorithm, we grow the optimal tree from the root, subject to the perfomance of a subband decomposition at each iteration. A training sequence consisting of random binary PAM constellation points is used at the input to the synthesis bank of the WPT. At the receiver, the SIRi statistic is measured on each subchannel and the average subchannel SIR is computed from (5.44). One by one, each subchannel of the current tree is decomposed using a node split and the average

SIR is recomputed each time. The node which gives the largest increase in SIRav is chosen for actual splitting, and the algorithm repeats from the beginning. No splits are allowed on nodes which are at maximum tree depth; in this case we split the node which gives maximum increase in SIRav, but which is not at maximum depth di = D. If no node-split will increase average SIR then the optimal tree has been achieved and the algorithm terminates. The algorithm is formalized in Table 5.3.

It is worthwhile to make some comments on this approach. Firstly, the maximum number of SIRav N 2 measures required before ﬁnding the optimal tree is less than 2 . Quite often the number of SIR

135 CHAPTER 5. WAVELET PACKET MODULATION

TABLE 5.3: TREE GROWING ALGORITHM FOR MAXIMUM SIR.

INITIALIZE: Set τ = θ0 i.e. start with single tone modulation. Measure SIR using random binary PAM training sequence.

STEP 1 FOREACH terminal node which has depth di < D, perform a node split and measure the SIR.

STEP 2 Split the terminal node which gave the greatest increase in SIR. If there is no node split which increases SIR, then terminate the algorithm.

STEP 3 REPEAT from Step 1. tests required will be well short of this number, as will be seen in the simulations. Thus, the algorithm terminates within a ﬁnite, known time. Furthermore, the algorithm is self-terminating; once the optimal tree is reached, it goes no further. Another advantage of this algorithm is that there are no arbitrary thresholds to be deﬁned before it can be used, unlike the channel mimicking algorithm recommended by Akansu et al. [329]. Node splitting is based entirely upon a measure of interference.

As regards optimality of the selected tree, the question remains as to whether some other tree θk could out-perform it, particularly by decomposition along some node which had been terminated at some earlier point in the algorithm. It is difﬁcult to provide a rigorous answer to this question, since all trees cannot be tested, as mentioned previously. However, if we assume that the subchannels have some degree of frequency selectivity (which we hope they do in a well designed multitone system), then it is postulated that the wavelet packet described by a terminated node approximately spans the space of the wavelet packets described by its subnodes. This is true in the case of an ideal channel. In a dispersive channel, orthogonality is destroyed and this may not hold. However, with good frequency selectivity ICI is minimized, and it can be assumed that if a particular node decomposition does not improve SIR, than neither will further subnode decompositions at that node (for a given step in the algorithm). Although this argument is not mathematically rigorous, it is seen to hold up under the simulation conditions tested.

The optimal tree τopt will, in all likelihood, have fewer subbands than the full tree θD. As has been claimed in [329], transceiver complexity can be signiﬁcantly reduced because of this. As shown in Fig. 5.18, the WPT divides the time-frequency plane into tiles or time-frequency atoms. Closer inspection reveals that these atoms are all the same area, which ensures that the entire time-frequency plane is spanned by the wavelet packet basis, regardless of the tiling. This is preservation of sam-

136 CHAPTER 5. WAVELET PACKET MODULATION

pling density and respects Heisenberg’s uncertainty principle which puts a lower limit on the accu- racy with which a signal can be speciﬁed in space and time, by ∆t∆f 4π, where ∆t and ∆f are ≥ the dimensions of the time-frequency atom [289]. The number of equal-area time-frequency atoms generated by any PR-WPT τ Θ is 2D [332]. Thus the communication bandwidth is equal for ∈ D all trees. This can be better explained by considering each tile of the time-frequency plane as a “subchannel”. Rather than having N frequency division multiplexed subchannels (multitone) or N time division multiplexed subchannels (single tone TDM), WPM gives N subchannels which are multiplexed in both time and frequency. Fast WP transforms based on polyphase lattice repre- sentations of the regular tree have implementational complexity N log N , which is the same O { } as for an N-channel FFT (note again that N = 2D for the full tree [333,334]). This is true of both the forward and inverse WP-transforms, required at the analysis and synthesis banks respectively [335]. However, for pruned subtrees, implementational complexity is reduced, since spectral decomposition is avoided on a number of nodes. Thus, average subchannel SIR is improved together with a possible reduction in implementation complexity in the transceiver. Obviously, the penalty for this improved performance is the added training complexity, but this can be done off-line at startup for a stationary channel, or at regular intervals for a slowly varying channel such as the DSL.

5.2.3 Simulation Results

We have an algorithm which returns a WPT that is allegedly optimal in the average SIR metric we have deﬁned. It would be instructive to test whether the returned tree is in fact optimal by comparing it, in terms of SIR performance, to all the other possible trees τ Θ . However, for a ∈ D realistic tree depth such as D = 7 (a 128-tile WPT) we have already seen that there would be over 210 billion trees that would have to be tested. Not only would this be prohibitively time consum- ing but generating that many distinct tree structures without repetition is, in itself, a challenging combinatorics problem. For these reasons we compared our fast tree-selection algorithm with a brute force method for the managable case of maximum tree depth D = 3.

We generated the library of 26 WPT’s in Θ3 shown in Fig. 5.21. The fundamental WP-bases were chosen as the Haar wavelets. 100 WP-symbols, generated by uncorelated binary PAM signalling at the appropriate rate on each subchannel, were communicated over the simple 6 tap FIR test 1 1 1 1 1 1 channel with impulse response h(n) = [ 1000 100 1 10 100 1000 ]. We chose this channel for its relatively low ISI levels and uncomplicated frequency response. The synchronization delay at the receiver is trivially seen to be 3 samples for this channel. The received signal was scaled to have the same power as the transmitted signal. Wavelet edge effects are removed in MATLAB by using the `per' option of the two-channel discrete wavelet transform function dwt.m, corresponding to periodic extension of the wavelet packet data. Thus, the ﬁrst two WP-symbols must be discarded at the receiver in order to remove the transmultiplexer transient, as described in [292]. In

137 CHAPTER 5. WAVELET PACKET MODULATION

1 2 3 4 5 6

7 8 9 10 11

12 13 14 15 16

17 18 19 20 21

22 23 24 25 26

FIGURE 5.21: THE SET OF TREES OF MAXIMUM DEPTH D = 3: Θ3. order to accommodate different sampling rates on each subchannel, cell arrays were used in MAT- LAB (using the cell.m function) to store the WP-symbols, as opposed to conventional matrices, which require the same number of elements in each row or column. The average subchannel SIR was measured for each tree at the receiver using (5.44). It was found that tree number 14 from the library gave maximum average SIR. The average subchannel SIRs for each of the alternative

WPTs are given in Table 5.4. It can be seen that the best tree (τopt = τ14) gives 1.9 dB increase in average SIR compared to the full tree (θ3 = τ26) and 3.6 dB increase over unmodulated single tone data (τ1). The SIR values are all relatively high, because the test channel is only moderately dispersive.

TABLE 5.4: AVERAGE SUBCHANNEL SIR FOR EACH OF THE 26 WPTS IN THE SET Θ3.

WPT 1 2 3 4 5 6 7 8 9 10 11 12 13 SIR (dB) 18.3 19.1 21.6 21.0 20.0 21.8 21.0 21.2 20.5 20.7 20.4 20.5 20.7

WPT 14 15 16 17 18 19 20 21 22 23 24 25 26 SIR (dB) 21.9 21.0 21.3 20.2 21.6 21.0 19.8 20.8 20.7 20.5 20.9 20.3 20.0

We then applied the fast tree-selection algorithm of Table 5.3 to the same test channel. The same

138 CHAPTER 5. WAVELET PACKET MODULATION

1 1 Test Channel Subband 0 Subband 1 Subband 2 0.8 0.9 Subband 3 Subband 4

0.6 0.8

0.4 0.7 Channel Magnitude Squared

SubChannel Magnitudes Squared 0.2 0.6

0 0.5 0 0.2 0.4 0.6 0.8 1 Normalized Frequency

FIGURE 5.22: TEST CHANNEL RESPONSE AND OPTIMAL TILING tree, number 14 from Fig. 5.21, was returned as optimal. To lend further credence to the result, Fig 5.22 shows the test channel magnitude response, together with the frequency tiling achieved by WPT τopt = τ14 from our library. This has ﬁve subbands, four of which are concentrated in the lower half of the allocated bandwidth. It can be seen that average SIR is maximized by a tiling which mimics the channel magnitude response. This would support the work of Akansu et al. [329], but has been achieved by a more theoretically rigorous approach. We then considered the more realistic example of optimizing for Very high Speed Digital Sub- scriber Loop (VDSL). We optimized for test loops VDSL#1, VDSL#3 and VDSL#7 as described in detail in Chapter 2. Using the same growing algorithm as before, with maximum depth D = 7, we obtained the subband decompositions shown in Fig’s 5.23–5.25 for each of the three test channels. These plots also show the normalized channel magnitude squared frequency response superimposed on the WP-tiling. In all cases the tiling is seen to mimic the spectral characteristics of the channel. The optimal WPT strucures for each channel are shown in Fig. 5.26. Although it is unfeasible to test if the optimal tree outperforms all other trees in each case, it is worthwhile to compare the SIR performance of the selected tree with the extreme cases of single tone data and

128-tone DMT corresponding to the full tree θ7.

On channel VDSL#1 the optimal tree was seen to give 18.9 dB increase in average subchan- • nel SIR over unmodulated single tone data and 1.8 dB SIR improvement over 128-channel multitone data.

On channel VDSL#3 the optimal tree was seen to give 15.5 dB increase in average subchan- • nel SIR over unmodulated single tone data and 1.5 dB SIR improvement over 128-channel multitone data.

139 CHAPTER 5. WAVELET PACKET MODULATION

0 0

−5 −20

−10 −40

−15 −60 Channel Magnitude Squared (dB) SubChannel Magnitudes Squared (dB)

−20 −80 0 0.2 0.4 0.6 0.8 1 Normalized Frequency

FIGURE 5.23: VDSL#1 CHANNEL RESPONSE (DOTTED) AND OPTIMAL WAVELET PACKET TILING

0 0

−5 −20

−10 −40

−15 −60 Channel Magnitude Squared (dB) SubChannel Magnitudes Squared (dB)

−20 −80 0 0.2 0.4 0.6 0.8 1 Normalized Frequency

FIGURE 5.24: VDSL#3 CHANNEL RESPONSE (DOTTED) AND OPTIMAL WAVELET PACKET TILING

140 CHAPTER 5. WAVELET PACKET MODULATION

0 0

−5 −40

−10 −80

−15 −120 Channel Magnitude Squared (dB) SubChannel Magnitudes Squared (dB)

−20 −160 0 0.2 0.4 0.6 0.8 1 Normalized Frequency

FIGURE 5.25: VDSL#7 CHANNEL RESPONSE (DOTTED) AND OPTIMAL WAVELET PACKET TILING

On channel VDSL#7 the optimal tree was seen to give 24.9 dB increase in SIR over unmod- • ulated single tone data and 1.5 dB SIR improvement over 128-channel multitone data.

These ﬁgures suggest that in the single tone case subchannel interference is severe. This is correct since there is only one highly dispersive subchannel. By using a maximum number of subchannels the ISI has been greatly reduced, with the cost of ICI spectral leakage. The optimal tiling is seen to minimize the overall combination of ISI and ICI and is thus seen to perform marginally better than the full WP-decomposition.

5.2.4 Noise Effects

The results of this chapter have been developed for interference minimization with the intention of improving the performance of the equalizer in a multi-carrier receiver. However, in a realistic communication environment, receiver noise will affect the choice of WPT made by our tree growing algorithm. In order to demonstrate noise effects on the tree selection algorithm, we repeated 1 1 1 1 1 1 the experiment for the same test channel h(n) = [ 1000 100 1 10 100 1000 ] used previously, subject to various receiver noise levels. In the initial experiment we retained a training sequence of 100 WP-symbols for each measurement of SIR; obviously, the performance metric is now Signal to Interference plus Noise Ratio (SINR) and not just SIR. AWGN of different variances was added to the receive signal to provide SNRs in increments of 10dB, with the following results:

141 CHAPTER 5. WAVELET PACKET MODULATION

FIGURE 5.26: OPTIMAL TREE STRUCTURING τopt FOR VDSL TEST LOOPS #1, #3 AND #7

TABLE 5.5: TREE SELECTED BY THE GROWING ALGORITHM FOR DIFFERENT NOISE LEVELS. REFER TO FIG. 5.21

SNR (dB) 10 20 30 40 50 60 >60

WPT τopt τ9 τ7 τ19 τ6 τ14 τ14 τ14

142 CHAPTER 5. WAVELET PACKET MODULATION

It can be seen that at high receiver SNRs the tree selection is unaffected by receiver noise as is to be expected. For the SNR value of 40 dB, τ6 is chosen as the optimal WPT, which still mimics channel spectral characteristics (cf. Fig’s 5.21 and 5.22). However, at lower SNR values, noise spectral leakage between the subchannels leads to unpredictable behaviour of the algorithm. Noise spectral leakage occurs due to the break-down of subband orthogonality in a dispersive channel. Since the subbands are not brick-wall in shape, out of band noise is passed into the subband, particularly where the subband sidelobes are highest. This is a problem for WPM, which is noted to have particularly poor stop-band performance compared to DMT or DWMT (DMT sidelobes are 13 dB down, DWMT sidelobes can be as low as 80-100 dB below the main lobe, wheres WPM tends to have sidelobes only 5 dB below the main lobe, depending on the wavelet).

In order to see if a longer training sequence could provide more predictable results, we re-ran the algorithm with a training sequence of 1000 WP-symbols for each SINR measure, with the results shown in Table 5.6. Once again, high SNR values did not affect the algorithm unduly. However,

TABLE 5.6: TREE SELECTED BY THE GROWING ALGORITHM FOR DIFFERENT NOISE LEVELS. REFER TO FIG. 5.21

SNR (dB) 10 20 30 40 50 60 >60

WPT τopt τ13 τ24 τ21 τ6 τ14 τ14 τ14 at lower SNR values the optimal tree was again seen to deviate from the SIR maximizing solution. The returned tree would in fact maximize the SINR value at the receiver. However, equalizers which exploit this fact are beyond the scope of this thesis and are suggested as a topic for future research. It is concluded that the tree growing algorithm is best suited for use in high-SNR environments, where signiﬁcant improvement in SIR can be achieved and exploited. Typical usable receiver SNRs in DSL range upwards of 20 dB (as seen in the simulations of Chapter 3) so the algorithm usefulness may be somewhat limited. However, further study of equalization techniques or different choice of wavelet may yield situations where the algorithm can be employed at lower SNRs.

143 CHAPTER 5. WAVELET PACKET MODULATION

5.3 Chapter Summary

In this chapter, we have shown how non-uniform bandwidth multi-carrier schemes as implemented by wavelet packet modulation can be optimized for a given communication channel. We presented a fast growing algorithm to achieve this optimum. Over a simple test channel, the tree growing algorithm was seen to extract the same tree as a brute force method which tests all possible WPTs. In the VDSL testbed the optimal trees were seen to mimic the spectral characteristics of the test loops.

Since the optimal wavelet packet tiling minimizes channel interference effects, the task of channel equalization should be simpliﬁed at the receiver. Channel equalization for WPM has not been studied in this thesis and is suggested as a worthwhile topic for future research. It is suggested that combination of our fast tree-growing algorithm together with appropriate channel equalization and fast wavelet packet transforms, could form the basis for an efﬁcient bandwidth optimal multichannel modem.

Uniform bandwidth schemes such as DMT and DWMT are a subset of WPM and, lacking the degree of freedom to arbitrarily tile the time-frequency plane, should not perform as well over ISI channels. However, the stopband performance of WPM has yet to be improved to the levels achievable by cosine modulated ﬁlterbanks and further research is needed on this topic. This chapter is not intended to be a comprehensive study of WPM, but rather a stepping stone on the way to optimization of multi-carrier schemes via this relatively new technology.

144 Chapter 6

Conclusion

In this thesis we have considered three distinct aspects of multi-carrier communication for the digital subscriber loop; channel shortening, bit-loading and wavelet packet modulation. Channel shortening is used to reduce the effective channel impulse response duration of the discrete-time equivalent channel, in order to reduce the amount of redundancy that must be incorporated in the multitone signal to achieve ISI-free transmission. Bit-loading is used to maximize the data-rate for a given power-budget and target error probability. It is an indirect form of the water-pouring solution for optimal transmit spectral shaping. Wavelet packet modulation employs a dyadic tree- structured ﬁlter-bank to achieve a non-uniform subband decomposition of the allocated spectrum. An intelligent spectral decomposition can allow reduced interference at the receiver. New results have been presented on all three topics and are summarized in the next section.

6.1 Summary of Novel Results

6.1.1 Channel Shortening

Chapter 3 presented a review of channel shortening techniques with the conclusion that all existing schemes in the literature fall into one of four categories, namely minimum mean-squared error, maximum shortening signal to noise ratio, maximum bit-rate or minimum-ISI. As an extension to existing methods, we developed the DIR-optimized MMSE-TEQ. In this approach the MSE cost function is expressed entirely in terms of the TEQ coefﬁcients, by an optimal substitution for the set of desired impulse response coefﬁcients. It was shown that this solution encompases MMSE, MSSNR and min-ISI solutions as special cases, and furthermore outperforms them all in the general case in terms of maximizing the bit-rate. Short of actual non-linear optimization to maximize the bit-rate, this is considered the best solution to the channel shortening problem,

145 CHAPTER 6. CONCLUSION

since the TEQ is provided by eigen-solution of a constrained quadratic problem. As a benchmark, the DIR-optimized TEQ was seen to offer 99% of the matched filter bound on the achieveable data rate in the ADSL test environment. The DIR-optimized TEQ was also seen to perform better than other methods in the event of imperfect channel estimation, also confirmed by simulation in the ADSL testbed. Finally, a time-domain adaptive LMS algorithm was provided which achieved the DIR-optimized MMSE solution for the TEQ by use of a training sequence and appropriate tap updates. This was easily derived, since the gradient of the quadratic cost function is trivial in terms of the TEQ coefficients. An alternative, simpler unit energy constraint was provided for the adaptive algorithm, than has been used in previous adaptive solutions. Furthermore, this algorithm does not require parallel computation of a set of DIR-coefficient updates as do existing schemes. It was seen that the LMS algorithm gets very close to the eigen-solution, even in a noisy environment.

6.1.2 Bit-Loading

Chapter 4 reviewed water-pouring and some fundamental information theoretic aspects of pulse- and quadrature-amplitude modulation. This review leads to an interesting interpretation of the modulation gap approximation, which the author has not previously seen presented. Convention- ally, the gap has been illustrated pictorially on the BER curves for a given modulation scheme at a given error rate. This is some-what erroneous, since the gap is then seen to change signiﬁ- cantly with error-rate. Our illustration, which shows the modulation gap on a rate vs. SNR curve, clearly emphasises the independence of the capacity gap from error rate at SNRs above about 10 dB. In Chapter 4 we also provided a review of existing optimal and near-optimal loading schemes, together with a new near-optimal loading scheme which optimizes subchannel rates and error-performance for maximum overall rate. It was seen that this algorithm completes the set of possible rate-maximizing loading schemes. The different loading algorithms were compared in terms of data-rate and algorithmic complexity. This comparison has not been made before, despite the proliferation of loading solutions available in the literature. While there was little variation between the rate achieved by each of the schemes, it was seen that a default scheme which does not require non-linear optimization gives the best value for complexity. In other words, there is very little to be gained by throwing large amounts of complexity at a sophisticated loading scheme, at least for the cases of the DSL channels considered in this thesis.

6.1.3 Wavelet Packet Modulation

Chapter 5 reviewed the concept of Wavelet Packet Modulation and identiﬁed structured-WPM for maximization of SIR in stationary channels as an open problem in the topic. Since WPT structuring by a brute force method (tests all possible structures) was seen to be unfeasible, we presented a

146 CHAPTER 6. CONCLUSION

tree-growing algorithm which isolates the optimal tree in a ﬁxed known time for a prescribed maximum tree depth. While the optimality of the selected tree is not proved mathimatically, a sound argument is provided to suggest that no other tree could perform better. This has been supported by brute force simulations for a simple test channel. Further simulations were undertaken in the VDSL test environment. In all cases, the selected tree was seen to give a frequency decomposition which approximately mimiced the channel spectral characteristics, corresponding to similar results in the literature which take a more heuristic approach to the problem. However, further work is needed on the topic of equalization for WPM in order to exploit the SIR improvements offered in this thesis. An important result of this study is that the optimal tree may be signiﬁcantly less complex than the regular tree, which would offer implementational savings. This comes at the cost of the training stage for the WPT.

6.2 Future Research

As a ﬁnal note, this section indicates worthwhile directions for research in multi-carrier communications.

6.2.1 Channel Shortening

Channel shortening, which originated for reduced complexity MLSE about 30 years ago, has been comprehensively studied at his point in time and this thesis has provided a generalised framework for the subject. Accordingly, it is not envisaged that further research in this area would be particularly beneﬁcial. A possible exception is the fractionally spaced TEQ, although preliminary work already exists on this topic [257, 258].

6.2.2 Bit-loading

Optimal multi-carrier loading was a closed topic by 1997, and we have seen that most near-optimal schemes in the literature get close to the optimal solution. Therefore, multi-carrier loading would probably not be a fruitful topic of further research. A novel variation which implements “band- loading” has recently been considered with some improvements [336]. Instead of optimizing subchannel power or rate or error-performance, this scheme optimizess the subchannel bandwidths. This paradigm is still relatively under-examined and may warrant further investigation.

147 CHAPTER 6. CONCLUSION

6.2.3 Wavelet Packet Modulation

Of the three topics in this thesis, WPM surely provides the richest pickings in terms of open problems and further research. While we have considered SIR maximization by appropriate subband tiling, there is need for an equalization method to exploit this reduced interference. Furthermore, whether the maximum SIR WPT corresponds to the maximum bit-rate WPT remains an open problem. Design of the fundamental wavelets for stop-band minimization is also a continuing theme in WPM research.

6.3 Concluding Remarks

Multi-carrier systems (not speciﬁcally for DSL) apply the age-old divide and conquer solution to the problem of communicating over highly dispersive channels. Rather than solving the problem of how to achieve reliable communication over a single malign channel, we attempt to solve a number of simpler problems, by communicating over many channels which are not nearly as bad. The wealth of signal processing techniques which are available to throw at this multi-variable problem render it open to many interpretatations and solutions. The ethos of this thesis has been a methodical gathering of proposed multitone techniques, comparing them under identical operating conditions, and isolating the best solutions in a clear and concise manner. It is hoped that the results of this work should tie up some loose ends of the very well researched topic of multicarrier communications.

148 Appendix A

DSL Test Loop Responses

ADSL Test-Loop Frequency Responses

−10

−20

−30

−40

−50

−60 Magnitude Squared (dB)

−70

−80

−90

−100 0 0.2 0.4 0.6 0.8 1 Frequency (MHz)

FIGURE A.1: MAGNITUDE SQUARED FREQUENCY RESPONSES FOR THE EIGHT ADSL TEST LOOPS.

149 APPENDIX A. DSL TEST LOOP RESPONSES

Sampled VDSL Impulse Responses

Test Loop VDSL3: Impulse Response 0.04

0.02 Amplitude (V)

0 100 200 300 400 500 600 700 800 900 1000 Sample

FIGURE A.2: SAMPLED IMPULSE RESPONSE FOR VDSL TEST LOOP NUMBER 3.

Test Loop VDSL4: Impulse Response 0.03

0.02

Amplitude (V) 0.01

0 100 200 300 400 500 600 700 800 900 1000 Sample

FIGURE A.3: SAMPLED IMPULSE RESPONSE FOR VDSL TEST LOOP NUMBER 4.

−3 x 10 Test Loop VDSL7: Impulse Response 6

Amplitude (V) 2

0 100 200 300 400 500 600 700 800 900 1000 Sample

FIGURE A.4: SAMPLED IMPULSE RESPONSE FOR VDSL TEST LOOP NUMBER 7.

150 APPENDIX A. DSL TEST LOOP RESPONSES

Sampled ADSL Impulse Responses

−3 x 10 Test Loop ADSL1: Impulse Response 5

Amplitude (V) 1

−1

−2 0 100 200 300 400 500 600 Sample

FIGURE A.5: SAMPLED IMPULSE RESPONSE FOR ADSL TEST LOOP NUMBER 1.

−3 x 10 Test Loop ADSL2: Impulse Response 6

2 Amplitude (V) 1

−1

−2 0 100 200 300 400 500 600 Sample

FIGURE A.6: SAMPLED IMPULSE RESPONSE FOR ADSL TEST LOOP NUMBER 2.

−3 x 10 Test Loop ADSL3: Impulse Response 6

Amplitude (V) 2

−1 0 100 200 300 400 500 600 Sample

FIGURE A.7: SAMPLED IMPULSE RESPONSE FOR ADSL TEST LOOP NUMBER 3.

−3 x 10 Test Loop ADSL4: Impulse Response 4

1 Amplitude (V)

−1

−2 0 100 200 300 400 500 600 Sample

FIGURE A.8: SAMPLED IMPULSE RESPONSE FOR ADSL TEST LOOP NUMBER 4.

151 APPENDIX A. DSL TEST LOOP RESPONSES

−3 x 10 Test Loop ADSL5: Impulse Response 6

Amplitude (V) 2

−1 0 100 200 300 400 500 600 Sample

FIGURE A.9: SAMPLED IMPULSE RESPONSE FOR ADSL TEST LOOP NUMBER 5.

−3 x 10 Test Loop ADSL6: Impulse Response 5

2 Amplitude (V)

−1 0 100 200 300 400 500 600 Sample

FIGURE A.10: SAMPLED IMPULSE RESPONSE FOR ADSL TEST LOOP NUMBER 6.

−3 x 10 Test Loop ADSL7: Impulse Response 5

Amplitude (V) 1

−1

−2 0 100 200 300 400 500 600 Sample

FIGURE A.11: SAMPLED IMPULSE RESPONSE FOR ADSL TEST LOOP NUMBER 7.

−3 x 10 Test Loop ADSL8: Impulse Response 5

2 Amplitude (V)

−1 0 100 200 300 400 500 600 Sample

FIGURE A.12: SAMPLED IMPULSE RESPONSE FOR ADSL TEST LOOP NUMBER 8.

152 Appendix B

Solution to the Generalized Eigen-Problem

The generalized eigen-problem is1

T T αopt = min α Aα s.t. α Bα = 1. (B.1) α

We manipulate (B.1) into the familiar form:

µ = min µT Cµ s.t. µT µ = 1 (B.2) opt µ as follows. The matrix B is in general symmetric and positive semi-deﬁnite. In the case that it is strictly positive-deﬁnite it can be factorized by a unitary similarity transformation [12, pp. 165]:

B = QΛQT (B.3) T = Q√Λ Q√Λ (B.4) T = √B√B (B.5) where we take √B = Q√Λ. The columns of Q consist of the orthonormal eigen-vectors of B and Λ is a diagonal matrix whose entries are the corresponding eigenvalues. Letting

T µ = √B α (B.6) we get

1 T 1 αT Aα = µT √B − A √B − µ (B.7) 1The notation in this mathematical appendix is self contained; the matrices and vectors used here do not refer to any of the multitone parameters described in the thesis, even if there appears to be a conﬂict of nomenclature.

153 APPENDIX B. SOLUTION TO THE GENERALIZED EIGEN-PROBLEM

= µT Cµ (B.8) which must be minimized subject to

T αT Bα = αT √B√B α (B.9) = µT µ (B.10) =! 1. (B.11)

The solution is 1 T − αopt = √B ξmin (B.12) where ξmin is the unit eigenvector corresponding to the minimum eigenvalue λmin of the matrix

1 T 1 C = √B − A √B − . (B.13) If we wish to solve the problem

T T αopt = max α Aα s.t. α Bα = 1 (B.14) α we perform the identical manipulations to those above, but this time

1 T − αopt = √B ξmax (B.15) where ξmax is the unit eigenvector corresponding to the maximum eigenvalue λmax of the matrix C.

In the case that B is not full rank, the eigen-solution is still tractable, but involves decomposition of B into its null-space and its range-space . Cholesky decomposition yields N R 2 T Σ 0κ ν Φ B = × , (B.16) Φ Θ T "0ν κ 0ν ν # "Θ # h i × × where Σ2 is a diagonal matrix of the non-zero eigenvalues of B. The columns of Φ are the orthonormal eigenvectors which span the range-space , and the columns of Θ are the eigenvectors R spanning the null-space . κ = dim Σ2 , ν = q κ and 0 represents an all-zero matrix. N − By this decomposition, we can write

1 α = ΦΣ− φ + Θθ, (B.17) where φ and θ are arbitrary vectors of length κ and ν respectively. Applying the UEC we write

T 1 T 2 T 1 α Bα = (ΦΣ− φ + Θθ) ΦΣ Φ (ΦΣ− φ + Θθ)

154 APPENDIX B. SOLUTION TO THE GENERALIZED EIGEN-PROBLEM

= φT φ (B.18) =! 1. (B.19)

We choose φ and θ to minimize

1 T 1 (ΦΣ− φ + Θθ) A(ΦΣ− φ + Θθ) (B.20) subject to φT φ = 1. Differentiating eqn.(B.20) with respect to θ, and equating to zero yields

T 1 T 1 θ = (Θ AΘ)− Θ AΦΣ− φ, (B.21) − which can be back substituted into eqn.(B.20) to give

1 T 1 T 1 T ΦΣ− φ Θ(Θ AΘ)− Θ AΦΣ− φ A − 1 T 1 T 1 ΦΣ− φ Θ(Θ AΘ)− Θ AΦΣ− φ (B.22) × − =φT Dφ, where D is appropriately deﬁned. The solution is

T 1 T 1 α = (I Θ(Θ AΘ)− Θ A)ΦΣ− ζ , (B.23) opt − min where ζmin is the eigenvector associated with the minimum eigenvalue λmin of D in eqn.(B.22).

As before, if we wish to solve the problem

T T αopt = max α Aα s.t. α Bα = 1 (B.24) α we perform identical manipulations to those above, but choose

T 1 T 1 α = (I Θ(Θ AΘ)− Θ A)ΦΣ− ζ , (B.25) opt − max where ζmax is the unit eigenvector corresponding to the maximum eigenvalue λmax of the matrix D.

155 List of Figures

1.1 N-channel ﬁlterbank transceiver: the synthesis bank transmits and the analysis bank receives. Also known as a transmultiplexer...... 6

1.2 Spectral interpretation of Filterbank based Multitone Modulation: (a) baseband spectrum of ith-subchannel data; (b) accordion effect of upsampling; (c) ith subchannel ﬁlter response; (d) modulated subchannel data...... 7

1.3 Block- and lapped-transform filterbank spectral responses: better stopband attenuation is seen for the DWMT filterbank (although there are twice as many subchannels in the DWMT case, spectral efficiency is equivalent, since these subchannels are confined to real co-efficients)...... 10

1.4 Uniform vs. nonuniform subband decomposition. The uniform case has the same sampling ratio M on each subchannel. The nonuniform case has a different sam-

pling ratio Mi on each subchannel...... 13

1.5 A 3-channel nonuniform synthesis ﬁlterbank, based on the recombination technique. 14

1.6 A 6 channel nonuniform synthesis bank base on the heirerarchical lapped transform. In this case the root tree is a 2-channel transmultiplexer, known as a quadrature mirror ﬁlter and the resulting structure is called a wavelet packet tree. . . . . 15

1.7 Hybrid Equivalent Circuit: a simple electrical bridge in which one branch is the twisted pair loop coupled through a line transformer. If the impedence ratio on the right and bottom of the bridge matches that on the left and top, there will be no echo [133]...... 17

1.8 a) Sample Timing: incorrect sampling phase will reduce SNR, as shown dotted. b) Multitone Symbol Timing: incorrect symbol alignment will reduce in block processing of a different set of samples, as shown dotted...... 18

2.1 Bridge-tap in a DSL loop...... 23

156 LIST OF FIGURES

2.2 Conﬁguration of the eight standard CSA loops [201]. Numbers represent length/thickness in feet/gauge. Vertical lines represent bridge-taps...... 24

2.3 Four of the ANSI VDSL test loops [25]. Vertical lines represent bridge taps. . . . 24

2.4 Transmission line schematic. Blown-up section shows incremental section of line at distance x...... 26

2.5 Incremental resistance R vs. frequency in the range 0–100 MHz, for wire-types AWG#24, AWG#26, DW10 and Flat-Pair...... 28

2.6 Incremental inductance L vs. frequency in the range 0–100 MHz, for wire-types AWG#24, AWG#26, DW10 and Flat-Pair...... 29

2.7 Incremental conductance G vs. frequency in the range 0–100 MHz, for wire-types AWG#24, AWG#26, DW10 and Flat-Pair...... 29

2.8 Incremental capacitance C vs. frequency in the range 0–100 MHz, for wire-types AWG#24, AWG#26, DW10 and Flat-Pair...... 30

2.9 Real part of propagation constant γp vs. frequency in the range 0–100 MHz, for wire-types AWG#24, AWG#26, DW10 and Flat-Pair...... 31

2.10 Imaginary part of propagation constant γp vs. frequency in the range 0–100 MHz, for wire-types AWG#24, AWG#26, DW10 and Flat-Pair...... 32

2.11 Magnitude of characteristic impedence Z0 vs. frequency in the range 0–100 MHz, for wire-types AWG#24, AWG#26, DW10 and Flat-Pair...... 32

2.12 Phase of characteristic impedence Z0 vs. frequency in the range 0–100 MHz, for wire-types AWG#24, AWG#26, DW10 and Flat-Pair...... 33

2.13 The analogue channel, comprising of the DAC, the splitters, the AA-ﬁlters, the DSL and the ADC...... 34

2.14 a) Filtering effect of the DAC. The discrete-time multitone signal is represented as a continuous-time stream of Dirac-pulses at the input to the DAC. b) Low- pass anti-alias ﬁltering removes signal components above the prescribed bandwidth. . 35

2.15 Magnitude Squared Frequency responses for the VDSL test loops...... 37

2.16 Sampled impulse response for VDSL test loop number 1. The sampling rate is

fs = 35.328 MHz ...... 37

157 LIST OF FIGURES

2.17 Typical noise PSD in a VDSL system. It is difﬁcult to distinguish the noise types, because they are superimposed. Notice, however, the 10 RFI spikes across the band, the ﬂat AWGN N = 140 dBm/Hz, and the various crosstalk disturbing 0 − signals in the ADSL and HDSL bands...... 41

2.18 a) Analogue FDM implemented by a bank of oscillators. b) DMT implemented by IFFT. The two systems are identical...... 42

2.19 DMT signals: a) The lth transmit block. b) The lth receive block. c) Consecutive transmit blocks. d) Consecutive receive blocks, with IBI. e) Consecutive transmit blocks with guard band. f) Consecutive receive blocks with guard band. IBI is reduced...... 45

2.20 Cyclic Preﬁx Insertion. The last L samples of each DMT block are reproduced at the beginning of the block...... 47

2.21 DMT system ...... 48

3.1 a) Discrete-time equivalent communication channel including sampled receiver noise and TEQ b) Equalized channel and noise c) Desired impulse response. . . . 51

3.2 Shortened channel impulse response and target window. Imperfect shortening causes tails outside the window which contribute to ISI...... 55

3.3 a) Actual discrete-time system (channel, noise and TEQ). b) Equivalent signal path. c) Equivalent ISI path. d) Equivalent noise path...... 58

3.4 CSA-loop #1 channel frequency response (including the splitter) and noise power spectral density...... 72

3.5 TEQ frequency responses for CSA-loop #1 equalized with the TEQ-optimized MMSE, gen-MMSE, MSSNR, min-ISI, and DIR-optimized MMSE algorithms. The number of TEQ taps is q = 17. It is assumed that the channel impulse response is known at the receiver...... 72

3.6 A three tap TEQ is trained using the proposed LMS algorithm to shorten ADSL test loop #1. The top chart shows MSE and the bottom chart shows the convergence of the TEQ coefﬁcient vector...... 75

3.7 TOP: Original and shortened channel impulse response for ADSL loop #1, using the DIR-optimized MMSE eigen-solution for the TEQ. The number of TEQ taps is q = 3 and the channel impulse response is assumed known at the receiver. BOTTOM: Same plots for the TEQ obtained by time-domain LMS...... 77

158 LIST OF FIGURES

4.1 Water-Pouring for maximum capacity ...... 82

4.2 Pulse amplitude modulation and Gray coding for 2 and 3 bit constellations. . . . 85

4.3 Gaussian probability distribution function, and graphical representation of tail error probabilities ...... 87

4.4 Pulse amplitude modulation symbol error probability curves...... 88

4.5 Quadrature amplitude modulation constellation for bi = 2, 4...... 89

4.6 Quadrature amplitude modulation constellation for bi = 5...... 90

4.7 ANSI quadrature amplitude modulation constellation for bi = 1, 3...... 90

4.8 ETSI quadrature amplitude modulation constellation for bi = 3...... 90

4.9 Quadrature amplitude modulation symbol error probability curves...... 91

4.10 Modulation gap Γ vs symbol error probability per dimension...... 93

4.11 The top two graphs show the maximum PAM constellation size possible for a given SNR and target error rate. The ﬁrst graph uses the approximation gap to

determine bi. The second uses exact numerical inversion of qi via the MAT- LAB function fminbnd. It is seen that the graphs are almost identical and the approximation gap holds very well. The bottom graphs show the same curves for QAM. Again the approximation is seen to be very close. For illustrative purposes, 6 the “gap” to capacity Γ is illustrated on the third graph for q˜i = 10− ...... 94

4.12 The number of possible QAM bits per symbol as a function of the symbol error rate for different SNRs...... 103

4.13 Willink’s test channel...... 105

4.14 Willink’s optimal loading of her test channel; results in non-uniform power, nonuniform bit-rates and non-uniform error-rates across the tones...... 107

4.15 Campello’s near-optimal loading of Willink’s test channel; results in non-uniform power, non-uniform bit-rates and uniform error-rates across the tones...... 108

4.16 New loading algorithm applied to Willink’s test channel; results in uniform power, non-uniform bit-rates and non-uniform error-rates across the tones...... 109

4.17 Default loading of Willink’s test channel; results in uniform power, non-uniform bit-rates and uniform error-rates across the tones...... 110

4.18 Willink’s optimal loading of ADSL test loop no. 2...... 112

159 LIST OF FIGURES

4.19 Campello’s near-optimal loading of ADSL test loop no. 2 ...... 113

4.20 New loading algorithm applied to ADSL test loop no. 2...... 114

4.21 Default loading of ADSL test loop no. 2 ...... 115

5.1 a) Uniform Subchannel Allocation b) Nonuniform Subchannel Allocation . . . . 118

5.2 A Two-Channel Transmultiplexer ...... 118

5.3 Spectral interpretation of Two-Channel Transmultiplexer: a) baseband spectrum of 0th-subchannel data. b) Effect of upsampling by 2. c) Filter response. d) Modulated subchannel data...... 119

5.4 a) Upsampling followed by ﬁltering. b) Polyphase implementation...... 120

5.5 a) Filtering followed by downsampling. b) Polyphase implementation...... 121

5.6 Polyphase Implementation of Two-Channel Synthesis Bank ...... 123

5.7 Polyphase Implementation of Two-Channel Analysis Bank ...... 124

5.8 Polyphase Implementation of Two-Channel Transmultiplexer ...... 125

5.9 The Haar ﬁlterbank frequency response ...... 126

5.10 Two channel transmultiplexer and schematics for the synthesis and analysis banks 127

5.11 a) Iterated two channel transmultiplexer, split on subchannel 0. b) Arbitrary wavelet packet tree...... 127

5.12 Possible arrangements of a 4-band WPT...... 127

5.13 θ and an example of τ Θ for D = 3 ...... 128 D ∈ D 5.14 Equivalent branch ﬁlter representation...... 130

5.15 The Noble identities...... 130

5.16 Sample Wavelet Packet Tree, with 3 Subchannels and equivalent branch ﬁlter representation...... 131

5.17 Subchannel magnitude frequency responses for the sample 3-channel WPT. . . . 131

5.18 Time-frequency plane tilings for different wavelet packet tree conﬁgurations. a) The full-tree (or regular-tree) corresponds to the short time fourier transform (Ga- bor transform). b) The logarithmic tree only splits the high-pass channel at each stage; known as the discrete wavelet transform. c) An arbitrary wavelet packet tiling.132

160 LIST OF FIGURES

5.19 Effect of impulse noise and a jammer (shown by arrows) on a) regular multitone b) single tone and c) intelligently structured wavelet packet modulation. Fewer symbols are hit by interference, by optimal time-frequency tiling...... 133

5.20 Transmitted binary PAM symbol Xi and interference corrupted receiver estimate

Xˆi...... 135

5.21 The set of trees of maximum depth D = 3: Θ3...... 138

5.22 Test Channel Response and Optimal Tiling ...... 139

5.23 VDSL#1 channel response (dotted) and optimal wavelet packet tiling ...... 140

5.24 VDSL#3 channel response (dotted) and optimal wavelet packet tiling ...... 140

5.25 VDSL#7 channel response (dotted) and optimal wavelet packet tiling ...... 141

5.26 Optimal Tree Structuring τopt for VDSL test loops #1, #3 and #7 ...... 142

A.1 Magnitude Squared Frequency responses for the eight ADSL test loops...... 149

A.2 Sampled impulse response for VDSL test loop number 3...... 150

A.3 Sampled impulse response for VDSL test loop number 4...... 150

A.4 Sampled impulse response for VDSL test loop number 7...... 150

A.5 Sampled impulse response for ADSL test loop number 1...... 151

A.6 Sampled impulse response for ADSL test loop number 2...... 151

A.7 Sampled impulse response for ADSL test loop number 3...... 151

A.8 Sampled impulse response for ADSL test loop number 4...... 151

A.9 Sampled impulse response for ADSL test loop number 5...... 152

A.10 Sampled impulse response for ADSL test loop number 6...... 152

A.11 Sampled impulse response for ADSL test loop number 7...... 152

A.12 Sampled impulse response for ADSL test loop number 8...... 152

161 List of Tables

1.1 Residential and commercial applications requiring digital inter-connectivity [1] . 2

1.2 Voice-band digital modem standards. The V-series modems are CCITT/ITU-T standards. The bis sufﬁx on some standards stands for second. This table is compiled from sources [1–7]...... 3

1.3 xDSL ﬂavours and rates ...... 5

2.1 VDSL test loop descriptions [25] ...... 25

2.2 Line constants found by curve ﬁtting, for the wire-types encountered in the ADSL and VDSL test loops [25]...... 27

2.3 Crosstalk power spectral densities in a DSL binder [25]. Nd is the number of disturbers. (f) is the PSD of the disturbing signal. l is the length of the loop in Sd feet...... 39

2.4 The internationally recognized amateur radio bands [25, 217]...... 39

3.1 Summary of Existing Channel Shortening Algorithms (modiﬁed from [250]). Those marked with a dagger are simulated in this thesis...... 62

3.2 Achievable bit rates for the eight CSA-loops equalized with the TEQ-optimized MMSE, gen-MMSE, MSSNR, min-ISI and DIR-optimized MMSE algorithms, as

a percentage of the Matched Filter Bound RMFB. The number of TEQ taps is q = 17. It is assumed that the channel impulse response is known at the receiver. 71

3.3 Figures of merit for CSA-loop #1 equalized with the TEQ-optimized MMSE, gen- MMSE, MSSNR, min-ISI and DIR-optimized MMSE algorithms. The number of TEQ taps is q = 17. It is assumed that the channel impulse response is known at the receiver...... 73

162 LIST OF TABLES

3.4 Achievable bit rates for the eight CSA-loops equalized with the TEQ-optimized MMSE, gen-MMSE, MSSNR, min-ISI and DIR-optimized MMSE algorithms, as

a percentage of the Matched Filter Bound RMFB. The number of TEQ taps is q = 3. It is assumed that the channel impulse response is known at the receiver. . 73

3.5 Achievable bit rates for the eight CSA-loops equalized with the TEQ-optimized MMSE, gen-MMSE, MSSNR, min-ISI and DIR-optimized MMSE algorithms, as

a percentage of the Matched Filter Bound RMFB. The channel impulse response is estimated using a training sequence...... 74

3.6 Achievable bit rates as a percentage of the Matched Filter Bound RMFB for the eight CSA-loops equalized with the DIR-optimized MMSE algorithm. The TEQ is implemented by time-domain LMS algorithm and by eigen-solution. In the case of the eigen-solution, results are presented for both the case where the channel is estimated at the receiver and the case where the channel is known at the receiver. The number of TEQ taps is q = 3...... 76

4.1 Willink’s optimal loading algorithm ...... 99

4.2 Campello’s near-optimal loading ...... 101

4.3 Krongold’s near-optimal loading. A bisection algorithm is used to ﬁnd λ. . . . . 102

4.4 Different loading algorithms based on optimization over the bitloading b, the power-loading P and the eroor-rate loading p...... 104

4.5 Relative performance of four loading algorithms when used to load Willink’s test channel...... 106

4.6 Comparison of 4 loading algorithms for the 8 ADSL test channels deﬁned in Chap- ter 2 ...... 111

5.1 The Catalan numbers ...... 128

5.2 Cardinality of the set of trees ΘD of maximum depth D...... 129

5.3 Tree growing algorithm for maximum SIR...... 136

5.4 Average subchannel SIR for each of the 26 WPTs in the set Θ3...... 138

5.5 Tree selected by the growing algorithm for different noise levels. Refer to Fig. 5.21 142

5.6 Tree selected by the growing algorithm for different noise levels. Refer to Fig. 5.21 143

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