M-ARY ORTHOGONAL MODULATION

USING WAVELET BASIS FUNCTIONS

A Thesis Presented to

The Faculty of the

School of Electrical Engineering and Computer Science

Fritz J. and Dolores H. Russ

College of Engineering and Tech~~olog~.

Ohio University

In Partial Fulfillment

of the Requirements for the Degree

Master of Science

by

Xiaoyun Pan

No\~ember,3000 THIS THESIS ENTITLED

"M-ARY 0:RTHOGONAL MODULATION

USING WAVELET BASIS FUNCTIONS"

by Xiaoyun Pan

has been approved

for the School of Electrical Engineering and Computer Science

and the Russ College of Engineering and Technology

Jerrel R. Mitchell, Dean :/ ,I Fritz J. and Dolores H. Russ College of Engineering and Technology Acknowledgement

I would like to express my sincere gratitude to my advisor, Dr. Jeffrey C. Dill, for his instruction and guidance during the development of this thesis. His knowledge, patience and insightful direction and continuous encouragement greatly contributed to the completion of this research.

I also wish to thank all the thesis committee members, Dr. David Matolali, Dr.

Joseph Essman and Dr. Thomas Hogan for their interest in this thesis, their suggestions and instructions. 'Their willingness to be part of the review process is greatly appreciated.

I would like to take this opportunity to express my deepest appreciation to my parents and my husband, Ming, for the love and encouragement they have given me o\ er the years. They are always the indispensable support in my emotion and spirit.

Special thanks are given to Jim, my good friend for helping nle check the grammar and spelling of this thesis. His continuous friendship and encouragement remain in my mc!noiy along ith thc two year 2nd eight nlonth's stiidy life in Athcns. Table of Contents

... TABLE OF CONTENS ...... 111

LIST OF TABLES ...... vj . . LIST OF FIGURES ...... VII

CHAPTER 1 TNTRODUCTION ...... 1

1. 1 History of Multicarrier Ivlodulation ...... 1

1.2 Wavelets and Their Application in MCM ...... 2

1.3 Outline of the Thesis ...... 6

CHAPTER 2 WAVELET IvIODULATION ...... 9

2.1 Mathematical Fundamentals of Orthonornlal Dyadic 'viiavelet ...... 9

2.1.1 WaveletsandMRA ...... 9

2.1.2 14{rt\~elets2nd FilterBank ...... 16

2.1.3 Perfect Reconstn~ction ...... ?!

2.3 r\;lr~lticarrierR4odulation ...... 26

2.2.1 Introduction ...... 26

2.2.2 Iniplt.ii~ent:~tion...... -17

2.3 M7a\;c?t:t Modulation System ...... 29 2.3.1 Waveform Development ...... 29

2.3.2 Wavelet Modulation ...... 32

CHAPETR 3 IMPLEMENTATION OF WAVELET MODULATION SYSTEM

3.1 Wavelet Development ...... 37

3.1.1 Wavelet Design ...... 38

3.1.2 Meyer Wavelets ...... 41

3.1.3 Meyer MR4 and Square Root Raised Cosine Function ...... 43

3.1.4 Approximation to Scaling ]Function ...... 46

3.2 Filter Bank Design ...... 43

3.2.1 Design Prccedure ...... 44

3.2.2 Filter Order ...... 50 -. 3.2.3 'Transmultiplexers ...... 3.3

3.2.4 Group Delay ...... 59

CHAPTER 4 SIMULATION SYSTEM DESlGN ...... 65

4.1 Design of Two-Channel Tsans;m~~ltipleser...... 65

4.1.1 Filters ...... 05

4.1.2 Deci~natorand Expander ...... 71

4.2 Transmitter ...... 72

4.2.1 M-ary Sigrlaling ...... 72

4.2.2 Ptilse Gencl-atos ...... 73 4.2 Receiver ...... -76

4.3 Channel ...... -78

CHAPER 5 SYSTEM SIMULATION AND PERFORMANCE ANALYSIS ...... 80

5.1 BER Performance ...... 80

5.2 Properties of Transmission Signal ...... 88

5.3 Narrowband Interference ...... 92

CHAPER 6 CONCLUSION AND FUTURE STUDY ...... 99

6.1 Summary ...... 99

6.2 Future Study ...... 100

REFERENCE ...... 102

APPENDIX ...... 105 List of Tables

Chapter 5

Table 5.1 Bit error probability of simulated wavelet modulation system using

diagonal matrix as orthogonal sequences ...... -84

Table 5.2 Bit error performance of simulated wavelet modulation system using

Hadamard orthogonal sequences ...... -86

Table 5.3 The results of DWT with {depthof 6 for a tone noise with frequency

of 4 Hz ...... 96

Table 5.4 The results of DWT with depth of 6 for a tone noise with frequency

of0.4 Hz ...... 97

Table 5.5 The results of DWT with depth of 6 for a tone noise with frequency

of 0.04 Hz ...... 98 List of Figures

Chapter 1

Figure 1.1 The signal approximatiolls and details at different resolution levels ...... 5

Chapter 2

Figure 2.1 The six level decomposition of a signal using Daubechies wavelets ...... 12

Figure 2.2 Two-channel analysis filter bank ...... 18

Figure 2.3 Two-channel synthesis filter bank ...... 18

Figure 2.4 Wavelet transform using filtering followed by subsampling ...... 20

Figure 2.5 Inverse wavelet transform using interpolation followed by filtering ...... 21

Figure 2.6 Wavelet decomposition and reconstruction filter bank ...... 22

Figure 2.7 Basic multicarrier no-dem" ...... 28

Figure 2.8 al;' on time-frequency plane ...... 31

Figure 2.9 Block diagram of wavelet modulation system ...... 33

Figure 2.10 Generation of the transrniltted sequence using synthesis filter bank ...... 34

Figure 2.1 1 Generation of the transmitted signal ...... 34

Figure 2.12 Rcco~ery of the data symbol using analysis filter balk ...... 36 Chapter 3

Figure 3.1 Haar scaling and wavelet functions ...... 39

Figure 3.2 Shannon scaling and wavelet functions ...... 39

Figure 3.3 Daubechies scaling filter with order of 40 ...... 40

1 Figure 3.4 The Fourier Transform of Meyer scaling function. ,B = - B(x) = x ...... 43 3 . 1 Figure 3.5 Square root raised cosine scaling function. ,B = - ...... 44 3 1 Figure 3.6 Spectrum of SRRC wavelet function. ,B = - ...... 45 3 1 Figure 3.7 SRRC wavelet function. P = - ...... 46 3 Figure 3.8 Successive approximation to SRRC scaling function ...... 48

Figure 3.9 Even order wavelet decomposition and reconstruction filter bank ...... 51

Figure 3.10 The general structurc of transmultiplexer ...... 53

Figure 3.1 1 Two-channel odd order transn~ultiplexer ...... 55

Figure 3.12 Two-channel even order transmultiplexer ...... 58

Figure 3.13 Non-unifonn tree structured transmultiplexer ...... 61

Figure 3.14 Modified transmultiplexer with uniform integer

group delays ...... 03

Chapter 4

Figure 4.1 Coefficic~ltsof square root raised cosine wavelet filter bank ...... 68

Figure 4.2 Frrclue~~cyrcsponsc ol'sclu:u-e root raised cosine wavelet tiltcr bank ...... 00

Fig111-e4.3 Inputs into even order ti\. o-cllanncl transni~~ltiplem...... 70 1);

Figure 4.4 Outputs of transmultiplexer with signal in Figure 4.3 as input ...... 70

Figure 4.5 Outputs of transmu1tiple:xer with inputs as in Figure 4.3, when

decimation and interpolsationare not performed consistently ...... 72

Figure 4.6 Correlator receiver ...... -77

Chapter 5

Figure 5.1 Simplified block diagram of wavelet modulation system ...... 80

Figure 5.2 Bit error performance of M-ary orthogonal signaling system

in AWGN channel ...... 83

Figure 5.3 Bit error performance of simulated wavelet modulation system using

diagonal matrix as orthogonal sequences ...... 85

Figure 5.4 Bit error performance of simulated wavelet modulation system using

Hadamard orthogonal sequences ...... , ...... 87

Figure 5.5 An example of transmitted signal of wavelet n~odulationsystem using

diagonal matrix as orthogonal sequences (M=64) ...... 89

Figure 5.6 An example of transmitted signal of wavelet modulation system using

JValsh-Hadanlard matrix as orthogonal sequences (M=64) ...... 89

t Figure 5.7 No~nlalizedpower spectrum of pulse shape $(-) ...... 90 T

Figure 5.8 Norrilalized PSD of one signal in 64-ary wavelet modulation

S~~stem...... 9 1

F~gure5.0 Average nom~alizcdPSD of 64-ary \\favelct motlulatio~~system . . . , , .9 I Figure 5.10 Dyadic wavelet with non-uniform tree structure ...... 93

Figure 5.1 1 Frequency response of the matched filter ...... 95 Chapter 1 Introduction

1.1 History of Multicarrier Modulation

The principle of a inodulation format that divides the transmitted data into several

interleaved bit streams and uses these to modulate several caniers, was put forward as

early as 1957 [I]. This idea has been introduced in orthogonally multiplexed

con~munications,and the technique is also referred to as multicar-rier modulation (MCM)

or ortliogonal frequency division mulitiplexing (OFDM). There are two important reasons

fbr the interest in h4CM [2]. First, an MCM signal can be processed in a receiver withoui

!he etlha~icementof noise or interference that is caused by linear equalizaticn of a singlz

c~niersignal; secondly, the loilg symbol time used in MCM produces a much greater

imnlurlity to impulse noise and frequency selective fading. Because a multicarrier signal

is integrated 01-er a long symbol period, the effect of impulse noise is much less than for a quadrature amplitude n~odulation(QAM) signal; indeed, this was one of the original

~notivarionsfor MChI. As ~nentioneclbefore, a QAM systenl is sensitive to ~~iipulscsill the time domain; since interesting timc/frequency duality is involved here, one may think

that hlCM systems might be sensitive to impulses in the frequency domain. say, tone

interference. The rid\,antage vf an MCh4 systeili lies in the fact that the sources of thcse

interkrcnces are usually stable (\vh~lcthc nccur-rence of imptilscs In limc dor,r;i111mlglit be quite random). Therefore, these tone interferences can be recognized during training

[I]. Indeed, this modulation scheme was proved to have superior performance in terms of combating the channel impairments, when being used with variable coding and equalization strategies.

In the MCM system proposed earlier, the main processing in the transmitter and receiver is done with an Inverse Fast Fourier Transform (IFFT) and a Fast Fourier

Transform (FFT) [2]. Hence, it is necessary to discuss the transform domain (TD) briefly here. The signal processing is said to be performed in the transform domain if the signal is transformed to the frequency domain using a Fourier transform. Transform domain processing is desirable if one needs to utilize it to suppress interference and, consequently, to irr~provesystem performance. After TD operation, the signal then is transiormed back into the time donlain using an inverse Fourier transform. When the inverse transfonn is implemented in the transmitter and the transfo~moperatioxi is placed in tile receiver, as presented in [2], transform domain processing can be used in MCM systems. This implementation can be realized quite efficiently by using the fast Fo~lrier transfoim [3]. Applications of this technique have primarily been for bandwidth efficiency on telephone grade channels. In recent years, MCM has been proposed for applications in the spread spectrum systc~n[3][5], which is growing fast in \x,irclc.ss colnmu~iicationsand makes this technique more interesting and attractive.

1.2 \Vavelets and Their Application in MCM

The con\rcrzcnce of computing and communicatior~shas encouraged rllc ~t\coi new mathematical 1-net11odsand high-l~er-fot-m:~~~~~digital signal processing tc.c.hr~iq~~c~~111 3 conlmunication systems. In early implementations of MCM (sometime known as discrete niultitone transmission, DMT), discrete Fourier transform was proposed for use in implementing multicanier modulation [6]. Along with the emergence and gradual completeness of wavelet analysis, discrete wavelet transform shows a potential trend to replace the FFT process in MCM system. Superior multicanier and multiresolution signals can be generated for modulation by networks of wavelet filters. By varying the wavelet filter and the wavelet filter network, one can design waveforms with selectable timeifrequency partitioning for multi-user applications. This partitioning can be used to improve multi-user capacity or to provide robust performance in difficult channel conditions [6].

The reasons for the interest in wavelets, generally speaking, can be attributed to their superior features over The Fourier transform. Orthonormal wavelets can represent any finite-energy signal in L~(R),and the basis functions (called wavelets) are not unique. They are obtained by scaling and translating a prototype wavelet function. In contrast, Fourier expansion can only represent signals in L2(0, 27r) with complex exponentials as basis functions. In addition, measuring the time variations of nearly instantaneous frequencies is an important application of wavelets, which can analyze the properties of a time-varying power spectrum for non-stationary processes.

Wavelets are a set of basis functions $!fi,k (x) in continuous time. The remarkable property that is achieved by many wavelets is orthogonality. In L2(Z?), a wavelet 1 x-2'n orthonomal basis is a family of linearly independent functions -ry( & 22' )",n''''3 which can be used to produce all admissible functions. All basis functions ~,,~(x)are built by dilating and translating a unique function ry(x), called the wavelet function [S].

Multiresolution approximations (MRA), introduced by Mallat [S], took the theory of wavelets to a new level of development by producing the framework not only for understanding wavelets but also for constructing a basis. MRA is a sequence of embedded subspaces of LI(R), denoted Vl , whose elements are approximations to

1 t-2'n signals in L? (R) . The set of functions -#(?) ,., is an orthonormal basis of & 2

V, for all j E 2.At a given resolution of a signal, the resolution level is set by j, and the time step at that level is 2-' . The approximation of the signal at resolution level j is

1 t-2'n represented by scaling functions -- 4(------),,,, while the detail at this level is 7 2'

1 x-2ln represented by the wavelets -y( ), ,, . Then the smooth signal plus the 6 2' details, p' s and v's,combine into a multiresolution of the signal at the finer level j-1 [9].

Figure 1.1 shows the relationship among the approximations and details at different resolution levels, where S is the original signal, a, is the approximation of S at level j, and d, is the detail of S at level j. Approximation a, plus the detail d, , combine into the approximation of S at the finer level j-I. Mallat demonstrated that [8] the difference of information between two approximations at the resolution levels J - 1 and J is extracted by decomposing the function in a w,avelet orthonormal basis, and can fonn the orthogonal complement (W,.) of coarse resolution subspace (Vj) in the finer resolution subspace

(lfi-,).That is to say, scaling the scaling function by 2' and taking all integer shifts constitute an orthonormal basis for Vj; scaling the wavelet function by 2' and taking its integer shifrs form an orthonormal basis for Wj. Finally, taking all dyadic scales and integer shifts of the wavelet function produces an orthonormal basis for L'(R).

Figure 1.1 The sig~lalapproxiniations and details at different resolution lelmels

Wavelet orthonormal bases provide an inlportant new tool in functional analysis.

Indeed, before then, it had been believed that no construction could yield simple

01-thononnalbases of I,' ( K) \1~110seelenicnts had snotl loc:rli7;1tion properties in hot11 tlic spatial and Fouricr domain [XI. 6

In [8], Mallat also showed that the computation of the wavelet representation may be accomplished with a pyramidall algorithm based on convolutions with quadrature mirror filters. The signal can also be reconstructed from a wavelet representation with similar pyramidal algorithm. This has converged wavelets, MRA and filter banks, which had been used independently in the fields of mathematics and signal processing, to form a new single theory. The wavelet representation of a discrete signal can be computed through an algorithm known as the discrete wavelet transform (DWT). The wavelet series expansion coefficients at a particular resolution can be produced by filtering the expansion coefficients at a finer resolution with digital low pass and high pass filters and decimating their outputs. The algorithm can be performed recursively on the decimated output of the lo\v pass filter. This not only made wavelet processing practical but produced a connection to a major branch in signal processing, the filter bank, also.

Namely, the D'A'T clln be seen as an alialysis filter bank and the inverse DWT (IDWT), of course, is the corresponding synthesis filter bank. In this research, we are particularly interested in the perfect rzconstruction FIK filter bank. It was proven that by appropriately choosing the arlalysis and synthesis filter banks, the reconsti-ucted output from the synthzsis filter bank is identic~lto the original input to the analysis filter bank.

Perkct rcconstructioi~ filter brinks :!llow thc practical application of ~r;l\c!cts in corni~~urticationsystems.

1.3 Outline of the 'Thesis

111 this thcs~s,;in orthn~o11;llmultiplexe~! comn?unic;rtion svstcn~using ~i;~telct lxrsc~ant1 tlic, i~nplenicntatioiiof th~s~~stc~n using pelfect reco11s11-IIC~~UI~cl~gi!;~lliltcl- banks are developed based on the general framework presented in [4]. Important system design issues and system simulation results are discussed. The well-modeled si~nulation program, written in MATLAB, is presented in the Appendix. The purpose of this thesis is to study the implementation of wavelet modulation in MCM systems, and provide a platfomi (the simulation system) on which extensive work involving wavelet nlodulation and its performance analysis could be conducted. This thesis also focuses on the analysis of transmission signal spectral characteristics and system performance under tone interference.

The outline of the following chapters is presented below. Chapter 2 will briefly review some mathematical fundamentals of wavelets. Since wavelet analysis has reached a certain level of maturity as a well-defined mathematical discipline, containing nurnerous theories and applications, only those topics that are related to this research, such as mu1:iresolution approxjn~ation,perfect reconstr-uction and filter banks, will be reviewed. Additionally, the fundamental theory of MCM and the frameu~orkof wavelet motiulation systems, proposed by Jones [4], will be introduced in this chapter also.

Chapter 3 will discuss implementation of wavelet modulatiori in orthogonally

~nultiplexcommunication systems, concentrating on detailed development of efficient implementation of this new nmodulation fornut. The practical design proccd111-c filtcr banks and the approximation method of scaling fiinction will be provided. In addition, some practical implementation issues regarding iilter banks, such as filter ortfer and group delay, dons ith pulse generator, will be discussed. Chapter 4 can be used as a imanual of the simulation program. All the important functions will be introduced and interpreted. The simulation steps and procedures are presented for detailed reference.

Chapter 5 will present the system simulation results and performance analysis.

First the bit error rate (BER) perfornnance of wavelet modulation system using two types of orthogonal sequences (diagonal identity matrix and Walsh-Hadamard matrix) is provided. The characteristics of the transmitted signals in the wavelet modulation system are studied in both the time domain and the frequency domain. Also, a brief analysis of the anti-narrowband interference property of the wavelet modulation system is included in this chapter.

Chapter 6 consists of conclusions and summary remarks in regard to the research psrfonned in this thesis. In addition, a number of future research areas 2re suggested in this chapter. Chapter 2 Wavelet Modulation

2.1 Mathematical Fundamentals of Orthonormal Dyadic Wavelet

Over the last decade, wavelet analysis, as a new tool, emerged from mathematics and was quickly adopted by diverse fields of science and engineering. It has begun to play a serious role in a broad range of applications. In the brief period since its creation in 1987-

1988, it has reached a certain level ocmaturity as a well-defined mathematical discipline

The purpose of this chapter is to develop the fundamental theories of MCM system implenlented by orthononnal dyadic wavelets, so only relevant theories within walreitl analysis are reviewed.

2.1.1 \Yavelets and MRA

Most of the work related to multiresolution approximation (MRA) for constructing orthononnal wavelet bases was done by Mallat. A11 the following definitions, theories and their proofs can be found in [7] [8] [lo]. The notation will follow his devclopmcnf

;1lso.

-Uefirlitiotl 2.1 (Rqultiresolu tions)

;I sc.rluctlce (I., } ,c, of closcrl szrh.sl~iic.esqf 1: (R) is u 1i1ztltr/-csolrttror1rrp/;ro\ rrrlrrlro~l

!f rlic~~~olloit~rtlgG~I-~~~PI~~I'L.J. c11.c scrti.jic.r/: lim Vi = nV; = (0) ;+m -m

+m Iim Vj= Closure ( U V, ) =L2(R) ;-+-a?

There exists B such that (B(t - n)),,, is a Riesz basis*for V,.

Here Z is the set of integers, R is th.e set of real numbers. L'(R) is the vector space of finite energy signals defined on R. The energy equivalence property of a Riesz basis guarantees that signal expansions over any Riesz basis are numerically stable.

This definition specifies the mathematical properties of multiresolutio~lspaces. Tlre approximation of a function f at a resolution 2-' is defined as an orthogonal projection onto the space Vi c L2(R). The space Vj regroups all possible approximations at resolution 2-.' . The orthogonal projection of f is the function fjE Vj that minimizes

' A ihrfiily of\-ectors fe, is said to be a Riesz basis ot'a space II I!',[ 1s linearly indepe~lde~~tanti there exist '4 > 0 and 13 > 0 s~lchth31 for any y E H one can find A[rz] n,~th Property (2.1 .l) means that T", is invariant by any translation proportional to the

scale 2 ' . The inclusion (2.1.2) is a causality property which proves that an approximation

at resolution 2-' contains all the necessary information to compute an approximation at

coarser resolution 2-'-' . Property (2.1.3) means that dilating a function in Y, by 2

enlarges the details by 2 and produces an approximation at a coarser resolution 2-'-'.

When the resolution 2-j goes to 0, equation (2.1.4) implies that we lose all the detail of f: On the other hand, when the resolution 2-' goes to infinity, property (2.1.5) implies

that the signal approximation converges to the original signal. Figure 2.1 shows the six

level decomposition of a signal. It is can be found easily that combining one level's

approxinlation and detail can get thc: approximation of the finer level. The detai! at one

level is just the information lost whein from finer level's approximation to coarser level's

approximation.

-Theorein-- --. -- 2.1- The original signal

DetaiIs 5

Figure 2.1 The six l~\~eldccomposition of a signal using Daubechies vs\ clcts The approximation of f at the resolution 2-' is defined as the orthogonal projection on

V,. To compute this projection, we must find an orthononnal basis of V, . The above theorem defines how to orthogonal.ize the Riesz basis (B(t-n)),,, and construct an orthogonal basis of each space Vj. Namely, by dilating and translating a single function

4 called the scaling function.

A multiresolution approximation is entirely characterized by the scaling function 4 that generates an orthogonal basis of each space V,. The multiresolution causality

1 t property (2.1.2) imposes that Vj c Vi-,. In particular -4(-) E c Vo. Since (qJo,n lnGL J5 2 is an orthononnal basis of V, , we can. decompose

with

This scaling equation relates a dilation of 4 by 2 to its integer translations. The sequence

11[11] will be interpreted as a discrete iilter.

The Fourier transform of (2.1.8) yields

1 Q(2w)= --JT I-l(w)@(co) (2.1 10)

\I here H is the discrete Fourier trnllsfor!li of /:[ll] . Thc foilo\\ it12 tbcorcii~intpoics iorlr~ important constraints oil N. Theorem 2.2 (MALLAT, MEYER)

Let 4 E L~(R)be an integrable scaling junction. The Fourier series of'

2 E R, IH(co)~~ + IH(O + n)l = 2 and

Discrete filters whose transfer functions satisfl (2.1.1 1) are called conjugate mirror filters. They make it possible to decompose discrete signals into separate frequency bands with filter banks.

Definition 2.2

Let W, be the orthogonal conzplement of VJ in Vl-,, that is, V,., = V, Q W, and

V,IW,.

Theorem 2.3

Let 4 he a scaling function and h the corresponding conjugate mirror .filter. Let

t,v be the.fiinction whose Fourier transfbrm is

lww Y(w)= --(?(-)@(-) , fi2 2 with

Let us denote For any scale 2', (ry,,, ),,, is an orthonormal basis of W,. For all scales,

I(,,N~Z~ is an orthonormal basis of L~(R) .

Equation (2.1.13) implies that I,M must satisfy

with

ry is called the wavelet function.

Equation (2.1.14) defines the relationship between g[n] and theconjugate mirror filter h[n] in the Fourier domain. In the time domain, it can be derived that

g[n]= (-I)'-" h[l -n]. (2.1.18)

The mirror filter plays an important role in the fast wavelet transform algorithm.

The following theorem gives necessary and sufficient conditions on G (discrete

Fourier transform of g[n]) for designing an orthogonal wavelet.

Theorem 2.4

The.fumily (y,,,Inez is an orthonormal basis of W, if and only if G(CI)H*(CI)+ G(OI + T)H*(O+ xi.)= o (2.1.20)

From above theorems and definitions, it can be seen that the scaling and wavelet functions are orthogonal at different coarse scales (theorem 2.1, definition 2.2, theorem

2.3). Additionally, the wavelet function is orthogonal to itself at different scales but the scaling function is not orthogonal to itself at different scales due to the embedded vector space property.

2.1.2 Wavelets and Filter Bank

Theorem 2.3 implies that it is possible to construct a wavelet orthononnal basis from any conjugate mirror filter h[n]. 'This gives a simple procedure for designing and building wavelet orthogonal bases, and relates wavelets to the filter bank also.

The following theorem imposes another relation between h[?zjand g[?z].It can be easily derived from above theorems.

Theorem 2.5

Let ~[II]and g[?z]he the series coef~cie?ltsof :he scalirzg arid MI~I\>C~C~fiulctioi~.~ ill or1 A4RA. Tlierz their transfolnls srrtis/S~

The relation in (2.1.21) is commonly referred to as the power conlplen~cntzry property.

t Rccall relations in (2.1 .S) and (2.1 .lG). By substituting - with t , it is casy to yct 2 and

These expansions clearly indicate that for subspace V, , both the scaling and wavelet functions can be considered as a series expansion in the basis function for V,.

Additionally, one can see that wavelet function t,v is actually determined by the scaling function for the MRA. Therefore, from the scaling hction q3 and the conjugate mirror filter h[n],it is not difficult to build a corresponding orthonormal wavelet basis. In the next step, the wavelet will be related to the filter bank directly. Some basic concepts about filter banks are reviewed here.

A filter bank is a set of filters, linked by sampling operators and sometimes by delay.

In a two-channel filter bank, the analysis filters usually have two filters, lowpass and highpass. They separate the input signal into different frequency bands. These subsignals can be compressed much more efficiently than the original signal. Usually, it is not necessary to preserve the full outputs from the analysis filters, and they are downsampled. Only the even components of the lowpass and highpass filter outputs are kept for further decomposition or reconstruction.

Figure 2.2 and 2.3 show the general structure of two-channel analysis filter bank and synthesis filter bank, where H, , F, are lowpass filters and H, , 4 are highpass filters. input analysis decimator

Figure 2.2 Two-channel analysis filter bank

expander syithesis output

Figure 2.3 Two-channel synthesis filter bank

The downsampling operators are decimators, the upsanipling operators are expanders. These two processes are not time-invariant operations. However, by appropriately choosing lowpass and highpass filters, it is possible to acllicve perfect

I-cconstl-~iction.This n ill be covered in the nest section.

1,ct's ctefine a f~inctionfin the square integrable space L2(K).Its projcctior15 into the space I,'(, can be expressed as Since (@(r - 11) j .,, is orthonormal,

= (f (t),$(f - 11)). (2.1.25)

Each a,[n] is thus a weighted average off in the neighborhood of n, so U,[II] can be

considered as the discrete signal off or the approximation off at scale 0 (V,) .

Let f,. denote the approximati~onoff at scale j. Since (4j,n)n,Zis an orthonormal 1 basis of Vj, f,, is characterized by

The discrete wavelet coefficients of a, are defined to be wavelet coefficients off:

d,[n]=(f, Y,.) (2.1.28)

Let f,, denote the detail infomiation off at scale j, i.e., the information lost frcm

approxinlation at scale j to coarser re~~olutionj+l. It is characterized by (2.1.28) or

i4 fist \\.a\.elet tsansfoml decomposes successi\.ely each approximation /'I illto ,I

coarser approsinintio~~,f, -, plus the wavelet coefficients carried by -6,. I11 the othcr ,+I

disectioii, the I-econsti-~~ctionfrom \~ravcletcoeftjcicnts recovers each f; fro111 L ll~ld i,, +, . The following theorem illustrates this successive decomposition and reconstruction procedure.

Theorem 2.6

At the deco~zlposition

Theoretn 2.6 proves that a ;+,[n]and dj+,[n]are computed by taking every other sample of the convolution of uj[n];md dj[n]with 1z[-n] and g[-121 respectively. This procedure can be represented by a lcascade of filtering followed by downsampling, as illustrated in figure 2.4.

Two-channel analysis filter bank

Figure 2.4 Wo\,elet transform using filtering follo~\edby slrbsanlpling The filter A[-n] removes the higher frequencies of the inner product sequences a, lvhereas g[--n] is a highpass filter which collects the remaining highest frequencies.

Similarly, the reconstruction (2.1.32) can be represented by a cascade of interpolation that inserts zeros to expand a,?,[n] and dj+,[n], followed by filtering and adding the output together, as shown in Figure 2.5.

I

Two-channel synthesis filter bank

Figure 2.5 Inverse wavelet tran!sform using interpolation followed by filtering

Comparing Figure 2.4, 2.5 with Figure 3.2 and 2.3, it can be easily seen that one stage of the discrete wavelet transform is equivalent to two-channel analysis filter bank with lowpass filter 11, = h[-n] and highpass filter It, = g[-111. One stage of the discrete inverse wavelet transform is equivalent to t\vo-channel synthesis filter bank with f, = Ii[n] and S, = g[n].

2.1.3 Perfect Reconstruclio~~

Connect analysis filter bank and synthesis filter bank, as shown in Fi~u~e2.6. 'l'he fast discrete \i~a\.clettransforni deconil~osessignal into lowpass and highpass components subsampled by 2; the inverse transform performs the reconstruction by filtering the zero

- - expanded signals with a dual lowpass filter h and a dual highpass filter g .

Figure 2.6 Wavelet decomposition and reconstruction filter bank

- For perfect reconstruction, the recoilstructed output ao[rz] froin the synthesis filter bank should be identical to the original input a,[??]to the analysis filter bank with, possibly, a fixed integer delay. However, as mentioned before, the downsampling process is not a time-invariant operation, in fact it introduces aliasing. One has to choose rhe filters appropriately so that they can jointly achieve the perfect reconstruction propel-ty.

We use the above filter bank as an example to discover the conditions for perfect reconstruction. This two-channel filter bank can then be used as a building block in [he multistage ~vaveletdecomposition ancl reconstruction.

Jl'ith iz,,[lr] as input and taking Fourier transform of both sides in Figure 2.6, the output is gi\.en by where T(w) is the distortion transfer function incorporating phase and amplitude distortions and A(w)4(w+ n)is the aliased component resulting from the decimator.

(The capital letters represent Fourier transform of corresponding signals.)

Definition 2.3

-4 two-channel filter bank having the ii~put/outputrelationship in (2.1.33) is said to lzave perfect reconstrziction ifthe alias componeizt is zero and T(w) reduces to e-'ON for so~nepositive integer N.

Theorem 2.7

Thefilter bankpeiforins an exact reconstruction for any input signal fund only if

- This theorem proves that the reconstruction filter h and g are entirely specified by the decomposition filter 11 and g .

Theorem 2.8

Perfect t.eco~zst?-zrction filters sati~fi) 24

Theorem 2.9

- Let N be odd. Let H satisjj the power complenzenta~yproperh;. Then the filter bank of Fipre 2.6 will have perfect reconstruction provided the remainingfilters are related as folloi~~s

- G(w) = - ~I(O+ t) (2.1.39)

Theorem 2.9 is given by Smith and Barnwell [I 11 as the necessary and sufficient conditions for obtaining perfect reconstruc!ion orthogonal filters with a finite impulse response. Expressions (2.1.37)--(2.1.39) were in the form of the Z-transform originally, while tiley are converted to the Fourier transform here for consistency.

Using the Fourier transfolm properties, in the time domain (2.1.37)-(2.1.39) can be rewritten 2 5

- If we impose that the reconstrucltion filter h is equal to the conjugate mirror filter h , then (2.1.36) is the condition of Smith and Barnwell [I I] that defines conjugate mirror filters

IH(w)~~+ IH(W + n)I2= 2 (2.1.43)

It is identical to the filter condition (2.1.1 1) that is required in order to synthesize orthogonal wavelets.

By now, it can be easily seen that there is a close relationship between wavelets and perfect reconstruction filter banks.

One should notice that the expressions (2.1.37)-(2.1.42) hold only under the condition that N is odd, where N is indeed the order of filter h[n].It is possible that an even order filter could be used as the conjugate mirror filter, and in this case, the conditions (2.1.37)-(2.1.39) for perfect reconstruction must be modified. This problem along with other practical issues abou~tfilter bank will be treated in great detail in Chzpter

3.

Following the essential research of MRA, various kinds of wavelets are developed, such as the dyadic wavelet, M-band .wavelet and wavelet packet. Amon2 these wavelet structures, this thesis only focuses on the dyadic wavelet. The M-Band wavelet ant1

~va~~clctpacket are bcyond the scope of this research, so thcir pertinent thcories arc not covered i17 this thesis. 2.2 hlulticarrier Modulation

2.2.1 Introduction

Multicarrier modulation is a class of orthogonal frequency modulations, and indeed can be thought of as a foml of frequency division modulation. It was used more than 40 years ago in Collins Kineplex Syste~n[2]. The principle of MCM is that the transmitted data is split into several bit streams and used to modulate several caniers. Thus the spectrum is divided into parallel orthogonal and narrowband subchannels. Each carrier occupies one subchannel [9].

Multicanier modulation can be interpreted as a transmultiplexer (a transmultiplexer is a filter bank with synthesis first) tlhat takes time-division multiplexed (TDM) data and transfomls it to frequency-division niultiplexed (FDM) data. With this view, MCM provides an efficient means to access multiple multiplexed data streams. This is potentially very attractive, since high-speed, broadband networks often involve multiple data sources that are multiplexed during transmission. For many channels, MCM approxinlates a constant transfer function in each subchannel. MCM can reduce the effect of impulse noise as a consequence oi' its longer symbol durations. Also, MCM provides an effective means for combating narrowband interferences. The subchannels which are affected by the narrowband interferences can eas~iybe identified and their use ~~rhih~tcd.

So MCM has the flexibility of not transmitting in the comipted subcl~anl~elsin case of liarso\\ band interference and the flexibility of transmitting important data in subchannels with high SNRs [6][9]. In addition, IVCM uses the transmission band efficiently because modulation and coding techniques can be employed independently in the subchannel to approach high capacity.

2.2.2 Implementation

In contemporary implementations of multicarrier modulation the generation and modulation of several subcarriers are accomplished digitally, using an orthogonal transformation on each sequence of a block of data stream. The receiver performs the inverse transformation on segments of the sampled waveform to demodulate the data.

Therefore, the subchannels overlap spectrally. However, as a consequence of the orthogonality of the transformation, if the distortion in the channel is not severe, the data in a subchannel can be demodulated with a very small amount of interferences from the other channels.

In the early stage of MCM (also known as discrete multitone trans~nission,DMT), discrete Fourier transform was proposed to be used to implement multicarrier modulation. The DFT exhibits the desired orthogonality, and it can be implemented with a fast DFT algorithm. A simplified block diagram of multicarrier transmitter and receiver is shown in Figure 2.7 [2]. The main processing in the transmitter and receiver is done with an IFFT and FFT.

To maiiltain the orthogonality of' the transform, DMT schemes employ rectangular pulse for data modulation. Consequen~tly,a given subchannel has significant overlap i~rith a large number of its spectrally neiglibosing subchannels, and, without con1penc;~tion. subchannel isolation is achievcd only for channels \\ hich liave very little distortion In recent years, some researchers have focused their attention on wavelet transforms, and tried to replace the FFT and IFFT operation in DMT system by the discrete wavelet transform. With the wavelet transfonn, the pulses for symbols in different data blocks

Tx Serial-to- DaLFI=FrFFbi=-lI-hbuffer

Rx Data

Figure 2.7 Barsic multicarrier "mo-dem"

overlap in time, and, in gencral, their cnvclopes are not rectang~~larin shape. The basis fi~nctions'pulsescan be designed so that a su1~ch:trlnelis narrowband and ha? significa~~t spectral o\rerIap \vith only a small nulnber of its spectral neighbors, \vl111c~ICS~J-\~II~~ the

01 tllogonal~ty of lhc transfo~inat~on. 2.3 Wavelet Modulation System

Jones has developed the fundamental theory of orthogonally multiplexed comnlunication system using orthonormal dyadic wavelet basis functions as the orthogonal signals on which the QAM sequences are placed [4]. The dyadic wavelets provide a non-uniform decomposition of the time-frequency plane. This allows for greater effectiveness against channel impairments. In addition, efficient digital filter banks exist for the implementation 'of these systems. Some fi~ndamentaltheories about wavelet and filter bank are reviewed in section 2.1.

2.3.1 Waveform Development

A quadrature amplitude modulation (QAM) digital signal, generally, can be defined as

where 4 is the pulse shape, E is thc average symbol energy, T is the symbol duration and d, are complex-valued QAM symbols. Usually, it is reasonable to assume that the data symbols are identically distributed with zero mean and unit variance.

Let 4 in (3.3.1) be a scaling function in an MRA. Because of the orthogonality of 4, it is then determined that the signal defined by (2.3.1) is free of intersymbol i~lterferencc.

Therefore, MKAs can provide a broad new class of p~ilseshapes for use in conventional data communications.

Sincc 4 is tl~cscaling fi~nctionof an MI

So,

where J is a positive integer. It follows that 4 can be expressed in terms of the basis functions for the subspaces on the light hand side of (2.3.2). Now the QAM signal in

(2.3.1) can be rewritten as the multidimensional signal

where a: are complex-valued QAM symbols for j = 1, ...... , J . Indeed, a: also can be thought of as the decomposition co'efficients of m(t) at different basis functions. To illustrate this idea more clearly, we can put a: on the time-frequency plane, as shown in

Figure 2.8.

The tiles in figure 2.8 represents the essential concentration in the time-frequency plane of a given basis function. So this figure is known as tiling diagram, indicating the concentration of energy in the time-frequency plane of each symbol.

The expression (2.3.3) is another fonn of QAM signal, if we choose a scaling function as the pulse shape. It is also referred to as n~ultiscalenlodulation (MSM), since data syrnbol a,: has been placed at different scales, as illustrated in Fig~ire2.8. 1 he orthogonality of the individual basis fiinction prevents ISI, fui-then?~ore,their rnuttlal orthogonality prer.ents interference across scales. time Figure 2.8 61: on time-frequency plane

The expression (2.3.3) goes back to (2.3.1) if J is set to 0. A general form of quadrature-quadrature phase shift keying (Q'PSK ) modulation is obtained for J = 1,

A gencral fo1-111 for fractal modulation is obtained \vhen ir: = 0 and a:)= (1,: = tr,, for

i$.j.

fractal niodulation: iii(i) =

I-]-on)FI~L~I-e 2.8, it call be easily see11 that 11ie niodulatlon format 1s co~npos~clcil 3 2 long, low frequency pulses as well as short, high frequency pulses. The ability to decompose the time-frequency plane enables time and frequency selective processing for greater effectiveness against channel impairments.

2.3.2 Wavelet Modulation

Jones proposed a wavelet modulation system to realize the MSM signal in (2.3.3)

[dl. He makes use of the discrete wavelet transform instead of using a set of matched filters which are matched to each pulse shape in (2.3.3). The block diagram of this system is shown in Figure 2.9

At the transmitter, source data are demultiplexed into J symbol streams at dyadic sub-rate of the source. Then an IDWT of these data is computed generating sequence .u, at the source rate. This sequence is transmitted across channels using a pulse shape defined by a scaling function 4 and transmitted across channels. At the receiver, the receit-ed signal is niatched filtered and sampled. Taking DWT of these salrlple points produces the recovcred data symbols ulj' , based on which decisions are made and fi~rthcr processing are performed.

The transmitted signal t)z(t) can be written as

M liere X-, is a complex sequeljce detcnnined by data sy~iibol(I,: .

Figure 2.10 ~llustratesthe re1;ition between s, and tllc data symbols. S~ncccccl11cncc

I-, IS thc in\,eisc \\a\ elet trn~lsfol~nof tfata sjn~hol (I,: , ~t can 1x2 pl ocl~~cccihj nj>l~l\i~~~v (2.1.32) recursively. Namely, combine the symbol on the scaling function pulse with the symbol on the j = 1 wavelet pulse in accordance with (2.1.32). Then apply this equation again to this new sequence and the symbol on the j = 2 wavelet function pulse. This procedure can be realized by using synthesis filter bank.

Serial-to- Pulse shape TxData parallel IDIT generator buffer ' ' t ,(,I

a,:-' +

Matched Sample D\VT -b filter - at t t = kT 4(-

Figure 2.9 Block diagram of wavelet modulation system

Before transmission, a scaling function pulse shape is applied to the sequence x, .

I Figure 2.11 shows how to realize this function. Note that the filter is scaled by to ,I 1. keep the ortl~ononnalityof the scaling fi~nction. Figure 2.10 Generation of the transmitted sequence using synthesis filter bank

xk Impulse 772, (t) 772 (t) Generator + Gain

Figure 2.11 Generation of the transmitted signal

The output of the ilnpulse generator is

where 6 is the Dirac delta function. The output of the pulse generator is

??I, (t)= x(2) * I$4 (---): --A w, t-kT " t = kT m,(t) = x, ((----I = CxA 41- - k) L =- m T k=-z T

So the transmitted signal really has the same format as that in (2.3.6).

The received signal is -

where n(t) is the additive noise. The output of the matched filter is

- 1 t I-([)= r(i)* Iz(t) and lz(t) = --@(--I 1/T 7' where Iz(t) is the inlpulse response of matched filter, thus

'I lleli the signals :ire sampled at t = 111T. UI = I.?,...... , and the sampled scquc~~ceI; is Because of the orthonormality of the scaling function 4, (2.3.12) reduces to

A discrete wavelet transform is performed on this sampled sequence using corresponding analysis filter bank, as shown in Figure 2.12. If there is no noise, the received signals are the exactly same as the transmitted data symbols as long as we use the perfect reconstruction filter bank.

Figure 2.12 Recovery of th~edata symbol using analysis filter bank Chapter 3 Implementation of Wavelet Modulation System

In Chapter 2, we reviewed some fundamental theories about wavelet transform and perfect reconstruction filter banks. A framework of multicarrier modulation system using orthononnal dyadic wavelet basis functions, proposed by Jones [4], was also introduced.

In this chapter, we will discuss some important practical issues in the implementation of this wavelet modulation system. These practical problems, arising from implementing filter bank and pulse shape generator, include the waveform design and filter bank design methods. We will treat filter bank siructure in great detail emphasizing the filter order and nonzero group delay problenls.

3.1 Waveform Development

Based on theorem 2.3 and the relationship between wavelet transform and filter banks, it can be derived that from a given cot~jugatemirror filter h[rz],it is possible to construct a wavelet orthononnal basis. This gives a simple procedure for designing and building wavelet orthogonal bases. Moreover, using wavelets in orthogonally multiplexed conimunication provides considerable flexibility in systeln design, since a large class of ivavelet bases eyists to choose from. The wavelet development in this research follows this design idea. Ho\\ c\ c'r, OIIC sllo~ldt~otice that not all filtcr hank deslg11

Several commonly used wavelets with different temporal and spectral characteristics are reviewed below. One of the main purposes for presenting these wavelets is to find an appropriate wavelet design in cornrniunication scenario.

3.1.1 Wavelet Design

Haar Wavelet The oldest and most basic wavelet is the Haar wavelet. Haar basis is obtained with a multiresolution of piecewise constant functions. The scaling function is

1 Oltll ((t)=lo otherwise

1 The filter h[n] given in (2.1.8) has two nonzero coefficients equal to -J5 at n = 0 and

n = 1. Hence

ItW 1 -v(-) = (-1)'-" h[l - n]((t- n)= -(((t - 1) - ((t)) 45 2 ,,=-a J5

SO @t<1 otherwise

The Haar scaling and wavelet functions are plotted in Figure 3.1

Shannon Wavelet The Shannon wavelet is constructed from the Shannon

multiresolution approximation, whiclh approximates functions by their restriction to low

frequency intervals. It corresponds to @(w) = 1 on the interval [- x, x], and the y(t) is and

2 sin(^) @(lo)= 1 a 4(t)= t

The Shannon scaling function and wavelet function are shown in Figure 3.2.

(a) Haar scaling functio~i (b) Haar wavelet functio~~

Figure 3.1 Haw scaling and wavelet functions

-10 -5 0 5 10 -1 0 -5 0 5 :0 t t (a) Sl~annonscaling fiisiction (bj Shannon wavelet function

Figure 3.2 Shannon scaling 2nd wavelet functions 40

Daubechies Wavelet Daubechies wavelets have a support of minimum size for any given number p of vanishing ]moments. The resulting wavelet has p vanishing moments, and the supports of 4 and cy are respectively [0, 2p - I] and [- p + I, JI].

Figure 3.3 displays the Daubechies scaling filter with order of 40.

N

Figure 3.3 Daubecchies scaling filter with order of 40

The Haar wavelet has the shortest support among all orthogonal wavelets. The functions in the Haar MRA have excellent time localization but poor frequency localization. In addition, Haar functions are not evenly continuous. Shannon wavelet is not compactly supported, although it has a slow asymptotic time delay. Both the scaling and wavelet functions in Daubechies wavelets are compactly supported, but they do not have good frequency characteristics desirable in communication scenario. In addition, the

Daubechies' functions are not symmetric.

In communication problems, the desirable functions are those which have both compact support and symmetry with respect to the center of its support. However, Daubechies has shown that it is im~possibleto achieve compact support and symmetry

simultaneously except for Haar wavelets, which has poor frequency property in real- valued case [13].To obtain a symmetric wavelet, the conjugate mirror filter h[n]must be symmetric, which means H(w) has a linear phase. Complex conjugate mirror filters with

a cornpact support and a linear phase can be constructed, but they produce complex wavelet coefficients whose real and imaginary parts are redundant when the signal is real

[lo].

Among these existing wavelets, the Meyer MRA has been found as a fairly good candidate for the design of wavelet nlodulation system [4], since the functions are symmetric and decay very fast to be considered as being approximately compact support.

In addition, the connections between Meyer MRA and familiar communication wavefornls have been made [4]. So, the rest of this section will concentrate on Meyer hlliA only.

3.1.2 Meyer Wavelets

A Meyer wavelet is a fi-equency band-limited function whose Fourier transform is relatively smooth. This smoothness provides a much faster asymptotic decay in time.

The scaling function a( f) has a compact support and is defined as and

B(x) + (Q(1- X) = 1 (3.1.8)

Setting B(x) = x for x E [0,11 does not lead to the loss of generality, since both (3.1.7) and (3.1.8) are satisfied. P is usually set in the interval

The conjugate mirror filter h[n] can be derived from the scaling function. Recall

(2.1.10),

thus

Because the compact suppon of

H(f) == h-al(2.f)

By applying the Fourier transfo~mproperties, \\ e i1a1.e 43

Since the scaling function has compact support in frequency domain, the resulting wa~~eformin time domain has infinite support. Consequently, the coefficients of conjugate mirror filter h[n] are infinite. This problem will be treated later together with other implementation issues.

1 Figure 3.4 The Fourier Transform of Meyer scaling function, P = - , B(x) = x 3

3.1.3 Meyer MRA and Square Root Raised Cosine Function

S~ibstitutingthe function 19 in (3.1.6) with B(x) = x , the Fourier transfolm of Meyer scaling function is reduced to Taking inverse Fourier transform, we eventually obtain the Meyer scaling function which is identical to the classically used con~municationwaveform -- square root raised cosine

(SRRC) function,

1 The waveform of SRRC scaling function is plotted in Figure 3.5 for P = ;. 3

i Figure 3.5 Square root rnised cosine scaling function, fl = - .? The Meyer wavelet function can then be derived from this square root raised cosine function. Its Fourier transform is defined by

otherwise I O and the spectrum is shown in Figure 3.6.

Taking in\ cosine \i.avelct fun

1 Figure 3.7 SKRC wavelet function, P = - 3

3.1.4 Approximation to Scaling Function

The conjugate mirror filter h[n] can be obtained from (3.1.1 I), but as mentioned before, this results in an infinite number of filter coefficients, which is not practical, because the implementation of the filter bank requires on FIR filter. Jones has shown that this problem can be solved simply by truncating this infinite series symmetrically about zero [4]. Since h[nJ filter coefficients are the expansion coefficients for the scaling function, truncating this series gives the best L~ approximation to the scaling function. The generation of the conjugate mirror filter leads to not only the design of filter bank which will be addressed in more detail in the next section, but also the approximation to scaling function. Sometimes the wavelet system design procedure starts from h[n], i.e., given the minimum sufficient conditions as constraints in an approximation, use the remaining degrees of freedom to choose h[n] that gives the best signal representation and decomposition. In this case, we use (2.1.8) in a reverse manner, compared with the way it was used previously -- use h[n] to calculate the scaling function. Although usually one doesn't use the scaling function explicitly in most applications, i.e., one only uses the scaling coefficients rather than the analytical form, it is necessary to calculate 4(t) in this research, since we need to use it as pulse shape.

Reference [I21 introduces an approximation algorithm for calculating the scaling function, which is called successive approximation. This algorithm is used theoretically to provide existence and uniqueness of 4(t), and can also be used to actually calculate them. This can be done in the time domain to find $(t) or in the frequency domain to find the Fourier transform of +(t), i.e.,

Again, recall the basic recursive equation (2.1.8),

now tlie filter coefficients of ~[II]has finite le~igihN. Rewrite (2.1.8)as We propose an iterative algorithm that will generate successive approximations to #(t).

If the algorithm converges to a fixed point, then that fixed point is a solution to (3.1.16). the iterations are defined by

for the kt'' iteration where an initial ql'"(t) must be given. This procedure can be viewed as applying the same operation over amd over to the output of the precious application.

The estimated square root raised cosine scaling function using successive

approximation is plotted in figure 3.8. The result of the 8'" iteration is quite close to the theoretical scaling function. Thus this successive approximation algorithm is proven to be an effective method.

(a) Successive approximation of (b) Successive appro\;imation of _.?ill ~tcration 8"' i tc~-a:ion

1;igul-e 3.8 Successive ;~ppl-osirnntio~~to SRIIC scaling l't~nctio~l 3.2 Filter Bank Design

3.2.1 Design Procedure

The wavelet modulation system is implemented by using filter bank, so the filter bank design is the key part of the system design.

In the last section, the connection between Meyer MRA and the square root raised cosine function has been made. Th.e square root raised cosine function is indeed the scaling function, with known analyti~calexpression. Therefore, the conjugate mirror filter h[n] can be obtained by applying (3.1.11)

It was also mentioned that in order to get an FIR filter for practical use, we simply sample and truncate $(t) symmetrically about the center of support, i.e., zero in this case.

In Chapter 2, the necessary relations between filters for a filter bark to have perfect reconstruction were introduced. Rewrite (2.1.40) - (2.1.42) below,

g[?l]- g*(IV - 11) (3.2.3) Jt is easy to find that the remaining three filters can be derived directly from corljugate rnirror filter II[II].Rearrange (3.2.1) -- (3.2.3) to illustrate these relations nlore clearly, g[n]= (-l)n h(n) (3.2.6)

Theoretically, the filter bank design is quite straightforward after the conjugate mirror filter tz[n] is obtained. However, one must pay close attention to the implementation problems that may arise in practice. 'The remaining issues of this section will concentrate on these problems.

3.2.2 Filter Order

Filter order N is essential for determining the filter bank structure and the relations among filters in a perfect reconstruction filter bank. As mentioned in Chapter 2, the equations (2.1.37) - (2.1.39) describe the relations between filters in a perfect reconstruction filter bank, where all the filters have odd order. However, the even filter is more desirable in communication applications, because even order FIR filter has linear phase and the resulting scaling function is symmetric. It is shown [13] that symmetry is a desirable property, since it is more robust to nonlinear quantization error in signal colllpression schemes. Lack of pulse symmetry increases loop noise in a widely used timing recovery scheme [14]. Additionally, from a practical perspective, symmetry in digital filter can simplify computations. For an even order filter bznk, the conditions

(2.1.37) - (2.1.39) are no longer feasible for perfect reconstruction; indeed, lhcy can cause serious distortion. Therefore, wc: need to derit e a set of new conditions for an ever1 order filer bank to have perfect reconstruction. 5 1

Theorem 3.1 - Let N be even. Let H satisfi the power co~npler?leiztaryproper~.Then thefilter bank

\till have perfect reconstruction provided the remainingfilters are related as follows

H(w) = e-'" G(o+ x) und

Notice that the term e-'" in (3.2:.7) - (3.2.9) implies that there is one sample delay - associated with filter h[n] and g[n].Therefore, we may change the filter bank structure - by separating this one sample delay from h[n] and g[iz].The new filter bank structure is illustrated in Figure 3.9.

I Delay I li I- - - I - by 1

- Delay g by 1

uo[ll]

Figure 3.9 Even order wavelet clecompositio~~and reconstruction filter bank

The ecluations (3.2.7)- (3.2.9) are changed to H(w)= G(o + ii)

- Again, let the reconstruction filter h be equal to the conjugate mirror filter h, and apply the Fourier transform properties, in the time domain (3.2.10) - (3.2.12) can be rewritten as

and

- g[.] = g*(~- 12) . (3.3.15)

These equations are identical to (3.2.1) - (3.2.3). So, it is interesting to note that in the even order case, the filter design procedure is the same as that for odd order filter, while the filter bank structure has been slightly changed in terms of inserting delays at proper positions.

Regarding filter order, another issue is what order one should choose. This is dependent on how much signal to distortion ratio (SDR) is acceptable in a system Due to the truncation effect. the filter bank, which results from the design procedure as described above, can only be achieved as an asymptotic perfect rcconstruction. Thcrefbrc, a certain

Icvel of distortion is inc\itCible. Jones' \zotk slio\\s that tllc signal to disto~tiolilatic)

Increases with increasing filter order. Also, :t li~glicrSIjK can bc achic\~ed\\/lien Llslng greater ,B. Indeed, for filter order greater than 16, the SDR exceeds 30dB, which implies that the distortion noise will be much less than the thermal noise if -Eb is set at typical No values. For this reason, we do not need to discuss this problem in more detail, as long as we choose a filter order which is both high enough and practicable.

3.2.3 Transmultiplexers

When implementing the multican-ier modulation system using orthonormal wavelet basis functions, we take inverse discrete wavelet transform at the transmitter and the discrete wavelet transform at the receiver (see Figure 2.8). This means if the filter banks are implemented for wavelet transfbrm, the synthesis filter bank has to be used in the transmitter, and the analysis filter bank should be placed in the receiver. This structure is called transmultiplexer. Its general fi~rmis shown in Figure 3.10.

11("1 +-I-~~'T' *!f

Figure 3.10 Thc gene~.itlstr-t~cti~re of transniultiplexer. M signals come in, and they are interpolated and passed through the synthesis filter bank

and combined into one signal. Then the received signal is filtered and downsampled to the original rates. The output xn (n) may suffer from distortion and crosstalk because of the decimation, interpolation and non-ideal filtering.

When the filter bank is used in a reverse order, the perfect reconstruction remains the essential criteria. The design solutioris for the perfect reconstruction transmultiplexer are closely related to those in the perfect reconstruction filter bank. Use H, (2) and F, (Z) as the analysis and synthesis filters of a perfect reconstnlction filter bank, where H,(Z) and F, (2)are the Z-transform of filter 11, (tz) and f,(n) (we will use the Z-transform format to represent a filter in the rest of this section.). Reference [15] shows that the filters H, (2) and Z-IF, (2)can yield a perfect reconstruction transmultiplexer.

Since we are using dyadic w,avelet basis function in this research, ~nlythr: transmultiplexer with M = 2 will be considered. The proof of the feasibility of the above design solutioll for a two-channel tran,smultiplexer is presented below.

Proof Recall the odd order two-channel filter bank and the conditions fbr perfect

I-econstnlction(all in Z-transfonn format).

Structure: Conditions for perfect reconstruction (Smith-Barnwell design):

HI(Z) = -z-"H,(-z-])

where N is the filter order and must be odd.

Structure of corresponding transmultiiplexer is

Figure 3.1 1 Two-ch~annelodd order transmultiplexer

The outputs of the expanders are

The conlbined signal is

'The output of upper decimator is The crosstalk term is

Using the relations in (3.2.16), (3.2.20) becomes

and the term inside the parenthesis is zero. So the crosstalk term is reduced to zero.

The distortion term is

Using (3.2.16), this term becomes

Since N is odd, we get

Recall the definition of conjugate mirror filter in theorem 2.2, and reillrite (2.I .I 1) in Z- transfonn foniiat, assuming the coefficients of filters are real \~alues,

If /;;,(%) is chosen to be the colij\igatc: mirror filter, \\,c h:i\rc 1 Substitute each Z in lefi hand side with z', the value in right hand side does not change.

Now we have

Finally, the distortion term becomes

That means

Similarly, we can prove that

N+1 -(--) X,(Z) = Z X,(Z) (3.2.30)

Therefore, we have proved that this two-cl~almelodd order transmultiplexer is distortion free and crosstalk cancelled, i.e., it has perfect reconstmction.

Following the same steps as presented above, it is not difficult to prove that a two- cl~annel--even order transmultiplexer with the structure illustrated in Figure 3.12 and filter relations expressed in (3.2.33) can perfectly reconstruct the orisinal signals ~)mvidcd i;,(%)is chosen to be the conjugate mirror tilter. 'The transfer filnctions of the reconstl-ucted signals are The structure of a two-channel even order transmultiplexer is

Two-channel synthesis filter bank I

Two-channel analysis filter bank

Figure 3.12 Two-channel even order transmultiplexer

The relations of filters are

HI(Z) = Z-.~H,(-:Z-')

F, (Z)= H,(-Z) = z-.'H, (z-')

Since these t\\.o-channel trat~smultiplexerswith structures illustrated in Figure 3.1 1

2nd 17igl~re3.12 are proven to have perfect reconstruction, they can be used as buil(ling block in tt-ee-structured dyatlic IDWT and D\\T anti yield perfect rcconstr-~~ctioti;!lso. 3.2.4 Group Delay

In Chapter 2, we have mentioned that a perfect reconstruction filter bank can have its output signal identical to the original input signal, but possibly, with a fixed integer delay.

Equation (2.1.33) is the Fourier transform of the output of a two-channel filter bank as illustrated in Figure 2.6. In the case of perfect reconstruction, the aliasing component

A(Go)&(w + n) in (2.1.33) should bt: zero, and distortion term T(w) should be equal to

- j0.V , which implies that the distortion indeed is only a delay in time. It turns out [4] that if the filter satisfies the conditions for perfect reconstruction (described in (2.1.37) -

(2.1.39)), then a two-channel odd order perfect reconstruction filter bank, as plotted in

Figure 2.6, has N delays, where N is the filter order. If the filters have even order, then the two-channel even order perfect reconstruction filter bank, as illustrated in Figure 3.9, has h7i-I delays, where N is still the Glter order. Similarly, equations (3.2.29) - (3.2.32) reveal that the tu.0-channel transmultiplexer has a non-zero group delay also, which are

N+1 N and - for odd order and even order respectively. Fortunately, the group delay is 2 2 an integer in both cases, even though the decimators are placed at receiver.

When these tn.0-channel filter bank and transmultiplexer are used as building b!ocks in tree-structured filter banks or transmultiplexers, group delay may cause problems because the sig~lalsin different paths are experiencing different group delays. Especially, a non-integer goup delay exists potentially becausc signals at some paths need to pass though a series of decimators, and a signal with a given group delay will have this delay divided in half when passed through each decimator.

Jones shows his solution to the problem of different group delays in a tree-structured filter bank in [4], so this issue will not be repeated here. Since the transmultiplexer is indeed implemented in this research, the solutions to the problem caused by non-zero group delay in a tree-structures transmultiplexer are needed.

The structure of a non-uniform tlransmultiplexer is illustrated in Figure 3.13.

The deepest level in the tree structured transmultiplexer is indeed a simple two- channel transmultiplexer. Thus we have

and

assuming all filters have even order (in the case of odd order filters being used, replace each N+2 by N+l). The output Yo of lowpass filter of the analysis filter bank will pass

h'+Z through another analysis filter bank. If me multiply Y,(Z) by zT, Y, becomes exactly the same as the input X, to this two-channel transmultiplexer. There is no longer any group delay, and collsequently, no potential non-integer group delay will be produced in the output of the next analysis filter bank. Furthenore, since Yo is exactly the same as

,Yo,the perfect reconstruction of V,, and V, is guaranteed at the output of thc next analysis filter bank, i.e.. Two-channel Synthesis 0 J Filter bank

Two-channel Analysis \ Filter bank

Fi,qo~-e3.13 Nan-u~~ifo~rnl tree structured transn~altiplexe~. The same operation can be performed at the output Wo of the next analysis filter bank, and so on so forth. Therefore, the non-integer group delay problem is solved and perfect reconstniction remains unchanged at :the same time. The operation of multiplying Yo by

.A'+? N+2 can be performed by discarding the first -outputs and keeping track of the zT 9

N+2 signal from the (-- + 1) th output, which is not difficult to perform digitally 2

Because of the non-unifarm of the tree-structured transmultiplexer, the signals at different paths suffer different group delays. The signals at upper levels pass through more filter bank blocks than the signal at deeper levels, so the different group delays need to be adjusted by judiciously inserting delays in the deeper paths to insure that ali signals experience the same group delays. The modified system is shown in Figure 3.14. Figut-e 3.14 Rlodified tr;insniurltipIeser \\it11 ui~iforniinteger gl-0111) tlclnys The outputs a,' are the delayed version of the original signals a,', and the delay is

If all filters have odd order, only a slight change needs to be made in Figure 3.14.

Replace N+2 in each delay block by N+l, and the system has no non-integer group

delays and non-uniform group delays as in the even order case. The relation between

output and input is Chapter 4 Simulation System Design

A wavelet modulation simulation system has been developed in this research. This system is intended, first, to simulate the implementation of multicamer modulation system using orthonormal dyadic wavelet basis function and to study practical issues which will aid in system design for application. Second, it will serve as a platform for future study of system characteristics and performance analysis as long as this simulation system is well modeled and easy to use.

In this chapter, the design of the sinrdation system is introduced and interpreted, and one may use it as a manual for the simulation program. The source codes developed for the sin~ulationsystem in this thesis are written in MATLAB language and presented in the Appendix of this thesis.

4.1 Design of Two-Channel Transmultiplexer

4.1.1 Filters

Meyer MRA is used as the wavelet basis. As shown in Chapter 3, Meyer scaling function is indeed the square root raised cosine function when B(x) is set to be equal to

X.

Since c\.cn ordci- filtcr is more dcsirablc in communication applications, onl~,c.\ CJI 6 6

order filters are considered here. The structure of even order two-channel perfect

reconstruction transmultiplexer is shown in Figure 3.12

The reconstruction filter f, is chosen to be the conjugate mirror filter, and its

coefficients can be obtained by sampling and truncating the scaling function (SRRC

function) 4(t) as expressed in (3.1.6)1,

Since the truncation is performed sy~nmetricallyabout zero, particular attention must be paid when sampled at n = 0.Rewrite the equation of square root raised cosine function,

When t is set to zero, the denominator and numerator are all equal to zero, and one must apply L'Hospital law to evaluate $((I).Taking limitation on both sides of (4.1.2) as I goes to zero and applying L'Hospital law, we get

n(1-P)t+4Pt.l - ?r(l-p)+4P lim $(t) = [+o n(1- 0'). t n so $(O) is equal to I(1 - P) + -Jril . The remaining filters were determined from (32.4)-(3.2.6), i.c.,

J[II]= (-l)n ft[~- /I] (3.1 4)

11, [?I]= fC,*[!v - I,] (3 15)

11,[/I] = J;' [A' - 111 (A I 0) 67

The filters studied in this research all have real-valued coefficients

(.f '[n]= f [n]),and the filters resulted from square root raised cosine scaling function are symmetric (f[N-n]=f[n]). Therefore, the equations (3.2.4) - (3.2.6) can be rewritten as

ACnI = (-1)"fobI (4.1.7)

ho [nI = fo [nI (4.1.8)

A, [nI= f, [nl (4.1.9)

The resulting filter f, in equation (4.1.7) is indeed the Quadrature Mirror Filter (QMF) of fo. So, in the case of symmetric, real-valued filter, the Smith-Bamwell filter bank is actually a quadrature mirror filter bank.

The coefficients and the frequency response of the reconstruction and decomposition lowpass, highpass filters resulted from square root raise cosine scaling function with

1 ,9 = - are illustrated in Figure 4.1 and Figure 4.2 respectively. The filter order is chosen 3 to bc 36.

It was sho\vn in previous chapters that a two-channel even order transmultiplexer,

\vlth structure illustrated in Figure 3.12 and filters designed using relations expressed in equations (3.1.6) and (3.2.4) - (3 2.6), have pel-fect rcconstruction. The following example practically proves this characteristic. Figure 4.3 shows two inp~itsignals so and

.v, into a two-channsl trans~~~ultiplexcr~vith 36''' order filtcrs. I;igu~-e4.4 shows the output signal jt0 alid j*,fi-om transmi~ltiple.;ir. Comparing 1:igul-c 4.3 and 4.4, it can bc obsen~edthat for all practical purpose perfect reconstruction has been achieved. No

0 10 20 30 40 n n (a) Coefficients of reconstructio~nlow- (b) Coefficients of reconstruction high- pass filter f, pass filter f,

0 10 20 30 40 n n (a) Coefficients of decon~positionlow- (b) Coefficients of decornj?osilion high- pass filter h, pass filtcr h,

Vignl-e 1.1 Coefficierlts of square root nised cosine ~\;l;~cletfilter h;~nI< -100 l 100 1 0 0.5 1 0 0.5 1 Norm a lized frequency Normalized frequency

(a) Frequency response of reconstruction (b) Frequency response of the reconslruction lowpass filter fo highpass filter J;

-100 l 0 0.5 1 Normalized Ireqiiency

(3) FI-~LISIIC)I-cspo~ise of decomposition (b) Frcq~~c~icyresponse of the dcco~nposition lo\\ pass filter 11, highpass filter h,

Figure 4.2 Frequency response olf scju;ire root I-aised cosine \\:~\.elctfiltcr t);111k aliasing or non-integer delays have occurred due to the proper filter bank design and the structures of transmultiplexer. The delay in the output is a group delay of 19

((36 + 2) I2 = 19) samples as predicted by (3.2.31) and (3.2.32).

The filter bank design is performed in function SRRCWA VE (see Appendix).

15 14

12

10

8

T- X I '6 4

2

0 0 0 5 10 15 20 0 5 'I 0 15 20 n n Figure 4.3 Inputs into even order two-channel transmultiplexer

0 10203040 0 10 20 n n Figure 1.1 Olrtputs of transmu1til)lcser wit11 signal in Figul-e 3.3 its inp111 4.1.2 Decimator and Expander

One can use the MATLAB build-in function DYADDOW and DYADUP to implement the decimator and expander in the dyadic wavelet filter bank.

The dyadic interpolation is accomplished by inserting zeros into a series of samples.

One can choose to insert the zeros as even or odd indexed elements by using different input parameters when calling functiion DYADUP. Similarly, whether the downsampled result of decimation contains the even or odd indexed samples of original signal depends on the value of the input parameters in function DYADDOWN. One can't choose these input parameters casually. For example, if we keep only the even components of the

lowpass filter outputs, the downsamlpled result is x[O], x[2], x[4] -.. . . a. If we delay all components of x by one time unit, the output from downsampling is totally different. The new sanlples x[--11, x[l], x[3]...... are entirely sepzrate and independent from the original samples x[O], x[2], x[4]...... These two subsampled signals are two "phase' of x, not conllected [9]. In a two-channel transmultiplexer, the interpolation is performed first in synthesis filter bank, while decimation is performed afterward in analysis filter bank. It's necessary that the decimation must be consistent with the interpolation for achieving perfect reconstruction. 111 particular, if zeros are inserted as even-indexed elements at expander, the decimator must skip out every even-indexed element; if the expander inserts zeros at odd-indexed positions, the downsampled result must contnin only the even-indexed elements. Otherwise, the perfect reconstruction c:~!l;~othc achieved. Figure 3.4 shows the outp~~tsof a t\\ o-channel transmultiplexer \I hcl-c tl~e decimation and implementation hale this relation usiiig signal in Figur-c 3.7 ,IS il~l)~~t\ 72

Obviously, the perfect reconstruction is achieved. Figure 4.5 shows the outputs of another transmultiplexer, where the decimator is not performed consistently with interpolation, using the same input as in Figure 4.3. The output signals are completely distorted as compared to the input signals.

n n Figure 4.5 Outputs of transmultiplexer with inputs as in Figure 4.3, when

decimation and interpolation are not performed consistently

The two-channel even order synthesis filter bank and analysis filter bank in a transmultiplexer are performed in functions S9FB and il2FB respectively (see Appendix).

4.2 Transmitter

4.2.1 hl-ar), Signaling

The actual wavelet modt~lntionsystem irsccl in oirr cimi~lationis snn~c\vl~;ltdrfft>ii rlt

fioin the psol)ost'd stt-uct~~rcin Figi11-c 2.8. 111 tllc tl,~l~s~~littet-,iliste;~d 01' simply con1 cr tr~rg a serial bit stream to several parallel signal streams before inverse wavelet transform, we use M-ary orthogonal signaling to produce the sequences used in IDWT. The transmitter considers k bits at a time, then produ~cesone of M = 2k digital sequences. This sequence is demultiplexed into (k + 1) subsequences at dyadic sub-rates, and an IDWT is performed on these subsequences to generate a sequence for transmission. Here we use an example to illustrate this process.

Example

1) Assume k = 4.

2) M (A4 = z4 = 16) orthogonal bit sequences are constructed as So - S,j.

3) Collect k bits at a time. Assume the collected four bits are 1, 0 , 0, and 1 at one

time.

4) Use these four bits as a binary number, its corresponding decimal value is 9, and

any four digit binary number has a decimal value between 0 - 15.

5) Choose sequence S, to represent these four bits. Any four digit binary number

can be represented by a sequence S, , i = 0 - 15, and i is equal to the deciximl

value of a four digit binary number composed of these 4 bits.

6) Assuming S, = [s,01' s, s; ...... sdj ] , \\,here s:, ( i = 0 - 15 ) is a binary bit.

Demultiplex S, into 5 subsequences at dyadic sub-rates. Two different types of M-ary signaling are used in this system simulation. One of then] is the simplest orthogonal signaling, every sequence in it has zeros at all positions except one position is set to 1. The position index of this non-zero bit is identical to the index of the sequence it belongs to. T'he following is an example of this M-ary orthogonal signaling with M = 8.

S, =[000~00010] S, =[000(30001]

Another M-ary orthogonal signaling is Walsh-Hadamard sequences. We use MATLAE build-in function HADAA4AHD to produce an M x A4 Hadanlard matrix, where each row

1s orthogonal to others and can be used as an orthogonal sequence.

4.2.2 Pulse Generator

7 I 7 hc seiluence, generated by IDWT, is applied by pulse shape bcfnl-e bcing tr-iunsn~ittedacross the channel. The str-ucture of pi~lscgcncr~ktor- is illustr-atctl in Figrir-e

2.0. First tlic scclucncc passes tliro~~glian imp~ilscgcncratot-, and a continuous si,y;rn;ll is produced based on the discrete sequence. This continuous signal has an inlpulse at every

T unit time, where T = -k Tb (T, is the bit duration of the original bit stream). Then this M continuous signal wavefonn is passed through an analog pulse shaping filter whose

t impulse response is the scaling funct.ion scaled by T, i.e., $(-). The output signal of the T analog filter is the signal that really propagates across the channel.

In the system simulation, the above process has to be implemented digitally. First, the impulse generator is replaced by interpolation by N. This means (N - 1) zeros are inserted into every two adjacent slymbols in the sequence. The resulting sequence approximates the sampled version olf the continuous impulse signal. The analog pulse shaping filter is replaced by digital :filter with coefficients resulting from sampling and

t truncating the dilated scaling function 4(-) T

One nlust pay attention to choose an appropriate sampling rate. Interpolating the sequence by N is indeed equivalent to sampling the continuous signal at rate N. Say, N samples are made within every syrrtbol interval T. Consequently, same sampling rate

t r~lustbe used to sample the dilateld scaling function 4(-) . Otherwise, this digital T implementation cannot approximate the effect of an analog pulse generator correctly

The actual pulse shape itsed in this digital simulatiol~syste~ii is a scaled, saitipled and tn~ncatedversion of the scaling function. The duration of this pulse shape is much longcr than the sj~mbolduration T, and this resulrs in \va\icform overlap bct~vccn ,idl<~ccnt synibols (see Figurc 5.0). l-io~vever,this \\on'[ cause aliasing pro\lidcd thc Icccr\ cr i~scs 7 6 matched filter and samples the siignal at t = kT, because the scaling function is orthogonal to its shifted version if they are at same scale (equation (2.3.12))

The pulse generator is implemented in function WA VEMOD (see Appendix).

4.3 Receiver

At the receiver, the received signal is matched filtered and sampled at I = iT to demodulate the signal from the applied pulse shape. The matched filter is matched to the analog pulse shaping filter. Again, in our simulation, it is replaced by a digital filter with coefficients resulted from sampling aind truncating the flipped scaling function.

In every k bit durations, the matched filtering and sampling yield a received signal vector r,

?'=(r, r2 ...... 5,) 7 where M = 2" Then, dyadic discrete wavelet transform is performed on this symbol sequence and this generates several subsequences at dyadic sub-rates. These subsequences are multiplexed into a single signal sequence, and a decision is made to determine which signal of M = 2k orthogonal sequences is actually sent. The receiver applies a correlation demodulator to make the signal decision. Figure 4.6 illustrates the structure of a co~~elatorreceiver. ilf = 2' orthogonal sequences are used as reference signals. The inner product of I- and each reference sequence Si are calculated. Assume

that reference sequence S, generates the largest inner product value ~vitliI-, then [he dccision can be ~nadethat tlic trallsll~ltted scq~~c~iccis th~ ~th SC~LICIICC. rl~~,lll~. \IC convert the decimal value ofj to the (correspondingk digit binary number, and these k bits are the original signal.

If M x M diagonal identity matrix is used as M orthogonal sequences, the decision could be made even more easily. We just find the largest symbol and its position index in vector r, then convert the value of its position index to the corresponding k digit binary number, and these k bits are the origiiial signal

The functions of receiver are included in WA VEDEMOD. -

b DWT L Matched GI X 7- r-(t> filter -.-*n0 -r--:B -- Z

b -

1. Compute + f, 4-J z0 decision variable iz,i Decision 1 (,4 + 2. Find the largest b value in ) : Z s ,

3. Convert j to k -

, digitnumber binary

S,,, 4.4 Channel

The system performance analysis is made in a simulated Additive White Gaussian

Noise (AWGN) channel. The MATLAB build-in function RANDN is used to generate random noise with normally distributed probability density function. One should be careful when adjusting the level of generated random noise in order to get the correct

E Assume the desired value of >-is . We make use of the property of white NO' Gaussian noise, that is

No where o is the standard deviation, - is the two-side power spectral density of 2

Gaussian noise.

Since we want to have The noise vector generated by RANDN has standard deviation o of 1, so the noise vector

E has to be multiplied by in order to have the effective No be equal to the

desired -.E b No Chapter 5 System Simulation and Performance Analysis

The simulation system design and some practical implementation problems were treated in detail in Chapter 3 and Chapter 4. A simulation program based on previous discussion is written in MATLAB language. In this chapter, the bit error rate (BER) of this simulation system is presented to verify if this system is correctly constructed and works properly. Some characteristics of the transmitted signal are studied, and the anti- tone interference property is discussed.

5.1 BER Performance

We can simplify the structure of a wavelet n~odulationsystem arid represent it as

Figure 5.1.

Orthogonal Modulation Channel Signaling (IDWT-tpulse shaping)

\Vavelet of M-ary Orthogonal Democi~ilatiori signals (matched filtering + sampling + 1D\\-T) Figure 5.1 Simplified block di:tgram of w;~velctmotlul:ltion system 8 1

The source data passes through an M-ary orthogonal signaling block and are mapped

to one of the M orthogonal sequences. Then the inverse wavelet transform of the

sequence is taken the transformed d,ata are transmitted across channel using the chosen pulse shape. At the receiver, the wavelet demodulation including matched filtering,

sampling and wavelet transform is performed on the received signal, then the coherent

detection of orthogonal signals is aplplied to produce the decision variable for making a

decision. If there is no noise in the channel, the recovered signal after demodulation (at

Y) should be the same as that before: modulation (at X) provided perfect reconstruction

filter bank and matched filter are used. Since the well-designed wavelet modulation and

denlodulation operation can reconstruct the signal perfectly, it is reasonable to deduce

that the error performance of this system in the AWGN channel should be the same as a

coherently detected M-ary orthogonal: signaling system.

The bit error rate (BER) is one of the essential judgment criteria for the performance

of a comn~unicationsystem. The probability of bit error for M-ary orthogonal signaling

system in white Gaussian noise [16] is given by

1 '.' P(&) = --Ep(h/ j) = p(&/j), M ,j=1

where

/-2lthough the analytical expression is provided, the integral inside it requires nulnerical

c\*aluationin order to obtain numerical values. Here \jncmake use of the nun~cricalrcsults 82 of expression (5.1.2) presented in [ 1 :7] to plot the curves of probability of bit error versus

---Eb for M-ary orthogonal signaling system in white Gaussian noise channel (shown in No

Figure 5.2).

The error performance of the s;i:mulation system is studied in this research. The transmitted signal of wavelet modultition system passes through an approximate Additive

White Gaussian Noise (AWGN) chanr~el.The method of adjusting noise level to produce

E desired 2 value is discussed in Clt~apter4. The received signal is demodulated and No detected, and the recovered signal is compared with the source signal to count the number of bit errors.

First, the bit error performance of the simulated wavelet modulation system using diagonal identity matrix as orthogonal sequences is studied. Table 5.1 lists the simulation results and Figure 5.3 illustrates the c:orresponding curves of bit error probability versus

Then the IYalsh-Hadanlard matrix is used as the orthogonal sequences. One might reasonably conclude that the system performance would be worse than using d~agonal identity matrix. Since the diagonal .;c.quences llave zeros at all positiorls except O~C

?articular position. the correspondin3 transmitted signals have all the energy concentrated in a sl~orttime duration. and the signal level is nearly zero most of the time; \vliile using

\\']-I ~n;itri\,a11 scclucnccs ha\ c cithcs 1 or 1 at all positions, and aftct- appl1 Ing p~~lsc Figure 5.2 Bit error performance of coherent M-ary orthogonal signaling

system in AWGN channel shape, the transmitted signal overlaps from symbol to syi~~boland the ener2-j is distributed evenly over all time dural.ions. It seems that this could make the delectlon more difficult than using diagonal or-thogonal sequences. Nevcrtheless, tlie sirnul 21 t' on results re~vealthat these two systems ,act~~allyha\.e the same bit error perfomlancc, i.c., the \\a\ clet modulation system llslrig complicated 01-thogonal sequences (\r~c.l~;is M'f I matrix) still can be reduced to a M-aiy orthogonal signaling system because of the perfect reconstruction property of wavelet :modulation and the orthogonality of the scaling function (i.e., pulse shape). Table 5.2 and Figure 5.4 illustrate the bit error performance of the simulation system using WH matrix as orthogonal sequences.

Table 5.1 Bit error probabilliity of simulated wavelet modulation system

using diagonal maikrix as orthogonal sequences lo"

Figure 5.3 Bit error performance of simulated wavelet modulation system

using diagonal matrix as orthogonal sequences

Cornpare Figure 5.3, 5.4 with Figure 5.2, we will find that the bit error performance of the simulated wavelet modulation system in a AWGN channel is very close to that of

M-ary orthogor~alsignaling system. So it is reasonable to say that this simulatiorl system is constructed correctly and works properly. Table 5.2 Bit error performaince of simulated wavelet modulation system using Hadamard orthogonal sequences

- - 2 0.2157 0.1731 0.1724 0.1813 0.1686

-1.5 0.2003 0.1666 - 0.1558 0.1517 0.1540 - 1 0.1870 0.1502 - 0.1361 0.1324 0.1269 -0.5 0.1666 0.1319 - 0.1231 0.1171 0.1039 0 0.1585 0.1216 0.1082 0.0874 0.0785 0.5 0.1411 0.1024~- 0.0918 0.0691 0.0.568 1 0.1318 0.0935 -- 0.0654 0.0495-- ,O.0486- 1.5 0.11.73 0.0727 0.0541 0.0413 0.0307 2 0.1034 0. 063.5 0.0432 0.0272 0.0205 -- 2.5 0.0911 0.0499 0.0328 0.0200 -0.0150 3 0.0814 0.0390 0.0237 0.0131 0.0082 4 3.5 0.0657 0.0306 0.0159 0.0099 0.0052 4 0.0585 0.0209 0.0101 0.0043 0.0027 4.5 0.0502 0.0154 0.00637 0.0025 0.00118 5 0.0409 0.0097 0.00320 0.00086 0.00041 5.5 0.0305 0.0068 0.00185 0.00044 0.00017 G 0.0219 0.0040 0.00105 0.00005 6.5 0.0181 0.0027 - 0.00044 0.00003 7 0.0139 0.0015 0.00018 7.5 0.0105 0.00064 0.00007 0.00607 0.00030 8.5 0.00440a.fi0~-L.J~ 0.000180.00003 p-9.5 0.00128 10 0.00084 -- 10.5 0.00031 -- .-.~ ?! C. 92017 .. .- .- - 11.5 0.00006 .. -. O.O3i'0? ------.-- .. -. Figure 5.4 Bit error performance of' simulated wavelet motiulation system

using Iladamard orthogonal sequcriccs 5.2 Properties of Transmissilon Signal

For an M-ary orthogonal signaling wavelet modulation system, there are only M possible transmitted waveforms. Thus, the transmitted signal has relatively stable properties in time and frequency domain.

As mentioned in the last section, the transmitted signals using diagonal orthogonal sequences have all energy concentrated in a short time duration and the waveform is quite simple. Figure 5.5 illustrates an example of the transmitted signal when using diagonal matrix as orthogonal sequences. If the Walsh-Hadamard matrix is used as the orthogonal sequences, the transmitted signals hiave lower level energy evenly distributed and the waveform is more complicated, because the applied pulse shape on one symbol overlapped with neighborhood syrnbols. Figure 5.6 illustrates an example of the transmitted signal when using WH m,alrix as orthogonal sequences.

It is shown in [4] that the power spectral density for wadelet modulation system is given by

where E is the average symbol energ.y., O(f) is the Fourier transfoni~of scaling filnction

q4 (t), T is the symbol duration. Applying the properties of Fourier transfomi, we car] find

t that O(fr)is indeed the Fourier transjfonn of pulse shape 4 (-) . Equation (5 2. I ) is true T o~llyunder the assumption that the symbols at different scales, i.e., cr; in figlire 2. 8. arc indepcntlent and ~dcntlcallydistr~butcd ~~thzero mean and ~111itvariance. I h~scoi~tl~l~on cannot bc satisfied strictly in our sccn;ll.io, so the pol\ el- spcct~altlclisity of t~.a~~srnit~c.cl Figure 5.5 An example of transmitted signal of wavelet modulation system using diagonal matrix as orthogonal sequences (M=64)

Figure 5.6 An esample of transmitted signal of wavelet niodulation system using \lralsh-Hadarnal-d matrix as orthogonal sequc~lccs(.11=64) signal in this simulation system can only be approxin~atelyequal to the scaled energy t spectrum of pulse shape 4 (-) . Figure 5.7 illustrates the normalized power spectrum of T t pulse shape 4 (-), where the x-axis represents the ratio between the absolute frequency T 1 and the symbol rate - because the absolute frequency is determined by the symbol T duration T.

t Figure 5.7 Normalized power spectrum of pulse shape #(-) T

Figure 5.8 sho~vsan example of the nornlalized power spectral density of the transmitted signal in a 63-ary orthogonal signaling wavelet n~odulatiollsystem using

Walsh-Hadanlard orthogonal sequences. As mentioned before, there are hi' possible transmitted sig~ialsin an .4f-:11-y ortllogonal signaling systcm, so tlic a\ c~,,gedpo\\ cr sl~cctr~~ldensity of tllcsc hl posxible t~,~nsmittccisignals can scpsescnt tlic po\ics sl,ectiiil density of the system. The averaged normalized power spectral density (PSD) of the above 64-ary wavelet modulation systlem is shown in Figure 5.9. Compare Figure 5.9 and

Figure 5.7, it is evident that they have quite similar narrowband characteristics.

Frequency x1lT

Figure 5.8 Normalized PSD of one signal in 64-ary wavelet modulation system

Figure 5.9 Average normalized IJSD of 64-ary wavelet niodulatior~system 5.3 Narrowband Interference

One of the advantages of using a multicarrier communication system is that the tone interference can be recognized during training as long as the sources of these interferences are discrete and their frequencies are stable.

After the dyadic wavelet transform, the frequency band is split in an octave manner.

Figure 5.10 illustrates an example: of the non-uniform filter bank structure and its corresponding frequency decomposition effect with decomposition level of 3. The lower half of the spectrum is split into two equal bands at each level of the tree, while the high half- band component of the signal at any level of the tree is decomposed no further. This results in a non-uniform octave band structure.

In the wavelet modulation systeai, a wavelet transform is performed at the receiver side. Therefore, the received signals; including additive narrowband interferences are decomposed into different frequency bands after being taken DWT, as shown in Figure

5.10. One received signal vector produces several symbol streams at dyadic sub-rates, which belong to different subbands. I[f a narrowband interference with stable frequency exists in the channel, the received signal is a mixture of the transmitted signal and interference. Taking DWT of the received signal extracts the narrowband interference by decomposing the received signal into different subbands and placing narrowband interference in a particular subband based on its frequency range. Therefore, the subband, which is affected by the narrowband interference, can easily be identified, and the interference can be suppressed in the transform domain. Usually, the narrowband interferences in a specified channel have a relatively stable frequency range compared Analysis Filter bank 2:-channel Analysis Filter bank y; 2-channel Analysis Filter bank b Y4 r I

(a) Non-uniform wavellet transform filter bank for depth L = 3

(b) Octave bands resulted from dyadic wavelet transfonl~

Figure 5.10 Dyadic wavelet lvith lion-uniform tree structure with the quite random occurrence of iimpulses in time domain. So the wavelet modulation system can provide an effective means for combating narrowband interference by identifying affected subbands and suppressing interferences in the transfom~domain based on the knowledge about interferences, which can be obtained during some necessary trainings.

The following example shows the basic ideas of anti-tone noise property of the wavelet modulation system. First, we notice that the matched filter at receiver side matches the pulse shape generator, which is indeed a low pass filter. Because of the

t symmetry of the pulse shape 4 (-), the matched filter is a low pass filter also, with the T same frequency response as that of pulse shape. Figure 5.1 1 illustrates the frequency response of the matched filter, where the x-axis represents the normalized frequency, with

1.0 corresponding to half the sample rate F, . It can be observed easily from this graph

that any narrowband interferei~cewith frequency higher than 0.12 x (:) - wii be suppressed first by the matched filter without further processing. The narrowband interferences n.ith lower frequency will be placed at different subbands after the DWT according to t!ieir frequency range. Table 5.3 - 5.5 list the symbol streams in different subbands after DMIT with a depth of 6 for narrowband interfcrenccs with different fscquciicics. l'hc syinbol rate is assumcd to bc 1 1 Iz and tlic sampling rate is I0 I IL. The cutoff fi-cqucncy of the m;itchcci filter is about 0.6, bccaus~ Fbrmalized frequency

Figure 5.11 Frequency response of the matched filter wavelet transform with depth of 6 of a tone noise with the frequency of 4Hz. Since this tone noise is already suppressed by matched filter before being taken DWT, all the subbands are not affected by it, i.e., tlhe signal levels in all the subbands are nearly zero.

Table 5.4 and 5.5 contain the results of DWT of the tone noise with frequencies of 0.4 Hz and 0.04Hz respectively. 'The narrowt)a,nd noise with higher frequency is concentrated in higher subband yj at transform domain, as shown in Table 5.4, while the narrowband noise with lon.er fi-equency is concentrated in lo\$,er subbands y, and I,, at transform domaiil, as shonn in Table 5.5.

This example shows that through training it is easy to find the subbands, c~hlcharc affected by narro\xrband interference ~;ithstable fi-equency range. We either suppress tlic noise Iciel in thc transforni clolnain or inhihit the transmission at thcsc \~il?i~a~ltlsto combat nan-o\$~band~nterlkrence. Table 5.3 The re!sults of DWT with depth of 6 for

a tone noise with frequency of 4 Hz Table 5.4 The results of DWT with depth of 6 for

a tone noise with frequency of 0.4 Hz Table 5.5 The re!snlts of DWT with depth of 6 for

a tone n~oisewith frequency of 0.04 Hz Chapter 6 Conclusions and Future Study

6.1 Summary

This thesis research is intended to study the application of wa\lelet basis flclnctions in orthogonal multiplexed communication system and the implementation of this system using perfect reconstruction digital filter banks. Based on the general structure of ~kavelet modulation system, proposed in [4], detailed system design issues are discusses and presented in a more practical way. A simulation system, written in MATLAB, is developeld for future extensive study. The transmission signal's spectral characteristics and system performance under narrowband interErst7caemeun~&~.fundamentals of orthonormal dyadic wavelets are revien~ed, including the definition of multiresolution analysis (MRA), condition for perfect reconstruction and the relationship between wavelets and digital filter banks, folloned by a short description of multicarrier ~~lodulationand ~ts lrnplementation. Next, the fundamental theory of orthogonally n~ultiplexed cn:nmunication system using oi-thononmal dqad~c\ravelet, developed by Jones [ill IS introduced. Then, the thesis focuses 011 the in~plelnentation of this wavelet rncxlulation sqstcm. Rlcycl- wavclct; arc c!loscii to bc ~~scdin \\a\cfolm ~1i.srgn becat~se o t' 111e1r-suitable tcnlpo~al :Ind sl~ect~-,ilcllaractcristics ;untl thc coil~~cct1011 with familiar communication wavefiorm. Some practical issues regarding filter bank

design which is essential in implementing wavelet modulation system, including

filter order, perfect reconstruction and group delay, are analyzed. The detailed

design procedure is given. Based on this implementation study. a simulated M-ary

orthogonal modulation system using dyadic wavelet basis hnctions is developed in

this research. A detailed description of how the simulation system, including

transmitter, channel and receiver designed and used in MATLAB programming

language, is provided. The BER performance analysis of this simulation system in

AWGN channel is conducted and compared with other coherent M-ary orthogonal

modulation system. It is shown th,ai. this simulation system is constructed correctly

and can be used in the future study. Finally, the spectral characteristics of

transmission signal are studied and show the consistence with the analytic form presented in [4]. The anti-narrowband interference property of the wavelet

modulation system is discussed in a veiy brief way at the end of this thesis.

6.2 Future Study

This thesis focuses on the practical issues of how to implement the MCM system

using dyadic wavelet basis functiol~s.Follo~ving is some recomn~ended potential

research topics that can be taken from this point.

,kf-lxtrid 11 n~lelet fi~nctions: Dyadic wavelet functions achieve a non-unifom~

partition of the data bandwidth. Soine applications requlre a unifonn partition of

thc ba~id\vidtli. nhich can be achic~icd by using M-band wavelet basis fi~nctions

C'onscquently, unifo~n~digital filtes bank should bc implementcd. Bz1sc.d on Ilic work in this research, it is not difficult to switch to the M-band wavelet modulation from the current dyadic wavelet nnodulation system implemented by tree-structured filter banks.

Wavelet packet: Both dyadic wavelets and M-band wavelets can be treated as special cases of wavelet packet, which is generated by generalizing the link between multiresolutio~ approximation and wavelets. One could expect that even better effectiveness against channel impairments could be achieved with this general decomposition of the data bandwidth using wavelet packet.

Detailed study of anti-narrowba'nd interference property: This property is studied in a very brief way in this research by showing the non-uniform frequency decomposition effect after wavelet transform of narrowband interferences. A more detailed and advanced study could be conducted for both dyadic and M-band wavelets by analyzing the BER performance using real signal with intentional narrowband interferences and comparing it with other techniques.

More practical n~odel: Througl~out this research, synchronization has been assumed and only the simplest channel model - AWGN channel is considered. In order to produce a working protoltype, the approach to synchronization must be developed, the system design and ilnplelnentation in fading channel should be studied and the corresponding performance in fading channel could be analyzed. M. L. Doelz, E. T. Heald, and D. L. Martin, "Binary Data Transmission Technoques for Linear Systerns," Pro. /RE,vol. 45, pp. 656-661, May 1957.

J. Bingham, "Multicarrier motlulation for data transmission: an idea whose time has con~e,"IEEE Conznlunication Magazine, pp. 5-14, May 1990.

B. Hiorsal:i, "&I orthogonally multiplexed QAM system using the discrete Fourier transform," /EEE Truwsactions on Communications, vol. COM-29, No.7, Juiy 1981

W. W. Jones, "A hified Approach to Orthogonally Multiplexed Communication Using 'CVavelet Bases and Diirital Filter Banks", P1l.D. Dissertation, School of Electrical Engineering aiid Coimputer Science, Russ College of Engineering and Technology, Ohio University, Athens, OH, August 1994.

S. Kondo, L. Milstein, "On the use of inuiticanier direct sequence spread spectrum systems,.' Puoc. IF93 IEEE MILCOM Conference, October 11-14, pp. 52-56 Boston, MA.

H. L. Resnikoff, R. 0. U7ells,Wavelet Analysis, Springer, 1998.

S. Mallat, "A theory for m~~ltiresolutionsignal decomposition: the wavelet representation," IEEE T?.cznsactions on Pattern Analysis and Machine Intelligence, vol. 11, No. 7, July 1989.

S. Mallat, "Multiresolution approsiniations and wavelet ortho~~ormalbases of L' ," fia~isaction,~on Anzericun Math. Socie!y, June 1989.

G. Strang, T. Nguyen, p-.p-----Wa\-elet and Filter Banks, \Vcllesley-Cambl-idge Press, 1996. M. Smith, T. Bamwell 111, "l?xact reconstruction techniques for tree-structured subband coders," IEEE Transactions on Acoustics, Speech and Signal Processing, pp. 434-441, June 1986.

C. S. Burrus, R. A. Gopinath, H. Guo, Introduction to Wavelets and Wavelet Transforms, Prentice Hall, 1998.

Daubechies, Ten Lectures on VJavelets, Academic Press, 1992.

M. Oerder, H.Meyr, "Digital filter and square timing recovery," IEEE Transactioizs on Communicatic~ns,vol. 36, No. 5, pp. 605-61 2, May 1988.

R. D. Koilpillai, T. Nguyen, P'. P. Vaidyanathan, "Some results in the theory of crosstalk-free transmultiplexers," IEEE Transactions on Signal Processing, vol. 39, No. 10, pp. 2174-21 83, October 1991.

G. R. Cooper, C. D. McGillem, Modem Communications and Spread- Spectrum, McGraw-Hill, 1986.

W. C. Lindsey, M. K. Simon, Telecommunications System Engineerix, Prentice- Hall, 1973.

P. P. Vaidyanathan, Multirate System and Filter Banks, Prentice-Hall, 1993.

B. Sklar, Digital Communications:-- Fundamentals and Applications, Prentice-Hall, 1988.

C. Herley, M. Vetterli, "'Wavelet and recursive filter banks," IEEE fiu~isnctionson Signal Processing, vol. 4 1, No. 8, pp. 2536-2556, August 1 993.

hI. Vetterli, C. Herley, ''V1avt.lets and filter banks: theory and design," IEEE Tt-ansuctto~~sof the Sigtz~111'1-OCCSS~II~, vo1. 40, NO. 9, pp. 2207-2232, Scptc1:lbcr- 1992.

0. Rioul, RI. Vcttcrli, "Wa\.clcts and signal psocessi~~g,"/LEE Sigtli~l PI-occssingL.I/f~~gcrzine, pp. 14-38, October 199 1.

[23j J. Psoakis, Di~italCommnnications, - .------h4cC;raw Hill, 1983 [24] A. Cohen, R. D. Ryan, --Wavelet and Multiscale Signal Processing, Chapman & Hall. 1995.

[25] C. Blatter, Wavelets: A Primer:, A K Peters, 1998. Appendix

function [LoF- D, HiF- D, LoF- R,. HiF- R, scaling] = srrcwave () Bsrrcwave even order filter bank and waveform design using 0 square root raised cosine pulse. ,

0 U [LoF- D, HiF- D, LoF R, HiF- R, scaling] = srrcwave() returns 'i, the filters coefficients by symmetrically truncating the B square root raised cosine scaling function. B o o LoF D is the lowpass decomposition filter. , H~F-Dis the highpass dec:omposition filter. 0 LOF-R is the lowpass rec:onstruction filter. o H~F~Ris the highpass reconstruction filter. O scaling is the pulse shape.

% check arguments. if errargn (mfilename, nargin, [O] , nargout, [O:51 ) ,error ( '+') , end nl= [-18:1: -I] ; n2=[1:1:18]; tl=n1/2; t2-n2/2; beta=1/3; phil=(sin(pi* !l-beta)*ti)+4*beta*tl.*~os(i*(+bt)*t)... . / (pi*(1- (4*beta*tl). A2). *tl) ; phi2= (sin(pin(1-bet-2) +4*bet33t22*cos(pi* beta 2) ) . . . ./(pi*(1- (4*beta*t2) .,'2) .*t2); phiO= (pi*(::beta) +4*beta)/pi; phi= [phil phi3 phi21 ;

6 filter design Lo7 R=pni/sqrt (2);

~li.~-~=~rnf-. (LcF-R) ; HiF D=wrev (SiF R) ; - -. LclF ~-KI-~L- (; :,? F,) ; - .-

B pulse sl~ai;?edesign tl=[-10:o. ?:-?.I]; t2=[0.1:0.i:10]; ph.il= (sin(ri* (1-beta) *t1) t4*beta*tl. *cos (pi*(beta) 1) ) . . . . / (pi*(1- (4*beta*tl.) . "2) . +tl); phi2=(sinipii (1-beta)*t2) +4*bet:ii*t2.*~0~ (pi (ltta) 2) ) . . .

. , ' (pi*(1- (4*bcta7t:L!) . -2).':.-:: ; scsli rig= [phi1 ;:hi O phi 21 ; xral i r!q=-'.r?l i nj/srjrt- (sum (scal iric~.*:cal in!\ ) ; function [s] = s2fb(y0, yl, fO, £1) ?s2fS ti.:--chan~elsynthesis fi:Lter bank.

C) s2fb performs a one-level 1-D wavelet synthesis using C, two-channel filter bank. p 0 8 [s] = sZfb(y0, yl, fO, fl) returns the 1-level wavelet 9 C reconstruction signal s from yo and yl. yo is the % approximation and yl is the detail, yo and yl must have % the same length. The length of s is two times of the length p O of yo and yl. B fO is the lowpass filter, and fl is the highpass filter. , They should have the same length.

8 check arguments. if errargn(mfilename, nargin, [4], nargout, [O:l]),error('*'),end

% yo and yl musr have the same length. if length (yo)-=length (yl), error('Error: the approximation and detail coefficients must have same length! I),, end

8 fO and £1 must have the same length. if length(f0)-=length(fl), errori'Error: the lowpass filter and highpass filter must have same length! '), end v0= [dyadup(yo, 0) 01 ; vO= [0 -JO (I:(length (v0) -1) ) ] ; vl= [dyadup(yl, 0) 01 ;

1-length (v0); s=[O, conv(v0, iO) 1 +[Of conv(v1, fl)1 ; function [yo, yl] = a2fb(x, hO, hl) 9oa~fb - - two-channel analysis filter bank. 6 a2fb performs a one-level 1-D wavelet analysis using % two-channel filter bank. 3 0 3 o [yo, yl] = a2fb(x, hO, hl) returns the 1-level wavelet 3 a decomposition of signal x. yo is the output 8 of lowpass filter, yl is the output of highpass filter. 3 o Filter bank is defined kmy hO, the lowpass filter, and 3 o hl, the highpass filter. 3 % The length of yo and yl are same and should be the half 3 of the length of x. 3 0 3 o hO and hl should have the same length.

% check arguments. if errargn(mfilename, nargin, [3], nargout, [0:2]),error('*'),end

% hO and hl must have the same length. if length (h0)-=length (hl), error('Error: che lowpass filter and highpass filter must have same length! ' ) , end

xO=conv (x, hC) ; xl=conv (x, hl); function [c,l] = wanaly (x, J, 120, hl) Ewanaly Multi-level 1-D wavel!?-: decomposition. 3 C. wanaly performs a multi-level 1-D wavelet analysis 3 C using specific wavelez decomposition filters. f ? [c,l] = wanaly(x, J, hO, hl) returns the wavelet 5 decomposition of the s:~gnal x at level J. "i % J must be a strictly positive integer. The output decomposition structure contaFns the wavelet % decomposition vector c and the bookkeeping vector 1. e 6 hO is the decomposition low-pass filter and % hl is the decomposition high-pass filter.

00

0 ,2 The structure is organized as: % c = [app. coef. (J)l det. coef. (J)I.. . idet. coef. (1)] 9. O l(1) = length of app. coef. (J) 0 O 1 (i) = length of det . coef. (J-i+2) for i = 2, . . . , J+1 0 O 1 (J+2) = length (x).

% check arguments. if errargn (mfilename, nargin, [4] , nargout, [O: 21 ) , error ( ' " ' ) , end if errargt (mfilename, J, ' int ' ) , error ( ' * ' ) , end

% check if the decomposition level is too large. if J>log2 (length(x) ) error('Error: too many levels! ') end

6 Inirialization c=[l; l= [length (x)] ; N=length (hO); order=N-I;

for i=J:-l:l [x,dl =a2fb (x, hO, hl) ; a=2" (i-1)* (order+2)/2; b=length (d)- (2"(i-1) *order/2) ; d=d(a+I:b) ; a= (order+2)/2; b=lenqth (x)--order/2; x-x (at1:b); 17- [d, C] ; l= [length(d) I] ; 2-..d r,

" I ~ctz~rr?sin?tl?? <= [x C] ; l= [length (x) 11 ; function [x] = wsynth (c,1, f 0,f 1) %wsynth Nul~i-level 1-D wavelec reconstruction. % wsynth performs a multi-level 1-D wavelet reconstruction p o using specific recons.truction filters (Lo-R and Ei- R). E: x = wsynth(c,l,fO,fl) reconstructs the signal X 0 O based on the multi-level wavelet decomposition structure 3 c' [c,l] (see wanaly) . 3 % fO is the reconstruction low-pass filter and 8 £1 is the reconstruction high-pass filter.

8 check arguments. if errargn (mfilename, nargin, [4], nargout, [0:ij ) ,error ( '* ') ,end

% check the conresponding relation of decomposition vector and S bookkeeping vector. if length (c)-=1 (length(1) ) errorilError: bookkeeping vector does not conrespond to decomposition vector! I); end

% check the length of c. if iog2 (length(c) ) -=round(log2 (length(c) ) ) error('Error: the length of c is not the power of 2! ') end

J=length(l)-2; % reconstruction level order=length ( f 3)-1; yO=c (1:l (1)) ; %initialize for i=J:-l:l yl=detcoef (c,1, i) ; if i==j N-0; else N=2^ (J-i)-1; end; yi- [zeros(1, N* (order+2)/2) yl zeros (1, N*order/2) 1 ; y0=s2fb(yOf yl, fO, £1); end x-yo; function [y, N] = wavemod(x, I;,, size-block) Ewavenod Kavelet modulation. , wavemod performs wavelet modulation using :6 two-channel filter bank. Take a block of signal

0 o and apply inverse wavelet: transform, then take C O the next block, and so on. The resulting sequence C O is transmitted across the channel using a pulse C C shape defined by square root raised cosine function

>

* [y, N] = wavemod(x, L, size block) returns the ?< modulated signal y and the block size after

C nodulation. x is the original signal, L is the , decompssition and reconstruction level and must ' , be an integer. size block is the size of a block, O it must be the power of two.

% check argu~.ents. if errargn(mfilename, nargin, [3J, nargout, [O:Z]),error('*'),end if errargt (mfilename, L, ' int ' ) , error ( ' * ' ) , end

?the block slze must be the power of two. sb=slze block; 15 loq2isb)-=round(log2(sb)) errsr('Errcr:the block size should be =he power of two1'! end

6 check if the reconstruction level is too large. if L>log2 jsb) error('Error: too many level.^!') end

% get filter Sank and pulse shape. ;Lor-D, HiF-D, LoF- R, HiF- R, ph.i]=srrcwave;

% wevelets reconstruction l=:[sb]; t emp=sb; for i=1:1:5 t emp=t err.^ / 2 ; l=[renp 11; er:d; 1-[terrLp 11;

', i:nr;ulse 35: 2rzter st=zeros (l?,, lenjth (xk)) ; for -1 : 1 eno:.k. >:k)

:;;: '\ \ ; end; .vl:=-:":r. , : 1 ' ; 2 apply pulse shape 7=20nL7(xt, phi) ; function [y] = wavedemod (mx, size-block) gwavedemod Wavelet demodulation.

o wavedemod performs wavelet demodulation using two-channel filter bank. The received signal is 9. c, match filtered and sampled. Then take the wavelet 0 transform, and produce the soft decision of the 3 transmitted symbols. ,

[y] = wavedemod(mx, L, si.ze block) returns the , demodulated signal y , I is-the decomposition and B reconstruction level and must be an integer. size- block is the size of a block after modulation.

% check arguments. if errargn(mfilename, nargin, [3], nargout, [O:l]),error('*'),end if errargt (mfilename, L, lint' ) , error ( '*' ) , end sb=size block; % check-if the decomposition level is too large if L>logZ(sb) error('Error: too many levels!') end

% get filter bank and pulse shape. [LoF- D, HiF- D, LoF- R, HiF- R, phi]=srrcwave;

R match filter phi-wrev (phi); xtl-conv (mx, phi) ; len=lenyth (xtl)-2' (length(phi) -1) ; xtl=wksep (xtl, len) ;

S sample xkl=[xtl(l) ] ; for j=l:length (xtl)/lo-l xkl=[xkl xtl(jxlO+l)]; end;

% wave1et.s demodulate

[c, ll=wanaly(xkl, L, LoF- D, HiF D); y=c ' . function signal = signaling (L) %signaling create M-ary ortnogonal slgnallng, dhere L=logZ(M).

% check arguments. if errargn (mfilename, nargin, [ij , nargout, [0:l] ) ,error [ ' * ' ) , end

S check parameter if errargt (mfilename, nargin, ' int ' ) , error ( ' * ' ) , end

for i=l:L-1 A=[A A A -A] ; end; % Main function to test the M-ary orthogonal modulation using % dyadic wavelet basis functions. tic; 8 Initialization 6 signal length N=input('Please input the length of the signal: '); 'r M-ary

M=input('Please input M-ary: I);

E the M-ary must be the power of two. if log2 (M)-=round (log2(M) ) error('Err0r:the M-ary shou:Ld be the power of two! ') end k=log2 (M); % k bits are one sjrmbo?

% reconstruction and decomposition level L=k;

% M-ary symbol (k bits are one symbol, one skmbol are M bits) msg= [ 3 ; if k==l rns g=bj.t s ; else for i=l:k:N if (i+k-1.)>N spboi= [bits (i:N) zeros (1, i-ik-1-N)J ; else sy-x.bol=bits(i:itk-1) ; end nsg= [insg bi2de (symbol)] ; end erid drnsg=zeros (M, length (msg)) ; [row, col]=size(dmsg); for i-i : col ck-.sq(:, i)=islynal(msg(i)+?,:)! '; 2nd

per fzrm= [ ] ; I wavelet modulation jmx, sb]=wavemod (drnsg ( : , j ) ' , L, M) ; Kx=mx. /sqrt (sum(mx.*mx) ) ; C=l . U L, Eb=E/log2 (M); 5 AWGN channel n=randn(l, length(mx))*sqrt(~b/(lO~(DB(index)/lO rx=mx+n; F wavelet demodulation rds=wavedemod (rx, L, sb); rs= [rs ras] ; end;

[rowl, coll]=size (rs); rmsgZ [I ; rbits= [I ; for i=l:coll correl=signal*rs ( : , i) ; [maximun,j ] =max (correl); rmsg= [rmsg, j-11 ; rbits-irbits, de2bi(j-1, ]

% performance analysis [sys- error, sys-ratio] =syrr,err (r~sg', rmsg (1:length (rnsg)) ' ) [bit- error, bit -ratio] -symerr (bits', rblts (1:ler,c;th (bits)) ' i perform= [perfcrm bit.-ration] ; end tcc %load chirp; Isound(y) ; % Function to test the perfect reconstruction property of the two- % channel &gital filter bank using Meyer wavelets.

x=randint(l, 10, 16); [a,5, c, dl = srrcwave; crder=length(c); xO=conv (x,a) ; xl=conv (x,b) ; vO=dyaddown (x0, 1 ) ; 5 insert proper delays vl=dyaddown( [O xl(1: (length(x1)-1))I, 1);

yO=dyadup (v0, 0) ; % insert proper delays yO=[O yO(1: (length(y0)-I))]; yl-dyadup (vl, 0) ;

s=conv (yo, c)+coriv (yl, d) ; plot (x); hold on plot (s(order+l: (length(x) +order) ) , 'r') grid on zoom on % Function to plot the curve of bit error rate versus E~/NOusing % curve fit technique.

XI=-13:1:12; y1=[33.05 35.51 34.22 32.27 30.42 29.33 25.84 23.89 ... 21.64 18.36 16.21 13.23 10.38 7.76 5.76 3.74 2.33 . 1.29 0.55 0.29 0.079 0.027 0.0071; yl=y1/100; -. yl=lSqlo (yl); pl=polyfit (xl,yl, 5) ; pic~r-~~e=polyval(pl, xl) ; p?curve=lO.A(plcurve); ~emllogy :XI, piczrve, 'x') ; hold on semiioqyix2, p2curver '+I); holci on semilogy(x3, p3curve, '0'); hold on . . se~~~~ogy(x4, p4curve, ' --' ) ; kold 3n semi logy (x5, p5curver ' -. ' ) ; k.old on semilogy (x6, p6curve, I-') ;

axis ! [-7.5 20 0.00001 11 ) xlabel ( ' EbiNO (dB)' ) ylabel('Bit error probability") grid ZOOIT function p = psa (h, kk) % calculate samples of' the scaling function phi(t) = p % by kk successive ap~~roximationsfrom the scaling coefficients % h. Initial iteration is a constant. if nargin==l kk-11; end; h2=h*2/sum (h); K=length ih2)-1; S=128; p- [ones( 1, S*K),0] /K; N-length (h2) ; hu-zeros (S, N) ; for i=l:N hu(1, ij=h2 (i); eild; hu=l:u ( : ) ; for i=l: kk p-conv ( hu, p) ; p=dyaddown (p, 1); end; [value, index] =max (p); p-pivalne*1.09; t=1: 1~1igt1-I(p) ; t= (t-izdex)*1/128; save t; save p;

7, ~LP!:(~,p; grid on zoom on axis([-10 10 -0.2 1.21)