Modulation and Channel Effects in Digital Communication

Home , Communication channel

: LICENTIATE T H E SI S

Sara Sandberg

Luleå University of Technology Department of Computer Science and Electrical Engineering, Division of Signal Processing

:|: -|: - --  ⁄ --  Modulation and Channel Effects in Digital Communication

Sara Sandberg

Dept. of Computer Science and Electrical Engineering Lulea˚ University of Technology Lulea,˚ Sweden

Supervisor: James P. LeBlanc ii To Marcus and Miriam iv ABSTRACT

This thesis investigates three possible methods to increase the performance of digital communication systems, with focus on wireless systems, by accounting for some of the channel effects that may occur. The modulation scheme plays an important role in the impact of different channel effects on system performance and this work considers both a single-carrier system and orthogonal frequency-division multiplexing (OFDM). The first work investigates effects of the channel estimation errors resulting from blind channel estimation. The performance of a communication system in terms of throughput may be increased by using blind channel estimation instead of non-blind. This will allow more useful information to be sent through the system, but the channel estimation will be less reliable. The effects of the channel estimation errors on the performance of separation in a multiple- input multiple-output (MIMO) system are investigated for a specific blind channel estimation method. This work quantifies the expected performance reduction, in terms of cross-channel power, due to channel estimation errors. The second and third work consider the OFDM framework, which enables simple equalization and has been adopted in several standards. However, OFDM is sensitive to frequency- selective fading and introduces a large peak-to-average power ratio (PAPR) of the transmitted signal. These problems can be alleviated by pre-multiplying the OFDM block by a spreading matrix, e.g. the Walsh-Hadamard matrix. It is shown that spreading by the Walsh-Hadamard matrix reduces the PAPR of the transmitted signal and increases the frequency diversity. Sur- prisingly, with a joint implementation of the spreading and the OFDM modulation, the spread OFDM system requires less computations than the conventional OFDM system. An alternative to PAPR reduction is to allow clipping of the signal in the transmitter. Clip- ping will however introduce losses due to the clipping distortion of the signal. In the thesis, receiver methods to mitigate such clipping losses are investigated. It is shown that for an OFDM system with low-density parity-check (LDPC) coding, the cost of completely ignoring the clipping effects in the receiver is minimal.

v vi CONTENTS

INTRODUCTION 1 1 Introduction ...... 1 2 Low-DensityParity-CheckCodes...... 3 3 Modulation ...... 6 4 ChannelEstimation...... 7 5 SummaryofContributions...... 10 6 Conclusions...... 11 7 FutureWork...... 12

PAPER A 19 1 Introduction ...... 21 2 SystemModel...... 22 3 SeparationwithKnownMixingMatrix...... 23 4 BlindIdentiﬁcationBasedonCumulantSubspaceDecomposition...... 24 5 Cost of Blindness ...... 25 6 SimulationResults...... 25 7 Conclusions...... 28

PAPER B 31 1 Introduction ...... 33 2 TheOFDMSystem...... 34 3 LDPCcodesforOFDMandSOFDM...... 36 4 Results...... 39 5 Conclusions...... 41

PAPER C 45 1 Introduction ...... 47 2 SystemDescriptionandChannelModel...... 48 3 CharacterizationofClippingNoise...... 50 4 BayesianEstimation...... 52 5 ResultsandDiscussion...... 53 6 Conclusions...... 54 viii ACKNOWLEDGMENTS

TheﬁrstpersonIwouldliketoexpressmygratitude to is my supervisor Professor James LeBlanc. Thank you for convincing me that I would ﬁnd the Ph.D. studies fun and for the guidance and support that you have given me. Because of your enthusiasm and expertise, becoming a Ph.D. student is a choice I have never regretted. Thanks also for always thinking of what is the best for me and my future and giving that the highest priority. Also, many thanks go to my assistant supervisor Professor Bane Vasic from University of Arizona, that has supported me with his coding expertise and interesting ideas. I really look forward to visit you and your group this autumn. I would also like to thank all my colleagues in the signal processing group. All together you make up a friendly and inspiring atmosphere that makes it enjoyable to go to work. Es- pecially, Martin Sehlstedt, that has always taken the time to answer my questions about the life as a Ph.D. student in general and computer problems in particular, deserves extra thanks. Acknowledgments also to the European Commission for co-funding this work, that is part of the FP6/IST project M-Pipe, and to the PCC graduate school. Finally, I would like to express my sincere gratitude to my husband Marcus. Without your encouragement and never ending support at home this thesis would not have been written. Thanks also to my parents for your support and belief in me.

Sara Sandberg Lule˚a, August 2005

ix x Part I xii INTRODUCTION

1 Introduction

This thesis addresses methods to reduce the impact on system performance of some channel effects that may occur in a digital communication system. To set the stage for the research presentation, this section gives a short introduction to the principles of a digital communication system. This will serve as the overall picture when more details of the communication system and the research are presented in the following sections. There are many textbooks on the subject of digital communication that the reader can refer to for a thorough introduction, see for example [1][2][3]. Figure 1 shows the main elements of a digital communication system. The information source is assumed to be in digital form and possibly encoded by a source encoder which is omitted in this presentation. The information bits (or digits) are fed to the channel encoder, which introduces redundancy in the information sequence. This redundancy can be used by the channel decoder to reduce the impact of channel effects as noise and interference and the result is increased reliability of the received data. One common way of adding redundancy is block coding where information bits at a time are mapped to a unique sequence of bits, called a codeword, with . The amount of redundancy added is found from the relation between the number of information bits and the number of codeword bits and the code rate is deﬁned as the ratio . In this thesis the focus is on one type of block codes called low- density parity-check (LDPC) codes, [4], that have gained much interest in the latest decade. These codes are discussed in more detail in Section 2. The codeword bits from the channel encoder are passed to the digital modulator, which maps the bits to appropriate signal waveforms. The simplest form of modulation is to map each binary zero to one waveform and each binary one to some other waveform that is easy to

1 2INTRODUCTION

Information Channel Digital source encoder modulator

SISO/MIMO channel

Demodulator, Output Channel separator and decoder signal equalizer

Channel estimator

Figure 1: A basic digital communication system. distinguish from the waveform representing the zero. This is called binary modulation and is one form of single-carrier modulation. In the last decade, multi-carrier systems have become more popular and especially orthogonal frequency-division multiplexing (OFDM) has received much attention. With multiple carriers the superposition of several waveforms representing several bits are transmitted at the same time. Section 3 describes modulation and especially OFDM more. In wireless communication, the modulated waveforms can be transmitted into the communication channel by one or several transmitting antennas. A system with one transmitting antenna and one receiving antenna is called a single-input single-output (SISO) system, while systems with several transmitting and receiving antennas are called multiple-input multiple- output (MIMO) systems. Multiple transmitting and/or receiving antennas will increase the spatial diversity and can be used to combat channel effects without increasing the bandwidth of the transmitted signal. The communication channel model represents the physical medium that connects the transmitter with the receiver. This medium can be the atmosphere as well as wire lines, optical fibers, etc. All received waveforms will be more or less corrupted due to thermal noise from electronic devices, non-linear distortion in the high power amplifier, interference from other transmissions, atmospheric noise, fading, etc. At the receiver side of the digital communication system, there are one or more receiving antennas. Each antenna receives a weighted and possibly filtered sum of the different transmitted waveforms. The digital demodulator processes these signals and produces a binary stream again. In MIMO systems, a separator reconstructs the transmitted signals from the weighted 2. LOW-DENSITY PARITY-CHECK CODES 3 sums of signals, transmitted by different antennas, that is received. Remaining filtering may be reduced by an equalizer. The demodulated signal is passed to the channel decoder that uses the redundancy added by the channel encoder to reconstruct the information sequence. With more redundancy added, the decoded output signal is more likely to equal the transmitted information sequence. The overall aim of the digital communication system is to transmit as much information as possible from the transmitter side to the receiver side with as few errors as possible. The performance measure of the digital communication system is the average frequency or rate with which errors occur (the bit-error-rate) and the rate with which information can be transmitted. Channel estimation is not required by the digital communication system as an output, but a channel estimate is often of use internally in the system, for example in the separator and the channel decoder. With a good channel estimate, both separation and decoding can perform better. This thesis investigates three possible methods to increase the performance of digital communication systems, with focus on wireless systems, by accounting for some of the channel effects that may occur. The modulation scheme plays an important role in the impact of different channel effects on system performance and this work considers both a single-carrier system and orthogonal frequency-division multiplexing (OFDM). The research presented in the thesis is separated into three parts, each represented by one research paper. One paper relates to the modulation while the other two investigate how channel effects (clipping and channel estimation errors) affects the receiver (channel decoder and separator). The thesis introduction is organized as follows. Section 2 describes the LDPC codes. Mod- ulation and OFDM is discussed in Section 3, where especially the drawbacks that make the OFDM system sensitive to some of the channel effects are considered. In Section 4, the importance of channel estimation and some channel estimation methods are discussed. The three papers that are included in the thesis are summarized in Section 5. Finally, conclusions and ideas for future work are given in Section 6 and Section 7 respectively.

2 Low-Density Parity-Check Codes

In this section the channel coding is discussed in more detail. The thesis focuses on LDPC codes, which can be decoded by a simple iterative algorithm and have been shown to achieve information rates up to the Shannon limit, [5]. LDPC codes were originally discovered by Gallager [4], in 1962. He introduced block codes specified by a very sparse parity-check matrix created in a random manner. The main contribution of Gallager was however a non-optimum but simple iterative decoding scheme for the LDPC codes, with complexity increasing only linearly with block length, that showed promising results. In the following years there were very few papers in this field, but Tanner introduced the bipartite graph to this problem in 1981 and used it both for code construction and for decoding (with a generalization of Gallager’s iterative algorithm), [6]. Around 1996, 4INTRODUCTION the LDPC codes were independently rediscovered by MacKay and Neal, [7], and Wiberg, [8], and in the latest decade much research has been devoted to this area. However, the practical performance of Gallager’s original work from 1962 would have broken practical coding records up to 1993, [5]. All LDPC codes are specified by a very sparse parity-check matrix. As suggested by Tan- ner, the LDPC codes can be represented by bipartite graphs. One class of nodes that corresponds to the elements of the codeword are called variable nodes, while the other class that corresponds to the parity-check constraints are called the check nodes. Figure 2 shows an example of the bipartite graph for a regular LDPC code of length 10, with the parity-check matrix given by

(1)

Every edge in the graph connecting a variable node with a check node corresponds to a one in the parity-check matrix. Both the bipartite graph and the parity-check matrix give a full speciﬁcation of the code on its own. The parity-check matrix given by (1) for the example code of length 10 is not really sparse and in practice the codeword length ranges from a few hundred to several thousand bits. The construction of the code can be random (like Gallager’s) or structured. Irregular LDPC codes constructed randomly under certain conditions (e.g. with a given degree distribution of the nodes) have been shown to perform at rates very close to the Shannon capacity, [9]. Several families of structured codes have been suggested recently, see for example [10]. In both random and structured constructions of the code, effort is spent to avoid short cycles in the bipartite graph. The length of the shortest cycle in the graph, the girth, affects the performance under iterative decoding.

The main advantage of the LDPC codes is the simple iterative decoding algorithm. In fact, turbo codes, that can also be decoded iteratively, have been shown to be low-density parity- check codes, [5]. One iterative algorithm that may be used to decode the LDPC codes is a message-passing algorithm known as the sum-product algorithm. A good tutorial paper on the sum-product algorithm is written by Kschischang et al., [11]. In short, the sum-product algorithm updates the a posteriori probabilities that a given bit in the codeword equals 0 or 1, given the received word. In the initialization each variable node is assigned the conditional probability of the codeword bit being a 0 or 1, given only the received value corresponding to that node. In order to calculate the conditional probabilities exactly some statistics of the channel must be known, for example the distribution of the noise and clipping distortion. The initial probabilities are then updated according to the sum-product rule to incorporate the structure of the code. In graphs with no cycles, the sum-product algorithm will yield the correct probabilities, that a given bit in the codeword equals 0 or 1 given the whole received word, when the 2. LOW-DENSITY PARITY-CHECK CODES 5

Variable Check nodes nodes

Figure 2: Bipartite graph describing a regular LDPC code of length 10.

algorithm terminates. However, if there are cycles in the graph, the algorithm will not come to a natural termination and the updated probabilities will not be the exact probabilities. Still, the sum-product decoding of LDPC codes, in which the underlying graph will have cycles, has been shown to perform very well for long codes, see for example [7].

One drawback with LDPC codes has been the relatively high encoding complexity. While turbo codes can be encoded in linear time, LDPC encoding in general has quadratic complexity in the block length. However, Richardson and Urbanke showed that the constant factor in the complexity expression is very low, and therefore practically feasible encoders exist even for large block lengths, [12]. They also give examples of optimized codes that can actually be encoded in linear time. Another example of a code that lends itself to linear time encoding is the irregular quasi-cyclic code suggested in [13].

LDPC codes have been used in two of the three papers included in this thesis. The main reason is that any real system today includes channel coding and the LDPC codes are known as good practical coding methods. Channel coding is one way of reducing the impact of channel effects, but the performance of the coding can also be affected by other methods for reducing the impact of channel effects. By investigating the performance of systems with LDPC coding, these effects are made visible. In this work, LDPC codes have been considered in the framework of OFDM, which will be described in the following section. 6INTRODUCTION

AWGN

IFFT P/S channel S/P FFT equalizer N N N N N

Figure 3: The OFDM system.

3 Modulation

The modulator is the element of the communication system that maps the digital information into analog waveforms, matching the characteristics of the channel. The waveforms may differ in amplitude, phase and/or frequency. Signal waveforms corresponding to multidimen- sional signal vectors can be transmitted by increasing the number of dimensions in either the time-domain or the frequency-domain. An N-dimensional signal vector may be transmitted in a time-interval T by dividing the time-interval in N shorter intervals. Alternatively, the frequency band may be divided into N frequency slots and the amplitudes of the N corresponding subchannel carriers can be modulated. One popular modulation method that uses multiple carriers in this way is orthogonal frequency-division multiplexing (OFDM). The remaining part of this section focuses on OFDM and the reader is referred to digital communication textbooks, for example [1] or [3], for more details on single-carrier modulation.

3.1 OFDM

OFDM has already been included in several standards, e.g. digital audio/video broadcasting (DAB/DVB) and the local area mobile wireless networks (802.11a). The main idea with OFDM is to convert an intersymbol interference (ISI) channel into parallel ISI-free subchannels. Each subchannel will have gain equal to the channel’s frequency response at the corresponding subchannel frequency. OFDM is implemented by an inverse fast Fourier transform (IFFT) at the transmitter side and an FFT at the receiver side, see Figure 3. A cyclic prefix of length no less than the order of the channel is inserted between successive blocks to prevent interblock interference. OFDM enables simple equalization since each subchannel can be assumed to have a frequency- flat impulse response. However, there are several drawbacks with the OFDM system, such as large peak-to-average power ratio (PAPR) of the transmitted signal and sensitivity to channel fades due to reduced diversity. The large PAPR makes power backoff necessary unless techniques to control the non-linear distortion in the power-amplifier are incorporated. These techniques can be either PAPR-reduction algorithms, [14][15], or time-domain amplitude clipping, [16][17]. PAPR-reduction algorithms often require side information to be transmitted and the resulting PAPRs are still higher than those of serial single-carrier transmission. Time-domain clipping on the other hand introduces clipping noise to the system. In [18], Wang et. al. compare OFDM and single-carrier block transmission over frequency- 4. CHANNEL ESTIMATION 7

AWGN

spreadingIFFT channel FFT Wiener N N N filter N

Figure 4: The spread OFDM system. selective fading channels. They conclude that single-carrier block transmission is superior to OFDM in the uncoded case, at the cost of slightly increased complexity. In the coded case, OFDM is to be preferred for low code rates. The sensitivity to channel fades can be alleviated by channel coding, which may correct errors on fading subchannels. Extra diversity can also be added by introducing dependence among symbols on different subcarriers. This is performed by linear precoding, also called spreading, which was initially introduced by Wornell, [19].

3.2 Spread OFDM

In spread OFDM (SOFDM), the vector to be modulated by the IFFT is first multiplied by a spreading matrix, Figure 4. Wiener filtering may be implemented at the receiver side to reduce the intercarrier interference introduced by the precoder, resulting in a minimum mean square error (MMSE) receiver. For a unitary spreading matrix, the Wiener filter is shown to be simply scalar channel equalization followed by the inverse of the spreading matrix, [20]. In [21], channel independent precoders that minimize the uncoded bit error rate (BER) in an OFDM system are derived. It is shown, for QPSK signaling and an MMSE receiver, that the class of optimal precoders is the unitary matrices with all elements having the same magnitude. Examples of this class are the discrete Fourier transform (DFT) matrix and the Walsh-Hadamard matrix. Us- ing the DFT matrix as a precoder gives back the single-carrier system, but with a cyclic prefix. This system still enables simple equalization, but to the cost of implementing both the IFFT and FFT in the receiver. The Walsh-Hadamard matrix has been shown to reduce the PAPR of the signal, [22], as well as increasing the frequency diversity. Debbah et. al. proposed a new method to reduce the intercarrier interference in the receiver for WH spreading, [23]. They achieve near maximum-likelihood performance, still with a reasonably low complexity.

4 Channel Estimation

Channel estimation is of importance for many of the elements in a digital communication system. If the channel is known to the transmitter, the water-pouring strategy can be applied to concentrate the transmit power to the frequencies where the channel has relatively little attenuation, [1]. This is necessary to approach capacity. At the receiver side, the channel information is utilized for channel equalization, in iterative decoding with soft information (for 8INTRODUCTION example in the sum-product algorithm for LDPC decoding), for separation of MIMO systems, etc. The channel estimation methods can be divided into two groups: Non-blind methods where a training sequence is inserted into the transmission and blind methods that exploit various statistical properties of the transmitted signals to carry out channel estimation in the receiver without access to the symbols being transmitted. Different non-blind and blind channel estimation methods are discussed in the following subsections. In the last subsection, channel estimation of MIMO systems is discussed separately. Estimating the channels of a MIMO system is usually a much more difﬁcult problem than estimation of a single channel, since the number of parameters to estimate is increasing linearly both with the number of transmit and receive antennas.

4.1 Non-blind Methods

In digital communication systems operating on time-varying frequency-selective fading channels, the data to be transmitted is often organized in blocks preceded by a known training sequence. The training sequence at the beginning of the block is used to estimate the channel, train an adaptive equalizer, etc. There exist several methods that can be used to retrieve a channel estimate from the received version of the training sequence. For example, least sum of squared errors (LSSE) channel estimation was suggested by Crozier et. al.,[24], where the solution for the channel estimate that minimizes the sum of squared errors between the received signal and its approximation using the channel estimate is given. The length of the required training sequence is in the order of two to three times the order of the channel, [24]. Another way to make parts of the transmitted signal known to the receiver is to use pilot symbols. Pilot-symbol-aided channel estimation for systems with fading channels has been analyzed in [25]. The transmitter periodically inserts known symbols, from which the receiver derives its amplitude and phase reference. By introducing known pilot symbols in the transmit stream, there is no need to change the transmitted pulse shape and the peak-to-average power ratio stays the same. However, there are two main drawbacks with the pilot-symbol-aided channel estimation. One is the delay in the receiver that is required to obtain enough pilot samples for a good channel estimate. The other is that the receiver interpolates the channel measurements provided by the pilot symbols and the interpolation coefﬁcients depend on the position within the frame of the sample whose amplitude and phase is to be determined. Pilot- symbol-aided channel estimation has also been investigated for OFDM systems, [26]. Another method suggested for channel estimation of OFDM systems is based on the singular value decomposition, [27]. This estimator can be used either with a training sequence or with known symbols (pilots) inserted into the transmitted data stream and is a low-complexity approximation to a linear minimum mean-squared error (LMMSE) estimator that exploits the frequency correlation of the channel. The number of multiplications required is reduced by applying optimal rank reduction, achieved by the singular value decomposition. 4. CHANNEL ESTIMATION 9

4.2 Blind Methods Blind channel estimation (and/or channel equalization) do not require a training sequence or pilot symbols. Instead, the statistical properties of the transmitted signals are exploited to carry out the estimation or equalization at the receiver without access to the symbols being transmitted. The equalization problem is equivalent to identifying the inverse of the channel. The idea of blind (self-recovering) adaptive equalization was first proposed by Sato, [28], then further developed by Godard, [29], and Treichler and Agee, [30]. Shalvi and Weinstein, [31], explored blind equalization further and derived necessary and sufficient conditions for blind equalization and proposed several optimization criteria with the respective stochastic gra- dient algorithms. Tong et. al., [32], proposed a channel identification and equalization method that exploits the cyclostationarity of oversampled communication signals. In this case, second- order statistics of the channel output contain enough information to identify the channel. Apart from non-blind and blind channel estimation methods, there are also methods in between, called semi-blind methods, [33]. The semi-blind methods exploit information from both training and statistical properties of the transmitted signal, which makes them more robust than the blind methods while they still require less training overhead than the non-blind methods.

4.3 Channel Estimation in MIMO Systems In a wireless multiuser communication system, the receiver needs to recover the desired user signal by simultaneously suppressing the intersymbol interference (ISI) introduced from tem- poral distortion and the multiuser interference (MUI) resulting from spatial mixture. One way to do this is to estimate the channels, use the channel estimates to separate the different users and then apply any of the equalization methods discussed above on each users signal. The channel estimate can of course also be used for equalization since it is already known from the previous steps. Blind channel estimation of MIMO systems has been studied by Liang and Ding, [34], Hua et. al., [35], etc. Liang and Ding developed an algorithm to identify the MIMO system from the common nullspace of a set of fourth-order cumulant matrices of the channel outputs. Hua et. al. exploit second order statistics to decorrelate subchannels of the system. 10 INTRODUCTION

5 Summary of Contributions

The three papers that are included in the thesis are summarized here.

Paper A - Performance Degradation Due to Blindness in Separation of MIMO-FIR Systems over COST207 Channels

Authors: S. Sandberg and J. P. LeBlanc Reproduced from: Proceedings of the 13th European Signal Processing Conference (EU- SIPCO) 2005, Turkey.

In this paper, the sensitivity to channel estimation errors in separation of MIMO ﬁnite impulse response (FIR) systems is investigated. The cost of blindness is considered in terms of the difference in remaining cross-channel power after separation based on blind channel estimation and perfectly known channels. Simulations for a speciﬁc blind channel estimation method verify that the remaining cross- channel power is much higher when the channel is estimated blindly compared to when the channel is perfectly known. However, the blind channel estimation method is not very sensitive to channel order underestimation in terms of remaining cross-channel power.

Paper B - Performance of LDPC Coded Spread OFDM with Clipping

Authors: S. Sandberg, C. de Frein,´ J. P. LeBlanc, B. Vasic and A. D. Fagan Reproduced from: Proceedings of the 8th International Symposium on Communication Theory and Applications (ISCTA) 2005, UK, pp. 156-161.

This is the ﬁrst paper in the thesis that considers LDPC codes and OFDM. Spreading, also called linear precoding, is investigated with the aim to both increase the frequency diversity of the OFDM system and reduce the PAPR of the modulated signal. This can be done by spreading with the Walsh-Hadamard matrix. The paper includes simulation results for an LDPC coded OFDM system with and without clipping and spreading, using the ETSI channel model for HIPERLAN/2. It is shown that there is a performance gain by spreading, especially in systems with clipping. Because of the very simple structure of the Walsh-Hadamard matrix, the spreading can actually reduce the computational complexity of the overall OFDM system. 6. CONCLUSIONS 11

Paper C - Receiver-oriented Clipping-effect Mitigation in OFDM - a Wor- thy Approach?

Authors: S. Sandberg, J. P. LeBlanc and B. Vasic Reproduced from: Proceedings of the 10th International OFDM-Workshop 2005, Hamburg.

The work considering LDPC coded OFDM systems is continued in this paper. The performance of two methods for clipping mitigation in the receiver, which might be of interest if the clipping for some reason can not be avoided in the transmitter, is investigated. One method is Bayesian estimation of the transmitted signal from the the clipped and noisy signal. The other method is to ﬁnd the exact distributions of the clipping noise and use this as input to the LDPC decoder. Surprisingly, it is shown that the cost of completely ignoring clipping in the receiver is minimal, even though it is assumed that the receiver has perfect knowledge of the channel. It seems like the LDPC codes can handle the clipping noise in the receiver and that clipping mitigation methods should be concentrated to the transmitter.

6 Conclusions

In a digital communication system with a wireless link, the channel effects can heavily degrade the system performance since the wireless link is time-varying and may experience multipath fading and interference. Channel effects like deep fades might be hard to avoid, while for example clipping in the high power ampliﬁer can usually be avoided by using a high-performance ampliﬁer or by power backoff. However, this may not always be the preferred way and we seek other solutions to the channel effect problems. Channel estimation plays an important role when seeking to minimize the degradation due to channel effects, since many receiver methods for handling channel effects require knowledge of the channel. The framework of OFDM, and especially the sensitivity of OFDM to two channel effects, is considered. An increased PAPR of the OFDM modulated signal leads to an increased probability of clipping and the frequency diversity is reduced since each symbol is transmitted over only one subchannel. This work includes investigation of receiver clipping mitigation methods and spreading as a mean to reduce the PAPR and increase the frequency diversity. Also, the sensitivity to channel estimation errors has been investigated for separation of MIMO systems. The papers included in this thesis show thatspreadingbytheWalsh-Hadamard matrix is a good way to increase the performance and reduce the sensitivity to clipping and fading in an OFDM system with LDPC coding. Walsh-Hadamard spreading can be implemented with less computations than the conventional OFDM system and gives increased performance to reduced complexity. Receiver clipping mitigation methods on the other hand have been shown to give a negligible performance gain in systems with LDPC coding, even though they require heavy computations. 12

7 Future Work

The research presented in this thesis has been concentrated on investigating performance losses due to channel estimation errors and performance gains of methods for reducing channel effects, for some specified methods and systems. LDPC codes have been included in some of the systems under investigation, but has not received much attention. Future work will be directed more towards the LDPC codes and their sensitivity to channel estimation errors and other channel effects. The channel information needed by the LDPC decoder with a frequency-flat channel is limited to the noise power, that is, only one parameter. For an LDPC coded OFDM system with a frequency-selective fading channel, the noise power of each subchannel must be fed to the decoder for calculation of the initial likelihoods of the different bits. In this case the number of parameters to estimate becomes much higher. The sensitivity to estimation errors is likely to be higher in such a system since errors will affect the relative reliability of the different bits in the first iteration of the decoder. The LDPC decoder considered so far has been the sum-product algorithm that needs channel information. There exists however also hard-decision decoding algorithms that do not need channel information. One question is at what point it is better to disregard estimates of the channel and use a hard-decision decoder instead of a soft decoder (e.g. the sum-product algorithm). Investigation of existing hard-decision algorithms and their connections to the sum- product algorithm might give insight into the problem of designing a soft LDPC decoder that do not need channel information. REFERENCES

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[23] M. Debbah, M. de Courville, and P. Maill, “Multiresolution decoding algorithm for Walsh-Hadamard linear precoded OFDM,” 7th International OFDM-Workshop 2002, Hamburg, Germany, Sep. 2002.

[24] S. N. Crozier, D. D. Falconer, and S. A. Mahmoud, “Least sum of squared errors (LSSE) channel estimation,” IEE Proceedings-F, pp. 371–378, Aug. 1991. 15

[25] J. K. Cavers, “An analysis of pilot symbol assisted modulation for Rayleigh fading channels,” IEEE Transactions on vehicular technology, pp. 686–693, Nov. 1991.

[26] Y. Li, “Pilot-symbol-aided channel estimation for OFDM in wireless systems,” IEEE Transactions on vehicular technology, pp. 1207–1215, July 2000.

[27] O. Edfors, M. Sandell, J.-J. van de Beek, S. K. Wilson, and P. O. Brjesson, “OFDM channel estimation by singular value decomposition,” IEEE Transactions on communications, pp. 931–939, July 1998.

[28] Y. Sato, “A method of self-recovering equalization for multilevel amplitude-modulation systems,” IEEE Transactions on communications, pp. 679–682, June 1975.

[29] D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Transactions on Communications, pp. 1867–75, Nov. 1980.

[30] T. R. Treichler and M. G. Agee, “A new approach to multipath correction of constant modulus signals,” IEEE Transactions on Acoustics, Speech, and Signal Processing, pp. 459– 472, April 1983.

[31] O. Shalvi and E. Weinstein, “New criteria for blind deconvolution of nonminimum phase systems,” IEEE Transactions on Information Theory, pp. 312–321, March 1990.

[32] L. Tong, G. Xu, and T. Kailath, “Blind identiﬁcation and equalization based on second- order statistics: a time domain approach,” IEEE Transactions on information theory, pp. 340–349, March 1994.

[33] H. A. Cirpan and M. K. Tsatsanis, “Stochastic maximum likelihood methods for semi- blind channel estimation,” IEEE Signal processing letters, pp. 21–24, Jan. 1998.

[34] J. Liang and Z. Ding, “Blind MIMO system identiﬁcation based on cumulant subspace decomposition,” IEEE Transactions on Signal Processing, pp. 1457–1468, June 2003.

[35] Y. Hua, S. An, and Y. Xiang, “Blind identiﬁcation of FIR MIMO channels by decorrelat- ing subchannels,” IEEE Transactions on Signal Processing, pp. 1143–1155, May 2003. 16 Part II 18 PAPER A Performance Degradation Due to Blindness in Separation of MIMO-FIR Systems over COST207 Channels

Authors: Sara Sandberg and James P. LeBlanc

Reformatted version of paper originally published in: Proceedings of the 13th European Signal Processing Conference (EUSIPCO) 2005, Turkey.

19 20 PAPER A PERFORMANCE DEGRADATION DUE TO BLINDNESS IN SEPARATION OF MIMO-FIR SYSTEMS OVER COST207 CHANNELS

Sara Sandberg and James P. LeBlanc

Abstract

This paper considers the performance penalty of a blind, compared to a non-blind, separation technique of a MIMO-FIR channel. In the blind method the mixing filters are first identified, while they are assumed to be known in the non-blind case. The blind system identification is performed using a recently proposed method based on cumulant subspace decomposition. Separation is then achieved by the FIR part of the mixing system inverse, which minimizes the cross-channel power. The performance penalty due to blindness is investigated for the case when the channel order is underestimated. Results of average residual cross-channel power of the wireless COST207 channel model are included.

1 Introduction

Blind source separation techniques in wireless communications have been under active research due to the achievable gains in system performance and capacity. In a commercial cellu- lar communication system there will be different types of interference, from multiuser interference in a single cell to interoperator interference. Blind source separation (BSS) can be used to lower the impact of interference on transmission quality as well as increase the capacity of the system in the uplink, [1, chap. 8.8]. Much of the work in the area of BSS addresses the case of instantaneous mixtures. However, in wireless communications multipath is usually present, which yields convolutive mixtures. Also, only mixing with sufficiently narrowband signals can be approximated as instantaneous, while the more common wideband case leads to convolutive mixtures, [1]. The work herein focuses on the separation of multiple-input multiple-output (MIMO) systems. The remaining ISI can be removed by a number of blind single-channel equalization methods, such as the constant modulus algorithm [2], [3], or by minimization of any of the criteria proposed in [4]. In BSS with convolutive mixtures, the sources are separated and equalized without knowledge of the mixing system or usage of training sequences. Blind methods therefore offer po- tential improvement in system capacity by eliminating the training sequences which carry no information. However, the performance of the separation is likely to degrade when less apri- ori information is exploited. This work investigates the cost of blindness in terms of residual 22 PAPER A cross-channel power (that is, after separation). Blind source separation methods can be divided into direct and indirect approaches. In direct approaches the separated signals are extracted without explicit identification of the mixing system, while indirect methods identify the unknown channels before separation and equalization. A recently proposed method for system identification, [5], exploits second order statistics and decorrelates subchannels of the system. In [6], higher order cumulant matching is used to estimate the channel and a Wiener filter separates the independent sources. Another recent method uses higher order statistics and subspace analysis to identify the MIMO finite impulse response (FIR) system assuming a known channel order, [7]. This is the method considered further here. This paper investigates the separation performance degradation when the mixing system is blindly estimated and compares it with the performance when the mixing system is known (in terms of remaining cross-channel power). The separation system is the FIR part of the mixing system inverse (known or estimated). The performance loss due to blindness is also investigated when the channel order (that must be known or estimated both for the blind identification and the calculation of the separating system) is underestimated, a common case of practical interest.

2 System Model

We focus on MIMO-FIR systems as in Figure 1, where observed signals, x , are the output of a linear channel x A s n (1)

The source signals, s , are assumed to be statistically independent and the number of source signals equals the number of observed signals. The additive Gaussian noise, n , is zero- mean and the noise components are mutually independent as well as independent of the source signals. The vectors x , s and n are length vectors. The channel order of the mixing system is denoted by and the mixing matrix A can be written as a matrix polynomial

A A A A (2) where each matrix A is .

The separating matrix B is deﬁned as

B B B B (3) where each B is and the separating ﬁlter is of order . The output of the system can be written as y B A s n (4) 23

sxy + A + B Mixing Separation

Figure 1: A MIMO-FIR system with N sources and N sensors. and we deﬁne the overall system H B A (5) where H H H H (6)

3 Separation with Known Mixing Matrix

Assuming that the mixing matrix A is known or may be estimated, we can ﬁnd a separating matrix B by taking the FIR part of the inverse of A . The inverse is given by

A A (7) A where is the adjoint (the transposed cofactor matrix). If the mixing filters in A are FIR, the filters in A as well as the A will be FIR too. The A in this equation can be seen as the separating matrix and the factor A as the IIR filter that equalizes all channels after separation. Full separation can be achieved if the order of the separating system is at least as large as the order of the filters in A ,thatis,if

(8)

Remember that full separation still leaves ISI in each separated source. If the order of the separating system is lower than the order of A , there will be residual cross-channel power denoted by and deﬁned as

(9)

where is element of H . If full separation can not be achieved, we choose the filters of the separating system to be the first taps of the filters in A . This selection of B minimizes the cross-channel power. 24 PAPER A

4 Blind Identiﬁcation Based on Cumulant Subspace Decom- position

If the mixing matrix A is not known aprioriit can be blindly estimated from the observed signals by a recently proposed method based on cumulant subspace decomposition, [7]. The proposed algorithm can identify a MIMO-FIR system where

the source signals are independent, stationary, temporally i.i.d. processes with zero- means and nonzero fourth-order kurtosis

the channel noises n are mutually independent zero-mean Gaussian stationary processes and independent of the source signals

the number of sensors are no less than the number of sources

the channel order is the same for all sources

there exists a complex point such that A has full column rank

the fourth-order kurtosis of all source signals have the same sign.

In [7], some of these assumptions may be relaxed and complemented by other conditions, but this is not considered in the summary given here. The main idea is to identify A given only the observed signals x by employing the fourth-order cumulants. These higher order statistics of x give enough information to identify the mixing ﬁlters up to an arbitrary scaling and permutation and are also not affected by additive white Gaussian noise. The MIMO cumulant subspace - joint diagonalization (MIMOCS- JD) algorithm described in [7] is summarized in three main steps.

1. A set of cumulant matrices containing the fourth-order cumulants with ﬁxed third and fourth argument are deﬁned and the nullspace of this set is estimated.

2. It is shown that a matrix containing the mixing parameters is orthogonal to this nullspace. The estimated mixing ﬁlters are obtained via the orthogonality principle.

3. Remaining ambiguities are reduced to scaling and permutation by joint diagonalization of a set of matrices.

A number of simulation examples are given in [7]. For example, the performance is shown for a case where the true channel order is two, but overestimated to three and four. As the authors point out, this algorithm (like other subspace based algorithms) is sensitive to channel order overestimation. However, the interesting case when the channel order is underestimated is not investigated. 25

Calculation of x Blind system A B Cost separation identification calculation matrix

A Comparison Cost of blindness

Calculation of A B Cost separation calculation matrix

Figure 2: The cost of blindness is estimated by comparing a blind system with a non-blind.

5 Cost of Blindness

This paper investigates how sensitive the source separation is to errors in the estimated mixing matrix A. The measure of performance used is residual cross-channel power deﬁned in (9). The non-blind case where a separation matrix B is calculated directly from the known mixing matrix A is compared with the blind case where the mixing matrix is estimated to A using the blind identiﬁcation method described in section 4. A separation matrix B is calculated from A and the cross-channel power of the non-blind system B A is compared with the cross-channel power of the blind system B A , Figure 2.

It is often the case in communications that the true channel order is higher than the channel order that is tractable to assume for blind identiﬁcation, [8]. Therefore, we herein investigate the cost of blindness when the channel order is underestimated (when ), to give an understanding of the robustness of the blind identiﬁcation method to channel order underestimation.

6 Simulation Results

This section presents simulations to demonstrate the performance of the blind versus the non- blind system (the cost of blindness) when the channel order is underestimated. The system model has two sources and two sensors and the channel order is for the four channels. The channels are randomly generated based on a modified COST207 bad urban wireless channel model [9], where the symbol rate has been increased to give channels with an order of .The source signals are mutually independent, temporally i.i.d. QPSK signals and the channel noises are zero-mean, complex, additive white Gaussian processes. The MIMOCS-JD algorithm, [7], is used for identification of the mixing system. In [7], the sample support used for identifying channels of order is . With a higher channel order, there are more parameters in the cumulant matrix that need to be estimated in the blind identification method. A sample support of would keep the ratio of samples to parameters in 26 PAPER A

14 Blind Non−blind 12

Cross−channel power 4

0 0 1 2 3 4 5 6 7 Estimated channel order

Figure 3: Cross-channel power after non-blind and blind source separation. The true channel order is and samples are used for the blind identiﬁcation. The bars show ,where is the standard deviation. the cumulant matrix constant, when the order of the channels is . The channel is assumed to be constant during the time needed to collect the samples. Other parameters are the same as in [7]. The order of the separating system, , is chosen equal to , since this should give perfect separation when there are sources and sensors, see (8). The simulations are performed at an SNR of . Figure 3 shows the average cross-channel power, ,when is varied from to the true channel order ,for different COST207 channels. The bars show ,where is the standard deviation. As a comparison it can be noted that the average cross-channel power of the mixing system,

(10) for these channels was .

As expected, there is a signiﬁcant residual cross-channel power due to use of the blind technique and it also exhibits greater variance. However, the blind method, which was originally introduced as needing the exact channel order, in fact, is not very sensitive to channel order underestimation for this class of mobile communication channels. The cost, in terms of cross- channel power difference between the blind and non-blind approach, is almost constant when the channel order is underestimated. The relative cost due to channel order underestimation 27

Blind 30 Non−blind

Cross−channel power reduction (dB) 5

0 0 1 2 3 4 5 6 7 Estimated channel order

Figure 4: Cross-channel power reduction achieved by blind and non-blind source separation respectively. The last value for the non-blind case is omitted since non-blind separation with the true channel order is perfect and the cross-channel power reduction is inﬁnite. is also small compared to the cost of blindness, at least for reasonably good estimates of the channel order. To further interpret the results, we deﬁne the cross-channel power reduction, ,asthe cross-channel power of the mixing system over the residual cross-channel power after separation, that is, (11)

Figure 4 shows the separation gain in terms of cross-channel power reduction. The last value for the non-blind case is omitted since non-blind separation with the true channel order is perfect, , and the cross-channel power reduction is inﬁnite. This measure of separation performance demonstrates more clearly the performance penalty due to blindness. The cross- channel power reduction achieved by the blind method does not vary much with estimated channel order. Even if the estimated channel order is just half the true ( ), the cross- channel power reduction is almost as high as we can possibly get using the blind method. The difference is less than . This is again suggesting that the blind method is not especially sensitive to channel order underestimation. In practice, the channel may rarely be so slowly varying that it can be assumed to be constant for a support of samples. To shed some light on results applicable to the small sample support case, Figure 5 shows the remaining cross-channel power having support of only samples. With this short sample support the parameters of the cumulant matrices 28

14 Blind Non−blind 12

Cross−channel power 4

0 0 1 2 3 4 5 6 7 Estimated channel order

Figure 5: Cross-channel power after non-blind and blind source separation. The true channel order is and samples are used for the blind identiﬁcation. in the blind method are not very accurately estimated for a high channel order, and the cost of blindness increases with an increasing estimated channel order. For this speciﬁc simulation a channel order of gives the lowest cross-channel power when the true channel order is . This result suggests that channel order underestimation might give lower cross-channel power in cases of limited sample support.

7 Conclusions

This work investigates the performance penalty of blind separation of a MIMO-FIR system for a class of wireless mobile communication channels. Non-blind separation is used in con- junction with a recently published method for blind system identiﬁcation to explore the performance degradation due to blindness. Previous work on the blind method investigates its sensitivity to channel order overestimation. Herein the effects due to channel order underestimation are considered, which is important since some form of under-modeling is often the case. Our simulations based on the COST207 channel model show the expected performance degradation due to blindness. Interesting to note is that the performance of the blind method is not very sensitive to channel order underestimation. If the sample support is short, channel order underestimation might even give a lower residual cross-channel power in the blind case. This suggests that even if the true channel order is known, blind identiﬁcation with a lower channel order than the true may increase the performance of the communication system in terms of reduced cross-channel power. REFERENCES

[1]S.H.ed.,Unsupervised adaptive ﬁltering, Vol. 1: Blind source separation. Wiley, 2000.

[2] D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Transactions on Communications, pp. 1867–75, Nov. 1980.

[3] T. R. Treichler and M. G. Agee, “A new approach to multipath correction of constant modulus signals,” IEEE Transactions on Acoustics, Speech, and Signal Processing, pp. 459–472, April 1983.

[4] O. Shalvi and E. Weinstein, “New criteria for blind deconvolution of nonminimum phase systems,” IEEE Transactions on Information Theory, pp. 312–321, March 1990.

[5] Y. Hua, S. An, and Y. Xiang, “Blind identiﬁcation of FIR MIMO channels by decorrelat- ing subchannels,” IEEE Transactions on Signal Processing, pp. 1143–1155, May 2003.

[6] J. K. Tugnait, “On blind MIMO channel estimation and blind signal separation in unknown additive noise,” IEEE Signal Processing Workshop on Signal Processing Advances in Wireless Communications, pp. 53–56, April 1997.

[7] J. Liang and Z. Ding, “Blind MIMO system identiﬁcation based on cumulant subspace decomposition,” IEEE Transactions on Signal Processing, pp. 1457–1468, June 2003.

[8] J. Treichler, I. Fijalkow, and C. J. Jr., “Fractionally spaced equalizers,” IEEE Signal Pro- cessing Magazine, pp. 65–81, May 1996.

[9] M. Benthin, Vergleich koharenter¬ und inkoharenter¬ Codemultiplex-Ubertragungskonzepte¬ fur¬ zellulare Mobilfunksysteme (Ph.D. thesis). VDI Verlag, 1996.

29 30 PAPER B Performance of LDPC Coded Spread OFDM with Clipping

Authors: Sara Sandberg, Cormac de Fr´ein, James P. LeBlanc, Bane Vasic and Anthony D. Fagan

Reformatted version of paper originally published in: Proceedings of the 8th International Symposium on Communication Theory and Applications (ISCTA) 2005, UK, pp. 156-161.

31 32 PAPER B PERFORMANCE OF LDPC CODED SPREAD OFDM WITH CLIPPING

Sara Sandberg, Cormac de Fr´ein, James P. LeBlanc, Bane Vasic and Anthony D. Fagan

Abstract

We present the performance of OFDM systems with coding, spreading, and clipping. Low- Density Parity-Check (LDPC) codes give coding gains and spreading by the Walsh Hadamard transform gives gains in terms of increased frequency diversity as well as reduced peak-to- average power ratio (PAPR) of the transmitted OFDM signal. By evaluating both the IFFT transform (OFDM) and the Walsh Hadamard transform in a single step, the number of operations needed for the spread OFDM system is actually less than for the conventional OFDM system. Reducing the PAPR is important in systems with clipping since it is related to the probability of clips. Each clip introduces clipping noise to the system which reduces the performance. Results of a clipped OFDM system with LDPC coding and spreading for an ETSI indoor wireless channel model are presented and compared with other systems. It is shown that there is a gain by spreading for LDPC coded OFDM systems, and especially for systems with clipping.

1 Introduction

In wireless communications, the channel is often time-varying due to relative transmitter- receiver motion and reflections. This time-variation, called fading, reduces system performance. With a high data rate compared to the channel bandwidth, multipath propagation becomes frequency-selective and causes intersymbol interference (ISI). A multicarrier OFDM system is known to transform a frequency-selective fading channel into parallel flat-fading subchannels if a cyclic prefix is used for preventing inter-block interference. The receiver complexity is thereby significantly reduced, since the equalizer can be implemented as a number of one-tap filters. In such a system, the data transmitted on some of the carriers might be strongly attenuated and could be unrecoverable at the receiver. Lately, spread spectrum techniques have been combined with the conventional OFDM to better exploit frequency diversity, [1][2]. This combination implies spreading information across all (or some of) the carriers by precoding with a unitary matrix and is in the following referred to as spread OFDM (SOFDM). Another way to resist data corruption on fading subchannels is to use error-correcting codes. In [3], joint precoding and coding of the OFDM system is suggested with convolutional codes or turbo-codes and it is shown that there is a significant performance gain offered by introducing precoding to the coded transmission. In the last decade low-density parity-check (LDPC) 34 PAPER B codes, first invented by Gallager in 1962 [4], have attracted attention, see e.g. [5]. Serener et al. investigate the performance of SOFDM with LDPC coding in [6][7]. One of the major drawbacks with the OFDM system is its high peak-to-average power ratio (PAPR). A high PAPR corresponds to a high probability of clipping in the power amplifier in the transmitter or, alternatively, a large input power backoff. This implies reduced signal power, degrading bit error rate and for clipping even spectral spreading. There has been much research in the area of reducing the PAPR for OFDM systems, [8][9]. It is shown in [10] that precoding by the Walsh Hadamard (WH) matrix reduces the PAPR of the OFDM signal and the associated reduced probability of clipping distortion will increase the performance of the system. This precoding scheme has also been suggested for spreading, [1]. Surprisingly, the joint WH spreading and OFDM modulation can be performed by one single transformation that requires less operations than the IFFT alone, [11]. In this paper, the total performance gain of the WH spreading is investigated for an OFDM system with LDPC coding and clipping. In particular, the gain in bit-error-rate performance is analyzed for an ETSI channel model and results for clipped OFDM signals are provided. The conventional and SOFDM system as well as the channel model are described in Section 2 and the application of LDPC codes to OFDM in Section 3. Results of the WH spreading applied to clipped and non-clipped OFDM signals follow in Section 4. Finally, Section 5 gives some concluding remarks.

2 The OFDM System

The OFDM modulation is obtained by applying Inverse FFT (IFFT) to message subsymbols, where is the number of subchannels in the OFDM system. The baseband signal can be written (1) where is symbol number in a random message stream, also called the frequency domain signal. Thus the modulated OFDM vector can be expressed as

x X (2)

A block diagram describing the conventional OFDM system is shown in Figure 1. is the channel impulse response and is a vector of uncorrelated complex Gaussian random variables with variance . The output from the FFT is

(3) where the diagonal matrix gives the frequency domain channel attenuations and is the FFT of the noise . The elements of are still uncorrelated complex Gaussian random variables with variance due to the unitary property of the FFT. The zero- forcing equalizer is considered for conventional OFDM, but Wiener equalizers are also used. A 35

X x y Y IFFT P/S h S/P FFT EQ N N N N N

Figure 1: Conventional OFDM. practical implementation of OFDM usually uses a cyclic preﬁx in order to avoid inter-symbol interference (ISI).

In SOFDM, the frequency domain signal is multiplied by a spreading matrix before it is fed to the IFFT, Figure 2. The spreading considered here is the WH matrix that can be generated recursively for sizes a power of two [12]. The ( ) WH matrix is given by

(4)

The ( ) WH matrix is given in terms of the ( )WHmatrix,

(5)

In the following isassumedtobethe( ) WH matrix. The WH transform is an orthogonal linear transform that can be implemented by a butterfly structure as the IFFT and since the WH and IFFT transforms can be combined and calculated with less complexity than the IFFT alone this means that the system complexity is reduced by applying WH spreading. At the receiver side, Wiener filtering is performed and the output of the Wiener filter is the vector given by . is defined as

(6) where is a diagonal matrix,

(7) and denotes conjugation. This means that the receiver consists of scalar channel equalization followed by the transpose of the spreading matrix, which in the case of unitary spreading equals the inverse. In the following, the noise power and the frequency domain channel attenuations areassumedtobeknown.

The wireless channel model used in this work is the ETSI indoor wireless channel model for HIPERLAN/2, [13]. Wireless channels can be modeled as Rayleigh fading channels. When a signal is sent from the transmitter across a wireless channel, it travels via many paths (due to reﬂections, refractions or diffractions) to the receiver. The different path lengths result in time delays or phase differences between the multipath components, which leads to constructive 36 PAPER B

X IFFT h FFT G N N N N

Figure 2: Spread OFDM. and destructive interference. The received signal can be written where is the amplitude and is the phase. The received amplitude is Rayleigh distributed with the probability density function

(8) and the received phase is uniformly distributed with

(9) where is the mean power of the waveform. A sample wireless channel response is shown in Figure 3. Two deep spectral nulls are apparent in this example. In Conventional OFDM, each subchannel is assigned one subsymbol. In the example channel given above, the subsymbols transmitted on the subchannels in the close vicinity of the two deep spectral nulls, will have a high probability of error. However, using the spreading employed in this work, the information transmitted on each subchannel will be a linear combination of the original subsymbols. This means that instead of a few subsymbols being severely affected by spectral nulls, several subsymbols are lightly affected. This approach leads to improved BER.

3 LDPC codes for OFDM and SOFDM

Error control codes used in this paper belong to a class known as low-density parity-check (LDPC) codes [4]. An LDPC code is a linear block code and it can be conveniently described through a graph commonly referred as a Tanner graph [14]. Such a graphical representation facilitates a decoding algorithm known as the message-passing algorithm. A message-passing decoder has been shown to virtually achieve Shannon capacity when long LDPC codes are used. In the next paragraph we will describe a speciﬁc class of LDPC codes used here. For more details on message passing decoding the reader is referred to an excellent introduction by Kschischang et al. [15]. Consider a linear block code of length deﬁned as the solution-space (in )ofthe system of linear equations ,where is an binary matrix. The bipartite graph representation of is denoted by . contains a set of variable nodes and a set of check nodes, i.e. nodes corresponding to equations in . A variable node is connected with 37

−10

−20 Subchannel Power Gain (dB)

−30 8 16 24 32 40 48 56 64 Subchannel Index

Figure 3: Example of a wireless channel response following the ETSI indoor wireless channel model for HIPERLAN/2. a check node if it belongs to a corresponding equation. More precisely, The -th column of corresponds to a variable node of the graph ,andthe -th row of the matrix corresponds to a check node of . The choice of a parity check matrix that supports the message-passing algorithm is a problem that has been extensively studied in recent years, and many random [16] and structured codes have been found [17]. We have chosen codes from a family of rate- compatible array codes, [18][19], because they support a simple encoding algorithm and have low implementation complexity. The general form of the parity check matrix can be written as

(10) ...... where each of submatrices ,isapowerofa permutation matrix (see [19] ). Notice that the row and column weights of vary, i.e. the code is irregular. A column in a set of leftmost columns has the weight , while the rest of the columns have weight .

Figure 4 shows a block diagram describing the SOFDM system with LDPC coding, denoted by LDPC-SOFDM. The message bits are ﬁrst encoded to codewords of length and modulated to QPSK symbols. The modulation symbols are ,where is the 38 PAPER B

X LDPC LDPC Modu− IFFT FFT G Encoding lation h Decoding N N N N

Figure 4: SOFDM with LDPC coding. symbol energy. The modulated codeword is partitioned into blocks of samples, where we assume that , and each block is multiplied by the spreading matrix . The spread signal is sent through the OFDM system and the Wiener filter. Both soft information from the output of the Wiener filter and the SNR for each subchannel are fed to the decoder. Since spreading averages SNR of different subchannels, a theoretical SNR taking the spreading into account is computed and used for the decoding. The theoretical SNR can be calculated from the output of the Wiener filter, which is

(11)

Component of can be written

(12) where (13)

(14) and (15)

is element in the WH matrix, ,and .Forlarge the interference-plus-noise at the output of the Wiener ﬁlter can be considered to be Gaussian, [2], and approximated as noise. The theoretical SNR of subchannel with spreading is Var (16) Var Var

(17)

For conventional OFDM ( ), the SNR is simply

Var (18) 39

−1 10 Conv. OFDM SOFDM LDPC−OFDM −2 10 LDPC−SOFDM

−3 10 BER −4 10

−5 10

−6 10 10 15 20 25 SNR (dB)

Figure 5: Performance of conventional OFDM, SOFDM, LDPC-OFDM, and LDPC-SOFDM for the case of 64 subchannels. 4 Results

To show the performance of LDPC-SOFDM, simulations are performed with rate 0.8 lattice codes and a codeword length of 1024. The maximum column weight of the parity-check matrix is 3. The performance is an average over different channel realizations of the ETSI indoor wireless channel model and the channel realizations are normalized to have energy ,thatis,

(19)

Figure 5 shows the BER performance of different OFDM systems without clipping. The OFDM system has 64 subchannels and the channel is assumed to be constant during the transmission of one codeword. Both SOFDM and LDPC coding give a large gain compared with conventional OFDM, but there is also a gain by spreading of about 2.7 of the LDPC coded system at bit-error-rate. Figure 6 shows that the performance does not change much with the number of subcarriers, which is also the size of the precoder. However, with a large number of subchannels, like 512, interleaving is necessary for LDPC-OFDM to get this performance.

A comparison to the work in [2] for convolutional codes is shown in Figure 7. The frequency domain channel attenuations in [2] are assumed to be independent identically distributed circular complex Gaussian random variables with variance 1. A convolutional encoder with constraint length 7 is used and the code rate is . The spreading and equalizer is the 40 PAPER B

−2 10 N=64 LDPC−OFDM N=64 LDPC−SOFDM N=512 LDPC−OFDM −3 10 N=512 LDPC−SOFDM

−4 10 BER

−5 10

−6 10 10 11 12 13 14 15 SNR (dB)

Figure 6: Performance of LDPC-OFDM and LDPC-SOFDM for both 64 and 512 subchannels. same as in this paper and an OFDM system with 64 subchannels is used. We compare this with our LDPC-SOFDM system with rate 0.8 for the same channel parameters as in [2]. Figure 7 shows that the LDPC-SOFDM system performs better than the system with the convolutional code.

Figure 8 shows the reduction of PAPR that is the result of the WH spreading. The instantaneous PAPR for the th OFDM block, and the overall PAPR are deﬁned, respectively, by

(20)

(21) where is the th block of the time domain signal vector. However, in practice it is more useful to know the distribution of the instantaneous PAPR. In Figure 8 the overall PAPR has decreased by 1.1 and the mean instantaneous PAPR has decreased by 0.8 by spreading. In many systems clipping occurs in the power ampliﬁer and a reduction of PAPR reduces the number of clips. If denotes the input complex signal, the clipping of the baseband signal can be modeled as , with

for (22) for 41

0 10 Conv−SOFDM LDPC−SOFDM

−1 10

−2 10 BER −3 10

−4 10

−5 10 5 6 7 8 9 10 11 SNR (dB)

Figure 7: Comparison with a convolutional coded SOFDM system. where is the maximum output amplitude. The clipping ratio is deﬁned as

(23)

Figure 9 shows the BER in a case where the signal is clipped with a clipping ratio of 2 .The total gain of spreading is around 4 at bit-error-rate, compared to 2.7 when there is no clipping, since the PAPR reduction by spreading has reduced the number of clips. The effect of clipping noise to the SNR is not taken into account in the decoder. This result suggests that spreading is of extra value in systems where there is a high probability of clipping.

5 Conclusions

In this paper the BER performance of LDPC-SOFDM is investigated. The spreading considered is the WH transform which actually can reduce the complexity of the system. Results for the ETSI indoor wireless channel model show that using LDPC-SOFDM instead of LDPC- OFDM in a system with clipping gives a gain of 4 at a bit-error-rate of . The gain is due to increased frequency diversity as well as reduced PAPR. The performance is also investigated for different number of subchannels and the results show that systems with different number of subchannels perform almost the same. However, a large number of subchannels increases the PAPR which in turn increases the probability of clips. Our results conﬁrm that spreading enhances the performance of the OFDM system for the ETSI channel model and show that especially the performance in a system with clipping is increased, while the complexity is reduced. 42

(hist(PAPR)) 2 10 log 0 0 5 10 15 20 25 30 35 (a) PAPR 6

(hist(PAPR)) 2 10 log 0 0 5 10 15 20 25 30 35 (b) PAPR

Figure 8: Histograms of the PAPR for an OFDM system with 64 subchannels, with (a) and without (b) spreading.

−2 10 LDPC−OFDM LDPC−SOFDM LDPC−OFDM 2 dB clip LDPC−SOFDM 2 dB clip −3 10

−4 10 BER

−5 10

−6 10 10 11 12 13 14 15 16 SNR (dB)

Figure 9: Performance of LDPC-OFDM and LDPC-SOFDM with 64 subchannels. The performance is shown both for no clipping and for a clipping ratio of 2 . REFERENCES

[1] Y.-P. Lin and S.-M. Phoong, “BER minimized OFDM systems with channel independent precoders,” IEEE Transactions on signal processing, pp. 2369–2380, Sep. 2003.

[2] M. Debbah, P. Loubaton, and M. de Courville, “Spread OFDM performance with MMSE equalization,” IEEE International Conference on Acoustics, Speech, and Signal Process- ing, 2001., pp. 2385–2388, May 2001.

[3] Z. Wang, S. Zhou, and G. B. Giannakis, “Joint coding-precoding with low-complexity turbo-decoding,” IEEE Transactions on wireless communications, pp. 832–842, May 2004.

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[7] A. Serener and D. M. Gruenbacher, “LDPC coded spread OFDM in indoor environments,” Proceedings of the 3rd IEEE International Symposium on Turbo Codes & Related Topics, France, pp. 549–552, Sep. 2003.

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[9] K. Yang and S.-I. Chang, “Peak-to-average power control in OFDM using standard arrays of linear block codes,” IEEE Communications letters, pp. 174–176, April 2003.

43 44

[10] M. Park, H. Jun, and J. Cho, “PAPR reduction in OFDM transmission using Hadamard transform,” ICC 2000 - IEEE International Conference on Communications, pp. 430– 433, June 2000.

[11] P. Mart-Puig and J. Sala-lvarez, “A fast OFDM-CDMA user demultiplexing architec- ture,” Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, 2000. ICASSP ’00., pp. 3358–3361, June 2000.

[12] N. Ahmed and K. R. Rao, Orthogonal transforms for digital signal processing. Springer Verlag, 1975.

[13] ETSI, “Channel models for HIPERLAN/2 in different indoor scenarios,” COST 256 TD(98), April 1998.

[14] M. Tanner, “A recursive approach to low complexity codes,” IEEE Transactions on information theory, pp. 533–547, Sep. 1981.

[15] F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Transactions on information theory, pp. 498–519, Feb. 2001.

[16] D. J. C. MacKay, “Relationships between sparse graph codes,” Proc. of IBIS 2000, Japan, available online at http://www.inference.phy.cam.ac.uk/mackay/ abstracts/ibis.html, 2000.

[17] B. Vasic and O. Milenkovic, “Combinatorial constructions of low-density parity-check codes for iterative decoding,” IEEE Transactions on information theory, pp. 1156–1176, June 2004.

[18] E. Eleftheriou and S. Olc¨ ¸er, “Low-density parity-check codes for digital subscriber lines,” Proceedings, IEEE International Conference on Communications, pp. 1752–1757, Apr. 2002.

[19] A. Dholakia and S. Olc¨ ¸er, “Rate-compatible low-density parity-check codes for digital subscriber lines,” Proc., IEEE International Conference on Communications, pp. 415– 419, Jun. 2004. PAPER C Receiver-oriented Clipping-effect Mitigation in OFDM - a Worthy Approach?

Authors: Sara Sandberg, James P. LeBlanc and Bane Vasic

Reformatted version of paper originally published in: Proceedings of the 10th International OFDM-Workshop 2005, Hamburg.

45 46 PAPER C RECEIVER-ORIENTED CLIPPING-EFFECT MITIGATION IN OFDM - A WORTHY APPROACH?

Sara Sandberg, James P. LeBlanc and Bane Vasic

Abstract

The high peak-to-average power ratio (PAR) in Orthogonal Frequency Division Multiplexing (OFDM) modulation systems can signiﬁcantly reduce performance and power efﬁciency. A number of methods exist that combat large signal peaks in the transmitter. Recently several methods have emerged that alleviate the clipping distortion in the receiver. We analyze the performance of two receiver clipping mitigation methods in an OFDM system with Cartesian clipping and low-density parity-check (LDPC) coding. Surprisingly, the cost of completely ignoring clipping in the receiver is minimal, even though we assume that the receiver has perfect knowledge of the channel. The results suggest that clipping mitigation strategies should be concentrated to the transmitter.

1 Introduction

It is well known that OFDM signals may suffer from a high peak-to-average power ratio (PAR), tending towards a Gaussian distribution for a large number of sub-carriers. There is a vast research literature on transmit-oriented signal processing and coding methods to mitigate problems associated with clipping due to these high PAR values, see [1][2][3]. However, many such methods aimed at lowering PAR come at the cost of increased transmitter complexity or lowering transmission rate. Thus, one may consider allowing such occasional clips due to high PAR transmit signals and suffer the associated loss due to clipping. Of course, one would still seek receiver methods to minimize such clipping loss. A number of recent papers document the loss of untreated clipping, [4][5], etc. Further- more, there is on-going research on understanding the clipping effects in the received signal, as well as ways to mitigate their effects, [6][7]. In this paper we make an attempt to quantify in the best practical sense how much the clipping loss can be mitigated. Surprisingly, even when using near optimal methods which involve signiﬁcant calculations (i.e. they lie beyond near-term practicality) we ﬁnd that the mitigation ability of such extreme processing is rather limited. To evaluate the limits of receiver oriented processing, we choose to combine best known practical coding methods, low-density parity- check (LDPC) codes, with the exact distribution of the clipping distortion. This combination could be expected to represent the best practically achievable system performance. It is as- 48 PAPER C

m QPSK X Z QPSK r LDPC LDPC x yzFFT Modu− IFFT Demodu− Decoding Encoding lation lation N N

Figure 1: LDPC coded OFDM system with clipping. sumed throughout the paper that the variance of the additive white Gaussian noise (AWGN) is perfectly known. Another way to combat the clipping distortion in the receiver is to estimate the signal before clipping from the clipped signal. In [7] a Bayesian estimator for the Cartesian clipper is derived. In this paper we show that for an OFDM system with LDPC coding, the ability to counter clipping gives an improvement of only 0.1 dB at bit-error-rate. To wit, the cost of completely ignoring clipping in the receiver appears to be minimal. These results are intended to give insight into the potentially unrecoverable nature of the clipping phenomena and imply directing attention to transmit-oriented strategies.

2 System Description and Channel Model

The system model used throughout the paper is an OFDM system with Cartesian clipping and AWGN. LDPC codes are utilized for error-correction and the encoded bits are QPSK- modulated, see Figure 1. In this section the OFDM system, the channel model and the LDPC codes will be discussed in turn.

The OFDM modulation is obtained by applying Inverse FFT (IFFT) to QPSK modulated message (or codeword) subsymbols, where is the number of subchannels in the OFDM system. The complex baseband signal (also called the time-domain signal) can be written

(1) where is subsymbol number in the message stream, also called the frequency domain signal. Thus the OFDM modulated signal vector can be expressed as

x X (2)

The time-domain signal can be represented by ,where and are the in- phase/quadrature (I/Q) components. A practical implementation of OFDM usually uses a cyclic preﬁx in order to avoid inter-symbol interference (ISI). In this paper we focus on the Cartesian clipper that clips and separately and in the following denotes either or ,sothat is always a real sequence. The distorted signal 49 is modeled as the output from the ideal clipper

(3) where is the clipping level and is the clipped I or Q component. The complex clipped signal is . Following the clipper, complex white Gaussian noise with variance (each noise component has variance ) is added to the signal. In the receiver, the clipped and noisy signal is demodulated by the OFDM FFT demodulator. Error control codes used in this paper belong to a class known as LDPC codes [8]. An LDPC code is a linear block code and it can be conveniently described through a graph commonly referred to as a Tanner graph [9]. Such a graphical representation facilitates a decoding algorithm known as the message-passing algorithm. A message-passing decoder has been shown to virtually achieve Shannon capacity when long LDPC codes are used. In the next paragraph we will describe a speciﬁc class of LDPC codes used here. For more details on message passing decoding the reader is referred to an excellent introduction by Kschischang et al. [10]. Consider a linear block code of length deﬁned as the solution-space (in )ofthe system of linear equations ,where is an binary matrix. The choice of a parity check matrix that supports the message-passing algorithm is a problem that has been extensively studied in recent years, and many random [11] and structured codes have been found [12]. We have chosen codes from a family of rate-compatible array codes, [13][14], because they support a simple encoding algorithm and have low implementation complexity. The general form of the parity check matrix can be written as

(4) ...... where each of the submatrices is a power of a permutation matrix, see [14]. Notice that the row and column weights of vary, i.e. the code is irregular. In the message-passing decoder, log-likelihood ratios (LLRs) are updated and passed between nodes. The wanted LLRs are the a posteriori probabilities that a given bit in c equals 0 or 1 given the whole received word r. The initial LLR for the th node (or codeword bit) is calculated as (5) where denotes the conditional probability of given . When the LLRs are updated by the message-passing algorithm, they will better approximate the wanted LLRs. 50 PAPER C

3 Characterization of Clipping Noise

The description of the clipping distortion in this section mainly follows the characterization given in the recent paper [6]. The time-domain OFDM signal is the sum of several statistically independent subcarriers and the I/Q components can be approximated by Gaussian processes, invoking the central limit theorem, if the number of subcarriers is large. The pdf of is therefore assumed to be Gaussian with mean zero and variance ,where is the power of each complex frequency-domain symbol . Knowing that the time-domain signal is approximately Gaussian, the Bussgang theorem can be applied, [15]. It states that the output of the clipper can be expressed as , where is uncorrelated with the clipper input signal and is an attenuation. The notation denotes as before either the I or Q component of a complex signal. The attenuation is dependent only on the input backoff ( ) of the clipper and for the Cartesian clipper deﬁned in (3) the attenuation is given by

(6) where the Q-function and the is deﬁned as

(7)

(8)

The part of the clipped signal that has no correlation with the input signal, , is called the clipping noise. The pdf of the I/Q component of the time-domain clipping noise is derived in [6] and is given by

(9)

where and

exp (10) is the Gaussian pdf with mean and variance . The pdf of the I/Q component of the clipping noise can be re-written as a function of the clipping parameters and only. Before LDPC decoding, the clipped and noisy received signal is demodulated by the FFT. The AWGN after the FFT will have the same variance as before the FFT, due to the unitary property of the FFT. This will not be the case for the clipping noise since it is not 51 assumed to be Gaussian. The frequency-domain clipping noise over the th subchannel, which is the output of the FFT when the input is the time-domain clipping noise, is

(11)

The (I or Q component) is a sum of approximately i.i.d. random variables and its pdf is the convolution of probability density functions. To avoid the convolutions the pdf of can be calculated as the product of the characteristic function of in different points. The characteristic function of (the Fourier transform of its pdf) is given by, [6],

(12) where denotes the real part. The characteristic function of the frequency-domain clipping noise can now be written as

(13) using (11) and (12), [6]. The pdf of is the inverse Fourier transform of , but it is usually easier to ﬁnd the cumulative distribution function by a numerical method called the Fourier series method, [16]. Since the clipping noise pdf is only dependent on the of the clipper, this pdf must be calculated only once for a system with a given and then a lookup table can be created. In the system discussed here, the clipping noise pdf is convolved with the AWGN pdf to give the pdf of the total noise for subchannel , denoted . If the channel is frequency ﬂat, the pdf of the total noise will be the same for all subchannels. The initial log-likelihood ratios for the decoder are calculated from this pdf according to (5), while accounting for the attenuation . The conditional probability where is either 0 or 1 can be written

(14) where is the pdf of the received signal and is the conditional density of the received signal given the corresponding codeword bit. In the case where 0 and 1 are equally likely to be transmitted, the LLR for the th node is

(15) 52 PAPER C

The LLRs can be calculated from the pdf of the total noise and the attenuation by

(16) since each codeword bit is represented by before applying the IFFT.

4 Bayesian Estimation

Another way to counter the clipping in the receiver is to estimate the time-domain signal from the clipped and noisy signal . In [7], a Bayesian estimator for signals clipped by the Cartesian clipper and distorted by AWGN is derived. The Bayesian estimator is the optimal estimator of given in the Minimum Mean-Square Error (MMSE) sense, given by

(17) where is the conditional pdf of the I/Q component given . A straight forward deriva- tion (see [7]) leads to a closed form expression of the Bayesian estimator for Gaussian input signals,

(18) where

(19)

(20)

The time-domain signal is estimated from the clipped and noisy and the output from the estimator is then demodulated by the FFT. The log-likelihood ratios that are needed for LDPC decoding should be calculated from the pdf of the estimation error after the FFT ( ) and the power of the AWGN, but in this case we assume that the log-likelihood ratios can be approximated by the ratios calculated only from the AWGN. It will be shown in the next section that the performance loss of ignoring the clipping effects when calculating the LLRs is minor, and this validates the approximation. 53

10−1 Clipping ignored Exact noise dist. 10−2 Bayesian est. No clipping

10−3 BER 10−4

10−5

10−6 1 2 3 4 5 6 7 SNR (dB)

Figure 2: Performance of LDPC coded OFDM with Cartesian clipping ( ).

5 Results and Discussion

The performance of the LDPC coded OFDM system with Cartesian clipping is shown in Figure 2foran of 4 dB. The rate of the LDPC code is 0.8, the codeword length is 1024 and 64 OFDM subchannels are used. The simulation shows that the performance gains of the two investigated receiver clipping mitigation methods are negligible. The best performance is obtained for the Bayesian estimation, but the improvement is only around 0.1 at bit-error-rate and it seems like the LDPC decoder is not very sensitive to having the exact log- likelihood ratios. The signal-to-noise ratio (SNR) used in this paper is the ratio of transmitted signal power (after clipping) per message bit to the power of the AWGN, that is,

(21) where is the rate of the LDPC code and the power of the transmitted clipped signal is , with ([6])

(22)

Figure 3 shows the pdf of the clipping noise, the pdf of the AWGN and the pdf of the total noise (the convolution) for an SNR of 6 and an of 4 dB. It is seen in the ﬁgure that the 54 PAPER C

3 Clipping noise AWGN 2.5 Clipping noise + AWGN

1.5

Probability density 1

0.5

0 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Noise value

Figure 3: Pdf of clipping noise, AWGN and the sum of the clipping noise and the AWGN for an of 4 and an SNR of 6 . clipping noise does not affect the total noise much, and the total noise has almost the same pdf as the AWGN. The log-likelihood ratios that are the input to the LDPC decoder are calculated from the total noise pdf and the attenuation according to (16), and in this case where the total noise pdf and the AWGN pdf looks almost the same, there will be a minor improvement in performance when the exact LLRs are used instead of just ignoring the effect of the clipping noise. The attenuation is 0.89 for this example. The reason why the effect of the attenuation is minimal is that with a scaling, all LLRs are affected in the same manner.

In systems without error correction, an improvement by the receiver clipping mitigation methods is observed for high SNR, see [6][7]. When using LDPC codes, these high SNRs are never used since even a moderate SNR gives low enough bit-error-rate, at least in a wireless system. It seems like the redundancy added by the LDPC code is effective also in mitigating the clipping distortion. The results suggest that at the receiver side in a system with LDPC coding, the clipping distortion can be assumed to be more AWGN with just a negligible performance loss.

6 Conclusions

Unexpectedly, it is shown that for bit-error-rates of down to that are reasonable in a wireless system, the clipping mitigation methods using extensive calculations and aprioriinforma- tion give little improvement on the performance of an LDPC coded OFDM system, compared to just ignoring the clipping effects. These results can be explained in part by noticing that the 55 changes in the initial log-likelihood ratios due to the clipping mitigation methods are small. Our results should imply directing attention to transmit-oriented clipping mitigation strategies. 56 REFERENCES

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