Modulation and Channel Effects in Digital Communication991367/...Modulation and Channel Effects in...

LICENTIATE T H E S I S

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

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Modulation and Channel Effectsin Digital Communication

Sara Sandberg

Modulation and Channel Effects in DigitalCommunication

Sara Sandberg

Dept. of Computer Science and Electrical EngineeringLulea University of Technology

Lulea, Sweden

Supervisor:

James P. LeBlanc

To Marcus and Miriam

ABSTRACT

This thesis investigates three possible methods to increase the performance of digital communi-cation systems, with focus on wireless systems, by accounting for some of the channel effectsthat may occur. The modulation scheme plays an important role in the impact of differentchannel effects on system performance and this work considers both a single-carrier systemand orthogonal frequency-division multiplexing (OFDM).

The first work investigates effects of the channel estimation errors resulting from blindchannel estimation. The performance of a communication system in terms of throughput maybe increased by using blind channel estimation instead of non-blind. This will allow more use-ful 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 estimationmethod. This work quantifies the expected performance reduction, in terms of cross-channelpower, due to channel estimation errors.

The second and third work consider the OFDM framework, which enables simple equal-ization 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 transmittedsignal. These problems can be alleviated by pre-multiplying the OFDM block by a spreadingmatrix, e.g. the Walsh-Hadamard matrix. It is shown that spreading by the Walsh-Hadamardmatrix 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 spreadOFDM 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 anOFDM system with low-density parity-check (LDPC) coding, the cost of completely ignoringthe clipping effects in the receiver is minimal.

v

CONTENTS

INTRODUCTION 11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Low-Density Parity-Check Codes . . . . . . . . . . . . . . . . . . . . . . . . 33 Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

PAPER A 191 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Separation with Known Mixing Matrix . . . . . . . . . . . . . . . . . . . . . . 234 Blind Identification Based on Cumulant Subspace Decomposition . . . . . . . 245 Cost of Blindness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

PAPER B 311 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 The OFDM System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 LDPC codes for OFDM and SOFDM . . . . . . . . . . . . . . . . . . . . . . 364 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

PAPER C 451 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 System Description and Channel Model . . . . . . . . . . . . . . . . . . . . . 483 Characterization of Clipping Noise . . . . . . . . . . . . . . . . . . . . . . . . 504 Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

ACKNOWLEDGMENTS

The first person I would like to express my gratitude to is my supervisor Professor JamesLeBlanc. Thank you for convincing me that I would find the Ph.D. studies fun and for theguidance 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 thinkingof what is the best for me and my future and giving that the highest priority. Also, manythanks go to my assistant supervisor Professor Bane Vasic from University of Arizona, that hassupported me with his coding expertise and interesting ideas. I really look forward to visit youand your group this autumn.

I would also like to thank all my colleagues in the signal processing group. All togetheryou 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 thelife 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 ofthe 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 yourencouragement 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 SandbergLulea, August 2005

ix

Part I

INTRODUCTION

1 Introduction

This thesis addresses methods to reduce the impact on system performance of some channeleffects that may occur in a digital communication system. To set the stage for the researchpresentation, this section gives a short introduction to the principles of a digital communicationsystem. This will serve as the overall picture when more details of the communication systemand the research are presented in the following sections. There are many textbooks on thesubject of digital communication that the reader can refer to for a thorough introduction, seefor example [1][2][3].

Figure 1 shows the main elements of a digital communication system. The informationsource is assumed to be in digital form and possibly encoded by a source encoder which isomitted 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 bythe channel decoder to reduce the impact of channel effects as noise and interference and theresult is increased reliability of the received data. One common way of adding redundancy isblock 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 relationbetween the number of information bits and the number of codeword bits and the code rateis 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, whichmaps the bits to appropriate signal waveforms. The simplest form of modulation is to mapeach binary zero to one waveform and each binary one to some other waveform that is easy to

1

2 INTRODUCTION

Informationsource encoder

Channel Digitalmodulator

SISO/MIMOchannel

Channeldecodersignal

Output

Channelestimator

Demodulator,separator and

equalizer

Figure 1: A basic digital communication system.

distinguish from the waveform representing the zero. This is called binary modulation and isone form of single-carrier modulation. In the last decade, multi-carrier systems have becomemore popular and especially orthogonal frequency-division multiplexing (OFDM) has receivedmuch attention. With multiple carriers the superposition of several waveforms representingseveral bits are transmitted at the same time. Section 3 describes modulation and especiallyOFDM more.

In wireless communication, the modulated waveforms can be transmitted into the com-munication channel by one or several transmitting antennas. A system with one transmittingantenna and one receiving antenna is called a single-input single-output (SISO) system, whilesystems with several transmitting and receiving antennas are called multiple-input multiple-output (MIMO) systems. Multiple transmitting and/or receiving antennas will increase thespatial diversity and can be used to combat channel effects without increasing the bandwidthof the transmitted signal.

The communication channel model represents the physical medium that connects the trans-mitter with the receiver. This medium can be the atmosphere as well as wire lines, opticalfibers, etc. All received waveforms will be more or less corrupted due to thermal noise fromelectronic devices, non-linear distortion in the high power amplifier, interference from othertransmissions, atmospheric noise, fading, etc.

At the receiver side of the digital communication system, there are one or more receivingantennas. Each antenna receives a weighted and possibly filtered sum of the different transmit-ted waveforms. The digital demodulator processes these signals and produces a binary streamagain. 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 bereduced by an equalizer.

The demodulated signal is passed to the channel decoder that uses the redundancy addedby 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. Theoverall aim of the digital communication system is to transmit as much information as possiblefrom the transmitter side to the receiver side with as few errors as possible. The performancemeasure of the digital communication system is the average frequency or rate with which errorsoccur (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, buta channel estimate is often of use internally in the system, for example in the separator andthe channel decoder. With a good channel estimate, both separation and decoding can performbetter.

This thesis investigates three possible methods to increase the performance of digital com-munication systems, with focus on wireless systems, by accounting for some of the channeleffects that may occur. The modulation scheme plays an important role in the impact of differ-ent channel effects on system performance and this work considers both a single-carrier systemand orthogonal frequency-division multiplexing (OFDM). The research presented in the the-sis is separated into three parts, each represented by one research paper. One paper relatesto the modulation while the other two investigate how channel effects (clipping and channelestimation 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 theOFDM system sensitive to some of the channel effects are considered. In Section 4, the im-portance of channel estimation and some channel estimation methods are discussed. The threepapers that are included in the thesis are summarized in Section 5. Finally, conclusions andideas 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 LDPCcodes, which can be decoded by a simple iterative algorithm and have been shown to achieveinformation rates up to the Shannon limit, [5].

LDPC codes were originally discovered by Gallager [4], in 1962. He introduced blockcodes specified by a very sparse parity-check matrix created in a random manner. The maincontribution of Gallager was however a non-optimum but simple iterative decoding schemefor the LDPC codes, with complexity increasing only linearly with block length, that showedpromising results. In the following years there were very few papers in this field, but Tannerintroduced the bipartite graph to this problem in 1981 and used it both for code constructionand for decoding (with a generalization of Gallager’s iterative algorithm), [6]. Around 1996,

4 INTRODUCTION

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 practicalperformance of Gallager’s original work from 1962 would have broken practical coding recordsup 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 corre-sponds to the elements of the codeword are called variable nodes, while the other class thatcorresponds to the parity-check constraints are called the check nodes. Figure 2 shows an ex-ample of the bipartite graph for a regular LDPC code of length 10, with the parity-check matrixgiven by

(1)

Every edge in the graph connecting a variable node with a check node corresponds to a onein the parity-check matrix. Both the bipartite graph and the parity-check matrix give a fullspecification of the code on its own. The parity-check matrix given by (1) for the examplecode of length 10 is not really sparse and in practice the codeword length ranges from a fewhundred to several thousand bits.

The construction of the code can be random (like Gallager’s) or structured. Irregular LDPCcodes constructed randomly under certain conditions (e.g. with a given degree distributionof 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]. Inboth random and structured constructions of the code, effort is spent to avoid short cycles in thebipartite graph. The length of the shortest cycle in the graph, the girth, affects the performanceunder 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 amessage-passing algorithm known as the sum-product algorithm. A good tutorial paper on thesum-product algorithm is written by Kschischang et al., [11]. In short, the sum-product algo-rithm updates the a posteriori probabilities that a given bit in the codeword equals 0 or 1, giventhe received word. In the initialization each variable node is assigned the conditional proba-bility of the codeword bit being a 0 or 1, given only the received value corresponding to thatnode. In order to calculate the conditional probabilities exactly some statistics of the channelmust be known, for example the distribution of the noise and clipping distortion. The initialprobabilities are then updated according to the sum-product rule to incorporate the structure ofthe code. In graphs with no cycles, the sum-product algorithm will yield the correct probabil-ities, 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

Variablenodes

Checknodes

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 cometo 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, hasbeen shown to perform very well for long codes, see for example [7].

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

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

6 INTRODUCTION

N NP/SIFFT

NS/P FFT

NN

AWGN

channel equalizer

Figure 3: The OFDM system.

3 Modulation

The modulator is the element of the communication system that maps the digital informa-tion into analog waveforms, matching the characteristics of the channel. The waveforms maydiffer 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 thetime-domain or the frequency-domain. An N-dimensional signal vector may be transmitted ina time-interval T by dividing the time-interval in N shorter intervals. Alternatively, the fre-quency band may be divided into N frequency slots and the amplitudes of the N correspondingsubchannel carriers can be modulated. One popular modulation method that uses multiple car-riers in this way is orthogonal frequency-division multiplexing (OFDM). The remaining part ofthis 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 OFDMis to convert an intersymbol interference (ISI) channel into parallel ISI-free subchannels. Eachsubchannel will have gain equal to the channel’s frequency response at the corresponding sub-channel frequency. OFDM is implemented by an inverse fast Fourier transform (IFFT) at thetransmitter side and an FFT at the receiver side, see Figure 3. A cyclic prefix of length noless than the order of the channel is inserted between successive blocks to prevent interblockinterference.

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 aslarge peak-to-average power ratio (PAPR) of the transmitted signal and sensitivity to channelfades due to reduced diversity. The large PAPR makes power backoff necessary unless tech-niques to control the non-linear distortion in the power-amplifier are incorporated. These tech-niques 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 theresulting PAPRs are still higher than those of serial single-carrier transmission. Time-domainclipping 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

NN N

AWGN

IFFTspreading channel FFTN filter

Wiener

Figure 4: The spread OFDM system.

selective fading channels. They conclude that single-carrier block transmission is superior toOFDM 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 correcterrors on fading subchannels. Extra diversity can also be added by introducing dependenceamong symbols on different subcarriers. This is performed by linear precoding, also calledspreading, 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 aspreading matrix, Figure 4. Wiener filtering may be implemented at the receiver side to reducethe intercarrier interference introduced by the precoder, resulting in a minimum mean squareerror (MMSE) receiver. For a unitary spreading matrix, the Wiener filter is shown to be simplyscalar channel equalization followed by the inverse of the spreading matrix, [20]. In [21], chan-nel independent precoders that minimize the uncoded bit error rate (BER) in an OFDM systemare derived. It is shown, for QPSK signaling and an MMSE receiver, that the class of optimalprecoders is the unitary matrices with all elements having the same magnitude. Examples ofthis 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 IFFTand FFT in the receiver. The Walsh-Hadamard matrix has been shown to reduce the PAPR ofthe signal, [22], as well as increasing the frequency diversity. Debbah et. al. proposed a newmethod to reduce the intercarrier interference in the receiver for WH spreading, [23]. Theyachieve 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 communicationsystem. If the channel is known to the transmitter, the water-pouring strategy can be appliedto concentrate the transmit power to the frequencies where the channel has relatively littleattenuation, [1]. This is necessary to approach capacity. At the receiver side, the channelinformation is utilized for channel equalization, in iterative decoding with soft information (for

8 INTRODUCTION

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 wherea training sequence is inserted into the transmission and blind methods that exploit various sta-tistical properties of the transmitted signals to carry out channel estimation in the receiverwithout access to the symbols being transmitted. Different non-blind and blind channel es-timation methods are discussed in the following subsections. In the last subsection, channelestimation of MIMO systems is discussed separately. Estimating the channels of a MIMOsystem is usually a much more difficult problem than estimation of a single channel, since thenumber of parameters to estimate is increasing linearly both with the number of transmit andreceive antennas.

4.1 Non-blind Methods

In digital communication systems operating on time-varying frequency-selective fading chan-nels, the data to be transmitted is often organized in blocks preceded by a known trainingsequence. The training sequence at the beginning of the block is used to estimate the chan-nel, train an adaptive equalizer, etc. There exist several methods that can be used to retrieve achannel estimate from the received version of the training sequence. For example, least sum ofsquared errors (LSSE) channel estimation was suggested by Crozier et. al.,[24], where the so-lution for the channel estimate that minimizes the sum of squared errors between the receivedsignal and its approximation using the channel estimate is given. The length of the requiredtraining 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 pilotsymbols. Pilot-symbol-aided channel estimation for systems with fading channels has beenanalyzed in [25]. The transmitter periodically inserts known symbols, from which the receiverderives its amplitude and phase reference. By introducing known pilot symbols in the transmitstream, there is no need to change the transmitted pulse shape and the peak-to-average powerratio stays the same. However, there are two main drawbacks with the pilot-symbol-aidedchannel estimation. One is the delay in the receiver that is required to obtain enough pilotsamples for a good channel estimate. The other is that the receiver interpolates the channelmeasurements provided by the pilot symbols and the interpolation coefficients depend on theposition 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 sin-gular value decomposition, [27]. This estimator can be used either with a training sequence orwith known symbols (pilots) inserted into the transmitted data stream and is a low-complexityapproximation to a linear minimum mean-squared error (LMMSE) estimator that exploits thefrequency correlation of the channel. The number of multiplications required is reduced byapplying 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 orpilot symbols. Instead, the statistical properties of the transmitted signals are exploited tocarry out the estimation or equalization at the receiver without access to the symbols beingtransmitted. 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 forblind equalization and proposed several optimization criteria with the respective stochastic gra-dient algorithms. Tong et. al., [32], proposed a channel identification and equalization methodthat 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 be-tween, called semi-blind methods, [33]. The semi-blind methods exploit information from bothtraining and statistical properties of the transmitted signal, which makes them more robust thanthe 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 usersignal by simultaneously suppressing the intersymbol interference (ISI) introduced from tem-poral distortion and the multiuser interference (MUI) resulting from spatial mixture. One wayto do this is to estimate the channels, use the channel estimates to separate the different usersand then apply any of the equalization methods discussed above on each users signal. Thechannel estimate can of course also be used for equalization since it is already known from theprevious steps.

Blind channel estimation of MIMO systems has been studied by Liang and Ding, [34], Huaet. al., [35], etc. Liang and Ding developed an algorithm to identify the MIMO system fromthe common nullspace of a set of fourth-order cumulant matrices of the channel outputs. Huaet. 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 ofMIMO-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 impulseresponse (FIR) systems is investigated. The cost of blindness is considered in terms of the dif-ference in remaining cross-channel power after separation based on blind channel estimationand 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 thechannel is perfectly known. However, the blind channel estimation method is not very sensitiveto 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, alsocalled linear precoding, is investigated with the aim to both increase the frequency diversityof the OFDM system and reduce the PAPR of the modulated signal. This can be done byspreading with the Walsh-Hadamard matrix.

The paper includes simulation results for an LDPC coded OFDM system with and withoutclipping and spreading, using the ETSI channel model for HIPERLAN/2. It is shown thatthere is a performance gain by spreading, especially in systems with clipping. Because of thevery simple structure of the Walsh-Hadamard matrix, the spreading can actually reduce thecomputational 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 perfor-mance of two methods for clipping mitigation in the receiver, which might be of interest if theclipping for some reason can not be avoided in the transmitter, is investigated. One method isBayesian estimation of the transmitted signal from the the clipped and noisy signal. The othermethod is to find the exact distributions of the clipping noise and use this as input to the LDPCdecoder.

Surprisingly, it is shown that the cost of completely ignoring clipping in the receiver isminimal, 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 clippingmitigation methods should be concentrated to the transmitter.

6 Conclusions

In a digital communication system with a wireless link, the channel effects can heavily degradethe system performance since the wireless link is time-varying and may experience multipathfading and interference. Channel effects like deep fades might be hard to avoid, while for exam-ple clipping in the high power amplifier can usually be avoided by using a high-performanceamplifier or by power backoff. However, this may not always be the preferred way and weseek other solutions to the channel effect problems. Channel estimation plays an importantrole when seeking to minimize the degradation due to channel effects, since many receivermethods 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 prob-ability of clipping and the frequency diversity is reduced since each symbol is transmitted overonly one subchannel. This work includes investigation of receiver clipping mitigation methodsand spreading as a mean to reduce the PAPR and increase the frequency diversity. Also, thesensitivity to channel estimation errors has been investigated for separation of MIMO systems.

The papers included in this thesis show that spreading by the Walsh-Hadamard matrix isa good way to increase the performance and reduce the sensitivity to clipping and fading inan OFDM system with LDPC coding. Walsh-Hadamard spreading can be implemented withless computations than the conventional OFDM system and gives increased performance toreduced complexity. Receiver clipping mitigation methods on the other hand have been shownto give a negligible performance gain in systems with LDPC coding, even though they requireheavy computations.

12

7 Future Work

The research presented in this thesis has been concentrated on investigating performance lossesdue to channel estimation errors and performance gains of methods for reducing channel ef-fects, for some specified methods and systems. LDPC codes have been included in some ofthe systems under investigation, but has not received much attention. Future work will bedirected more towards the LDPC codes and their sensitivity to channel estimation errors andother channel effects.

The channel information needed by the LDPC decoder with a frequency-flat channel islimited to the noise power, that is, only one parameter. For an LDPC coded OFDM systemwith a frequency-selective fading channel, the noise power of each subchannel must be fed tothe decoder for calculation of the initial likelihoods of the different bits. In this case the numberof parameters to estimate becomes much higher. The sensitivity to estimation errors is likelyto be higher in such a system since errors will affect the relative reliability of the different bitsin the first iteration of the decoder.

The LDPC decoder considered so far has been the sum-product algorithm that needs chan-nel information. There exists however also hard-decision decoding algorithms that do not needchannel information. One question is at what point it is better to disregard estimates of thechannel and use a hard-decision decoder instead of a soft decoder (e.g. the sum-product al-gorithm). 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 thatdo not need channel information.

REFERENCES

[1] E. A. Lee and D. G. Messerschmitt, Digital Communication. Massachusetts: KluwerAcademic Publishers, 2nd ed., 1996.

[2] J. B. Anderson, Digital transmission engineering. New York: IEEE Press, Prentice Hall,1999.

[3] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 4th ed., 2001.

[4] R. G. Gallager, “Low density parity check codes,” IRE Transactions on information the-ory, pp. 21–28, Jan. 1962.

[5] D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEETransactions on information theory, pp. 399–431, March 1999.

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

[7] D. J. C. MacKay and R. M. Neal, “Good codes based on very sparse matrices,” Cryptog-raphy and coding. 5th IMA conference (Lecture notes in computer science), pp. 100–111,1995.

[8] N. Wiberg, Codes and decoding on general graphs. Linkoping, Sweden: Dissertation no.440, Dept. of Elect. Eng. Linkping Univ., 1996.

[9] T. J. Richardson, M. A. Shokrollahi, and R. L. Urbanke, “Design of capacity-approachingirregular low-density parity-check codes,” IEEE Transactions on information theory,pp. 619–637, Feb. 2001.

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

13

14

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

[12] T. J. Richardson and R. L. Urbanke, “Efficient encoding of low-density parity-checkcodes,” IEEE Transactions on information theory, pp. 638–656, Feb. 2001.

[13] S. J. Johnson and S. R. Weller, “A family of irregular LDPC codes with low encodingcomplexity,” IEEE Communication letters, pp. 79–81, Feb. 2003.

[14] B. S. Krongold and D. L. Jones, “PAR reduction in OFDM via active constellation ex-tension,” IEEE Transactions on broadcasting, pp. 258–268, Sep. 2003.

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

[16] H. Ochiai and H. Imai, “Performance analysis of deliberately clipped OFDM signals,”IEEE Transactions on communications, pp. 89–101, Jan. 2002.

[17] K. R. Panta and J. Armstrong, “Effects of clipping on the error performance of OFDMin frequency selective fading channels,” IEEE Transactions on wireless communications,pp. 668–671, March 2004.

[18] Z. Wang, X. Ma, and G. B. Giannakis, “OFDM or single-carrier block transmissions?,”IEEE Transactions on communications, pp. 380–394, March 2004.

[19] G. W. Wornell, “Spread-response precoding for communication over fading channels,”IEEE Transactions on information theory, pp. 488–501, March 1996.

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

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

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

[23] M. Debbah, M. de Courville, and P. Maill, “Multiresolution decoding algorithm forWalsh-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 chan-nels,” IEEE Transactions on vehicular technology, pp. 686–693, Nov. 1991.

[26] Y. Li, “Pilot-symbol-aided channel estimation for OFDM in wireless systems,” IEEETransactions 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 chan-nel 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-modulationsystems,” IEEE Transactions on communications, pp. 679–682, June 1975.

[29] D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional datacommunication 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 mod-ulus 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 phasesystems,” 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 subspacedecomposition,” 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.

Part II

PAPER A

Performance Degradation Due toBlindness in Separation ofMIMO-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 BLINDNESSIN 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, separationtechnique 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 identificationis 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 minimizesthe cross-channel power. The performance penalty due to blindness is investigated for the casewhen the channel order is underestimated. Results of average residual cross-channel power ofthe wireless COST207 channel model are included.

1 Introduction

Blind source separation techniques in wireless communications have been under active re-search 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 interfer-ence in a single cell to interoperator interference. Blind source separation (BSS) can be used tolower the impact of interference on transmission quality as well as increase the capacity of thesystem in the uplink, [1, chap. 8.8]. Much of the work in the area of BSS addresses the caseof instantaneous mixtures. However, in wireless communications multipath is usually present,which yields convolutive mixtures. Also, only mixing with sufficiently narrowband signals canbe approximated as instantaneous, while the more common wideband case leads to convolu-tive 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-channelequalization methods, such as the constant modulus algorithm [2], [3], or by minimization ofany of the criteria proposed in [4].

In BSS with convolutive mixtures, the sources are separated and equalized without knowl-edge 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 noinformation. However, the performance of the separation is likely to degrade when less a pri-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 di-

rect approaches the separated signals are extracted without explicit identification of the mixingsystem, while indirect methods identify the unknown channels before separation and equaliza-tion. A recently proposed method for system identification, [5], exploits second order statisticsand decorrelates subchannels of the system. In [6], higher order cumulant matching is usedto estimate the channel and a Wiener filter separates the independent sources. Another recentmethod uses higher order statistics and subspace analysis to identify the MIMO finite impulseresponse (FIR) system assuming a known channel order, [7]. This is the method consideredfurther here.

This paper investigates the separation performance degradation when the mixing systemis 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 themixing system inverse (known or estimated). The performance loss due to blindness is alsoinvestigated when the channel order (that must be known or estimated both for the blind iden-tification and the calculation of the separating system) is underestimated, a common case ofpractical interest.

2 System Model

We focus on MIMO-FIR systems as in Figure 1, where observed signals, x , are the outputof a linear channel

x A s n (1)

The source signals, s , are assumed to be statistically independent and the number of sourcesignals 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 sourcesignals. The vectors x , s and n are length vectors. The channel order of the mixingsystem 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 canbe written as

y B A s n (4)

23

++

Mixing Separation

A B

n

s x y

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 separatingmatrix B by taking the FIR part of the inverse of A . The inverse is given by

AA

A (7)

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 equationcan be seen as the separating matrix and the factor A as the IIR filter that equalizesall channels after separation. Full separation can be achieved if the order of the separatingsystem is at least as large as the order of the filters in A , that is, 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 filtersof the separating system to be the first taps of the filters in A . This selection ofB 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 a priori it can be blindly estimated from the observedsignals by a recently proposed method based on cumulant subspace decomposition, [7]. Theproposed 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 pro-cesses 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, butthis is not considered in the summary given here.

The main idea is to identify A given only the observed signals x by employing thefourth-order cumulants. These higher order statistics of x give enough information to iden-tify the mixing filters up to an arbitrary scaling and permutation and are also not affected by ad-ditive 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 andfourth 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 diagonalizationof a set of matrices.

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

25

calculationCost

calculationCost

Comparison

Calculation of

matrixseparation

Calculation of

matrixseparationBlind system

identification

blindnessCost ofA

A B

A Bx

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 mixingmatrix 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 mixingmatrix A is compared with the blind case where the mixing matrix is estimated to A usingthe blind identification method described in section 4. A separation matrix B is calculatedfrom A and the cross-channel power of the non-blind system B A is compared with thecross-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 channelorder that is tractable to assume for blind identification, [8]. Therefore, we herein investigatethe cost of blindness when the channel order is underestimated (when ), to give an under-standing 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 systemmodel has two sources and two sensors and the channel order is for the four channels. Thechannels are randomly generated based on a modified COST207 bad urban wireless channelmodel [9], where the symbol rate has been increased to give channels with an order of . Thesource signals are mutually independent, temporally i.i.d. QPSK signals and the channel noisesare zero-mean, complex, additive white Gaussian processes.

The MIMOCS-JD algorithm, [7], is used for identification of the mixing system. In [7], thesample 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 identi-fication method. A sample support of would keep the ratio of samples to parameters in

26 PAPER A

0 1 2 3 4 5 6 70

2

4

6

8

10

12

14

Estimated channel order

Cro

ss−

chan

nel p

ower

BlindNon−blind

Figure 3: Cross-channel power after non-blind and blind source separation. The true channel order isand 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 tobe 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 perfectseparation when there are sources and sensors, see (8). The simulations are performed atan SNR of .

Figure 3 shows the average cross-channel power, , when is varied from to the truechannel order , for different COST207 channels. The bars show , where is thestandard deviation. As a comparison it can be noted that the average cross-channel power ofthe mixing system,

(10)

for these channels was .

As expected, there is a significant residual cross-channel power due to use of the blind tech-nique and it also exhibits greater variance. However, the blind method, which was originallyintroduced as needing the exact channel order, in fact, is not very sensitive to channel orderunderestimation 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 whenthe channel order is underestimated. The relative cost due to channel order underestimation

27

0 1 2 3 4 5 6 70

5

10

15

20

25

30


Cro

ss−

chan

nel p

ower

red

uctio

n (d

B)

BlindNon−blind

Figure 4: Cross-channel power reduction achieved by blind and non-blind source separation respec-tively. The last value for the non-blind case is omitted since non-blind separation with the true channelorder 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 thechannel order.

To further interpret the results, we define the cross-channel power reduction, , as thecross-channel power of the mixing system over the residual cross-channel power after separa-tion, that is,

(11)

Figure 4 shows the separation gain in terms of cross-channel power reduction. The lastvalue for the non-blind case is omitted since non-blind separation with the true channel orderis perfect, , and the cross-channel power reduction is infinite. This measure of separationperformance demonstrates more clearly the performance penalty due to blindness. The cross-channel power reduction achieved by the blind method does not vary much with estimatedchannel 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. Thedifference is less than . This is again suggesting that the blind method is not especiallysensitive to channel order underestimation.

In practice, the channel may rarely be so slowly varying that it can be assumed to be con-stant for a support of samples. To shed some light on results applicable to the smallsample support case, Figure 5 shows the remaining cross-channel power having support ofonly samples. With this short sample support the parameters of the cumulant matrices

28

0 1 2 3 4 5 6 70

2

4

6

8

10

12

14


Cro

ss−

chan

nel p

ower

BlindNon−blind

Figure 5: Cross-channel power after non-blind and blind source separation. The true channel order isand samples are used for the blind identification.

in the blind method are not very accurately estimated for a high channel order, and the costof blindness increases with an increasing estimated channel order. For this specific simulationa 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 powerin cases of limited sample support.

7 Conclusions

This work investigates the performance penalty of blind separation of a MIMO-FIR systemfor 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 per-formance degradation due to blindness. Previous work on the blind method investigates itssensitivity to channel order overestimation. Herein the effects due to channel order underes-timation are considered, which is important since some form of under-modeling is often thecase. Our simulations based on the COST207 channel model show the expected performancedegradation due to blindness. Interesting to note is that the performance of the blind methodis not very sensitive to channel order underestimation. If the sample support is short, chan-nel order underestimation might even give a lower residual cross-channel power in the blindcase. This suggests that even if the true channel order is known, blind identification with alower channel order than the true may increase the performance of the communication systemin 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 datacommunication 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 modu-lus 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 phasesystems,” 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 un-known additive noise,” IEEE Signal Processing Workshop on Signal Processing Advancesin Wireless Communications, pp. 53–56, April 1997.

[7] J. Liang and Z. Ding, “Blind MIMO system identification based on cumulant subspacedecomposition,” 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-Ubertragungskonzeptefur zellulare Mobilfunksysteme (Ph.D. thesis). VDI Verlag, 1996.

29

PAPER B

Performance of LDPC CodedSpread OFDM with Clipping

Authors:Sara Sandberg, Cormac de Frein, 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 OFDMWITH CLIPPING

Sara Sandberg, Cormac de Frein, 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 Hadamardtransform 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 IFFTtransform (OFDM) and the Walsh Hadamard transform in a single step, the number of oper-ations needed for the spread OFDM system is actually less than for the conventional OFDMsystem. Reducing the PAPR is important in systems with clipping since it is related to theprobability of clips. Each clip introduces clipping noise to the system which reduces the per-formance. Results of a clipped OFDM system with LDPC coding and spreading for an ETSIindoor wireless channel model are presented and compared with other systems. It is shownthat there is a gain by spreading for LDPC coded OFDM systems, and especially for systemswith 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 perfor-mance. With a high data rate compared to the channel bandwidth, multipath propagation be-comes frequency-selective and causes intersymbol interference (ISI). A multicarrier OFDMsystem is known to transform a frequency-selective fading channel into parallel flat-fadingsubchannels if a cyclic prefix is used for preventing inter-block interference. The receivercomplexity is thereby significantly reduced, since the equalizer can be implemented as a num-ber of one-tap filters. In such a system, the data transmitted on some of the carriers might bestrongly attenuated and could be unrecoverable at the receiver. Lately, spread spectrum tech-niques 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 byprecoding 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 codesor turbo-codes and it is shown that there is a significant performance gain offered by introduc-ing 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 etal. 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 amplifierin the transmitter or, alternatively, a large input power backoff. This implies reduced signalpower, degrading bit error rate and for clipping even spectral spreading. There has been muchresearch in the area of reducing the PAPR for OFDM systems, [8][9]. It is shown in [10] thatprecoding by the Walsh Hadamard (WH) matrix reduces the PAPR of the OFDM signal andthe associated reduced probability of clipping distortion will increase the performance of thesystem. This precoding scheme has also been suggested for spreading, [1]. Surprisingly, thejoint WH spreading and OFDM modulation can be performed by one single transformationthat requires less operations than the IFFT alone, [11]. In this paper, the total performance gainof the WH spreading is investigated for an OFDM system with LDPC coding and clipping. Inparticular, the gain in bit-error-rate performance is analyzed for an ETSI channel model andresults for clipped OFDM signals are provided.

The conventional and SOFDM system as well as the channel model are described in Section2 and the application of LDPC codes to OFDM in Section 3. Results of the WH spreadingapplied to clipped and non-clipped OFDM signals follow in Section 4. Finally, Section 5 givessome 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 bewritten

(1)

where is symbol number in a random message stream, also called the frequency domainsignal. 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 thechannel impulse response and is a vector of uncorrelated complex Gaussian random variableswith variance . The output from the FFT is

(3)

where the diagonal matrix gives the frequency domain channel at-tenuations and is the FFT of the noise . The elements of are still uncorrelated complexGaussian 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

N NP/S EQFFT

Y

N N Nh yx

IFFTX

S/P

Figure 1: Conventional OFDM.

practical implementation of OFDM usually uses a cyclic prefix in order to avoid inter-symbolinterference (ISI).

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

(4)

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

(5)

In the following is assumed to be the ( ) WH matrix. The WH transform is an orthog-onal linear transform that can be implemented by a butterfly structure as the IFFT and sincethe WH and IFFT transforms can be combined and calculated with less complexity than theIFFT alone this means that the system complexity is reduced by applying WH spreading. Atthe 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 equaliza-tion followed by the transpose of the spreading matrix, which in the case of unitary spreadingequals the inverse. In the following, the noise power and the frequency domain channelattenuations are assumed to be known.

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

36 PAPER B

X

N N N N

FFTIFFT Gh

Figure 2: Spread OFDM.

and destructive interference. The received signal can be written where isthe amplitude and is the phase. The received amplitude is Rayleigh distributed with theprobability 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 inFigure 3. Two deep spectral nulls are apparent in this example.

In Conventional OFDM, each subchannel is assigned one subsymbol. In the example chan-nel given above, the subsymbols transmitted on the subchannels in the close vicinity of thetwo deep spectral nulls, will have a high probability of error. However, using the spreadingemployed in this work, the information transmitted on each subchannel will be a linear com-bination of the original subsymbols. This means that instead of a few subsymbols beingseverely affected by spectral nulls, several subsymbols are lightly affected. This approachleads 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 describedthrough a graph commonly referred as a Tanner graph [14]. Such a graphical representationfacilitates a decoding algorithm known as the message-passing algorithm. A message-passingdecoder has been shown to virtually achieve Shannon capacity when long LDPC codes areused. In the next paragraph we will describe a specific class of LDPC codes used here. Formore details on message passing decoding the reader is referred to an excellent introduction byKschischang et al. [15].

Consider a linear block code of length defined as the solution-space (in ) of thesystem of linear equations , where is an binary matrix. The bipartite graphrepresentation of is denoted by . contains a set of variable nodes and a set of checknodes, i.e. nodes corresponding to equations in . A variable node is connected with

37

8 16 24 32 40 48 56 64−30

−20

−10

0

10

Sub

chan

nel P

ower

Gai

n (d

B)

Subchannel Index

Figure 3: Example of a wireless channel response following the ETSI indoor wireless channel model forHIPERLAN/2.

a check node if it belongs to a corresponding equation. More precisely, The -th column ofcorresponds to a variable node of the graph , and the -th row of the matrix corresponds toa check node of . The choice of a parity check matrix that supports the message-passingalgorithm 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 havelow implementation complexity.

The general form of the parity check matrix can be written as

......

......

(10)

where each of submatrices , is a power of a permutation matrix (see [19] ). Noticethat 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, denotedby LDPC-SOFDM. The message bits are first encoded to codewords of length and modulatedto QPSK symbols. The modulation symbols are , where is the

38 PAPER B

N N N NEncoding

LDPC FFTIFFT GhXModu−

lation Decoding

LDPC

Figure 4: SOFDM with LDPC coding.

symbol energy. The modulated codeword is partitioned into blocks of samples, where weassume that , and each block is multiplied by the spreading matrix . The spreadsignal is sent through the OFDM system and the Wiener filter. Both soft information fromthe output of the Wiener filter and the SNR for each subchannel are fed to the decoder. Sincespreading averages SNR of different subchannels, a theoretical SNR taking the spreading intoaccount is computed and used for the decoding. The theoretical SNR can be calculated fromthe 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 . For largethe interference-plus-noise at the output of the Wiener filter can be considered to beGaussian, [2], and approximated as noise. The theoretical SNR of subchannel with spreadingis

VarVar Var

(16)

(17)

For conventional OFDM ( ), the SNR is simply

Var(18)

39

10 15 20 2510

−6

10−5

10−4

10−3

10−2

10−1

SNR (dB)

BE

R

Conv. OFDMSOFDMLDPC−OFDMLDPC−SOFDM

Figure 5: Performance of conventional OFDM, SOFDM, LDPC-OFDM, and LDPC-SOFDM for thecase of 64 subchannels.

4 Results

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

(19)

Figure 5 shows the BER performance of different OFDM systems without clipping. TheOFDM system has 64 subchannels and the channel is assumed to be constant during the trans-mission of one codeword. Both SOFDM and LDPC coding give a large gain compared withconventional OFDM, but there is also a gain by spreading of about 2.7 of the LDPC codedsystem at bit-error-rate. Figure 6 shows that the performance does not change much withthe number of subcarriers, which is also the size of the precoder. However, with a large numberof 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 fre-quency domain channel attenuations in [2] are assumed to be independent identically dis-tributed circular complex Gaussian random variables with variance 1. A convolutional encoderwith constraint length 7 is used and the code rate is . The spreading and equalizer is the

40 PAPER B

10 11 12 13 14 1510

−6

10−5

10−4

10−3

10−2

SNR (dB)

BE

RN=64 LDPC−OFDMN=64 LDPC−SOFDMN=512 LDPC−OFDMN=512 LDPC−SOFDM

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 withour LDPC-SOFDM system with rate 0.8 for the same channel parameters as in [2]. Figure 7shows that the LDPC-SOFDM system performs better than the system with the convolutionalcode.

Figure 8 shows the reduction of PAPR that is the result of the WH spreading. The in-stantaneous 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 moreuseful to know the distribution of the instantaneous PAPR. In Figure 8 the overall PAPR hasdecreased 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 reducesthe number of clips. If denotes the input complex signal, the clipping of thebaseband signal can be modeled as , with

forfor

(22)

41

5 6 7 8 9 10 1110

−5

10−4

10−3

10−2

10−1

100

SNR (dB)

BE

R

Conv−SOFDMLDPC−SOFDM

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 . Thetotal gain of spreading is around 4 at bit-error-rate, compared to 2.7 when thereis no clipping, since the PAPR reduction by spreading has reduced the number of clips. Theeffect of clipping noise to the SNR is not taken into account in the decoder. This result suggeststhat 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 consid-ered is the WH transform which actually can reduce the complexity of the system. Resultsfor 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 dueto increased frequency diversity as well as reduced PAPR. The performance is also investigatedfor different number of subchannels and the results show that systems with different numberof subchannels perform almost the same. However, a large number of subchannels increasesthe PAPR which in turn increases the probability of clips. Our results confirm that spread-ing enhances the performance of the OFDM system for the ETSI channel model and showthat especially the performance in a system with clipping is increased, while the complexity isreduced.

42

0 5 10 15 20 25 30 350

2

4

6

(a) PAPR

log 10

(his

t(P

AP

R))

0 5 10 15 20 25 30 350

2

4

6

(b) PAPR

log 10

(his

t(P

AP

R))

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

10 11 12 13 14 15 1610

−6

10−5

10−4

10−3

10−2

SNR (dB)

BE

R

LDPC−OFDMLDPC−SOFDMLDPC−OFDM 2 dB clipLDPC−SOFDM 2 dB clip

Figure 9: Performance of LDPC-OFDM and LDPC-SOFDM with 64 subchannels. The performance isshown 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 independentprecoders,” IEEE Transactions on signal processing, pp. 2369–2380, Sep. 2003.

[2] M. Debbah, P. Loubaton, and M. de Courville, “Spread OFDM performance with MMSEequalization,” 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-complexityturbo-decoding,” IEEE Transactions on wireless communications, pp. 832–842, May2004.


[5] D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEETransactions on information theory, pp. 399–431, March 1999.

[6] A. Serener, B. Natarajan, and D. M. Gruenbacher, “Performance of spread OFDMwith LDPC coding in outdoor environments,” IEEE Vehicular Technology Conference,pp. 318–321, Oct. 2003.

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

[8] P. V. Eetvelt, G. Wade, and M. Tomlinson, “Peak to average power reduction for OFDMschemes by selective scrambling,” Electronics letters, pp. 1963–1964, Oct. 1996.

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

43

44


[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 SignalProcessing, 2000. ICASSP ’00., pp. 3358–3361, June 2000.

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

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



[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.


[18] E. Eleftheriou and S. Olcer, “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. Olcer, “Rate-compatible low-density parity-check codes for digitalsubscriber lines,” Proc., IEEE International Conference on Communications, pp. 415–419, Jun. 2004.

http://www.inference.phy.cam.ac.uk/mackay

PAPER C

Receiver-oriented Clipping-effectMitigation 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-EFFECTMITIGATION 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. Anumber of methods exist that combat large signal peaks in the transmitter. Recently severalmethods have emerged that alleviate the clipping distortion in the receiver. We analyze theperformance of two receiver clipping mitigation methods in an OFDM system with Cartesianclipping and low-density parity-check (LDPC) coding. Surprisingly, the cost of completelyignoring clipping in the receiver is minimal, even though we assume that the receiver hasperfect knowledge of the channel. The results suggest that clipping mitigation strategies shouldbe 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 re-search literature on transmit-oriented signal processing and coding methods to mitigate prob-lems associated with clipping due to these high PAR values, see [1][2][3]. However, manysuch methods aimed at lowering PAR come at the cost of increased transmitter complexity orlowering transmission rate. Thus, one may consider allowing such occasional clips due to highPAR transmit signals and suffer the associated loss due to clipping. Of course, one would stillseek 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 clip-ping loss can be mitigated. Surprisingly, even when using near optimal methods which involvesignificant calculations (i.e. they lie beyond near-term practicality) we find that the mitigationability of such extreme processing is rather limited. To evaluate the limits of receiver orientedprocessing, we choose to combine best known practical coding methods, low-density parity-check (LDPC) codes, with the exact distribution of the clipping distortion. This combinationcould be expected to represent the best practically achievable system performance. It is as-

48 PAPER C

NEncoding

LDPC

NModu−

QPSK

lation

QPSK

Demodu−FFTxIFFT

y zXm Z

lationDecoding

LDPCr

Figure 1: LDPC coded OFDM system with clipping.

sumed throughout the paper that the variance of the additive white Gaussian noise (AWGN) isperfectly known.

Another way to combat the clipping distortion in the receiver is to estimate the signal be-fore clipping from the clipped signal. In [7] a Bayesian estimator for the Cartesian clipper isderived. In this paper we show that for an OFDM system with LDPC coding, the ability tocounter clipping gives an improvement of only 0.1 dB at bit-error-rate. To wit, the cost ofcompletely ignoring clipping in the receiver appears to be minimal. These results are intendedto give insight into the potentially unrecoverable nature of the clipping phenomena and implydirecting attention to transmit-oriented strategies.

2 System Description and Channel Model

The system model used throughout the paper is an OFDM system with Cartesian clippingand 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 LDPCcodes will be discussed in turn.

The OFDM modulation is obtained by applying Inverse FFT (IFFT) to QPSK modulatedmessage (or codeword) subsymbols, where is the number of subchannels in the OFDMsystem. 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 domainsignal. 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 acyclic prefix in order to avoid inter-symbol interference (ISI).

In this paper we focus on the Cartesian clipper that clips and separately and in thefollowing denotes either or , so that 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 clippedsignal is . Following the clipper, complex white Gaussiannoise with variance (each noise component has variance ) is added to the signal. In thereceiver, 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]. AnLDPC code is a linear block code and it can be conveniently described through a graph com-monly referred to as a Tanner graph [9]. Such a graphical representation facilitates a decodingalgorithm known as the message-passing algorithm. A message-passing decoder has beenshown to virtually achieve Shannon capacity when long LDPC codes are used. In the nextparagraph we will describe a specific class of LDPC codes used here. For more details onmessage passing decoding the reader is referred to an excellent introduction by Kschischang etal. [10].

Consider a linear block code of length defined as the solution-space (in ) of thesystem of linear equations , where is an binary matrix. The choice of aparity check matrix that supports the message-passing algorithm is a problem that has beenextensively studied in recent years, and many random [11] and structured codes have beenfound [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]. Noticethat 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 be-tween nodes. The wanted LLRs are the a posteriori probabilities that a given bit in c equals0 or 1 given the whole received word r. The initial LLR for the th node (or codeword bit) iscalculated as

(5)

where denotes the conditional probability of given . When the LLRs are updatedby 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 characterizationgiven in the recent paper [6]. The time-domain OFDM signal is the sum of several sta-tistically independent subcarriers and the I/Q components can be approximated by Gaussianprocesses, invoking the central limit theorem, if the number of subcarriers is large. The pdfof is therefore assumed to be Gaussian with mean zero and variance , where is thepower of each complex frequency-domain symbol .

Knowing that the time-domain signal is approximately Gaussian, the Bussgang theoremcan 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 isdependent only on the input backoff ( ) of the clipper and for the Cartesian clipper definedin (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 theclipping 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 theclipping 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 theFFT. The AWGN after the FFT will have the same variance as before the FFT, due to theunitary 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, whichis 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 isthe convolution of probability density functions. To avoid the convolutions the pdf ofcan be calculated as the product of the characteristic function of in different points. Thecharacteristic 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 clippingnoise can now be written as

(13)

using (11) and (12), [6]. The pdf of is the inverse Fourier transform of , but it isusually easier to find the cumulative distribution function by a numerical method called theFourier series method, [16].

Since the clipping noise pdf is only dependent on the of the clipper, this pdf must becalculated only once for a system with a given and then a lookup table can be created. Inthe system discussed here, the clipping noise pdf is convolved with the AWGN pdf to give thepdf of the total noise for subchannel , denoted . If the channel is frequency flat, thepdf of the total noise will be the same for all subchannels. The initial log-likelihood ratios forthe 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 ofthe received signal given the corresponding codeword bit. In the case where 0 and 1 are equallylikely 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 signalfrom the clipped and noisy signal . In [7], a Bayesian estimator for signals clipped by theCartesian clipper and distorted by AWGN is derived. The Bayesian estimator is the optimalestimator 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 inputsignals,

(18)

where

(19)

(20)

The time-domain signal is estimated from the clipped and noisy and the output fromthe estimator is then demodulated by the FFT. The log-likelihood ratios that are neededfor 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-likelihoodratios can be approximated by the ratios calculated only from the AWGN. It will be shown inthe next section that the performance loss of ignoring the clipping effects when calculating theLLRs is minor, and this validates the approximation.

53

1 2 3 4 5 6 710

−6

10−5

10−4

10−3

10−2

10−1

SNR (dB)

BE

RClipping ignoredExact noise dist.Bayesian est.No clipping

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 Figure2 for an of 4 dB. The rate of the LDPC code is 0.8, the codeword length is 1024 and64 OFDM subchannels are used. The simulation shows that the performance gains of thetwo investigated receiver clipping mitigation methods are negligible. The best performanceis obtained for the Bayesian estimation, but the improvement is only around 0.1 atbit-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 transmittedsignal 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 totalnoise (the convolution) for an SNR of 6 and an of 4 dB. It is seen in the figure that the

54 PAPER C

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

Noise value

Prob

abili

ty d

ensi

tyClipping noiseAWGNClipping noise + AWGN

Figure 3: Pdf of clipping noise, AWGN and the sum of the clipping noise and the AWGN for an of 4and an SNR of 6 .

clipping noise does not affect the total noise much, and the total noise has almost the same pdfas the AWGN. The log-likelihood ratios that are the input to the LDPC decoder are calculatedfrom the total noise pdf and the attenuation according to (16), and in this case where the totalnoise pdf and the AWGN pdf looks almost the same, there will be a minor improvement inperformance when the exact LLRs are used instead of just ignoring the effect of the clippingnoise. The attenuation is 0.89 for this example. The reason why the effect of the attenuationis 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 mitigationmethods is observed for high SNR, see [6][7]. When using LDPC codes, these high SNRs arenever used since even a moderate SNR gives low enough bit-error-rate, at least in a wirelesssystem. It seems like the redundancy added by the LDPC code is effective also in mitigating theclipping 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 performanceloss.

6 Conclusions

Unexpectedly, it is shown that for bit-error-rates of down to that are reasonable in a wire-less system, the clipping mitigation methods using extensive calculations and a priori informa-tion give little improvement on the performance of an LDPC coded OFDM system, comparedto 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.

REFERENCES

[1] B. S. Krongold and D. L. Jones, “PAR reduction in OFDM via active constellation ex-tension,” IEEE Transactions on broadcasting, pp. 258–268, Sep. 2003.

[2] M. Breiling, S. H. Muller-Weinfurtner, and J. B. Huber, “Peak-power reduction in OFDMwithout explicit side information,” 5th International OFDM-Workshop 2000, Hamburg,Germany, Sep. 2000.


[4] K. R. Panta and J. Armstrong, “Effects of clipping on the error performance of OFDMin frequency selective fading channels,” IEEE Transactions on wireless communications,pp. 668–671, March 2004.

[5] H. Ochiai and H. Imai, “Performance analysis of deliberately clipped OFDM signals,”IEEE Transactions on communications, pp. 89–101, Jan. 2002.

[6] H. Nikopour and S. H. Jamali, “On the performance of OFDM systems over a cartesianclipping channel - a theoretical approach,” IEEE Transactions on wireless communica-tions, pp. 2083–2096, Nov. 2004.

[7] P. Banelli, G. Leus, and G. B. Giannakis, “Bayesian estimation of clipped guassian pro-cesses with application to OFDM,” Proceedings of the European Signal Processing Con-ference (EUSIPCO), Sep. 2002.



57

58


[11] 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.


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

[14] A. Dholakia and S. Olcer, “Rate-compatible low-density parity-check codes for digitalsubscriber lines,” Proc., IEEE International Conference on Communications, pp. 415–419, Jun. 2004.

[15] A. Papoulis, Probability, random variables, and stochastic processes. McGraw-Hill,3rd ed., 1991.

[16] J. Abate and W. Whitt, “Fourier-series method for inverting transforms of probabilitydistributions,” Queuing Systems, vol. 10, pp. 5–88, 1992.

http://www.inference.phy.cam.ac.uk/mackay

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