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Performance and receiver structures for OFDM on Rayleigh fading channels Burchill, William S. 1997

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P e r f o r m a n c e a n d R e c e i v e r S t r u c t u r e s f o r O F D M o n R a y l e i g h F a d i n g C h a n n e l s by William S. Burchill B. Eng. (Electrical) Carleton University, 1981 M. Eng. (Systems and Computer) Carleton University, 1985 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES ELECTRICAL ENGINEERING ,We accept this thesis as conforming to the required standard n THE UNIVERSITY OF BRITISH COLUMBIA March 1997 © William S. Burchill, 1997 in presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract This thesis is a study of performance and receiver structures for Orthogonal Frequency Division Multiplexing (OFDM) on land mobile radio channels characterized by flat or frequency-selective Rayleigh fading. The type of OFDM considered is a parallel modulation scheme in which 'N data symbols are assembled and used to simultaneously modulate N frequency-orthogonal tones, forming the OFDM symbol or block. Relative to a conventional serial modulation scheme of comparable bandwidth, OFDM has an extended signaling interval which allows for channel averaging in fading conditions. However, the channel variation over the duration of an OFDM block impairs the orthogonality of the modulated tones, causing intersymbol interference. ' A new matched filter bound is derived which does not require that the channel remain constant over the signaling interval, and has Doppler frequency as a parameter. Pulse shape, diversity order, and interray correlation may also be varied. The matched filter bound is used to establish analytic performance limits on the probability of bit error for any receiver of uncoded OFDM on flat or frequency-selective Rayleigh fading channels. In contrast to the AWGN channel, the optimal pulse shape used in the receive correlator is time-varying and the transmitter pulse shape affects the probability of error. Using the Maximum Likelihood Sequence Estimation (MLSE) criterion, an optimal receiver for OFDM on flat fading channels is derived. This turns out to require a constraint length L = N — 1, which is generally infeasible due to complexity constraints. However for BPSK OFDM a suboptimal truncated version of the MLSE receiver is able to approach the MFB to within 1 dB for a wide range of Doppler rates. For QPSK OFDM simple truncated MLSE is found to be impractical due to the required constraint length, which is greater than for BPSK OFDM. i i The Minimum Mean Square Error (MMSE) criterion is used to derive optimal linear and nonlinear decision feedback equalizing receivers for OFDM on flat fading channels. A continuous-time analysis is used to show that the optimal linear MMSE receiver requires a sampling rate in excess of N samples per OFDM block, and that the optimal weighting function applied to the received signal to mitigate channel fading is tone-dependent. Approximations are considered to remove the tone-dependency, yielding a result consistent with previous work. An MMSE-based criterion is proposed and used to derive a method for modifying the channel impulse response to an optimal desired impulse response having a specified constraint length. This limits error propagation with decision feedback and reduces the complexity of sequence estimation, making the latter feasible for QPSK OFDM as well as BPSK OFDM. The resulting nonlinear receiver structures have probability of error performances which improve on previously published results for the same modulation and channel. Finally optimal and suboptimal receivers for OFDM on frequency-selective Rayleigh fading channels based on MLSE and MMSE criteria are derived. The additional complexities of receiver design arising from the presence of delay spread are studied analytically and evaluated by simulation. It is shown that for the MLSE receiver, long blocklength OFDM is relatively insensitive to the distribution of signal strengths between the rays of the two-ray frequency-selective Rayleigh fading channel model. iii Table of Contents Abstract » Glossary ix List of Figures • • xii Acknowledgment . xv 1 Introduction 1 1.1 OFDM Modulation 3 1.1.1 Modulation Constellation 6 1.1.2 Bandwidth of OFDM 7 1.1.3 Channel Measurement 7 1.1.4 Effect of Fading on OFDM 8 1.2 Previous Work 9 1.3 Thesis Contribution 10 1.4 Thesis Organization • • • ^ 2 The Land Mobile Fading Channel 14 3 Matched Filter Bounds for Fast Fading Rayleigh Channels . . . . 19 3.1 Introduction 19 3.2 System Model . 20 3.3 Analysis 21 3.4 Extension to A'-Channel Diversity . . 27 IV 3.5 Some MFB Results 28 3.5.1 Flat Rayleigh Fading 28 3.5.2 Frequency-Selective Rayleigh Fading 32 3.6 Conclusion 37 4 A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels 39 4.1 The Transmitter and Channel Model 40 4.2 Derivation of a Maximum Likelihood Receiver for OFDM 40 4.3 FFT Evaluation of Un and Vn 44 4.4 Truncated MLSE 47 4.5 Pulse Shaping 47 4.6 Perfect Channel Knowledge 50 4.6.1 Simulation Framework 50 4.6.2 Rectangular Pulse Shape 51 4.6.3 Raised Cosine Pulse Shape , 53 4.7 Effect of an Imperfect Channel Estimate . 55 4.7.1 Noisy Reference Simulation 59 4.8 QOFDM 62 4.9 Conclusion . 63 5 Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 65 5.1 An Optimal Linear Equalizer 66 5.1.1 Transform Approximation 69 5.1.2 Aliasing Approximation , 72 5.2 A Discrete Ts Spaced Linear Equalizer 73 5.2.1 Optimal Linear Mean Squared Error . . . 74 5.3 Classic Decision Feedback Equalization 75 5.4 Frequency Domain Zero-Forcing Decision Feedback Equalization . . . . . . . 76 5.5 Obtaining a Desired Impulse Response (DIR) . .11 5.6 A Zero-Forcing Desired Impulse Response Decision Feedback Equalizer (ZFDIRDFE) 79 5.6.1 Optimal Mean Squared Error 82 5.6.2 Algorithm Summary 82 5.6.3 Structural Comparison to FDZFDFE . 82 5.7 A Minimum Mean Squared Error Desired Impulse Response Decision* Feedback Equalizer (MSEDIRDFE) 83 5.7.1 Minimum Mean Squared Error 90 5.7.2 Algorithm Summary' 91 5.7.3 Structural Comparison 91 vi 5.8 Combined Mean-Squared Error Desired Impulse Response Shaping and Sequence Estimation (MSEDIRSE) 92 5.9 MSE Performance Surfaces 93 5.10 Simulation Results 95 5.10.1 Feedback Data 96 5.10.2 Comparison of ZFDIRDFE and MSEDIRDFE 96 5.10.3 Evaluation of MSEDIRDFE . 97 5.10.4 Comparison of MSEDIRDFE to Previous Work: FDZFDFE . . . . . 100 5.11 MSEDIRSE Simulation Results 103 5.12 Conclusion 105 6 OFDM on Frequency-Selective Rayleigh Fading Channels . . . . 107 6.1 A Frequency-Selective Rayleigh Fading Channel Model 108 6.2 Derivation of the MLSE Receiver 109 6.2.1 Evaluation of Un and Vn Ill 6.2.2 Pulse Shaping and the Tone Correlations 113 6.2.3 Simulation Results 115 6.3 Derivation of an MMSE Linear Receiver . . .121 6.3.1 Direct Implementation 124 6.3.2. Alternate Implementation 124 6.3.3 Evaluation of Um and Vm . . 125 6.3.4 Relation between Integral Equation and Matrix Formulations 126 6.3.5 An Efficient Matrix Formulation for Rate N/To Sampling . . . . . . . 129 6.3.6 Simulation Results 133 vii 6.4 Conclusion 139 7 Conclusion and Suggestions for Future Work 142 7.1 Conclusion • 142 7.2 Suggestions for Future Work. 143 Appendix A MFB Details 146 A.l Evaluation of an Integral . 146 A.2 Mean and Variance of y in Flat Rayleigh Fading 148 A.3 Selection of M '. 151 Appendix B The DFFT and the iDFFT 152 Appendix C Derivation of a Metric Recursion 153 Appendix D Matrix Notation Conventions . . . . . . . . . . . . . . . 154 Bibliography 156 viii Glossary Acronyms AWGN Additive White Gaussian Noise BOFDM BPSK OFDM BPSK Binary Phase Shift Keying DIR Desired Impulse Response DFE Decision Feedback Equalization DFFT Discrete Frequency Fourier Transform DFT Discrete Fourier Transform FEC Forward Error Correction FFT Fast Fourier Transform GSM Groupe Special Mobile iDFET inverse DFFT ISI Inter-Symbol Interference MFB Matched Filter Bound MLSE Maximum Likelihood Sequence Estimation MMSE Minimum Mean Square Error MSE Mean Square Error kbaud kilosymbols/sec OFDM Orthogonal Frequency Division Multiplexing PAM Pulse Amplitude Modulation PCS Personal Communications Systems QOFDM QPSK OFDM QPSK Quadrature Phase Shift Keying SNR Signal to Noise Ratio ZF Zero-Forcing ix Symbols ak data symbol Ea energy of a data symbol Eb energy of a data bit fc carrier frequency ID maximum Doppler frequency / spread IN, normalized maximum Doppler frequency fs sampling frequency M-) zeroth order Bessel function of the first kind L sequence estimator constraint length N number of data symbols in an OFDM block N0 power spectral density of AWGN Pb probability of a bit error Pe probability of a bit or symbol error Ps probability of a symbol error P(t) transmitted pulse shape received pulse shape r(t) received signal s(t) transmitted OFDM block / signal At sampling interval To nominal OFDM block period Un matched filter output v„ tone correlation w(t) output of a noiseless channel z(t) complex Gaussian random process z(a,t) impulse response of a linear channel Greek Symbols a rolloff factor of a raised cosine pulse amplitude scaling factor of fading waveform 6(t) delta function fin unit sample function noise to signal ratio No/Ea p correlation coefficient a2 variance T delay Tmax maximum delay spread A Viterbi metric / eigenvalue xi L i s t o f F i g u r e s ^ Figure 1.1 Parallel QAM and FFT based OFDM transmitters 4 Figure 1.2 Definitions of To and T . 6 Figure 2.1 BPSK on AWGN and Rayleigh channels 15 Figure 3.1 System Model 21 Figure 3.2 Equivalent system in terms of an AWGN channel. 23 Figure 3.3 Effect of normalized Doppler rate and pulse shape for fast flat Rayleigh fading 31 Figure 3.4 Effect of rolloff on Pj 32 Figure 3.5 Effect of delay spread and correlation between rays, for two-ray very slow Rayleigh fading 35 Figure 3.6 Effect of delay spread and correlation between rays, for two-ray fast Rayleigh fading. Normalized Doppler rate = 0.64.. . . . . 37 Figure 4.1 OFDM transmitter and channel model 41 Figure 4.2 Variation of P& with a for N = 32 tones and various rolloff factors. Matched filter reception, no MLSE applied 50 Figure 4.3 Ph of MLSE vs. L compared to the MFB for fN = 0.32 . . . . 52 Figure 4.4 Pb of MLSE vs. L compared to the MFB for fN = 1.28 . . . . 52 Figure 4.5 Effect of pulse shaping on the P& of an MLSE for /jy = 0.32 and L = 3 . . . . . . 54 Figure 4.6 Effect of pulse shaping on the Pj of an MLSE for = 1-28 and L = 3 54 Figure 4.7 Variation of d\k with Nj for several values of (f,P,jft^- • • • 59 Figure 4.8 Variation of SNR' with / for various (7^ , /? ) pairs 60 Figure 4.9 Effect on Pj, of a noisy vs. ideal channel reference! 60 xii Figure 4.10 Limitation of truncated MLSE applied to QOFDM for fN = 0.32 .62 Figure 5.1 Classic DFE structure 75 Figure 5.2 Forcing a DIR of fixed length before DFE 78 Figure 5.3 DIR obtained by concatenating ZF and FIR filters . . . . . . . 78 Figure 5.4 Normalized optimal mse surfaces for LMSE, ZFDIRDFE, and MSEDIRDFE at N = 64 and e = 0.1; shallow fades . . . . . . 94 Figure 5.5 Normalized optimal mse surfaces for LMSE, ZFDIRDFE, and MSEDIRDFE at N = 64 and e = 0.001; deep fades 94 Figure 5.6 Variation of Ps with B for ZFDIRDFE and MSEDIRDFE, fN = 1.28, EB/N0 = 16 dB, QOFDM 97 Figure 5.7 Comparison of MSE, ZFDIRDFE, and MSEDIRDFE, JN = 1.28, QOFDM 98 Figure 5.8 Variation of Ps with B for MSEDIRDFE, fN = 0.32, EB/N0 = 22 dB .99. Figure 5.9 Variation of Ps with B for MSEDIRDFE, 'fN = 1.28, EIJNQ = 16 dB 99 Figure 5.10 Comparison of LMSE and MSEDIRDFE, B = l,fN = 0.32. 101 Figure 5.11 Comparison of LMSE and MSEDIRDFE for fN = 1.28.... 101 Figure-5.12 Comparison of LMSE and MSEDIRDFE to MLSE, B = 1, fN = 0.32. 102 Figure 5.13 Comparison of LMSE and MSEDIRDFE to MLSE, fN = 1.28. 102 Figure 5.14 Comparison of MSEDIRDFE to FDZFDFE, for the parameters of,fN = 2.05, N = 512, QOFDM. : 103 Figure 5.15 Performance of MSEDIRSE, B = 1, fN = 0.32. 104 xiii Figure 5.16 Performance of MSEDIRSE, B = 2, fN = 1.28 . . 104 Figure 6.1 Effect of Delay Spread with Ideal MLSE for BOFDM, TV = 4 116 Figure 6.2 Effect of Delay Spread with Ideal MLSE for QOFDM, N = 4 116 Figure 6.3 Effect of Delay Spread on Constraint Length for fjy = 0.32 . 117 Figure 6.4 Effect of Pulse Shaping on Constraint Length for /yv = 0.32 119 Figure 6.5 Effect of Pulse Shaping on Constraint Length for — 1.28 119 Figure 6.6 Effect of Power Distribution Between Rays for fx = 0.32 . 120 Figure 6.7 Effect of Power Distribution Between Rays for = 1.28 . 120 Figure 6.8 Effect of Delay Spread for Ideal MSE, for fN = 0.04 and BOFDM 134 Figure 6.9 Effect of Delay Spread for Ideal MSE, for fN = 0.04 and QOFDM 134 Figure 6.10 Effect of Delay Spread for Ideal MSE, for fN = 0.32 and QOFDM . 135 Figure 6.11 Proximity of the Matrix Formulation Results to the Ideal MSE Results 136 Figure 6.12 Effect of Delay Spread for fN = 1.28 136 Figure 6.13 Comparison of the MFB to the IEMSE and MLSE Receivers for fN = 0.04. . 138 Figure 6.14 Comparison of the MFB to the MSE and MLSE Receivers for fN = 0.32 and fN = 1.28. 139 Figure 7.1 IOFDM 144 Figure 7.2 Comparison of BIOFDM with ideal interleaving to BOFDM with fx = 1.28 on a flat fading Rayleigh channel 144 xiv Acknowledgment This thesis is dedicated to my parents, William Joseph and Joyce Eleanor (Meikle) Burchill for getting things started. Many friends and colleagues in Ottawa and Vancouver who encouraged my graduate studies were influential. In particular I would like to thank my thesis advisor Dr. Cyril Leung for his constant patience and critical thinking. Thanks also go to the members of my thesis examination committee, and to the external examiner for their time and effort. This work was supported in part by a B.C. Science Council Great Award, and NSERC grant #OGP0001731. XV Chapter 1 1 Introduction We are at the dawn of the era of universal wireless personal communications systems (PCS). The goal of PCS is to provide access to a global communications network without regard to location or mobility. These systems represent the third generation of public wireless services and will be completely digital [1]. It is unlikely that any single technology can be applied to all areas of PCS because of the fundamentally different economic and technical issues of low-tier and high-tier PCS situations. Low-tier applications are characterized by low power and low complexity, for example a hand held cordless telephone for pedestrian use. High-tier applications are characterized by large macrocells and high-speed mobility, the most important example being communication with a vehicle over land mobile radio channels. Communications devices may be optimized for a particular tier [2] but intercommunicate when appropriate, such as having a pedestrian phone relay its information through a high-tier device when used inside a vehicle. From the perspective of modulation and detection, it is the high-tier situation, and in particular the vehicular digital mobile radio, which presents the most interesting and challenging technical problems. This is due to the severe and random nature of the channel distortion encountered. Land mobile radio channels suffer from random signal fading which corrupts both the amplitude and phase of the received signal, causing dropouts whose frequency varies with the vehicle's speed, and there can be significant delay spread introducing random frequency selectivity. Assuming that the transmitter cannot anticipate the random channel1 some form of diversity is required to obtain error rates comparable to those attainable on an additive white Gaussian noise (AWGN) channel. 1 In certain situations, e.g. if the channel varies slowly relative to the rate at which its state can be assessed and communicated from the receiver back to the transmitter, some anticipation is possible. Introduction 2 Some early forms of diversity [3] took the form of multiple transmissions or recep-tions in the time, frequency, or less often, polarization planes2. These methods effectively halve the transmission rate. Another approach is to combine forward error correction (EEC) with any non-diversity modulation method. Although not generally viewed as a diversity technique, FEC provides an effective diversity by delocalizing the information content of a specific data symbol. Its bandwidth cost is 1 — Rc, where Rc is the code rate. More recently some researchers have been advocating spread-spectrum modulation which has inherent frequency diversity in combination with FEC [4]. Its bandwidth cost or gain remains a subject of debate [2]. A novel method for dealing with the impairments of the land mobile radio channel was proposed in [5]. The idea was to use a version of orthogonal frequency division multiplexing (OFDM) with signaling elements chosen long enough to enable channel averaging, a type of inherent time diversity essentially without bandwidth cost, except for a small portion reserved for channel measurement. Simulation results presented in [5] showed substantial gains for channel-averaging OFDM over conventional modulation techniques without FEC in fast fading conditions. Motivated by the results of [5] and the importance of the land-mobile radio channel, this thesis is concerned with the further development of OFDM theory. In particular we seek to establish the best performance achievable with this technique. We begin by deriving a new matched filter bound (MFB) for fast Rayleigh fading channels. This bound is unique in that it does not assume the fading channel remains constant over the duration of a signaling element, and establishes analytic limits on the probability of bit error (Pj) performance of any uncoded OFDM receiver with the maximum fading rate as a parameter. In subsequent chapters, several new receiver structures are derived via maximum-likelihood sequence estimation (MLSE) and minimum mean-square error Space diversity is does not halve the transmission rate. Introduction 3 (MMSE) optimality criteria. Their Pj, or probability of symbol error (Ps) performances are evaluated by simulation for mutual comparison and for comparison to the bounds based on the MFB work. This is done for both flat (i.e. non-frequency-selective) and frequency-selective Rayleigh fading channels. The Ps performances of the receiver structures derived in this thesis meet, or in most cases substantially exceed previously published results on channel-averaging OFDM. The rest of this introduction is organized as follows. OFDM is introduced in Section 1.1 and the effects of fading on it are outlined. The research contributions of this thesis are summarized in Section 1.3. Section 1.4 presents the organization of the rest of this thesis. 1.1 OFDM Modulation OFDM can be.described as a parallel modulation in which 7Y data symbols are assembled and simultaneously modulated onto N orthogonal carriers. The modulator outputs are summed and the resulting waveform is further shaped by the unit energy pulse p(t) to form s(t), the transmitted waveform of a single OFDM block. This is N-l s(t)=p(t)J^ake:>2tkt (1.1.1) where ak is a data symbol, generally complex, and p(t) is a pulse shaping function. The nominal duration of the block is To = NTS, where Ts is the data symbol interval in a comparable serial scheme having the same signaling rate. The frequency interval between adjacent complex tones is 1/Tb- This is shown in Figure 1.1. The bank of modulators need not appear explicitly in a digital implementation. An efficient method for generating an OFDM signal takes advantage of the inverse Fourier transform relationship between ak and s(t) seen in (1.1.1). If we sample the time-domain waveforms at intervals At, letting xm = x(mAt), then a discrete version of (1.1.1) is N-l akeJTo . (1.1.2) k=0 Introduction 4 Figure 1.1 Parallel QAM and FFT based OFDM transmitters Choose N' At = To, N' > N, and define = 0, k > N. Then sm can be written as N'-l Sm=PmJ^ a ^ k m (1-1.3) k=0 which is clearly an inverse discrete Fourier transform (DFT) [6] of the data post-multiplied by a shaping function. Since N' can be chosen for convenience, such as a power of 2, the fast Fourier transform (FFT) [6] algorithm can be used to evaluate (1.1.3) very efficiently. The choice of At affects the accuracy of the signal representation and is considered in Chapter 4. Previous work on land mobile radio channels [5], [7] has used At — TQ/N with rectangular pulse shaping. Introduction 5 We will also need to consider the more general time-domain root raised cosine pulse shape p(t) given by Pit) 0<t<aTQ (1.1.4) „ (t\ - J i /r 0 , OCTQ < t < T0 Io < t < J o ( l + a ) otherwise where 0 < a < 1 is a rolloff factor describing the percentage of the pulse duration beyond To. The total time interval for which the pulse is nonzero is To(l + a), where To is the time interval between the inflection-points of its leading and trailing edges as shown in Figure 1.2. As subsequent blocks of N data symbols are formed into OFDM blocks, they are transmitted serially over the channel at intervals of T, yielding s.3(t) = Y,s(t-nT) .v-i (1.1.5) n k=0 where ank is the kth data symbol of the nih block. This is also shown in Figure 1.2. We will assume throughout that T > To(l •+ a) so that interblock interference (IBI) is not an issue. Additionally, on delay spread channels we will assume that T>T0(l+a) + rmax (1.1.6) where rmax is the maximum value of delay spread for the same reason. Typically Tmax < 0.01 To so that (1.1.6) is a mild requirement for OFDM. With IBI precluded we can henceforth confine our attention to the transmission and reception of a single OFDM block. To a close approximation3 the bandwidth of OFDM is the same as that of a conventional serial modulation transmitting data symbols at the same rate. This follows 3 At this point we are'not considering overhead for channel measurement, or effects of a and Tmax-Introduction 6 T Figure 1.2 Definitions of To and T from noting that the bandwidth expansion due to transmitting N data symbols on frequency orthogonal carriers is compensated for by the bandwidth reduction due to each data symbol being transmitted N times more slowly. This extension in time of the data symbols is the key to the channel averaging ability of OFDM. We note that the preceding description of OFDM as a parallel modulation scheme is convenient but not required. OFDM could equally well be viewed as a serial modulation with a nominal OFDM symbol duration of To and possibly a very large alphabet of -MN OFDM symbols, where M is the size of the data symbol alphabet. 1.1.1 Modulation Constellation In principle the individual tones comprising the OFDM signal could be modulated with points from any quadrature amplitude modulation (QAM) constellation. However the El/No required to obtain a specified.Pj, where Ej, is the energy per bit and No is the noise power spectral density, increases with constellation size [8]. Since we are primarily interested in low Ei/No environments, we will confine our attention to BOFDM and QOFDM. BOFDM denotes OFDM with binary phase shift keying (BPSK) modulation of the OFDM tones and QOFDM denotes OFDM with quadrature phase shift keying (QPSK) modulation of the tones. Introduction 7 1.1.2 Bandwidth of OFDM The bandwidth of OFDM is readily derived from first principles [9]. The general approach is to first determine the autocorrelation function Of the modulated waveform, which turns out to be a cyclostationary process, and then remove the time dependency by averaging over one period of the process. The data symbols are assumed to be mutually independent and of equal energy. A Fourier transform of the time-averaged autocorrelation function gives the desired power spectral density as Fourier transform of the shaping pulse. Note that T does not appear in (1.1.7); this is a consequence of the independent data assumption. An interesting consequence of (1.1.7) is that the bandwidth of OFDM is only mildly influenced by p(t) when N is large. To illustrate, if we consider two serial modulations, one with pi(t) = smc(t/Ts) and the,other with p2(t) = rect(t/Ts). pulse shaping, and measure bandwidth as the distance between the mainlobe's first nulls, then the bandwidth in the first case is l/TS which is only half that in the second. Similarly, we compare two OFDM modulations, one with pi(t) = sinc(i/Tb) and the other with P2(t) = rect(£/Tb) pulse shaping. The bandwidths for these two cases are N/TQ and (N + 1)/TQ respectively, a factor of (N + l)/N change. Since (N.+ l)/N w 1 for N large, the bandwidth for OFDM is approximately 1/TS in both cases, showing low sensitivity to the pulse shape. (1.1.7) 2" p j 2?r ^ where Ea = E \a].\ is the energy of a data symbol and P(f) = J p(t)e 3t° dt is the 1.1.3 Channel Measurement The type of OFDM system we have been describing is a type of coherent amplitude and phase modulation applied to a channel with randomly time-varying amplitude and Introduction 8 phase characteristics. Therefore optimal detection of the data requires knowledge of the channel parameters at the receiver. We assume this knowledge is obtained by allocating a small portion of the available bandwidth to channel estimation. The idea is to insert known pilot tones or pilot symbols into the transmitted signal. At the receiver these can be used to estimate the channel's impulse response. The amount of bandwidth which must be allocated for channel measurement purposes follows from Nyquist's criterion, which states that the channel must be sampled at a rate higher than This represents only about 2% of the data bandwidth since typically ID < 0M/Ts, e.g. fD = 100 Hz and fs = 1/TS = 10 kbaud. This could be considered a lower limit since it makes no provision for guard intervals to protect pilots from Doppler spread data nor does it provide for mutual pilot signal orthogonality in an interference environment. These issues have been addressed in the literature. Comparative studies of pilot symbol and pilot tone techniques appear in. [10] and [11] in the context of serial modulation and flat fading channels. The Groupe Special Mobile (GSM) standards [12] specify a 26 symbol pilot sequence for estimating frequency-selective channels in an interference environment. The original OFDM for land mobile radio study [5] presents pilot tone based techniques for estimating flat and frequency-selective fading channels. In this thesis we do not develop the channel estimation problem further. The channel is assumed known to the receiver, and the focus is then on optimal detection. An exception occurs in Chapter 4 where an analysis is performed to estimate the deterioration in Pj due to using a noisy versus perfect channel estimate, and the result is verified by simulation. 1.1.4 Effect of Fading on OFDM The extended symbol interval of OFDM that was introduced to provide channel averaging also causes it to suffer a serious impairment on fading channels. Fading impairs the orthogonality of the tones comprising the OFDM signal. The result is a type Introduction 9 of convolution of the data symbols within an OFDM block with a channel-dependent waveform [5], [7]. The convolution introduces intersymbol interference (ISI) amongst the data symbols. It is shown in subsequent chapters that the exact type of convolution depends on the signal representation. Similarly, there is also intrasymbol interference4 when the data symbols are complex. This is generally not a problem for serial modulation schemes because the channel is approximately constant over a symbol's duration. On frequency-selective fading channels the problem is compounded by additional ISI arising from multiple arrivals of the same block, each of which suffers from its own pattern of ISI. 1.2 Previous Work Historically, many names have been used for OFDM, e.g. multichannel modulation, multicarrier modulation, or multitone modulation, depending upon the author. The basic idea is to subdivide a channel into a set of frequency-orthogonal subchannels. The complex exponential tones of the preceding OFDM description are the subchannel carrier frequencies of a multichannel description. An early work on OFDM considered data communication on telephone channels [13]. Use of an FFT to construct the OFDM block was introduced in [14]. Application of OFDM to fading channels was proposed in [15] and [16]. The former considered HF band communications which are slowly fading such that Doppler spread induced ISI was of little importance. The latter considered digital audio broadcasting on land mobile radio channels, but used OFDM blocklengths much shorter than the channel's coherence time. Thus in these previous works the channel is essentially constant during the OFDM block and the motivation for OFDM is not channel averaging but simplified equalization. The simplified equalization follows from observing that individual OFDM subchannels are much narrower in bandwidth than the entire signal, therefore there is relatively less 4 Interference of the in-phase and quadrature components. Introduction 10 change in the channel over the bandwidth of a subchannel as compared to the entire channel bandwidth. In this work the OFDM blocks exceed the coherence time of the channel and OFDM is considered for its channel averaging ability. Only a few publications have been found to date (1997) which explore this approach. The first is [5] which makes the original proposal for OFDM on land mobile radio channels. This work related time-domain fading to frequency-domain ISI5 and proposed simplified equalization structures for flat-fading and frequency-selective fading channels. A channel-averaging variation of OFDM in combination with FM transmission on flat-fading channels was considered in [17]. This work also presented a type of parallel decision feedback called decision feedback correction (DFC). Conventional decision feedback equalization (DFE) for OFDM of the type proposed in [5] was studied in [7]. In this thesis OFDM refers exclusively to coherent OFDM of the channel-averaging type. We study OFDM without FEC so that the effects of fading channels on OFDM can be observed without the artifacts of a particular code choice. 1.3 Thesis Contribution The main contributions of this thesis are listed below: 1. A new matched filter bound for fast flat and frequency-selective Rayleigh fading channels is derived. This bound does not assume that the fading channel remains constant for the duration of a signaling element. It establishes analytic limits for the Pb performance of any receiver for uncoded OFDM, with maximum Doppler rate as a parameter. Arbitrary power levels and correlations between multiple rays due either to delay spread or diversity can be accommodated. Previous results for slow6 fading Rayleigh channels are special cases of this result. 5 What we call ISI was also called ICI in [5]. 6 The term slow is used to refer to a channel whose characteristics remain approximately constant over an entire symbol interval. Introduction 11 2. An optimal receiver for OFDM on flat Rayleigh fading channels is derived using the principles of maximum likelihood sequence estimation (MLSE). A suboptimal version of the MLSE receiver is introduced to reduce complexity and its performance evaluated by simulation. An approximate analysis of the effects on Pj of a noisy channel estimate is presented and verified by simulation. 3. The theory of equalization as modified for OFDM in flat Rayleigh fading is developed, resulting in the derivation of a new decision feedback equalization (DFE) structure having substantial complexity and performance benefits over the current state of the art. A method is devised for modifying the channel impulse response to a desired impulse response (DIR) which could also be applied to conventional serial modulation schemes. 4. Optimal and suboptimal receivers for OFDM on frequency-selective Rayleigh fading channels based on MLSE and MMSE criteria are derived. The additional complexities of receiver design arising from the presence of delay spread are studied analytically and evaluated by simulation. 1.4 Thesis Organization In Chapter 2 background material on the land mobile fading channel, is presented, and the modeling of flat and frequency-selective Rayleigh fading channels with one and two-ray models is described. In Chapter 3, matched filter bounds for fast Rayleigh fading channels are presented. The MFB system model for time-varying channels with AWGN is presented in Section 3.2. An analysis to determine P& is presented in Section 3.3. Section 3.4 extends the results to accommodate if-channel diversity. Examples of the MFB evaluated for flat and frequency-selective Rayleigh fading channels are given in Section 3.5. In Chapter 4, the MLSE receiver for OFDM in flat Rayleigh fading is derived and evaluated. Section 4.1 describes the transmitter and channel model. Section 4.2 presents . Introduction 12 the theoretical derivation of an optimal maximum likelihood receiver, in the sense of sequence estimation. Evaluation of two key parameters, matched filter outputs and tone correlations, by FFTs in a digital implemention and determination of the required sampling rate is examined in Section 4.3. Section 4.4 describes the truncation of the constraint length used by the sequence estimator to make it less complex and Section 4.5 presents a criterion for choosing a non-rectangular pulse shape. Section 4.6 presents simulation results for Pf, as a function of E^/No, of BOFDM for a variety of parameters and compares them to the MFBs of Chapter 3. Section 4.7 develops an approximate analysis to predict the effect of using an imperfect channel estimate. Section 4.8 shows a limitation of the truncated MLSE receiver when applied to QOFDM. In Chapter 5, linear and decision feedback receivers for OFDM in flat Rayleigh fad-ing are derived and evaluated. Section 5.1 derives the optimal linear MMSE equalizer for OFDM in flat Rayleigh fading. Section 5.2 derives a discrete symbol-spaced equalizer, also by the MMSE criterion. Section 5.3 reviews classical DFE as developed for serial modulation and Section 5.4 describes the application of this approach to OFDM in previ-ous work. The idea of reshaping the overall impulse response of the transmitter, channel, and front-end filter7 to some desired impulse response (DIR) is reviewed in Section 5.5. Section 5.6 derives a DIR DFE based on the zero-forcing criterion (ZFDIRDFE) and in Section 5.7, an improved DIR DFE based on the MMSE criterion (MSEDIRDFE) is derived. The combination of a Viterbi-type sequence estimator in combination with MSEDIR shaping is considered in Section 5.8. A theoretical comparison in terms of MSE performance surfaces for a simplified fading channel model appears in Section 5.9, and simulation results for the receivers are presented in Sections 5.10 and 5.11. In Chapter 6 some of our previous work for flat fading channels is extended to frequency-selective channels. Receivers for OFDM on frequency-selective Rayleigh The term front-end filter is defined in Section 5.3. Introduction 13 fading channels are derived and evaluated, based on the optimization of MLSE and MMSE performance criteria. Section 6.1 describes the channel modeling. Derivation of optimal and truncated MLSE receivers is presented in Section 6.2. Methods for evaluating matched filter outputs and tone correlations are given in Section 6.2.1, and the effect of pulse shaping on the tone correlations is considered in Section 6.2.2. Performance results obtained by simulation of the MLSE receivers are presented in Section 6.2.3. In Section 6.3 optimal linear MMSE receivers are derived. Direct and alternate implementations are described in Sections 6.3.1 and 6.3.2. Methods for evaluating matched filter outputs and tone correlations are given in Section 6.3.3. The relationship between the integral equation and matrix formulations of the estimation problem is examined in Section 6.3.4, and an efficient matrix formulation for rate N/To sampling is outlined in Section 6.3.5. The performance results obtained by simulation of the MMSE receivers are presented in Section 6.3.6. In Chapter 7 the conclusions of this thesis are summarized, and some suggestions for future work are presented. Chapter 2 14 The L a n d Mobi le Fading Channel In this chapter background material on the land mobile fading channel is presented, and the modeling of flat and frequency-selective Rayleigh fading channels with one and two-ray models is described. In the case of the land mobile radio channel there is generally no line of sight path, and the transmitted signal propagates to the vicinity of the receiver over multiple paths via reflection and diffraction. Each of the paths has a potentially different attenuation and delay. Time variations of the multipath channel arise from motion of the receiver which alters the significance of different paths. Fading occurs when the receiver moves through an area of destructive interference. The fading is commonly described as either flat, meaning that all frequencies in the signaling bandwidth are similarly affected, or frequency-selective, meaning that different frequencies may undergo different fading. Useful characterizations are the coherence time Atc, a measure of how long the channel stays approximately constant, and the coherence bandwidth Afc, a measure of the bandwidth over which signals spaced in frequency will be similarly affected. Alternatively the Doppler spread and delay spread may be used. The Doppler spread fjy is a measure of the width of the received spectrum when a single sine wave is transmitted through the channel, and the delay spread r is a measure of the width of the received signal in the time-domain when a single impulse excites the channel. It can be shown [9] that Atc « l/fD and.A/c w 1/r. A rule-of-thumb for categorizing a channel as flat or frequency-selective for a digital signal relates the symbol interval to the rms delay spread [18]. Channels with rms delay spread less than about 10% of the symbol interval are considered flat and conversely those with a greater rms delay spread are considered frequency-selective. This is based The Land Mobile Fading Channel 15 10l BPSK rectangular pulse shaping perfect channel knowledge 10' 0 2 4 6 8 10 12 14 16 18 20 22 24 Figure 2.1 BPSK on AWGN and Rayleigh channels on work which shows Pb < 10 - 3 can be obtained without equalization for the channels categorized as flat fading. Fading can be quite detrimental to the average Pb performance of a standard modula-tion format as compared to Pb on an AWGN channel. For example, Figure 2.1 compares the Pb of BPSK on an AWGN channel to its Pb on a Rayleigh flat fading channel. At Pb — 10 - 3, a difference of over 17 dB difference is seen. We will focus on the land mobile radio application with AWGN as the dominant impairment. Although this is somewhat unrealistic in the context of cellular systems which tend' to be interference limited, it does provide insight and has the important advantages of being mathematically tractable and broadly understood. The output of a linear channel with AWGN may be written in terms of its input and impulse response as (2.1) The Land Mobile Fading Channel 16 where s(t) is the channel input, n(i) is AWGN, and w(t) is the channel output. Equation (2.1) is in terms of the complex lowpass equivalents of bandpass waveforms [9], which are used throughout this thesis. The channel impulse response z(a, t) is defined as the response of the channel at time t to an impulse applied at time t — a. In this section, expressions for z(a, t) are sought which model both flat and frequency-selective Rayleigh fading channels. The approach follows that of [9]. The mobile receiver typically receives the transmitted signal as the sum of several attenuated and delayed versions resulting from paths with reflectors and diffractors [3]. The effect of delay on a bandpass signal appears as a delay and phase rotation of its lowpass equivalent [9]. Neglecting noise for the moment, the received signal from a multipath channel is M*)=Yi'Mt>(t-<t)>~i2TfeTn{t) (2-2) n where fc is the carrier frequency of the bandpass signal, and an(t) and rn(t) are the attenuation and delay of the nth path at time t respectively. Comparing (2.1) and (2.2) it is apparent that z{a,t) = Y,Mfy~i2xfcMM°-Tn{t)). (2-3) n Equation (2.3) could be written as a general integral expression to accommodate a continuous distribution of paths, but for simplicity we consider mainly one and two-ray models. Suppose that several of the path delays are closely distributed about a mean delay ri and that the remainder of the path delays are closely distributed about r 2 . By closely distributed it is meant that the differences in path delays are small relative to a symbol interval. The Land Mobile Fading Channel 17 Writing " rni(t) = ri + Arni(t) • (2.4) Tn2(t) = T2 + Arn2(t) and indexing the paths grouped about T\ with n\ and those about T 2 with n 2 we have z(a,r) = e~^f^ ^ a n i ( t ) e - ^ A ^ W % - rx - Arni(t)) (2.5) + e - ^ T s ^ a B 2 ( t ) e - W ^ W % - T 2 - A r B j ( t ) ) . • where f i is the mean of rni(i) over ni and r 2 is the mean of T„ 2(£) over rc2 at some arbitrary time within the duration of an OFDM block. In order to simplify our model, it would be convenient to drop the Ar„.(t) terms in (2.5). To ensure this is reasonable, we upper bound the magnitude of ATni(t). The geometry which causes the greatest change in delay spread during the reception of an OFDM block is the same as that which causes the maximum Doppler spread: motion of the vehicle directly in line with the direction of arrival of the signal. In this situation the path length will change by vT$ over the duration of an OFDM block, where T 0 is the duration of an OFDM block. This establishes an upper bound on |Arn i(i)| of A r m a i = ^Tb, where c is the speed of light. Since A r m a i is several orders of magnitude smaller than To, Arn i(i) can be dropped from the delta functions in (2.5). However because of the large factor fc in the complex exponentials, those terms remain. It is assumed that the phase distribution over the paths is uniform and that there are several terms in each of the summations. Setting T\ = 0 and r = r 2 , and applying the Central Limit Theorem [19], (2.5) approaches . z(a, t) = Zl(t)6(a) + e~^Tz2{t)8(a - r) (2.6) where z\(t) and £ 2(i) are independent complex Gaussian random processes. The ex-ponential rotation factor in (2.6) is a constant; it can be included in the definition of z2(i) without altering its statistics. Additionally, it is convenient to normalize The Land Mobile Fading Channel 18 E 2 L z\(f)\ = E |^ 2(*)l = 1-0 a n d use the explicit factors a\ and a2 to indicate the relative strengths of the rays, yielding for our two-ray model z(a,t) = a1z1{t)S(a) + a2z2(t)6(a-T). . (2.7) The channel described in (2.7) is in general both time varying and frequency-selective; specialized to a Rayleigh flat fading model it requires only one term z{<x,t) = z1(t)8(&). (18) Equations (2.7) and (2.8) are widely used for modeling channels with serial modulations. The purpose of outlining their derivations is to ensure that the assumptions leading to their final forms are not invalidated by the extended duration of the OFDM block. Additional characterizations of Rayleigh fading are derived in [3]. We will find use for both the power density spectrum $Zi(f) and autocorrelation function </>Z,(T) of the process zi(t). They are * « ( / ) = ^ ( 1 - ( Z 7 r ) y , / a . > 9 ) and ^,(r)^J0(2irfDr) (2.10) where Jo(-) is the zeroth-order Bessel function of the first kind. Chapter 3 19 Matched Filter Bounds for Fast Fading Rayleigh Channels 3.1 Introduction Many analyses of digital signaling on a Rayleigh fading channel assume that the Doppler rate is small compared to the symbol rate, allowing the channel to be treated as constant for the duration of a single symbol. This chapter presents a matched filter bound (MFB) analysis of BPSK on a Rayleigh fading channel in which the normalized Doppler rate is unrestricted, including the slowly fading model as a special case. It is shown that the optimal matched filter receiver is time varying, and in contrast to matched filter reception on the AWGN or slowly fading channels, it is found that the probability of error depends on the transmitted pulse shape. Compared to a single carrier modulation of similar bandwidth and bit rate, the duration of an OFDM block period is increased in proportion to the number of carriers used. Previous studies [5], [17] have used simulation to show gains of several dB due to the channel averaging of OFDM on Rayleigh fading channels, where the savings increase with the Doppler rate. The maximum Doppler rate fry encountered in land mobile radio systems operating near 900 MHZ is typically about 100 Hz. For a transmission rate 1/T of about 10,000 symbols/sec, the normalized Doppler rate fx = JDT% is on the order of 1%. At such a low rate the channel characteristics may often be treated as constant over a single symbol period. We will refer to channels for which fpT < 0.01 as slow fading channels. Conversely, a fast fading channel will be defined as one which changes significantly during a symbol period, i.e. j^T > 0.01. Since OFDM systems may use hundreds of 8 We are using T as opposed to To, because at this point we are not necessarily considering just an OFDM block, but rather any symbol of duration T. Matched Filter Bounds for Fast Fading Rayleigh Channels 20 tones, an analysis applicable to fast fading is required. For OFDM systems having up to 512 tones we are interested in a normalized Doppler rate in the range 0 < /jy < 5.12. As a first step, this work considers the transmission of a single pulse over fast fading Rayleigh channels of arbitrary normalized Doppler rate, where it is assumed that the receiver is a matched filter with perfect channel state information. The resulting error rate gives the Matched Filter Bound [20]. Although developed in terms of a matched filter bound, the results presented for cases of zero delay spread are equivalent to a probability of bit error (Pj) analysis for continuous BPSK signaling with perfect channel state information available at the receiver. The analysis also allows for delay spreads due to multiple rays and K channel diversity, where the correlations between rays and between channels are arbitrary and each channel may have a different impulse response. While previous works [20], [21] with the slow fading Rayleigh channel have con-sidered delay spread and channel diversity, the extension to arbitrary Doppler rates as well as the examination of correlations and differing impulse responses, are believed to be new contributions. 3.2 System Model Figure 3.1 shows the system model in terms of the complex baseband equivalents of passband signals. Although we consider BPSK modulation for simplicity, our method is valid for any pulse amplitude modulated (RAM) scheme with minor modifications. The modulator sends a single pulse +Ap(t) over a Rayleigh fading channel whose complex impulse response is given by z(o~, t), which denotes the response of the channel at time t to an impulse applied at time t — a [9]. The pulse pif) has duration To and unit energy, therefore the energy per bit Ei is A2. The real and imaginary components of z\o~,t) are independent zero-mean Gaussian processes of identical variances, and the maximum Matched Filter Bounds for Fast Fading Rayleigh Channels 21 modulator n(t) fading channel demodulator Figure 3.1 System Model delay spread is Tmax. We assume that z(o~,t) is .wide-sense stationary so that we may use its autocorrelation function defined as ',{o-a,ab;ta -h) = E[z(aa,ta)z*(ab,tb)]. (3.2.1) The received signal is given by Tmax r(t) = A J z(cr, t)p(t - a)da + n(t) (3.2.2) where n(t) is zero-mean complex AWGN of variance a\ = N0. The receiver uses r(t) to form the decision variable = Re J r(t)g(t)dt (3.2.3) where g(t) is the pulse shape used in the receiver's correlator, and the integration is performed over the interval for which g(t) is nonzero. 3.3 Analysis Denoting the signal component of by y = Re 'max j j z(a,t)p(t — a)da g(t)dt (3.3.1) Matched Filter Bounds for Fast Fading Rayleigh Channels 22 and the noise component by n\ = Re J n(t)g(t)dt (3.3.2) we have x = Ay + n\. (3.3.3) Given g(t), n\ is a zero-mean Gaussian random variable with variance The probability density function (pdf) of x conditioned on y is Gaussian with mean Ay, Px{x | y) = 1 . *°t. (3.3.5) The probability Pc of a correct decision is the probability that x is greater than zero oo oo f I - 1 -J J y/2Trani -(x-AyY e 2 a n i py{y)dy dx. (3.3.6) 0 —oo Changing the order of integration and recognizing the error function ***)•= ± j o we have for the probability of bit error Pb e~* dt (3.3.7) (3.3.8) To proceed further we must specify the pulse shape g(t) used in the demodulator correlator. It is well known [9] that the optimal demodulator for a pulse transmitted on a nonfading AWGN channel is a matched filter demodulator, where the demodulator filter is matched to the transmitted pulse shape. From the receiver's viewpoint, we observe Matched Filter Bounds for Fast Fading Rayleigh Channels 23 modulator n(t) A W G N channel demodulator Figure 3.2 Equivalent system in terms of an AWGN channel that the fixed-shape pulse Ap(t) entering the fading channel of Figure 3.1 is equivalent to a time varying transmit pulse, Aq(t) = A f z(a,t)p(t — a)da, entering a nonfading. o AWGN channel as shown in Figure 3.2. Then it is clear that the. optimal receive correlator Tmax pulse shape is also time varying, g(t) = J z*(a,t)p*(t — a)da. • o For the matched filter case (3.3.1) and (3.3.2) become Tmax y=.J- J z(a,t)p(t - a)da o = J \l(t)\2dt dt (3.3.9) and ni = Re Tmax j n(t) j z*(a,t)p*(t - a)da dt (3.3.10) where the noise variance is N0 (3.3.11) Note that the variance a\x is itself a random variable. The expression for P& in (3.3.8) can be written as (3.3.12) Since y is nonnegative from (3.3.9), the lower limit in the integral has been replaced by 6. Matched Filter Bounds for Fast Fading Rayleigh Channels 24 We will refer to Eb/N0 as the transmitted signal to noise ratio (SNR). The quantity Eby/N0 is the received SNR at the output of the matched filter, shown with its equivalent implementation as a matched correlator in Figure 3.2. Equation (3.3.12) states that the Pi, for our analysis is given by the average over all received SNRs of the Pb for a static AWGN channel. This expression has the same general form as that which results from a slow fading analysis [9], except that the pdf of y is harder to obtain. The effects of fast fading, correlations, and diversity are reflected solely in how they affect the pdf of y. The average received SNR is given by jbr = E[y]Eb/N0. Taking the expectation of (3.3.9) yields . ' max 1 max Efe/] = J J J <t>z{cra,crb;0)p(t - aa)p*(t - ab)daadabdt. (3.3.13) o o Using the autocorrelation function of the pulse p(t) f(*b - *«) = JP(t ~ <?a)p*(t - crb)dt , (3.3.14) we have ' max ' max Eb] = J J <f>z{aa,o-b-:0)f(ab - aa)dcradab. (3.3.15) 0 0 ' Equation (3.3.15) will be used to obtain more specific results in Section 3.5. To derive the pdf of y, we approximate (3.3.9) arbitrarily closely using a finite summation M y = AtJ2h\2 (3.3.16) i=\ where # = q((i - \) At) and M = T o ( 1 + ^ + r m a * 9 . The |g8|2 are correlated random variables with a chi-squared degree 2 pdf. Defining the vector q H = \f~A~i \q\, q%,q*M] (3.3.16) can be written as y = qHq. (3.3.17) 9 ['J means the integer part of. Matched Filter Bounds for Fast Fading Rayleigh Channels 25 To obtain an expression for y as a sum of uncorrelated random variables, we apply a discrete version of the Karhunen-Loeve [22] expansion to q as q H = U H V A P h (3.3.18) where = I (3.3.19) uuH uH = [uj, u^], E A = diag(Ai,A2,...,AM) (3.3.20) and P = [pi,p 2,...,PM], P H P = L (3.3.21) Defining the Hermitian matrix Q = E qqH P A P H (3.3.22) the Aj-s are recognized as the eigenvalues of Q and are also real and nonnegative [23]. We have for the (n, m)th element of Q ' max I max Q n > m = At J J '(j)z(o-a,o-h;(n - m)At)p{n At - aa)p*(m'At - ab)daadab o o ' . -(3.3.23) where n' = n — 1/2, m' = m — 1/2 and L < n,m < M. Using (3.3.18) in (3.3.17) we can write y as a weighted sum of independent chi-squared random variables of degree 2 R y = J2^Hi)\2 (3-3.24) i=l where R is the number of nonzero eigenvalues of Q, which is equal to its rank. Equation (3.3.24) indicates that the characteristic function of y , $y{v), expressed in terms of the characteristic function of |u;| , $^u.p(v), is R Matched Filter Bounds for Fast Fading Rayleigh Channels 26 The characteristic function for degree 2 chi-squared random variables is given in [9] as 1 *M'(V) = (1 - J2V**) (3.3.26) 1. where we set of = 1/2 in order to normalize E |u;| The simple form of $\u.f(v) allows the pdf of y to be found analytically. Combining (3.3.25) and (3.3.26) R i=i J 1 (3.3.27) Equation (3.3.27) may be rewritten using a partial fraction expansion [24] as D n. •»(») = E E A ik i=l k=l i1 ~ 3Xiv) (3.3.28) where the constants A,-jt are given by 1 Mk = (ni-k)\(-jXi)ni ni—k (3.3.29) D is the number of distinct X[s, n{ is the order of A;. Equation (3!3.28) is easily inverted by recognizing the inverses of its individual terms as Inverse < Ak ( i - i W The pdf of y is then o, otherwise. (3.3.30) D n, py{y) 0, otherwise (3.3.31) and combining with (3.3.12) for Pj yields R m / \ P h = \ l . l f f ^ i * [ed(<[^)y'-h->*dy. 2 2 ^ ( * - l ) ! A ? / \y N0)y (3.3.32) Matched Filter Bounds for Fast Fading Rayleigh Channels 27 The integral in (3.3.32) is evaluated in Appendix A.l. The resulting expression for Pb is R ni k—1 p 1 A ^{2l-l)W ' XiEb/N0 (3.3.33) 2 * u t r K u (2o" V ( W i v 0 + i f + i • An expression having the same form as (3.3.33) was given in [21] for diversity reception on slowly fading channels. However, the formulation in [21] does not allow the fast fading channel case to be studied. In contrast, our model includes an arbitrary Doppler frequency as a parameter. 3.4 Extension to it-Channel Diversity We can extend our technique for deriving the A,-'s to include if-channel space, frequency or time diversity reception by considering the channel impulse response z{a,t) in (3.3.9) to be the concatenation in a of the K diversity channel impulse responses, which may in general be correlated. The vector q becomes the concatenation of vectors O J qi,q?, (3.4.1) where q; is due to the ith channel impulse response and where the kth element of is q?k= J ^ ^ i - i J A i j p * ^ f c . - ^ A t - a j d a . (3.4.2) Defining the matrices Qik = E (3.4.3) we have Q Qn Q 1 2 Q 2 1 Q 2 2 _ Q K I Q K 2 QlK Q 2 K Q K K (3.4.4) Matched Filter Bounds for Fast Fading Rayleigh Channels 28 When the impulse responses of the different channels are uncorrelated, Q simplifies to a block diagonal matrix "Qn 0 .. 0 0 Q 2 2 0 Q' 0 0 . . Q K K It is easily shown that the matrix of eigenvalues of Q' is (3.4.5) A' A n 0 0 A 2 2 0 0 (3.4.6) 0 0 . . A K K . where An is the diagonal matrix having as its elements the eigenvalues of Qn. Thus the eigenvalues of Q' can be found by finding the eigenvalues of the smaller matrices Qii, 1 < i < K independently. If we further assume that all diversity channels have the same autocorrelation function, then we need only find the eigenvalues of Q n , since the eigenvalues of Q' will be those of Qn, but with their orders increased by a factor of K. 3.5 Some MFB Results 3.5.1 Flat Rayleigh Fading In flat Rayleigh fading there may be a fixed distortion of the power spectral density of the received signal, but during fades all frequency components are attenuated and phase shifted equally. This implies that the channel's impulse response is separable in the delay and time variables. In the simplest case, there is no fixed distortion and the channel's impulse response and autocorrelation function may be written as z{a,t) = S(a)Zl{t) (3.5.1) (f>z{o-aiO-b]ta - th) - 8(aa)8(ab)cj)Zl(ta - th). We will use the channel model of (3.5.1) to see how Pb is affected by the normalized Doppler rate fx, transmitted SNR, and pulse shape in flat Rayleigh fading. We begin by considering the limiting cases of very slow (/# —>• 0+) and very fast ( /JV —> oo) fading. Matched Filter Bounds for Fast Fading Rayleigh Channels 29 It has been shown [3] that the channel autocorrelation function for a mobile whip antenna moving through a field with isotropic scattering is given by <j>z((n - m)At) = J0(u!D(n - m)At) , (3.5.2) where Jo (a;) is a zeroth order Bessel function and WQ = 2wfj) is the maximum Doppler frequency in radians. For this case the elements of the matrix Q as given by (3.3.23) simplify to Q n , m = AtJ0(uD(n - m)At)p(n'At)p* (m'At): (3.5.3) For very slow fading, Jo(wj)(n — m)At) in (3.5.3) is essentially constant, and so Q has only a single eigenvalue, which is A 0 = 1.0. This follows from using (3.3.15) with (3.5.1) to show that E[y] = f ( 0 ) ^ ( 0 ) = 1.0. The pulse shape has no effect on the Pb since it does not influence AQ. The P& is given by the R = 1, n\ = 1 case of (3.3.33), i.e. as is well known [9]. At the opposite extreme as /jy —> oo it can be shown (Appendix A.2) that the variance a2 approaches 0 for any unit energy pulse. Ultimately as a2 —» 0, the pdf oi y becomes a delta function at E[y] = 1.0, i.e. lim p(y) = S(y-1.0). (3.5.5) Using this in (3.3.12) we find lim Ph = \ - 1 - erf(^/E^/K). (3.516) The Pj expression given by (3.5.6) is identical to that for BPSK signaling on a nonfading AWGN channel, as might be expected. Thus the use of an extremely long (relative to the Matched Filter Bounds for Fast Fading Rayleigh Channels 30 coherence time of the channel) signaling pulse of arbitrary shape has essentially made the fading channel look like a nonfading channel. Between these extremes we might suspect that the pulse shape would be significant since the effective channel averaging or variance of y depends on the pulse shape. The matrix Q given by (3.5.3) was calculated for different pulse shapes, and for a range of normalized Doppler rates and Ejy/N0 values. The pulse shapes chosen were a rect pulse defined as a half sin pulse defined as p . ( ( ) = ^ ' - n ( ' ' / i i ) . ' o < t < r , F v ; ' 0, otherwise and the root raised cosine pulse given previously in (1.1.4) as (3.5.9) ' ^ ( l + - s p y ^ ) ) , 0<t<aT0 p (t)= I V r o , aT0 < t < T 0 3 9 ^ ( i + c o s ( ^ ) ) , r 0 <*<To(l + a) • • \ 0, otherwise. A computer package [25] was used to find the eigenvalues resulting from each combination of pulse shape and normalized Doppler rate. In every case all R eigenvalues were found to be distinct, so that (3.3.29) and (3.3.33) simplify to 1 _ l / Eb/N0 p _ -. 2 2 V E„/N0+l' . i l - l I 2 - 2 L, 11 J^Xi V XiEb/No+l' 2 = 1 ( — 1 Figure 3.3 shows the P& curves for the rect pulse shape in solid lines for a range of normalized Doppler rates up to 5.12. A couple of curves for the half sin pulse shape are shown in dashed lines for comparison. The curve for the limiting case of very fast Matched Filter Bounds for Fast Fading Rayleigh Channels 31 Figure 3.3 Effect of normalized Doppler rate and pulse shape for fast flat Rayleigh fading fading (/jy —• oo) is also shown. As noted previously, this curve is identical to the Pi curve for BPSK optimally received on a nonfading AWGN channel. For P& < 0.1, substantial improvements over the slowly fading case of several dB are seen for all the curves presented. Below Pj, — 10~2, at least half of the achievable gain is obtained by increasing the pulse duration such that the normalized Doppler rate is 0.64 for the reel pulse or 1.28 for the sin pulse. Above these rates diminishing returns are clearly shown. Each succeeding curve corresponds to a doubling of the normalized Doppler rate, but the incremental improvement decreases with each curve. At Pj, = 10 - 2, both pulse shapes are within approximately 1 dB of nonfading AWGN channel performance, for /jy > 5.12. For finite positive values of /jy, the P& for the half sin pulse is worse than that for the rect pulse, due to its inferior channel averaging. Figure 3.4 shows the effect of varying the rolloff parameter a for the root raised cosine pulse defined by (3.5.9) at f^ = 0.32 and fx = 1.28. There is very little change Matched Filter Bounds for Fast Fading Rayleigh Channels 32 10° 10"' io-2 10"3 10"4 IO"5 10"7 0 2 4 6 8 10 12 14 16 18 20 22 24 Figure 3.4 Effect of rolloff on in the curves for a < 0.25, curves for a < 0.125 are practically coincident over much of the graph's range. For a > 0.5 loss in channel averaging ability is more significant. These curves indicate the tradeoff between pulse smoothing and channel averaging which is an important consideration in' Chapter 6. 3.5.2 F r e q u e n c y - S e l e c t i v e R a y l e i g h F a d i n g In frequency-selective Rayleigh fading the fading at different frequencies of the power density spectrum is not completely correlated.. This implies a significant delay spread compared to the pulse duration. A common example [20] is when the transmitted pulse and a delayed version arrive at the receiver after having travelled over different fading paths. We model this channel impulse response with z(a,t) = aiS(cr)zi(t) + a2S(a - T)z2(t) (3.5.11) Matched Filter Bounds for Fast Fading Rayleigh Channels 33 where T\ = 0 and r = r2 — T\. Its autocorrelation function is (/>z(o-a,o-b;ta - tb) = al8(aa)S(ab)4>Zl(ta - th) + aia2$(o-a)6(o-b - r)(j)ZlZ2(ta - tb) (3.5.12) + a\a28(o-a — T)8(o-b)cf)Z2Zl(ta—tb) + al8(o-A-T)8(crB-T)<f)Z2(ta-tb). We would like to consider the general case where the processes z\(i) and 2 2 ( £ ) may be correlated. To facilitate comparison with the flat Rayleigh fading case, we assume that they each have the same, to within a constant, autocorrelation function as used in the one-ray case <f>Zl((n - m)At) •=• <f)Z2{(n - m)At) = Jo(uD(n - m)At) (3.5.13) and <f>ZlZ2((n - m)At) = pJ0(ooD(n- m)At) (3.5.14) where p is a normalized cross-correlation coefficient having \p\ < 1. The elements of Q for this case are Q n , m = AtJ0(ujD(n - m)At)[p(n'At)p* (m At) +app(n'At)p*(m'At-T) (3.5.15) -\-ap*p{n At — r)p*[m At) + [a2p(n'At-r)p*(m'At-T)]. Matched Filter Bounds for Fast Fading Rayleigh Channels 34 Normalization Evaluating (3.3.15) for our two-ray model we obtain % ] = ^i (0 ) ( (« i + o$)f(0) + 2a1a2Re[pf(r)}) (3.5.16) = 1 + 2a1a2Re[pf(T)} assuming a\ + a\ — 1.0. Assuming for the moment that both rays have equal power we see that the average received SNR 7 j r depends on the correlation between the rays and the amount of pulse overlap, and different combinations can yield the same 7&r. Thus there are several plausible ways for comparing Pe plots. If diversity occurs because extra antennas are being added at the receiver then it might be appropriate to have %r increase with each additional ray received. On the other hand, if we have a fixed amount of total power to radiate and want to compare the merits of sending it with or without diversity then it would likely be more insightful to assume the power per ray is reduced as the number of diversity channels is increased. A third option is to compare on the basis of a constant 7fcr. The choice made in producing Figures 3.5 and 3.6 is to assume that each of the two rays has a variance of a\ = c?2 = 1/2 and this remains fixed independently of how the correlation and pulse overlap are varied. This makes two independently faded pulses with 100% overlap equivalent to the transmission of a single pulse. To limit the number of curves presented, the relative attenuation factor, a2ja\, between the two rays was not varied. Two-ray slow Rayleigh fading Figure 3.5 shows some P& curves for two-ray slow Rayleigh fading for several combinations of the normalized delay spread r and normalized correlation p between rays. The results for independent rays are the same as those in [20], while the results for nonzero p are new. A rect pulse shape was used in the calculation of Q by (3.5.15). The two Matched Filter Bounds for Fast Fading Rayleigh Channels 35 10° 10-1 lb" 2 10"4 10- 5 0 2 4 6 8 10 12 14 16' 18 20 22 24 EJN0 r • Figure 3.5 Effect of delay spread and correlation between rays, for two-ray very slow Rayleigh fading. coincident curves with (r, p) = (0.0,0.0) Sz (1.0,1.0) are equivalent to the transmission of a single pulse on a single ray with the same /N0 arid have the poorest performance for p. > 0.0. Several connections between r and p are apparent. For the cases where p = 1, the greater the overlap of the received pulses the better the performance, but the maximum difference is only 3 dB. For the p = 0 cases, the better performance occurs with smaller overlap and the dB advantage can be significantly greater depending on the point of comparison, for example > 9.5 dB at Pi, = 10 -3. The reasons for the gain occurring in each case are also different. In the former, the gain occurs because the noise added to each pulse is independent and so combines noncoherently, but the pulse components are able to combine coherently. In the latter case, the gain occurs because of the more important effect of diversity, where it is unlikely that both pulses fade simultaneously. (t.P): Matched Filter Bounds for Fast Fading Rayleigh Channels 36 For intermediate values of ray correlation there ought to be some point where the gains and losses due to overlap oppose each other and the-trend is not so clear, but as the curves show, most of the gain obtainable is achieved even for values of (r,p) as high as (0.5,0.5). Additionally, each (r,p) pair is seen to have a dual in the sense that there is more than one (r, p) combination that produces the same curve. The Q matrices produced by the duals are generally different but their eigenvalues are of course identical. For p > 0 the dual to {r,p) is given by (1 — 'p, \ — r). The combinations (0.0,1.0), (1.0,0.0) and (0.5,0.5) are their own duals. The Pb curves were calculated by (3.5.10) except when (r,p) = (1.0,0.0). This is equivalent to diversity with two independent channels as described in Section 3.4. Therefore there are R/2 distinct eigenvalues of order 2. In this case (3.3.33) simplifies to Pb where 9 9 ^ i=l , , , / \Eb/N0 1 / XiEh/No A 2XT f - i i I T — - — ( i - M/Xi) t U2 f _ 1 1 = n—-;_n (1 — X, 1=0 2 • (3.5.17) (3.5.18) Two-ray fast Rayleigh fading Figure 3.6 shows several Pf, curves at a normalized Doppler rate of = 0.64 for two-ray fast Rayleigh fading for several combinations of the normalized delay spread r and normalized correlation p between rays. The variations of Pj with (r, p) are similar to the slow fading case except the correlation between rays seems less important. Comparing the cases for r = 1.0 in Figure 3.6 with those of Figure 3.5 for slow fading, the dB spread is seen to be much reduced. This is because in slow fading the only source of diversity Matched Filter Bounds for Fast Fading Rayleigh Channels 37 10° 10"' 10"2 10-4 10" 10"6 0 2 4 6 8 10 12 14 16 18 20 22 24' E „ ' N o Figure 3.6 Effect of delay spread and correlation between rays, for two-ray fast Rayleigh fading.' Normalized Doppler rate = 0.64. is the extent of the two rays' independence, while in fast fading there is diversity within a single pulse. 3.6 C o n c l u s i o n A matched filter bound analysis was presented for B P S K signaling on fast Rayleigh fading channels. The effects of delay spread and correlation between the fading processes for multipath reception were included in the analysis, and several examples given. The bit error rate performance was shown to improve with increasing fu10, eventually approaching the performance of B P S K on the nonfading AWGN channel. This result does not depend on the specific pulse shape. For Pj — 10 - 2 and = 5.12, the Ei/N0 (0.5, -0.5) (0.5, 0.5) (1.0, 0.0) -(1.0, 0.5) -(1.0, 1.0) -(\ P): (0.0, -0.5) (0.0, 0.0) (0.0, 0.5) (0.0, 1.0) This is consistent with previous work. Matched Filter Bounds for Fast Fading Rayleigh Channels 38 required is within 1 dB of that required on a nonfading channel for both the rect and half sin pulse shapes. In contrast to the limiting case, for any finite normalized Doppler rate high enough that the channel gain cannot be assumed constant over the pulse duration, the pulse shape affects the Pb. This applies even in the case of a channel tracking receiver which ideally matches the impulse response of its correlator to the received pulse shape. Delay spread was found to result in a lower Pb for both the slow and fast fading cases, when p < 0.5. The reduction was smaller in the latter case due to the inherent diversity present within a single pulse. Chapter 4 39 A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels The MFB analysis presented in Chapter 3 obtained performance bounds on the achievable for OFDM in part by neglecting the ISI introduced by the channel fading. Since this ISI may be substantial, it remains an open question as to how. closely these bounds may be approached by a receiver when ISI is not neglected. In this chapter we derive an optimal receiver for OFDM based on the principles of maximum likelihood sequence estimation. A multiplicative fading channel model with AWGN is assumed in the derivation. The receiver's performance in flat Rayleigh fading is evaluated by simulation and compared to the MFB results derived in Chapter 3. This chapter is organized as follows. Section 4.1 describes the transmitter and chan-nel model for which the receiver is being derived. Section 4.2 presents the theoretical derivation of an optimal maximum likelihood receiver, in the sense of sequence esti-mation, for OFDM in flat Rayleigh fading. Evaluation of two key parameters, matched filter outputs and tone correlations, by FFTs in a digital implemention and the required sampling rate is examined in Section 4.3. Section 4.4 describes truncating the constraint length used by the sequence estimator to make it less complex and Section 4.5 presents a criterion for choosing a non-rectangular pulse shape. Section 4.6 presents simulation results for Pb as a function of Eh/No, assuming perfect channel knowledge, of BOFDM for a variety of parameters and compares them to the MFBs of Chapter 3. Section 4.7 develops an approximate analysis to predict the effect of using an imperfect channel es-timate, and compares this result to a simulation with a noisy channel reference. Section 4.8 shows a limitation of the truncated MLSE receiver when applied to QOFDM and Section 4.9 summarizes the conclusions of this chapter. 39 A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels 40 4.1 The Transmitter and Channel Model Figure 4.1 shows the transmitter and channel model in terms of the complex lowpass equivalents for bandpass signals. N data symbols are assembled and simultaneously modulated onto N frequency-orthogonal carriers. The modulator outputs are summed and the resulting waveform is further shaped by the unit energy pulse p(t) to form s(t), the transmitted waveform of a single OFDM block. The energy of an OFDM block is EQ = NEa, where Ea is the energy of a data symbol. Both the OFDM block and the data symbols comprising it have duration (1 + a)To. The channel introduces multiplicative fading z(t) and AWGN n(t) of spectral density No. The channel fading is modeled as a complex Gaussian random process with a power spectrum appropriate for land mobile radio [3]. The AWGN is statistically independent of both the data and the fading. 4.2 Derivation of a Maximum Likelihood Receiver for OFDM In order to determine the best possible performance attainable with an OFDM signal in Rayleigh fading, we derive an optimal receiver structure which assumes knowledge of the fading waveform z(t) is available as side information at the receiver. In practice z(t) might be estimated by reserving some of the OFDM tones for a pilot signal. Since the fading assumed is flat, the phase and amplitude changes impressed upon the pilot will be identical to those impressed upon the data tones, neglecting the effects of AWGN. We will not develop the estimate of z(t) here, but merely assume that a perfect estimate (4.1.1) and autocorrelation function = Vo(27r/j,r) (4.1.2) A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels 41 O F D M transmitter channel Figure 4.1 OFDM transmitter and channel model is available. Later in Section '4.7 we will consider the effect of noise on the channel estimate, to verify that this assumption is not too brittle. As previously noted, an OFDM block consists of N data symbols, and we are interested in the optimal detection of a single OFDM block. There are different possible interpretations of what is optimal. We could for example seek the minimum probability of a data symbol error or we could seek to minimize the probability of an OFDM block error. These criteria are not quite identical. In the latter case the problem is to find the optimal data symbol sequence composing an OFDM block. Experience with serial modulation [26] on ISI channels has shown that optimization by the sequence estimation criterion tends to perform well by the symbol optimization criterion as well. Related work in [27] shows that the symbol optimization criterion generally leads to a more. computationally complicated receiver. With these points in mind, we choose to derive an optimal receiver in the sense of sequence estimation. A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels 42 Consider a signal w(t,a), dependent on a vector of data symbols a, transmitted on an A W G N channel, where the noise is statistically independent of w(t, a). The received signal is r(t) = w(t,a) + n(t). (4.2.1) It is well known from the principles of maximum likelihood sequence estimation [28] that the log-likelihood function for a is J \r(t) -w(t,a)\2dt (4.2.2) assuming the integral is finite. Equation (4.2.2) must be minimized over all possible choices for a. Expanding (4.2.2) and retaining only the terms dependent on a we can show that minimizing (4.2.2) is equivalent to maximizing the metric A = JRe[r(t)w*{t,a)} - ^\w(t,a)\2dt (4.2.3) which is the form we shall work with here. For the flat fading model, the received signal is given by r(t) = z(t)s(t) + n(t), 0<t<T0{l+a) N-l = z{t)p(t) ^2 ane3^nt + n(t) where N-l 71 = 0 0 < t < Tb(l + a). Comparing with (4.2.1), it is apparent that N-l (4.2.4) (4.2.5) w(t, a) = z(t)p(t) ane3^nt. (4.2.6) n=0 A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels. 43 Substituting (4.2.6) into (4.2.3) we have N-l A Re r(t)*W(*)-£ a» e ra=0 dt N-l N-l (4.2.7) \ I i*(oi2ip(*)i2EEa» Z J n n > m e J T » V " "',ldt where the interval of integration is the interval over which p(t) is nonzero, i.e. 0 < t < Tb(l + a). Interchanging the orders of the integrations and summations 'N-l A = Re X>»/ r(t)z*(t)p*(t)e-^ntdt n=0 J - ^ E E a » ^ / K * ) l 2 b ( * ) | V ^ (4.2.8) dt n=0 m=0 and defining and Un = J r(t)z*(t)p*(t)e-J%ntdt V n = j W)\2W)\2^ nt dt we can write for the metric A = Re 'N-l n=0 T V - l i V - l (4.2.9) (4.2.10) (4.2.11) re=0 m=0 Some insight into the meaning of Un and K can be obtained with a little manipulation of (4.2.9) and (4.2.10). Let the equivalent pulse shape after transmission over the channel be q(t) — p(t)z(t), and let P(f) — Jp(t)e~i2T:ftdt. Using this and applying Parseval's relation11 to (4.2.9) yields oo Un= j R{f)Q*(f-^df —oo-= { £ ( / ) ® Q V / ) } l / = * ' (4.2.12) -f- oo . -11 / a(t)b'(t)dt= f A(f)B*(f)df. A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels 44 where © is a convolution operator. Un can be interpreted as the output U(f) of a filter matched to Q(f) operating in the frequency-domain, evaluated at / = U/TQ. We will refer to Un as the matched filter outputs. Using q(t) in (4.2.10) we have Vn = J \q(t)fe~j^ntdt. (4.2.13) Vn is the correlation between two complex tones spaced n/To apart in frequency after shaping by the equivalent pulse. We will refer to Vn as the tone correlations. Once we have obtained Un and Vn, (4.2.11) gives A in a form maximizable by the Viterbi algorithm. Specifically, it may be shown (Appendix C) that A may be found recursively by defining • 71-1 An = Re[a* Un] - Re and maximizing k=n—L -\\an\2V0 (4.2.14) N-l X=^2\n. (4.2.15) 71 = 0 Equation (4.2.14) results in an optimal sequence selection only for the selection of L = n, or equivalently if Vn = 0 for n > L. The Viterbi algorithm is a search for an optimal path through a trellis with ML states, where M is the number of symbols in the alphabet of an. Although much more efficient than an exhaustive search when L < N, it is still exponentially complex in L, and thus L must be fairly small in actual use or approximations become necessary. 4.3 FFT Evaluation of Un and Vn Both the matched filter outputs Un and the tone correlations Vn, which are defined by continous-time integrals, must be evaluated from discrete samples of r(t) and z(t) in a digital implementation. While numerical integration techniques could be used, it is much A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels 45 more efficient to take advantage of certain relations between continous-time and discrete-time Fourier transforms. If we define U(f) as the Fourier transform of r(t)z*(t)p*(t) 2 2 and V(f)' as the Fourier transform of \z(t)\ \p(t)\ , then Un and Vn are seen to be these transforms evaluated at / = n/Tb. This leads to a criterion for determining the sampling rate which will allow all N values of Un and Vn to be evaluated accurately and simultaneously using FFTs. If we have,a discrete signal sm related to a continuous signal s(t) by sampling as sm = s(t)y~2fi(t ~ mAt), the relation between S(f) = £ sme-^xfmAt and S(f) = m m f sfte-iWdt is [6] n _ which states that the spectrum of the discrete signal is a periodic version of that of the continuous signal. When (4.2.9) and (4.2.10) are converted to DFTs by sampling as in Un = Y,r^mPm^nmAt . -(4.3.2) m K = ^ | , m | 2 | p m | 2 e - ^ A < (4.3.3) m we are actually calculating periodic approximations to Un and Vn. Equation (4.3.1) indicates one period of AtUn = Un and one period of AtVn = Vn if Un and Vn are bandlimited and is small enough to avoid aliasing. The bandwidth of s(t) depends on how many sidelobes are considered significant components of P(f). For rectangular pulse shaping, even considering only the first sidelobe on either side and choosing At to avoid these aliasing requires At < To/(N + 3). The bandwidth of the received signal is greater because of the multiplicative fading, and the bandwidth of the argument of (4.2.9), r(t)z*(t)p*(t), exceeds that of s(t) by twice the Doppler frequency. Since the Doppler frequency is assumed to be on the order of 1 % of the signal bandwidth only a slight decrease in At is required. The implication is that despite the invertibility of the A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels 46 DFT, an iV-point transform (corresponding to At = TQ/N) is inadequate to represent an N tone OFDM signal on a fading channel12. To allow an adequate guard interval, and because it is convenient to work with factors of 2 when using FFTs we use At = ^fr, and our DFT equation for Un is T N'-l u* = ^Y,r™z*™p™e~mnm- ( 4 3 - 4 ) m=0 The upper limit N' must be large enough to cover the time interval 0 < t < Tb(l + a), requiring N' greater than 2N for a > 0. Again choosing a factor of 2 for convenience 4JV—1 Un = £ rmz*mP*me-^nm, (4.3.5) m=0 where we have rescaled Un to remove the leading factor of This rescaling is also applied to Vn below and does not affect the optimal path selection as can be seen from (4.2.11). Equation (4.3.5) can be calculated from two 2iV-point FFTs by using the periodicity of discrete complex exponentials. It is easily shown 2N-1 Un= 2^ {rmZmpm + rm+2NZm+2NPrn+2N)e 2 N . ( 4 3 „ n=0 \ • • ) 0 < n < N - 1. In (4.2.10) for Vn, \z(t)\ \p(t)\ has a much narrower bandwidth than that of r(t)z*(t)p*(t), permitting a much larger A t than.that required to evaluate Un. However, if a shorter FFT is used an interpolator is required because the frequency samples Vn are spaced too far apart. Since an FFT is an efficient interpolator, we use the following expression for Vn 2N-1 Vn = £ (\zmf\pm\2 + \zm+2N\2\Pm+2N\2)e-^nm, 71 = 0 0<n<L. AT-points are adequate for AWGN channels, because the Nyquist criterion need not be satisfied. A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels 47 4.4 Truncated MLSE We have noted in Section 4.2 that the complexity of the Viterbi algorithm used to evaluate the metric of (4.2.14) and (4.2.15) varies as ML. From (4.2.14) L is potentially as large as TV—1, or 1 less than the number of tones used. Even for' M = 2, a binary signaling alphabet, the complexity of the optimal receiver rapidly becomes unsupportable for more than perhaps 10 tones. Optimal receivers for L = 64 or 128 are clearly impractical, yet we are interested in large numbers of tones for the channel averaging they provide. Thus the maximum value of L must be reduced, and the simplest means is to set a limit for it. Specifically, in the calculation of An by (4.2.14), we allow L = n to increase with n for n = 0,1, ...,Lmax, and then hold it fixed at Lmax for n = Lmax + 1, Lmax + 2, ...,7V- 1. By so limiting L the MLSE receiver is made suboptimal. The indication that L may be limited while losing only a small portion of the attainable performance follows from the interpretation of Vn as the correlation between two tones. As the sidelobes of Q(f) decrease with increasing distance from the mainlobe, it is apparent that inter-tone correlations will decrease as the frequency spacing between the tones increases. How rapidly this decorrelation occurs depends on the type of pulse shaping used and on the fading waveform during the signaling interval. The selection of the pulse shape.is affected in part by a tradeoff between channel averaging and rapid sidelobe reduction. A raised cosine pulse, for example, has more rapidly decreasing sidelobes than a rectangular pulse, but is less effective at channel averaging. In the following sections the required Lmax for BOFDM is found by simulations of the suboptimal receiver for both rectangular and raised cosine pulse shapes. 4.5 Pulse Shaping Without foreknowledge of the fading waveform the most effective channel averaging in the interval To is obtained with a rectangular pulse shape. A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels 48 However in practice a pulse without discontinuities is desirable because of bandwidth limitations, •• and additionally it seems possible that a pulse with more rapidly decaying sidelobes might cause less ISI and therefore require a smaller L. A criterion for choosing p(t) is to require that the basis functions of s(t) remain orthonormal after shaping and matched filtering in the absence of fading: . 8n_m = J \p(t)\2e-^n-m)tdt. (4.5.1) Let Psq(t) = \p(t)\2 and Psq{f) = Jpsq{f)e-^^dt, then &n = Psq(f)\f=n/T0 (4.5.2) — P Transforming both sides with an inverse discrete frequency Fourier transform (Appendix B) and scaling by 1 /To yields - nTo) = ± (4.5.3) n which is merely Nyquist's criterion for zero-ISI pulse shaping with the traditional roles of time-domain and frequency-domain expressions reversed. An important consequence of (4.5.3) is that there is no pulse shape of duration less than or equal to To (other than the rectangular) which satisfies the orthonormality criterion. Thus alternative pulses must have their duration extended relative to the rectangular pulse. We consider the raised cosine family for psq(t) since it is known to satisfy (4.5.3) and its spectral sidelobes decay approximately as l / / 3 . The raised cosine pulse is given by ' ^ ( l + c o s ( ^ i ) ) , 0<t<oT 0 p (t)= ) V ^ o , oiT0 < t < T0 (4 5 4) ^ ( l + c o s ( ^ ) ) , To<t<T0(l + a) k 0, otherwise where 0 < a < 1 is a rolloff factor describing the percentage of the pulse duration beyond To. The pulse p(t) satisfying (4.5.1) is p(t) = yjpsq(t). A Maximum Likelihood Sequence Estimation Receiver for 49 OFDM on Flat Rayleigh Fading Channels The need to project p(t) onto complex exponentials eJ^n< of duration greater than TQ leads to an interesting observation. Redefining the exponentials to include a linear phase shift, the shaped basis functions are p(i)e i^n(t~*), 0<t'<(l+a)To. ' (4.5.5) The Pb, in contrast to the AWGN channel, is affected by the choice of a. Figure 4.2 shows the sensitivity of Pb to the choice of a for BPSK modulation of N — 32 tones and a = 0.125. These errors are due entirely to ISI caused by flat Rayleigh fading as there is no AWGN. Matched filter detection is employed without MLSE. The optimal a occurs at a — <xTo/2. The effect occurs because p(t) has been defined to start at t = 0. This causes pulse shapes with nonzero a to have a delay of TJ = aXb/2 relative to the r e c t pulse. Consider a single modulated tone from an OFDM block as in (4.5.6) anp{t)e3^nt. (4.5.6) Time delay of the pulse, as shown in (4.5.7) = (a„e J f nTd)p{t')e^nt' .... . (4.5.7) = anp(t)ejTo where t' = t — T& and a'n = ane>TonTd, effectively alters the transmitted constellation by introducing a tone dependent rotation of the data symbol. There is no effect from rotation on Pb for an AWGN channel because the received tones remain orthogonal. But Pb is increased when orthogonality is lost as on a fading channel, because the a'n can be closer in signal space than the an. The variation in Pj with a is greater than 2 : 1 without compensation, but can be entirely removed by prerotating the an by e~Jwna when a ^ 0. This is equivalent to A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels 50 1.4e-02 1.2e-02 1 .Oe-02 ft? 8.0e-03 6.0e-03 4.0e-03 -6 -4 -2 0 2 4 6 8 10 a Figure 4.2 Variation of Pj with a for N = 32 tones and various rolloff factors. Matched filter reception, no MLSE applied. setting a = aTo/2 in (4.5.5), and was done in producing the simulation results which follow. 4.6 Perfect Channel Knowledge 4.6.1 Simulation Framework The probability of a bit error, P j , was evaluated by simulation for several important cases. The OFDM blocks were generated as specified by (4.2.5) with an = ±y/E^. This is independent BPSK modulation of each of the N tones comprising the OFDM block. Thus we have Ea = Pj, . and the probability of a symbol error P , = Pj,. The simulation implements the transmitter and channel model shown in Figure 4.1, and the truncated MLSE receiver with suboptimal constraint length L derived in this chapter. The simulations were made long enough to obtain 95% confidence intervals of approximately ±10% of P j . The transmitted signal s(t) is subjected to flat Rayleigh fading and AWGN as specified by (4.2.1) and (4.2.6). The received signal r(t) is processed to produce the matched filter !•——•0=0.125 r > A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels 51 outputs and tone correlations given by (4.3.6) and (4.3.7) respectively. These are then passed to a Viterbi algorithm which maximizes the metric X given by (4.2.15). The figures presented are annotated with //y since it is this parameter upon which the Pj, of the MFB of Chapter 3 depends. A maximum Doppler rate of fry = 0.01 times the data symbol rate is assumed and the sampling rate and number of samples per block is such that //y = foTo = /D-/V . Therefore we always have N = 100/jy data symbols per OFDM block. 4.6.2 Rectangular Pulse Shape We begin by considering a rectangular pulse shape pit) = ^ T rect(£/Tb) which has a sinc(Tb/) amplitude spectrum. The sidelobes fall off only inversely with / and the first sidelobe is only 6.6 db down from the main lobe. It therefore seems possible that after channel fading has altered the special structure required to obtain orthogonality that L might be quite large, but our simulation results in comparison to the MFB indicate this is not always the case. Figure 4.3 shows some simulation results for the MLSE receiver with various values of L. The OFDM signal uses BPSK modulation of 32 tones with rectangular pulse shaping on a flat Rayleigh fading channel. The normalized Doppler rate is //y = 0.32, and the MFB for /jy = 0.32 is also shown for comparison. We observe that the truncated MLSE receiver is able to approach the MFB to within approximately 1 dB over a Ph range exceeding 1 0 - 2 to 1 0 - 4 , and that the discrepancy between the MFB and MLSE receiver increases only slightly with EI/NQ. This is for a modest value of L = 3. If L is restricted to only 1 there is less than a 1 dB loss at 1 0 - 3 ; increasing L to 5 results in negligible improvement over L — 3. The L = 0 curve is simply matched filtering without sequence estimation. It exhibits an error floor slightly below 1 0 - 2 . This is due to the large amount of uncompensated ISI. The effect becomes more visible .with increasing EI/NQ because the proportion of i A. Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels MF£ • o L=0 * »L =; =3 =5 • J * * L - f„=0J2 1 1 1 0 2 4 6 8 10 12 14 16 18 20 22 24 EbIN0(dB) Figure 4.3 Pb of MLSE vs. L compared to the MFB for fN = 0 . 3 2 MFI »L=0 i-• — < L=2 >L=3 iL=5 i s H V 5 ' L=7 \ N i— - fN=1.28 I I I 0 2 4 6 8 10 12 14 16 18 20 22 24 Figure 4.4 Pb of MLSE vs. L compared to the MFB for fN = 1 . 2 8 A Maximum Likelihood Sequence Estimation Receiver for 53 OFDM on Flat Rayleigh Fading Channels error due to ISI relative to AWGN increases as well. Presumably error floors also exist for L = 1, 3, and 5, but at levels too low to measure by simulation. Figure 4.4 shows results for the same parameters as Figure 4.3 except there are now 128 tones and /jv = 1.28. Again values of L = 3 or 5 are within 1 dB of the MFB at 10 - 3, although the discrepancy between these curves becomes significant at lower error rates. Additional points show the inadequacy of L = 2 and a negligible improvement for L = 7 over L = 5. 4.6.3 Raised Cosine Pulse Shape Here we compare the required L for raised cosine pulses of various a to the required L for a rectangular pulse. The pulse p(t) is specified by (4.5.4) and (4.5.1). In comparing pulses of different a there is an unavoidable discrepancy in the overall signaling rate. This is because the orthonormality constraint of (4.5.1) requires a pulse of length Tp(tj = To (1 + a), and longer pulses imply fewer OFDM blocks can be transmitted in a given time. Figure 4.5 shows Pb results for fx = 0.32 with L — 3 for a = 0.0 and a = 0.125. It was shown previously that there is negligible improvement beyond L = 3 for this fx. Not too surprisingly, the change in a has a negligible effect. Figure 4.6 compares the Pb for various a at fx = 1.28 and L = 3. Since L = 3 is noticeably worse than L > 5 at this fx, it was thought that the reduced spectral sidelobe levels of the pulse shapes for a > 0.0 might result in a lowered Pb relative to the a = 0.0 case. However in the simulation results there is very little dependence on a from which we conclude that it is the ISI from nearby mainlobes which causes most errors over the simulated Pb range, and that the effect of ISI from sidelobes is relatively unimportant. A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels a. - } R • • a=0.0 • • cx=0.125 -fN=0.32 L=3 1 1 1 0 2 4 6 8 10 12 14 16 18 20 22 24 ETW 0(dB) Figure 4.5 Effect of pulse shaping on the Pj, of an MLSE for f^ = 0.32 and L 10"4 I VU-Ji x=0.0 A * ( )t=0.125 x=0.25 • • ( y -V n V-> s i -fN=l-28 L=3 I I I 0 2 4 6 8 10 12 14 16 18 20 22 24 Figure 4.6 Effect of pulse shaping on the P& of an MLSE for = 1.28 and L A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels 55 4.7 Effect of an Imperfect Channel Estimate In deriving and simulating the MLSE receiver we have so far assumed the availability of a perfect channel estimate z(t) at the receiver. The problem of channel estimation by means of pilot signals for Rayleigh fading has been previously studied [5], [10], [11]. In this section we provide an approximate analysis of the effect of using a noisy estimate on the receiver's Pj performance and verify our result by simulation. An imperfect channel estimate affects the MLSE receiver by introducing errors into the computation of the metric parameters Un and Vn. We will focus on the matched filter outputs Un since the results of Section 4.6 show the MLSE receiver to be tolerant of small errors in Vn. . We model the channel estimate zm with zm = zm + nim (4.7.1) where nZm is AWGN with variance a\, and is statistically independent of zm. Consider the term rmz*m from the argument of the summation in (4.3.5) used to calculate Un. Substituting for rm we have (4.7.2) — wmzm -f- nm where wmZm is a signal component and n'm = nmz*m is an AWGN component of variance NQ. With the noisy estimate we have instead = wm[z*m + nlm] +nm[Z*m + nlJ (4.7.3) ~ * , / , / / • ~ wmZm -t- nm -t nm where n" = wmn\ is an additional Gaussian noise term and the second order noise term nmn*~m has been dropped. Equation (4.7.3) is approximately equal to (4.7.2) with A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels 56 the addition of an extra noise term n ^ . Its variance is E I Wm n*~ 1 2 = Eaal . (4.7.4) which at first appears proportional to the energy per data symbol Ea. We note n'm and n'^ are independent, since they are assumed jointly Gaussian and their cross-correlation is . = <E[nm]E[4JE[naJ . . (4.7.5) = 0. Therefore the noise powers add, and the ratio of signal energy to noise power in (4.7.3) is given by SNR' = 9 E a ? EJN0 ( 4 7 - 6 ) where we have used E[z m 2£j = 1. The variance a\ depends on the type of estimator used to form zm. Suppose the channel is estimated by transmitting a pilot tone of energy E\ per data symbol interval. The received pilot is [El lPm = \ 7p zm + nPm (4.7.7) V -Is where nPm is independent AWGN of spectral density iVo- lPm is bandpass filtered by an ideal rectangular filter of unity gain and bandwidth Bp. Bp is chosen wide enough to pass lPm undistorted except for band Umiting the noise spectrum; i.e. Bp > 2frj with equality in the absence of frequency offsets. Adequate guard intervals (in frequency) are assumed to protect the pilot from ISI effects. The output of the bandpass filter after sampling is [E~i lm = J7fT zm + nXm (4.7.8) V -t s A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels 57 where a2ni = N$BP. These outputs are further processed by a Wiener filter. Without loss of generality we can consider the estimate of ZQ by a Wiener filter as zo = hHl (4.7.9) T J where () is the Hermitian operator, combining transposition and conjugation. For an r i T iV/-point estimate, the tap input vector 1 = I Nt-i,7_i, Zo, h, ifr± • It is well L 2 2 -known from Wiener filter theory [23] that the optimum (in an MMSE sense) choice for the tap weight vector h is h = R - 1 g where R = E[llH] is the autocorrelation matrix of the tap input vector and g• = 'Ef^l] is the cross-correlation between the desired response and tap input vector. It is not difficult to show that the variance of the estimation error is E \zo — ZQ\ = l - gHR" 1g (4.7.10) For the present problem we have R = E = EiE where zz H + N0E n i r i ! H i T Z N l - l , Z-l, Z0, Zl, ...,Z N,-l 2 2 and n i = m N ! , . . . ,7 i / _ 1 ,n / 0 ,n / 1 , . . . ,n /„_ 1 The {i,j)th element of R is given by Ei Rij = ^Jo^foli - i|Ai) + NoBpsmc(Bp{i - j)At) and g = E V -t s (4.7.11) (4.7.12) (4.7.13) (4.7.14) (4.7.15) A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels 58 The ith element of g is given by gi = J^-M2irfD\i\At). (4.7.16) s Let us assume At = Ts and normalize the pilot bandpass filter bandwidth by the duration of a data symbol 3 = BPTS. We define the two Ni x Ni matrices 3o(3) with (ij)th element JoK)(B) = J0(27r/?|i — j\), Sinc(/?) with (i,j)th element Sincjj(/?) = sinc(/?(i — j)), and the TV; x 1 matrix jo(3) with ith element jo;(3) = JQ(2-K3I). Then we can write < = (l - J?()9)(jo()9) + ^ |^Sinc(/?)) 1j0(/?)j . (4.7.17) Let / = Ei I Es be the fraction of the total energy transmitted per data symbol which is allocated to the pilot. The total energy transmitted per data symbol is Es = Ea + E\ giving Ea = Es(l - f) and Ex = Esf. Using these in (4.7.17) < = ( l - (jo(/3) + |^Sinc(/?)) lj0(f3^j (4.7.18) and substituting in (4.7.6) yields for SNR' of (4.7.3) . —(1 - f) SNR' = ^ — — - . (4.7.19) 1 + / K 1 - /') (X - J S ( / ? ) ( J O ( / ? ) + f f Sinc(/?))~ j 0 ( f l The variation of cr2. with TV; for several values of is shown in Figure 4.7. It can be seen that <r2. is nearly independent of Ni, suggesting that N} = 1 is an appropriate choice. This is because the noise components at the pilot filter output are strongly correlated. As Ni is increased to obtain less correlated noise components, the correlations of the pilot components with the desired component zo decrease, and these effects tend to cancel each other. With Ni = 1 (4.7.18) simplifies to 2 f* A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels 59 -10 -15 -20 -25 ii . • II : II — — , » o •——•(0.1, 0.02, 7) • •(0.1,0.03, 7) • A (0.1, 0.04, 7) 4 •(0.1, 0.02; 13) •4 *(0.2, 0.02, 13) *• •(0.3, 0.02, 13) Legend: (f, p, £ t/N„) • 1 1-1.0 3.0 w, 5.0 Figure 4.7 Variation of cr^ . with JVj for several values of (^f,(3,j^J. and (4.7.19) to SNR' (4.7.21) / ( r - / ? ) + /?(i + § ) ' Figure 4.8 shows the variation of SNR' with / for various (jft,/^ pairs. It can be seen that the optimal allocation of power to the pilot signal is only 12.5% at 8 = 0.02 and increases only slightly as 8 is increased. There is very little variation with ES/NQ. Compared to an ideal system having perfect knowledge of zm at the receiver without having to expend power to measure it, the curves of Figure 4.8.at 8 = 0.02 predict a cost of approximately 1.3 dB for a system which must allocate power to channel measurement and subsequently use the noisy estimate which results. 4.7.1 Noisy Reference Simulation Figure 4.9 compares the measured by simulation for a noisy pilot reference verses an ideal reference, for = 0.64 and L = 3. The MFB and ideal (noiseless) reference curves are the same as appeared in Figure 4.3. A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels (13, 0.02) (13, U.U4) ( 7, 0.02) ( /, 0.04) Legend: (EJN 0, p) I . . . . I . . . . r . . . 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 f Figure 4.8 Variation of SNR' with / for various (j^, 0) pairs. a. 10-4 Ml • • ideal reference \ \ 1 = P=0.02 fn=0.64 f=0.125 L-3 I I I I I I I I 0 2 4 6 8 10 12 14 16 18 20 22 24 . £„/N0(dB) Figure 4.9 Effect on Pj, of a noisy vs. ideal channel reference. A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels 61 The curve for the noisy reference was obtained with the same simulation code as for the ideal reference case, except the ideal reference was replaced with a noisy reference as specified by (4.7.1), and having reference noise variance given by the first line of (4.7.20). The noisy reference was used by the MLSE receiver in calculating the matched filter outputs and the tone correlations, which are the only places a channel reference is required. The Ei/No numbers of the horizontal axis consider the total energy used to transmit a single bit; the actual Ei of a single bit was reduced by / = 12.5% to model the energy allocation to a pilot signal. About 0.3 dB higher than predicted by the analysis, the energy required to measure the channel and the subsequent use of a noisy reference result in approximately a 1.6 dB penalty relative to an ideal, costless reference. A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels 62 4.8 QOFDM I D = ' • BOFDM L=5 QOFDM L=i QOFDM L=4 ! fN=0.32 1 1 1 0 2 4 6 8 10 12 14 16 18 20 22 24 EJN0(dB) Figure 4.10 Limitation of truncated MLSE applied to QOFDM for fN = 0.32 Thus far we have considered results only for BOFDM where we found the MFB at Pb > 10 - 4 c a n D e approached within 1 dB for practical values of L ranging from 3 to 5. Unfortunately, as shown in Figure 4.10, the required L for QOFDM is impractically large. There are two reasons for this. The first is the exponential dependence on L. of the number of states which must be extended with each iteration of the truncated MLSE algorithm. For a given L, the number of states is squared for QOFDM compared to BOFDM. Secondly, there is more interference with QPSK symbols as opposed to BPSK symbols. This is because in addition to the ISI from nearby tones, there is also intrasymbol13 distortion of the QPSK symbols. inphase-quadrature (IQ) A Maximum Likelihood Sequence Estimation Receiver for 63 OFDM on Flat Rayleigh Fading Channels The truncated MLSE receiver in its present form is clearly inadequate for use with QOFDM. A more sophisticated approach, proposed in the context of serial modulation on an ISI channel, reduces the required MLSE constraint length with a linear prefilter [29]. Variations of this idea are developed in Chapter 5, which ultimately make the use of MLSE with QOFDM feasible. 4.9 Conclusion In this chapter we have derived and simulated a truncated MLSE receiver for OFDM on flat Rayleigh fading channels. The metric used by the sequence estimator is evaluated by the Viterbi algorithm. Key quantities input to the algorithm can be interpreted as frequency-domain matched filter outputs and tone correlations. These quantities can be efficiently evaluated using FFTs. For BOFDM, with the assumption of perfect channel knowledge, it is possible to approach the MFBs of chapter 3 to within 1 dB for bit error rates as low as P& = 10 - 4 using constraint lengths of only L = 3 — 5 for normalized Doppler rates of = 0.32 — 1.28. Compared to uncoded serial schemes which are unable to take advantage of channel averaging, there are gains of several dB for OFDM at Pb rates of 10 - 2 and below. These gains are of course contingent on the normalized Doppler rate being high enough to allow channel averaging to occur, but not so high as to require an excessive constraint length. The examples simulated covered the range from fN - 0.32 - 1.28. The use of pulse shaping of the OFDM block to reduce the spectral sidelobes of its tones was investigated but found to have little effect on Ps. The implication is that the sidelobes themselves are relatively unimportant at the simulated bit error rates, and that the visible error is due to a combination of mainlobe ISI resulting from the flat Rayleigh fading and AWGN. While useful for comparison purposes, an ideal channel reference is unobtainable in practice. An analysis ,was presented to determine the dB loss of using a noisy as A Maximum Likelihood Sequence Estimation Receiver for OFDM on Flat Rayleigh Fading Channels 64 opposed to an ideal reference, resulting in an estimated loss of about 1.6 dB for the example considered. Application of the receiver to QOFDM proved impractical due to the required constraint length, which is longer than that required for BOFDM. Further work with QOFDM is deferred to Chapter 5. Chapter 5 65 Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels In the previous chapter a truncated MLSE receiver was derived for channel-averaging OFDM which was able to approach the MFB quite closely for a range of normalized Doppler rates. A serious limitation of this approach is its complexity, which increases exponentially with the channel constraint length L. For serial modulations, the traditional alternatives have been linear and nonlinear decision feedback equalization techniques whose complexity is only linearly dependent on L. While there has been substantial activity in this area, particularly since the early seventies, to date (1997) there are few publications, [5], [7] and [17] in which equalization techniques have been applied to channel-averaging versions of OFDM. In this chapter, we look quite closely at the theory of equalization as modified for OFDM in flat Rayleigh fading, ultimately resulting in the derivation of a new decision feedback equalization structure having substantial complexity and performance benefits over the current state of the art. Section 5.1 derives the optimal linear MMSE equalizer for OFDM in flat Rayleigh fading. Section 5.2 derives a discrete symbol-spaced equalizer, also by the MMSE criterion. Section 5.3 reviews classic DFE as developed for serial modulation and Section 5.4 describes the application of this approach to OFDM in previous work. The idea of reshaping the overall impulse response of the transmitter, channel, and front-end filter14 to some desired impulse response (DIR) is reviewed in Section 5.5. Section 5.6 derives a DIR DFE based on the zero-forcing criterion (ZFDIRDFE) and in Section 5.7 an improved DIR DFE based on the MMSE criterion (MSEDIRDFE) The term "front-end filter" is defined in Section 5.3. Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels. 66 is derived. The combination of a Viterbi-type sequence estimator in combination with MSEDIR shaping is considered in Section 5.8. In Section 5.9 MSE performance surfaces are defined for a simplified fading channel model and used to compare MSE (LMSE) equalization to ZFDIRDFE and MSEDIRDFE. Simulation results for the receivers are presented in Sections 5.10 and 5.8. Section 5.12 summarizes the conclusions of this chapter. 5.1 An Optimal Linear Equalizer We consider an OFDM system in which the transmitted signal is represented as i i V _ 1 s(t) = —=y^aneJ%nt, 0<t<T0. (5.1.1) In (5.1.1) the pulse shaping p(t) has been assumed rectangular primarily to simplify the notation to follow, but also because the investigation of other pulse shapes in Chapter 4 showed no advantage in terms of Pb-The received signal is given by ^ " = ° (5.1.2) N-l = q(t) E a n ^ n t + n(t) where z(t) is the fading waveform, q(t) = z(t)/y/To and n(t) is independent AWGN. We require for the equalizer a linear structure whose estimate of the nth data symbol is given by To . an = -^= [ r(t)cn(t)e-j2tntdt (5.1.3) V-io J Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 67 and where cn(t) is a weighting function to be determined by the MMSE criterion. The MSE is MSE = E = E \CLn Q>n \a„.\ 2E[Re[ana*]] + E f f l. (5.1.4) Expanding r(t) it can be shown that T o N-l E ^ / q(t)cn(t) £ e - m ^ d t r, m=0 0 T0 (5.1.5) x J q*(u)c*n(u)e^n-^udu + ^J \cn(t)\2dt and To . 2E[Re[anal]} = ^ J (q(t)cn(t) + q\t)cn(t))dt. ' (5.1.6) o Combining (5.1.4), (5.1.5) and (5.1.6) we obtain the MSE as a function of cn(t) To N _ x MSE = ^ fq(t)cn(t) J2 e ^ - ^ d t 0 { m=0 0 To X / \ jl? ( i i-m ) t i , . dt (5.1.7) To J {q(t)cn(t) + q*{t)c*n{t))dt + Ea. The derivative of (5.1.7) w.r.t. cn(x) is To J V _ 1 dMSE Ea 2dcn{x) T0 I q(t)cn(t) £ e - ^ ( B - m ) t d * Q m=0 X g ( x j e ^ ' + — cn(x) J-o (5.1.8) Ea q*(x). Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 68 Setting dMSE _ s c a]j ng by To/Ea and substituting x = u we obtain the integral equation whose solution yields c(t) n2cn(u) = q*(u) - ^Mq*(u)Y^e^-m> j q ^ c ^ e ' ^ - ^ d t (5.1.9) m=0 Q where ry2 = No/Ea is the noise to signal ratio. Defining • To B„-m = j q(t)cn(t)e-3%in-m)tdt (5.1.10) 0 (5.1.9) can be written as N-l n2cn{u) = q\u) - j%q\u) E B n - m e ^ { n - m ) u . (5.1.11) m=0 If we apply the linear operator To L[-] = '-J q(u)[-]e~j^n~l)udu (5.1.12) o to (5.1.11) and define To Vn_j = J \q(u)\2e-^n-^wdu ' (5.1.13) o we obtain N-l m=0 0 < n-l < N - 1 (5.1.14) a system of linear equations which can be solved for Bn. Equations (5.1.11), (5.1.13) and (5.1.14) specify the continuous linear MSE equalizer for calculating an. The solutions to (5.1.14) can be concisely expressed in matrix form. Defining b n = [ B n , B n - i , B n - N + i ] , v n = [Vn, Vn-i, • • •, K-JV+i] and V a square Hermitian Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 69 Toeplitz matrix with (m,l) element V m i = Vm-\, 0 < l,m < N — 1, (5.1.14) can be written Vbn = V^bvn - bnV (5.1.15) which has for its solutions bn = v ^ V n t V + T ? 2 ! ) - 1 . . (5.1.16) Equation (5.1.16) indicates that bn and consequently the weighting function cn(t) depend on n, the index of the data symbol being estimated. This dependency arises because ISI experienced by a given data symbol depends on its location (tone) within the OFDM block. For example, centrally located data symbols have several adjacent channel interferers on either side but data symbols at the edges (n = 0 and N — 1) have interferers on only one side. As can be expected from physical considerations, symmetries can be used to reduce the calculation required. The Vn of (5.1.13) are recognized as the tone correlations defined in Chapter 4. We have Vn = Vln and combined with definition of v n it follows that v* = v^_j_ n , where the superscript B indicates the vector's elements are reversed in order. Accordingly, we also have b* = b^_ln. In general, solution of (5.1.16) requires an N x N matrix inversion followed by N/2 row vector by square matrix multiplications. Thus the exact solution for the linear MMSE equalizer is fairly tedious unless N is restricted to be quite small, possibly N < 16. For the larger values of N used in this work we consider approximations to the optimal solution to reduce the complexity of the equalizer. 5.1.1 Transform Approximation A very useful simplification would be to eliminate the explicit matrix inversion required by (5.1.16). For this purpose we will apply the discrete frequency Fourier Linear and Decision Feedback Receivers for OFDM on Fiat Rayleigh Fading Channels 70 transform (DFFT)15 denned by Xn = J x{t)ej2tntdt (5.1.17) and its inverse (iDFFT) defined by x{t) = YJXke^ki. (5.1.18) k This is done in Appendix C along with some relevant properties. Taking an iDFFT with respect to the index n of (5.1.14) yields , 2 £ i W ^ = V ^ £ K > ^ (5.1.19) n n m=0 n or J V - l r)2e-J^H(t) = e-J^H^/Tov(t) - b(t) £ Vm-i<T3^ m=0 -^ ft-^ t5(0-C-^ 't6(t)^ "Vl:e-^ fct (5.1.20) = e J T o k=-l where the tilde accent (•) indicates a periodic or time-aliased function as in x(t) — r 0 £ x ( i - n T o ) . n The last summation in (5.1.20) is not quite an iDFFT of V^. However if we assume L i—kt Vj. has significant values only over the range — L < k< L then v(t) — YI Vke 0 • k=-L Comparing with (5.1.20), it is apparent that provided L < I < N — 1 — Iwe can write v(t) + W2 Note that (5.1.21) reveals b(t) to be a smoothed function relative to v(t), thus the restriction on the significant values of V}. imply a similar restriction on the values of 1 5 Not to be confused with the DFT or FFT. Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 71 Bk. Therefore we assume that the significant values of B^ are also restricted to the range —L < k < L and applying this to (5.1.11) yields 772c(i) = Vnq*(t)-q*(t)b{t) (5.1.22) from which with (5.1.21) it is readily shown that V(t) + T]2 VT0q*(t) (5.1.23) T o £ k ( * - ™ r 0 ) r + 772 For the q*(t) used in (5.1.2) this reduces to c ( ( ) = A O * * « 0 < * < T 0 (5.1.24) x(t) where x(i) = |z(£)|2 + " 2 has been introduced due to its frequent appearance in what follows. The function x(t) is the inverse Fourier transform of the channel autocorrelation function in the frequency-domain, i.e. a time-domain version of the channel's power spectrum. Equation (5.1.23) is an optimal weighting function only for use in estimating those data symbols ak whose indices are restricted to L < k < N — 1 — L, i.e. the data symbols transmitted near the band edges of the OFDM waveform optimally require different weighting functions determined from the more complicated expressions given previously, even with the restrictions on the significant values of V^. Practically though, c(t) as given by (5.1.23) is expected to be a useful approximation for estimating all data symbols, since although the data symbols near the band edges are suboptimally detected, they also suffer from the least ISI. We see the difference between exact equalization with cn(t) and approximate equalization with c(t) is mainly a matter of edge effects. Thus there should be little difference in averaged over all N tones, when L •< N. Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 72 Note that except for a-domain change (5.1.23) has the familiar form of a continuous matched filter followed by a filter having a periodic response implementable as an infinitely long tapped delay line [30]. However, since in this case the OFDM block has finite duration To, it is convenient to determine c(t) directly in the time-domain using (5.1.23) or (5.1.24). 5.1.2 Aliasing Approximation The two previous works, [5] [7]16 concerning the equalization of channel averaging OFDM in fading both used a discrete model of the OFDM signal. In the notation of this work17, the model for the transmitted signal is sm = -= E anej2£nm, 0 < m < i V - l (5.1.25) v n=0 and for the received signal rm = Zm.Sm + nm, 0<m<7V —1. (5.1.26) These equations may be interpreted as the result of sampling the waveforms of the continuous model at rate N/TQ and then setting the scaling factor l/y/To = 1/y/W. As discussed in Chapter 4 this is a sub-Nyquist sampling rate and results in an aliasing effect which causes all tones to have the same pattern of ISI. Viewed in the frequency-domain, the linear convolution of the channel impulse response with the data is modified by the sub-Nyquist sampling into a circular convolution of an aliased impulse response with the data. Thus the rate N/TQ sampling increases the level of ISI irreversibly, but following the arguments of the Transform Approximation this effect should be quite small on the error rate averaged over all tones, provided L <C N. The principal benefit of the aliasing approximation is that the optimal (given the sampling rate) MMSE equalizer 1 6 A third, [17] was concerned with an F M version of O F D M , O F D M - F M . 1 7 The transform scaling has been adjusted to normalize energies; Es = Ea. Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 73 has a weighting function which is independent of the data symbol's index as shown in Section 5.2.. This follows directly from the fact that a finite discrete sequence and its DFT can be viewed as periodic functions. 5.2 A Discrete Ts Spaced Linear Equalizer Accepting a sampling interval of TQ/N = TS seconds at the input to a discrete equalizer, we derive a linear MMSE (LMSE) equalizer which is optimal from that point on. The desired estimate is formed as , • • N-l 1 *r—> ; • 2-nmn an = —= y rmcme 1 " . (5.2.1) m=0 Substituting (5.1.25), (5.2.1) and (5.1.26) into (5.1.4) it can be shown that the normalized MSE for the discrete equalizer is A , c r N-l o N-l , N-l \cm\ \qm\ + 2 ^ l C m l ~ ~7t? (Cmqm + S ' m ) + l- (5.22) a n N - n vN „ m=0 m=U m=U Setting oEdc = 0 yields an equation for c\ cMl\2+ ^~qi =0 (5.2.3) which has the solution 'Nqf ci = N\qi\l + V2 * I z? I2 i 2 (5.2.4) L , 0 < I < N - 1 for g/ = zi/y/N and x; = |g'|2+i72. Equations (5.1.24) and (5.2.4) are deceptively similar in form, yet the approximations each represents are quite distinct. Again, with a-domain change, the equalizer can be recognized as a matched filter followed by a tapped delay Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 74 line, but now and without further approximation the delay line is finite with N elements used to circularly convolve the outputs of the matched filter. While in this case it is more efficient to implement the equalizer directly in the time-domain as. specified by (5.2.4) and (5.2.1), the frequency-domain interpretation becomes important when the equalizer is generalized to include decision feedback. The most important point concerning (5.2.4) is that like (5.1.24) it does not require an inversion of an JV x N matrix, as does the expression in (5.1.16). An expression essentially similar18 to (5.2.4) was presented in [5]. In that work, the error criterion was defined as the difference between an estimate of the transmitted waveform and the actual transmitted waveform — Sm ' (5.2.5) instead of the estimated and actual data as defined by (5.1.4). Thus (5.2.4) is not a new result, but merely establishes the equivalence of the two criteria for this case. 5.2.1 Optimal Linear Mean Squared Error Unfortunately it is not possible to get an exact expression for Pj, thus we resort to the more tractable MSE. The normalized MSE resulting from use of the optimal c\ as given by (5.2.4) reduces to a simple form. Rearranging (5.2.2) yields N-l MSE 1 • * 1 m=0 A T ^ ] cmxmcm „T2Re Ea W m N and applying (5.2.4) yields N-m=o E +1 (5.2.6) MMSE . l ^ o ' , r o ^ m=0 or 2 N-l , where the superscript (-)° designates optimal. 1 8 An adjacent channel interference term was used in place of the noise term here. MMSE 77z ^ 1 Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 75 5.3 Classic Decision Feedback Equalization As with linear equalization, nearly all work with decision feedback equalization (DFE) has occurred within the context of serial modulation on an ISI channel in the time-domain. Consequently, the following description is presented within that context. The structure of the classic DFE [31] is shown in Figure 5.1. IV r(t) C(f) + decis ions sampler D(f) A Figure 5.1 Classic DFE structure It is designed to work with a received signal of the form oo r(t) = . E anh(t - nTs) + n(t) (5.3.1) where h(t) is the combined response of the transmitter pulse shaping filter and the channel impulse response. The noise n(t) is generally assumed to be AWGN. There are two basic sections, the feedforward section containing the feedforward filter C(f) operating on the received waveform, and the feedback section containing the feedback filter D(f) operating on previous decisions. The impulse response of either filter may be finite or infinite in length. The feedforward section is a linear equalizer optimally consisting of a matched filter followed by a sampler and a tapped delay line with spacing equal to the symbol duration Ts, or optionally may consist of a fractionally spaced equalizer operating above the Nyquist rate [9] which incorporates matched filtering as part of its overall response. The feedback filter has the structure of a tapped delay line with spacing Ts and is used to form an estimate of the ISI due to the previously detected Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 76 data. ISI from future data is minimized by C(f). The impulse responses of C(f) and D(f) are generally chosen by either the MMSE or zero-forcing (ZF)19 criteria. The idea of the DFE is to subtract this calculated ISI from the sample entering the decision element before making the current decision. Assuming previous decisions are correct, the DFE makes a decision based on a sample with less residual ISI than a comparable purely linear equalizer, and should therefore have a lower error rate. It has been shown [30] that with the perfect decision assumption, the MSE of the MMSE DFE is upper bounded by the MSE of the purely linear LMSE equalizer.' A potential problem that arises with DFE is error, propagation due to incorrect decisions in the feedback section. While the derivation of a DFE requires the assumption of ideal decisions for tractability, to be useful a DFE must feed back data from previous decisions, some of which will be in error. Thus there is a tradeoff regarding the length of the feedback section. A long section removes more ISI when decisions are correct, but risks greater error propagation when an error is made. 5.4 Frequency Domain Zero-Forcing Decision Feedback Equalization A technique called frequency-domain zero-forcing equalization (FDZFDFE) was developed for OFDM and presented in [7]. The basic idea is to immediately convert the received signal after sampling into the frequency-domain, and then to apply the classic [9] matrix-based DFE structure, originally posed in the time-domain for serial modulation, to the resulting signal. In the notation of this work (5.1.26) is transformed into N-l (5.4.1) ZF: elimination of ISI at the sampling instant over a certain range. Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading. Channels 77 where Zk = z m e - ^ m k and NK V - l £ n m e - ^ m k . The FDZFDFE has m=0 m=0 the form (5.4.2) where the coefficients CK were determined by the inversion of an (F + 1) x (F + I) channel dependent matrix and the actual data, rather than the decisions, was used in the feedback section [32]. F + 1 is the number of CK tap coefficients and B the number of feedback coefficients. In [7] F was chosen as 10 and B as 15 for N = 512 and a normalized Doppler rate of fjy = 2.05. Simulation results showed FDZFDFE required 6.5 dB less E^/NQ than linear ZFE, and 2 dB less than an approximate method presented in [5], both at PS = 10~3. Thus these results represent the best PS performance obtained for OFDM prior to the work being reported in this thesis. 5.5 Obtaining a Desired Impulse Response (DIR) With the classic DFE structure the causal20 component of the combined transmitter impulse response and channel impulse response may be so long as to preclude complete cancellation of ISI from previous decisions, either due to the required filter length or error propagation effects. Typically a workable length is found through simulation which is a compromise between error rate and complexity. An alternative to the simple truncation described above is to reshape the combined impulse response to some desired impulse response (DIR) of fixed length. An important but not obvious point is the selection of the DIR. Simple examples include the preceding truncation, and the unit sample function which reduces to linear ZFE. More sophisticated examples, in the context of serial modulation, have appeared in [29], [33], and [34]. The first two publications are principally concerned with obtaining a 2 0 causal: that part defined for i > 0. Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 78 shortened impulse response to simplify a subsequent MLSE, but are of course applicable in the context of DFE as well. In [33], the DIR chosen was a truncated version of the original response and was shown to be an improvement over simple truncation without shaping. K DIR of fixed length H Combined Transmitter and Channe l 1 r ( " . Reshap ing D F E — • A an n(t) Figure 5.2 Forcing a DIR of fixed length before DFE In [29] an MSE criterion in combination with a total energy constraint on the DIR is used to select the DIR, which is found by determining the eigenvector corresponding to the minimum eigenvalue of a certain channel dependent matrix. Included comparisons showed error rate improvements over the shaped truncation method of [33]. - y ( t ) Reshap ing Figure 5.3 DIR obtained by concatenating ZF and FIR filters Subsequently [34] suggested modifying a ZFE to obtain a finite impulse response for DFE. The idea is to obtain the DIR by concatenating a linear ZF filter with an FIR filter of fixed length. The FIR is determined by fixing the 0th tap weight to a constant, and then selecting the remaining coefficients to minimize the noise variance at the equalizer output. Noise variance estimates indicated that most of the attainable improvement required a DIR length of only 4 or 5 for the channel investigated. Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 79 5.6 A Zero-Forcing Desired Impulse Response Decision Feedback Equalizer (ZFDIRDFE) In the following we apply the approach of [34] in order to obtain a ZFDIRDFE for OFDM. The estimate of the nth data symbol is given by 1 N-l x B a n = — 7 = E CkRn-k "=  E DkUn-k (5.6.1) V k=0 ^ k=l N-l .27r where Rk = -j^ r m e ~ ^ r n k is the DFT of rm given by (5.1.26), and Ck and Dk m=0 are the coefficients of the equalizer filters. Note that it is rather ambiguous to refer to Ck as a feedforward filter in this context for two reasons. The first is due to the circular nature of the ISI as viewed in the frequency-domain. The index k of Zk increments and decrements modulo N, thus there is no clear distinction between past and future ISI. The second reason is that Ck will be designed to remove ISI due to an_k for B < k < N — 1 when the estimate an is being formed. To avoid this ambiguity we will refer to Ck as the front-end filter, and continue to refer to Dk as the feedback filter. Rather than perform the N. circular convolutions indicated by the first term of (5.6.1) directly, it is more efficient to define a new sequence Vm = rmcm (5.6.2) where and form the estimate N-l C™• = VKF E C k e ^ k m (5.6.3) k=0 1 B ln = Yn /= E Dk^n-k (5.6.4) ^ k = l where all N Yn are obtained simultaneously from an FFT of yr Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 80 Of all the possible ZFE's, it is shown in [35] that the ZFE having a matched filter (MFZFE) is the ZFE which maximizes the SNR at the equalizer output. The Output of a matched filter applied to the received signal (5.1.26) is V-m = f m z m \Zm\ Sm ~\~ ^mZjr, (5.6.5) Jzm^m i <0m^m where (f>zm = \zm\2. (5.6.6) Applying u m to a filter with frequency response l/(f>Zm yields „. ' „ i n m Z m ,c (• n\ um = sm + — . . (5.6.7) Pirn The sample u'm can be interpreted as the MFZFE estimate of sm. It is applied to a filter with response dm to obtain a DIR of dm, i.e. . cm = = (5.6.8) • 4>zm which yields for yr ym — °mu'm \ , Oz,„ (5.6.9) * A where nv = Urnlm m is a colored Gaussian noise term. Its average variance is AT N ~ l U I2 • ^ 4 V ^ - . (5.6.10) The noise variance of (5.6.10) is minimized by the selection of the frequency-domain coefficients Dk where 1 3 drn = ^JlDk^mk (5-6.11) Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 81 and Do — viV. Note this method has the peculiar feature that Dk must have some constraint or the minimum variance solution would Dk — 0, 0 < k < B, i.e. no output at all. Denning hm = l/<f>Zm ' (5.6.12) (5.6.10) can be written as a™ = ~ E dmhmdm (5.6.13) N m=0 which has the equivalent frequency-domain expression B B < = ^kY,D*ED?H'-n (5.6.14). where HK = i ^ J^e"^™*. v m=0 Defining d H = [Du D2,... ,DB]*, h H = [HUH2, • • • HB]* and B x B Hermitian Toeplitz matrix H with (I, n)th element Hi n = B\\_n (5.6.13) in matrix form is B U - (NH0 + 2v/iVRe dHh + d HHd). (5.6.15) a2 da2 The value of d for which (5.6.15) attains its minimum is determined by setting = 0. This yields No NVN which has the solution (V/Vh + H d ) = 0 (5.6.16) d = - V W H - 1 ! ! . (5.6.17) Using (5.6.17) in (5.6.15) we obtain a frequency-domain expression for the optimal noise variance ^ ^ ( t f o - h ^ H - V ) . (5.6.18) Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 82 5.6.1 Optimal Mean Squared Error Taking a DFT of (5.6.9) to obtain Yn yields Yn = - = > j Dkan_k + NYri (5.6.19) mn is the DFT of nVm. Using (5.6.19) in (5.6.4) yields an = an + NYn (5.6.20) which shows that the ZFDIRDFE has no residual ISI when previous decisions are correct. Therefore for this equalizer the MSE is equivalent to the noise variance (5.6.18), and the normalized MSE, for the optimal choice of Dk is 5.6.2 Algorithm Summary 1. matrices h and H are formed using (5.6.6) and (5.6.12) 2. H is inverted to obtain d from (5.6.17) 3. (5.6.11) yields dm, (5.6.8) cm, and (5.6.2) with an FFT yields Yn 4. an is obtained from (5.6.4) The first three steps need only be performed once per OFDM block. The final step is repeated once per data symbol. 5.6.3 Structural Comparison to FDZFDFE An interesting structural difference between ZFDIRDFE and FDDIRDFE concerns the pattern of ISI which is cancelled. The ZFDIRDFE exploits the circularity of the ISI to cancel all ISI except that from B previous (in a circular sense) data symbols. The FDZFDFE does not exploit the circularity; consequently less ISI is cancelled by the front-end filter and B must be larger for FDZFDFE than ZFDIRDFE. (5.6.21) Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 83 Of course, the FDZFDFE could be modified to provide the same ISI cancellation pattern as does the ZFDIRDFE. However this could pose a problem for large F because the complexity of the required matrix inversion varies with (F + 1) . 5.7 A Minimum Mean Squared Error Desired Impulse Response Decision Feedback Equalizer (MSEDIRDFE) While the ZFDIRDFE is interesting for its structure and performance, it is reasonable to question whether the ZF criterion is a very good choice to use for the basis of a nonlinear DIRDFE. Certainly in the context of linear equalization for serial modulations it is well understood that ZF generally causes increased noise enhancement relative to a comparable LMSE. The LMSE results in a lower error rate than the ZFE when the combination of noise and residual ISI is less damaging than no ISI in combination with enhanced noise, and this is generally found to be the case21. Motivated by this observation we apply a minimum MSE criterion in the following to the selection of a DIR to derive an MSEDIRDFE for OFDM. Aside from changes to accommodate OFDM in fading, the formulation here differs from [29] in that there is no energy constraint placed on the DIR, and the DIR is not obtained in terms of an eigenvector. In order obtain the same structure for the MSEDIRDFE as found for the ZFDIRDFE, the form of the estimate is unchanged. It is repeated here for convenience. The estimate of the nth data symbol is given by 1 N-l B T T T / . CkRn-k TTT / Dk^n-k (5.7.1) N-l where Rk = -4= r m e * **mh is the DFT of rm given by (5.1.26), and and Dk m=0 are the equalizer coefficients. In the time-domain cm is again given by (5.6.3), but the 2 1 Constant amplitude, purely phase distorting channels are an exception since there is no noise enhancement problem. Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 84 expression for dm in the context of MSEDIRDFE is modified to 1 B dm = n=YjDke^hm. (5.7.2) Rather than perform the iV circular convolutions indicated by the first term of (5.7.1) directly, it is more efficient to define a new sequence Vm — rmcm (5.7.3) and form the estimate 1 B an = Y n - —= y^-Dkan-k (5.7.4) where all N Yn are obtained simultaneously from an FFT of ym. We begin with a coupled equation approach [36] to obtain some insight into the nature of the front-end filter Ck and to derive some expressions for the MSE of relatively simple form. Defining and (5.7.1) in matrix form is C H = - ^ [ C O , C I , . . . , C A T - I ] , (5.7.5) dE = -L=[DuD2,...,DB], (5.7.6) V-/V rn = [-^w-O? Rn-1, • • • , Rn-N+l], (5.7.7) l n = [an-0, O-n-1, • • • , an-B\ (5.7.8) a„ = c Hr n - dHa n fHt (5.7.9) Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 85 where fH = [cH,-dH] and t T = [r^aj . It is well known [23] that the MMSE solution for f satisfies where and $f = p E[rnr*] E [r n aH] E (a n rH] E[ana* E [ r n <] E [a n <] Evaluating the moments E[Rn^iR*n_k E„ En and where E[an_/a*_fc] = EaSk-i x N-l ®Zk = — 7 = E Zm+kZm. v m=0 (5.7.10) (5.7.11) (5.7.12) (5.7.13) (5.7.14) (5.7.15) (5.7.16) Expanding (5.7.10) yields a pair of coupled which must be simultaneously satisfied; N-l / , x 1 B Y(^Zk_l + VHk_l)Ct--mY,Z*-iDi = Z-'' 0<1<N-1 (5.7.17) k=0 and ^ t i • N - l ^Y^zi-kCt-pt = VNSh \<1<B. (5.7.18) The form of Ck can be deduced by expanding in (5.7.17) and defining Qi, the linearly equalized fading samples (frequency-domain) as N-l Qi — —f^ ^2 Zi_kCk (5.7.19) £=0 Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 86 which yields B-l , N-l 2 TCt = Z-» + -±= E - "7*7 £ - < & r i z * . * < ' < * (5.7.20) V i v jfc=l-J V i v jfc=0 from which it is apparent N-l 'N ^ = 77*7 E Z*W*+»- (5.7.21) k=0 Equation (5.7.21) shows that the front-end filter can be viewed as the cascade of a filter matched to the channel fading waveform followed by an additional feedback dependent filter specified by the coefficients Wk. The Wk expressed in terms of the DIR and linearly equalized fading samples are N - Q Q V2. W h = D* Q \ \<k<B (5.7.22) n Wk = ^S±, B + l < k < N - l . T We shall return to (5.7.22) with further developments. Using (5.7.21) and applying the linear operator to (5.7.17) yields • 1 N-l ' N-l 2 -^-1 B — S^$7 $7 W* ,1-1- — $7 V^$7 D* = $ 7 k=0 m=0 V i m=0 V J m=l 0 <p < i V - 1. (5.7.24) Transforming (5.7.24) and dividing by <pZm yields 1 < + . i / 2 < - C - l = 0 (5.7.25) Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 87 from which we obtain an explicit time-domain expression for the optimal wm in terms of the feedback filter (5.7.26) or equivalently cm = —7 " j — 2 ~ : (5.7.27) Equation (5.7.27) shows that the time-domain response of the front-end filter for MSEDIRDFE is the product of the responses for the LMSE equalizer and a factor based on the DIR. This factor has the frequency-domain transform (dm + l)^\y/N,DuD2;...,DBy (5.7.28) which is the DIR. augmented with a coefficient Do = \/N. Defining the frequency-domain sequence in (5.7.28) as the augmented DIR, we see that MSEDIRDFE determines an augmented DIR which is effectively constrained to have DQ = y/N. Further insight is obtained by rewriting (5.7.18) as J V - l where r .0, I < k < B undefined, otherwise. Transforming and conjugating (5.7.29) <t>zmwm - d*m ~ 1 = 1-k (5.7.31) and subtracting from (5.7.25) yields wm = . (5.7.32) T Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 88 or with a further transform Wk = - 1 k n2 (5.7.33) From the definition of Tk in (5.7.30) and assuming ry2 > 0, (5.7.33) shows that Wk = 0, I < k < B, precisely the range where the DIR is allowed nonzero values. This in turn implies, with (5.7.22), that Dk = Qk, 1 < k < B, i.e. the DIR coefficients are exactly equal to the linearly equalized fading samples. There still remains the problem of determining the DIR. Recalling the MSE |2" MSE = E =. E \an an KI 2E[Re[a„<]] + Ea (5.7.34) we seek the joint minimization of (5.7.34) with respect to Ck and Dk. For the first term in (5.7.34) E 1 N N-l N-l 7=0 k=0 1 N-l B 1=0 k=l B B /=! fc=l Evaluating the moments in (5.7.35) and defining Xk-i = $z f c _, + V^V r}26k_t (5.7.35) (5.7.36) yields E N-l N-l En = T= V V CtCtXk-i - 2Re = V V CjDtZk-i N-l B + ^EiAi a . (5.7.37) Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 89 Similarly, the second and third terms of (5.7.34) are N-l 2Re[arag*] 2Re * E c>z-> 1=0 (5.7.38) and E Ea = 1. (5.7.39) Combining (5.7.34), (5.7.37) and (5.7.38) and transforming to the time-domain yields the normalized MSE A / r o c 1 N ~ l 1 N ~ l MSE _1_ ^ * f * *^ l En N m=0 m=0 j AT-1 ' j N-l - £ 2 R e [ 4 ^ ] + + 1 (5.7.40) m=0 m=0 in terms of cm and dm. Expanding the first term of (5.7.40) with (5.7.27) and expanding d*m with its transform yields N-l N-l ^ ] cmxmcm ~ ^ ] ( z m ~t~ zm,dm){zm ~\~ zmdm) m=0 m=0 = E 5 1 (z*m + zmdm) [ zm + Zm^Z Dte-^lm (5.7.41) — This expression can now be differentiated w.r.t. Dk to obtain N-l N-l d 2dDi E m=0 7^ E (9m+9mdm)e j 2 N k r - Gk + —7= E DiGk-I B (5.7.42) where 5m (5.7.43) Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 90 Similarly it can be shown for the remaining terms d N-i • B — — ]T 2Re[cmzmdm] = Gk + ^ =Y, DiGk-h (5.7.44) m=0 v <"=1 ' 'N-l 8 2 R e t c ™ 4 J = <?*, (5.7.45) 2dDk n and K m=0 Combining (5.7.40)-(5.7.46) yields an implicit expression for the optimal Dk i B . -= Y Gk-iDi = D k - Gk. (5.7.47) N i=i Defining the B x B Hermitian Toeplitz matrix G with (k,l)th element G^i = d T = [Di, D2,... ,DB] and g T = [G\, G2, • • •, GB], (5.7.47) in matrix form is 1 which has for its solution Gd = d - g (5.7.48) 1 d = ( I - ^ = G ) g. (5.7.49) The elements of d as determined from (5.7.49) are the desired D°. 5.7.1 Minimum Mean Squared Error A frequency-domain expression for the MMSE is readily obtained from [23] MMSE = Ea - fHp. (5.7.50) Evaluating fHp and normalizing by Ea yields Ea • N 1 - -U< k=Q (5.7.51) =1 'TV Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 91 Alternatively, applying Parseval's relation to (5.7.51) yields J V - l MMSE _ J _ o Ea * N L>CmZm m = ° (5.7.52) N-i, , , o + l ) a pair of time-domain expressions in terms of the front-end and feedback filters respec-tively. Note that the first line of (5.7.52) is identical in form to (5.2.7) found for linear equalization, but differs due to the choice of cm . It can also be seen that in the absence of feedback (5.7.52) reduces to (5.2.8) as found for linear equalization. 5.7.2 Algorithm Summary 1. matrices g and G are formed using (5.7.43) 2. (l - --J=G) is inverted to obtain d from (5.7.49) 3. (5.7.2) yields dm, (5.7.27) cm, and (5.7.3) with an FFT yields Yn 4. an is obtained from (5.7.4) The first three steps need only be performed once per OFDM block. The final step is repeated once per data symbol. 5.7.3 Structural Comparison The MSEDIRDFE has the same structure as the ZFDIRDFE, the difference between the two lies in the selection of the DIR. In addition both require essentially the same amount of computation when each has the same number of feedback taps B. Thus the comparison of Section 5.6.3 between ZFDIRDFE and FDZFDFE applies as well between MSEDIRDFE and FDZFDFE. A key advantage of MSEDIRDFE over ZFDIRDFE carries over from their linear counterparts. An MSEDIRDFE exists in cases where a ZFDIRDFE does not, specifically Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 92 when some values of zm are 0 as can be seen from (5.6.12) and (5.6.17).. Due to finite wordlength effects it is not necessary for zm to be exactly equal to 0 to encounter numerical difficulties with the latter. 5.8 Combined Mean-Squared Error Desired Impulse Response Shaping and Sequence Estimation (MSEDIRSE) The truncated MLSE of Chapter 4 was quite successful when applied to BOFDM, but its use with QOFDM was precluded by the required constraint length. We are now in a position to rectify that situation by setting the constraint length to a desired value using the methods derived for MSEDIRDFE, and then processing the resulting controlled ISI sequence with a truncated MLSE based on the DIR. Assuming we have obtained the sequences DN and YN as per Section 5.7, we express YN in terms of the DIR as 1 B Y n = ~7T7 E D^n-k + Nn (5.8.1) where DQ = VN and Nn is a Gaussian noise term with correlation function • N N-l fc=0 Equation (5.8.2) indicates that the noise terms of (5.8.1) are generally correlated in the presence of fading. However, we shall treat the Nn as if they were AWGN to obtain a suboptimal but simplified sequence estimator (simulation results presented in Figure 5.16 validate this approach). With this assumption it is easily shown that the ML metric to be maximized w.r.t. an, 0 < n < N — l is N-l • *= - E 71=0 Equation (5.8.3) may be evaluated by defining 1 B 1 k=0 n—k (5.8.3) Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 93 for which B Am — A r Ym r-r-r ^ ^ Dk®"m — k Nk=0 (5.8.5) and determining the sequence an which maximizes A m , 0 < m < N — 1, recursively via the Viterbi algorithm. 5.9 MSE Performance Surfaces The preceding MSE expressions provide a basis for a theoretical comparison of the equalizers. These expressions have been used to generate sets of performance surfaces which present the mse ° for the LMSE, ZFDIRDFE, and MSEDIRDFE as functions of the noise to signal ratio n2, the number of feedback taps B, and the particular channel fading waveform. The fading waveform is modeled with a deterministic vector z to make the MSE evaluations tractable. It is T r i /•** = !, & 4,5,6,42,43 Z =[^'--'^]' {Zk = e, ^ = 4,5,6,42,43 (5.9.1) 0 < e < 1. Selection of e < 1 models fading at the indexes designated in (5.9.1). Two sets of surfaces are presented in Figures 5.4 and 5.5, to illustrate the effects of fade depth. In Figure 5.4 e = 0.1 and in Figure 5.5 e = 0.001. The normalized optimal MSE (mse°) for a specific equalizer is represented by the vertical height of its performance surface. The horizontal axes, pictured as a plane orthogonal to the vertical dimension, show the variation of mse° with B and n2. Note that for equalizers designed by the MMSE criterion, mse0 = MMSE/Ea. Figure 5.4 shows that with the shallow fades, for most of the explored ranges of B and rj2, that mse° for ZFDIRDFE is lower than for LMSE, and it is lowest of all for MSEDIRDFE. This statement has been verified by plotting a version Of Figure 5.4 (not Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 94 Figure 5.5 Normalized optimal mse surfaces for LMSE, ZFDIRDFE, and MSEDIRDFE at N = 64 and e = 0.001; deep fades Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 95 shown) without the LMSE surface so as to render the entire ZFDIRDFE surface visible. It is interesting that for small B and high rj2 there are regions where the performance of ZFDIRDFE is actually worse than for LMSE. In these areas the ZFDIRDFE is approaching a ZFE and the noise enhancement characteristic of ZFE is significant. At rj2 « —20 dB there is very little difference between ZFDIRDFE and MSEDIRDFE, and in this region most of their improvement relative to LMSE is at-tained for only B = 2. As ry2 increases the performance of ZFDiRDFE deteriorates more rapidly than that of MSEDIRDFE, increasing the discrepancy between the two methods. Figure 5.5 shows that for deeper fades, the performance differences between the methods increase. Again MSEDIRDFE has at every point the lowest mse ° and attains most of the improvement to be had relative to LMSE for 73 = 2 or 3. The situation is quite different for ZFDIRDFE. It can be seen that increasing B mitigates its noise enhancement problem, but we must have B = 6 at least, to do as well as LMSE, and this is adequate only for small rj2. Although increasing B improves its performance, at B = 20 it is still worse than MSEDIRDFE with only B = 4. 5.10 Simulation Results The Ps performance of the equalizers derived in this chapter was evaluated by Monte Carlo simulation. The simulation framework, concerning OFDM in flat Rayleigh fading, is identical to that described in Chapter 4, except that the sampling rate was reduced to take advantage of the structural simplifications afforded by the aliasing approximation. As discussed in Chapter 4, perfect knowledge of the channel impulse response is assumed available at the receiver. The simulations were made long enough to obtain 95% confidence intervals of approximately ±10% of Ps. We begin with a comparison of ZFDIRDFE and MSEDIRDFE, continue with an evaluation of MSEDIRDFE for a range of parameters, and then compare its performance to previous work. Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 96 5.10.1 Feedback Data To include the effects of error propagation in the results which follow, all the simulations use the actual data symbol decisions in the feedback loop as opposed to the ideal (correct) decisions previously assumed to make the derivations tractable. An exception occurs with the initial decisions made for each OFDM block. Due to the circularity of the ISI resulting from the aliasing approximation, the detection of every data symbol requires knowledge of B previous, in a circular sense, data symbols. In order to preclude an error propagation problem damaging the first decision, the first decision is made with B correct data symbols in the feedback loop. This decision is then shifted into the feedback tapped delay line, and only B — 1 ideal data symbols remain. Clearly, after.the B initial decisions are made, all subsequent decisions involve only data symbols from actual previous decisions in the feedback loop. In the following, B turns out to be quite small, e.g. B = 3 for N = 128 and fr> = 0.01, an overhead of less than 3%. 5.10.2 Comparison of ZFDIRDFE and MSEDIRDFE Figure 5.6 shows the variation of Ps with B for both ZFDIRDFE and MSEDIRDFE. The curves are for QOFDM at Eb/N0 = 16 dB and fN = 1.28. It can be seen that for both DFE's there is significant reduction in Ps as B increases, and essentially all the available improvement is achieved for only B — 3 feedback coefficients. It is also clear that for a given value of B MSEDIRDFE has a lower Ps than ZFDIRDFE. Figure 5.7 shows explicitly the discrepancy in Eb/No, again for QOFDM and fry = 1.28, required for the two DFE's. The curves were produced with B = 3 as suggested by Figure 5.6. It is clear that MSEDIRDFE has a consistent advantage of approximately 1 dB over ZFDIRDFE. Since there is no complexity penalty for MSEDIRDFE relative to ZFDIRDFE, there is little motivation to pursue the latter beyond this point. In fact, if the Ps achievable by ZFDIRDFE with B > 3 was considered adequate, essentially the same performance could be achieved by MSEDIRDFE with 5 = 1. In the latter case, the matrix inversion Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 97 to obtain the feedback coefficients reduces to a simple scalar inversion. We also note that for B > 2 with MSEDIRDFE, there is no value of B which can be used with ZFDIRDFE to obtain as low a Ps. Consequently subsequent comparisons focus on MSEDIRDFE. A Ps curve for the LMSE equalizer of Section 5.2 also appears in Figure 5.7. It is included in this and subsequent figures for comparison between linear and decision feedback equalization. It can be seen that the advantage of MSEDIRDFE over LMSE increases with increasing Eb/No, being near 3 dB at Ps = 1 0 - 3 . 10-4 • • MSEDIRDFE • • ZFDIRDFE I QOFDN 'I f„='-28 EbIN0=]6 dB 0 1 2 3 4 5 6 7 8 B Figure 5.6 Variation of Ps with B for ZFDIRDFE and MSEDIRDFE, fN = 1.28, Eb/N0 = 16 dB, QOFDM 5.10.3 Evaluation of MSEDIRDFE Figure 5.8 compares the variation of Ps with B for BOFDM and QOFDM, at /iv = 0.32 and E0/No = 22 dB. Nearly all the available improvement is attained with B = 1 for both cases. Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 98 » » LMOD " • ZFDIRDFE * » MSEDIRDFE Q OI M V A I fN=l-28-1 1 1 0 2 4 6 " 8 10 12 14 16 18 20 22 24 EJN0W) . Figure 5.7 Comparison of MSE, ZFDIRDFE, and MSEDIRDFE, fN = 1.28, QOFDM The Ps for QOFDM would be very close to twice that for BOFDM if the channel were nonfading AWGN, since then there would be no ISI and QOFDM is equivalent to two BOFDM signals in quadrature. Figure 5.8 in contrast shows a ratio of approximately 3. This indicates, as found in Chapter 4, that QODFM suffers more from ISI than does BOFDM. A key difference however, is that here the significant contributors to ISI have been reduced to a number B which is small for QOFDM as well as BOFDM. This is important for the feasibility of MSEDIRSE with QOFDM. Figure 5.9 makes the same comparison as Figure 5.8, but at /jy = 1-28 and Eb/No = 16 dB. Most of the improvement is obtained for small B, possibly B = 2 or 3 for BOFDM and 73 = 3 for QOFDM. • Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 99 • • B O F D M • • Q O F D M x \ i 1 i i fN=0.32 MSEDIRDFE EbIN0-22 dB 0 1 2 3 4 5 6 7 8 B Figure 5.8 Variation of Ps with B for MSEDIRDFE, fN = 0.32, Eb/N0 = 22 dB. o," •10-4 • • BOFDM ' • • Q O F D M f„=1.28 MSEDrRDFE EJN0=16 dB 0 1 2 3 - 4 5 6 7 8 B Figure 5.9. Variation of Ps with B for MSEDIRDFE, fN = 1.28, Eb/N0 = 16 dB. Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 100 PS Comparison of LMSE and MSEDIRDFE Figures 5.10 and 5.11 compare MSEDIRDFE to LMSE for BOFDM and QOFDM, at fN = 0.32 and fN = 1.28. The B used to generate each curve is indicated on the graph legends. At fx = 0.32 there is little to be gained with MSEDIRDFE compared to LMSE, less than. 1 dB at PS = 10 - 3 for BOFDM. For QOFDM the improvement is slightly better, about 1.5 dB. We note though that obtaining this gain is simple since B = 1 for these curves. At higher //y the advantage of MSEDIRDFE is more apparent. Figure 5.11 shows gains of about 1.5 dB for BOFDM and 2.7 dB for QOFDM at PS = 10"3. It can also be seen that the reduction in PS with increasing E^/No is greater for MSEDIRDFE than LMSE, and that the relative improvement for QOFDM is again greater than for BOFDM. PS Comparison of MSEDIRDFE to MLSE and the MFB Figures 5.12 and 5.13 compare curves for MSEDIRDFE to the MFB and MLSE results of Chapter 4 for BOFDM. In both cases, fN = 0.32 and fN = 1.28, the MSEDIRDFE curve is slightly closer to the MLSE curve than the LMSE curve, indicating that over half the loss of LMSE relative to MLSE is recoverable with MSEDIRDFE, and the gain of MSEDIRDFE over LMSE increases with E^/NQ and //y. These curves also show the gain of MLSE over LMSE, about 1.7 dB at //y = 0.32 and 2.9 dB at fN = 1.28, both at P3 = 10 -3. 5.10.4 Comparison of MSEDIRDFE to Previous Work: FDZFDFE Figure 5.14 shows a comparison between FDZFDFE from [7] and MSEDIRDFE developed here. The comparison was made by simulating on the basis of the OFDM parameters used in that work, //y =. 2.05 and N = 512, since the exact equations for calculating the front-end and feedback filter coefficients it used were not given. The simulation of [7] used ideal decisions in the feedback loop with B — 15. The MSEDIRDFE simulation uses B = 4 and two curves are presented, one where ideal Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 10"' 10"4 10"5 L MS >E, B 3F Dl\ I • • L M S E , Q O F D M 0 » MSEDIRDFE, B O F D M , B=l h A MSEDIRDFE, Q O F D M , B=l -i -- fN=0.32 I I I 0 2 4 6 8 10 12 14 16 18 20 22 24 .• EJN0 (dB) Figure 5.10 Comparison of LMSE and MSEDIRDFE, B = 1, fN = 0. • — — « L M S E , B O F D M • • L M S E , Q O F D M * * MSEDIRDFE, B O F D M , B=2 A A MSEDIRDFE, Q O F D M , B=3 -. i - fN=l-28 1 1 1 N • 0 2 4 6 8 10 12 14 16 18 20 22 24 EbIN0 (dB) Figure 5.11 Comparison of LMSE and MSEDIRDFE for fN = 1.28. Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 102 10 10" 10 10~ 10 =r 10" Figure 5.12 Comparison of LMSE and MSEDIRDFE to MLSE, B = 1, fN = 0.32. 10 10" 10"' 10-10 =f 10"' 0 2 4 6 8 10 12 14 16 18 20 22 24 £„W0(dB) Figure 5.13 Comparison of LMSE and MSEDIRDFE to MLSE, fN = 1.28. Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 103 decisions were used in the feedback loop and one where the actual decisions were used. Thus a fair comparison is between the left and rightmost curves. We see that MSEDIRDFE, B=4, ideal dec MSEDIRDFE, B=4, act dec | ' r \J£- tri Z, L U , I U u c u Q O F D M / „ = 2 . 0 5 N=512 1 \ 1 1 1 0 2 4 6 8 10 12 14 16 18 20 22 24 EJN0(AW> Figure 5.14 Comparison of MSEDIRDFE to FDZFDFE, for the parameters of [7], fN = 2 .05, N = 512, QOFDM. MSEDIRDFE has about 3 dB gain over FDZFDFE at Ps = 1 0 ~ 3 and this increases with increasing Ei/N0. The middle curve, for MSEDIRDFE with actual decisions in the feedback loop loses only 1 dB compared to the ideal case, and still maintains a 2 dB advantage over ideal FDZFDFE at Ps = 1 0 - 3 . 5.11 MSEDIRSE Simulation Results This section presents simulation results for the MSEDIRSE described in Section 5.8. Since both MSEDIRSE and MSEDIRDFE use the same front-end filter, a lower bound on Ps for MSEDIRSE given that filter is obtained from a simulation of MSEDIRDFE Linear and Decision Feedback Receivers, for OFDM on Flat Rayleigh Fading Channels 10"4 -• • MSEDIRSE 1SED1RDFE, ideal dec QOFDM . B=l - fN=0.32 I I I 0 2 4 6 8 10 12 14 16 18 20 22 24 £ tW 0(dB) Figure 5.15 Performance of MSEDIRSE, B = 1, fN = 0.32. 4= • MSEDIRSE, B=2 » MSEDIRDFE, ideal dec, B=2 QOFDfi 1 -1 1 1 0 2 4 6 8 10 12 14 16 18 20 22 24 EJNc(fiB) Figure 5.16 Performance of MSEDIRSE, B = 2, fN = 1.28. Linear and Decision Feedback Receivers for OFDM 105 on Flat Rayleigh Fading Channels with ideal decisions in the feedback loop. This is because MSEDIRSE can do no better than to make no errors due to ISI from the DIR. Figures 5.15 and 5.16 present PS versus E^/NQ curves for MSEDIRDFE with both the actual and ideal decisions for comparison to MSEDIRSE. From the spread of the MSEDIRDFE curves with actual and ideal decisions it is apparent that the potential improvement at PS = 1 0 - 3 is limited to about 0.8 dB for / /y = 0.32 and about 1 dB for / /y = 1.28. As can beseen, nearly all of the potential gain over MSEDIRDFE with actual decisions is achieved by MSEDIRSE with only 4 and 16 state sequence estimators respectively. Although the addition of a sequence estimator to obtain 1 dB or less available gain on a fading channel is not likely to be a desireable engineering tradeoff, these results validate the derivation of MSEDIRSE, and are in stark contrast to the poor performance. of truncated MLSE with QOFDM shown in Chapter 4. 5.12 C o n c l u s i o n We have examined linear and decision feedback equalizer theory in the context of OFDM modulation on flat Rayleigh fading channels. The aliasing approximation was identified as a convenient means of obtaining simplified equalizer structures. This was combined with the idea of modifying the overall pulse response to obtain a desired impulse response, resulting in two new DFE structures for OFDM, ZFDIRDFE and MSEDIRDFE. The former is a translation of a previously existing idea for serial modulation to the OFDM problem here. The latter, MSEDIRDFE, is thought to be a new formulation with application broader than the OFDM context which inspired it. A third structure, MSEDIRSE, combines the use of MSEDIR shaping with an ap-proximate MLSE, and performs essentially as well as MSEDIRDFE with ideal decisions in the feedback loop. Linear and Decision Feedback Receivers for OFDM on Flat Rayleigh Fading Channels 106 Of the three, MSEDIRSE has the best performance, but MSEDIRDFE performs nearly as well and is slightly less complex, thus it may be preferable. Comparisons of MSEDIRDFE with previous best results show a significant improvement in terms of Ps without increased complexity. Chapter 6 107 OFDM on Frequency-Selective Rayleigh Fading Channels Although the flat fading model of previous chapters is frequently encountered in practice, it is not always applicable. As noted in Chapter 2, frequency selectivity is a function of both the transmitted signal and the delay spread of the channel. In urban environments the typical worst case excess delays22 for arrivals which are within 10 dB of the maximum signal are less than 25 usee [18]. For the delays near 25 Msec, this corresponds to the channel appearing frequency-selective for a signaling rates of about 4 kbaud and above. The delays can be even greater in suburban environments. For example the IS-54 [37] digital cellular channel model uses a two-ray model with a 40 Msec delay between the rays, which corresponds to one symbol period. Thus the applicability of any modulation proposed for land mobile radio to delay spread or frequency-selective channels is an important consideration. In this chapter we extend some of our previous work for flat fading channels to frequency-selective channels. We derive receivers for OFDM on frequency-selective Rayleigh fading channels, based on the optimization of MLSE and MMSE performance criteria. The chapter is organized as follows. Section 6.1 describes the channel modeling. Derivation of optimal and truncated MLSE receivers is presented in Section 6.2. Methods for evaluating matched filter outputs and tone correlations are given in Section 6.2.1, and the effect of pulse shaping on the tone correlations is considered in Section 6.2.2. Performance results obtained by simulation of the MLSE receivers are presented in Section 6.2.3. with a probability of 0.99 OFDM on Frequency-Selective Rayleigh Fading Channels 108 In Section 6.3 optimal linear MMSE receivers are derived. Direct and alternate implementations are described in Sections 6.3.1 and 6.3.2. Methods for evaluating matched filter outputs and tone correlations are given in Section 6.3.3. The relationship between integral equation and matrix formulations of the estimation problem is examined in Section 6.3.4, and an efficient matrix formulation for rate N/TQ sampling is outlined in Section 6.3.5. The performance results obtained by simulation of the MMSE receivers are presented Section 6.3.6. Section 6.4 summarizes the conclusions of this chapter. 6.1 A Frequency-Selective Rayleigh Fading Channel Model The model introduced in Chapter 3 for a delay spread and time varying channel impulse response had a received signal r(t) given by Tmax r(t)= J z(a,t)s(t - a)da + n(t) • (6.1.1) 0 • • where z(a,t) is the response of the channel at time t to an impulse applied at time t — a, and Tmax is the maximum delay spread. The time variation models the fading process and the delay spread gives rise to the frequency selectivity. Although not much more difficult to work with than a flat fading model, the generality of (6.1.1) makes it difficult to obtain insight into some of the quantities23 which arise in the following sections. For this purpose it is helpful to specialize the model slightly by considering a received signal consisting of NB Rayleigh faded rays and AWGN. This model is still quite general because the rays may be correlated, and by assuming suitable weighting coefficients a ; for each ray, arbitrary delay spread profiles may be approximated. The model of the received signal is then .v«. r(t) = J2aiZi(t-Ti)s(t-Tiy+n(t) (6.1.2) .• i=i 2 3 i.e. the matched filter outputs and tone correlations OFDM on Frequency-Selective Rayleigh.Fading Channels 109 where Nj, is the number of rays and T; is the delay, associated with the i ray. The channel impulse response implied by (6.1.2) and (6.1.1) is Nb . ".. . z(a,t) = YJaifi('cr ~ Ti)zi(t - (6.1.3) Setting Ni = 2 in (6.1.3) gives the two-ray Rayleigh fading channel model as z(a,t) = a18(a)z1(t) + a26(a-T)z2(t-T). (6.1.4) Although the analyses are performed with the more general models, (6.1.4) is used to produce the simulation results of this chapter since it has the fewest parameters while still remaining able to model frequency-selective Rayleigh fading. 6.2 Derivation of the MLSE Receiver The first problem encountered in generalizing the results of Chapter 4 to delay spread channels is in obtaining a discrete and sufficient statistic from the continuous received waveform. This statistic could then be processed by discrete structures similar to the MSEDIRDFE or MSEDIRSE derived previously. Although truncated MLSE can be expected to suffer the same limitations on frequency-selective channels as were found for flat fading channels, its derivation is useful because such a statistic is obtained as an intermediate result. The derivation is similar to that of Chapter 4 but generalized to consider a received signal given by (6.1.1). Recalling the derivation of the MLSE receiver in Chapter 4, the signal dependent parameter w(t, a) is generalized to w(t,a) = J' z(a,t)s(t - a)da. (6.2.1) Equation (4.2.3) gave the metric A as a function of w(t,a). It is repeated here for convenience A = j Re[r(t)w*(t,a)} - ^\w(t,a)\2dt. (6.2.2) OFDM on Frequency-Selective Rayleigh Fading Channels 110 Combining (6.2.1) and (6.2.2) and simplifying yields [N-l A = Re n=0 J J da dt N-l N-l - \ E E <** j I - nw^-^d*! n n J " (6.2.3) n=0 m=0 x j z*(a2,t)p*(t-a2)e~3^n{t-a2)da2 dt from which the quantities and Un = J r(t)J z*(a,t)p*{t-a) Vm!n=J J z{a^t)p{t - c r i ) e - % e - ^ n ^ d a dt (6.2.4) (6.2.5) x Jz*(a2,t)p*(t-a2)e can be defined. As per the observations of Section 4.2, Un and Vm^n will be referred to as the matched filter outputs and tone correlations respectively. The metric can now be written as r /V-l 1 N-l N-l A = Re E<^ ra=0 (6.2.6) n=0 ~m=0 A„ = Re[a* Un] — Re (6.2.7) which has the same general form as (4.2.11) except that Vm,n can no longer be written as Vn-m- However the essential property for efficient recursive evaluation of (6.2.6) is that Vm)n = V*m, which can be seen to hold from (6.2.5). Using this property it may be shown (Appendix C) that A may be found recursively by defining n - l an E akVk>n k=n—L and maximizing N-l A = E V (6-2.8) ra=0 Equation (6.2.7) results in an optimal sequence selection only for the selection of L = n, or equivalently if Vn — 0 for n > L . The matched filter outputs Un, 0 < n < N — 1 are the desired sufficient statistic. OFDM on Frequency-Selective Rayleigh Fading Channels 111 6.2.1 Evaluation of Un and Vn We obtain expressions for Un and Vn by specializing the channel impulse response to the JVfc-ray model. Matched Filter Outputs Using (6.1.3) in (6.2.4) we obtain Un = r(t) / Y arf{<J - Ti)zi(t ~ Ti)p*{t - <7 i=l = J r(t) Y ~ ri)p*(t - r , - ) e - ^ B ( * - r i ) e - ^ ^ d a d t I i=l Nb dt i=l Nb " (6.2.9) i=l i=l where qi(t) = zi(t)p(t). Equation (6.2.9) has been manipulated into the form of a single Fourier transform of its argument in the square brackets. The interval of integration is over the range of non-zero values of its argument. In this case an appropriate range is 0 < t < To(l + a i ) + T m where a is the pulse rolloff and r m is the maximum delay spread. There are N values of Un required to calculate the data estimates by (6.2.6), but we note that approximation of integrals with the form of a Fourier transform via an FFT method was previously discussed in Section 4.3. Using this method all N values of Un are calculated simultaneously with one FFT. This calculation is essentially similar to that for the flat fading model of Chapter 4. Specialized to the two-ray case (6.2.9) is Un = j [air(t)ql(t) + a2r(t + r)q*(t)]e-j^ntdt. (6.2.10) OFDM on Frequency-Selective Rayleigh Fading Channels 112 Tone Correlations The tone correlations are defined in (6.2.5). Expanding this with (6.1.3) as above yields Vm,n = J j z { a u t ) V { t - a i y % m ^ d a l a2)e-3^n{t-<T2)da2 dt x J z*(a2,t)p*(t r Nb 2 = / E - Ti)z(t - T{)p(t - a ^ t ^ - ^ d a t Nb 2 X E«fc%2 - rk)z*(t - rk)p*(t - a ^ e - ^ - ^ d ^ d t k=l f Nb Nb (6.2.11) »'=1 k=l Comparison of (6.2.11) with (4.2.13) shows the key difference between the flat fading and frequency-selective fading cases. In the former Vm,n = Vm-n, i.e. the tone correlations do not depend on m and n individually but only their difference. As can be seen above this is no longer the case for frequency-selective fading. The dependency of the Vm,n on the individual values of m and n complicates their evaluation; they can no longer be found from a single Fourier transform as in the flat fading case. However it is still possible to express it as a sum of Fourier transforms. Upon collecting terms with a common index and moving exponential rotations outside the integral sign (6.2.11) becomes Vr. l'n~I E a ^wi _ » = i Nb Nb + E E « i « * e - ' ' £ m V f r " * / qi(t - Ti)qt(t - rk) 1=1 k=l J (6.2.12) dt. Defining the transforms £«?i*(*)is Li=l e To dt (6.2.13) OFDM on Frequency-Selective Rayleigh Fading Channels 113 and Ii,kn = <Xi<Xk J q%(t - Ti)ql(t - Tk)e~3^nidt (6.2.14) we can write (6.2.12) as V™,„ = Iu_m + ' (6.2.15) which is a sum of Nb(Nb — 1)+1 Fourier transforms. Noting I{^n is conjugate symmetric, this can be reduced to Nb(Ni — l)/2 + 1 transforms. As with the matched filter outputs, these transforms can be evaluated with the FFT method of Chapter 4. Equation (6.2.15) specialized to the two-ray case is , T/ r i i^F~nT T i — jir-mT T (6.2.16) i-n—m 1 - tl,^n—m 1 ^ l , Z m —n which requires only two transforms. 6.2.2 Pulse Shaping and the Tone Correlations In Chapter 4 a criterion for pulse shaping was introduced as (4.5.1) which was shown to be satisfied by the root raised cosine family with rolloff parameter a. The specific criterion was to require that in the absence of fading the shaped basis functions should be orthogonal over the duration of p(t), i.e. J \p(t)\2e-^n-m»dt = 6n-m (6.2.17) and that this should hold for any choice of a. Equivalently, (6.2.17) states that the tone correlations in the absence of fading are Vn-m — fin-m- (6.2.18) The effect of flat Rayleigh fading is to modulate the pulse shape at frequencies at and below the maximum Doppler frequency. Since the tone correlations (6.2.19) OFDM on Frequency-Selective Rayleigh Fading Channels 114 are samples of the frequency spectrum of this modulated pulse shape it is apparent that the effect of the fading on Vn is to spread it out from a unit sample function by convolving the spectrum of \p(t)\ with the power spectrum of the fading process. However since the fading is lowpass and of much smaller bandwidth than that of the OFDM signal, the most significant values of the tone correlations are clustered on either side of any particular tone. It is this effect which allowed the use of truncated (L < N — 1) MLSE for use in detecting BOFDM, with little degradation of P&. Simulation results showed that pulse shaping with moderate (a < 0.25) values of a had negligible effect on Pf,, as might be expected from (6.2.17) and (6.2.18). The situation for the case of a delay spread channel is markedly different. This can be seen by comparing (6.2.19) and (6.2.15) of the tone correlations for the flat and frequency-selective cases respectively. The principal difference is the presence of cross-terms between the rays in the latter. Considering any one of these cross-terms Ii,kn = a i a k J qi(t - Ti)ql(t - Tk)e~3^ntdt (6.2.20) it can be seen that even in the absence of fading where (6.2.20) is Ii,kn=<xi<xk j'Pi{t-Ti)pftt-Tk)e~3lZntdt (6.2.21) we will find that in general /,-,*„ ± $n (6.2.22) and consequently Vm,n 7^  Sn-m- (6.2.23) The important point of this is that pulse shaping is much more important on frequency-selective fading channels than flat fading channels. In particular the short constraint lengths used with rectangular pulse shaping and flat fading result in poor performance on frequency-selective channels. Fortunately (6.2.21) also indicates that pulse shaping can be used to control the spreading of Vmjn and thereby make truncated MLSE feasible. OFDM on Frequency-Selective Rayleigh Fading Channels 115 6.2.3 Simulation Results The performance of the receivers developed in the preceding sections has been evaluated by simulation. Perfect channel knowledge is assumed available at the receiver and thus the results bound the best attainable performance for both receiver categories, nonlinear MLSE and linear MMSE respectively. As in previous chapters, the figures presented are annotated with //y since it is this parameter upon which the P0 of the MFB of Chapter 3 depends. A maximum Doppler rate of fjj = 0.01 times the data symbol rate is assumed and the sampling rate and number of samples per block is such that //y = JDTQ = /z>iV. Therefore we always have N = 100//y data symbols per OFDM block. The two-ray channel model was used to model frequency-selective Rayleigh fading. The delay spread r is normalized by Ts = To /N, the data symbol interval. For example r = 0.5 indicates a difference in arrival times, or delay spread, of 0.5TS between the two rays. The other channel model parameter is a\, the proportion of total received power in the first ray. In the figures it is designated by 7 to avoid confusion with a, the rolloff parameter of the root raised cosine pulse shape. Figures 6.1 and 6.2 show the effect of delay spread on the symbol error rate, Ps, of an ideal MLSE receiver. By ideal MLSE we mean an optimal MLSE receiver as derived in Section 6.2 without truncation or other modification of the constraint length. This requires L = N — 1; thus simulation is practical only for small N due to complexity constraints. Figure 6.1 shows the results for BOFDM with N = 4 and similarly Figure 6.2 for QOFDM. Both show results for three values of delay spread. The case of r = 0.0 is equivalent to flat fading.. For r = 0.5 and r = 1.0, independent Rayleigh fading of two equal power rays was applied with amplitudes scaled so as to make the total received power of the two rays equal to that of the one ray flat fading case. Nevertheless, both figures show substantial gains in the presence of delay spread, of about 5 and 6 dB at OFDM on Frequency-Selective Rayleigh Fading Channels 10-5 I I I I I I I I I I I I I I I I I I I I I I I I I I. 0 2 4 6 8 10 12 14 16 18 20 22 24 Figure 6.1 Effect of Delay Spread with Ideal MLSE for BOFDM, TV MI .SI 2,1 =0 .0 • • MLSi i , X=0.5 . a t =i .u = fN=0.04 Q O F D M : 0 2 4 6 8 10 12 14 16 18 20 22 24 EbIN„(<lB) Figure 6.2 Effect of Delay Spread with Ideal MLSE for QOFDM, TV OFDM on Frequency-Selective Rayleigh Fading Channels 111 Ps = 10~3 respectively. This is because such short blocks have negligible time diversity, so there is a large diversity gain due to the second ray. It is interesting that the MLSE receiver is able to attain such gain despite the ISI which must be substantial. At r = 0.5 the second arrival overlaps the first by 7/8 of the OFDM block length and for T = 1.0 the overlap is 3/4. 10° 10"' 10"2 10"3 10"4 io-5 0 2 4 6 8 . 10 12 14 16 18 20 22 24 Figure 6.3 Effect of Delay Spread on Constraint Length for /jv = 0.32 Figure 6.3 shows the effect of delay spread on constraint length for //y = 0.32. In this case ideal MLSE is not feasible24 and so truncated MLSE with L = 3 is compared for flat fading and r = 0.5. Rectangular pulse shaping was used. The deterioration of the latter case is remarkable, about 5 dB at Ps = 10~3. As discussed in Section 6.2.2, this effect is due to the emergence of cross-terms which are not present in the flat fading case. One solution to the cross-term problem, the use of root raised cosine pulse shaping to modify the required constraint length is evaluated in Figure 6.4 for = 0.32 and 2 4 Ideal MLSE would require L = N — 1; it is precluded by the exponential complexity of the Viterbi algorithm ML .SI i,l ?,i =0 .0 • • M L S E , L=5, X=0.5 = a=0.0 BOFDM fN=0.32 OFDM on Frequency-Selective Rayleigh Fading Channels 118 Figure 6.5 for /jy = 1.28. Values of L = 3 and 5 respectively were used to be consistent with the flat fading simulations of Chapter 4. Results are shown for various values of rolloff parameter a with r = 0.5. For /JV = 0.32 about 5 dB at Ps = 10 - 3 is gained with a — 0.125 compared to a = 0.0, which is a substantial improvement. For fa = 1.28 the improvement is not very significant above Ps — 10 - 4, where it is only about 1 dB. In both cases, increasing a to 0.25 provides little benefit. . The relatively small improvement for the longer (N = 128) blocklength is because there is relatively little deterioration to improve upon. This is likely, because r = 0.5 is a proportionately smaller delay spread for longer OFDM blocks. Figures 6.6 and 6.7 show the effect of various power distributions between the rays, for a = 0.125 and r = 0.5. In both cases, //y = 0.32 and fx = 1.28, the effect of the power distribution is seen to be nearly negligible. This is in contrast to the MFB results which show maximum benefit for equal power distribution. It appears that the additional factor of ISI, which also varies with the power distribution, is counterbalancing diversity gain from delay spread. OFDM on Frequency-Selective Rayleigh Fading Channels 119 nr 1 • • M L S E , a=0.0 MLSE, 0=0.125 MLSE, a=0.25 = L=3 x=0.5 ri. /1Y1 0 2 4 6 8 10 12 14 16 18 20 22 24 Figure 6.4 Effect of Pulse Shaping on Constraint Length for //y = 0.32 10"4 MI .SI x=( J.0 • " M L S E , <X=0J25 VII x = j E fN=1.28 L=5 T=0.5 r ^5 1 i • 0 2 4 6 8 10 12 14 16 18 20 22 24 £„/JV0(dB) Figure 6.5 Effect of Pulse Shaping on Constraint Length for //y = 1.28 OFDM on Frequency-Selective Rayleigh Fading Channels 120 10"' 10"5 n • • MLSE, 7=0.9 • • M L S E , 1 1=0.5 = fH=0.32 L=3 1=0.5 a=0.125 1 1 1 Dr\T7T\A(T " 0 2 4 6 8 10 12 14 16 18 20 22 24 £ t / N „ ( d B ) Figure 6.6 Effect of Power Distribution Between Rays for fx = 0.32 .a. 10"4 10"5 • • M L S E , 1=1.0 • • M L S E , 7=0.9 E fN=1.2S L=5 1=0.5 a=0.125 \ 1 1 B O F D M : 0 2 4 6 8 10 12 14 16 18 20 22 24 .E„IN0(.dB) Figure 6.7 Effect of Power Distribution Between Rays for fx — 1.28 OFDM on Frequency-Selective Rayleigh Fading Channels 121 6.3 Derivation of an MMSE Linear Receiver In this section we derive the optimal linear receiver for frequency-selective (delay spread) channels based on the MMSE criterion. This allows us to compare the optimal linear receiver to the truncated MLSE receiver for BOFDM, and to obtain useful results for QOFDM of long blocklength. The latter was not possible for MLSE due to complexity constraints. In addition, simulation of the receiver from this derivation enables us to quantify the effect of the aliasing approximation used in Chapter 5, verifying the assumptions made there. The approach taken is a generalized version of that used for the flat fading. Again we seek to minimize the MSE given by |2 MSE = E = E and we assume a linear receiver structure of the form 2 an\ - 2Re[ana*n] + Ea (6.3.1) an = j r(t)cn(t)e~J%ntdt (6.3.2) where the interval of integration is over the time of the OFDM block being demodulated. The problem is to determine the weighting functions cn(t), 0 < n < N — 1, that are optimal in the MMSE sense. The difference compared to the flat fading case lies in r(t) now having a more complicated form. We can obtain an implicit expression for cn{t) using a variational argument [22], or more concisely by differentiation with respect to a function. Using (6.3.2) in (6.3.1) yields MSE = E[aB j r\t)cn{t)e^ntdt - an J r*(t)cn(t)eJ^ntdt-- a n j r{t)cn(t)e-3%nidi\ + Ea (6.3.3) OFDM on Frequency-Selective Rayleigh Fading Channels 122 and taking the derivative of (6.3.3) with respect to cn(u) yields dMSE = E anr [uje To E anr 2dcn(u) Setting (6.3.4) to 0 and again using (6.3.2) for an yields (u)e3-To (6.3.4) / E[r*(u)r(t)}cn(t)e-J^ntdt = E[anr*(u)}. (6.3.5) Equation (6.3.5) is the desired implicit solution for cn(t) in terms of a general received signal r(t). Expanding s(t) in (6.1.2) and using qi(i) — zi(i)p(i) yields Np N-l r(t) = Y «.-«.•(* - rt) E a m e ^ m ( < - r i ) + n(t). m=0 (6.3.6) i=i Using (6.3.6) it is straightforward to evaluate the expectations in (6.3.5), which are E[r*{u)r(t)\ Ea N-l Np N„ m=0 i=l fc=l +.?72^-u) JV-1 = £ / m ( * ) / m M + '7 2 *(*-«) (6.3.7) m=0 and E[awr*(u)] = / » » (6.3.8) where for ease of manipulation we have defined Nv fm(t) = Ea»'9»'(*-T«')eJ3b j * £ m ( t _ T i ) (6.3.9) i=i Upon substituting these expectations into (6.3.5) we obtain N-l . q2cn{u)e-3%™ = f*(u) - Y /m(«) / fm{t)cn(t)e-'%ntdt. (6.3.10) m=0 Equation (6.3.10) is in the form of a Fredholm integral equation [38] of the second kind with a degenerate kernel. OFDM on Frequency-Selective Rayleigh Fading Channels 123 Denning the constants Bm,n = J fm{t)cn{t)e-32tntdt (6.3.11) (6.3.10) can be written N-l V2 c n ( u ) e - ^ n u = f*(u) - Y,B™,nfm(u) (6.3.12) m=0 which indicates that cn(u) is expressible as N-l cn(u) = -Y/Cm,nfm(u)ej2tnu, (6.3.13) a weighted sum of the fading functions defined in (6.3.9). C m i „ and 5 m , n are related by Cm,n — 8m—n Bm^n. (6.3.14) We can obtain a matrix form by defining the linear operator LN = J fi(u){-)du (6.3.15) and applying it to (6.3.12) to yield N-l ' , V2Bl,n= I fl(u)rn(u)du-J2Bm,n fl(u)fm(u)du. (6.3.16) m=0 Defining the constants V]<n as Vi,n = J Mu)f*n(u)du ' (6.3.17) (6.3.16) becomes (6.3.18) N-l V2Blin = Vl>n - E Vl,mBm,n, m=0 0 < /. n < ;V - 1. Making some additional definitions, b£ = [B0)m • • •, -B/v-i,n] > v n = [Vb,n,^i,n,..-,VAr-i,n]» and square matrix V having {l,m)ih element V i i i n = V;,m, 0 < /, m < N — 1, (6.3.18) can be written in matrix form as 7/ 2b n = v n - V b n (6.3.19) which has the solution b n = (V + 77 2 l ) _ 1 v n . (6.3.20) OFDM on Frequency-Selective Rayleigh Fading Channels 124 6.3.1 Direct Implementation It can be seen from the preceding that the operation of the optimal linear receiver is fairly complicated for large N. The sequence of operations is as follows. Channel measurements are used with (6.3.9) and (6.3.17) to obtain the V\im, and these are then used in (6.3.20), requiring an N x N matrix inversion, to obtain the Bm,n, which with (6.3.14) and (6.3.13) yield cn(u). Finally cn(u) is used in (6.3.2) to obtain the data symbol estimate an. The matrix inversion is performed only once per OFDM block, but the matrix multiplication in (6.3.20) and subsequent operations must be performed for each data symbol, or N times. 6.3.2 Alternate Implementation Its possible to circumvent explicit evaluation of cn(t) by combining (6.3.13) and (6.3.2) to obtain ^ = ~ E G * > " / r ( < ) / m ( * ) ^ - ( 6 - 3 - 2 1 ) ^ m=0 ^ Defining the matched filter outputs Um =J r(t)fm(t)dt (6.3.22) and applying to (6.3.21) yields 1 N ~ 1 ® n = ~2 E Cm,nUm. (6.3.23) ^ m=0 Equation (6.3.23) expresses the data estimates as weighted sums of the matched filter outputs. To use this expression the C m , n are calculated as in the direct implementation and the Um are determined from (6.3.22). There is little difference in complexity compared to the direct implementation; (6.3.23) was derived to obtain the relationship between an and Um explicitly. OFDM on Frequency-Selective Rayleigh Fading Channels 125 6.3.3 Evaluation of Um and V7 m Matched Filter Outputs Using the definitions of Um in (6.3.22) and the fading func-tions fm(t) defined in (6.3.9) yields U„ J r(t)fm(t)dt i e T° dt (6.3.24) which is identical to (6.2.9). We see that both the MLSE and MMSE analyses prescribe calculation of the same matched filter outputs. Tone Correlations The tone correlations are defined in (6.3.17). Expanding this with (6.3.9) as above yields Vl,n = J fl(t)f*n(t)dt , Nb Nb = / E E a«'Q !*ft( i - T«)9fc(* - Tk) I i=l k=l Nb-2 i - J ^ ( » - 0 * -»=1 Nb Nb dt + E E e - ' ^ ^ o i a , j qi(t - rl)qt{t - r ^ ^ d t i=l k=l. J and using the definition (6.2.14) this can be written Nb Nb (6.3.25) (6.3.26) which is identical to (6.2.15). As with the matched filter outputs, both the MLSE and MMSE analyses prescribe calculation of the same tone correlations. OFDM on Frequency-Selective Rayleigh Fading Channels 126 6.3.4 Relation between Integral Equation and Matrix Formulations The preceding analysis based on solving the integral equation (6.3.5) determined that the optimal linear MMSE estimates have the form 1 N ~ l ^n = ~ E Cm,nUm. (6.3.27) ^ m=0 An alternative solution method for the Cm >„ can be obtained by assuming (6.3.27) as an optimal form for the solution and applying matrix based MMSE estimation theory. It is instructive to compare the solutions obtained by the two methods. We obtain (6.3.27) in matrix form by defining the vectors — [Co,n,Ci,„,CN-i,n] and u T = [Uo,Ui,UN-I], then K - - 4 c n « - (6.3.28) T It is well known that the MMSE solution for an equation having the form of (6.3.28) is given by [23] C n = ^ 2 R _ 1 * P n (6.3.29) where (6.3.30) uuH R = E and P n = E[ua;]. (6.3.31) The (l,k)th element of R is Ri^ = Ri^. It is determined from (6.3.30) and (6.3.22) yielding Rhk = E[UhU*k} = J J E[r(t)r*(uM(t)fk(u)dtdu. Expanding ~E[r(t)r*(u)} with (6.3.7) and using (6.3.17) it is readily shown (6.3.32) N-l ' Rl>h = EaJ2 Vk,mVm,i + NoVk-j (6.3.33) m=0 OFDM on Frequency-Selective Rayleigh Fading Channels 127 or R = £ a V T V T + JV 0VT. (6.3.34) Similarly it can be shown Pn = £ a V * . (6.3.35) Combining (6.3.34) (6.3.35) and (6.3.29) yields c n as - l * f n • ' (6.3.36) -1 c n = r ? 2 ( v T V T + 7 ? 2 V T ) v n = n 2 ( W + »72V) v n where we have also used V = V H = V T * to obtain the last line of (6.3.36). The matrix based solution for c n in (6.3.36) has a different form than that from the integral equation solution, and so it is not obvious that they are the same. Using the definition of c n and combining with (6.3.20) we have from the latter method c n = d n - ( V + /7 2l) _ 1v n • , • . (6.3.37) where d n has a 1 in the nth position and 0's elsewhere. We demonstrate the equivalence of (6.3.36) and (6.3.37) by contradiction. Assuming inequality V2(yy + * ? 2 V ) - 1 v n ^ d n - (V + 772X)~1vn (6.3.38) from which it follows u\V + ^rfV-ivn ± d n - (V + 77 2 l ) _ 1 v n A ^ V n ^ V n + ^ d n - V n (6-3.39) and > ? 2 V n A 2 v „ . (6.3.40) Equation (6.3.40) is a contradiction, establishing the equivalence of (6.3.36) and (6.3.37). OFDM on Frequency-Selective Rayleigh Fading Channels 128 Stability Comparison In terms of numerical stability (6.3.37) from the integral equation approach is a more desirable form than (6.3.36). This can be seen by comparing the condition numbers of the matrices which must be inverted in each case. The condition number cn(M) of a matrix M is defined as the ratio of the magnitude of its minimum to maximum eigenvalues [39] Xmin cn(M) = Xmax (6.3.41) where Xmin and Xmax are the minimum and maximum eigenvalues respectively.- By definition cn(M) ranges between 0 and 1. A condition number close to zero implies the matrix is nearly singular [25] and problems of numerical stability can be expected with its attempted inversion. Since V = V H it is possible to perform an eigenvector decomposition and express it as N-l (6.3.42) k=0 where the et are the eigenvectors of V. Using (6.3.42) we can write the relevant matrix of (6.3.37) as N-l {V + q2l) = ^2xkeke$ + q2I k=0 N-l = E ( A f c + 7 ? 2 ) e k e k (6.3.43) which reveals the condition number of (V + q2l) to be ^min ~\~ q cn(V + ?72I) = (6.3.44) OFDM on Frequency-Selective Rayleigh Fading Channels 129 Similarly, the condition number of (VV + V72V) is cn(VV + 772V) = cn(V(V + u2l)) /N-l /N-l ^ = cn E Afcekeg £ (A,- + q^ef .k=0 i=0 /N-l N cnl Afc(Afc + n2)ekejj \k=o J (6.3.45) which shows cn(VV + ?72V) cn(V + 772l). (6.3.46) Two conclusions that can be drawn from (6.3.44)-(6.3.46) is that (V + q2l) cannot be exactly singular for non-zero u2, but it can be close for q2 small. In contrast the matrix (VV + 572V) can be singular for non-zero q2 if \min = 0, and will in any case be closer to singularity than (V + q2t) by the factor | A m j n / A m a i | . 6.3.5 An Efficient Matrix Formulation for Rate N/TQ Sampling The transmitted OFDM signal s(i) is given by N-l s(*) = PW E a n e ^ n \ 0<t<T0{l + a). (6.3.47) n=0 (6.3.48) Using the two-ray Rayleigh fading model, the received signal is given by r(t) = ctizi(t)s(t) + a2z2(t - r)s(t - r) + n(t), 0 < t < T 0 ( l + a) + T. In Chapter 5 we reasoned that the aliasing introduced by sub-Nyquist sampling at rate N/TQ should not noticeably increase the error rate for large N. This suggests we could model equations (6.3.47) and (6.3.48) sufficiently accurately by sampling at rate OFDM on Frequency-Selective Rayleigh Fading Channels 130 1/Ai = N/To to obtain a discrete model which is then suitable for a matrix description. The discrete versions of the transmitted signal and received signal are then .' N-i w=0 where sm = Pm E a» '^"B. -0 < m < 7Ya - 1 T 0(l + a) At = \N(l+a)\ and where Oi\Z\msm + a2Z2m_Tsm-T + n m , 0 < m < NT - 1 JVr = [N(l +a) + r/AtJ respectively. Defining the following matrices and (6.3.49) can be written S T ' = [50, S l , . . .67V a _ i ] , . P = diag(p0,.Pi, • • • ,PNa-i), D = [{Dni,n}], D m n = e^""\ 0 < n < N - 1 • 0 < m < iVa - 1 a T = [oo, ai , . . . , a/v-i] (6.3.49) (6.3.50) (6.3.51) (6.3.52) (6.3.53) (6.3.54) (6.3.55) (6.3.56) s = PDa. (6.3.57) OFDM on Frequency-Selective Rayleigh Fading Channels 131 Similarly defining and (6.3.51) is { ai*im, 1 = m , « 2 ^ 2 m , . l = m + r/At 0, l ^ m 0 < m < Na - 1 0 < / < NT - 1 r T = [ro,r-i,... ,rjvT] n T = [7io,ni,...,rijtfr] r = Zs + n = ZPDa + n = QDa + n where Q = ZP. (6.3.58) (6.3.59) (6.3.60) (6.3.61) (6.3.62) If we form the estimates of a with a = C H r (6.3.63) then it is readily shown that the linear MMSE choice for the weighting matrix C H is found from - l E ra* (6.3.64) C = E ra - l "]) = ( Q Q H + ? ? 2 I ) X Q D or C H = D H Q H ( Q Q H + 772l) - l (6.3.65) OFDM on Frequency-Selective Rayleigh Fading Channels 132 which yields for the estimate a = D H Q H ( Q Q H + ? 7 2l)~ 1r. (6.3.66) Equation (6.3.66) is a more efficient way to obtain a than to explicitly determine C H and use it in (6.3.63). By first premultiplying r by (QQH + ?72l) \ premultiplying the result by Q H and then by D H , the rectangular by rectangular matrix multiplications are replaced with rectangular matrix by vector multiplications. This requires approximately N times less computation. Further, with the computation organized in this way D H is multiplying a vector s = Q H ( Q Q H + r , 2 l ) _ 1 r (6.3.67) which can be interpreted as an estimate of the undistorted signal s. Combining (6.3.67) and (6.3.66) a = DHs (6.3.68) which is efficiently evaluated by an inverse FFT of s. Comparison to Integral Equation Solution The advantage of this formulation over that derived with the integral equation approach can be seen from the definitions of Z and Q in (6.3.58) and (6.3.62) respectively. These fading matrices are two-diagonal and consequently (QQH + q2l) is three-diagonal. By re-diagonal it is meant that all matrix entries are zero except possibly those on n diagonals, where the diagonals are not necessarily adjacent. Therefore (QQH + q2T) can be inverted with only O(N) operations [25] as opposed to (3 (TV3) operations for a general matrix inversion. For the more general Nb-ray fading model the complexity of the matrix inversion varies as Nb2N with Nb typically 6 or less and independent of N. The principal limitation of this formulation is that the complexity of the matrix inversion varies with the sampling rate OFDM on Frequency-Selective Rayleigh Fading Channels 133 as can be seen from (6.3.50). In (6.3.58) it is assumed that r is an integer multiple of At. More generally the sampling rate must be increased to obtain finer resolution of r since this resolution is limited to At. With the integral equation formulation the size of the matrix which must be inverted is independent of At. 6.3.6 Simulation Results In this section we evaluate by simulation the performances of the various optimal linear MMSE receivers developed in the previous sections, and compare them to the MLSE receivers. As in the flat fading case, the complexity reduction of MMSE relative to MLSE receivers allows simulation of QOFDM as well as BOFDM. For this work we also produced the first simulation of the ideal linear MMSE receiver based on the integral equation formulation. Consequently we are able to verify empirically certain comments made previously about the significance of the Transform Approximation made in Chapter 5 and about the relationship between the integral equation and rate N/TQ matrix formulations made in Section 6.3.5 of this chapter. Figure 6.8 shows the effect of delay spread on the ideal linear MMSE receiver de-termined by the integral equation approach (IEMSE) and also includes the corresponding results from the ideal MLSE for comparison. Blocklength N — 4 is chosen because it is the only blocklength for which ideal MLSE is feasible with QOFDM; this is shown in Figure 6.9. The performances of the two receivers are remarkably close for both modulation formats, within a fraction of 1 dB at Ps = 10 - 3 in each case. As might be expected25, the greater difference is for BOFDM. The closeness of the results indicate that ISI poses little problem at such a short blocklength. This is because the channel changes little during the duration of the block, therefore there is little intrablock ISI, and the small number of symbols means there are few contributors to interray ISI. Further, In Chapter 4 we showed B O F D M suffers less than Q O F D M from ISI on flat fading channels OFDM on Frequency-Selective Rayleigh Fading Channels m vlS E , 0.0 • • IEMSE, 1=0.5 e © OBMSE, x=l.C MLSE, x=0.0 MLSE, x=0.5 MLSE, x=1.0 - • • 0 0 \ \ 3 = f^O.04 \ BOFDM 0 2 4 6 8 10 12 14 16 18 20 22 24 EJN„(dB) Figure 6.8 Effect of Delay Spread for Ideal MSE, for fN = 0.04 and BOFDM E J E , X=l 9.0 • • 1EMSH, X=1.U 0 G M L S E , x=0.0 a • M L S E , x=1.0 A. i fN=0.04 1 J O F D M 0 2 4 6 8 10 12 14 16 18 20 22 24 E„IN0(dB) Figure 6.9 Effect of Delay Spread for Ideal MSE, for fN. = 0.04 and QOFDM OFDM on Frequency-Selective Rayleigh Fading Channels 135 10° 10"' 10"2 10"3 1CT4 10"5 0 2 4 6 8 10 12 14 16 18 20 22 24 • 2yW„(dB) Figure 6.10 Effect of Delay Spread for Ideal MSE, for fN = 0.32 and QOFDM since the equalized block is essentially ISI free, the Ps for QOFDM is very nearly twice that for BOFDM, indicating a lack of intrasymbol ISI. Figure 6.10 shows the effect of delay spread for IEMSE for QOFDM with fx = 0.32. For this longer block, the accumulated effects of the various types of ISI make the delay spread case with r = 1.0 noticeably poorer than the flat fading case, causing the appearance of an error floor near Ps = 10 -3. Some deterioration can also be anticipated with longer blocks, such as fx = 1-28. Unfortunately, as discussed in the derivations, the complexity of the matrix inversion required for the IEMSE receiver varies with JV3, and is largely impractical for TV — 128, as the simulations used to produce the results of this section process from 50,000 to 100,000 blocks. Using the insights developed concerning pulse shaping and constraint length in the context of the MLSE work, it is clearly possible to develop an approximation to the IEMSE of complexity proportional to L2TV which would then allow simulation of • • IEMSE, X=0.0 — — m i KT _ i n -™ L C I c, V — = fN=0.32 QOFDM OFDM on Frequency-Selective Rayleigh Fading Channels 10" 10" 10" 10" 10 10" 0 2 4 6 8 10 12 14 16 18 20 22. 24 £„/JV0(dB) Figure 6.11 Proximity of the Matrix Formulation Results to the Ideal MSE Results 10" 10 10" 10" 10 ' 10"4 = / W - 2 « 0 2. 4 6 8 10 12 14 16 18 20 22 24 £/JV„(dB) Figure 6.12 Effect of Delay Spread for '/jy = 1-28 OFDM on Frequency-Selective Rayleigh Fading Channels 137 any blocklength of interest. An alternative approach, in which the frequency-selective channel was treated as a set of parallel flat fading channels was presented in [5]. However, we pursue instead means of determining the performance of the optimal linear MMSE receiver. The approach we take is to establish that the efficient matrix formulation of Section 6.3.5 produces results which are very close to those of the IEMSE receiver for TV > 32, and may thus be used to determine the linear MMSE performance bound for fx = 1.28. In Chapter 5 we reasoned that the use of sub-Nyquist sampling at rate TV/To was an aliasing approximation which replaced a linear convolution with a circular convolution, and further that this effect would be insignificant for large values of TV. We note that the only deviation of the matrix formulation of Section 6.3.5 (MTXMSE) from the ideal MMSE linear receiver is the use of this same aliasing approximation. In Figure 6.11 results from the IEMSE are compared to the MTXMSE for QOFDM, for TV = 4 and 32, and for both flat and delay spread fading channels. For the IEMSE results, the symbols identifying the curves are small and opaque. The corresponding curves for the MTXMSE use the same symbol shapes, but transparent and enlarged. In this way it can be seen that while there is a very small difference when TV = 4, the curves for TV = 32 are essentially coincident. The same coincidence would be expected for any TV larger. Thus we obtain in Figure 6.12 the effect of delay spread for fx = 1-28, and find about 1.2 dB of deterioration relative to the flat fading case at Ps = 10 -3. Figure 6.13 shows comparisons of the ideal MLSE and IEMSE receivers to the MFB for IN .= 0.04. MFB's are given for three values of delay spread, with the amount of delay spread increasing from the top to the bottom of the graph. The results for the'MLSE receiver are almost coincident with the MFB at all delay spreads, indicating negligible loss from ISI. The IEMSE receiver performs nearly as well for r = 0.0 and r = 1.0, but has about 1 dB deterioration at Ps = 10 - 3 when r = 0.5. OFDM on Frequency-Selective Rayleigh Fading Channels 138 10° 10"' 10"2 10-3 10-4 io-5 0 2 4 6 8 10 12 14 16 18 20 22 . 24 £ „ / N 0 ( d B ) Figure 6.13 Comparison of the MFB to the IEMSE and MLSE Receivers for fN = 0.04. The discrepancies found between the MFB, MLSE, and MMSE receivers are greater for //y = 0.32 and fx = 1.28. This is expected because long blocklengths suffer substantial ISI which is neglected in the MFB formulation. Figure 6.14 shows a comparison of the MFB to the MLSE receiver for these two cases. The MLSE receiver, which at these blocklengths is the truncated MLSE receiver with a = 0.125, still performs quite well, with about 0.7 dB discrepancy from the MFB for fx = 1.28 and about 1.8 dB for /jv = 0.32, both compared at Ps = 10 -3. As shown in Figures 6.6 and 6.7, the MLSE receiver has essentially the same performance on delay spread as flat fading channels. The MFB in contrast has a larger diversity gain due to delay spread at /jy = 0.32 than at //y = 1.28. This accounts for the relatively poorer performance of the MLSE receiver in the former case. MTU, 1=U.U M F B , 1=0.5 = f ^ FB,T=7.0 /TLSE, 1=0.0 ALSB, 1=0.5 • > • J • M L S E , 1=1.0 0 IEMSE, 1=0.0 D IEMSE, 1=0.5 O IEMSE, 1=1.0 [ \ = V K i j fN=0.04 B O F D M = OFDM on Frequency-Selective Rayleigh Fading Channels 139 10° 10-' io-2 10"3 10"4 10"5 • 0 2 4 6 8 10 12 14 16 18 20 22 24 EJN0(£B) Figure 6.14 Comparison of the MFB to the MSE and MLSE Receivers for = 0.32 and fN = 1.28. Figure 6.14 also shows results for a linear MMSE 2 6 receiver for /jy = 1.28. Its performance is about 3 dB worse at Ps = 10 - 3 than the MLSE receiver. This is due to the inherent inferiority of linear compared to maximum likelihood estimation techniques in the presence of ISI. 6.4 Conclusion We have derived optimal and suboptimal versions of an MLSE receiver for OFDM on frequency-selective fading channels, and evaluated their performance by simulation. In contrast to our previous results for the flat fading channel we found that pulse shaping has a large effect on the Ps of the truncated MLSE receiver when it is used on channels with delay spread. This is due to the presence of cross-correlations between the received OFDM blocks in the latter case. This is designated M S E on the figure, and was simulated by M T X M S E . OFDM on Frequency-Selective Rayleigh Fading Channels 140 We also derived an optimal linear MMSE receiver for frequency-selective fading channels, based upon an integral equation formulation of the estimation problem. This approachallowed us, with Ps results generated by simulation, to quantify the effect of the aliasing approximation first discussed in Chapter 5 in the context of fiat fading, as well as for the delay spread cases of this chapter. The effect was shown to be negligible for N > 32, and only a fraction of 1 dB for N as low as 4. This led to an MSE receiver form using the aliasing approximation whose complexity is only linear in N, with essentially optimal performance for long blocklengths. We compared the solution form obtained from the integral equation approach to the solution form obtained from a conventional matrix analysis approach, and showed them to be theoretically equivalent when finite precision effects are neglected, but having the important difference that the integral equation form is more stable numerically. This difference was quantified in terms of the eigenvalues of a certain channel dependent tone correlation matrix. For the short blocklength N = 4 it was shown that diversity gains are achieved by the MLSE and MMSE receivers which are very close to the MFB gains. For the longer blocklengths, it was found that the diversity gains were almost balanced by the increase in ISI with blocklength and diversity, yielding performance basically unchanged from the flat fading case. This result is based on the normalization choice which compares the flat fading and delay spread fading on the basis of same total received power. If we were to ask a slightly different, but equally valid, question by considering the performance on a flat fading channel, and then ask what would be the effect of the arrival of a second ray, we would then see a net gain due to the increase in total received power. A key difference between the flat-fading and delay spread channels is revealed by comparing the results for QOFDM with N = 4, 32, and 128. These show that in contrast to the flat-fading case, on delay spread channels longer block lengths do not OFDM on Frequency-Selective Rayleigh Fading Channels 141 necessarily have a lower Ps. This suggests that deliberately introduced delay spread via transmitter diversity in combination with short blocks might be more effective than channel averaging with long blocks. Chapter 7 142 Conclusion and Suggestions for Future Work 7.1 Conclusion The performance and receiver structures for channel-averaging OFDM on land mobile radio channels characterized by flat and frequency-selective Rayleigh fading have been studied. A new matched filter bound (MFB) applicable to fast fading was derived to establish lower limits on the Pj, of any receiver structure applied to uncoded OFDM. on these channels, and to provide a benchmark for receiver performance. The theory of equalization was developed in the context of OFDM using integral equation and matrix formulations. New nonlinear sequence estimating and decision feedback receiver structures were derived using MLSE and MMSE optimality criteria. The performance of these receivers was evaluated by simulation and found to equal or in most.cases exceed previous results for the same modulation and channels. Several insights were obtained into the nature of the ISI in OFDM caused by fading. BOFDM was found to be much less sensitive to ISI than QOFDM. With BOFDM, a simple truncated MLSE receiver can approach the MFB to within 1 dB, but it is effectively useless for long blocklength QOFDM due to the required constraint length. The channel impulse response is amenable to reshaping by the MSEDIR technique which was devised to reduce the constraint length of the ISI, making sequence estimation feasible and improving the performance of DFE substantially. ISI on frequency-selective fading channels is much worse than for flat fading channels. The problem was shown to be the appearance of certain cross-terms arising in the tone correlations due to the interaction of multiple received versions of the transmitted OFDM block, but controllable with by proper pulse shaping. We conclude that the channel averaging ability of OFDM gives it a substantial Pj, performance advantage over any other uncoded modulation on fast flat Rayleigh fading Conclusion and Suggestions for Future Work 143 channels, and that most of the gain indicated by the MFB is attainable by the nonlinear receiver structures devised in this thesis. On frequency-selective Rayleigh fading channels a comparison with conventional serial modulation schemes is less positive. This is mainly because of diminishing returns with increasing diversity, also predicted by the MFB. The presence of a second uncorrected ray in the delay spread case causes a large diversity improvement for short blocklength OFDM and serial modulation schemes which is hot available for long blocklength OFDM2 7. By comparing the results for N = 4 and jV = 32 was found that given an equally sophisticated receiver, the performance of a serial modulation scheme would be comparable to OFDM if the powers in the two rays, were similar. Of course, this is a very special case. Long blocklength OFDM maintains an important advantage when the relative strengths of the received rays are unknown a priori because it is fairly insensitive to the power distribution among the rays. 7.2 Suggestions for Future Work 1. Interleaved OFDM Interleaved OFDM (IOFDM) is a variation of OFDM originated during this investigation. The idea is shown in Figure 7.1. A serial data stream ak is partitioned into short blocks of N = 4.' These are transformed by a 4-point inverse FFT to obtain a new set of symbols bk. The bk stream is applied to an interleaver, and the output of the interleaver feeds a conventional serial modulator. At the receiver the process is reversed. The inverse FFT time-diversifies the information over the bk, and the interleaver decorrelates the samples of a particular OFDM block. This method provides the diversity gain of a much longer block with a block length short enough to allow ideal MLSE. Some preliminary results for BPSK IOFDM (BIOFDM) on a flat Rayleigh fading channel, plotted in Figure 7.2, show a comparison to BOFDM with fj) = 0.01 and N = 128. Ideal interleaving unless Tmax approaches N/2, an unpractically large value. Conclusion and Suggestions for Future Work 144 (ideal interleaving results in uncorrelated received samples for fa > 0.0) and perfect channel estimation were assumed in the simulation. FFT -1 Interleaver pilot. MOD channel' r(t) r(t) A Ck A DEMOD Deinterleaver FFT A at Figure 7.1 IOFDM 2. DFE for frequency-selective channels The MSEDIRDFE equalizer developed for flat fading could be extended to the frequency-selective fading case. A possible approach 10" 10" 10 10" 10 10" AWGN BIOFDM, N=4 BOFDM, MLSE, L=5,f„=1.28 0 2 4 6 8 10 12 14 16 18 20 22 24 EJN,(dB) Figure 7.2 Comparison of BIOFDM with ideal interleaving to BOFDM with fa = 1.28 on a flat fading Rayleigh channel. Conclusion and Suggestions for Future Work 145 might be to subdivide the OFDM block into smaller units that experience relatively flat fading, and then apply the flat fading techniques to each subunit. 3. Trellis-coded blocks The effectiveness of channel averaging obtained by trellis-coding of conventional serially-modulated data blocks should be compared to that of OFDM. Since trellis-coding requires a larger signal constellation to maintain the same data rate, it is unlikely that the two methods would have the same sensitivity to channel estimation noise, and this would be an important point of comparison. 4. Channel estimation Channel estimation for flat fading channels is well understood, but much remains to be done in the area of frequency-selective fading channels. Pilot tones, data sequences, and frequency chirps have all been applied ([5],[10], [11], [12], ) but a comparative analysis of the costs and merits of each would be quite useful not only for OFDM, but other coherent techniques as well. It is also important to quantify the robustness of channel estimation methods in an interference limited environment. 146 A p p e n d i x A M F B D e t a i l s A.l Evaluation of an Integral In Chapter 3 the following integral occurs: Pb = 1 --YY A k h J e v i ( ^ ) y k ^ e - y ^ d y . 2 2kh^-i)^l \yN°Jy (AAA) A closed form can be found by expanding the erf(-) function within the integral as Setting oo v I=2=J. J e-^dty^e-y^dy. 0 0 (A.1.2) and it is easily shown and Integrating by parts k-i I -l dt dv = yk-xe-ylX'dy 0 - 1 / 2 -Eby/N0 du = — -=—y '<-& V7T dy k-i ( k - i y - ^ \ . k - i ' f-' (k-l - l)\ 1=0 y e 1=0 -. y=oo y=0 + VEb/N0 (k - 7 fc-f-J -(Eb/N0-+l/\i)yd t0(k-i-iy.JY (A.1.3) (A.1.4) (A.1.5) (A.1.6) (A.1.7) 147 and noting the remaining integral above is in the form of the gamma function 1 ^ ( f r - l ) ! A ^ r ( f c - / - l ) f Eb/N0 ~ (*- i -0 ! V W ^ + I/AO2^-0-1' where T() is the gamma function defined by [40] \T(x) = Jl*-1 e dt. (A. 1.9) Equation (A. 1.8) may be simplified by modifying the summation index to m = k — I to obtain and using the identity [40] r(m + |) = ^^m)!^'' Y^ds Jfc-1 < 2 m ) ! V (WJVo + l ) 2 m + 1 where (•)!! is the double factorial function [40] defined by /„\M - 2 ) ( ' n - 4 ) - - ' - C 1 ) . n o d d (\ i i o \ ~ 1 - 2)(n - 4) • • • (2), n even ' l^-iaz, Combining (A.l.11) with (A.l.l) yields the desired expression for Pb: R nt k — 1 P s"5-5g£-4"S^rV(A.w+ 1r1' ( A ' U 3 ) 148 A.2 Mean and Variance of y in Flat Rayleigh Fading For the flat Rayleigh fading model the matched filter output y is given by / |p(*)l2ki(*)l 2dt. (A.2.1) The mean of y is • rny = j \p{t)\2E[\Zl{t)\2]dt = 2a2 (A.2.2) zr where we have used E \zi(t)\2 = 2a2 r for a Rayleigh pdf and where cr2r is the variance of the real part of z\(t). Finding the variance of y is more involved. We have a2y = E[y2]-Aa2zr. (A.2.3) To simplify the notation let v\ = \z(t\)\ and v2 = 1 ^ ( ^ 2 ) 1 - Solving for the second moment E[y2]= f fWi)\M^)\2E[v\vl]dtxdt2. (A.2.4) We seek E ^ = J J * 0 r f ^ < i $ ^ h * » - , (A'2:5) o o . In (A.2.18) we have used the joint pdf for two Rayleigh distributed random variables [22] where p is the normalized cross correlation between two samples of the real part of the process zi(t) or equivalently two samples of the imaginary part. The real and imaginary parts are assumed to be independent. IQ() is the zeroth order modified Bessel function of the first kind. Expanding io() as [40] 149 and integrating term by term, it can be shown that E[v2v2] =4cr4 r(l + p2). (A.2.8) A result for the univariate case [9] is E\vk] =(2a2zryv(l + t) L J V 2 / ' : (A.2.9) =$> E [v2] = 2a2zr and E [v*] = 8<74r. This provides two consistency checks of (A.2.21) by setting p to 0 or 1. We assume [3] .P(T) = JO{UDT) (A.2.10) where Jo() is the zeroth order Bessel function of the first kind, u>j) is the maximum Doppler frequency in radians, and r = ti — t2 is the time interval between samples. Combining (A.2.16),(A.2.17),(A.2.21) and (A.2.23) we have v2y = J yb(<i)|2b(<2)|2J02(^(<i-t2))^id<2. (A.2.11) Note that the evaluation of (A.2.24) can be reduced to at most the evaluation of two one dimensional integrals by rewriting it in terms of r = t\ — t2 as °2y = I J ^tWi-r)\2dh Jl{uDr)dr. (A.2.12) The inner integral is the autocorrelation of the squared pulse magnitude Cp(r)= j\P(t)\2\P(t-r)\2dt. (A.2.13) Cp(r) could be considered the autocorrelation of the pulse power.. In many cases, CP(T) can be determined analytically. For example, if p(t) is a rect pulse we have To CPT(T) = I vect2(t)rect2(t - r)dt o (A.2.14) 150 Using (A.2.26) in (A.2.25) we obtain ' To a2y= J Cp{r)Jl{ojDr)dr. (A.2.15) -To A Limiting case Noting the normalized Doppler frequency in radians is LON = 2TTJN — WJ)TQ, we can obtain the limiting variance of y as / A T —> oo by holding w£f constant in (A.2.28) and taking the limit as To —• oo. The pulse p(t) has been defined as having unit energy and duration T 0 . These features constrain the behavior of Cp(t) and hence cr2 as To increases. Writing p(t) in terms of pu(t), a pulse having unit energy and also unit duration p(t) = -j=Pu(t/To) ' (A.2.16) and combining this with (A.2.26) at r = 0 we have To 1 x)\^dx o c where c is some finite constant denned as I o Noting an autocorrelation function has its peak at lag 0, Using (A.2.32) in (A.2.28) yields (A.2.17) c = J \Pu(x)fdx. • (A.2.18) CP(T) < (A.2.19) To • 2c j Jl(ior)T)dT <y\ < ° m : — (A.2.20) 151 and applying L'Hopital's rule lim al = lim 2cJ02(ujI)T0)dT = 0. (A.2.21) To—•oo To—>oo The variance a2 approaches 0 as the pulse duration To or equivalently as the normalized Doppler frequency fjy increases without limit for any unit energy pulse. A.3 Selection of M Assuming fast fading, we first require that At in M y = AtJ2\qi\2 (A.3.1) !=1 be much less than the coherence time tc = 1 / frj of the channel so that M . i=i is a close approximation, i.e. At < tc = (A.3.3) JD In (A.3.37) At was chosen so that /oAt = 0.01 was used to calculate the fast fading results of Section 3.5, and JD = 0.0 was used for the slow fading results. We have fN = fDTo = fDMAt (A.3.4) and for fast fading the normalized Doppler rate was set by selecting M = -pL- (A.3.5) ID At For the slow fading examples M was selected just large enough to allow an accurate representation of the desired delay spread, e.g. for normalized delay spread r = 0.25, M = 4 was selected. 152 A p p e n d i x B T h e D F F T a n d t h e i D F F T The discrete frequency Fourier transform (DFFT) is defined by ' Xk = J x(t)e~3%ktdt (B.l) and its inverse (iDFFT) by k where x(i) is a periodic function in time related to x(t) by x(t) = To^x(t-nTo). ' (B.3) n Equations (B.2) and (B.3) are readily established from the definition (B.l) using the conventional continuous Fourier transform X(f) '— J x(t)e~j2Tftdt and the Fourier transform pair £ S ( t - nT0) ^  ± £ 8(f - ^) . We have Xk = X(f) x £ £ (/ -n k V ' k V which must then have for its inverse Xk ^x{t)®T0 J2$(t-nTo) n = r 0 J > ( i - n r 6 ) CB-4). n = x(t) where ® is the convolution operator. Another useful relation is found by applying the orthogonality property of complex exponentials to (B.2) yielding / x{t)e-^mtdt = Y*k f e ^ h t e - ^ m t d t I k I (B.5) = ToXm or X k = kS ~x(t)e~3¥°hi • ( B - 6 ) To 153 A p p e n d i x C D e r i v a t i o n o f a M e t r i c R e c u r s i o n The MLSE receivers of Chapters 3 and 5 are implemented with a Viterbi algorithm which requires a recursive metric. Details of the derivation are given below. The original application of the Viterbi algorithm to ISI problems required a noise-whitening filter before the sequence estimator [26]. We use the approach of [41] which is able to avoid this requirement. Consider the metric A; below for which we seek a recursive expression Xi = Re ln=0 I 1 n=0 m=0 and expand the double summation as follows i i i / i-i E E anamVm,» = E alamVm,l + E ana m ,n n=0 m=0 m=0 \ n=0 Isolating the terms for which m = I the R.H.S. above can be written l-i i-i / l-i n=0 WlfVij + Ea"a'y'-« + E a^amVm,l + E vm n=0 m=0 V and using Vm j„ = the preceding is equivalent to \ai\2 + 2Re Using (C.4).in (C.l) yields l-i a i ^ a n V i , n=0 1-1 1-1 n=0 m=0 A, =Re[tfU}} \ai\2yi,i Re l-i al E a"Vl'n n=0 1-1 .1=0 +Re from which it is apparent A; = A/_! + ReltfUi] - Re l-i l-i ££<•* n=0 m=0 l-l a'Efl»^>n n=0 \al\2vl,l (C.7) (C.8) (C,9) (CIO) (C.11) (C.12) 154 A p p e n d i x D M a t r i x N o t a t i o n C o n v e n t i o n s Matrix notation, when clearly denned, provides an elegant representation for systems of linear equations. We rely on it.from Chapter 3 onwards. Unfortunately there is no one consensus on how to identify matrices and define their elements, therefore the conventions adopted for this work are explained below. All matrices are represented by boldface characters. An uppercase boldface character represents a two-dimensional matrix, and a lower case boldface character represents one-dimensional matrix, which is a column vector. For example Q is a two-dimensional matrix and q is a column vector. The (i, k)th element of Q is Q;^  and the ith element of q is qi. Note the subscripts are not bold. Examples: Q 351,1 X2A a? 1,2 £2,2 $M,\ XM,2 X2,N XM,N " XI ' X2 q = \ -XM . (D.13) where Qi^ = X{^ and OJ = x{. There is no relation between the case of a matrix and the case of its elements. We do this because it is helpful when checking the syntax of matrix equations to be able to distinguish two-dimensional matrices from column vectors at a glance; this determines the case of the matrix. Yet, depending on the situation, we need to assemble matrices of scalars defined in either the time-domain or frequency-domain. Following the well-established convention for Fourier Transforms, we consistently represent frequency-domain scalars with uppercase characters and time-domain scalars with lowercase characters. Thus d — X2, * * * XM] and d — T T [Xi,X2 • • • ,XM] are both valid constructs. The superscript (•) denotes transposition. 155 Occasionally we need to define several related vectors. We do this with a bold subscript, e.g. di, d2, • • •, represents ./V different column vectors. The ith element of the kth vector is d^. Note k is bold, i is not. It follows an alternative way of describing Q above would be to first define qk = [xi,k,X2,k,-• • iXM,k]. and then define Q = [qi,q2,---,qN]-Another convenient way to define the same matrix Q is: Q = [{Qi,k}] > Qi,k = xi,k-> where 1 < i < M, 1 < ~k < N. The [{Qi,k}] denotes the set of elements Q^ k are arranged in a matrix. Note the order of the subscripts in Qi^ = x^^ can be reversed to obtain Q T , e.g. Q T = [{Qi,k}] , Qi,k = xk,i-Analogous to the vector case, a two-dimensional matrix can also be defined as an array of two-dimensional sub-matrices, e.g. Y = [{Xji}], where the (i,k)th element of Xj.1 is given by Xj,! = X j > l i k . We also use the Hermitian, (-)H, to denote conjugate transpose, e.g. Q H = Q T*. Conclusion and Suggestions for Future Work 156 B i b l i o g r a p h y [1] J. E. Padgett, C. G. Gunther, and T. Hattori, "Overview of wireless personal communications," IEEE Communications Mag., vol. 33, pp. 28-41, Jan. 1995. [2] D. C. Cox, "Wireless network access for personal communications," IEEE Communi-cations Mag., vol. 30, pp. 96—115, Dec. 1992. [3] W. C. Jakes, Microwave Mobile Communications. N.Y.: Wiley, 1974. [4] A. J. Viterbi and R. Padovani, "Implications of mobile cellular CDMA," IEEE Communications Mag., vol. 30, pp. 42-49, Dec. 1992. [5] L. J. Cimini Jr., "Analysis and simulation of a digital mobile channel using orthogonal frequency division multiplexing," IEEE Trans. Comm., vol. COM-33, pp. 665-675, July 1985. [6] A. V. Oppenheim and R. W. Schaffer, Digital Signal Processing. Englewood Cliffs, N.J.: Prentice-Hall, 1973. [7] J. Ahn and H. S. Lee, "Frequency domain equalisation of OFDM signals over frequency nonselective Rayleigh fading channels," Electronics Letters, vol. 29, pp. 1476-1477, Aug. 1993, [8] J. M. Wozencraft and I. M.Jacobs, Principles of Communications Engineering. N.Y.: Wiley, 1965. [9] J. G. Proakis, Digital Communications. N.Y.: McGraw-Hill, 1989. [10] J. K. Cavers, "An analysis of pilot symbol assisted modulation for Rayleigh fading channels," IEEE Trans. Veh. Tech., vol. 40, pp. 686-693, Nov. 1991. [11] J. K. Cavers, "Performance of tone calibration with frequency offset and imperfect pilot filter," IEEE Trans. Veh. Tech., vol. 40, pp. 426-^ 134, May 1991. [12] M. Mouly and M.-B. Pautet, The GSM System for Mobile Communications. 49 rue Louise Bruneau, F-91120 Palaiseau, France: Mouly & Pautet, 1992. Conclusion and Suggestions for Future Work 157 [13] J. Holsinger, Digital Communication over fixed time continuous channels with memory. PhD thesis, M.I.T., 1964. [14] S. B. Weinstein and P. M. Ebert, "Data transmission by frequency division multiplexing using the discrete Fourier transform," IEEE Trans. Comm. Tech., vol. COM-19, pp. 628-634, Oct. 1971. [15] P. Bello, "Selective fading limitations of the kathryn modem and some system design considerations," IEEE Trans. Comm., vol. 13, pp. 320-333, Sept. 1965. [16] J. Rault, D. Castelain, and B. LeFloch, "The coded orthogonal frequency division multiplexing (COFDM) technique, and its application to digital radio broadcasting towards mobile receivers," in IEEE Global Communication Conference (Globecom), pp. 428-432, 1989. [17] E. F. Casas and C. Leung, "OFDM for data communication over mobile radio FM channels-part 1: Analysis and experimental results," IEEE Trans. Comm., vol. COM-39, pp. 783-793, May 1991. [18] J. Andersen, T. Rappaport, and S. Yoshida, "Propagation measurements and models for wireless communications channels," IEEE Communications Mag., vol. 33, pp. 42-49, Jan. 1995. [19] A. Papoulis, Probability, Random Variables, and Stochastic Processes. N.Y.: McGraw-Hill, 1984. [20] J. Mazo, "Exact matched filter bound for two-beam Rayleigh fading," IEEE Trans. Comm., vol. COM-39, pp. 1027-1030, July 1991. [21] M. Clark, L. Greenstein, W. Kennedy, and M. Shafi, "Matched filter performance bounds for diversity combining receivers in digital mobile radio," IEEE Trans. Veh. Tech., vol. 41, pp. 356-362, Nov. 1992. [22] W. Davenport and W. Root, An Introduction to the Theory of Random Signals and Noise. N.Y.: McGraw-Hill, 1958. [23] S. S. Haykin, Introduction to Adaptive Filters. N.Y.: MacMillan, 1984. [24] A. V. Oppenheim, A. S. Willsky, and I. T. Young, Signals and Systems. Englewood Cliffs, N.J.: Prentice-Hall, 1983. Conclusion and Suggestions for Future Work 158 [25] W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes in C. Cambridge: Cambridge University Press, 1988. [26] G. D. Forney, Jr., "Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference," IEEE Trans. Inf. Theory, vol. IT-18, pp. 363-378, May 1972. [27] G. Ungerboeck, "Nonlinear equalization of binary signals in Gaussian noise," IEEE Trans. Comm. Tech., vol. COM-19, pp. 1128-1137, Dec. 1971. [28] R. E. Blahut, Digital Transmission of Information. N.Y.: McGraw-Hill, 1982. [29] D. D. Falconer and F. R. Magee, Jr., "Adaptive channel memory truncation for maximum likelihood sequence estimation," Bell System Technical Journal, vol. 52, pp. 1541-1562, Nov. 1973. [30] J. Salz, "Optimum mean-square decison feedback equalization," Bell System Technical Journal, vol. 52, pp. 1341-1373, Oct. 1973. [31] C. A. Belfiore and J. H. Park, Jr., "Decision feedback equalization," Proc. of the IEEE, vol. 67, pp. 1143-1156, Aug. 1979. [32] J. M. Ahn, "private communication." re: ideal decisions in DFE feedback loop, July 1996. [33] S. U. H. Qureshi and E. E. Newhall, "An adaptive receiver for data transmission over time-dispersive channels," IEEE Trans. Inform. Theory., vol. IT-19, pp. 448-457, July 1969. [34] D. G. Messerschmitt, "Design of a finite impulse response for the Viterbi algorithm and decision feedback equalizer," in Rec. Int. Conf. Communications, ICC-74, (Minneapolis, MN), June 1974. [35] D. G. Messerschmitt, "A geometric theory of intersymbol interference," Bell System Technical Journal, vol. 52, pp. 1483-1519, Nov. 1973. [36] P. Monsen, "Feedback equalization for fading dispersive channels," IEEE Trans. Inform. Theory., vol. IT-17, pp. 56-64, Jan. 1971. [37] D. D. Falconer, F. Adachi, and B. Gudmundson, "Time division multiple access methods for wireless personal communications," IEEE Communications Mag., vol. 33, pp. 42-49, Jan. 1995. Conclusion and Suggestions for Future Work 159 [38] A. Jerri, Introduction to Integral Equations with Applications. N.Y.: Dekker, 1985. [39] G. Strang, Linear Algebra and its Applications. San Diego: Harcourt, 1988. [40] M. Abramowitz and I. E. Stegun, Handbook of Mathematical Functions. Washington, D. C.: National Bureau of Standards, 1972. [41] G. Ungerboeck, "Adaptive maximum-likelihood receiver for carrier-modulated data-transmission systems," IEEE Trans. Comm., vol. COM-22, pp. 624-636, May 1974. 

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