Technical Report No. 195 3 674- 18-T INTERSYMBOL INTERFERENCE IN BINARY COMMUNICATION SYSTEMS by Christopher V. Kimball Approved by::c __ek. T. G. Birdsall COOLEY ELECTRONICS LABORATORY Department of El'ectrical Engineering The University of Michigan Ann Arbor, Michigan for Contract No. Nonr-1224(36) Office of Naval Research Department of the Navy Washington, D.C. 20360 August 1968 Reproduction in whole or in part is permitted for any purpose of the U. S. Government.

Christopher Vernon Kimball 1968 All Rights Reserved ii

ABSTRACT When a binary communication system transmits symbols through a bandlimited channel, the received symbols will generally overlap in time, giving rise to intersymbol interference. In the presence of noise, intersymbol interference produces a significant increase in the system probability of error. The problem of intersymbol interference and noise is considered here for known, linear, time invariant channels and with added white Gaussian noise. Although a particular underwater acoustic channel is used as a source of motivation, the results presented are equally applicable to other communication channels. Traditional approaches to the intersymbol interference problem — spectrum and transversal (time) equalization are examined. A basis for the comparison of intersymbol interference problems using the concept of phase equalization, is given. A major assumption which limits the interference to that caused by adjacent symbols is made. This assumption is shown to be equivalent to restricting the transmitter to reasonable signalling rates relative to the bandwidth of the channel power spectrum. All subsequent analysis and evaluation are done under this assumption. Several linear filter receivers prevalent in the literature —the matched filter receiver, the transversal filter receiver, and the optimized linear filter receiver —are reviewed and evaluated. Two easily implemented nonlinear receivers, the switched-mode receiver and iii

the iterated switched-mode receiver, are considered as alternatives to the more complex optimized linear filter receivers. The iterated switchedmode receiver, which is described for the first time here, is shown to perform better than any optimized linear receiver when intersymbol interference is moderate. Finally, the optimum (likelihood ratio) receiver is described and evaluated to provide an absolute lower bound on error probability for a given intersymbol interference problem. A comparison of the error performance of the receivers shows that if intersymbol interference is reduced to moderate amounts by proper choice of signalling rate, then the easily implemented iterated switched-mode receiver gives near intersymbol interference free performance. For higher signalling rates and consequently larger amounts of intersymbol interference more complicated receivers are required to achieve near optimum performance and even the performance of the optimum receiver is significantly worse than intersymbol interference free performance. iv

FOREWORD This report considers a practical problem in underwater communications- - intersymbol interference. Four major contributions are presented. First, the extent of intersymbol interference in a given situation is shown to be dependent on the autocorrelation function of the received symbol instead of the received symbol itself. Examination of the received symbol usually indicates more intersymbol interference than is actually present. Second, several traditional and proposed receivers are compared on a consistent basis, indicating the trade-offs between system error performance and system complexity. The optimum (likelihood ratio) receiver is included in the comparison to provide a lower bound on error performance. Third, an easily implemented nonlinear receiver whose performance is close to that of the optimum receiver in many practical cases is described. Finally, a rule-of-thumb is given which relates transmission rate, error performance, and system complexity. The above contributions should be of considerable importance to a designer of an underwater communication system. v

ACKNOWLEDGMENTS I would like to thank the members of my committee, Professor T. G. Birdsall (Chairman), Professor W. A. Ericson, Professor K. B. Irani, Professor A. B. Macnee, and Professor M. P. Ristenbatt for their helpful suggestions. Credit is also due to my colleagues at the Cooley Electronics Laboratory, The University of Michigan, for their many comments and suggestions during the course of the research. The research reported in this dissertation was supported by the Office of Naval Research, Department of Navy, under Contract No. Nonr-1224(36), "Acoustic Signal Processing." Particular thanks are due to Mrs. Dianne Bohrmann and Miss Heather Sawler for their fine preparation of the manuscript. vi

TABLE OF CONTENTS Page ABSTRACT iii FOREWORD v ACKNOWLEDGMENTS vi LIST OF ILLUSTRATIONS ix LIST OF TABLES xiii LIST OF APPENDIXES xiv LIST OF SYMBOLS xv CHAPTER I: INTRODUCTION1 1.1 The Mimi Channel 2 1.2 Assumptions 10 1.3 Intersymbol Interference 16 1.4 Summary and Contributions 18 CHAPTER II: INTERSYMBOL INTERFERENCE 23 2.1 Traditional Approaches 23 2. 2 Characterization of the Problem 35 2.3 The M= 1 Assumption 42 CHAPTER III: LINEAR FILTER RECEIVERS 51 3.1 General Discussion 52 3.2 Matched Filter and Transversal Filter 63 Receivers 3.3 Optimized Linear Filter Receivers 78 CHAPTER IV: TWO SIMPLE NONLINEAR RECEIVERS 90 4.1 The Switched Mode Receiver 91 4.2 Iterated Switched Mode Receiver 97 4. 3 Comparison with Linear Filter Receivers 101 vii

TABLE OF CONTENTS Cont. Page CHAPTER V: THE OPTIMUM (LIKELIHOOD RATIO) RECEIVER 109 5.1 Operation of the Receiver 110 5. 2 Evaluation of the Optimum Receiver 132 5. 3 Comparison of Optimum and Suboptimum 152 Receivers CHAPTER VI: CONCLUSIONS AND FUTURE STUDIES 170 6.1 Conclusions 170 6.2 Future Studies 172 APPENDIX A 174 APPENDIX B 177 REFERENCES 186 DISTRIBUTION LIST 188 viii

LIST OF ILLUSTRATIONS Figure Title Page 1.1 Physical configuration of the Mimi 4 channel. 1.2 Typical complex impulse response and 7 spectrum for the Mimi channel. 1.3 Typical autocorrelation of the complex 8 impulse response and power spectra for the Mimi channel. 1.4 Time waveform and spectrum of a 40 ms 17 rectangular symbol. 1. 5 Received time waveform and spectrum of 19 a 40 ms rectangular symbol passed through the typical Mimi channel of Fig. 1.2. 2.1 Block diagram of channel and linear filter 25 receiver 2.2 A receiver using spectrum equalization 28 2. 3 Typical transversal filter response to 30 received symbol. 2.4 Seven tap transversal filter receiver. 31 2. 5 Unequalized and phase equalized systems. 36 2. 6 Response of Mimi channel to 40 ms symbol 38 with and without equalization. 2.7 Time response and power spectrum of 40 ms 40 symbol through Mimi channel and through equivalent ideal bandpass channel. 2.8 Phase equalized impulse response of Mimi 45 channel. 2.9 Received symbol duration based on r1 as a 46 function of r7. ix

LIST OF ILLUSTRATIONS Cont. Figure Title Page 2.10 Plot of 60 ms word perfect transmitted symbol. 48 2.11 Phase compensated 60 ms perfect word 49 received symbol. 3.1 General linear filter receiver. 51 3. 2 Decision plane of a linear filter receiver. 55 3.3 Canonical linear filter receiver. 60 3.4 Heuristic phase equalized received symbol 63 (M= 1). 3. 5 Noise free signal components in the 64 interval (kT, (k+2)T). 3.6 Vector space representation of signal 67 components in (kT, (k+2)T). 3.7 Decision plane for matched filter receiver. 71 3. 8 Decision plane for transversal filter receiver. 74 3.9 Comparison of MFR and TFR for d = 10. 77 3. 10 Canonical admissable receiver with 2q+l 80 taps. 3. 11 Optimum tap weights for OLFR3, OLFR5, 85 OLFR7, d- 10. 3.12 Probability of error for OLFR, OLFR 86 OLFR5 OLFR7 d - 10. 3.13 Decision planes for the MFR, TFR, and 89 OLFR(2T). 4.1 Heuristic implementation of the switched 91 mode receiver. 4. 2 Simplified implementation of the switched 97 mode receiver. x

LIST OF ILLUSTRATIONS Cont. Figure Title Page 4.3 An implementation of the ISMR. 102 4.4 Comparison of SMR with MFR, TFR, and 104 OLFR3, 5. 4.5 Comparison of ISMR with MFR, TFR, and 106 OLFR3, 5' 5. 1 Updating the log odds ratio L.113 5. 2 Procedure used to determine I4 during 127 observation. 5. 3 Updating process used by TOOR (9=3). 129 5.4 Division of the reception x into Xk and 133 Xk' 5. 5 Representation of xk in the plane defined 141 by UO u1. 5.6 Plot of y(L 1 Lk1) for r(T) =.25, d= 10. 145 k+l1 k 5.7 Plot of p(Lk I Lk bk =+1) for Ir(T)=.25, 148 d= 10. 5.8 Plot of p(L1 Ibk +1)k = 0, 1, 2, 3 r(T)=.5, 149 d = 10. 5.9 Plot of p(L 2 Ibk= 1) for Ir(T) I =.0,.1,.2, 150.3,.4,.5, d= 10. 5.10 Heuristic two pass implementation of the optimum 153 receiver. 5.11 Probability error P versus I r(T)l,d = 5. 155 e 5.12 Probability error P versus I r(T)l, d = 7.5. 156 xi

LIST OF ILLUSTRATIONS Cont. Figure Title Page 5.13 Probability error P versus I r(T) I, d = 10. 157 5.14 Probability error P versus Ir(T)I, d 12.5. 158 5.15 Probability error P versus I r(T) I, d = 15. 159 5.16 Probability error P versus I r(T) I, d =.1. 162 5.7 Probability error P versus r(T), d =.2. 163 5.18 Probability error P versus I r(T) I, d =. 3. 164 e 5.18 Probability error P versus I r(T) I, d.3. 164 5.x19 Probability error P versus I r(T),iid.4. 165 e 5. 20 Probability error P versus I r(T) I, d -.5. 166 e xii

LIST OF TABLES Table Title Page 4.1 Results for system using Mimi channel 108 (Fig. 1.2) and a 60 ms perfect word symbol (Fig. 2.10), d = 10, r(T) = -. 2. 5.1 Equations for the log likelihood ratio 124 In lk(xj). 5. 2 Results for system using Mimi channel and 169 a 60 ms perfect word symbol, Ir(T)I =.2, d= 5,10, 15. xiii

LIST OF APPENDIXES Page APPENDIX A: PROOF OF THE DECREASE IN P (k) WITH 174 ADECREASE IN I h 1. APPENDIX B: DERIVATION OF EQUATIONS 5.19, 5.22, 177 5.26, AND 5.81. xiv

LIST OF SYMBOLS Symbol Definition Page A(w) magnitude spectrum of p' (t) 26 bk value of the kth transmitted symbol 51 Bk set of m dimensional vectors b0...bi... 54 bn i k b.=~1 n 1 b k-q(bkql*...*bib+l) i k 81 b (b, bj) 116 bkco "large" magnitude with sign of bk 123 IBkl any set {be Bk I be IBkI = -b I Bk } 175 CW continuous wave 5 C(c) channel (complex) spectrum 24 C. weight of jth delay in canonical form of 59 1 the linear filter receiver c0, c1 coefficients used in TFR 73 dk decision value on the kth symbol 51 Dk hyperplane through the origin for kth symbol 54 d index of signal detectability 2E/NO for a 68 single symbol dk "first guess" decision on kth symbol for ISMR 98 1 dk final decision on kth symbol for ISMR 99 i dk "ith guess" decisions made by ISMR 101 E energy in a single noise free symbol 26 EF(wc) equalizer filter (complex) spectrum 27 xv

LIST OF SYMBOLS Cont. Symbol Definition Page e(t) response of transversal filter to p'(t) 32 E(w) Fourier transform of e(t) 32 E[ ~ ] expected value 56 e0' e1'2 orthonormal unit vectors used to represent xk 66 f frequency (Hz) 7 FL linear filter 23 FLOPT(W) complex spectrum of the optimum post 27 equalizer filter FL(w) overall transfer function of a linear filter 29 receiver FLMF () complex spectrum of a matched filter 32 FLTF(W) complex spectrum of a transversal filter 32 F, F1 Fourier transform, inverse Fourier transform 39 G0(xj, Lk1) see Eq. 5.19 119 G1(xj) see Eq. 5.22 120 G2(x L1 (+1),L 1 (-1)) see Eq. 5.26 121 G0(x0) see Eq. 5.34 123 -1 0 k k- - G01 inverse (given xk, Lk 1 Lk ) of the GO 143 mapping h impulse response of linear filter 52 H(w) complex spectrum of linear filter in a 52 linear filter receiver h h(kT - t) 53 xvi

LIST OF SYMBOLS Cont. Symbol Definition Page H1 subspace spanned by {p I i- 0...m} 58 H2 orthogonal complement of H1 58 el-/k rak rak.s 5k h, h components of h in the Hi, H subspaces 58 1 2 2 ISMR iterated switched mode receiver 97 J(xk'xk) Jacobian of the Go mapping with xk 143 as the auxiliary variable k. transversal filter tap weights 31 K normalizing constant for p(x I bjlbj) 118 L2 Hilbert space of finite energy waveforms 23 Lk decision variable for the kth symbol 51 LL; ~ ~ output of the first adder for canonical linear 61 receiver L, ~ output of the matched filter for nonlinear 92 receivers lk(x) likelihood ratio of the reception x for the 111 kth symbol 4Lk ~ log odds ratio for the kth symbol given X. 112 lnlk(xj) log likelihood ratio of xj for the kth symbol 11 lnlk(xj) log likelihood ratio of x. for the kth symbol 113 lnl (X.) log likelihood ratio of X. for the kth symbol 114 k J ] Li-l(bk) conditional log odds ratio (Eq. 5. 25) 122 lnlI(x. bk) conditional log likelihood ratio (Eq. 5. 29) 122 nxj k JLk reverse log odds ratio, the log ratio of X. 133 for the kth symbol 3 xvii

LIST OF SYMBOLS Cont. Symbol Definition Page M degree of intersymbol interference 42 m m +1 is the number of transmitted symbols 52 MFR matched filter receiver 69 N noise power 12 IN" (o) 2 noise power spectrum at the output of equalizer 27 filter n(t) noise waveform 52 No noise power density 52 n! internal noise waveform of canonical linear 59 receiver OLFR. optimized linear filter with i taps 83 OLFR(2T) optimized linear filter receiver for impulse 88 response having duration 2T PR psuedo random 5 PEF(o) complex spectrum of phase equalizing filter 35 P probability 54 P (k) probability of error for the kth symbol 57 e P average system probability of error 58 P (w) probability of error for OLFR with tap 82 coefficients given by w P0, P1 conditional probabilities of error for SMR 93 P (a, f3) probability of error when Lk has the form 94 e of Eq. 9.7 xviii

LIST OF SYMBOLS Cont. Symbol Definition Page P probability of error for the "first guess" 98 decision of ISMR P00, P01, P10, P1l conditional probabilities of error for the 99 evaluation of ISMR Pe probability of error for ISMR final decisions 99 P e probability of error for ith decision level in the 101 logical extension of ISMR p probability density function 111 Q ratio of center frequency to bandwidth 3 q1(t), q2(t) binary symbol waveforms 10 q(t) transmitted binary simplex signal 24 Q(w) Fourier transform of q(t) 24 ~ binary receiver 23 R(t) autocorrelation function of the noise free 68 received signal r(t) normalized autocorrelation function 68 = R (t) / R(0).k k-1 R * p1 R(T) 183 S/N average signal-to-noise power ratio 5 SMR switched mode receiver 91 sb signal components in x. given b 118 S(u, u2) u10(u1) + u20(u2) 175 t time 7 xix

LIST OF SYMBOLS Cont. Symbol Definition Page T duration of the transmitted symbol 10 TFR transversal filter receiver 72 TOOR truncated observation optimum receiver 128 uW, u1 orthonormal vectors used to represent xk 139 Var variance 56 w. weight of the i th tap for OLFR2q+l 79 w (w...w ) 81 -q +q _ * w optimum tap coefficients for OLFR 83 w,w+1 coefficients used in OLFR(2T) 87 x total received waveform 52 x portion of the reception x in (kT, (k+2)T) 64 X. portion of the reception in the time interval 112 ~J ~ (0, jT) x. portion of the reception in the time interval 112 J (jT, (j+l)T) X. portion of the reception in the time interval 133 J] ~ (jT, (n+2)T) 0 1 XkXk components of xk in the u0, u1 directions 140 Xk representation of xU in plane defined by 140 u0, u1 XX

LIST OF SYMBOLS Cont. Symbol Definition Page,k k Y1 h1 * p 174 Y2 E bp h1 174 ik,-k 2 _-k 2 Y3 1/No (lhIl + I h2 )/2 174 Z(Sbw) Z2(I(,w) functions used in computing P (w) 82 for OLFR e a, B constants used in evaluation of SMR, ISMR 94 r(dkl) threshold for SMR 96 0 0 r(dk 1 dk+) threshold for ISMR 100 k k~1 Y(L1, Lk ) integration path for determination of 144 k+1 k p(qL ILk l,bk_ +1) Ak-lk interference component of xk given bk bk 65 bk-1bk k k"l k 1/2 [ L1 - 2 00 184 17 percentage of total energy of p(t) in an 44 interval of duration 2T k 1/2[Lk l + 2xp0] 184! I Pl(t),P2(t) noise-free channel outputs due to symbols 13 q1(t), q2(t) p (t) uncompensated, noise-free channel output 15 for simplex symbols P '(W) Fourier transform of p'(t) 26 po "(W) complex spectrum of the noise-free output of 27 the equalizer filter due to a single symbol xxi

LIST OF SYMBOLS Cont. Symbol Definition Page p noise-free phase-equalized received symbol 36 p1 p(t-iT) 56 kk i i PkPk pP0(t - kT), pl(t - kT) 64 P0= 0 p 0 t < T 64 = 0 elsewhere P1 pl - p T < t <2T 64 - 0 elsewhere p (t) a selected segment of p(t) 70 p p(iT - t) 61 k- i Pij component of p-i in the u. direction 139 ] T time center of a waveform 26 c TRMS RMS time duration of a waveform 26 0 (u) zero mean, unit variance Gaussian density 56 function @(u) cumulative distribution function of 0 (u) 57 4(w) phase spectrum of p (t) 26 W radian frequency 24 superscript * complex conjugate 27 operator L inner product 53 l- LL orthogonality relation 65 0 convolution operator 136 xxii

CHAPTER I INTRODUCTION This paper is the result of a study of the general underwater acoustic communications problem, using experimental results from a comparatively well-known channel. Because of the complexity of the general problem, a specific, but important aspect of the problem, is considered: intersymbol interference in binary signalling systems. Restriction to binary signalling systems provides considerable simplification in both analysis and implementation. Intersymbol interference occurs in such systems when the received symbols overlap one another in time, increasing the probability of error. Channel phenomena which produce irregularities in the received power spectrum, such as multipath and selective fading, are sources of intersymbol interference. Although underwater acoustic channels are the frame of reference for this study, application to other channels is easily accomplished. For binary communication systems in the absence of intersymbol interference, the classical theory of signal detectability provides the basis for the design of the optimum (likelihood ratio) receiver (Ref. 1). Only a limited number of studies have been made which consider the intersymbol interference problem, however. These studies deal with the determination of the optimum linear filter receiver for a given situation (Refs. 2, 3, 4). Aein and Hancock have demonstrated an easily 1

2 implemented nonlinear receiver which is superior in performance to their optimum linear receiver (Ref. 2). A major objective of this paper is to extend the theory of signal detectability to intersymbol interference problems by analysing and evaluating the optimum (likelihood ratio) receiver. A comparison of the above receivers with those used in practice is also made. In subsequent discussion, we often refer to experimental results from the Miami to Bimini test facility of the Michigan-Miami ("Mimi") project as a source of motivation (Ref. 5). Some significant aspects of the Mimi channel are sketched in the next section. We then discuss the working assumptions of the paper and ways in which these assumptions may be relaxed. Finally, the major conclusions and contributions of the thesis are summarized. 1. 1. The Mimi Channel The Mimi project is a joint effort by the Institute of Marine Science of the University of Miami and Cooley Electronics Laboratory of the University of Michigan. The Mimi test facility has two features which distinguish it from other facilities. First, both the transmitter and receiver hydrophones are in permanently fixed positions. This allows repetitive study of exactly the same physical channel. Second, very stable (1-2 parts in 10 ) oscillators are available at both the transmitting and receiving sites, which allow coherent averaging and analysis of the receptions. In this section we will sketch the

3 physical structure of the channel and several of its features which are relevant to the communications problem. The basic Mimi channel consists of a transmitting transducer and reflector at Fowey Rocks near Miami, Florida, and receiving hydrophones at North Bimini Islands, Bahamas, 43 miles to the east as shown in Fig. 1. 1. For the first 13 miles from Fowey Rocks the depth is sloping to 400 meters. Near the end of this shelf is the mean center line for the Gulf Stream, which is a source of turbulence. Beyond the first 13 miles the depth drops off abruptly to 800 meters until Bimini is reached. Ray path computations indicate bottom reflection combined with surface reflected and/or refracted modes of propagation in the channel. Two fundamental limitations are placed on the channel by the transmitting transducer and reflector. The first is that the nominal bandwidth of the transducer is 100 Hz centered at a 420 Hz carrier frequency. This limitation is due to the construction of the transducer and the beam forming reflector. One can view the bandwidth restriction by considering the "Q" (center frequency/bandwidth) of the system, which indicates the system is wide band, i.e., 1.0 MHz at 4. 2 MHz would be a wide band in HF radio. Alternatively, when typical information rates are considered, the system appears narrowband, i. e., a 100-wpm teletype requires a bandwidth of at least 80 Hz. We see that, relative to common information rates, bandwidth is a significant limitation.

4 25~ __ i______ ____ 50' ~ UJ s5o'. ' ATLANTI % MARINE LAB */ ~~aBIMI I ~'~VIRGINIA KEYt ATLANTI t W. 30',,/I.; 0'tE TO OE OCEA t / *.'I Z / 0 800 300 300 00' 790 30' w O 0-, ------------— ' ----------- I ------- 4T-LONG z - \ 8800 (VERTICAL SCALE = 37 x HORIZONTAL) 0 10 20 30 4 N. MILES (ONE TO ONE SCALE) Fig. 1.1 Physical configuration of the Mimi channel

5 The second limitation due to the transmitting transducer is a peak power limitation. Ceramic elements in the transducer are subject to fracture and/or fatigue at high power levels. Because of the great difficulty and expense involved in obtaining a replacement transducer, the system is normally operated well below its specified rating. Even if more durable transducers were available, cavitation in the water surrounding the transducer would provide a peak power limitation. The effect of the peak power limitation on the communications problem is to severely limit the waveforms which may be transmitted~ Noise is a major problem in the Mimi channel. In travelling the 43 mile distance a signal is attenuated by 105 to 135 db, yielding S/N ratios in the range -10 db to + 20 db in a 100 Hz band. The precise form of the noise is largely unknown at the present time. Nonthermal noise such as ship or biological noise is known to be present, and hence an accurate description of the noise would be difficult. Two types of signals have been used to measure the channel spectral characteristics: continuous wave (CW) signals and periodic pseudo-random (PR) signals. CW signals allow high S/N analysis of a single spectral line, usually at the 420 Hz center frequency. Results of these CW transmissions indicate very slow (less than 10 cycles per day) phase changes of the received signal relative to the coherent reference. During these same tests the amplitude of the reception was noted to fluctuate considerably, occasionally becoming undetectable. Two

6 tone (two simultaneous CW) tests indicate similar phase characteristics on both spectral lines, but with independent amplitude characteristics. Both forms of the CW experiment suggest the Mimi channel is amazingly phase stable and subject to a time variant frequency selective fading. The wide band PR signals can be employed to measure the spectrum of the Mimi channel. This spectral analysis is done by crosscorrelating the received signal with the original pseudo-random signal and then performing deconvolution techniques to obtain the complex channel spectra. Figure 1.2 shows a typical complex impulse response and corresponding spectrum based on a 5-minute coherent time average of data taken in February 1965. Figure 1.3 shows the autocorrelation of the impulse response and the corresponding spectrum, which is the channel power spectrum. The plotted data (which has been subjected to discretionary filtering) is believed representative of the form of the spectra to be expected in the Mimi channel. From the spectra of Figs. 1. 2 and 1. 3, the selective fading effects of multipath are apparent. Two distinct null frequencies in the effective 50 Hz bandwidth are indicative of selective fading. The linear In this and subsequent discussions of the channel, we represent the real bandpass waveform as a complex low pass waveform as is usually done. The complex low pass waveform is given by M(t) ej 0t) where M(t) is the magnitude waveform and 0(t) is the phase waveform. The physical bandpass waveform is given by M(t) cos [ 2r(420)t - 0(t)] ** The autocorrelation function is a conjugate symmetric complex waveform because of the complex low pass representation of the impulse. The autocorrelation of the physical bandpass waveform is given by one-half the real part of the complex autocorrelation.

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9 phase characteristic in the third (highest frequency) sub-band suggests that energy in this sub-band arrives later than that in the other two subbands. This assertion is borne out in the time domain by the delayed peaks in the impulse response. Such a late arrival may also be attributed to multipath. Another significant feature of the Mimi channel is the presence of time variations of different time orders. Long term variations dependent on the time of day; tides and weather have been observed in both the CW and PR sequence transmissions. Sequence transmissions have shown very slight changes in the channel spectrum over a 5-minute interval. Studies using receiving hydrophones close (3 miles) to the transmitting site indicate the presence of an amplitude modulation effect caused by wave height. The time order of this effect is of the order of 1 to 10 seconds. More rapid time variations in the channel may also be present, but are difficult to distinguish from the noise. Other interesting features of the channel are known to exist. Indeed, further analysis of the Mimi channel is a continuing project. We summarize the major features of the channel below: 1. Limited bandwidth: - 100 Hz 2. Transmitter peak power limitation 3. Low received S/N ratios: -10 to +20 db 4. Multipath/ Selective fading effects: Figs. 1. 2, 1. 3 5. Time variations of different durations: 5 min., 5 sec.

10 Although the Mimi channel is a very specific channel, the above effects are generally encountered in underwater acoustics and hence we expect to be able to generalize results obtained for the Mimi channel to other situations. 1.2. Assumptions In order to study the communication problem on a firm theoretical basis, several simplifying assumptions are made. These assumptions take on varying degrees of importance and many of them may either be relaxed or considered worst case assumptions. In this section we state the working assumptions for the Mimi channel discussed previously. Transmitter A major assumption of this work is that the transmitter is restricted to binary signalling. That is, in any one time interval, T the transmitter may transmit only one of two signals q1(t) or q2(t). The signals q1(t) and q2(t) considered will usually have low peak to average power ratios. More important, however, we assume that a detailed knowledge of the channel spectrum (beyond bandwidth and center frequency) is not available to the transmitter. This precludes the use of signals which are carefully chosen to smooth out irregularities in the channel spectrum. Restriction of the problem to binary signalling has two desirable effects. The first is to simplify the analysis by allowing the use of

11 likelihood ratio procedures. The second is that the implementation of the transmitter is significantly simplified. Furthermore, studies have shown that binary signalling is a reasonable method of signalling at the S/N ratios encountered in the Mimi channel (Ref. 6). The assumption that details of the channel spectrum are unknown to the transmitter is a realistic assumption in most underwater acoustic channels. In order to have the transmitter know the channel spectrum in detail, either the spectrum must not change at all with time or the communications system must have a feedback link. Even when the communications system is capable of two-way operation, the problem of maintaining an adequate, up-to-date knowledge of the channel at the transmitter is comparable to the original communications problem in difficulty. Channel The simplifying assumptions on the channel are comparatively numerous; however, most of them are acceptable from a practical viewpoint. We will assume that the channel is a time invariant, band limited, linear system with white Gaussian noise added at its output. Let us discuss these assumptions one by one. As mentioned in the previous section, the Mimi channel (and other underwater acoustic channels) has significant time variations occurring in it. These variations, however, occur at a much slower rate than the information rate at which a communication system

12 could be expected to operate. For example, the fastest time variation in the Mimi channel observed to date has a time order in the range of 1 to 10 seconds, while the duration of a single symbol at 100 wpm, for example, is approximately 20 ms. Thus, over the time duration of a single symbol the channel is expected to change very little. Time variations in the channel spectrum have been observed to have even a longer time order, nearly 5 minutes. The assumptions that the channel is bandlimited and linear are also borne out in practice. As mentioned earlier the transmitting transducer-reflector imposes a definite bandwidth restriction on the channel. The linearity assumption can be considered a reasonable approximation since, with the exception of the region immediately adjacent to the transmitter, signal energies throughout the physical channel are very small. The transmitter peak power limitation mentioned earlier also tends to reduce the chances of nonlinear behavior by the channel. The added white Gaussian noise assumption is more difficult to justify. Although only limited noise studies have been made to date, evidence is available to indicate the actual noise is both non-white and non-Gaussian. Because of the lack of a good description of the channel noise and the obvious advantages of a white Gaussian noise model, this assumption is made anyway. We note, however, that for a given noise power, N, white Gaussian noise is a "worst case" form of noise. That

13 is, if a system is designed on the basis of white Gaussian noise of power N, then its performance in different noise of the same power will be no worse than its computed performance. The potential of using peculiarities of the noise to obtain better performance is eliminated by this assumption. Nevertheless, the added white Gaussian noise assumption allows us to gain considerable insight into the overall problem. Receiver We will assume that the operation of the receiver is coherent and synchronized with respect to the transmitter. Moreover, we assume that the noise-free outputs of the channel, p'(t) and p'2(t) due to bothtransmitted symbols, q!(t) and q2(t) are known exactly. If the transmitted symbols are known, as they usually would be, this assumption is equivalent to a knowledge of the channel spectrum. The assumption that the noise-free received symbols are known exactly greatly simplifies subsequent analysis. Since the availability of highly stable oscillators is one of the key features of the Mimi channel, the coherency and synchronization assumption is immediately valid for the Mimi channel. The problem of incoherent communications is somewhat more difficult than the coherent problem; however, the results for the coherent problem provide a basis for the study of the incoherent problem. Synchronization is a wellstudied problem from other communications work. The assumption that the noise-free received symbols p{(t) and

14 p'2(t) are known exactly can be well approximated in practice through transmitted reference techniques similar to those used in HF radio (Refo 7). In a transmitted reference technique the transmitted signal has two components: an information component and an unchanging reference component. Usually the two components are made orthogonal. At the receiver, the reference component is processed over a long (relative to the symbol duration) time to achieve a high S/ N estimate of the noisefree channel response. In such systems a weighting factor is often introduced to allow the estimate to "track" slow time variations. Although more elegant methods of obtaining an estimate of the channel response warrant further study, the transmitted reference technique currently appears to be adequate. Incidentally, the estimate of the channel response afforded by this technique may in some systems (i. e., Mimi project) be of interest by itself. The above assumptions sufficiently simplify the basic communication problem to allow a rigorous analysis. In summary we list these assumptions and their motivations below: Transmitter: Binary signalling Easy to implement and efficient Peak power limitation Common transducer property No detailed knowledge of channel True unless feedback is used

15 Channel: Time Invariant True over short times Bandlimited Common transducer/ reflector property Linear First order approximation Added white Gaussian noise A "worst case" assumption Receiver: Coherent and Synchronized Available at Mimi facility Operation Exact knowledge of noise free Possible with transmitted symbol reference The objective of the systems considered here is to minimize the average probability of error at the receiver for long sequences of transmitted symbols. Two other relatively trivial assumptions are also being made for convenience in the analysis. The first is that the transmitted symbol values are equiprobable and independent, a very common and reasonable assumption in a communication system. The second is that the noise free input symbols to the receiver are binary simplex symbols; that is, the noise-free input symbol to the receiver is either p'(t) or -p'(t). If theactual received symbols P' (t), p'2(t) are not binary simplex, but are known exactly as assumed above, one can obtain simplex symbols P'l(t) + P'2(t) by subtracting. from the reception.

16 1. 3. Intersymbol Interference When binary symbols are transmitted through a band limited channel, the received symbols in general will overlap in time, giving rise to intersymbol interference. We will later see that irregularities in the channel power spectrum such as selective fading notches increase intersymbol interference beyond that encountered in an ideal bandpass channel. If noise is not present at the output of the channel, a simple linear filter often called an equalizer, can be used to reduce intersymbol interference to a large degree. On the other hand, if noise is present in the systems, the use of the equalizer may severely degrade system error performance. This second problem of intersymbol interference in the presence of noise is the key problem of the idealized Mimi channel described in the preceding section and is the subject of this study. To demonstrate the importance of the intersymbol interference problem, we will consider a heuristic example using the Mimi channel. Inspection of the power spectrum of Fig. 1. 3 suggests the useful bandwidth of the channel is approximately 50 Hz. By using biphase signalling with a signal duration of 40 ms, the major lobe of the transmitted signal spectrum will be passed by the channel. * Figure 1.4 depicts the time waveform and spectrum of a single 40 ms symbol on the same scale as the earlier figures. By multiplying the complex spectra of the 40 ms symbol * In biphase signalling, the transmitted symbols are pulses of carrier, 180~ degrees out of phase with each other.

17 0 0 Ln Ln M C 0 'J C0 t> — r --!- 1 co Ln Ps,., G ~) (-)i7.4 Q I t \ 00l0~ 00' 00"-O8-,OS'1 00' OS 00w' 0 0 j ^4-4 0 oo~~ 0 Cl)3~~~~~~ 0 a) S b0 0 a) 0 Sc a V) PH S H >Cl)~~~~~~~~C %o 0 IS 1 0 05 0C k 0 E O~~~~~~~~~~~~~~~~~~~~~f UD F OS' OO' OS 00100'" )' o

18 and the channel, the spectrum of the received symbol is found. Figure 1. 5 depicts the spectrum of the received symbol and the complex time waveform of the received symbol. From Fig. 1. 5 we note that the received symbol has significant energy over more than 200 ms. If the 40 ms symbol is used in the heuristic biphase modulation communication system, there will be portions of at least five symbols in every 40 ms interval at the receiver. Depending on the particular values of the interfering symbol, this overlap will bias the decision on the symbol in question one way or the other. The net effect, of course, is to increase the probability of error for the system beyond that expected from the added noise. An increase by a factor of ten or more in the probability of error is typical. In the second chapter we deal with the general intersymbol interference problem in detail. Until then the above discussion will provide an adequate background. 1. 4. Summary and Contributions In the next chapter, we discuss several traditional approaches to the intersymbol interference problem and their limitations. The notion of phase equalization is introduced to characterize the problem in terms of the power spectrum of the received symbol. The degree of intersymbol interference is defined and a major assumptioconcerning it is made. This assumption is shown to be equivalent to the limitation of signalling rates to reasonable values.

19 o 0 Ln LO — C-"'"'"'^ '^ ^ " - - ~'" 0 4-1 4-4 "o 0 o" S 0-^-^ C 0 rc — CD0 CD0)?( CO CQ CO y}, LIn n LI) oo OS' 00^ | ~~~o o ~~o= 4.J o. e0 o.Q E. OS', t 0'!1'00 0 bD C) — - 0' 0>______ ---^ o ^^. —^ Q cq ^ _-^g -^ ^ **s -^ in~~~~~~c S y ^ " h" oT ^^^ aT TH~~~~~~~~~~~~~O~r t.~, r^-^.~ bL~~~~~~~~~~~)?~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~S -8^ -rr^ ~~~~~~~~~~~~~~~~~~~~t C s "^-.-^~~~~~~~CZ ^ s -^ ^~~~~~3

20 The third chapter reviews the class of linear receivers using a convenient vector space notation which allows meaningful visualization of receiver operation. Three types of linear receiver prevalent in the literature are considered, the matched filter receiver derived from classical detection theory, the traditional transversal filter and the optimized receiver. The optimization involved in the third receiver is done for a number of different classes of varying degree of complexity in implementation. Finally, comparisons are made between the linear receiver types. In the fourth chapter two nonlinear receivers are described and evaluated. These receivers are of interest because of their ease of implementation and good performance. The first receiver is due to Aein and Hancock (Ref. 2), the second is described for the first time in this paper. The performance of the nonlinear receivers is compared with that of various linear receivers. In many instances the second nonlinear receiver performs better than the difficult-to-implement optimized linear receiver. The fifth chapter describes and evaluates the optimum receiver for the intersymbol interference in noise problem. This receiver computes the likelihood ratio for each received symbol and bases its decision on it. The optimum receiver is significant because of the absolute bound on performance and the insight it provides. In the sixth chapter the conclusions of the paper are summarized

21 and topics for future study are suggested. Contributions This study emphasizes the importance of the power spectrum of the received symbol as opposed to the complex spectrum of the received symbol on the intersymbol interference problem. By working with power spectrum and limiting the amount of intersymbol interference to a reasonable amount, a common basis for the comparison of traditional and proposed receivers is found. A new nonlinear receiver is suggested as a practical alternative to the complicated optimum linear receiver. This receiver is simple to implement and performs remarkably well. The major contribution of this thesis, however, is the design and evaluation of the optimum (likelihood ratio) receiver. The optimum (likelihood ratio) receiver is important for two reasons. First, its performance provides a lower bound on the error probability for any other receiver and hence provides an absolute measure how a given receiver performs. Secondly, the basic form of the optimum (likelihood ratio) receiver shows the necessity of using information on both past and future symbols in making each decision. This provides a basis for future design of easily implemented, near optimum receivers. A comparison of the optimum receiver with various linear and nonlinear receivers is made. This comparison provides a useful rule

22 of thumb for the system designer who wishes to trade off error performance, ease of implementation, and information rate.

CHAPTER II INTERSYMBOL INTERFERENCE The general problem of intersymbol interference in binary systems with noise is considered in this chapter. We first note several approaches to the intersymbol interference problem which are frequently used in practice. The important concepts of phase equalization and the dependence of receiver performance on power spectrum are introduced. A convenient characterization of the intersymbol interference problem and a measure of the degree of intersymbol interference are also given. Finally, a key assumption regarding the degree of intersymbol interference in practical systems is made and justified. In this and subsequent chapters it will be helpful to distinguish between the terms "filter" and "receiver. " By a filter, FL, we mean a linear, time invariant system which maps finite energy waveforms 2. 2 into finite energy waveforms, FL: L - L. The question of realizabiiity of the filters described here will be ignored, since adequate approximations are usually possible. A receiver, A, maps finite energy waveforms (the received signal) into binary decisions on symbol values?: L2 - I ~ 1 \ o Although a receiver may have a linear filter within it, a receiver is inherently a nonlinear device. 2. 1. Traditional Approaches Most of the traditional approaches to the intersymbol interference 23

24 problem involve the use of a single linear filter followed by a sampler and threshold device as the receiver. This class of receivers will be referred to as the linear filter receiver class. The third chapter deals with this class in detail. Figure 2. 1 shows a block diagram of a binary communication system in which the symbols ~ q(t) are transmitted through a channel, C(co), (meeting the assumptions of Section 1. 2) and are received using a linear filter receiver. It is easily shown that for the equiprobable binary simplex signals assumed here the best decision threshold is zero. Spectrum Equalization Spectrum equalization methods arose from applications where the reception was essentially noise-free; i. e., land-line teletype. Designers realized that variations in the amplitude and phase spectra led to distortion of the received symbol and, consequently, intersymbol interference. The obvious solution was to introduce filters, appropriately called equalizers, to flatten the amplitude spectrum and to linearize the phase spectrum. The design of such filters has been the subject of much study by circuit designers. The basic objective of traditional spectrum equalization techniques has been to reduce the time duration of the received symbol, and hence, intersymbol interference. This simple notion can be put on a quantitative We represent time waveforms using small letters and represent their Fourier transforms by corresponding capital letters. For example, the Fourier transform of the symbol q(t) is Q(co).

25 cq Co 0 ~ a) ~~oaa) a ~~~~~~~~~C ~~~~~~~~~~~~~~~~~~~~~~l Q) a) SE4'~~~~~~~~~~ I "0 k Q~~~~~~~~~~~~~~) +~ 0 Cd~~~~~~~~~~~~~~~~Q b. a) 0 4-j ~ ~ ~ ~ ~ ~ Cid HkE Co -I

26 basis by considering the RMS time duration, TRMS of the received symbol, p'(t). The time center r of p'(t) is defined by + 00 T - E t p'(t)I dt (2.1) - 00 + co E Ip'(t) dt (2.2) - 00 Then the RMS time duration of p'(t), TRMS is defined as the deviation 2 of Ip'(t)l about T normalized by E. -00 2 A 2 2 T RMS E (t -r ) p'(t) dt (2.3) -00 c Thus TRMS is a measure of how compact the energy of p'(t) is in time. We will choose our time origin so that T = 0 in the following discussion, A convenient equality derived from Fourier transform theory allows us to relate the RMS time duration of p'(t) and its complex spectrum, p '(o) = Q(a))C(w). Let A(co) be the magnitude spectrum of p'(wo), and let q(cvo) be the phase spectrum of p'(O). Using Parseval's theorem, one may show that: + 00 + o002 2 E 21 00 ) 2t 1 dA(w) + A 2(W) d4(w) dw 0 t2 lp(t) 12E \ d/ dt /o (2.4) where we have chosen the time origin so that rc = 0, (Ref. 8, page 62 ). 'E A2(c~~~~~~~~~~~~~~~~~~~~~~~o)

27 We see that by reducing the magnitude of the derivative dA( or the dw derivative dl(w the RMS time duration of p'(t) is reduced. This, dw of course, is the purpose of the traditional spectrum equalizer. Figure 2.2 shows a receiver using spectrum equalization. We will assume that intersymbol interference is completely eliminated at the output of the equalizer, EF(w). Then the classical theory of signal detectability indicates the (matched filter) form of the optimum post-equalizer receiver. Let the spectrum of the equalizer output be p "(W) = Q(cw)C(w)EF(w), then the transfer function of the optimum post-equalizer receiver, FLOPT(co) is given by FLPT() - P() (2.5) Yhr * 2 where indicates complex conjugate and IN"(co)! is the power spectrum of the noise at the input to the post equalizer receiver (Ref. 9). Since the 2 equalizer generally has a non-white spectrum IN"(w)t will not usually be white. Let us compute the overall transfer function FL(o) of the linear filter portion of the receiver shown in Fig. 2. 2. We have (/ "()) = [Q(c)C(co)EF(w) ] (2.6) and IN"(o)12 = EF(v)12 (2.7)

28 O " ) To a e. N N ^d o 0, 0 C Ce E Q ~ z I-N -a - as rt',,3 _ X 1 Cr ~ ^ -JIL-C~~~~~~~~~~~~~,.= Q) I E4.-S. -) I -4

29 since the noise at the input to the equalizer is white. The overall transfer function FL(w) is then FL(o) = EF(w)Q*(w)C*(w)EF*(w3/ EF(o) 2 (2.8) = Q*(w)C*(o) = [p' ()]* (2.9) The above reduced transfer function FL(co) is seen to be that of a filter matched directly to the output of the channel. The result given by Equation 2. 9 is very interesting. Even when the equalizing filter EF(o) is completely successful in eliminating intersymbol interference and when the optimum post equalizer receiver is used, the optimum post equalizer filter FLOPT(c) contains a factor i/ IN"()l2 which cancels out the effect of the equalizer. The overall transfer function after this cancellation, given in Equation 2. 9, is that of a filter matched to the channel output and does not contain any distinct equalization factor. Furthermore, the performance of the receiver given by Equation 2. 9 is very poor. Thus the spectrum equalization technique is not effective in reducing the effects of intersymbol interference in a communication system. Transversal (Time) Equalization Another approach to the intersymbol interference problem is to * This is shown in Section 3.2

30 use a transversal filter receiver (Ref. 10). Such a receiver evolves from consideration of the system time response rather than its spectrum and has the advantage that intersymbol interference is completely eliminated. As with spectrum equalization techniques, noise performance is not taken into account in the design. The basic notion of the transversal filter receiver is very simple. If receiver decisions are to be based on the filter output sampled every T seconds, then if the superimposed responses of consecutive symbols go through zero every T seconds, intersymbol interference is eliminated. Figure 2.3 depicts the response of a typical transversal filter to Response to received symbol, e(t) t -3T T -T 0 T 2T 3T Fig. 2.3. Typical transversal filter response to a received symbol

31 a received symbol. Since the superposition of such waveforms shifted by multiples of T has a non-zero component from only one symbol every T seconds, intersymbol interference has been completely eliminated. Transversal filter receivers are used because of their effectiveness in eliminating intersymbol interference and the ease with which they may be implemented using a tapped delay line. Figure 2.4 shows a simple seven-tap transversal filter receiver. The outputs of the delay Input Delay Line T i T i T i T + T + T jl) ^kJ) (k3) )(;) (W) (i;) (kJ W111,^<^^ Threshold i YJ- -At — Decisions Sampler Zero Zero Fig. 2.4. Seven tap transversal filter receiver line taps (spaced T seconds apart) are weighted with adjustable coefficients and then added. The tap coefficients may be set by cycling a noisefree received symbol through the delay line with T second delays and adjusting the coefficients to obtain zero output at all but one delay. Considerable study has been done to achieve "automatic equalization" by

32 having the receiver continuously adjust the tap coefficients to compensate for slow channel variations (Ref.10). Since the transversal filter receiver completely eliminates intersymbol interference, it is the optimum receiver in the absence of noise. One might be optimistic and hope that good intersymbol interference performance and good noise performance go hand-in-hand. Unfortunately, this is not the case and one must trade noise performance and intersymbol interference performance against each other in order to obtain the best overall system error performance. The following brief discussion illustrates the necessity of compromise. Consider the system shown in Fig. 2. 1. If no intersymbol interference is present, classical signal detection theory indicates the use of a matched filter receiver, that is, FLMF(W) = [Q(w)C()] (2.10) For a transversal filter receiver with a filter symbol response e(t) having the required zeros at all but one multiple of T seconds, we must have: E(w) = Q(w)C(w)FLTF(w) (2. 11) Thus the transfer function of the transversal filter, which is optimum in the absence of noise, is given by E(w) (2 FLTF(W) - Q(w)C(w) (2.12)

33 Comparing Equations 2. 10 and 2. 12 shows that the matched filter and the transversal filter are equal only if E(o) = IQ(w)C(w)12 (2.13) which implies that e(t) is the autocorrelation of the channel symbol response. This is an unlikely occurrence in realistic channels and, in general, FLMF(w) and FLTF(w) are different. Thus the optimum no-intersymbol interference receiver derived from detection theory and the optimum no-noise receiver derived above are different and a compromise between the approaches is needed. Transmission of Special Signals The above discussion suggests an effective but impractical method of handling the problem of intersymbol interference in noise. Suppose that the transmitted symbol, q(t), is carefully constructed so that Q()t2 E() (2. 14) IC(c)12 where e(t) has the desired zeros at all but one multiple of T seconds. Then Equation 2. 13 is satisfied and the optimum interference-free receiver FLMF(c) and the optimum noise-free receiver FLTF(w) are identical. Hence, we have achieved both optimum noise performance and optimum intersymbol interference performance.

34 Although appealing from a receiver design point of view, the above approach does not meet the specified assumptions given earlier. Contrary to our assumptions, this approach requires that the transmitter know the channel power spectrum and also that the transmitter be unrestricted in terms of peak power capability. These drawbacks eliminate this system from further consideration here. Transmission at Slower Rate Perhaps the simplest method of coping with intersymbol interference is to increase the symbol duration T to the point where the interference becomes tolerable in some sense. Although this is a simple and common method of avoiding the problem, it forces the system designer to accept a loss in rate without indicating the trade-off in error performance. Another disadvantage to this approach is that longer symbol durations may be contrary to other aspects of system design. For example, to reduce the effects of sudden channel fades it may be desirable to send one symbol in several short pieces separated in time (time diversity). The use of long transmitted symbols would impose a severe limitation on such a time-diversity system. A major result of this study is that intersymbol interference is essentially due to a bandwidth limitation on the received power spectrum and that the transmission rate should be chosen in the light of this bandwidth limitation. We will provide analysis, however, which indicates the initial trade off between transmission rate, intersymbol interference

35 and system error performance. 2. 2. Characterization of the Problem In this section we describe measures of the severity of a given intersymbol interference problem. We will show that the amount of intersymbol interference is directly related to the power spectrum of the received symbol, and hence, can be viewed in terms of a bandwidth limitation. We also define an integer M, known as the degree of intersymbol interference, to indicate the amount of overlap of the received symbols. Phase Equalization An important problem for the system designer is to determine the actual extent of intersymbol interference in a given practical situation. One approach to this problem is simply to measure the duration of the received symbol -- the longer the duration of the symbol the more severe the intersymbol interference problem would be expected to be. As we will soon see, this approach can be very misleading for a practical channel because it usually indicates more severe intersymbol interference than is actually present. To provide a consistent measure of the extent of intersymbol interference in a given situation, we consider the equalization of the phase spectrum of the received symbol. Let p'(w) be the complex spectrum of the received symbol. Then the phase-equalizing filter PEF(w) is defined by:

36 p'(w) PEF(c) = p'(w)l = p (w) (2.15) That is, the spectrum p(w) of the output p(t) of the phase-equalizing filter is equal to the magnitude spectrum lp '(w) of the input p'(t) to the filter. Since Ip'(w)l is always positive and real, the phase spectrum of p (w) has a constant value of zero and thus the term phase equalization is appropriate. Consider the two systems shown in Fig. 2. 5. The phase-equalizing filter PEF(O) in system No. 2 equalizes the phase spectrum of the received symbol, and hence, system No. 2 is called the phaseequalized system. Because the phase-equalizing filter has a white magnitude spectrum, the noise at the receiver input to system No. 2 has exactly the same statistics as that of system No. 1. On the other p(t) p'(t)+n pt Receiver Q &(w) I 1 C(w) + PFw System #1 (Unequalized) FPig) 2. 5. neualzeanpaseeqaliedn(t) ----— p'(t)+n - Q(W ) -- C(o) — 1 --- PEF() Reiver - #2 System #2 (Phase equalized) Fig. 2. 5. Unequalized and phase-equalized systems

37 hand, since the spectrum p (a) of the output of the phase-equalizing filter has a constant phase, d4'(w)/dcw is zero and, from Equation 2. 4, the RMS time duration of p(t) is less than or equal to that of p(t). Thus by using phase equalization we can reduce the RMS time duration of the received symbol without changing the noise problem. This is in contrast with the general spectrum equalization technique described earlier, in which the noise power spectrum is changed as the RMS time duration is reduced, Figure 2. 6 depicts the response of the Mimi channel to a 40 ms rectangular symbol with and without phase equalization. Analysis of the unequalized received symbol indicates the symbol has 90% of its energy within 165 ms and that there will be components of at least four symbols in each 40 ms time interval. The phase-equalized symbol has 90% of its energy within 50 ms indicating much less intersymbol interference than one would expect from inspection of the unequalized received symbol. By phase equalization the RMS time duration of the received symbol is reduced from 48 ms to 26 ms. The relationship between the unequalized received symbol p'(t) and the phase-equalized symbol p is interesting in that it relates the amount of intersymbol interference after phase equalization to power spectrum. Since the magnitude spectrum of PEF(o) is white, the power spectrum of p'(t), I p'(w)l2 is identical to that of p(t), [p(c)12I

38 0 0 co 2~ o o N 0 C0 I 0 o -. ---a 0~04. -- o _!., 0| w.S ^ _^D g ^ ' *.3-4 L-^ cr-1 q PI -' I ^ _ -? ) 0 0 ~I r: J' 0^ 0 o m o ~s..o 00 C.) 0 -- ~ '~ U 00'1 O' 00' I 'o~= o o I I" --- " ------------------------------------— ^~~~~~~~~~~~~~~~~~~E~

39 From Equation 2. 4 we see that any waveform for which d4i / do is zero will have the smallest possible RMS time duration of all waveforms having the same magnitude spectrum, or equivalently, the same power spectrum. Since di4(w)/ do is zero for a phase-equalized waveform, we conclude that among all waveforms having the same power spectrum, the phase-equalized waveform has the smallest possible RMS time duration. 2 From the power spectrum I p'(w) I of the received symbol p'(t), the waveform of the corresponding phase-equalized symbol p(t) can be found p(t) = F p;1 (O)2 ] (2. 16) where F -[Z(co)] is the inverse Fourier transform of Z(w). Since 2 12 ipl'()t2 is a power spectrum, Ipl'(w) is positive and real. The inverse transform of a positive and real spectrum is conjugate symmetric in time. By relating intersymbol interference to the power spectrum of the received symbol, considerable insight is gained. For example, consider the phase-equalized magnitude response and power spectrum of a 40 ms rectangular symbol passed through the Mimi channel shown in Fig. 2. 7. Figure 2.7 also shows these functions for a flat bandpass channel having the same nominal bandwidth. Because of the notches and irregularities in the Mimi channel power spectrums one would

40 o Ln V),t.,Ct N1 - 11 CL c ^,.. [.) Qo ID co _ o^~~ -^~~~~~~~~~~ o KE~ o ^^ N '0________ - 0' - o - OS 00 00 ' T OS' 00 00' 00 CO C I o C^ 0cu0 cI ) t QN T$ I 0 0 01 0 -a - *. ** | a r^. - ^ - \ - I - I - I -- I -- I -- I 1 1 OO'T OS' 00* OOT S'00

41 expect to achieve different system performance through the two channels. An intuitive way of expressing this is to say that the bandwidth of the Mimi channel is reduced by the notches and irregularities in its power spectrum. Because these same notches and irregularities in the channel power spectrum produce intersymbol interference, we are led to the intuitive, but productive conclusion that intersymbol interference (after phase equalization) is caused by signalling too fast for the bandwidth of the channel. Before proceeding further, it is important to point out that it is seldom necessary to use a separate phase-equalizing filter PEF(c) in a receiver. As will be seen later, most receivers perform a cross correlation of the reception with the noise-free received symbol p'(t) Since the noise-free output of such a correlation process is the autocorrelation function of the received symbol, the output waveform of the correlator is the same whether or not phase equalization is used. Thus in many instances it is not necessary to realize a separate phaseequalizing filter. In subsequent discussions we will assume that the received symbol is phase-equalized. This means that the received symbol p will have the minimum RMS time duration of all waveforms having the same power spectrum and that it will be conjugate symmetric in time. Degree of Intersymbol Interference In order to work with the intersymbol interference problem

42 conveniently, it is helpful to limit the amount of intersymbol interference due to a single received symbol. This may be accomplished by defining an integer M, known as the degree of intersymbol interference, as the smallest integer multiple of T completely contained in the "duration" of the phase-equalized received symbol. For example, if the phaseequalized received symbol duration is less than 3T but greater than 2T, the degree of intersymbol interference is two, M=2. The M=0 case corresponds to no intersymbol interference. Practically speaking, it is unlikely that a received symbol will have a finite time duration due to the bandlimited nature of the channels considered here. Because of this it is necessary to use some reasonable measure of symbol duration. Since signal energy plays such an important role in classical detection theory, a reasonable criterion is that some specified percentage of the energy, r7, be within the received symbol duration. The effect of the energy outside of the symbol duration under this criterion depends on the particular power spectrum and receiver under consideration. At present, a requirement that 90% of the symbol energy be contained in the symbol duration, 77 = 90%, is believed to be adequate. 2. 3. The M = 1 Assumption The remainder of this paper is concerned with intersymbol interference problems in which the degree of intersymbol interference is one (M =1). That is, the duration of the phase-equalized received symbol

43 is less than 2T. This is a major assumption; however, it can be considered equivalent to a restriction to moderate signalling rates through the channel. The relation between the M =1 assumption and signalling rate is discussed below. Suppose that we are to use binary simplex signalling to communicate through a given channel, such as the Mimi channel. At the outset, both the transmitted symbol duration, T, and the waveform of the transmitted signal, q(t), are free variables. Of course, restrictions such as bandwidth and peak power limitations impose constraints on these variables. It is important to note, however, that the symbol duration, T, and the bandwidth of q(t) can be adjusted practically independently of one another. For example, by using a block coded binary signal one can obtain a wide bandwidth signal having a long-time duration (Ref. 11). From the earlier discussion on phase equalization we know that the time duration of the received symbol, and consequently intersymbol interference, is reduced if the power spectrum of the received symbol is nearly white. Furthermore, from Equation 2. 16, we see that the duration of the phase-equalized symbol is reduced as the width of the power spectrum is increased. Thus the ideal power spectrum of the received symbol is wide band and white, independent of the duration of the transmitted symbol. Since the channel spectrum is unknown to the transmitter, the best transmitted signal also has a wide band and white power spectrum, again independent of its duration T. This is a wellknown result.

44 Let us temporarily neglect the constraints on the possible transmitted waveforms in order to examine the effect of the channel alone on the intersymbol interference problem. Suppose that we use an impulse function as the transmitted signal and that the transmitted signal duration (in this case the time between impulses) is T. Then the transmitted power spectrum is white and the power spectrum of the received symbol is simply the channel power spectrum. The phase-equalized channel impulse response is then the waveform of the phase-equalized received symbol, using the widest possible transmitted waveform. Figure 2. 8 depicts the phase-equalized impulse response for the Mimi channel. Given the phase-equalized impulse response for a channel and an energy criterion, 77, on the phase-equalized received symbol duration (i. e., 90% of the total energy is within the symbol duration), we can determine the "duration" of the received symbol. Figure 2.9 shows the duration of the phase-equalized impulse response for the Mimi channel as a function of the percentage of energy in the response duration. From the assumed duration of the received symbol found above, we can find the shortest transmitted signal duration for which the M = 1 assumption is valid. For example, consider the Mimi channel with a duration criterion based on 90% of the total energy being within the phase-equalized received symbol duration. From Fig. 2. 9 we see that the symbol duration is

45 0 4~~~~~~~t~ ~ W L N N o 0 __.._o-. u, <2,) 4 5 (l cr~ o'3 m n: (Y-):: (,. o) c - 0 OS'I 0 01' OS' 0 000 00 o 02 0Q I 9) 0.0~.. I I ~ I -^^ ' i OS*T 00'1 OS' 00'

46 Received symbol duration 200 based on r (ms) 150 200 50 0 10 20 30 40 50 60 70 80 90 100 Fig. 2.9. Received symbol duration based on r] as a function of q] for the Mimi channel

47 approximately 112 ms, and hence, under this criterion the M =1 assumption is valid, for T > 56 ms. The bandwidth of the channel power spectrum for the Mimi channel is less than 50 Hz, as can be seen from Fig. 1. 3. If this bandwidth were that of a flat bandpass channel, a reasonable signalling rate would be 25 symbols per second, which requires a 40 ms symbol. Thus we see that for this particular example, the M = 1 restriction is equivalent to requiring that the signalling rate be slightly slower than for a flat bandpass channel. Because of the severe notches in the actual channel power spectrum, one would not expect to signal at the ideal channel rate. The foregoing discussion was based on the use of ideal transmitted signals which cannot be used in practice. To go from this idealization to a practical peak power limited signal set is the relatively well studied problem of generating wide band, flat power spectrum signals having good peak-to-average power ratios (Ref. 11). Pseudo-random sequences are one convenient and common solution. Although we do not consider this problem in this thesis, we offer as a simple example, the three-digit, 60 ms "perfect word" symbol shown with its spectrum in Fig. 2. 10. The phase-equalized received symbol (after the channel of Fig. 1. 2) is shown in Fig. 2. 11. Signalling with this particular symbol corresponds to M = 1 with more than 94% of the energy within the symbol duration. In summary, limiting the intersymbol interference due to adjacent

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49 Ln ~ ~ ~ ~ ~~~~I N N in in cv 0 4-4~ ~~~~o 4 *- V; c CD~ CD Ln r I &, 0___ _ --- — ^^' Q- 0 - - 0 bfel U) 0 - ~ ~ Os Cr)~~~~~~~~~~~~~~~~r '0S 001" QQ W i E <D~~3 --- t^ '_ -^''. * <:^^.^+ inc 0 * o r( ~~~~o U) 3 Is~T rt = a St c Q, Sj P' B S~~~~~~~~~~~~~~~~~~~~~~~~a~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~a c,~~~~~~~~ 3 E: Q~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~) "0 ~ ~ ~ ~ ~ ~ ~~~~~~ o o tO 00T~ oS0 00 0 3!-^ ^~~~~~~t i~~~s ^^^ a ^~~~~~~Q f? -=:^^ i S~~~~~~~~~~`E ^^ ^~ <^ —^'- 3 ^~~~~~~~Q S^ _j~~ 'i- or <-.^ '~- v~~~ *^^^^^^ _^ -* ^^ 0~~~C ^ ^^-, s — ^^ s a~ 0 /.rt <^.1-4 *~~~~~~~~~~~~~~~~~~~~~~~~~'Ct ^ Zs ^ 6 ^-^ — o ^ fe~~~~~~~~~~~~~~~~~~Q >~ ~ L. -S_ 3 ~ ~~o ^ L~- to^ of I 'r I ^^ '1> e OS' 00* OS 00'

50 symbols (M = 1) is equivalent to a restriction on the signalling rate. The phase-equalized channel impulse response and the "enclosed energy" criterion r7, for the duration of the response provide the information required to calculate an upper bound on the signalling rate for which the M = 1 assumption is valid. We have seen that for the Mimi channel of Fig. 1. 2 and an "enclosed energy" criterion of 77 = 90%, the resulting upper bound on signalling rate is very close to the rate for a flat bandpass channel. Furthermore, the M=1 restriction may be maintained using peak power limited signals with a relatively minor decrease in rate.

CHAPTER III LINEAR FILTER RECEIVERS A linear filter receiver or correlation receiver is a receiver in which decisions are made on the basis of the sampled output Lk of a linear, time invariant filter. If Lk > 0 the receiver makes a dk=+l th decision indicating the value of the k symbol, bk, was +1, if L < 0 * a dk = -1 decision is made. Figure 3.1 depicts the general linear filter receiver. Input Sa ler x(t)_ H(w) Threshold Decisions dk O, T, 2T...at zero Fig. 3. 1. General linear filter receiver Linear filter receivers have been studied comparatively intensively in the literature (Refs. 2, 3, 4). The purpose of this chapter is to review these earlier efforts in a common frame of reference. We first consider the general operation of linear filter receivers and their * This is simply a sign convention. 51

52 performance. Special forms of linear filter receivers, such as the transversal filter receiver are considered under the assumption of a phase-equalized, unit degree of intersymbol interference (M = 1) symbol. Finally, we consider the optimization linear filter receiver for various optimization constraints. 3. 1. General Discussion In this section we consider the operation, error performance and canonical form for the class of linear filter receivers. Although we will quickly return to our assumption of a unit degree of intersymbol interference, the material presented in this section is for the general problem. Operation of the Receiver Let p(t) be the waveform of a single, noise-free received symbol due to the transmission of a symbol of value +1 at time t =0o Let b0. obm b. = ~ 1 be the transmitted symbol values in a finite duration transmission. Then the actual reception, x(t) is given by m x(t) = L b.p(t - iT) + n(t) (3. 1) 0 1 where n(t) is white Gaussian noise of noise power density N watts/Hz. Let h(t) be the impulse response of the time invariant linear filter H(w) which represents the linear filter portion of the receiver shown in Fig. 3. 1. We will not require that h(t) be physically realizable since an approximate realization of H(co) can be generally obtained,

53 Then the response of the filter to the reception x(t) at time kT, L, is given by + 00 Lk = f x(t)h(kT - t)dt (3.2) -00 Define -k h (t) = h(kT -t) (3.3) then Lk = J x(t)h (t)dt (3.4) k -0oo If x(t) and h(t) are of finite energy (a realistic assumption), then x(t) ~-k 2 and h (t) are vectors in the Hilbert space L which is referred to as signal space. In signal space Equation 3. 4 defines the dot (or inner) ~k product of x(t) and h (t). Suppressing the time variables t, T we can write Equation 3. 4 as Lk = x h (3.5) Equation 3. 5 gives the geometrical interpretation that the correlator output at time kT is proportional to the projection of the received wavek form vector x onto h o This geometrical idea provides a simple visualization of the operation of a linear filter receiver. Since the receiver makes a dk = +1 decision if Lk > 0 and a dk = -1 decision if

54 Lk < 0, the hyperplane Dk through the origin defined by k k -k x h = 0 (3.6) partitions the signal space into two half spaces. If the reception x lies ~k in the upper half space (x * h > 0) a dk= +1 decision is made, if ~k x lies in the lower half space (x h < 0) a dk = -1 decision is made. Figure 3.2 depicts a possible visualization of this operation. Subsequent decisions, d+.1., are obtained by comparing x with the corresponding hyperplane Dk+..., etc. Performance of the Linear Filter Receiver The probability of error on the kh symbol is the probability that Lk will be negative (and the resulting decision, dk = - 1) when, in fact, the transmitted symbol was positive, bk = + 1.- Let Bk be the set of all possible m dimensional vectors bobl.. b... bm i f k where b. = ~ 1. Then we can define P (k) = P[Lk < 0 I bk =+1 (3. 7) -2n P[Lk< I b..b, bbk++l] (3.8) positive when bk = - 1 also equals the probability of error. k B

55 I Dk Fig. 3. 2. Decision plane of a linear filter receiver > 0 Fig.~~~~~ 3. 2.Dcso lneo ierfitrrcie

56 since the symbol values, b, i / k are assumed equiprobable. The probabilities P[ Lk < 0 I b0...b bk = + 1] are easily computed as we see below. Suppose that the transmitted symbol values are b0... b where bk= +1. Because the input noise is Gaussian, the sampled filter output Lk is a random variable having a Gaussian density function. Its mean is given by k -~k [(k 'k 1 Eii E[Lklb.. b k +] = E (p + bi +n) h (3.9) where p denotes the L waveform, p(t - iT). Because the noise has a zero mean, we have F|[Lk I b b bk + 1] =h p + b.p ~h (3.10) The variance of Lk is given by Ok Var [Lkl b b =+ 1] = N Ih 12/ 2 (3 11) where NO is the one-sided noise power density of the white input noise. Let 0(u) be the zero mean, unit variance Gaussian density function, then

57 ( k~k k 0 u- h ~ p + bip o h P[ Lk < O b.b b=+1] =S 2 - du h h /2 - (3. 12) ~k ik ik h p + b.p h 2 NO h / 2 / 2 (3. 13) where e (u) is the cumulative distribution function of 0(u). Substituting Equation 3.13 into Equation 3. 8, we obtain ~k i+k. k k Ki h op + P h P (k) = - ' (3. 14) the noise power density, N. In general, P (k) will depend on where the kth symbol is located e

58 in the transmission. Since we are considering communications systems, it is reasonable to assume that the transmission consists of a very large 3 number of consecutive symbols; i. e., m 103. Under this assumption the effect of the beginning and end of the transmission on the system error probability (the average of P (k) over k) is negligible. Hence, we will consider the kth symbol to be located in the center of a very long sequence of transmitted symbols and that Equation 3. 14 therefore gives the system probability of error P Canonical Form of Linear Filter Receivers The foregoing discussion is sufficient to develop a canonical model of all linear filter receivers. This canonical model illustrates the effect of the filter impulse response h and suggests a convenient implementation for linear filter receivers. ^k 'k Let h be composed of two orthogonal components h 1 and -k ~ k h 2 where h 1 is in the m + 1 dimensional subspace H1 spanned by the time-shifted received symbols p = p(t - iT), i = 0... m, and k i where h 2 is in the orthogonal complement, H2, of the p. That is 'k p h 0 (i = O...m) (3.15) ~k if and only if h2 e H2. The correlator output Lk is then given by 2 2 k

59 L = x. (h + hk ) (3.16) Lk '1 2 m i k ~ k ~k Zb. h + n h + n h (3. 17) i=0 i 1 1 2 ~k The third term n * h is a zero mean Gaussian random variable with I.2 2 variance N h2 2. ~k We note that any vector h in HI can be presented as ~k n h1 = E C.p (3. 18) i= O since by definition, h 1belongs to the subspace spanned by the p. Consider the system shown in Fig. 30 3 in which a filter matched.th to p, p* (c) is followed by a tapped delay line. The output of the j delay line tap (which corresponds to a delay of jT seconds) is weighted by C j and the weighted outputs are summed in the first adder. The output of the first adder, plus a zero mean Gaussian noise waveform n'(t) having noise power N h /2 is sampled at time kT to give Lk which is compared to a threshold in the usual manner. The noise n' is included in the system shown in order to simulate receivers in which ~k h 2 is non-zero as we shall soon seeo The impulse response, h, of the matched filter-delay line-first adder section of the receiver is given by

60 CO 0 COl Q) cl uiz 0 To o *_r \^\C~~~~ 22~~~~~~CD \4 — ~ 4,-,m= -S * CO^ c=~o-.-.l lj ~C (- g >. \. y Y y co r^ ^^^ ^"r ^~~~~~~~~~~~~~~~~~~Y Q // ^T S~~~~~~~~~~~~b ^3// l ~ ^ s~~~~~~~~~~~~~~~~r rCI~~~~~/0 C~~~~~~~f;E: c~~~~~~~~~~~~~~~~~~~i E~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~*T5 cD ^

61 m h -= C p(-t + jT) (3.19) j=0 m-j = CMj (3.20) where p = p(jT - t). Then the output L" of this section of time kT due to the reception x is L" = f x(t)h(kT-t)dt (3.21) m =f x(t) 3 C p(t- (k-j)T) dt (3 22) j=0 m-j = x h (3 23) The sampled output of the second adder is then i k ~k L = 2 b p hl n h + n (3.24) ~k 2 Since n' is simply zero mean Gaussian noise of noise power NO h2 /2, Lk is equal to the Lk given by Equation 3.17. From Equations 3. 17 and 3. 24, we see that the noise process n' internal to the ~k canonical receiver simulates the contribution, n * h2, of the input noise in the H2 subspace. Thus by proper adjustment of the coefficients CO... C and the noise power associated m

62 with n' any linear filter receiver can be simulated in the form shown in Fig. 3. 3 We note in passing that no separate phase-equalizing filter is required with the canonical receiver. This is because the matched filter portion of the receiver p*(w) forms the autocorrelation of the noise-free symbol. Since the power spectrum (hence autocorrelation function) of a waveform and its phase-equalized version are identical, there is no need to precede the canonical receiver with a phase-equalizing filter. Nevertheless, the concept of phase equalization is confirmed — linear filter receiver performance depends on the power spectrum (autocorrelation function) of the received symbol. One would expect that in any reasonable receiver design the magni-?%.. -k tude h2 I would be zero since a non-zero Ih2 serves to add noise to the decision variable Lk. This is indeed the case. In Appendix A we show that k. k k if h * p is positive then the probability of error P (k) is reduced by dee I k N k k i ^ ke 1i creasing h2 to zero. Any receiver in which h p > 0 and h 2 = will be referred to as an admissible receiver. Since the probability of ^k k error for any non-admissible receiver with h p > 0 can always be reduced by making h2 =0, we will usually consider only admissible receivers in the remainder of this chapter. We note, however, that it is c* ok k If h ~ p is negative, the receiver decisions are of the wrong polarity, even in the absence of noise and intersymbol interference.

63 possible for a receiver with a well-chosen set of coefficients CO... C * O m |-k and a non-zero h2 to perform better than an admissible receiver with poorly chosen C... Cm 3. 2. Matched Filter and Transversal Filter Receivers Two simple linear filter receivers are considered in this section: the matched filter receiver and the traditional transversal filter receiver. Unlike the previous section, we will assume that the received symbol p is of duration less than 2T (M = 1). Figure 3.4 depicts a stylized received symbol that will be used repeatedly to clarify the meaning of the notation. v p(t) 0 2T Fig. 3. 4. Stylized phase-equalized received symbol (M = 1)

64 Note, however, that any realistic received symbol p will be time symmetric because of phase equalization. As a matter of convenience, the axis of symmetry for the symbol is t = T as indicated in Fig. 3. 4. Vector Representation Vector space representation of the reception x over an interval of length 2T is very helpful in understanding the matched filter and transversal filter receivers. Let po be equal to p for 0 < t < T and zero elsewhere and let p1 be equal to p for T < t < 2T and k zero elsewhere. Then the reception x in the interval (kT, kT + 2T) where k > 0 is given by (t) b k-i Fk ki k+1 (t) (t) xkt) = bk (t) + bk [ (t) + p(t) + k+ (t) + n(t) (3.25) where b. is the value ( 1) of the kh symbol. This is illustrated in 1 Fig. 3. 5 for the symbol shown in Fig. 3. 4. bk-1 bk bk+1 Fig. 3. 5. Noisefree components in the interval (kT, (k2)T) P 1 pk Fig. 3.5. Noise-free components in the interval (kT, (k+2)T)

65 Let us assume that bk = + 1. Then the noise free reception conk k k sists of a vector p = p + pl plus one of four "interference waveforms" dependent upon the values of bk 1 and bk+. Define the interference waveform for the kth symbol as Ak b b bk- P + bk+ IP (3. 26) k-l k+i k k k Both A 1 and A are orthogonal to p since k k k-i k k+1 k bOA b b bk- Pl *pO + bk+liO P (3.27) k-i k+I and k k-i k+i k Pg * Pk = Pk * Pi (3. 28) k k We write A A However, A and k arenot k k-1 k orthogonal to p unless pl * p = 0 k k-1 k A 11 =~2p1 * P0 (3.29) k k from Equation 3. 27. Furthermore A and A are orthogonal (Ak Ak and Ak -Ak )since +1+1 +1-1 -1-1 -1+1 ) since b b b b'k-i = bk- + +i p k- lbk+i k lbk+ K1 klk+ (3. 30)

66 and by symmetry due to phase equalization Pkl Pk1 (3.31) k-1 k+1 Since pl and p0 are orthogonal in time we have A *L b 2 k-1 2k+i 2 (3.32 bk b P + (3. 32) 2 = I [ (3. 33) that is, all the interfering waveforms Ab b have the same magnik+l k- l tude. The relations derived in the preceding paragraph can be represented very simply in 3-space. One convenient choice of orthonormal basis vectors e, e, and e2 is A k I k ^e P /I P 1 (3.34) e = +1-/ | +1-1 (3.35) - (3. 36) 2 A _ A o e o A +1-1 [^+1+1 e 0oI I +1+1r [+1+1 'O]xO_ [ +1+1 - 1] i 3. 36) Figure 3. 6 depicts the relations given above for a typical interference problem.

67 co, t- I o \ + \ O \ ra \ r+ Cl)k~~~~~~ S \\\ \^ <^ Y\\ -__|~~ _- \ \ 44 \ \ o ~C) C-4 \ ~ ~ I ^'

68 Before proceeding further, we will relate the quantities described above back to "real world quantities." The quantity Ipl2 is the energy of a single noise-free symbol, which is also the value of the autocorrelation function R(r) of the received symbol at zero. From Fig. 3. 5 k k-i k k+ (337) PO P1 =PI PG (337) = R(T) (3. 38) Thus all of the quantities describing the phase-equalized, M = 1 received symbol are given by the autocorrelation function R( r) evaluated at 0 and T. The normalized autocorrelation function r(r) is r(T) = R(T) / R(0) (3. 39) From time symmetry and the M = 1 assumption one can show that ir(T)l < 5. The case when r(T) = 0 corresponds to the no-intersymbol interference case. The effects of noise on the problem can be conveniently included by dividing the magnitudes of all waveform vectors by /N, making the resulting variance unity. Under this convention the magnitude of p becomes d where d = 2E/N0 (3. 40) is the index of signal detectability used in classical detection theory. For

69 binary simplex signals in the absence of intersymbol interference, the smallest probability of error is given by classical detection theory as P = <(- d) (3.41) This represents a lower bound on error probability for all systems with intersymbol interference. Matched Filter Receiver (MFR) The matched filter receiver (MFR) is the simplest possible solution to the intersymbol interference in noise problem. If intersymbol interference is not present, classical signal detection theory indicates the MFR to be the optimum (likelihood ratio) receiver. Because intersymbol interference is not considered in the design of the MFR, system performance is degraded when intersymbol interference is present. We will define the MFR as the linear filter receiver for which ~k k h p (3~ 42) In terms of the canonical receiver given in the preceding section, this means that all but one of the delay line tap coefficients is zero, or equiv-k alently, no delay line is used. Since h is entirely within the subspace ~k H1 h2 = 0 and no noise source is required either. From this ~k k k 2 and the fact that h p = p > 0 we see that the MFR is an admissible receiver. There is another way of interpreting the term "matched filter

70 receiver, " however. An intuitive approach is to carefully select a segment of the received symbol Ps(t) having a duration less than or equal to T, and match a filter to it. The segment would be chosen so as to have a large amount of signal energy and little intersymbol interference in it. In general, a filter matched to Ps(t) would not be an admissible receiver in the sense of the preceding section and we do not consider it here. The operation of the MFR in terms of the 3-space representation of xk is apparent. The equation of the decision plane is k k x k p 0 (3. 43) Note that the direction of the decision plane does not depend on the intersymbol interference, (see Fig. 3. 7). k Equation 3. 14 with h = p gives the equation for the system error probability. Because of the unit degree of intersymbol interferi.~k ence (M=1) assumption p ~ h = 0 for i< k-1 and i> k+1.,2, k-i k k+1 k = 1 E l pl+ bk-1P1 +bk k+lP p1 p = 1 ~, e 2m Bk Ipi vij (3. 44) Combining terms we obtain: al/4 l 1 pi + r(T) (bkl+bk+l t bk-(3. 45) (3. 45)

71 <~~~~~~~~~a~PT~~ ~oc) 0 O l) a) bL GV E

72 Writing the above explicitly in terms of d P 1/4 [-V 4(1+2 r(T))] + 1/24) [- + 1/4 [-D-d (1-2r(T))] (3. 46) Note that for r(T) near zero the system probability of error approaches that for the interference free receiver, as one would expect. Transversal Filter Receiver (TFR) The transversal filter receiver (TFR) represents a traditional approach to the intersymbol interference problem. The TFR completely eliminates intersymbol interference but at the expense of performance against noise. The defining characteristic of the TFR is that the linear filter output is uneffected by the interfering symbols, that is h x = Lk (3 47) does not depend on b, i k k. From Equation 3.1,,.k k ok — k ~k h x = bkp h + bip h + h ~ n (3.48) i~k In order to have the desired independence, we must have p * h =0 (3. 49) k k for all i / k o If p p I 0 for all i $ k, this implies that h

73 must have a non-zero component in the H2 subspace and hence the TFR is not an admissible receiver in the sense of the preceding section.* Since the TFR is a traditional and commonly used receiver, it will be analyzed even though it is not an admissible receiver. Since the probability of error can be expected to increase with -k the magnitude of the component of h in the H2 subspace, it is desirable to make this component as small as possible while satisfying ~k Equation 3. 49. This can be done by making h zero outside the inter~k val (kT, (k+2) T) and making h satisfy -k k-i h p* =0 (3. 50) and ~k k+i h P P = 0 (3.51) The preceding equations describe a tilted plane in the 3-space representation of x which is parallel to the plane of the interference vectorsFigure 3.8 depicts the decision plane for the TFR. k-i k-i Figure 3.8 depicts the decision plane for the TFR. k-1 k-1 ~k Equations 3. 50 and 3. 51 above allow the determination of h Let k k-i k+i h P= +1P1 +cOPO (3.52) then * i k From the definition of r(T) we see that p p = 0 for all i # k if and only if r(T) = 0, that is, if there is no intersymbol interference.

74, o > Q C) a) orl\ I _y\ A- o~~~~ \ \ Hi~~~~~~~

75 k- k k- k k- k- k- k-k-1 k+ h P- Pk.op1 + PI P1 + C1 p1 + cp1 p p (3. 53) k k-1 k-1 = Po PI + C PI o0 (3. 54) k I k- p k- I k+ I since. k-P and Pk- P k+ Solving for cl c1 = -2r(T) (3. 55) Similarly, k k. k-i k a nd Pp P + Co PO (30 56) and cO = -2r(T) (3. 57) The probability of error for the TFR is found directly from Equation 3. 14; however, because of Equations 3. 50 and 3. 51, the expression is very simple '-k -h. pk e I 0 /J 2 (3. 58) Now we have, from Equations 3. 54 and 3. 56 above

76 l-k.k k2 k-i k+1 k h k Ip -2r(T)pl p -2r(T)p0 p (3.59) = kpk2 [- 4(r(T))] (3. 60) thus P [ Vd (1 4(r(T))2] (3. 61) Note that as r(T) goes to zero, the probability of error approaches that of an interference free receiver, as one would expect. As r(T) approaches. 5, however, the probability of error approaches. 5, the worst possible error probability. Figure 3. 9 compares the error performances of the MFR and TFR as a function of lr(T)l for a d of 10. Also shown in this plot is the performance of an interference-free receiver operating with a d of 5 and 10. Error probabilities of both the MFR and TFR increase rapidly with r(T)l. With the exception of a very small region (Ir(T)I >.45) the TFR performance is superior to that of the MFR. A natural question arising from this comparison is why does the TFR, which is not an admissible receiver, perform better than the MFR which is an admissible receiver. The answer to this is that "admissible" ~k means that components of h which add only noise to Lk have been O-k eliminated. By eliminating the component of h in the H subspace, a TFR can be converted into an admissible receiver and the probability

77 0.10 MFR TFR Interference free receiver d= 5 0.01 / p e 0.001 '^^^^^ ~Interference free receiver d = 10 0.0001 I I I I 0 0.1 0.2 0.3 0.4 0.5 Ir(T)I Fig. 3.9. Comparison of MFR and TFR for d = 10

78 of error reduced. Unfortunately, this modification destroys the characteristic of eliminating intersymbol interference and the receiver loses its identity. A word of caution is necessary with regard to error probabilities. At first glance, it appears that the TFR gives a loss "of just 3 db" in performance at Ir(T)l =.35, since its performance at Ir(T)I =.35 is equal to that of an interference-free receiver operating with a d of 5. In many problems a depreciation in performance of 3 db in some sense is negligible. On the other hand, in communications systems such as the one considered here "just 3 db" may be a large factor. For example, suppose the binary simplex system is used to transmit 5-bit teletype characters. If the bit probability of error P is. 00078 then the probability of a character error is. 0039. If the performance of system is degraded by "just 3 db"' the probability of a character error becomes. 0614. Thus a "db down from ideal" measure of system performance is a crude measure for teletype systems. 3. 3. Optimized Linear Filter Receivers In Section 3.1 the canonical form of all admissible receivers was developed. For each set of tap weights C, a different admissible receiver is obtained using this canonical form. A very natural approach to the receiver design problem is to determine a set of C. 's which gives the smallest probability of error. The resulting receiver

79 would then be the optimum linear filter receiver. * Since the number of symbols in the transmission, m+l, is normally very large, it is not practical to use the m + 1 tap delay line of the canonical receiver. Instead, it is necessary to constrain the problem by limiting the number of taps the receiver is allowed to use. The receivers resulting from such constrained optimization are called optimized linear filter receivers and are studied below. Probability of Error Consider the admissible linear filter receiver with 2q+ 1 taps shown in Fig. 3. 10. Let the weighting coefficients of the taps be w q... w0...W+q w here w q is the weight of the tap representing the longest time delay. To avoid realizability difficulties, the time origin for the output of the adder is taken qT seconds after the time origin of the input. The system impulse response is then given by + q h = wip (3.62) i=-q Ni where p = p(i T - t) and the probability of error can be determined from Equation 3. 14. * Here we have made the assumption that the optimum linear filter receiver is an admissible receiver. Smace the probability of error for any non-admissible receiver (with h p. > 0) can be reduced by setting lh2 l = 0 this is a very reasonable assumption.

80 0 CQj i *rj 1 C~OC QQq~~~~~C) c! *r E ac \i... i-_..0 2u ^- 3- S - c-?X-;s s - X~~ / =t~~~* x /u ^^V/

81 Let ~k +q h w.i+k (3. 63) i=-q 1 then 'C'k C2 ~q L~~i+k j+k h = ~ i w. Wp ~ (3. 64) i=-q j=-q +q q-1 pt2 w+ k2p*pk' w.w (3.65) i=-q 1 i i=-q 1+ because of the phase equalization and M=1 assumptions, we have k *k+i 2. i=O (3.66) p. p = Ip! if i=o (3.66) k k+i k k-I (3.67) p = P p ifi=~ (3.67) k. k+i p p = 0 otherwise (3.68) from the preceding section. Let b = (b -q1*. bk+ ) and w = (w.W. w ) then one can show by a similar procedure ~q +qj

82 h. Pk + j b.pi+k. h = (bw) + p 0 p Z (b w) if k (3. 69) where i/k Z (b,w) = w + Z biw. (3. 70) i=-q+l Z2(b,w) = b_q_lwq q+ w q b.(w.i i+l) + w1 (3. 71) i=q-l W+1+ W1 bi(w. i- i+) + bqwq + bq+ wq +1 L i i-l i+l q q-1 q+1 q i =+1 Substituting Equations 3. 64 and 3. 69 into Equation 3. 14 and converting to the real world parameters d and r(T) yields the probability of error for the receiver with the tap weights given by w, Pe(w) Pw) 1 1 Zl1(b,w) + r(T)Z2(b,w) e (W) 2q+'1 Z1 -- 2 q+ allb +q q-1 w w + 2r(T) wi+ Li -q -q7 (3.72)

83 When r(T) is zero and w. = 0 for i, 0, the above equation gives the probability of error for an interference-free matched filter receiver as one would expect. The optimized 2q+l tap linear filter receiver, OLFR2q+i, is defined as the canonical receiver having 2q+ 1 taps with tap weights ---, -- * given by w which minimizes P (w). We note in passing that the MFR can be considered to be the OLFR1, the simplest admissible receiver possible. Determination of Tap Coefficients In order to construct and evaluate the OLFR q+ the optimum tap weights w must be determined as a function of both d and r(T). Equation 3. 72 is sufficiently intractable to preclude useful analytic expression for w o Both Aein and Hancock (Ref. 2) and Aaron and Tufts (Ref. 3) have used the calculus of variations to derive the necessary coefficients for their analogous equations. Unfortunately, in both cases, their approaches produced equations as horrendous as Equation 3. 72 and they resorted to numerical search techniques. We have therefore approached the problem directly by using a trial-and-error search technique to determine w Several simple observations make the direct search for w easier. A convenient normalization is to take w0 = 1. Because the * -* The implicit assumption that there is a unique w is borne out by the work of Aaron and Tufts, Ref. 3.

84 received symbols are symmetric due to phase equalization and their values bk are independent, the effects of bk+i and bki on the kt k k+i k-i symbol decision dk are the same. Hence the tap coefficients are symmetric, that is, w. = w. If the weights w are to help eliminate the interfering components of bk~ 1 the weights must be negative, or they will actually increase the intersymbol interference. Proceeding in this manner indicates that the tap weights w i must alternate in sign. Because the output of the tap corresponding to w0 represents a direct indication of the value of bk, whereas the other taps serve to eliminate intersymbol interference, one would expect the magnitudes of w. (i#0) to be less than w0 =1. These observations have been borne out by numerical analysis of Equation 3. 72 and serve to reduce the search space to a unit cube in q-dimensional space. Figure 3.11 depicts the optimum tap weights for OLFR3,- OLFR5, and OLFR7 as a function of tr(T)t for a d of 10. These weights were obtained by searching over the unit cube with an effective resolution of.025, that is, the coefficients are within.025 of their true value. This resolution is considered sufficient because of the smooth variation of P (w) with w and more elaborate techniques, such as the steepest descent technique do not appear necessary. Performance In Fig. 3. 12, the probability of error for the OLFR1, OLFR3, OLFR5, and OLFR7 receivers is shown as a function of Ir(T)I for a

85 1.0 /-wl OLFR(2T) 0.9- / 0.8 / / / 0.7/ / 0.6 - / 0.5 / / -w OLFR7 o/4 L -w* OLFR5 0.4 1 5 / 0.3 / /-w* OLFR3 0. 2 W* OLFR7 2 7 w* OLFR5 0.1 2 - w* OLFR7 3 ~/ 0 0.1 0.2 0.3 0.4 0.5 r(T)X Fig. 3. 11. Optimum tap weights for OLFR3, OLFR5, OLFR7 d= 10

86 0.10 - / OLFR3 / //oOLFR ~~~~- ~~ / OLFR7 //f/ 0.01 OLFR = MFR OLFR(2T) ) / 0.001- o 0. 0001, I l D D. J D 2 P,3!, I I: I r(T) I Fig. 3.12. Probability of Error for OLFR1, OLFR3, OLFR5, OLFR7 d = 10

87 d of 10. This figure shows that a very significant improvement in error performance is obtained in going from one tap (MFR) to 3 or more. There is little improvement between the 5iand 7-tap receivers, however. Another Constraint Another way of constraining the optimization is to limit the duration of the filter impulse response to some specified amount. This constraint may be a reasonable one in implementations of the linear fil-k ter which correlate a stored reference, h, with the reception. Receivers evolving from this constraint will not be admissible receivers but will have performances intermediate between that of the longest OLFR having an impulse response duration less than the constraint duration and the OLFR having two additional taps. We consider only the case in which the impulse response is limited to 2T o"k * Let h be of duration 2T and of the form ~k bk-l k k+ 1 h = w'p1 + + p + w (3 73) We wish to determine the constants w' which minimize the probability of error. From symmetry arguments similar to the ones given earlier, wl = w' and -1 < w' < 0. The optimum value of w' is +1 -1 - Pl I I1 shown as function of r(T)l is shown as a dotted line for d= 10 in * Ok One can show that h should have this form from arguments analogous to those for admissible receivers.

88 Fig. 3.11. The error performance of this receiver, known as the optimized linear filter receiver over a time duration 2T, OLFR (2T), is shown as a dotted line in Fig. 3. 12. As expected, its performance lies between that of OLFR1 and OLFR3. A major advantage to considering the OLFR (2T) is the insight Ok it provides concerning the MFR and TFR. Since h is of duration 2T, the decision plane of the OLFR (2T) can be completely represented in the 3-space shown in Figs. 3. 7 and 3. 8. Figure 3.13 depicts the decision plane for these three receivers. The intermediate position of the decision plane of the OLFR(2T) between those of the MFR and TFR, indicates the trade-off between intersymbol interference elimination by the TFR and good noise performance by the MFR.

89 + ~~~~4 ~ - Q,~~~~~~~~~~~~~~~Q C9 0 4-4 C~~~~~~~~~~~~~ r4~~~~~~~~~~~~~~~C v-4~~~~~~~~~~~~~~~~~~+~~~~~~~~~~~~~~~~~~ v-4O V-4~~~~~~~I b c~~~~~~~~~~~~~~~~~~~~~ Pf~~~~~~~~~~~~~~~~~~~~~~~~~~~~

CHAPTER IV TWO SIMPLE NONLINEAR RECEIVERS The two receivers considered in this chapter are nonlinear receivers because their decision variable Lk is not a linear function of the reception. There are many possible nonlinear receivers which could be studied. The two receivers studied here, the switched-mode receiver and the iterated switched-mode receiver, are important because of their ease of implementation and comparatively good performance (relative to the optimized linear filter receivers). The first of these receivers, the switched-mode receiver, is a direct extension of a receiver of the same name proposed by Aein and Hancock (Ref. 2). The second receiver, the iterated switched-mode receiver, is described for the first time here. Chapter V will show that the optimum (likelihood ratio) receiver is a nonlinear receiver and thus these receivers belong to a promising class of receivers. The underlying principle of both the switched-mode receiver and the iterated switched-mode receiver is very simple. Suppose that th the receiver is to make a decision on the k symbol. Then if the values of the interfering symbols b bk were known their k-i. ak+ih * Aein and Hancock (Ref. 2) appear to have been the first to make the important observation that a simple nonlinear receiver can perform better than a complicated optimized linear filter receiver. 90

91 "interference" could be subtracted out to give an interference free kth symbol. Since the objective of the receiver is to produce decisions do... d which are equal to the transmitted symbol values b0.... bm it seems reasonable to use symbol decisions to subtract out the interference to make even better symbol decisions. This sort of "boot strap" technique is the essence of both receivers. 4. 1. The Switched-Mode Receiver The switched-mode receiver (SMR) represents an elementary receiver using decisions to subtract out interference. Consider the receiver shown in Fig. 4. 1 which uses a matched filter, p *(w), Sampler Decisions Input x \ Threshold d ~1 - p* (w) ---- at ____ |_ 0, T,2T,... zero d k-1 k k-i pk k-1P1 P1 Fk-1 4k 1 H i mDelay of _ P1 'p T seconds dk-1 Fig. 4. 1. Heuristic implementation of the switched-mode receiver

92 and a decision circuit containing memory. At time kT, the sampled output, Lk of the matched filter is T b k-1 k k 2 k+1 k k Lk k-= p + bk P + bklPO p + n p (4.1) where we have used the notation introduced in Chapter III. The first and third terms in Equation 4. 1 represent the intersymbol interference due to the k-th and k+th symbols; if they could be eliminated, the receiver performance would be that of an interference-free receiver. Suppose that at time kT, the receiver has already made a decision dk_ on the (k-l)th symbol. Form a new decision variable Lk, which is dependent on the previous decision dk_1 k-k-i k Lk Lk -dk-P * (4. 2) k-1 k 2 k+I k (bk-1 - dk)pl p*~ p + bklpk + p + n p (4. 3) If the system is working with a low probability of error the first term in Equation 4. 3 will be zero most of the time. Thus we have succeeded in forming a decision variable Lk in which there is less intersymbol interference than in L. A receiver whose decision variable L is formed according to Equation 4. 2, such as the receiver in Fig. 4. 1, is called

93 a switched-mode receiver (SMR). We note in passing that since dk_1 = ~ 1, Lk is not a linear function of the reception and, hence, the receiver is nonlinear. The evaluation of the SMR's performance is straightforward. Let P0 be the probability of error for the kt symbol given that decision dk1 is correct and let P1 be the probability of error for the kth symbol given that dk_ is incorrect. Further let P (k) be e the probability of error for the kt symbol. Then P (k) P P (k - 1) + P0(1 - P(k-l)) (4.4) If the kth symbol is in the center of a long sequence of transmitted symbols, P (k) is the average probability of error, P, for the syse e tem. Using Pe(k) = Pe(k-l) = Pe (4.5) Equation 4. 4 can then be solved for P P0 P = (4.6) e 1 - (P1 - P0) If there were no intersymbol interference, P1 and P0 would be equal and Equation 4. 7 gives the probability of error for the optimum, interference free receiver. On the other hand, if intersymbol interference is severe, P1 becomes much larger than P0 and P increases 1 0 ~~~~~~e

94 considerably over P. We proceed in a general manner to compute the probabilities P0 and P1. Suppose that Lk is the output of the matched filter p *(w) k-1 k k k+1 added. with a constant term of the form a p p + k3p pk added. k-1 k k 2 k k+i k L ap P p b* + p + pp ' P0 +n p (4.7) k apI p +bkp The distribution of Lk, f(Lk I bk, a, ) is Gaussian since Lk conk sists of a zero mean Gaussian random variable, n * p, plus a constant. Lk- ( PI ' P bk p + p ' P f(Lkl bk,,3) = [ - (ak+b jkjIXo) (4. 8) Suppose that decisions are based on a comparison of Lk with zero in the usual manner. Then the probability of error P (a, 3 ) is given by k 2 Pk-1 k k+1 k Pe(a t) 4 - p -c (4. 9) p P 0 /2 Equation 4. 9 can be written very simply in terms of the parameters d and r(T) introduced in the preceding chapter. Pe(aI3) = <[-d- (l+(o+l3)r(T))] (4.10)

95 To determine P0, the probability of error for the kth symbol given that the decision dk 1 was correct, we use Equation 4. 3 2 L k bk k+1 kk (4. 11) b =1 \ p+ \ PO * P + n ~ p (4.11) and P0= Z P(bk+l)Pe(bk+l) (4.12) bk+lPO = 1/2 (-Vd(I+r(T))) + 1/2 (- dT-(1-r(T))) (4. 13) Similarly, P1, the probability of error for the kth symbol given that the decision dkl was incorrect can be found. Under the condition that dkL1 is incorrect, Lk is given by pk k-i k 'k k+1 k Lk 2bk-lp1 p +bk Pk 2 + bk+lp pO + n p (4.14) and P b - 2 (4 15) bk-lbk+ 1

96 P = 1/ 4 D (- f(1+3r(T) + (- f(l+r(T)) (4.16) ++ (-4(l-r(T + c(-V4(1-3r(T))) ( As noted earlier as the intersymbol interference is reduced (r(T)- 0), P1 approaches PO; PO in turn approaches the probability of error for an interference free receiver. The third section of this chapter compares the performance of the SMR with the various linear filter receivers. The implementation of the SMR shown in Fig. 4. 1 is certainly not the easiest one. Since the effect of the decision circuit is simply to bias L. one way or the other, the system is equivalent to one which has a variable threshold r(dk 1) at time kT where k-1 k r(dkl) = -dk 1 * P (4.17) = dklR(T) (4. 18) If a receiver gives a decision dk = + 1 if Lk > r(dk _) and a decision dk=- 1 if Lk <r (dkl), the decisions will be exactly the same as those of the SMR shown in Fig. 4. 1. Figure 4. 2 shows the simplified SMR. The fact that the threshold is switched by the preceding symbol decision in this realization gives the receiver its name.

97 Decisions Input Sam ler Disins x Threshold k ----- p(*() -—. b - — atr | O0,T,2T,... r(dk_1) dk-1 Delay of T seconds Fig. 4.2. Simplified implementation of the switched-mode receiver 4. 2. Iterated Switched-Mode Receiver Although the SMR discussed in Section 4. 1 uses earlier decisions to try to eliminate the intersymbol interference of the preceding symbol it does nothing to eliminate the interference due to the subsequent symbol. At first this seems like an inherent limitation on a decision oriented receiver, since the decision on the (k+1)th symbol is to be made after the decision on the kt symbol. This limitation is circumvented by making two decisions on each symbol: a "first guess" decision and a final decision. This idea of a double decision process is the foundation of the iterated switched-mode receiver (ISMR).

98 It is helpful to temporarily consider a complicated realization of the receiver before developing a simpler implementation. Suppose that the output Lk of a matched filter, p*(w), is stored for all values of k, 0 < k < m during a transmission. Then for each k, Lk is given by Equation 4. 1. Let the "first guess" decisions dj be made on the basis of a comparison of Lk with a zero threshold in the usual manner. The decisions d0 are identical to those that would be made by a MFR under the same conditions. From Equation 3. 46 the probability of error for the decisions d, is Po = 1/4 [|-V/d(l+2r(T))] + 1/2c ( —d) + 1/4 < [-I4-(1-2r(T)] (4. 19) Given the stored matched filter outputs Lk and the stored decisions d, subtract the interference from Lk in a manner analogous to the SMR. Let T o k-1. k o k+l1 k Lk = Lk - 1P P - k+P P (4. 20) o xk-i k k2 k+1 k k =(bk-1-dk-1)pl p +bk+ p +p (4. 21) If the decisions d0 have a low probability of error, the first and third terms in Equation 4. 18 are zero most of the time and intersymbol

99 interference is eliminated from L. The final decisions dk of the ISMR are made by comparing Lk to zero. Since Lk is not a linear function of the reception, the ISMR is a nonlinear receiver. We can determine the probability of error for the ISMR's final decisions easily. Let P0 be the probability that dL is in error given dk_1 and d +lare correct and let Pl be the probability that dk is in error given that dk+ and dk1 are incorrect. Further, let PD1 be the probability that dk is in error given that exactly one of,d I d0+1 is incorrect. Then the probability of error P for decision dk is simply P = P P [d-l' d both +correct] + P [one of d correct] e 00 - 1 +1 -l +1 - 0 -i (4. 22) - P P both of d 1 d+ incorrect Decisions d _1 and d+1 are independent of each other since they are t t based on Lk_ and Lk+1 which, in turn, correspond to disjoint time intervals. Thus Equation 4. 22 can be written in terms of the "first guess" probability of error P~ as follows e P P (1-P )2 + 2P01(1-P) + P1 ( 0)2 (4.23) e 0 1 e 11e The probabilities P00, P01, and P1l are easily computed from Equation 4. 21 under the appropriate condition and P (a, j ) given by e

100 Equation 4. 10. P0 o= Pe(, 0) (4. 24) = c ( —d) (4.25) P0 = 3 [b kl] Pe(0,2bk+l) (4.26) k+1 1 I/2 < (-d(l+2r(T) )+ 1/2 (-Vd(1l-2r(T)) (4. 27) P1 = b P [b k-lb Pe(2bk 2bk+) (4.28) k-1' k+1 i = 1/ 4 (-V (l+4r(T)) + 1/2c(-V-d) + 1/4 <(-/ (1-d-4r(T))) (4.29) The above probabilities can be substituted into Equation 4. 23 to give the probability of error for the ISMR. The implementation suggested by the above discussion is unnecessarily complicated for practical applications. Just as in the SMR, the receiver can be considerably simplified by using a variable threshold F (d-1,d+1 ). Let

101 i do/ k k-i 0 k k-1 d 4+30l r( -l'dk+l) = dk-O p + dk l * (4.30) = R(T) [d1 + d+] (4.31) Then a receiver which compares Lk to r(d,1 d +) makes exactly the same decisions as the complicated implementation given earlier. Note that there are actually only 3 distinct values for r(d_ ~, d+l): ~ 2R(T) and zero. Figure 4.3 depicts a possible implementation of the ISMR. Before continuing to the comparison of the ISMR with other receivers in the next section, a word of caution about the ISMR is necessary. From the preceding discussion of the ISMR it seems reasonable to consider a multi-decision process in which an entire sequence of decisions d... d is made on each symbol in a logical extension of the ISMR procedure. The hope of such a procedure would be that the final probability of error Pe is reduced by making more preliminary decisions. Unfortunately this is not the case, as the interdependence of errors produced by more than two decisions in each symbol leads to an actual increase in error probability over that of the ISMR. Thus this logical extension of the ISMR procedure is of no avail. 4. 3. Comparison with Linear Filter Receivers This section compares the nonlinear SMR and ISMR with the linear filter receivers discussed in the preceding chapter. The

102 VI 0 4,0 S | I 3 13 -- I — " ^ --— ^ ^-. " 0 *~ 1_ o a ~ E- Ca m E —i *rl Pc ^c g \ c^ ^ t3 ^ ^ ~~~~~~~~~~~~~bb I ^ *>-<~~~~~~~~~~~~~~~~~~~~r <= rLn

103 comparison is made taking into account the ease with which a receiver may be implemented. Switched-Mode Receiver Figure 4.4 depicts the probability of error for the SMR and the MFR, TFR and OLFR of the preceding section. In terms of error performance, the SMR is superior to the MFR for all Ir(T)I and is roughly comparable to the TFR and OLFR3. The OLFR5, the next more complicated optimized linear filter receiver, is noticeably superior to the SMR. The SMR offers several distinct hardware advantages. First, the SMR requires only a single digital delay; i. e., a flip-flop, to remember dk 1, whereas the TFR and OLFR receivers require an analog delay of at least 2T seconds. Second, the SMR has only a single variable, r(d kl) which must be adjusted whereas both the TFR and OLFR receivers require the adjustment of 2 or more tap coefficients. Finally, the single variable r(dk 1) of the SMR is simply the autocorrelation of p(t) evaluated at t =T. Both the TFR and OLFR require weighting coefficients which are relatively difficult to obtain. From the above considerations, the SMR may be an acceptable substitute for either the TFR or OLFR3 because of its ease of implementation. On the other hand, the SMR performance is noticeably worse than that of the more complicated OLFR5. The SMR does appear to be an improvement over the simple MFR.

104 0.10 OLFR3 MFR OLFR1 - TFR OLFR5 0.01 ~_ SMR e 0.001 - d= 10 0.0001 I I I I I_ 0 0.1 0.2 0.3 0.4 0.5 I r(T) I Fig. 4.4. Comparison of SMR with MFR, TFR, and OLFR3 5

105 Iterated Switched-Mode Receiver Figure 4. 5 depicts the probability of error for the ISMR and the MFR, TFR and OLFR. The outstanding feature of this figure is that for Ir(T)l <.25, the ISMR performs better than any linear receiver. For Ir(T)I >.25, the ISMR performance is nearly that of the OLFR3 By comparing Figs. 4. 4 and 4. 5, we see that the ISMR performance is superior to that of the SMR for Ir(T) <. 4. Thus the ISMR is an excellent performer in moderate intersymbol interference and an acceptable performer under more severe conditions. The ISMR shares with the SMR a significant ease of implementation. The ISMR can be implemented using one digital delay (i. e., flip-flop) and one analog delay of T seconds, which is easier to obtain than the long analog delays required by the TFR and OLFR. As with the SMR, only a single variable r(dk_, dk+l) must be changed as p(t) changes and this variable is easily computed from p(t) The ISMR represents an increase in complexity over the SMR. If a communication system is designed so that on the average, the intersymbol interference is only moderate, i. e., Ir(T) <. 25, the ISMR is an excellent choice of receiver. Since larger amounts of intersymbol interference lead to considerable increases in the probability of error for any receiver, one might use Ir(T)l <.25 as a reasonable restriction. Thus the ISMR is an important receiver in communications systems with intersymbol interference and noise. As a final comparison, consider a system for the Mimi channel

106 0.10 - MFR - OLFR //TFR _1 -/ / ISMR OLFR3 OLFR5 0.01 e 0.001 " — S ISMR 0. 0001 I I I I I 0 0.1 0.2 0.3 0.4 0.5 I r(T) I Fig. 4.5. Comparison of ISMR with MFR, TFR, and OLFR3 5

107 shown in Figs. 1. 2 and 1. 3 which uses the 60-ms perfect word symbol shown in Figs. 2. 10 and 2. 11. When this symbol is transmitted through the Mimi channel, approximately 94% of the signal energy is within 2T = 120 ms for the phase-equalized received symbol, and hence the M =1 assumption is valid. The value of r(T) found from the autocorrelation function of the received symbol is found to be -. 2. Assuming a d of 10, which corresponds to a received S/N of approximately 3 in a 50 Hz band (+ 4.0db), the probability of error for an interference-free receiver is. 00078. For a simple MFR the probability of error is. 0076, approximately 10 times that of the interference-free receiver; for a traditional TFR, the probability of error is.0019 or about two and a half times that of the interference-free receiver. In the same conditions the ISMR has a probability of error of only.00099. Table 4. 1 gives these results.

108 Receiver P % increase in number of errors over interference free receiver Interference Free.000783 0 MFR.007615 870 TFR.001876 140 OLFR (3 tap).001332 70 (5 tap).001191 53 SMR.002962 280 ISMR.000993 27 Table 4. 1. Results for system using Mimi channel (Fig. 1. 2) and a 60 ms perfect word symbol (Fig. 2. 10), d=10, r(t) = - 2

CHAPTER V THE OPTIMUM (LIKELIHOOD RATIO) RECEIVER Up to this point we have considered receivers which either represented reasonable approaches to the problem, such as the TFR or ISMR, or gave optimum system performance over a specified class, such as the OLFR. In so doing, we have neglected a well-known result from decision theory that states that optimum binary decisions (under any reasonable criteria) should be based on likelihood ratio. None of the receivers discussed so far base their decisions on likelihood ratio and, consequently, they are suboptimum in the absolute sense. The reason for the neglect of likelihood ratio in earlier receivers for the intersymbol interference in noise problem stems from the inherent difficulties of the general problem. By imposing the requirement of phase equalization and a unit degree of intersymbol interference (M = 1) analysis and evaluation are possible. Operating equations for the optimum receiver with M > 1 have been derived; however, the equations are quite complicated and offer no hope of evaluation. As mentioned in Chapter II, for reasonable signalling rates, the M = 1 assumption is acceptable and hence the results presented here have practical * If there is no intersymbol interference ( Ir(t)l = 0), any of the linear filter receivers bases its decisions on likelihood ratio, as can be seen from classical detection theory. 109

110 importance. Because of the difficulties inherent in implementing the optimum receiver, even for M =1, the subsequent analysis is, in a sense, a mathematical exercise. Only in the most critical applications could the complexity of the optimum receiver be justified. The major benefit obtained from the analysis is the absolute bound on system performance which it gives and the method of operation it suggests. The lower bound on probability of error derived shows to what extent intersymbol interference is a fundamental problem and the optimum receiver's method of operation provides guidelines for the development of practical suboptimum receivers. The following section reviews the concept of likelihood ratio and derives the operating equations for the optimum receiver. In the second section, the time symmetry produced by phase equalization is used to evaluate the receiver performance in a relatively simple manner. Finally the performance and operation of the optimum receiver is compared to the performance and operation of the receivers studied earlier. 5.1. Operation of the Receiver This section reviews the basic concept of likelihood ratio which provides the basis for the optimum receiver's design. A convenient transformation of the likelihood ratio is introduced which allows sequential operation and analysis of the receiver. Using this transformation,

1ll the operating equations for the receiver are derived. Likelihood Ratio Let x be the total reception, a waveform of duration (m + 2)T and let b be the value of the kth symbol in x. In the reception x k there are components of other symbols whose values are independent of bk, and added white, Gaussian noise. The likelihood ratio of the reception x for the kth symbol lk(x) is defined by p(x bk = + 1) k(x) p(xbk 1) (5.1) where p(x I bk= ~ 1) is the conditional probability (or probability density) of the waveform x given that the kth symbol has the specified value. A major result from binary decision theory states that the likelihood ratio lk(x) (or any monotone function of it) is the best possible indication of the value of the kt symbol. For the case in which the symbol values bk = 1 are equiprobable and in which both types of error are equally costly, as they are here, the classical theory requires that lk(x) be compared to a threshold of one. If lk(x) is greater than or equal to one, a dk=+ 1 decision is made, if lk(x) is less than one, a dk=-1 decision is made.** *More elegant definitions of likelihood ratio are available but unnecessary for our analysis. **The decision when lk(x) is equal to one is arbitrary, the decision indicated here is simply a convention.

112 The above results completely specify the optimum receiver, and from one theoretical point of view, the problem is solved. To provide useful results, however, a method of determining lk(x) from the reception x must be found. To do this conveniently, a sequential form of Equation 5. 1 and the log odds ratio transformation will be introduced. Let x. be the portion of the reception in the time interval (jT, (j + 1)T) and let X. be the portion of the reception in the interval (0, jT). Note that X+ = x. Then from two forms of P(bk,x.j Xj) we have P(bk (Xjxj) Xj +)p(xj Xj) = P(bk Xj)p(xjbk Xj) (5.2) Dividing Equation 5. 2 with bk =+ 1 by Equation 5. 2 with bk= - 1 we obtain P(bk =+1 Xj1) P(bk= +1Xj) p(x I bk =+1,Xj) P(bk = -I- Xj1) P(bk -l j) p(j bk, Xj) (5.3) Define the log odds ratio of the kth symbol given Xj, Li as j k P(bk =+1 Xj) L = In - (5.4) k P(bk - 1 Xj) and the log likelihood ratio of x. for the k symbol, as J

113 p(xj I bk = +,Xj) lnl (x.) = In ---— (5. 5) Ik l p(xj lbk = -l,Xj) Then Equation 5. 3 becomes simply Lk L + lnlk(xj) (5.6) That is, the log odds ratio at time j-1 is updated to give the log odds ratio at time j using the log likelihood ratio. Figure 5.1 is helpful in visualizing the above results. \b, b, b, b k-1 k k+1 (k-l)T kT (k+l)T (k+2)T -xk Xk ik +. nlk (Xk)- +1 Fig. 5. 1. Updating the log odds ratio Lk

114 We now show that for equiprobable symbols the log odds ratio after the entire transmission has been received, L +2 is identical with the log of the likelihood ratio of the observation lk(x). From two forms of P(bk, X+2 ) P(bk =+ 1IX +2)(Xm+2) = p(X+ k b= 1)P(b 1 - ) (5. 7) Divide Equation 5. 7 with bk = + 1 by Equation 5. 7 with bk =-1 and take the logarithm m+~2 PP(bk = +1) k lnlk(x) + In (5.8) P(bk =-1) The last term is zero since the symbols are equiprobable. Thus in our work the log odds ratio Lk and the log likelihood ratio lnlk(Xj) are the same. Since the log odds ratio is a monotone function of the likelihood ratio, we will base decisions directly on log odds ratio. It is easy to see that comparing the log odds ratio to zero (deciding dk = +1 if m+2 m+2 Lk > 0 and dk =1 if Lk < 0) is equivalent to comparing the likelihood ratio to one in the manner described earlier. It is easy to determine the probability of bk given X. from the J log odds ratio Lk k 5 (bk 1)L P(bkIXj) = ---- (5.9) 1+e k

115 In subsequent work it will be convenient to use the symbols Lk = ~ oo to indicate that P(bk IXj) takes on a value of one or zero; Lk =+o signifiesthat P(bk=+lIXj) = 1 and P(bk = -IIXj) =0. Derivation of the Log Likelihood Ratio lnlk(xj) The derivation of the log likelihood ratio lnlk(xj) is quite tedious and the resulting equations are discouraging at best. Inspection of these equations indicates the enormous difficulty involved in implementing the optimum (likelihood ratio) receiver and, at first, evaluation of receiver performance seems impossible. As we will see in the second section of the chapter, evaluation of receiver performance is actually not too difficult. Thus the equations derived here are important not because one would try to implement them, but because they provide a basis for the evaluation of the optimum receiver's performance. With the above comments well in mind, let us derive the equations for lnlk(xj). As a convenience we consider four separate cases, j < k, j = k, j = k+1, j > k+1. The first case, with j < k is the easiest since there is no energy corresponding to kth symbol in either X. or in x. for j < k. Thus lnlk(x). (j< k) (5.10) For the cases where j > k expand p(xj I bk = 1,Xj) in terms of the four possible signals, determined by bj bj, which are present i-i' j

116 in the reception x. Let b =(bj 1,bj), then p(xjIb,Xj) = P(blbk =~ 1,Xj)p(xjb,bk,X (5.11) bjW ik b. b. j-1 ] Since the symbol values b.j,b. are independent, the first factor in each term of Equation 5. 11 can be written as a product. Further, p(x. I b,bk = ~ 1 Xj) is completely specified by b so that the last two conditions are superfluous. Then from Equations 5. 5 and 5. 11 j P(bj_ bk =+l,Xj )P(bj bk=+l,Xj)p(xj b) b. lb. lnlk(x.) = In / P(bj_-lbk -1X j )P(bj = b 1,X j)p(xj Ib) b. b. i-1 J (j > k) (5. 12) If j =k 4 0, Equation 5, 12 simplifies considerably since P(bj Ibk X.) is either one or zero and P(bj_1 I bk Xj) is independent of bk. The case where j =k=0 will be treated later. 2 P(bj_ iIXj )p(xj l b 1' bj =+1) n(xJ) = In - (j=kjO) (5. 13) P(bj_ 1Xj)p(xj I bj 1 bj - 1) b. J-1

117 For j > k, Equation 5.12 also can be simplified. Since there.th is no energy corresponding to the j symbol in Xj, we have p(b I bkXj).5. For j=k+.5 p(x ib - +lb.) b. j j.5 Z p(x lb =- l,bj) b.s pcx 'bj-1= j b. For j > k + 1, Equation 5. 12 reduces to.5L P(bj 1lbk +lX) p(xjlbj 1,b.=+l)+p(xjlb 1,b.-1) bj-1JJJ J Inlk(x ) = In.-.......,5 P(b jllbk=-l,Xj) p(x lbj l,bj =+l)+p(xjlbj lb — 1) bj-1 (j > k+ 1) (5. 15) Equations 5. 10, 5. 13, 5. 14 and 5. 15 give the equation for the log likelihood ratio in each of the four possible cases. We must now write the probabilities in these equations in terms of the reception, x;. The determination of the probabilities p(xjlb) in the preceding equations leads to some deep mathematical questions. These questions concern the representation of a waveform by a finite dimensional vector having independent components (Ref. 12). We avoid a lengthy discussion of this well-studied problem and merely state its result. That is,

118 o1 1'2 b 12 Ixjl - 2x. sb + lSbs p(xj I bj 1bj) = Ke (5. 16) where Sb = j-A~1+ b jp (5. 17) and the dot product is the L2 inner product used in Chapter III. The constant K is the normalizing constant. The result given by Equation 5.16 is valid under what is essentially a bandwidth limitation.* Before proceeding further, we will normalize all waveforms in the problem by dividing by N/2. Under this convention, Equation 5.16 becomes -.5(lx.2 - 2x. * s + Is b 2) p(xj|b) = Ke b b 8) which is a unit variance Gaussian density. We can now determine the log likelihood ratio for each of the three remaining cases, through the use of Equations 5. 9 and 5. 18 and much manipulation. From Equation 5.13, we have for j = k O, * Equation 5.16 does not, however, require a Fourier series bandlimited assumption and hence does not lead to paradoxical results. Equation 5.19 is derived in Appendix B.

119 Ak cosh - + p- (x Inlk(xj) - 2x P + In(j=k 0) kLj P k-1 j-1 cosh(2 + P (x + )) (5. 19) k G0(xj, Lk_1) (j=k/O) (5.20) Equation 5.19 can be checked very simply. Suppose that the value of the k k-1th symbol is almost certain to be b, - = +1, (-1) then Lkk is of large magnitude and positive (negative) and the cosh functions approach an exponential. As the cosh functions approach exponential, the terms common to both their arguments cancel and we have lim Inl (xk) = 2p(x. pJ) Lk ~00 k o ( 1 ) (5. 21) ~k-~1 (J~k(j =k 0) This limiting form of lnlk(xk) is the same as if bk 1 were known exactly and the interference component bk l subtracted from x. k-1P1 subtrctedfo The interference free difference is then correlated with the signal p0 as one would expect. After some manipulation the log likelihood ratio for j =k+ 1 can

120 * be found from Equation 5. 14 cosh(P - p( J )) Inlk(xj) 2x. * p + In -(x j-1 (j=k+1) cosh ( + 1 )) (5.22) G1 (xj) (j=k+l) (5.23) Equation 5. 22 can also be readily checked. Suppose that bj =+1(-1) and J that the component of x. in the pO direction is large and positive (negative). This is equivalent to saying that the symbol value b. is pretty well known. Then Equation 5. 22 becomes lim lnlk (xj) = 2p 1(j pO) (j=k+l) (5.24) 0 X.- ~ 0o Thus the limiting form of lnlk(xk+1) corresponds to the correlation of P1 with the reception x minus the "known" interference. For j > k + 1, no particularly convenient simplification of Equation 5. 15 is possible. Define the conditional log odds ratio L! l(bk) as* Equation 5.22 is derived in Appendix B. *Note that the symbol for the conditional log odds ratio Lj^1(b) always has the condition in parenthesis, whereas the symbol for the (unconditional) log odds ratio LJ 1never contains parenthesis.

121 P(bjl = +1 IXbk) L. (bk) = n (b (5.25) j1- ~-1....bk) Then Equation 5. 15 becomes L (-1). ' L-1(+1) + 2x p}.5{L (+ ) + 2x *p} e cosh(p * (x- p)) + e cosh(p- 1 (xj+ pj)).5{ L (-1) + 2x. p}.5{ L (-1) + 2x. p e cosh (pj1 (x.j -) + e cosh(pi- (xj+pI)) (j>k+l) (5.26) For convenience write lnlk(xj) A G2(Xj,L l(+1), L (- )) (j>k+l) (5.27) We note in passing that lnlk(x ) is zero for j>k+l only if L 1(+) = L! 1(-1). This means that lnlk(xj) is zero only when a knowledge of * Equation 5.26 is derived in Appendix B.

122 the kt symbol would not affect the decision on j-1th symbol. One would expect as symbols j-1 and k are separated by increasing time amounts, lnlk(xj) would become small. Although Equations 5.10, 5. 19, 5. 22, and 5. 26 give lnlk(xj) for all four possible cases, in Equation 5. 26 a new variable, the conditional log odds ratio was introduced. In order to determine lnlk(xj) for j> k+ 1 we must also be able to determine L (b). In a manner analogous to that used in deriving Equation 5. 6 - (bk) - L- 1(bk) + lnlj1(xjllbk) (5. 28) where the conditional log likelihood ratio lnl l(xj 1 bk) is givenby p(xj_ 1 bJ =+1,X j-lk) Inl jl(xj1 I-bk) n In (5. 29) p(xi_ b j_1 = -1Xj_bk ) Because no energy corresponding to the j-1th symbol is contained in Xj- Lj(b) is always zero. Hence L j (b lnlI 1( b The conditional log likelihood ratio lnl. 1 (x.j1 bk) is derived in J-l J-llbk) k sderiven essentially the same manner as the unconditional log likelihood ratio given in Equation 5o 19. We have l]l(Xj- llbk) = G(xj-l ( (i~ 2(bk (5. 30)

123 If j =-k + 2, we interpret 1-1 k+ If j =k + 2, we interpret L (bk) = Lk (bk) as having the singular value bko, that is, of very large magnitude and of the same sign as bk Thus we have from Equation 5. 19 k+l k+1 k Lk (bk= 1) 2p0+ (x - bkP) (5.31) Using the above equation as the starting point, we can derive Li (bk) J-1 k for all j > k+ 1 as required by Equation 5. 26. One small problem must be cleared up before discussing the operation of the receiver. This problem occurs at the very start of the decision process when j =k =0. Since there is no preceding symbol, we have P(X0 t b0 = +1) inl0(xO) = ln ( (j=k = 0) (5. 32) 0 = 2x0 p0 (j=k=0) (5.33) G (x0) (j=k=O) (5.34) which is the log likelihood ratio for an interference free reception, as x0 certainly is. The foregoing results are summarized in Table 5. 1.

124 ji lnlk(xj) equation j<k 0 5.10 j=k=0 G(xo) 5.34 j-=k/0 G k(X, L 5.20 j =k il GG0(xj, k_ 1) j=k+1 G1(x) 5.23 j>k+l G2 xL. 1(bk 1), LJ (bk ) 5 27 where j=k+2, k+3,... L l(bk G (xjL -( 5 30 Lk (bk) bkc~ Table 5. 1. Equations for the log likelihood ratio lnlk(x) Operation of the Receiver The operation of the receiver is best understood by considering the operation relative to a single symbol, bk, in the interior of the transmission. To go from this single symbol operation to a receiver which updates Lk for all symbols simultaneously is not difficult. Later we discuss the memory requirements for such a receiver. Since Lk is zero for j < k

125 Lk1 = nlxk) (5.35) = GO(Xk, Lk1) (5.36) k The quantity Lk_1 is required to perform this computation. The log odds k ratio Lk1 can be computed inductively with Lo = G() (5.37) i+_ iLi = Go(x, Li i) i = 1,2,...k-1 (5. 38) From Equations 5. 36 and 5. 38 we note that Lk 1 is dependent on Xk, the entire reception up to time kT. k+1 k+2 Given Lk+ compute Lk+2 by adding the log likelihood ratio k k nl(xk+l) to Lk+1 k+2 k+i k+2 f Then + Gi(vk+l) (L c ) Then given Lk compute Lk for j=k+3, k+4,...m+2 inductively L Lk + GG xj( 6 LJ 2(+1) LJ- 1)) (5 40) k whee- where

126 Lj-2(bk) G(j2, L3(bk) (5.41) Lk(bk) = bk oo (5. 42) Figure 5.2 depicts the process used to determine Lk. The decision dk on the kt symbol is made by comparing the log odds ratio Lk2 to zero in the usual way. By operating in parallel many single symbol receivers of the form described above, decisions can be made for all of the transmitted symbols. It is easy to see that the memory requirements of such a receiver are enormous. At each stage j, the receiver must have in memory the updated log odds ratios L... L3 for the symbols which have been re-... j-1 ceived up to this time. Furthermore, the receiver must also retain in memory the conditional log odds ratios L (b.. LJ (b ) so j-l(b) Lj_ i j-2 that the total memory requirement at time jT is 3 j-2 memory variables. This memory requirement and the great amounts of computation required, make this implementation of the optimum receiver economically infeasible (in 1968) for transmission lengths encountered in communicationso A natural compromise is to base decisions on only a portion of the reception. That is, to base the decision dk on Lk+ (q<< m+l-k) instead of on Lk. Such an implementation requires 3q-2 memory

127 + i + + + -S Cl) X + ~ +~~+ I + X! \ oq '~C14 ~ ~ ~ ~ ~~~~~'- H +,E-I. + ccq "+ + 6 C).,'+ I / A 0 0-4 'o 0 + /%+ - ": 0 \ / I f 4O r-+H I X4 Id Q ^2 0 Cl)jU ~?r3 k c~"0 K m Cm~" H ~~ 05 '-4a 0 0

128 locations to process a reception of any length and gives the optimum decisions based on the truncated reception Xk+q. Receivers of this type are called truncated observation optimum receivers (TOOR). Figure 5. 3 depicts the operation of a TOOR with q =3. The author expects that the difference in performance between the optimum receiver and a TOOR with q -3 would be very slight. Even with the reduced memory requirement, implementation of a TOOR would be justified only in very critical applications because of the computations needed. In Section 5. 2 one method of operation for the optimum (likelihood ratio) receiver in which the number of computations is greatly reduced will be described. Unfortunately, this method has an even greater memory requirement than the optimum receiver described above. We note in passing that each computation of lnlk(xj) requires the quantities x. p and x. p. These quantities can be ob1hqatte j p a j 0 tained as the output of filters matched to pi and p0 respectively. Because the optimum receiver does not correlate the reception with the noise-free symbol pJ, it is essential that the optimum receiver be preceded by a phase-equalizing filter in order for the above results to be valid. This is in contrast with linear filter receivers in which the first element in the canonical form is a filter matched to pJ, making previous phase equalization unnecessary. Thus the optimum receiver has the additional disadvantage of requiring a separate phase-equalizing filter.

129 Time (k+2)T xk+2 Time (k+3)T k+2 k+3 L<+1 -GO = Lk'm+2 2k+2 k+2 b l k+3 (bk + +1 (bk+l-) - + k+2( k+2 + )-= k+2 GO k+3(b J+l (bk | 1+2(bk+2=-l) X1 k+3 _k+2b L k+3 +1) ]^^ Lk+l k J G 2 k+2 k+3 (bk 'k+2 (bk+l -1) -1 k k+3 1 I Lk+2 LkDe n Decision Decision onbkl on bk Fig. 5.3. Updating process used by TOOR (q = 3)

130 Insights The notation associated with the operating equations given above obscures the basic nature of the receiver's operation. To better understand what the receiver really does consider the following explicit equam+2 tion for Lk+ k coshk k' 1 k- k-1 k n2 -2- + PI ' (x -PO Lk =In Term A k k cosh (k1 + k * (x + p) 1 + k * + 2Xk+lPl Term B / (xk+1 k + 9k+2 E cosh p0 (xk+l - P1) l cosh(p I n | kk (5. 43) Term B in the above equation is recognized as simply the correlation of the reception (xk, xk+) containing k symbol with the noisek free symbol p. In the absence of intersymbol interference, the log odds ratio from classical detection theory is given by Term B.

131 The effects of Terms A and C can best be understood by considk+1 ering their limiting values as Lkl and po Xk+1 become large in magnitude. Neglecting the second term in Term C, we have from Equations 5. 21 and 5. 24 m+2 k k-i k k+l lim Lk = 2pO (x-bk-p ) + 2p (X-bk+lPo ) k L - b o00 Lk- k-l (5.44) k+1 b 00 P0 Xk k+1 k+ As the value of bk 1 becomes better known, ILk 1 — o, and as the k+1 component of Xk+1 in the p0 direction becomes large, the receiver subtracts the "known" interference from the reception and correlates the result with the noise-free received symbol. Terms A and C serve to subtract components due to the interfering symbols from the decim+2 sion variable Lk The notion of "subtracting out" interference components is, of course, the fundamental idea behind the nonlinear ISMR. In the ISMR, however, the amount subtracted can take on only one of three possible values. The optimum receiver, on the other hand, subtracts a continuous valued amount which depends, in a nonlinear manner, on the entire reception. Except for this difference, the operations of the optimum receiver and the ISMR are similar.

132 5. 2. Evaluation of the Optimum Receiver At first glance, evaluation of the optimum receiver's performance appears virtually impossible. By looking at the problem in a different way, however, the evaluation can be done quite easily. This section describes a method of evaluating the receiver's performance exactly on a digital computer. This method of evaluation also provides insight into the operation of the optimum receiver and suggests another possible implementation. The objective of evaluation is to determine the probability of error for a single symbol located interior to a long sequence of transmitted symbols. If the number of transmitted symbols, m + 1, is very large, as we have assumed, this probability of error will be the system probability of error P. Since the decision dI is based on Lk we must determine P = P(L+2 0 lbk = + 1) (5. 45) 0 P(L+2lbk + 1)dL m+2 (5.46) -00 The major problem of evaluation is to determine the density function p(Lk2 bk =+1) of the decision variable Lkm+ when bk =+1. Determination of p(Lk2 b =+1) The preceding section gave a sequential method of computing

133 Lk2 starting with x0 and working along in time until time (m+2)T. The computation of L+2 is not dependent on which direction in time the sequential operation proceeds. It is also possible to start with xm+ and work backwards in time until time 0. Let Xj be the portion of the reception in the interval (jT,(m+2)T). The total reception x is equivalent to the pair X. and X. as shown V~j in Fig. 5. 4. Define the reverse log odds ratio Lk as simply P(bk= +1 Xj) Lk= In -1IX (5.47) k P(b =-llXj) k-/ \+1 JkT (k+l)T (ha2)T (k+3)T c --- —---— kkT (k+l)T (k+2)T (k+ 3)T xFig. 5.4. D n of te rn x ito ad Fig. 5.4. Division of the reception x into Xk and Xk

134 From the discussion of the preceding paragraph, m+2 ~O Lk = Lk (5. 48) Reiterating, the log odds ratio for the kth symbol given the entire reception does not depend on whether it was obtained by a forward operating receiver or a backward operating receiver. Suppose that we conceptually "burn the candle at both ends" by k+1 operating in the forward direction to obtain Lk and then operating in the reverse direction to obtain L. We will now show that m- +2 k+l 'k+l k = + k (5.49) k+2 L k+1 and that given the value of bk and L are independent random variables. From Equation 5. 8, m+2 P(x I bk=+ 5I1) L 2 In P(xlb 1) (5. 50) P(X bk =+1) (Xk Xk bk=+1) k+1' k +1X k+l, k = = In (5.51) P(Xk+1 bk = 1) p( k1Xk+l 1bk =1) The only component that Xk+l and Xk+l have in common is the waveform due to the k symbol, since the noise is independent from interval

135 to interval. Hence the condition on bk makes the condition Xk+1 unnecessary, P(Xk+l Xk+1 bk) P(Xk+llbk) (5.52) Substituting Equation 5. 52 into Equation 5. 51 m+2 P(Xk+1 bk P) +(Xk+l lbk +1) L = In + In k+_ bk (5. 53) k P(Xk+1 bk=1) p(X+1 bk= -1) k+1 'bk+- =L + L (5.54) th k+1k+ Whenever the value bk of the k symbol is given Lk and Lk are statistically independent since their only common component is specified. Let us now continue our effort to determine p(Lm+2 bk =+). k+1 -k+l Under the condition that bk =+1, Lk and Lk are independent random variables. Since L+2 is the sum of two independent random k+l -k+l m+2 variables, Lk and L, the desired density function of Lk k k k+~ is the convolution of the densities p(Lk bk =+1) and p(Lk lbk =+l) pLm+2 lb + 1) p(L +k+l,.Lk+k lb p(Lk+2 bk = +1) = p(Lkk = P+1=+) (5.55) thWe can go another step further. If the k symbol is located in the We can go another step further. If the k symbol is located in the

136 k+1 I interior of a long sequence of symbols, the densities p(Lk+1 bk = +1) and p(Lk+ bk =+1) will be the same because of the time symmetry (about time (k+l)T) produced by phase equalization. That is, p(L bk = +) = Lk b = +1) (5. 56) and thus Equation 5.55 can be written p(Lk2 bk= +11) p(L =+1) p(L 1 bk = +1) (5. 57) k+1 The distribution p(Lk |bk = +1) is relatively easy to obtain, as we shall see. From Equation 5. 20, k+1 k Lk = GO(k, Lkl) (5.58) which is a mapping from the random variables representing the recepk-1 k k tion, x p1 and x * po and the random variable Lk_1 into a new random variable L. Let us suppose that the distribution of k kl k+1 ik Lkk+ conditional to Lk _,(L + Lk, bk =+1), is known. Later in this section we will derive an equation for this distribution. -<00 p(Lkk'jbl=+1) = f p(Lk+lLkbk = kl)p(Lk|b=l)dL^ (5. 59) ~~~* ~k+1 ' k+l Of course, the random variables Lk and Lk themselves will generally be different.

137 Since there is no energy corresponding to the kth symbol in the portion k of the reception from which Lk_1 is computed p(Lkbk = +1) = P(Lk) (5, 60) P(Lk- 1 - k (L bk = +1) + (L lb (5.61) 2 k_ k —l I k — +k L\- Because of the symmetry of the problem p(Lk- bk- -1) p(-Lk-bk- +1) (5. 62) From Equations 5. 60, 5.62 and 5. 63, +00 (Lk+ bk +1)= p(Lk+lLk bk +1) -00 k k P(Lk_1 bk-1 = + 1) + p(-Lk I bk1 - +1) d Lk From p(L = ) we can compue bk = + 1) iteratively and From p(L b +1) we can compute p(Lk 'bk _ +1) iteratively and k+ m+2 Ibk +) then convolve p(Lk Ibk = +1) with itself to obtain p(Lk |b = +1) The system probability of error Pe can then be found from Equation 5.46. Intuitively one would expect that for sufficiently large k, that

138 k+l1 p(Lk bk = +1) is stationary. That is, p(Lk Ibk = +1) = P(Lk1 Ibk = +1) (5. 64) Under this condition, then, Equation 5. 63 becomes k+l|+ 0 k+1| k p(Lk lbk=+l) = f p(Lk+l Lk-,bk=+l) - 00 (5. 65) p(k+l bk = +1) + p(-k+1 bk +) P(Lk -L k k -----— 2 d k-1 which can be reduced to a homogeneous Fredholm equation of the second k+1 I kind. Although the integral equation for p(L+ bk = +1) is interesting k+1k from a mathematical point of view, it is much easier to apply the inductive procedure on Equation 5. 63 to obtain p(Lkk bk = +1). The result of this inductive procedure is the solution to the above integral equation and the solution is obtained with 2 or 3 iterations. To initialize the induction process p(L1 |b0 = +1) is needed. From Equation 5. 33, L = 2x0 * p0 (5.66) L0 is the output of a linear filter matched to p o From either the discussion of Section 3. 1 or classical detection theory

139 p(LO|bk=+l) = 10(L-d721 (5. 67) p(L0 bk 0(L0o -J0(L0 where 0(u) is the zero mean, unit variance Gaussian density function. This is the initializing distribution p(L | bk = +1) for Equation 5.63 _k+l k,bk-+l) Determination of p(Lk Lk-1 b =+1) Let us now determine the conditional density function, pL k+1 Lk b = +1). Although the derivation is long, it provides P(Lk 'k_ 1 k insight into the operation of the receiver. Let the reception xk be represented in terms of its components k k-1 k+1 in the plane of the signal vectors pk and p. Since L k k lnlk(xk) depends only on the inner product of xk with po and p1 this representation is adequate. Define the orthonormal axes u0 and u1 as k k Uo = Po/ IP (5. 68) k-1 k-1 0 A P1 - (Pi U0) uo 1 k-1 k-1 ui = 1k-~~i - (ki A |Li~)Qj (5. 69) PI - (P1 U )Uo and let k-i P = Pi * u. i-0,1 (5.70) j - 0, 1

140 k-i be the component of the p. signal vector in the u. direction. Then i i~~~~~~~J we can represent the projection xk of the reception xk onto the plane defined by ^u0,. Let x be the component of xk in the u0 direction and let xk be the component of xk in the u1 direction, so that - 0/0 1A Xk =k uO + xk u (5. 71) Figure 5. 5 depicts the above relations for bk = +1. In terms of the real world quantities d and r(T) from Chapter IIn, PQO = Id/21 (5.72) 2 2 = P + P 2 (5.73) Similarly, k k- 1 O * P1 = POOP10 (5.74) = r(T) d (5. 75) Thus the signal components are completely specified by the quantities d and r(T) as in the receivers considered earlier. We now determine p(xk, x x, L 1, bk = +1). Given bk l, bk 0 1 the distribution of (xk, xk ) is bivariate normal with unit variance (because of the normalization of all waveforms by dividing by VN"0/2 2 ) and

141 A *1 Xk k-1 Xk I 0 k JP Fig. 5. 5. Representation of xk in the plane defined by u0, u1 Fig 5.. Rpreetto fx ntepaedfndb 0

142 k k-1 A mean bkPO + bk P1. Because the axes u0, u are orthogonal, 0 1 the components of the reception xk, xk are independent and p = (xk, k) 1 bk)= 0 - (o0 +bk-_lpo) ( - bklPl) (5.76) 0 1 Ik k The density P(x, Xk x Lk1 bk = +1) is then a weighted (by Lk_1) sum of the above densities. Using Equation 5.9, we have - k P(xk IL-l bk=+1) = P(bk1 Lk-l1) P(xkl, bk = bk-1 (5.77)! 0 (P 00 + p!0 ~k - Pll e Lk 0(xj (p0O + pi0)) 0(x - P11)L Lk-1 l+e + 0 - (Po PIo)) 0( P+ (5. 78) Thus the density function for the vector x consists of two bivariate Gaussian "hills" of generally unequal height as indicated by circles in Fig. 5. 5. From Equation 5. 20, Lk+l ( G/( 0,Xk) Lk.7 L =G L-) (5. 79)

143 k which, when Lk 1 is fixed, is a mapping from the two random vari0 1.Ltik+g 0 ables xk, xk into a single random variable, Lk+ Letting x be the auxiliary variable, one can show by the usual procedure that k+l k ' (Lk Lk- bk = +1) P(Lk k-i k 0 -1 -— k L k --- — bx+ S p k,xk GO (Xk' Lk- l k ) k- J(xk dxk ) (5.80) -1 k k+1 where GO is the inverse (given Lk_ and Lk+ of the G mapping an 0 1kthJos i Lk ) of the mapping and J(xk, Xk) is the Jacobian associated with G. The limits of the above integral can be found by inspection of the GO mapping, as will soon be seen. -1 By studying the function Gi we can gain insight into the operation of the receiver. After considerable manipulation of Equation 5. 19 * we obtain -10 k k+1 1 k 0 0 (xk, Lk-' Lk ) = 2P 1 Lk-1 2iOXk [ - Lk - - 0 sinhLp 0P0- k P 0o ] Lk O+1 sinh P. P. Lk 0 1 1TOO 10 ~2 Xk POO * Equation 5. 81 is derived in Appendix B.

144 -1 0 k k+i 12 Go (Xk, Lk_-l k ) Xk (5.82) k+l 1 _ _ _ One can show that xk is defined only in the interval p |- -POPO Xk J p. 2 -P0'10 P 00 Lk+l O 1 k < x < P 2 + PO Po which are the limits of integration for P00 the integral in Equation 5. 80. Similarly, one can show that lim xk =: oo (5. 83) L k+l 0 1 Lkl k+l '~< ' t 2: PO0 PlO k k+1 0 k The curve, y,(Lk, Lk ), of xk versus xk for fixed Lk_1 and Lk+ is S shaped and has an inflection point at _ k+1 k k x0= 2 x 1 L - Lk 1 Pi0 Figure 5.6 depicts k k+l k k+l y (Lk, Lk ) for selected values of Lk Lk k-1' k k-l' Lk One may interpret Equation 5. 80 as the integral of the density k k+1 k function of xk along the line Y(Lkl, Lk )' As Lkl becomes large k k+l in magnitude and with sign given by bk 1' y(Lk ' Lk+ ) is effectively k1 P2 P a vertical line at x^ =b.- -y + b ^ PQQ P\Q1. Evaluating

145 0'~0 CoD Ink ~ k, —4 < / \ I 11 1 \ s \b y^ S.c^l r, fc^

146 Equation 5. 80 along this line gives Lk+l p(Lk ILk:l, bk =+1) pk 2 + bk lP P0Ib k bk =+ k i Vk 1 2L k-1P00P0bP(Lk1 k1 *1 L 1 1 dxk -0 J(Xk' Xk) dx x (5.84) ' p(Lk 1bkl, bk = +1) (5. 85) k k+1 Ikk k+1 Thus as the magnitude of Lk becomes large, p(Lk+1 |Lk 1,bk-+1) approaches the distribution on Lk+ which would be obtained if bk1 kkk were actually known. Numerical Evaluation of the Optimum Receiver The foregoing equations for the evaluation of the optimum receiver are readily implemented numerically. This subsection briefly summarizes the numerical methods used to obtain the results presented in the next section and gives selected examples of the density functions involved in the computations. k+1 Lk The conditional density function p(Lk [ Lk lbk = +1) was

147 determined by performing the integration indicated in Equation 5. 80. To speed up the integration process different techniques were used on the k k+1 central and asymptotic portions of the y(Lk_1, Lk ) curve. In addition, the cases where lr(T)l = 0 or Ir(T) >.49 were treated separately k k+l due to the peculiar form of y(Lk 1 Lk ) involved in these cases. The resulting density functions resemble slightly distorted Gaussian densities as shown in Fig. 5. 7 for d = 10, Ir(T)I =.5. To determine the density p(Lk bk = +1), the initial density function p(L Jbk = +1) (Equation 5. 67) was represented by its values taken at 40 equally spaced points on the real line. A matrix of 1600 entries corresponding to p(Lk+ Lk b = +1) taken over the same k Ik-i1 k - points was computed. Then the iterative process given by Equation 5. 63 was performed five times, with the density p(Lk bk = +1) apparently converging to its stationary value, the solution of the integral Equation 5. 65, in two or three iterations. Figure 5.8 depicts p(Lk bk=+l) for k=0, 1, 2, 3 with d=10, r(T)=.5, a relatively slowly converging case. After the determination of p(Lklbk = +1) by the above procedure, the density was convolved with itself to obtain p(Lk |b = +1). Figure 5.9 depicts p(Lm+2 bk +1) for Ir(T)I =.0,.1,.2,.3,.4,.5, ~ k_ m+2 d= 10. A simple integration of p(Lk bk = +1) according to * The rapid convergence of p(Lklbk = +1) suggests that the performance of a TOOR with q =3 probably is very close to that of the optimum receiver.

148 + ii ' —4 ~~+ \\ I~~~~~~~ + < I I I I I~~~~~~~~~~~~~~~~~ ~ I I I I I I I \ \ lo v-4' Ills, FC O/ o oa + oo) Cl.4 0 V-c.4.4 v-4 C) CO tL- / c c Cl 0i 0 0 0 0 0 0 0n0 0 0 0 >^ / csl~~~~~~~~~~~~~ ~~~a~ "S \. ^^ ^- ^- ^- o d o i ^ cd i f o cs d

149 co Y-4 ii '-4 Cl >00 II '-.4 CII '-40 + r-4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'. Cl3 0 Ob 00^ -C< ^ O~ c~ 0 0 0C 0) 00 0 0 00 JL ^^ ^~~~~~~~~~~~~~G (N /^N Icut t ' ^^'^^ ^s^ ^ '0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ r 4^ 11 ^ ^^^S. ^< ln~~~~~~~~~~~~~~~~~~~~~~ -0^ ^^^^K\ ^~~~~~~~~~L o o~~oo t- o irs ^ CO CM J^\~~~~E v-4 0 0 00 0 0 0H 0 0 0 0000 000 ^\~~~~

150 c~ + c~ '.4 o o,^ ~ + Ct0 0 0 0I 0 0 0 00I0 0 0 X Wv _,, I I I I I I I i I X o -"4 ~ c~ a: c cD c~ c cO CJ c r4 O ~ O7 O~ O7 O * * * * * * *

151 Equation 5. 46 gives P. Representation of the density functions by 60 points instead of 40 showed no significant changes in any of the results. Because of the relatively small number of points used in the representation, no attempt to analyze receiver performance for values of d greater than 20 (P <. 000004) has been made. e Two-Pass Implementation of the Optimum Receiver The method of evaluation for the optimum receiver has the added advantage of suggesting what is perhaps the easiest implementation of the receiver. This implementation requires that the entire reception be stored and two separate analyses (passes) of the data are made. Although this implementation has a large memory requirement and does not allow real time analysis of the data, the operations performed by the receiver are relatively simple. The performance of this implementation is identical with that of the "one-pass" implementation given in Section 5. 1. Let us consider a heuristic implementation of the two-pass optimum receiver in which the reception x is stored on a long magnetic k+1 tape. On the first pass the receiver computes the log odds ratio Lk through Equation 5. 19 and records the result on a separate channel of the magnetic tape. During the second pass the receiver computes the *k+1 reverse log odds ratios Lk using exactly the same procedure as used k+1 on the first pass and adds the result to the recorded value Lk to obtain Lk+2. The receiver's decision dk is then based on Lk+2

152 Figure 5. 10 depicts this implementation of the two-pass processor. 5. 3. Comparison of Optimum and Suboptimum Receivers A major objective of the analysis and evaluation of the optimum (likelihood ratio) receiver was to provide an absolute basis for comparison for the suboptimum receivers. This section compares the error performance of four suboptimum receivers with the optimum receiver and each other. For the sake of clarity only four of the suboptimum receivers have been selected for comparison: MFR, TFR, ISMR and OLFR7. Each of these receivers is representative of one approach to the intersymbol interference in noise problem. The MFR and TFR were chosen because they represent the approaches which neglect either the intersymbol interference (MFR) or the noise (TFR). The OLFR7 represents very closely the best possible linear filter receiver which could be used. The ISMR was selected as a good example of an easily implemented nonlinear receiver. The results presented here are for values of signal detectability index d in the range from 5 to 15. The error probabilities for an ideal interference-free receiver operating with signal detectabilities in this range go from.012 to. 00005. The lower limit was chosen because it corresponds to what seems to be the highest probability of error one might tolerate in a communication system. The upper limit was selected because it represents a high signal-to-noise ratio case and

153 Two channel magnetic tape Read Write #1 # I Xk Lk k ti^^-1 Go I T Forward pass Delay of T seconds Two channel magnetic tape _ Il k+1l ~k+2,k1 l+1 Threshold decisi ns Delay of T seconds Reverse pass Fig. 5.10. Heuristic two-pass implementation of the optimum receiver

154 because the evaluation of the optimum receiver becomes difficult for higher values of d. We will later see that our results are easily extrapolated to other values of d. In one sense, the result of our comparisons will be obvious — the optimum (likelihood ratio) receiver always yields the smallest error probability. The real objective, however, is to compare receiver performances taking into account the cost of implementation. Such a comparison requires much subjective judgment and a detailed knowledge of the particular problem at hand. The subsequent discussion is the author's personal view relative to current technology and the Mimi channel. The reader is free to draw his own inferences from the results presented. Comparison for Constant d Figures 5. 11 through 5. 15 depict the error probabilities of the four suboptimum receivers and the optimum receiver as a function of I r(T) t. This can be viewed as showing the effects of increased intersymbol interference for a fixed received signal-to-noise ratio. As suggested in Chapter II, increased intersymbol interference is caused by signalling too rapidly relative to the bandwidth of the channel. The results show that the error probability for the simple MFR increases rapidly with Ir(T)I. When intersymbol interference is moderate, the probability of error for the MFR is many times that of the probabiliother receivers. 'For example, if Ir(T)I =.25 and d= 10, the probability of error for the MFR is about 15 times that of the optimum receiver, and

155 0.10 MFR ISMR / OLFR7 OPTR 0.01 L p e 0.001 0. 0001_____ 0 0.1 0.2 0. 0.4 0.5 I r(T) l Fig. 5.11. Probability error P versus Ir), d= 5

156 0.10 MFR ISMR OLFR7 TFR 0.01 OPTR Pe 0.001 0.0001 0 0.1 0.2 0.3 0.4 0. 5 I r(T)I Fig. 5.12. Probability error P versus I r(T), d 7.5

157 0.10 MFR = OLFR / TFR ISMR OLFR7 0.01 e 0 A / / A /I OPTR 0. 001 L0.0001 [I I I I. I 0 0.1 0.2 0.3 0.4 0.5 I r(T) I Fig. 5.13. Probability error P versus t r(T)t, d= 10 e

158 0.10 TFR MFR/ //ISMR OLFR 0.01 ~ / / //~ / OPTR 0 0.1 0.2 0.3 0.4 0.5. 0001 Fig. 5. 14. I r(T) 1 PeroPbabiliye versus ( r(T)I, dr 12.5

159 0.10 TFR MFR/ // ISMR OLFR7 0.01 0. 001 OPTR 0.0001 / 0. 00001 I _____ 0 0.1 0.2 0.3 0.4 0.5 I r(T) I Fig. 5.15. Probability error Pe versus Ir(T)I, d 15

160 about 10 times that of the ISMR. Even if the intersymbol interference is very slight, i. e., I r(T) I =. 1, the use of the MFR can lead to a twofold increase in the number of errors. I conclude, then, that a better receiver than the MFR is necessary even when intersymbol interference is not particularly severe. The TFR, OLFR7 and ISMR receivers offer improvement in performance with less complexity in implementation than that required by the optimum receiver. Among these three receivers, the relative error performance depends considerably on Ir(T)I and to a lesser extent, d. For Ir(T)I <.25 the ISMR is generally slightly superior to the TFR and OLFR7 in performance. On the other hand, if Ir(T)l >.25 the OLFR7 gives the best performance, the TFR and ISMR performances are roughly comparable. In terms of implementation, the ISMR is distinctly superior to both the TFR and the OLFR7. The optimum receiver performs significantly better than the suboptimum receivers such as the ISMR and OLFR7 only when Ir(T)l >.25. Even though a substantial reduction of the error probability is possible for tr(T)I >. 25, the complexity of the optimum receiver makes its use impractical. A reduction in signalling rate to reduce I r(T) I and allow the use of a simpler receiver such as the ISMR would be a wise alternative to implementing the optimum receiver. Comparison for Constant Ir(T)I Let us now fix the amount of intersymbol interference, as given

161 by Ir(T)l and examine the receiver performances as a function of d. This is equivalent to fixing the channel spectrum and signalling rate and observing the effects of varying signal-to-noise ratio. Figures 5.16 through 5. 20 give the performances of the four receivers as a function of d for Ir(T)I =.1,.2,.3,.4, and.5. The curve labeled IFR in each of these figures represents the performance of an interference — free receiver operating with the given value of d. The smoothness of the curves allows us to extrapolate our general results for values of d outside the range plotted. Figures 5. 16 through 5. 20 show that the performance of the MFR relative to the interference-free receiver actually deteriorates as d increases. The MFR is relatively more successful (in its limited way) at low signal-to-noise ratios than at high signal-to-noise ratios. This can be attributed to the fact that as the signal-to-noise ratio increases, the error producing effect of intersymbol interference over-shadows the effects of the noise. Because the TFR represents the optimum receiver in the absence of noise, one might hope that as d becomes large, the TFR performance would approach that of the optimum receiver. Figures 5.16 through 5. 20 indicate that this does not happen even for the larger values of d considered here. This suggests that "neglecting noise" and using a "noise-free" receiver design is a poor approach. For I r(T)l >. 25 the performance of the ISMR relative to the

162 0.10 MFR TPR - OLFR<0.01 r ISMR OPTR - IFR e 0.001 - 0.0001 I 0 5 10 15 d Fig. 5.16. Probability error P versus Id, I r(T) =.1 e

163 0.10 -MFR TFR OLFRI7 ISMR OPTR \ \ TTTI'D 'XXX \ Ik 0.01 IFR 0.001 IDEAL 0.0 0 5 10 15 d Fig. 5.17. Probability error P versus d, Ir(T)I =.2 e

164 0.10 - MFR TFR ISMR OLFR 7 OPT IFR 0.01 P e 0. 001 0.0001 \ O 0 5 10 15 d Fig. 5.18. Probability error P versus d, I r(T)I =.3 e

165 0.10 - TF ~- \^^ ^ ^ ^ ^ - -^ ^ M FR - ISMR \ ISMRNISMR 0.01 IFR \\ \ TFR P 0.001 0. 000 I I \\ I 0 5 10 d15 Fig. 5.19. Probability error P versus d, Ir(T)I - e

166 0.10 ISMR MOLFR___________________ OPTR IFR 0.01 - p e 0.001 -;P (MFR) >.1 -Pe(TFR) =.5 0.0001 0 5 10 15 d Fig. 5.20. Probability error P versus d, Ir(T)l =.5 e?

167 optimum receiver depreciates in essentially the same manner as the MFR as d increases. This can be attributed to poor "first guess" decisions used in the ISMR, which have the same probability of error as the MFR. Thus the ISMR is not a suitable receiver for problems with severe intersymbol interference and high signal-to-noise ratios. However, Ir(T)l <. 25 the ISMR is a good performer for all values of d. The performance of the OLFR7 depreciates slightly relative to the optimum receiver as d increases, with greater depreciation occurring when intersymbol interference is severe. The performance of the optimum receiver, on the other hand, actually improves relative to the interference-free receiver performance as d increases. This can be attributed to the receiver's better knowledge of the interfering symbol values at high signal-to-noise ratios. Conclusions The above results show that the amount of intersymbol interference determines which is the best suboptimum receiver. If lr(T)I <.25, the ISMR is the best suboptimum receiver, both in error performance and in ease of implementation. Further, this remains true for all d. If lr(T)l >. 25, however, the performance of the ISMR is inferior to that of the OLFR7 and depreciates relative to the ideal receiver with increasing d. The actual choice between the TFR, OLFR7 and ISMR

168 receivers for Ir(T)l >. 25 is a difficult one. We can draw still another important conclusion from the results presented. Since the system designer can usually control lr(T)I by varying the signalling rate, the above results suggest a good choice of signalling rate. When lr(T)|>. 25, two undesirable things happen. First, the receiver required becomes more complex, such as the OLFR7 or TFR. Secondly, the probability of error for this receiver is significantly larger than that for an interference-free situation. On the other hand, if the signalling rate is reduced so that Ir(T)I <. 25, the best practical receiver (ISMR) is easily implemented and gives near interference free performance. Furthermore, when Ir(T) I <.25, the M = 1 assumption on which the foregoing work is based is more likely to be valid. Thus, I suggest as a general rule of thumb, the signalling rate for a binary communication system should be reduced to give Ir(T) <. 25 in order to obtain both good error performance and a relatively simple receiver implementation. The 60-ms perfect word symbol described in Chapter II for the Mimi channel of Fig. 1. 2 provides an example of the application of this rule of thumb. For this symbol we have I r(T)I =. 2. Table 5. 2 below gives the performances of the five receivers for d =5, 1 0 and 15 for this example. The ISMR gives excellent performance.

169 C O o o cI 6 o a\ 0 0 0 0 0 =00 00 0 0 10- I00 r-. I1 s In 4 CD ) c C m q LO t o o 0 o0 ~o o -o o o * 0 0 0 0 0 0 CD C CD 0 0L ~ ~ ~ ~ ~ ~ r-( ^2 S " 0c o 0 o, I. c" 0 o 0 ) 0' CD 6 L C0 00 I _ o-o oo o I, ______l- O 0 0 k ' I' CD l- O tOC C) 0C) 0) '44 to Eto 0)) 0 t c C0 0 D [3 ^< (3 I0 0 to Lo Lo LoS e a a) tO C) Q) or a) O O H 0 r^ P 0 - r4 (

Chapter VI CONCLUSIONS AND FUTURE STUDIES In this chapter we state the major conclusions of this paper and the future studies which they suggest. 6.1 Conclusions Chapter II showed that the amount of intersymbol interference can be related to the power spectrum or, equivalently, on the autocorrelation function of the received symbol. This relationship was shown from the existence of a waveform of minimum RMS time duration which could be obtained through the use of a phasek equalizing filter. Ripples or notches in the power spectrum increased the RMS time duration of the received symbol and consequently increased intersymbol interference. Thus intersymbol interference could be viewed intuitively as a result of signalling too fast for the bandwidth of the channel. The dependence of intersymbol interference on power spectrum is important because the usual (unequalized) channel symbol response often appears to indicate much more intersymbol interference than is actually present. The dependence on power spectrum also indicates that the linear variations of different slopes in the channel phase spectrum believed to be caused by multipath effects do not produce intersymbol interference. Instead, the notches in the 170

171 power spectrum caused by multipath effects (selective fading) are the source of intersymbol interference. By examining the actual channel power spectrum, an intuitive idea of a reasonable signalling rate can be obtained. The optimum (likelihood ratio) receiver was derived and evaluated under the assumption of phase equalization and limited intersymbol interference. The results of this analysis are important not because we would ever implement the optimum receiver, but because of the insight into the design of suboptimum receivers which it provides. Since the optimum receiver is a nonlinear receiver, the receiver designer should look for good nonlinear suboptimum receivers instead of the best linear filter receiver. The performance of the optimum receiver also provides an important lower bound on the probability of error in a given situation. Several common or proposed suboptimum receivers have been compared with the optimum receiver on the same frame of reference: phase equalization and limited intersymbol interference (M = 1). From this comparison the system designer can determine the relative merits of the various receivers for his particular problem. The results of this comparison provide another important guideline to the system designer. If the received signal has I r(T)I <.25 then an easily implemented receiver (ISMR) can provide near ideal performance. On the other hand, if I r(T)I >.25 more

172 complicated receivers must be used and even with these receivers the probability of error is significantly increased. From a practical point of view, then it is very desirable to reduce the signalling rate so that I r(T)I <.25. An easily implemented nonlinear receiver, the iterated switchedmode receiver (ISMR), which performs very well under the condition I r(T)I <. 25, has been described for the first time. This receiver does not require the long tapped delay line of proposed linear filter receivers and does not require carefully computed tap coefficients in its operation. The ISMR represents an economical and flexible solution to the receiver design problem when intersymbol interference is not too severe. 6. 2 Future Studies At the conclusion of any theoretical study, the question arises as to whether to pursue the theory further, or to jump into the perilous experimental world for confirmation of the results. The most productive approach seems to be the latter. A carefully controlled implementation of a communication system through the Mimi channel would point out the truly significant problems of underwater communications and perhaps eliminate others from consideration. Although actual implementation of a communication system through the Mimi channel is the next logical step, several theoretical problems are of particular interest. The problem of simultaneous

173 channel measurement to provide a receiver with an up-to-date replica of the noise-free symbol response is important. While transmitted reference techniques suggested in Section 1. 2 can be used, a portion of the transmitted signal energy must be devoted to the reference. One could hope that if the probability of error is sufficiently low, a reference component could be reconstructed from a transmission consisting of only an information component. The development and analysis of such a technique would simplify both transmitter and receiver design. Another subject for future study would be to consider nonbinary communication systems such as M-ary signalling. By using a larger number of transmitted symbols, the transmitted symbol duration could be lengthened while maintaining the same data rate. Hopefully, the reduced symbol duration in such a system would reduce the effects of intersymbol interference. From a purely theoretical point of view, it would be interesting to consider receiver designs when intersymbol interference is more severe; that is, when M > 1. The operating equations for the optimum receiver are known for such problems but offer no hope of evaluation. Development and analysis of a good suboptimum receiver, analogous to the ISMR would provide a reasonable approach to the problem.

APPENDIX A PROOF OF THE DECREASE IN P (k) WITH A DECREASE IN h 1 ___________e h 2 Proposition k k If h p > 0 then the probability of error P (k) for the *^P~~~ c~e Nk canonical linear filter receiver is reduced by decreasing I h2 I to zero. Proof Ok i Since h is orthogonal to all of the p, i = 0,...,m Eq. 3.14 becomes l i k k k i. k e (k) 1 h p + bp h(A. 1) 2 f0-.,./k 2 V1%$k 2) N0(h I + I{h21 )/2 Let ~k k (A. y = h ') p (A. 2) i k Y2(b) = - biP * h1 (A. 3) 1 2 2 Y = 1/AN(lhl2+Ihl ) / 2 (A. 4) then Eq. A. 1 can be written as 174

175 (k) = 1 -3(Y1 + Y2(b)) (A. 5) 2 Bk Let b = (b...b... b) i # k, and let -b = (-b0.. -b... -b ), then y2(-b) -Y2(b) (A. 6) Define I Bkl as any set of 2 vectors be Bk such that if be I BkI,then -biBk. Equation A. 5 becomes p (k) = i [y3(y+y2(b) +4[-Y] (Y1 Y2(b] (A 7) 2 IBI Differentiating Eq. A. 7 with respect to y3 we have dP (k) dy3 2m (y1 y2()) [y 3(1 )] (A.8) I BI + (Y - Y2(b)) [-y3(1- Y2(5)] Each term in the summation of equation A. 8 is positive if Y1 > 0. This can be shown by considering the function S(u1, u2) S(Ul'U2) = U0(u1) + u20(2) (A.9)

176 which is strictly positive if u1 + u2 is positive. Let u1 = 3(Y1 + y2(b)), u2 - Y3(y1 - y2(b)) then U1 + u2 = YY3 > 0 (A. 10) if Y1 > 0. Since y1 > 0 by hypothesis from equation A. 8 we have that dP (k) dy-3 is negative, so that P (k) is reduced by increasing -k y3. Since y3 is increased by decreasing I h2 P (k) is reduced by decreasing I h2 I

APPENDIX B DERIVATION OF EQUATIONS 5.19, 5.22, 5.26 AND 5.81 Derivation of Equation 5.19 Substituting Equations 5.9 and 5.18 into Equation 5.13 gives for j=kO0 lnlk(j) = K 'txj2 -2x. k ej K ^\^-ll!s V +l l+l a2 k L k- 1 i+e - 1 - -- x - - -x - - _ K e 2 1 i2 2 K e Lk-1 -^xj2-2xj - l s l-l 2 k e e k- 1 L -k k S 2l 2 Lk- 1 - txj 2 - 2xj' s+l- + |S+~- 1 + k e LLk- 1 l+e (j =k 0) (B. 1) Canceling factors common to the numerator and denominator we have 177

178 1k 1 2 eJ S —+ 2 -1+1 Lk- j 'sI+1 + 2 1+1t lnlk(xj) = In 12 k 1 2 X.- S-1-1 2 -1- Lkl+x. s12 1s e +e (j =k O) (B.2) Squaring Equation 5.17 we have 2 2 IS b. 2 - i P + 2bj bj-pi O + PO (B.3) Inserting Equations B. 3 and 5.17 into Equation B. 2 and canceling common factors in the numerator and denominator gives: lnl,(xj) e * (p+Pe)-1 * po k-+ PO P1) -P1 oP In j-1 j-1 j k j-1 j- 1 1 j (-pP-p p + 1 )+p- e +e (j = k ) (B. 4) After multiplying numerator and denominator of Equation B. 4 by Lk /2+x. - P k-_ 1 j e we obtain k k j-1 __j-iLk-I L k-I 2x.pJI +P 1 P 2 PI (xj- p)+ 2 1 2x1. (X eP 1 -P 2(xjpi 21 e+ e lnlk(xj) = ln e....k 0..e.k +p- 'X +p0 Lk-i j j k-i e 2 +e (j =k#O 0) (B. 5)

179 Since U -u cosh u = +e (B. 6) 2 we may rewrite Equation B. 5 in the desired form (5.19) kL,cosh ~-1 (x j-p3)+ -nI ~p +In (5.19) lnlk(xj) = 2x.n 4 (5.19) cosh(1 1 (xjp) + 2 1 (j =k 0) Derivation of Equation 5.22 Substituting Equation 5.18 directly into Equation 5.14 we obtain for j=k+1: lnlk(x) l xjl-2x. e - t - s _n K, '2 2 - 2x. 1-1+ Is 121 2 Le -1 __ _-2 x. S _ I +. s s2 2+ X 13( +k1+ +1+1 [ + e (j=k+ 1) (B.7)

180 Substituting Equation B. 3 into Equation B. 7 and canceling common factors in the numerator and denominator gives j-1 j.s ^+1-i PO +l 1+1+1 1' Inl(x.) In e + e k( x. s -pj-1.pi x. s +p1 p ] 1V1-p1 p0 — 1 -1+1 e +e (j=k+1) (B. 8) By inserting Equation 5.17 into Equation B. 8 and multiplying by x. j-1 e J we obtain 2x. -i( -~o. (x.-f) p> (J.-pi1) lnlk(x.) - (X P Pn (j Inlk (X l n e p (X.+pf') -p. (X +-1 e +e (j =k+1) (B.9) We may rewrite Equation B, 9 in the desired form (5.22): =1. cosh[p (x -p ) lnlk(x) = 2x. p1 + n h3 (5. 22) cosh (X~ +7p ) (j=k+1) Derivation of Equation 5.26 From Equation 5.18 we have

181 jp(x jbj l, bj =+1)+p(xjbj_l bj-1) -Ixj lx - sb + s By substituting Equations B. 3 and 5.17 into Equation B. 10 and extracting common factors we obtain P(xj| b 1, b = +1) +p(x b 1' b =-1) - 2j. -12 2 - -1 2iKe xj 2 p1 + PO - 2bl 1xj p1} e 1 (B. 1 {p (xjbl, b-+ lp})+-p (xj -bj_1,pb=-l)1 l{x 12 + Ip 1 + IPO -2bj xj*Pl} cosh p * (xj- bj_ 1 P3) (B. 12) Equation 5.25 can be inverted to give

182 ~(bj + 1) L _(bk) P(bj_l Xjbk) = e(B.13) jl (bk) 1 +e Substituting Equations B.12 and B.13 into Equation 5.15 and canceling common factors in the numerator and denominator, we obtain L! (+1) l 1+e j-1 lnlk(x) = In 1 L (-1) 1ej-1 1+e J _ ~X p ~ ~ ~ ~~ O 1 -X j-1 Li (+l)+x.e X -p _ e i P1 cosh pI0(x +p + e cosh po(x -p p~) J -x.-P1 Lj ( -1)+x. 1 e cosh pi (xj.+p +e cosh p1 ex -P (j > k+l) (B.14) ~ L! (+1) By removing a factor of e from the numerator and a factor L (-1) of e - from the denominator and changing the resulting product into a sum of logarithms gives the desired form of Equation 5.26.

183 Lj (-1) L j-l lnlk(X) =. 5 (L (+1) - Lj (-1)) + In l+e 1+e 5 (Lj (+l) + 2xj * p -.5 (Lj 1(+) + 2x p) j-1 j 0 j- 1 j e cosh (p 1 * (xj p )) +e cosh (p (x +po)).5(L (-1)+ 2x. p) -.5(Lj (-1)+2x. p}) ]-1 j - j j-1 ) e cosh(p1 *(x.-pJ)) +e cosh(p 1 (xj+p)) (j >k+1) (5.26) Derivation of Equation 5. 81 We have, from Equation 5.70 P = p u (B.15) P0 -O 0 k- 1 A P1 P= 0 Ou+ P1 uP (B. 16) Then from Equations B.15, B. 16 and 5.71 we obtain Xk Pko = XkP0 (B.17) Xk' PO = xkPOO X P k- I 0, 1 (B 18) Xk P = xk P + xk P (B.18 Let k k- I R = PO pk- (B. 19) = P0 po (B. 20)

184 from Equations B.15 and B.16. Then Equation 5.19 can be rewritten (with j =k) as k coshk-1 +0 R (B.21) 53= 2 [lkP 0 + 2X (B.23) = a[Lkk- 2Po] (B.24) Then from Equations B. 21 and B. 22 we have 11 cosh(t + xkP11 - R) (B. +xkPil-R - -XkPl+R e + --- ------ ~ -e ----. ------ (B.26) '~ = ~ -1 +2R e (B.27) e e e

185 Taking the logarithm of both sides of Equation B. 27 and solving for xk gives Xk =\i -2 + In sinhR- (B. 28) k 2P1 sinh R+ The desired Equation 5.81 is obtained by inserting Equations B. 22, B. 23 and B. 24: k+1 xk 2- -L^k -2x~o +In o(PbOPiO- P(-O \: 2p _ _ I-k-1 xkPiO — k+1 sinh(pbO pio + -kx p0) 0 10 2 (5. 81)Poo (5. 81)

REFERENCES 1. W. W. Peterson, T. G. Birdsall and W. C. Fox, "The Theory of Signal Detectability, " IRE Trans. on Information Theory, IT-4, 1954. 2. J. M. Aein and J. C. Hancock, "Reducing the Effects of Intersymbol Interference with Correlation Receivers," IEEE Trans. on Information Theory, July 1963. 3. M. R. Aaron and D.W. Tufts, "Intersymbol Interference and Error Probability,' IEEE Trans. on Information Theory, January 1966. 4. D. C. Coll, A System for the Optimumization of Pulse Communication Channels, Defense Research Telecommunications Establishment Report No. 168, Ottawa, Canada, December 1966. 5. J. C. Steinberg and T. G. Birdsall, "Underwater Sound Propagation in the Straits of Florida, " Journal of the Acoustic Society of America, 39, pp. 301-315, 1966. 6. M. P. Ristenbatt, et al., Digital Communication Studies, Part I: Comparative Probability of Error and Channel Capacity, Cooley Electronics Laboratory Report No. 133, University of Michigan, Ann Arbor, Michigan, March 1962. 7. R. Price and P. E. Green, "A Communication Technique for Multipath Channels, " Proceedings of the IRE, March 1958. 8. A. Papoulis, The Fourier Integral and Its Applications, McGraw-Hill Book Co., New York, 1962. 9. C. W. Helstrom, Statistical Theory of Signal Detection, Pergamon Press, New York, 1960. 10. H. Rudin Jr., "Automatic Equalization Using Transversal Filters, " IEEE Spectrum, Vol. 4, No. 1, January 1967, pp. 53-59. 11. S. W. Golomb, et al., Digital Communications with Space Applications, Prentice Hall, Inc., Englewood Cliffs, N. J., 1964. 186

187 REFERENCES Cont. 12. U. Grenander, "Stochastic Processes and Statistical Inference, " Arkiv det Mat., 1, 1950, pp. 195-277.

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Security Classification DOCUMENT CONTROL DATA - R&D (Security claaification of title, body of abstract and indexing annotation muet be entered when the overall report is clasilied) 1. ORIGINATING ACTIVITY (Corporate author) 2Z. REPORT SECURITY CLASSIFICATION Cooley Electronics Laboratory UNCLASSIFIED The University of Michigan 2b. Rpou Ann Arbor. Michigan 48105.. 3. REPORT TITLE Intersymbol Interference in Binary Communication Systems 4. DESCRIPTIVE NOTES (Type of rport and inclusive datee) Technical Report No. 195 - 3674-18-T S. AUTHOR(S) (Laot name, fit name, initial) Kimball, Christopher V. 6. REPORT DATE 7I. TOTAL NO. OF PAGES 7b. NO. OF REFS August 1968 217 12 8a. CONTRACT OR GRANT NO. 94. ORIGINATOR'S REPORT NUMBER(S) Nonr-1224(36) b. PROJECT NO. 3674-18-T c. |. OTMIHER RpE:PORT NO(S) (Any other number that may be easigned ihse repor(' __________d. _____TR 195 10. AVA IL ABILITY/LIMITATION NOTICES Reproduction in whole or in part is permitted for any purpose of the U. S. Government. I1. SUPPLEMENTARY NOTES 12I. SPONSORING MILITARY ACTIVITY Office of Naval Research Department of the Navy........l____ [I_ _Washington, D. C. 13 ABSTRACT When a binary communication system transmits symbols through a bandlimited channel, the received symbols will generally overlap in time, giving rise to intersymbol interference. In the presence of noise, intersymbol interference produces a significant increase in the system probability of error. The problem of intersymbol interference and noise is considered here for known, linear, time invariant channels and with added white Gaussian noise. Although a particular underwater acoustic channel is used as a source of motivation, the results presented are equally applicable to other communication channels. Traditional approaches to the intersymbol interference problem —spectrum and transversal (time) equalization are examined. A basis for the comparison of intersymbol interference problems using the concept of phase equalization, is given. A major assumption which limits the interference to that caused by adjacent symbols is made. This assumption is shown to be equivalent to restricting the transmitter to reasonable signalling rates relative to the bandwidth of the channel power spectrum. All subsequent analysis and evaluation are done under this assumption. Several linear filter receivers prevalent in the literature are reviewed and evaluated. Two easily implemented nonlinear receivers are considered as alternatives to the more complex optimized linear filter receivers. The iterated switched-mode receiver is shown to perform better than any optimized linear receiver when intersymbol interference is moderate. DD JAN. J 1473 __ Security Classification

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