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UBC Theses and Dissertations

Bit error rate performance analysis and optimization of suboptimum detection procedures 1986

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BIT ERROR RATE PERFORMANCE ANALYSIS AND OPTIMIZATION OF SUBOPTIMUM DETECTION PROCEDURES By NORMAN CHARLES BEAULIEU B.A.Sc., The University of Bri t i s h Columbia, 1980 M.A.Sc., The University of Bri t i s h Columbia, 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 1986 • Norman Charles Beaulieu, 1986 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of E l e c t r i c a l Engineering The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date / H I ^ i i ABSTRACT Three suboptimum detection schemes are examined. The deterioration in performance, measured in the probability of error sense, of weighted partial decision, binary partial decision, and sample-and-sum detectors are analyzed. Even though these schemes are inferior to the d i g i t a l matched f i l t e r , they can be used in systems with more modest computational capabilities. Analytic expressions are obtained for the penalties. The effects on the peaalties of the signalling waveform employed, the number of samples processed, and the signal-to-noise ratio are considered in d e t a i l . Included are the penalties for the optimum weighted partial decision detector. The e f f e c t s of dependence among the samples on the detector losses are investigated. It i s shown that, in some cases, the losses of the suboptimum procedures can be reduced by processing more, dependent, samples. The amount of the l o s s that can be recovered depends on the p r e f i l t e r c h a r a c t e r i s t i c and the sampling rate, as well as the detection algorithm. The structure of the optimum detector for hard-limited data signals i s presented and i t s performance i s compared with those of some commonly used schemes. Performance in Impulsive as well as Gaussian noise environments i s considered. The optimum receiver for M-ary signalling based on received signal samples quantized to an arbitrary number of levels i s derived and compared to another common detector. The fundamental l o s s i n s i g n a l d e t e c t a b i l i t y due to hard- l i m i t i n g i n a sampled system operating i n Gaussian noise i s investigated. The r e l a t i o n of the l o s s to the signal-to-noise r a t i o and the number of samples i s analyzed. i v TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES v i i i LIST OF FIGURES i x ACKNOWLEDGEMENT x i i I INTRODUCTION 1.1. Motivation 1 1.2. Scope of the Thesis 2 1.3. Review of Relevant Work 4 1.4. Outline of the Thesis 7 II PERFORMANCE COMPARISON OF THREE SUBOPTIMUM SCHEMES FOR BINARY SIGNALLING 2.1. Introduction 9 2.2. The Sample-and-Sum (SAS) Detector 12 2.3. The Weighted P a r t i a l Decision (WPD) Detector 15 2.4. The Binary P a r t i a l Decision (BPD) Detector 17 2.5. Generalization to A r b i t r a r y S i g n a l l i n g Waveforms 18 2 . 6 . Discussion l q V Page II I EFFECTS OF OVERSAMPLING ON THE PERFORMANCE OF THREE SUBOPTIMUM DETECTION SCHEMES 3.1. Introduction 26 3.2. The Sample-and-Sum Detector with Dependent Samples 29 3.3. The Weighted P a r t i a l Decision Detector with Dependent Samples 39 3.A. The Binary P a r t i a l Decision Detector with Dependent Samples 47 3.5. Discussion 48 IV OPTIMAL DETECTION OF HARD-LIMITED DATA SIGNALS IN DIFFERENT NOISE ENVIRONMENTS 4.1. Introduction 49 4.2. Derivation of the Optimum Detector f o r Hard-Limited Samples 49 4.3. Optimum Weights for Low Signal-to-Noise , Ratios 52 4.4. Optimum Weights for High SNR's 53 4.4.1. Gaussian Noise D i s t r i b u t i o n 54 4.4.2. Laplace Noise D i s t r i b u t i o n 4.4.3. Cauchy Noise D i s t r i b u t i o n 5 5 4.5. Some Numerical Examples 55 4.6. Generalization to M-ary S i g n a l l i n g and M u l t i l e v e l Quantization 62 4.7. Conclusions 6 8 v i Page V PENALTIES OF WEIGHTED PARTIAL DECISION DETECTORS IN GAUSSIAN NOISE 5.1. Introduction 69 5.2. Problem Statement 6 9 5.3. Weighted P a r t i a l Decision (WPD) Detectors 72 5.4. The WPD Detector f o r a Piecewise Constant Amplitude S i g n a l l i n g Waveform 7 ^ 5.5. The Optimum WPD Detector f o r A r b i t r a r y S i g n a l l i n g Waveforms 78 5.6. The Weights = |s±I and u± = s i WPD Detectors 8 l 5.7. Discussion 81 VI PENALTIES OF SAMPLE-AND-SUM AND BINARY PARTIAL DECISION DETECTORS IN GAUSSIAN NOISE 6.1. Introduction 88 6.2i The Sample-and-Sum (SAS) Detector Loss ....... 89 6.3. The Binary P a r t i a l Decision (BPD) Detector Loss 9 4 6.4. Discussion of Results 100 VII CONCLUSION 7.1. Summary of Results 102 7.2. Suggestions f o r Further Research 104 APPENDIX A 105 APPENDIX B 106 v i i Page APPENDIX C 1 0 7 APPENDIX D 1 0 9 APPENDIX E 1 1 1 APPENDIX F 1 1 3 APPENDIX G 1 1 4 APPENDIX H 1 1 5 APPENDIX I 1 1 8 APPENDIX J 1 2 0 APPENDIX K 1 2 4 REFERENCES 1 2 6 GLOSSARY 1 3 0 V l l l LIST OF TABLES Table P a S e I Asymptotic losses f o r suboptimum schemes 20 ix LIST OF FIGURES Figure Page 2.1 Block diagram of the data receiver 10 2.2 The suboptimum detector penalties for a sinusoid with P(e) = 10"3 21 2.3 The suboptimum detector penalties for a sinusoid with P(e) = 10~7 22 2.A The suboptimum detector penalties for a _3 square-wave with P(e) = 10 23 2.5 The suboptimum detector penalties for a square-wave with P(e) - 10 24 3.1 , Block diagram of the data receiver 27 3.2 The normalized autocorrelation function of white noise f i l t e r e d by a Butterworth lowpass f i l t e r of order N 34 3.3 The normalized autocorrelation function of white noise f i l t e r e d by a cascade of N identical poles 36 3.4 The additional penalty T^ for the SAS detector with a Butterworth pref l i t e r of order N 37 3.5 The additional penalty for the SAS detector with pref l i t e r consisting of a cascade of N identical poles 38 X Figure Page 3.6 The normalized autocorrelation function of white noise after lowpass filtering by a Butterworth filter of order N and hard- limiting 43 3.7 The normalized autocorrelation function of white noise after lowpass filtering by a cascade of N identical poles and hard- limiting 44 3.8 The additional penalty of the WPD and BPD detectors with a Butterworth prefliter of order N *5 3.9 The additional penalty of the WPD and BPD detectors with prefilter consisting of a cascade of N identical poles 46 4.1 Error probabilities for the DMF detector and four different WPD detectors for a raised cosine pulse 56 4.2 Error probabilities for the DMF detector and four different WPD detectors for a half sinusoid pulse 58 4.3 Error probabilities for the detection of a raised cosine pulse in Laplace noise 60 4.4 Error probabilities for the detection of a raised cosine pulse in Cauchy noise 61 PAM error probabilities for the DMF and the WPD detectors with optimum and unity weights , Block diagram of the data communication system The penalty T y p p C if* 1."^ ) as a function of the signal-to-nolse ratio it A. The penalty Trrnr. ( s,M,— ) of the optimum WPD WPD O detector for a raised cosine The penalty ( s,M,— ) of the optimum WPD detector for a half sinusoid The penalty ( s,M,£ ) of the - |s ±| weights WPD detector for a raised cosine The penalty ( s,M,£-) of the a± - IsJ weights WPD detector for a half sinusoid A 2 The penalty ( s,M,- ) of the « i - e± weights WPD detector for a raised cosine A 2 The penalty ( s,M,— ) of the w± - s ± weights WPD detector for a half sinusoid The penalty r_A_( s,M ) for a half sinusoid The penalty r g A S( s,M ) for a sinusoid The penalty r B p D ( s,M,— ) for a raised cosine The penalty r B p D ( s.M.y ) for a half sinusoid ACKNOWLEDGEMENT I wish to thank my advisor Dr. C.S.K. Leung for suggesting the topic and for rendering assistance during the course of the work. I am g r a t e f u l to the Natural Sciences and Engineering Research Council of Canada f o r the f i n a n c i a l support received under t h e i r graduate scholarship program and from Grant A-1731. A s p e c i a l thanks i s given to MDI Mobile Data International for the generous assistance received from the MDI Mobile Data International Fellowship i n Communications Engineering. I am also indebited to the Science Council of B r i t i s h Columbia f o r the support provided under the auspices of the Graduate Research and Engineering Technology (GREAT) Award. Thanks are due to the B r i t i s h Columbia Telephone Company for g i v i n g me a B r i t i s h Columbia Telephone Company Graduate Scholarship and to the Un i v e r s i t y of B r i t i s h Columbia f o r providing some funds for conference t r a v e l . F i n a l l y , I wish to express my appreciation to the o f f i c e s t a f f f o r t h e i r w i l l i n g assistance, encouragement and cheerfulness. 1 I INTRODUCTION 1.1. Motivation A fundamental problem In communication theory and design is how to best detect information signals in noise. Optimal detection methods are known but may be expensive or d i f f i c u l t to implement. Suboptimum detection procedures are employed in a number of diverse areas. Data recovery c i r c u i t s i n modems are frequently based on suboptimal procedures where c o s t - e f f e c t i v e implementations trade o f f complexity against performance. In underwater sound detection systems, arrays of receivers that nay be quite large are used to detect weak signals. For large arrays, the processing required can be reduced significantly by employing suboptimal procedures that have more modest computational requirements than the optimal processing. Digital receivers for systems employing tine or frequency spreading techniques have large time-bandwidth products and oust process a large number of samples. Simplified versions of digital Batched filters offer flexibility, reliability, speed, compactness and cost efficiency. In order to assess, compare and choose a l t e r n a t i v e detection procedures for • given application, the communications engineer requires knowledge of the performance of the schemes. This is one of the principal objectives of this research. The performances of a number of ad hoc detectors are analysed. The results are useful for performance evaluation of existing designs as veil ss for initiating new designs in data recovery and signal detection systems. 2 Another principal objective i s to investigate the optimality of the schemes employed. In particular,, given certain processing constraints, what are the optimum detection schemes? The results of this investigation w i l l provide theoretical limits for the performance that can be achieved by systems operating with particular hardware or processing limitations. The thesis research has industrial as well as academic significance. In the short term, the results w i l l provide answers that w i l l enable product designers to construct improved receiver subsystems. In the long term, the development of theory of suboptimal detection procedures entails dealing with some fundamental issues regarding the properties and the components of the losses incurred by these processes. A better understanding of signal detection w i l l come from the theory developed. 1.2. Scope of the Thesis This thesis considers suboptimum detection procedures. Both theoretical and numerical results are presented. With the exception of section 4 . 6 , the work deals with binary systems. The emphasis i s on evaluating the performance of the detectors with the performance measured in the signal detectability sense. That i s , the probability of error i s used as the performance criterion. The thesis i s concerned throughout with the processing of sets of signal samples. The analysis and results are cast in the context of data recovery in d i g i t a l communication receivers. The mathematical techniques and the results, however, are quite general and apply to a wide range of problems. The signal samples may, for example, be from a diversity d i g i t a l communication system, or a radar or sonar signal detection system. 3 The effects of hard-limiting on signal detectabillty are investigat- ed in d e t a i l . The relations of the detector losses to the number of samples processed, the signal-to-noise ratio, and the processing algorithm ere of central Interest. In order to gain insight into the principal mechanisms and relations of the detector losses, some simplifying assumptions have been made. In particular, i t has been assumed that there 1B negligible intersymbol interference (ISI) and that bit synchronization i s available. ISI i s not present in pulse detection systems. Furthermore, many communication systems operate with l i t t l e or no ISI. Low speed modems for telephone channels and radio modems are two examples. The present results also apply to ISI channels when the received signal has been equalized. The method used to obtain b i t synchronization i s not of particular interest here. Timing recovery c i r c u i t s can provide bit synchronization with less than a few percent j i t t e r . The effects of this j i t t e r on the penalties of the detectors Is of minor Importance as consideration of the binary eye Indicates. Some of the results of this thesis have been reported earlier in the journal papers, "On the Performance of Three Suboptimum Detection Schemes for Binary Signalling" 1 7 ) , •'Optimal Detection of Hard-Limited Data Signals i n Different Noise Environments" [ 6 ] , "Penalties of Sample- and-Sum and Weighted Partial Decision Detectors in Gaussian Noise" [ 9 ] , and the conference papers, "A Comparison of Three Suboptimum Detectors for Binary S i g n a l l i n g " [ 3 2 ] , "On Hard-Limiting in Sampled Binary Data Systems" 1 3 3 ] , "The Optimal Hard-Limiting Detector for Data Signals in Different Noise Environments" [ 1 0 ], and "Penalties of Weighted Partial Decision Detectors In Gaussian Noise"[34]. A 1.3. Review of Relevant Work A review of work relevant to this thesis is given in this section. Brief comments on the methods used and the results obtained are given for each reference. The reader may find i t helpful to refer to the glossary for explanation of some of the terms. Tozer and Kollerstrom [1] have considered the penalties of hard decision in the detection of binary antipodal signals in additive white Gaussian noise. In this analysis, short subsections of the signal are detected, giving a number of hard binary decisions. The subsections of the signal are constrained to be of equal energy and are detected by appropriate matched f i l t e r s . The data polarity i s recovered by using majority rule on the subsection decisions. This detection scheme is compared to the optimum detection achieved by using an analogue matched f i l t e r over the entire signal duration. It i s shown that for a large number of independent subsections and small signal-to-noise ratios a penalty of 1.96 dB i s incurred. Milutinovic [2,3,35,36] has described a suboptimum detection pro- cedure based on weighting partial decisions. This work considers binary signals in additive white Gaussian noise. The detection algorithm is based on two counters, B o and B^. The received 6 i g n a l i s sampled M times in the duration of one signalling element. Each sample i s compared with a threshold value and, depending on the outcome, counter B or B, Is incremented by a weight which depends on the sample index, o 1 After M par t i a l decisions, the transmitted signal is determined by a comparison of the two counters. The weights are chosen to be proportional to the distance between the two transmitted signals at the sampling 5 instant. Also considered is an algorithm based on binary partial decisions. In this scheme, a l l samples are weighted equally. The performance of the two suboptimum detectors i s compared to that of a d i g i t a l matched f i l t e r . The penalty i s computed for a particular set of antipodal signals for three values of sample size M. The results are example specific. It i s found that the penalty of the weighted partial decision detector is about 2 dB for a l l three values of M and increases slightly with increasing signal-to-noise ratio. The penalty of the binary partial decision detector i s about 3 dB for low signal-to-noise ratios and increases sharply as the signal-to-noise ratio becomes large. Lockhart [4] has considered replacing analogue f i l t e r and analogue detector c i r c u i t s in data receivers by d i g i t a l networks designed from truth table specifications. A method of compiling the truth tables from received signal probabilities is presented. The technique i s illustrated by an example. The detection of binary antipodal raised cosine signals in the presence of Gaussian noise i s examined. The received 6 i g n a l samples are assumed to be independent. The performance of the d i g i t a l network i s compared to that achieved by using a single sample detector. An error probability versus signal-to-noise ratio curve i s presented for each scheme. It i s noted that the proposed scheme performs better. It i s also noted that the truth table i s valid for a l l values of signal-to-nolse ratio and can be derived more directly by considering a hard-limited received signal f i l t e r e d by a nonrecursive matched f i l t e r . 6 The algorithm presented by Lockhart is an application of the maximum aposteriori probability (MAP) rule to the situation where there are a number of independent hard-limited received signal samples and the transmitter signal set, as well as the noise stat i s t i c s are known. The independence assumption w i l l require that the noise be white or that appropriate f i l t e r i n g of the received signal be done. The computationally direct method of considering the hard-limited, nonrecursively matched fi l t e r e d signal samples i s the weighted partial decision algorithm described by Milutinovic [2,3]. For the signalling waveform and the number of samples considered the MAP rule and the weighted partial decision algorithm yield the same truth table. In general, however, the MAP rule truth table w i l l depend on the signal-to-noise ratio and the two procedures w i l l not be truth table equivalent. Chie [5] has investigated a simplified d i g i t a l detector which performs only additions on the noisy signal samples. In this analysis, the signalling waveforms are antipodal nonreturn-to-zero (NRZ) pulses and the additive noise i s white and Gaussian. The detector p r e f i l t e r i s assumed to pass the signal without distortion. Hence, a l l the signal samples have the same magnitude and there i s no ISI. It i s also assumed that perfect synchronization i s available. Numerical results are presented for a typical implementation example configured with sixteen samples and a four b i t analogue-to-digital converter (ADC). The sensitivity of the detector performance to the number of samples processed, the number of ADC bits, and the ADC loading i s examined^ each Individually. For this example, It appears that four bit quantization performs almost as well as quantization with an infinite number of bits and that very little can be gained in performance by using more than sixteen samples. Chang [6] has also investigated the sample-and-sum detector examin- ed by Chle 15]. Again, antipodal NRZ signals in white Gaussian noise are considered. The performance degradation of the detector is related to the bandwidth of the prefilter, the sampling rate, and the number of quantizat- ion levels. The distortion of a single pulse resulting from the prefilter is dealt with in the analysis, but the effects of ISI are assumed to be negligible. Perfect synchronization is assumed throughout. It is con- cluded that, for the examples investigated, a prefilter bandwidth on the order of twice the bit rate is adequate and that three or four bit quantization is almost as good as infinite quantization. 1.4. Outline of the Thesis In this section, an outline of the thesis is given. The principal results of each chapter are described in turn. Chapter two Introduces three suboptlmum detection procedures: the Sample-and-Sum (SAS), the Weighted Partial Decision (WPD) and the Binary Partial Decision (BPD) algorithms. The bit error rate performances of these detectors are analysed for large time-bandwidth product conditions. This is first done for binary antipodal signals and It is then shown that the results may be generalized to arbitrary binary signals. The relation among the losses of the three schemes is presented. In chapter three, the performances of the three detectors for large time-bandwidth systems with dependent samples are considered. It Is shown that, In some cases, the losses of the auboptimum schemes can 8 be reduced by processing more, dependent, samples. The amount of the loss recoverable i s related to the prefl i t e r shape and the sampling rate. The structure of the optimum, minimum probability of error, detector for hard-limited samples i s presented in chapter four. Whereas previous chapters have dealt with large time-bandwidth product conditions and Gaussian noise, the results of this chapter are general and apply to an arbitrary number of samples and most common noise environments. The optimum detector for M-ary signalling with each received sample quantized to an arbitrary number of levels i s also derived. Again, the result i s valid for most noise distributions. In chapter five, the penalty associated with the use of the WPD detector in Gaussian noise i s examined in d e t a i l . The effects on the penalty of the signalling waveform employed, the number of samples processed, and the signal-to-noise ratio are examined. Two common ad hoc choices of weights are considered as well as the optimum weights. The performance degradations of the SAS and BPD detectors are analysed for arbitrary SNR's in chapter six. The effects on the penalties of the signalling waveform employed and the number of samples processed are also considered in detail. The relationship among the losses of the SAS, BPD and WPD detectors for low SNR and f i n i t e sample sizes i s derived. Finally, chapter seven gives a more detailed summary of the results of the thesis research and suggests some topics for further research. 9 I I PERFORMANCE COMPARISON OF THREE SUBOPTIMUM DETECTION SCHEMES FOR BINARY SIGNALLING 2.1. Introduction In this chapter, the problem of detecting one of two equally l i k e l y signals using d i g i t a l techniques i s addressed. The case of antipodal signals i s f i r s t considered and i t i s then shown that the results are readily gener- alized to arbitrary signals. The model considered i s shown in figure 2.1. Depending on the message «c{0,l} to be transmitted, s signal +s(t) or -s(t) Is sent over the additive white Gaussian noise (AWGN) channel. The two-sided power spectral density of the noise process o(t) i s assumed to be NQ/2. The signal s(t) Is assumed to be non-zero only In the time Interval I0,T] sec. •nd bandlimited to B Hzt. The received signal r ( t ) i s f i l t e r e d to remove excess out-of-band noise producing the s i g n a l v(t) which i s then processed by the detector. The detector samples the signal v(t) at a rate of 2B samples per second, yielding a total of H - 2BT noisy samples of the transmitted signal. A consequence of this sampling rate Is that the (Gaussian) noise samples i n v(t) w i l l be Independent 111]. Furthermore, in order to ensure that the transmitted signal s(t) Is essentially undlstorted by the receiver f i l t e r , we require that M - 2BT » 1. (2.1) Exactly how large M should be depends i n part on the shape of s ( t ) . If »(t) i s f a i r l y smooth (e.g. • sinusoidal wave) then a value of 10 would suffice. t S t r i c t l y speaking, a time-limited signal cannot be completely bandlimited. Bbwever, for practical purposes, a l l the signal energy w i l l l i e within a frequency range of B » * . LOWPASS FILTER - B B v(t) DETECTOR •Cm) Figure 2.1. Block diagram of the data r e c e i v e r . 11 On the other hand If s(t) hae 6 h a r p transitions (e.g. a square wave), then a larger value of about 100 i s required. These values of M are sufficient to T / |s(t)-s(t)|dt keep p • —~ , where s(t) denotes the output of the f i l t e r , to / |a(t)|dt 0 less than 1.52. It Is well-known [ 1 1 ) that the optimum detector for minimizing the A * probability of error P(e) • Pr { m * m) i n the above problem i s the di g i t a l matched f i l t e r (DMF). Its operation can be described as follows: Let V i " S i + n i * 1 ' 1 , 2 M ( 2 ' 2 ) IT denote the M samples of v(t), i . e . » V "̂~M̂  which are to be processed. M IT The values { B I ) I M ^ denote the samples of s ( t ) , i . e . s^ - and the { n^ }^_^are Independent Gaussian noise random variables with means 0 and 2 variances o • BN . The DMF computes n o Don ' lml V , < 2 - 3 ) A A and decides m - 0 if D^«„ > 0: otherwise i t declares m • 1. The resulting OPT probability of error is given by [11] /2E~ P(e) - Q C / y - * ) o (2.4) 12 T " 2. where E & / s 2(t)dt Is the energy of s(t) end Q( a ) & - ~ / e~* dx. 8 0 ^5* a Equation (2.4) gives the error p r o b a b i l i t y for an analogue matched f i l t e r . I t i s also v a l i d for the DMF when M i s large. We note that the DMF requires M m u l t i p l i c a t i o n s and (M-l) additions. A number of suboptimal schemes I 1-10,12,13 1 have been proposed which have more modest computational requirements. These include the Sample-and- Sum (SAS), the Weighted P a r t i a l Decision (WPD) and the Binary P a r t i a l Decision (BPD) detectors. Previous analyses of these suboptimal schemes have been confined to s p e c i f i c s i g n a l l i n g waveforms s ( t ) and s p e c i f i c (small) values of M. In the following sections, an an a l y s i s of the penalty incurred by each of these schemes f o r an a r b i t r a r y s i g n a l l i n g waveform and large values of M i s given. The penalty i s defined as the increase i n s i g n a l energy required by a suboptimum detector i n order to achieve the same error p r o b a b i l i t y as a d i g i t a l matched f i l t e r ( i n the large sample case, t h i s i s the same as the error p r o b a b i l i t y of the analogue matched f i l t e r ) . Examples i l l u s t r a t i n g how the penalties vary with M are also included. Note that the model used implies a small s i g n a l - to^noise r a t i o c ondition. The r e s u l t s , therefore, are v a l i d f o r small signal-to-noise r a t i o s . This r e s t r i c t i o n i s removed in chapters 5 and 6. (2.5) 2.2. The Sample-and-Sum (SAS) Detector The SAS detector [ 5 , 6 , 1 3 ] computes the quantity D S A S M Y i •«nC*i) (2.5) f+1 i f *>0 < 0 i f * - 0 , L-i i f *<o where sgn(x) 0 if « - 0 , a n d declares a - 0 if D ^ > 0. Otherwise, * - 1 13 i s decided. Comparison of (2.3) and (2.5) shows that the SAS detector avoids multiplication by not weighting the samples of the received signal. The resulting penalty i s now analysed. Clven • - 0, I.e. 4«(t) Is transmitted, the nean of i s given by w j , Ki (2-6> and the variance of D_._ 16 given by aAs 2 u 2 ou - Mo ^SAS « MBN A ° - T ^ . C2.7) An error will be nade if D„ir. < 0. We note that since De._ is the sum of SAb a<va Gaussian random variables, It is Itself a Gaussian random variable. Thus, M I K r P ( e f m - O ) - Q ) *BAS Q f 1 1 ) . (2.8) • N /2T o By symmetry, P(e) - P(e|»-0) - P(e|«-1). Vote also that 14 M _ T lim I (s J - / js(t)|dt. Therefore, as M< M-M» i - l 0 He) - Q (/p<|s(t>|>) , (2.9) 1 T where <|s(t)|> - / js(t)jdt represents the average magnitude of the T 0 signalling waveform s ( t ) . If we define E o - 5- , (2.10) T{<|s(t)i>r equation (2.9) can be rewritten as '2E 8 ^ (2.11) PC) • 9(/^-} By comparison of (2.11) with (2.4), i t can be seen that to achieve the same value of P(e), the SAS detector uses a times the energy required by the DMF detector. For a constant s ( t ) , o - 1 as might be expected. (It can be shown using Schwarz'6 Inequality [11] that the minimum value of a i s 1.) However, 2 for a sinusoidal signalling waveform, a " i /8 or 0.912 dB. 15 2.3. The Weighted Partial Decision (WPD) Detector In the WPD detector (2], the necessity for multiplication i s avoided by ignoring the magnitudes of the received signal samples and using only their polarities. The decision i s based on implementation, two accumulators AQ and A^ could be used. If v^ > 0, A^ is incremented by 8^ and i f v^ < 0, then   i s incremented by s^. In the (unlikely) event that v^ - 0, neither accumulator i s incremented. After a l l N samples have been processed, the contents of A^ and   are compared to A determine m. We now proceed to calculate P(e) for the WPD detector. Since by symmetry, P(ejm-O) • P(e|m*l), we assume with no loss of generality that D - 0, i . e . +s(t) i s sent. In this case, we can rewrite (2.12) as (2.12) A A If D > 0, m - 0 i s declared; otherwise m - 1 i s declared. In an actual M (2.13) where the partial decision random variable D. i s defined by (2.14) It follows that P(D 1»1) - l- p 1 and P ^ —1) - where - Q i y i 1 ^ , ! ) . (2.i5) 2 The mean and variance of D^Js^j are given by (l-2p^)Js^| and 4p^(l-p^) Js^ | respectively. Under certain conditions which are satisfied i n this case, i t can be shown [14] that as M-̂ °°, D^^ has the asymptotically Gaussian distribution M M n( I (l-2p )|s |, I 4p (1-p )|s J 2 ) . 1-1 1 1 1-1 Using t h i s r e s u l t , i t i s shown i n Appendix A f o r small signal-to-noise r a t i o s that the p r o b a b i l i t y of error f o r the WPD detector as M** i s given by 4E 8 ^ (2.16) o It Is interesting to note that for large values of M, the WPD detector uses y tlaes the energy required by the DMF detector to achieve the sane value of P(e). This penalty of 1.96 dB Is independent of the specific signalling waveform used, i n contrast to the SAS detector i n which the penalty does depend on the shape of s ( t ) . 17 2.A. The Binary Partial Decision (BPD) Detector The BPD detector can be considered as a special case of the WPD detector i n which the Information regarding the magnitude of the sample M values {s^)^ m^ i s not used. This results In a simple implementation in which a counter ( I n i t i a l l y reset to 0) i s Incremented or decremented by 1 depending on the polarities of v^ and s^. Specifically, define i " l where Di - sgnCv^ • sgn ( s ^ , as in (2.1A). Then i f D B p D > 0 , m = 0 is A declared; otherwise m - 1 Is decided. It can easily be seen that the detector w i l l make an error i f and only i f a majority of the M samples have had their polarities reversed by the channel noise. Proceeding as i n section 2.3, i t can be shown that as M+«°, ̂ BPD * i a s t n e aBYmot-°ticaHy Gaussian distribution M M n( I (l-2p ) , I 4 p . ( l - P l ) ) . i - l 1 i - l 1 1 Using this result, i t i s shown i n Appendix B that the probability of error for the BPD detector as M+- i s given by p ( e > - o ( / n r . ) • o (2.18) 18 where a i s as defined in (2.10). Compared with the DMF detector, we see that the BPD detector Is 10 l o g 1 0 ( - ~ ) - 1.96 + 10 log 1 ( )a dB less e f f i c i e n t . We note that the penalty can be interpreted as consisting of 2 components: 1.96 dB Is lost because decisions are based only on the polarities of the received signal samples (not on their magnitudes) and 10 l o g ^ a dB i s lost because equal weights are being given to received signal samples even though the sample corresponding to a large js^ J i s less l i k e l y to be i n error than the sample corresponding to a small \ B ^ \ . These 2 components correspond to the losses resulting from the WPD and the SAS detectors respectively. 2.5. Generalization to Arbitrary Signalling Waveforms The results of the three preceeding sections can be easily generalized to arbitrary signalling waveforms. Let a Q(t) and 8j(t) denote any two finite-energy waveforms defined on [0,T]. This set of signals can be transformed into a set of binary antipodal signals by defining s (t) • a (t) . . ( t ) - . o ( t ) - _° and e n ( t ) + a (t) s j ( t ) - Bj(t) - - ^ j - ^ This transformation subtracts the arithmetic mean of the two signals from each signal. The results derived in the previous sections then apply directly to { *o^t^» }» where E G is to be interpreted as the energy in sl(t) or s!(t). The energy E of ( 2 . 4 ) , (2.11), (2.16) and (2.18) can u x s be related (2.19a) (2.19b) 19 to the signals SQCO and s^(t) by noting that T E - / [ s ' ( t ) ] 2 dt s ' 0 0 i T 4- [ E + E c - 2 J s n ( t ) s.(t) dt ], (2.20) 4 i s Q s 2 ' Q 0 1 where E , i-0,1, i s the energy of s.(t). Of course, i f s.(t) • - s n ( t ) , 6^ 1 1 U T E - E - E . On the other hand, i f E - E and / B-(t) s.(t) dt - 0 s s 0 6 l s 0 S j ' Q 0 1 E 80 corresponding to binary orthogonal signalling, E g " leading as expected to a los6 of 3 dB relative to binary antipodal signalling. 2.6. Discussion The asymptotic losses associated with the use of three suboptimum detection schemes have been analysed. Table I gives a summary of the results as applied to three specific signalling waveforms. 20 Losses relative to DMF detector (dB) Waveform SAS WPD BPD Square Sinusoid Raised Cosine 0 0.912 1.76 1.96 1.96 1.96 1.96 2.87 3.72 Table I - Asymptotic losses for suboptimum.schemes. As Indicated below, these asymptotic values are reached quite rapidly. The losses for the SAS, WPD and BPD detectors are plotted against the sample size M in figures 2.2 - 2.5. In figures 2.2 and 2.3, a sinusoidal signalling waveform Is assumed, whereas a square signalling waveform is used in figures 2.4 and 2.5. For each figure, a target value of P(e) is used. In figures -3 -7 2.2 and 2.4, this value Is 10 and in figures 2.3 and 2.5, it is 10 . The losses represent the Increase In E required to achieve the same 8 target value of P(e) using the suboptimum detector and equation (2.4) respectively. The plots in these figures were obtained numerically using a VAX- 11/750. Losses were calculated for each suboptimum scheme for different (odd) values of M. It should be noted that the distortion of the signalling waveform which would result for small values of M was taken Into account In these calculations. Details concerning the computation of P(e) are given in Appendix C. Recall that the samples are spaced TT sec. apart. The first O b i 1 1 10 100 1000 10000 SAMPLE SIZE M Figure 2.2. The penalty as a function of the (odd) sample s i z e M . The s i g n a l l i n g waveform i s a single sinusoid. Curves. B, D and F (A, C, E) are for the best (worst) choice of sampling s t a r t i n g time f or the BPD, WPD and SAS detectors r e s p e c t i v e l y . SAMPLE SIZE M Figure 2.3. The penalty as a function of the (odd) sample siz e M. The s i g n a l l i n g wave i s a single sinusoid. Curves B, D and F (A, C, E) are for the best (worst) choice of sampling s t a r t i n g time for the BPD, WPD and SAS detectors r e s p e c t i v e l y . Figure 2.h. The penalty as a function of the (odd) sample siz e M. The s i g n a l l i n g waveform i s a square-wave. Curves A and B are for the worst and best choice of sampling s t a r t i n g time respectively for the BPD and WPD detectors. Curves C and D represent the worst and best choice of sampling s t a r t i n g time respectively for the SAS detector. Figure 2.5. The penalty as a function of the (odd) sample s i z e M. The s i g n a l l i n g waveform i s a square-wave. Curves A and B are f o r the worst and best choice of sampling s t a r t i n g time respectively for the BPD and WPD detectors. Curves C and D represent the worst and best choice of sampling s t a r t i n g time respectively for the SAS detector. sample can be chosen anywhere In the interval (0 , - g ) . The difference i n losses obtained by selecting the best and the worst times for the f i r s t sample i s also indicated i n the figures. As would be expected, the time of the f l r 6 t sample has l i t t l e effect on the losses for large values of M. Figure 2.2 shows that with the sinusoidal signalling waveform and P(e)«10 , for M greater than about 15, the S A S , WPD and BPD detector losses are within 0.05, 0.03 and 0.25 dB of their asymptotic values. For P(e)«*10-7, figure 2.3 shows that the corresponding figures are 0.1, 0.2 and 1 dB respectively. For square wave signalling, the WPD and BPD detectors are _3 equivalent. From figure 2.4, i t can be seen that for P(e)=10 , and M greater than about 15, the SAS and WPD detector losses are within 0.5 dB of their asymptotic values. For a smaller value of T(.e)B10 \ figure 2.5 indicates roughly the same behaviour. 26 III EFFECTS OF OVERSAMPLING ON THE PERFORMANCE OF THREE SUBOPTIMUM DETECTION SCHEMES 3.1. Introduction In chapter 2, the penalty incurred in the use of three suboptimum detectors was analysed. The f i l t e r characteristics and the sampling rates used i n the analysis guaranteed the independence of the received signal samples. In this chapter, the effects of sample dependence on the penalties are examined. The effects of oversampling are analysed for Butterworth, Gaussian and ideal lowpass f i l t e r s as well as for a cascade of N identical poles. The receiver model i s shown in figure 3.1. The case of antipodal signals w i l l be analysed but the results can be generalized to arbitrary binary signals by using the transformation of section 2.5. Depending on the message me{0,l} to be transmitted, a signal +s(t) or -s(t) is sent over an additive white Gaussian noise (AWGN) channel. The two-sided power spectral density of the noise process is NQ/2. The signal s(t) Is assumed to be non-zero only in the time Interval [0,T] sec. and bandlimited to B H z. The received signal r(t) i s f i l t e r e d to remove excess out of band noise to produce the signal v(t) which i s then processed by the detector. The lowpass f i l t e r has 3dB cutoff frequency B. The detector samples the signal v(t) at a rate of 2cB samples per second yielding a total of M - 2cBT noisy samples « + of the transmitted signal where f - f ( l x ~ ^' 5^) » n d t h e ^ n i ^ i - i a r e Gaussian (not necessarily independent) noise random variables with means 0 and variances a 2 . The parameter c may be thought of as the oversampling factor, n Increasing c gives more, dependent, samples for processing. There are a LOWPASS FILTER V . - B B v ( t ) DETECTOR Figure 3 . 1 . Block diagram of the data r e c e i v e r . 28 maximum of M • 2BT independent samples available from an ideal lowpass f i l t e r corresponding to c • 1 and sampling frequency f • 2B [11]. The normalized sampling rate is c • f g/2B. In order to ensure that the transmitted signal i s essentially undistorted by the receiver f i l t e r , we require that M = 2BT » 1. (3.1) The ideal lowpass f i l t e r admits noise power 0 Q 2 • NQB. In general, °n 2 " W ( 3- 2 ) where Y n " 2T ^ |H(f)| 2df (3.3) is the normalized noise bandwidth of the f i l t e r and H(f) i s the amplitude response of the f i l t e r . One has [18,19] |H(f)| 2 - [1 + (f/B) 2 1 * ] * 1 (3.4a) N-th order Butterworth | H ( f ) l 2 - [1 + ( f / B ) 2 ( 2 1 / N - 1 ) ] " N (3.4b) cascade of N identical poles | H ( f ) | 2 - e-(f/B) 2ln2 ( 3 4 c ) Gaussian for the N-th order Butterworth, the cascade of N identical poles and the Gaussian f i l t e r respectively. Using (3.4a) - (3.4c) and (3.3) gives Y n = 2N sln[*/2N] ( 3 * 5 a ) N-th order Butterworth Y - 1'3»5. (2N - 3)u ( 3 > 5 b ) 1 1 / 2 1 / N - i V (H-1) I cascade of N identical poles Gaussian for the N-th order Butterworth, the N-pole cascaded and the Gaussian f i l t e r respectively. 3.2. The Sample-and-Sum (SAS) Detector with Dependent Samples The SAS detector with dependent samples computes the quantity M D » £ v sgn(s.) as described previously in section 2.2. The SAS ± m l I I penalty may be determined by proceeding as was done there. Given that m • 0 i s transmitted, the mean of D g A S is given by _ M DSAS " ^ ' 8 i l a n d t h e v a r i a n c e o f DSAS 1 8 8 I V E I 1 by 30 « V C - E « D S A S " DSAS>21 * v̂gnCê  - j IsJ)2] (3.6) SAS i«l i-1 where E[x] denotes the expected value of x. The square in (3.6) may be expanded and the terms rearranged to give M—1 M a 2 - o * M { l + ' ! E I ign(s . )r ( J - i ) j (3.7) SAS D i - 1 J-i+1 where r ( j - i ) " E[n .n ]/E[(n ) 2] is the normalized autocorrelation of the n j I I noise which is assumed stationary. If the signalling waveform s(t) is continuous on 0 < t < T and single- phase, i.e. s(t) > 0 for 0 < t < T, then (3.7) gives SAS i = l j - 1 Using the result [20] that . M-l i 11* £ I X rB(J) - I rn(j) M-~» " i - 1 j - 1 ° J - l whenever the series (3 . 9 ) 31 is convergent with (3.8) gives for large values of M S A S - a (3.10) when the series (3.9) converges. Let the term split-phase refer to a signal s(t) which is continuous on the interval 0 < t < T and which satisfies the conditions signum{s(tA)} • signum{s(t B)} - -1 for (0 < t>A < T/2, T/2 < t g < T) and s(T/2) - 0. Then result (3.10) i s valid for split-phase signals. This is proven from (3.7) by proceeding as in the single-phase case. The following analysis applies to a l l signalling waveforms which satisfy (3.10). Gaussian random v a r i a b l e s . Therefore, (3.1), (3.2), (3.10) and (2.10) give The random variable D S A S is Gaussian since i t i s the sum of jointly P(e) - P(e|m-0) - P(e|m-1) } (3.11) 32 Comparison of (3.11) with (2.4) shows that to achieve the same error probability as the optimum detector, the SAS detector with dependent samples requires aT^ more energy where r - Y U + 2 I r ( i ) } / c . (3.12) The loss represented by is in addition to the o loss described in section 2.2. One observes from (3.12) that T has two components. The factor Y n n . arises because a f i l t e r with f i n i t e r o l l - o f f admits more noise than an CD ideal f i l t e r with i n f i n i t e l y sharp cut-off. The sum £ r n ( l ) results from i - l the dependence of the samples. ( sin(ni/c) this case, Y - 1 and r (i) - — ; — . , x . Using the fact that the Fourier * 'n n v (m/c) ° sine series representation for the function f(x) - n - %, 0 < x < it, is [21] It i s interesting to consider for the ideal lowpass f i l t e r . In 8 in ix f(x) ~ 2 I l l f i - l gives for x - K/C with c > 1, . r s i n ( t t l / c ) _ . 2 J 1 O c i / c ) C 1 ' where convergence is guaranteed by a Fourier Theorem. Hence, T - 1 for a l l n values of c > 1 for the ideal lowpass f i l t e r . That i s , processing more than the maximum number of independent samples neither improves nor deteriorates the performance when (3.1) i s satisfied. Note also that the loss, a, incurred by not weighting the samples is not retrievable in whole or in part by oversampling. In the general case, the normalized autocorrelation r (t) of f i l t e r e d n white noise is related to the f i l t e r characteristic by the Wiener-Khintchine theorem [22]. That i s , r n ( t ) is the normalized inverse Fourier transform of the squared magnitude frequency response of the f i l t e r . Starting from (3.4a) - (3.4c) one may derive N r (T) - sin(n/2N) J exp[-2nB|t|sin ( n A=l 2A-1 2N Tt)] sin{ 2JL-1 2N n + 2itB|x|cos( 2JI-1 2N Nth order Butterworth (3.13a) r n < * > cascade of N identical poles (3.13b) r n ( T ) -T 2* 2B 2/Jln2 e Gaussian (3.13c) for the N-th order Butterworth, the cascade of N identical poles and the Gaussian f i l t e r respectively. Figure 3.2 presents r (T) as a function of 2BT Figure 3.2. The normalized autocorrelation function * n ( T ) of white noise f i l t e r e d by Butterworth lowpass f i l t e r of order N. The ideal lowpass f i l t e r corresponds to N =» «°. 35 BT for the Butterworth (N - 1-4) and Ideal lowpass filters. Figure 3.3 shows rn(t) for the cascaded pole (N • 1, 2, 4, 6) and Gaussian filters. The terms i n the sum of (3.12) occur at time instants - l/2cB. Therefore, equations (3.13a)-(3.13c) combined with (3.12) •nd (3.5a)-(3.5c) yield, after some manipulations, - — sin a. N sinfa,+ - cot i . ) - e C sin a. n ' 2cNsin(*/2N) l l + B l n i 2 N j * Z7^~~t ^ r * \ ' ' JM. cosh'̂— sin ŝ j - coŝ— cos â) 21-1 • A • ic N-th order Butterworth (3.14a) r - 1'3»5« ... (2N-3)n ... 2(N-1)1 r -ih n / i/N •» M (2N-2) 1 £ /21/N-1 2N(N-l)!c 1 1 " r 1 (2N-k-2)!(21h)k , h 1 k!(N-k-l)l >» C/2T7T7 ( 3' 1 4 b ) cascade of N Identical poles x 1 Gaussian (3.14c) The quantity Is plotted as • function of the normalized sampling rate f /2B - c in figures 3.4 and 3.5 for Butterworth and cascaded pole filters respectively. Also shown In figure 3.5 is the curve for the Gaussian filter Figure 3.3. The normalized autocorrelation function r n ( T ) of white noise f i l t e r e d by a cascade of N identical poles. The Gaussian f i l t e r corresponds to N = ».  3.0 r (dB) n 2 3 5 10 Normalited Sampling Rate f / 2 B Figure 3 . 5 . The additional penalty r n for the SAS detector with p r e f i l t e r consisting of a cascade of H identical poles. The Gaussian prefilter corresponds to N • ». u > OO 39 which corresponds to a cascaded-pole f i l t e r with N - ». In a l l cases, T decreases with c and lim T • 1. Using the Wlener-Khintchine theorem, i t n c+" n ° ' can be shown that lim r - 1 for any f i l t e r provided that r (T) i s c*» n n integrable. 3.3. The Weighted Partial Decision (WPD) Detector with Dependent Samples The WPD detector with dependent samples bases i t s decision on M M DWPD " I 8 g D ( V i ) , 8 i " I D i ' 8 i l ( 3 - 1 5 )  U i-1 1 1 i-1 1 1 as described in section 2.3. The error probability i s found by proceeding as previously. That i s , the mean and variance of D^^ are derived and a central limit theorem is used to approximate the error probability for large values of M. The par t i a l decision random variables D̂  are now permitted to be dependent. In order to proceed, we postulate that a central limit theorem holds for certain sums of dependent random variables. Many central limit theorems for dependent random variables formalise in some sense a heuristic notion that one expects a central limit theorem to hold i f the random var i - ables behave more like independent random variables the further they are separated [23]. The dependent random variables considered here behave in this fashion. Computer simulations are used to test the validity of the postulate and to i l l u s t r a t e how the penalty varies with M. When a general f i l t e r and dependent samples, as described by (3.1) and (3.2), are considered equation (2.15) must be replaced by p i = P r ( n i > l S i l ) " Q ^ y / ~ Y p ^ ^ i 1 1 ' ( 3' 1 6 ) The mean and variance of D^|s^| are again given by (l-2p^)|s^| and A p ^ ( l - p ^ ) | | 2 respectively. The mean of D^pp is found from (3.15) DWPD- X CX-2p±>|.±| (3.17) 1=1 and the variance of n y p D i s M M UWPD i - l i - l M M M I |. | 2 + I I |s ||s | {E[DD -DD ]} . (3.18) i - l " i 1 i - l J - l x J i * j It i s shown in Appendix D that for large values of M E[DtD ] - - -| arcsin { r n ( j - i ) } (3.19) where the noise process i s assumed to be stationary. Then (3.18) can be rewritten as 41 M 2 M M 1 - I |s±|2 + - I I |s ||s |arcsin{r (j-i)} . (3.20) WPD 1=1 1 1 71 1=1 j*»l n i * j By proceeding as in Appendix A with p^ given by (3.16), i t may be shown that for large values of M M ME I - J I - i l ' - T 1 ( 3 - 2 1 > i=l i and / 4cM \ * V n T D » /_-l£EL- E . (3.22) WPD / IN v  s Furthermore, i t i s proven in Appendix E that as M-*89 Combining (3.21) and (3.23) with (3.20) and (3.22) gives WPD WPD 4E c 8 1 (3.24) Applying a central limit theorem to the sum of (3.15) and using (3.24) gives the error probability P(e) 4E c 8 u l W 1 + " I arcsin{r n(i)}) (3.25) 1=1 Comparison of (3.25) with (2.4) shows that the penalty of the WPD detector with dependent samples is it/2 where r n " I I «csln{r n(l)}]/c . (3.26) The loss represented by Tn is in addition to the n/2 or 1.96 dB loss of the WPD detector with independent samples. Note that 2/it arcsin{r n(i)} i s the normalized autocorrelation function after hard-limiting of a random variable possessing normalized autocorrelation function r n ( i ) [24]. Equation (3.26) i s , therefore, analagous to equation (3.12). The function 2/ir arcsin{r n(BT)} i s shown in figures 3.6 and 3.7 for Butterworth and cascaded pole f i l t e r s respectively. The additional penalty r i s plotted as a function of the normalized sampling rate in figures 3.8 and 3.9 for Butterworth and cascaded pole f i l - ters respectively. The values r (i) are determined from equations (3.13a) - n (3.13c) with r (i) - r (i/2cB). Observe that for high sampling rates, the n n penalty for a low order f i l t e r i s less than that for a high order f i l t e r . For example, in figure 3.8 at f /2B • 15, T « -1.5 dB for N - 1 while s n T - -0.9 dB for N - °°. A similar observation was made in [25], in the n -0.25 I I I I I I 0 1 2 3 4 5 2BT Figure 3.6. The normalized autocorrelation function of white noise after lowpass filtering and hard-limiting. The filter used Is N-th order Butterworth. The ideal lowpass filter corresponds to N • 0 0. -p- 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 BT Figure 3 . 7 . The normalized autocorrelation function of white noise after lowpass f i l t e r i n g and hard-limiting. The f i l t e r used is a cascade of N identical poles. The Gaussian lowpass f i l t e r corresponds to K * •, Figure 3.8. The additional penalty r of the WPD and BPD detectors. The prefliter is H-th order Butterworth lowpass. The ideal lowpass filter corresponds to H • "• 46 r (dB) n Konnallred Sampling Kate f g / 2 B Figure 3.9. The additional penalty T of the WPD and BPD detectors. The p r e f i l t e r i s a cascade of F identical lowpass poles. The Gaussian lowpass corresponds to N • ". 47 context of polarity coincidence array detectors. 3.4. The Binary Partial Decision (BPD) Detector with Dependent Samples The BPD detector with dependent samples computes A M D„™ - I D . . (3.27) BPD / , i 1=1 The probability of error i s determined by proceeding as in section 2.4, For large M, PBPD / 4cT{<ls(t)|>} 2 m I 4cEs B ? D «csin{r n(i)}] / u N o y n a [ l + ^ I arcsin{r n(i)}] i-1 V i-1 (3.28) where a has been previously defined in (2.10). Assuming a central limit theorem holds for the sum in equation (3.27) and using (3.28) gives for the error probability r / *CEB W I P B P D ( e ) " Q 1 / r A r r n f ( 3 ' 2 9 ) 1/ - , o T n a t l + i : X « " l l l { r n ( 1 ) H J The penalty of the BPD detector with dependent samples Is found by comparing (3.29) with (2.4). It i s o w/2 T where T i s given by (3.26). That i s , the n n additional penalty i s the same for the BPD and WPD detectors and is independent of the signalling waveform. 3.5. Conclusions The asymptotic losses of three suboptimum detection schemes with dependent sampling have been analyzed. In order to verify the central limit theorem postulated in section 3.3 and to ill u s t r a t e how quickly the asymptotic values are reached, we have simulated the WPD detector with dependent samples. Coloured Gaussian noise samples were generated using the method of reference [26], The signal-to- _3 noise ratio required to attain P(e) • 1.00 x 10 i s determined from figure 3.8 and equations (3.25) and (2.4). Using a f i r s t order Butter- worth p r e f i l t e r , the simulated error probabilities for f s/2B • 1 were 1.13 x 10" 3 and 1.01 x IO - 3 for M - 9 and M * 31 respectively. For -3 f /2B - 3 and M « 27, 93 and 303 the P(e) values were 1.77 x 10 , s -3 -3 1.14 x 10 and 1.05 x 10 respectively. It was noted earlier that better performance can be achieved by using a f i r s t order lowpass f i l t e r rather than a fourth order f i l t e r . This has also been verified by simulation. Again, the SNR _3 required to achieve P(e) • 1.00 x 10 i s determined from the present results and used in the simulator. The analysis indicates that 0.48 dB greater SNR i s required for the fourth order f i l t e r when both f i l t e r s are Butterworth and f /2B = 15 for M = 1515. The s -3 -3 simulated error probabilities were 1.09 x 10 and 1.04 x 10 for the f i r s t and fourth order f i l t e r s respectively. Note that an -3 -4 increase of 0.48 dB decreases P(e) from 1.00 x 10 to 5.46 x 10 IV OPTIMAL DETECTION OF HARD-LIMITED DATA SIGNALS IN DIFFERENT NOISE ENVIRONMENTS 4.1. Introduction A number of digital techniques for detecting binary antipodal 6 ig rials are based on examining the polarities of the received signal samples and ignoring their amplitudes. The weighted partial decision (WPD) and binary partial decision (BPD) detectors analyzed in chapters 2 and 3 are two examples. In this chapter, the structure of the optimum detector DQJJ ̂  for the hard-limited samples is derived. Its performance is compared with those of some commonly used ad hoc detectors in both impulsive and Gaussian noise environments. For the Gaussian case, the performance is also compared with that of the optimum detector D̂p̂  which operates directly on the unquantized received samples. The generalization of the D0pT ̂  detector to M-ary signal- ling with each received sample quantized to an arbitrary number of levels is also examined. The optimum, minimum probability of error, receiver for this case is derived. 4.2. Derivation of the Optimum Detector for Hard-Limited Samples In this section, we derive the optimum processing for a number of hard-limited samples. For ease of discussion, it is assumed that the samples come from one of two antipodal signals that have been corrupted by additive channel noise.* Depending on the message The analysis Is extended to arbitrary signalling schemes in section 50 me{0,l} to be transmitted, a signal +s(t) or -s(t) i s sent over a noisy channel. The detector decides which message m, me{0,l} was sent on the basis of the hard-limited samples. The optimum detector D minimizes OPT , HI* the probability of error P(e) * Pr( m ¥ m ). Assume that the transmitted signals ±s(t) are time-limited to the interval 10,T]. If s(t) is sent (corresponding to message m-0), then v i " + + nj • 1 " lf2....»Ne (4.1) N .i~0 5 IT where {v^} 6 denote the Vg samples of v(t), i.e. - v — K * ' ), to be N N N i-1 processed. The values {s.} 6 denote the samples of +s(t) and {n.} s 1 i-1 1 i-1 represent outcomes of independent noise random variables (r.v.'s). The noise is assumed to possess an even probability density function. If the message Is m-1, then -s(t) Is sent, and V j - - S j + n̂, 1-1,2,...,Ng. Each sample is hard-limited by the detector according to {I if sgn ( V j ) - sgn(s1) # 0 if sgn(v1) - 0 or sgn(&1) - 0 (4.2) otherwise if x > 0 where sgn(x) - 0 if x - 0 if x < 0 The probability that sample v± Is of opposite polarity to the transmitted sample +ei or -s1 (ŝ O, v^O) Is given by P j - Pr̂ -llm-0) - Pr̂ -llm-l) - Pr^ > \ B ± \ ) . (4.3) The minimum P(e) detector corresponds to the maximum a posteriori (HAP) decision rule which states that m-0 is chosen i f N N Pr({d } 8 |m-0) Pr(m-O) > Pr ({d > 8 |m-l) Pr(m-l) (4.4) 1 1-1 1 1-1 where d^ denotes a particular outcome of the random variable D^. In the case of equally l i k e l y messages, Inequality (4.4) reduces to N N Pr({d.) 6 |m-0) > Pr({d,} 8 |m-l) . (4.5) X 1-1 1 i - l Let A 1 denote the set of a l l I's such that d^-1 and A_^ denote the set of a l l A i's such that d^—1. Then, the optimum decision rule Is to choose m-0 i f n (i-p.) n p > n p. n U-P.). (4.6) icAl 1 icA_ x 1 icAj 1 IcA_ 1 From (4.6), the optimum decision rule can be stated as follows: form the s t a t i s t i c D O P T , H L £ j ^ i * ' ( A ' 7 ) i f d-__ „. > 0, m-0 is declared; otherwise, m-1 Is declared. OFT,HL Various weighted partial decision (WPD) detectors for hard-limited signals have been proposed (2] which have the following general form. The . N s N s test s t a t i s t i c i6 d„__ - J d.u., where {u. } are weights assigned to the W t i - l 1 1-1 different samples. The WPD detector chooses m-0 i f & ^ > 0 and m-1 i f d w p D < 0. The weights Wj-1, w 1"l s 1l a n d w i " 8 i * a r e o f t e n u 8 e d » 1* c5 n ^e s e e n 5 2 from equation ( 4 . 7 ) t h a t the optimum weights are given by " An (-p-i) . ( 4 . 8 ) In the next three s e c t i o n s , the de t e c t o r DQPJ h l i s compared to these schemes and, i n the case of Gaussian noise, to the optimum de t e c t o r based on the N unquantized samples {v.} 1 i - 1 4 . 3 . Optimum Weights f o r Low S l g n a l - t o - N o i s e R a t i o s In t h i s s e c t i o n I t I s shown t h a t , f o r most common noise models, the optimum weights as given by equation ( 4 . 8 ) are w e l l approximated by = |s^| when the s i g n a l - t o - n o i s e r a t i o (SNR) i s low. Assume that the noise v a r i a b l e s N {NjJ 6 have an I d e n t i c a l p r o b a b i l i t y d e n s i t y f u n c t i o n (pdf) which Is even. 6 i - 1 T h i s holds f o r most commonly used noise models such as the Gaussian, Laplace or Cauchy p d f s. Let the cumulative d i s t r i b u t i o n f u n c t i o n (CDF) be denoted by Fn(*)» Suppose that F n ( • ) can be represented by a Ma c l a u r i n s e r i e s expansion, i . e . F n < " > - j 0 F n k < ° > £ < 4 ' 9 ) where F n (0) denotes the k-th d e r i v s t l v e of F n ( o ) evaluated at a-0. One may combine ( 4 . 3 ) , ( 4 . 8 ) and ( 4 . 9 ) to o b t a i n # F ( | s . | / o ) 5 3 1 r- k < l » i l / " > k JUiI — i — 1 (A.10b) I r k ( l 6 i l / 0 > k ' 4 - i O ( o ) 1 2 k-i n - k < i » , i M k 2 I F n ( 0 ) El - An ( 1 4 — ^ - ) CA.lOc) V i ? r " n n ( l ' ' | / o ) 2 - J / n ^ - k l - In (4,10b) the fact that the noise pdf i s an even function has been used and o is the noise scale parameter. For low signal-to-noise ratios, i.e. |s^|/o << 1, one has w* - A F n 1 ( 0 ) IsJ/c, (A.11) since An(l + x) - x - -| x 2 + -| x 3 + ..., for -1 < x < 1. In (A.11) i t has also been assumed that F A ( 0 ) ^ 0 , as i s the case for a Gaussian, Laplace n or Cauchy pdf. Finally, one notes that scaling of the weights by a * constant does not affect the decision rule. Hence, may be chosen to be approximately equal to |s i ' - 4.4. Optimum Weights for High SNR's In the case of high SNR's the optimum weights depend on the noise pdf. The Gaussian, Laplace and Cauchy distributions w i l l be considered i n turn. 54 4.4.1. Gaussian Noise Distribution For a Gaussian noise r.v. of variance a 2, it follows from equation n j wnere v\«U - • JTh a s 2 1' 1 1 — » 1, p. Is well approximated [31] by r——r e x P ( )• From (4.8) ti i ^ i i 1 i 1 2o 2 M\K 1 r" -X2/2 (4.3) that p. - Q M - where Q(a) - J e ' dx. When the SNR Is high, 1 °n m a n °n the optimum weights are approximated by s 2 * 1 w. . (4.12) 1 W Since scaling the weights by a constant doe6 not affect the decision rule * 2 defined by (4.7), can be chosen to be approximately equal to ŝ  . 4.4.2. Laplace Noise Distribution This distribution Is sometimes used as a model for Impulsive noise [11]. The Laplace pdf Is defined by f(o) - -j£ e"'°'/c, < a< with 1 - I 8 l ' / C variance 2c2. In this case, Pj --J « . For high SNR's, from (4.9), the optimum weights can be approximated by i - . (4.13) 1 c Since the decision rule is unchanged by scaling [u.} , u. can be chosen as i-1 4.4.3. Cauchy Noise Distribution The Cauchy distribution defined by the pdf f(o) --£7̂ .̂ < a< », Is used to model severe impulsive noise [27] • In this case for high SNR's, * the optimum weights are approximately given by " 1 as shown in Appendix F. 4.5. Some Numerical Examples In t h l 6 section a number of examples are presented. These examples Illustrate some of the Issues Involved In the selection of the weights and compare the detector performances for d i f f e r e n t choices. In a l l of the A A examples the signal-to-noise ratio is defined as 20 logi0 (—), 20 logi0 (-) n and 20 loĝ Q f^) for the Gaussian, Laplace and Cauchy distributions respectively, where A is the pulse amplitude. As a first example, we consider the detection of a taised cosine pulse, sampled Ng - 11 times according to (4.1), in Gaussian noise. The optimum detector for the unquantlzed samples In Gaussian noise Is the digital — * P(e) - Q(/ I •12/on). matched filter (DMF) for which P(e) - Q(/ I *42/o J. Figure 4.1 shows the i-l probability of error obtained using the DMF and the WPD detector with weights 1-P4 I-P* i n ( -). |s1|, s42 snd 1. The Inf——) and curves, though indistin- Pi ' ' 1 V p i gulshable, are not Identical. It can be seen that the use of the ŝ 2 weights Instead of the optimum weights results In little loss. However, the use of Figure U.l . Error p r o b a b i l i t i e s for the DMF detector and four d i f f e r e n t WPD detectors. A r a i s e d cosine pulse i n additive white Gaussian noise i s sampled 11 times. 57 tî -1 results in a substantial loss, e.g. a penalty of about 3.7 dB i s incurred at P(e)-10~1* relative to n Q p T For the same error probability an inherent penalty of about 2 dB results from hard-limiting the samples. It Is Interesting to note that In certain cases, the WPD detectors with weights 1 _ P i l n ( — - — ) , Iŝl and B ^ 2 are equivalent. Examples Include the case of a raised cosine In Gaussian noise sampled N - 3, 4 or 6 times. 6 For a second example, the detection of a half sinusoid pulse sampled Ng"3 times is investigated. Thl6 example shows that caution should be N 5 exercised when dealing with tie s . A tie occurs when d^p^ - £ d^ o^-O. If i-1 the weights | | are used, a tie w i l l occur i f the f i r s t and third samples have opposite polarity to the second sample. One option In this case Is to A choose message m based on the outcome of a f a i r coin toss. This may, however, lead to poorer performance than that obtained by using the optimum MAP decision rule. This Is Illustrated i n figure 4.2 which shows the error probability as a function of the SNR for the WPD detector with weights 1-P4 1-P4 w - Jtnf i ) , Is. I, and s 2 . In this case, the Jtnf " ) and s 2 weights are 1 v p ' - • l ' l " i equivalent. The |ŝ| weights detector with random t i e resolution performs poorer at high SNR'6, e.g., an Increase of about 1.5 dB In SNR Is required to maintain a target value of PCe)-!©"1*. In this case If the decisions corresponding to ties are properly chosen, the | | weights detector i s equivalent to D 0 P T ) H L - Also shown in figure 4.2 are the error performances of the DMF detector Figure U .2 . Error p r o b a b i l i t i e s f or the DMF detector and four d i f f e r e n t WPD detectors. A h a l f sinusoid i n additive white Gaussian noise i s sampled 3 times. oo 59 and the WPD detector with u ^ - l . It i s 6een that the WPD detector with optimum weights (or 6^* weights) performs poorer than the DMF detector by about 1*8 dB (actually a factor of 1.5). It performs, nonetheless, appreciably better than the equal weights detector; e.g., to achieve P(e)-10~\ an additional 2.0 dB i s required. An example involving the detection of a raised cosine pulse sampled N g»7 times in Laplace noise 16 now considered. As discussed in sections 4.3 and 4.4, the weights |s^| are nearly optimum for very low and very high SNR environments. This can be observed i n figure 4.3. The only noticeable l " p i difference between the A n ( ~ — ) *»d |s 1| curves occur for SNR's between -3 dB and 21 dB. The weights u -1 detector is significantly poorer, the difference being about 4 dB at P(e)«10~ 5. The • B ^ 2 detector performs almost as well as the optimum weights detector. The difference is less than 0.6 dB for P(e) < ~10~ 6. The last example of this section Involves the detection of a raised cosine pulse sampled N g«3 times In Cauchy noise. The error probabilities for 1-pj , . u>, • inf =•), Is. I and 1 are plotted i n figure 4.4. The weights \s. | 1 vp^ ' 1 detector outperforms the equal weights detector when the SNR is less than 15.1 dB. For higher SNR values, the unity weights detector has a signif- icantly better performance. In Appendix G i t i s shown that in this example with N g » 3 samples the P(e) of the optimum weights detector i s equal to the smaller of the P(e)'s for the weights Js^j and 1 detectors. For this example, i t can easily be seen that the weights |s.| and s. detectors are equivalent. P(e) -20 -10 0 10 20 SNR (dB) Figure U.3. Error p r o b a b i l i t i e s for the detection of a r a i s e d cosine pulse i n Laplace noise sampled 7 times. Three d i f f e r e n t WPD detectors are i l l u s t r a t e d . o 61 -20 0 20 40 SNR (dB) Figure k.k. E r r o r p r o b a b i l i t i e s f o r the detection o f a r a i s e d cosine pulse i n Cauchy noise sampled 3 t i n e s . Three d i f f e r e n t WPD detectors are i l l u s t r a t e d . However, there are signalling waveforms for which the weights Jsj| detector performs much better than the weights s 2 detector. 4.6. Generalization to M-ary Signalling and Multilevel Quantization In this section, the WPD detectors for M-ary signalling with each received sample quantized to an arbitrary number of levels are examined. These detectors can be viewed as generalizations of the binary signalling detectors of section 4.2. Thus, consider that the message m can now take on one of M values, m e{l,...,M}. Corresponding to n • J, the signal s^(t) Is 6ent and the N g received samples are v j " s j j + n 1 » 1 * l»2,...N f i, j e {l,2,...M} where s, - s. ft *~f; : ) and {n } 6 are Independent noise J.* J N s 1 i-1 samples. Let v-(vj,v2»••«v^ ) denote the vector of received samples. In the _ s case of M equally l i k e l y messages, the minimum P(e) detector corresponds to the maximum likelihood decision rule, I.e., choose m-j If Fr(v|m=j) > Pr(v|m-k) for a l l k # j . (4.14) Let each received sample v , i-l,2,...,N be quantized to one of q regions {R .} q and define the function d -A i f v e » . On the basis of d - ( d l t d 2 d N ), the WPD detector decides on the transmitted message m. "~ 6 A The minimum P(e) detector w i l l decide m-j i f Pr(d|m-j) > Pr(d|m-k) for a l l k # J . (4.15) Since the samples are Independent, Inequality (4.15) becomes 63 N N 6 S n Pr(d |m-j) > II Pr(d |m-k) , a l l k * j . (4.16) 1-1 1 i - l 1 Let A ^ denote the set of a l l I's such that d^-I. Then (4.16) can be rewritten as n Pr (d i-l|m-j) n Pr(d 1-2|m- j ) . . . n Pr (d^q |m«j ) > cA, ieA, leA 1 2 q (4.17) n Pr(d 1-l|iu-k) n Pr(dj-2|m«k)... n Pr (dj-q |m«k) a l l k * j . I E A J 1EA 2 ieA q Defining P i Jl|n " ^ ( d 1 " A l D " n ) » 1 " 1» 2»'*« N S» i " l » 2 , . . . , q , n - 1,2,...M and taking logarithms, (4.17) becomes l z q ^ L * P i . l | ^ + , L * P ^ 2 i l c + - + I *> P i , q | k ' 8 1 1 k * J ' i c A x ' 1 leA 2 ' 1 iEA q From (4.18) i t i s seen that one way of implementing the optimum WPD detector is as follows: Associate with each message an accumulator C , n=l,2,...,M. n Step 1. In i t i a l i z e a l l M accumulators to zero* Step 2. For each 1, i-l,2,...,N s increment C n by In ^ n i f and only If v i c R i , r Step 3. Determine the accumulator with the largest value, i.e., C. - max{C }; declare m-j. ^ n It can be verified that the above procedure when M-q-2 and sit±m~B2ti reduces to that of section 4.2. We conclude this section by looking at an example. Figure 4.5 shows the error probabilities for 2, 4 and 8 level pulse amplitude modulation (PAM) for the DMF and the WPD detectors with optimum and unity weights. In this example q=M and the term "unity weights" refers to incrementing accumulator C by 1 i f and only i f v. c R, • In this 6 n J i i,n example, the signalling waveforms are raised cosines and the noise is Gaussian. The M amplitude values are ±B, ±3B,..., ±(M-1)B where B i s the amplitude of the smallest energy pulse. The M-l decision thresholds for any sampling Instant are located at the midpoints of the Intervals between adjacent signals. In a l l cases the receiver processes 5 samples. The SNR Is defined as the average value of 20 log 1 0(A/o n) where A Is the amplitude of the pulse and the M messages are assumed equiprobable• The optimum detector for the N g unquantized samples makes Its decision according to (4.14). Furthermore, Pr(v|m«j) • P^ (y-_sj ) where _6j denotes the vector of signal samples {s. .} and P»(*) denotes the N-fold Gaussian density of the noise samples. Equivalently, the optimum decision i s to pick the signal j that l i e s closest In terms of Euclidean distance to the received vector. By making use of the fact that the signal vectors are collinear the probability of error for the DMF In this case can be shown to be where_£ u denotes the signal with amplitude B. One sees in figure 4.5 that the WPD detector with optimum weights suffers some loss relative to the optimum detection of the unquantized P(«) 0 10 20 30 SNR (dB) (•) Figure 4 . 5 . PAM error p r o b a b i l i t i e s f o r the DMF and the WPD detectorB with optimum and unity weights. The received signals are r a i s e d cosines In Gaussian noise sampled 5 times. The number of s i g n a l l i n g l e v e l s i s 2 , h and 8 for fi g u r e s (a), (b) and (c) r e s p e c t i v e l y . 0 10 20 30 SNR (dB) (b) Figure U.5. PAM error probabilities for the DMF and the WPD detectors with optimum and unity weights. The received signals are raised cosines in Gaussian noise sampled 5 times. The number of signalling levels i s 2, 1* and 8 for figures (a), (b) and (c) respectively. o ON Figure U.5. PAM error p r o b a b i l i t i e s for the DMF and the WPD detectors with optimum and unity weights. The received signals are r a i s e d cosines i n .Gaussian noise sampled 5 times. The number of s i g n a l l i n g l e v e l s i s 2, k and 8 for figures (a), (b) and (c) r e s p e c t i v e l y . <JN 68 samples. For P(e)«10~* there Is a penalty of about 2.2 dB for a l l 3 cases. The performance i s , however, significantly better than when equal weights are used. The differences are about 3.2, 3.4 and 3.5 dB for signalling with 2, 4 and 8 levels, respectively, for the same PCe)-!©"4* value. 4.7. Conclusions The problem of detecting binary antipodal data signals based on a number of hard-limited samples has been analyzed and the optimum detector has been derived for an arbitrary nol6e environment. The optimal processing Is characterized by a set of weights. In general, the values of the weights depend on the signal-to-noise ratio as well a6 on the shape of the signals and the probability distribution of the channel noise. The optimal weights are approximately |s^| for low SNR's for most noise environments. For high SNR's the weights *^2» |s^| and 1 are nearly optimum for Gaussian, Laplace and Cauchy noise respectively. In some Instances, a set of weights which Is Independent of the signal-to-noise ratio performs almost as well as the optimum weights for practical ranges of SNR. It Is interesting to note that i n these cases, nearly optimum processing of the hard-limited samples can be performed using only threshold decisions and additions. The optimum detector for M-ary data signals based on samples quantized to an arbitrary number of levels has also been derived. The processing requires forming sums of optimized weights, analogous to the binary antipodal case. V PENALTIES OF WEIGHTED PARTIAL DECISION DETECTORS IN GAUSSIAN NOISE 5.1. Introduction The performance of the WPD detector with weights - |s^| has been analyzed for low SNR conditions i n chapters 2 and 3. The optimum WPD detector for arbitrary SNR was derived in chapter A. In this chapter, the penalties of WPD detectors are analyzed for arbitrary signal-to-noise ratios. The weights * (ŝ J and ID • ŝ ^ detectors as well as the optimum weights detector are considered. The effects on the penalties of the signalling waveform employed, the number of samples processed, and the SNR are considered in d e t a i l . The optimum WPD detector i s the optimum detector for hard- limited samples and the d i g i t a l matched f i l t e r i s the optimum detector for the continuous amplitude samples. Hence, the penalty of the optimum WPD detector relative to the DMF detector represents the fundamental loss in signal detectability due to hard-limiting in a sampled system. 5.2. Problem Statement We consider the model of a data communication system shown in figure 5.1. Depending on the message me{0,l) to be transmitted, a signal +A s(t) or -A s(t) i s sent over the additive white Gaussian noise (AWGN) channel. The positive constant A i s a scaling factor. The received signal r(t) i s f i l t e r e d to remove out of band noise and the resulting signal v(t) i s sampled at some appropriate rate. TRANSMITTER NOISY FILTER • t (0, 1} * A 8(t) CHANNEL r(t) m e (0, 1} DETECTOR SAMPLER v(t) Figure 5.1. Block diagram of the data communication system. 7 1 Assume that the signal 6(t) is time limited to the Interval [ 0 , T J . If A s(t) i s sent (corresponding to m-0), then V i " + A 8 i + °i * i - l » 2 f - » M (5.1) ^ 1 1 — 0 5 I T where {v.} denote the M samples of v(t), i . e . v. - Ei~)» t 0 D e 1 i - 1 i n M M processed. The values {s.} denote the samples of + s(t) and {n } 1 i - 1 1 i - 1 represent outcomes of Independent Gaussian noise random variables, each with variance o 2. If m-1, then -A s(t) i s sent and v^ - -A s^ + n^, M 1 - 1,2,...,M. The detector bases i t s decision on {v } 1 i - 1 It i s v e i l known [ l ] that the optimum detector for minimizing the probability of error P(e) - Pr {m * m} i n the problem described above i s the di g i t a l matched f i l t e r ( D M F ) . The DMF computes D O P T " X V* 8* ( 5 ' 2 ) i - 1 and chooses m-0 i f DQ^ > 0; otherwise i t declares m-1. The resulting probability of error i s /M 1 ( 5 . 3 ) 72 where Q(a) • / e X ^ dx. It is convenient to assume that s(t) i s /fit a normalized so that i t s maximum value i s equal to 1. We define the A signal-to-noise ratio (SNR) to be 2 0 l o g 1 Q - ^ measured i n dB. In this chapter, the penalties of the weighted partial decision detect- ors are measured relative to the DMF detector. The penalty i s defined as the increase in signal-to-noise ratio required by a suboptimum detector in order to achieve the same target value of probability of error as the DMF detector. 5 . 3 . Weighted Partial Decision (WPD) Detectors In the family of WPD detectors, the received signal samples are f i r s t A A . . hard-limited. The decision a6 to which message m, B E { 0 , 1 J was sent Is based on the hard-limited samples. Let the random variables representing these samples be denoted by i f sgn (v4) - sgn (s 1 ) # 0 i f sgn (vA) - 0 or sgn (s 1 ) - 0 (5.4) otherwise * Then the general WPD detector forms the test s t a t i s t i c M A D - T D u . (5.5) "WPD * x ul w i ' y 1 M where {u^} are the weights assigned to the different samples. The detector chooses m-C i f D ^ > 0 snd m-1 i f D ^ < 0. Some caution should be 73 exercised in choosing m if DypD = 0 as shown in section 4.5. The optimum ( minimum P(e) ) weights to be used in (5.5) are given by (4.8) in(-1 " P. ) . (5.6) where p̂  i 6 the probability that the 1-th sample has its polarity reversed, i.e. Pj » Q(A|si|/a). The weights « 1, - 16̂  | and • 2 have also been previously suggested. Let B denote a subset of U - {l,2 M} and Bc its complement. Then the probability of error for a WPD detector can be written as PWPD(e) " E I ( B ) 1 1 p i 1 1 c ( 1 " p i ) " all B ieB 1 ieBc (5.7) where 1(B) ieB leB" ieB ieB- 0 , i f I itfj < I m1 IeB ieB- In ( 5 . 7 ) , it is assumed that if Dypn " 0, m is chosen according to the out- come of a fair coin toss. The probability of error P̂ ê) can be directly computed from equation ( 5 . 7 ) . This, however, may Involve considerable computational effort. In some cases, more computationally efficient methods can be used to compute P^pp(e) as described i n Appendix C. An expression f o r the pe n a l t y at high SNR values i s derived i n Appendix H, namely M r „ D ( . , « , . ) - if (5.8) i e B vhere B* C U and B* has the p r o p e r t i e s that I(B*) > 0 and X s. 2 < X s. * * * 2 " i ieB i e B f o r a l l B C U f o r which 1(B) > 0. The p e n a l t y f o r l o v SNR valu e s i s gi v e n by 2 2 M" 2 J 8 i 2 ^ ( s,M,0 ) ^ . (5.9) [ X K B ) { X K l - X l»ilH a l l B i e B i c B c This r e s u l t may be d e r i v e d by a p p l y i n g the procedure of Appendix K to ? y p D ( e ) as given i n (5.7). 5.4. The WPD Detector f o r a Piecewise Constant Amplitude S i g n a l l i n g Waveform For a piecewise constant amplitude s i g n a l l i n g waveform, the weights • 1, • | s 4 | , - S j 2 and - w* WPD de t e c t o r s are e q u i v a l e n t . With | s 1 | « 1, the pe n a l t y r e l a t i v e t o the DMF de t e c t o r I s defined i m p l i c i t l y by X (?) P 1 d-P)"" 1 • M odd -l * ( 5 - 1 0 ) M M X (?) P 1 (1-P) M _ 1 + 4 ( " ) [P( J-P)] 2 • M e v e n i» where p - Q(/2» m4 )%4 ) 8 n d t h e n o t a t l o n ?m < ±***0 ) is used to explicitly Indicate the dependence of the penalty Typ̂  on »• (s itS 2> • • • »ŝ) , M and ̂ . Equation (5.10) can be used to compute r w p D ( ) as follows. Defining y ̂ / IĴ C 2 »M»4 > 4 » o n e c a n r e w r l t e (5.10) as Q( & 7 ) " T(M,y) (5.11) / where M I (?) P1 (I-P)"'1 M+l 1 , M odd T(M,y) - / and p - Q(y). From (5.11), one has W x»M'i > ^ — \ - ( 5-1 2 ) [Q"1{T(M,y)}] Figure 5.2 shows a plot of J t M ^ ) against ~ o for M-1, 2, 3, 4, 10 and M v2 y 11 obtained by plotting 2 as a function of y • • It [(TMT(M.y)}]2 0T1{T(M,y)} can be observed that the penalty Is a non-decreasing function of and 16 upperbounded by 2 (3.01 dB). Explicit expressions for the penalty for small and large SNR values are now examined. In Appendix I, it Is shown that 4 M - 2 M - 4 M - 10 M - 11 M M - 1 -50 -25 25 50 20 l o g 1 ( ) ( A/o ) (dB) Figure 5.2. The penalty ( ) as a function of the signal-to-noise ratio for seven values of sample size M. 77 ,2M-2 M - 1 , M odd 2 ,2M M (I) , M even (5.13) U6ing Stirling's formula i t follows that lim r W p D ( i.»M»° ) " * Equation M-K» (5.13) also shows that r^a.i.o) < r W DQ,3,o) < ... < r^a.4,0) < r H P D(i,2 io) . (5.14) By using the approximation Q(a) * shown that -a2/2 e /2~* o , a » 1, in (5.10) i t can be rWPD( i ' M ' " } " 12 , M even \ 2 " Wl • M o d d • (5.15) It 16 also possible to determine the penalty T^^i .i»M»"o- ) f o r l a r 8 e v a l u e s UB A of M. The basic approach i s to derive an upper bound T^^i ) using the UB A Chernoff bound [28,29] and to show then that J..-*— ) 1 6 a l s o 8 lower A bound on Iyp^ ±»mrj, ) • The details appear In Appendix J. The curve rWPD( ) 1 6 P l o t t e d i n f i g ^ e 5.2. 5.5. The Optimum WPD Detector for Arbitrary Signalling Waveforms In the case of an arbitrary signalling waveform, the penalty * A *WPD̂  —'^'"o ̂  * o r t n e ° P t ^ n u m detector can be evaluated numerically by >ting Q(/T~7*A] equati (/ J s . 2 — ) to the right side of (5.7) with i - l ° P4 - Q(/r̂ pDC ) I s j l ) and - Jta£-—i). The procedure i s ^ 1 analogous to that followed in (5.10) to (5.12). The results for a raised cosine and a half sinusoid signalling waveform are plotted in figures 5.3 and 5.4 respectively. In both figures, for a fixed value of M, the penalty A increases with — even though i t i s essentially constant for small and large values of —. a It i s shown i n Appendix J that, for large values of M, the penalty i s upper bounded by C < > , ^ < • ' < ' > > — (5.16) W D ° < in{4Q(y|s(t)|)[l-Q(y|s(t)|)]} > where y - / rj^ D( B,»J± j± and < f ( t ) > &j jj f ( t ) d t i s the average of f(t) f ( t ) on the Interval te[0,T]. This bound i s plotted i n figures 5.3 and 5.4. For the examples considered here T^^i. ) close to Iy^ D( .£»•»•" ) when M > 10. As sn example, Jrjjp^ mT-A0 ) " r̂ pD< ^ . l O ^ )l < 0.14 dB for both -50 -25 1 M-1 25 50 20 l o g 1 0 ( A/o ) (dB) * A Figure 5.3. The penalty ( J . tM,— ) of the optimum WPD detector as a function of signal-to-noise ratio for a raised cosine signalling waveform. Six values of sample size M are illustrated. The curve for M B • is an upper bound. 4 M - 2 M - 4 M - • M - 5 M - 3 J J L l -50 -25 25 50 20 l o g 1 ( ) (A/o ) (dB) Figure 5.H. The penalty T* p D ( j»tM»~ ) o f t n e optimum WPD detector as a function of signal-to-no ratio for a half sinusoid signalling waveform. Six values of sample size M are illustrated. The curve for M = » is an upper hound. 81 the h a l f s i n u s o i d and r a i s e d c o s i n e . For l a r g e values of — , (5.16) becomes 5.6. The Weights u± - | s 1 | and - s±2 WPD Detectors The weights • | | and • s ^ 2 WPD d e t e c t o r s have been p r e v i o u s l y examined f o r low s i g n a l - t o - n o l s e r a t i o c o n d i t i o n s . I t has a l s o been shown that the optimum WPD de t e c t o r 16 equ i v a l e n t t o the | | and s ^ 2 weights d e t e c t o r s f o r low and high SNR c o n d i t i o n s r e s p e c t i v e l y . In t h i s s e c t i o n , the performance of these weighting choices f o r other values of SNR I s i n v e s t i g a t e d . The pen a l t y r w i ) ( ±»^r^ ) f o r the ( s J weights and the weights WPD d e t e c t o r s may be evaluated using the technique of the previous s e c t i o n . F i g u r e s 5.5 and 5.6 show £,Mr^ ) as a f u n c t i o n of s i g n a l - t o - n o i s e r a t i o f o r a r a i s e d c o s i n e and a h a l f s i n u s o i d s i g n a l l i n g waveform r e s p e c t i v e l y . In both f i g u r e s the • J s ^ J weights are used. There are p l o t s f o r the cases of M - 1, 2, 3, 4, 5, 10 and 11 samples. F i g u r e s 5.7 and 5.8 show r,_TX S,M,— ) versus SNR f o r a r a i s e d cosine and a h a l f s i n u s o i d WPD — a r e s p e c t i v e l y , f o r the same values of sample s i r e w i t h weights • s ^ 2 . In a l l cases, f o r f i x e d M, the p e n a l t y Increases w i t h ^ but i s approximately constant f o r small and l a r g e values of ~ • _ - y 2 < s 2 ( t ) > . 2 # < -y 2s 2(t ) / 2 > 5 . 7 . D i s c u s s i o n The performance l o s s e s f o r WPD d e t e c t o r s i n Gaussian n o i s e have been M - 10 •a <\t> m X a -50 -25 0 20 l o g 1 Q ( A/o ) (dB) 25 50 Figure 5 - 5 . The penalty ( £.M,^-) of the - IsJ weights WPD detector as a function of signal-to-noise ratio for a raised cosine signalling waveform. Seven values of sample size M are illustrated. 00 N J 5 r — to o 4 — 2 = 1 — -50 -25 25 20 l o g 1 ( ) ( A/o ) (dB) 50 Figure 5.6. The penalty T w p D ( B , H ~ ) of the u± - | 8 i | weights WPD detector as a function of signal-to-noise ratio for a half sinusoid signalling waveform. Seven values of sample size M are illustrated. oo 4 r— T j X •I 1 — -50 -25 25 50 20 log1() ( A/o ) (dB) Figure 5.7. The penalty ( £,M,£ ) of the u>1 - e* weights WPD detector as a function of signal-to-noise ratio for a raised cosine signalling waveform. Seven values of sample size M are illustrated. 00 4> 4 a .52 1 — -50 M - 2 M -25 25 50 20 l o g 1 0 ( A/o ) (dB) Figure 5.8. The penalty ( B _ , M , — ) of the - s^ weights WPD detector as a function of signal-to-noise ratio for a half sinusoid signalling waveform. Seven values of sample size M are illustrated. 00 investigated. Three choices of weights have been considered. Previous analyses of these suboptimal schemes have been confined to low signal- to-noise ratio conditions. In general, the penalty depends on the choice of weights, the samples of the signalling waveform (and their number) as well as on the signal-to-noise ratio. In a l l of the examples considered, the penalties of the WPD detectors are approximately constant for low SNR values, increase in a transition region, and are approximately constant for high SNR values. Since equations (5.8) and (5.9) are independent of the signal-to-noise ratio, the losses for the general WPD detector w i l l be approximately constant for low and high SNR values. Since the u « In — - — weights WPD detector i s the optimum detect- or for hard-limited samples and the DMF 'detector i s the optimum processor for continuous amplitude samples, the penalties of figures 5.2 - 5.4 represent the fundamental losses due to hard-limiting of independent samples. In a l l cases, this loss Is a non-decreasing function of signal- to-noise ratio and i s upper bounded by 2(3.01 dB). The value of w/2 - 1.96 dB is often cited as the loss due to hard- limiting [25,31,37], This result applies to an i n f i n i t e number of independent samples and a vanishingly small SNR. The work of this chapter has shown that the loss i s a function of the signal-to- noise ratio and of the number of samples. Independent samples are assumed in this work and the results are shown to agree with the previous result when the number of samples i s i n f i n i t e and the SNR is vanishingly small. However, the well known result of 1.96 dB as i t applies to an i n f i n i t e number of independent samples does not represent a physically r e a l i s t i c system. The results derived here deal with f i n i t e numbers of independent samples. This i s representative of r e a l systems. The u)^ = |s^| detector i s optimal for low SNR conditions. Figur 5.5 - 5.8 show that the penalty for t h i s detector may exceed 2(3.01 dB) 2 fo r high SNR values and i s greater than the penalty incurred by the s^ 2 weights detector. S i m i l a r l y , the s^ weights detector which i s optimal f o r high SNR's has greater lo s s at low SNR values than the |s^| weights detector. VI PENALTIES OF SAMPLE-AND-SUM AND BINARY PARTIAL DECISION DETECTORS IN GAUSSIAN NOISE 6.1. Introduction The performances of the SAS and BPD detectors were analyzed for low signal-to-noise (SNR) conditions in chapters 2 and 3 . In this chapter, the performances of SAS and BPD detectors are analyzed for arbitrary signal-to-noi6e ratios. The effects on the penalties of the signalling waveform employed, the number of samples processed, and the SNR are considered in detail. The SAS detector is examined first because its loss constitutes a part of the BPD detector's loss. The losses are compared to those of some WPD detectors and a relationship between the losses of the SAS, BPD and WPD detectors for low SNR and finite sample sizes is derived. The model considered here Is the same as in section 5 . 2 . Antipodal signals are used to communicate a message mt{0, l} over an additive white Gaussian noise (AWGN) channel. The signals 4A s(t) and -A s(t) are aent corresponding to m - 0 and m-1 respectively. The M M receiver processes M samples {•JJJ.J* v j " * A * i * n i w h e r e {'ili-i a n d Ŵ i-1 ° " e n o t e t h e • a n P 1 ' » o f t h e signal a(t) (normalised ao that its maxim amplitude Is equal one) and outcomes of Independent Gaussian noise random variables respectively. Signal-to-noise ratio (SNR) la defined to be 20 log 10-̂ measured In dB. The penalty is defined as the increase in elgnal-to-noise ratio required by the suboptimum detector in order to achieve the same target value of probability of error as the DMF detector. 6.2. The Sample-and-Sum (SAS) Detector Loss In this section, the penalty incurred in using the SAS detector is analyzed. This detector [5,6,13] forms the s t a t i s t i c M DSAS V i 8 * n <Si> ( 6 - A ) r + i , i f x > o jn(x) - < 0, i f x - 0 L-l, i f x < 0 where sg • and chooses m-0 i f D g A S > 0. Otherwise, m-1 i s chosen. Given that m-0, i.e. +A a(t) i s transmitted, the mean of Dg A S i s D S A S - A X |s I (6.2) S A S i-0 1 and i t s variance i s given by t,2 - M o 2 . (6.3) DSAS An error occurs i f D G A S < 0. To compute the probability of error, P S A g ( e ) » we observe that D„ _ i s a Gaussian random v a r i a b l e ( r . v . ) since i t i s SAS the sum 90 of independent^Gaussian r.v.'s. Also, because of the symmetry, PgAg(e|m=l) M A I |.,| 1 r° ~x 2 / 2 where Q(o) • J e dx . The penalty rcAC incurred by the SAS / 2 n detector relative to the DMF detector i s implicitly defined by comparing PSAS ( e J W l t h P D M F ( e ) a s 8 I V E N b v <5-3> namely, /~M N M a / 1 8 I 2 AX Q ( - ~ ) - Q ( - ^ i — ) . (6.5) From ( 6 . 5 ) , i t follows that M M I s,2 r s A s " ~ " • ( 6 - 6 ) i - i Note that T S A S i s independent of the channel noise power, but does depend on the samples {s } and the number of samples M. Where necessary, we w i l l 1 i - l use r.._ ( s,M ) to indicate this dependence e x p l i c i t l y . For large values of SAS — 91 M, I s * - | / |s(t)| dt (6.7a) i-1 1 1 0 and M T I l«J / l»(0| dt . (6.7b) i-1 1 1 0 Consequently, equation (6.6) can be rewritten as T T / B 2 ( t ) dt £ r s A s " - r ; 5 ; (6-8> [/ |s(t)| dt] T[< | B ( t ) | >] 0 vhere E « J s ^ t ) dt i s the energy of the signalling waveform s(t) and 8 0 A 1 T < |s(t)| > / |a(t)| dt i s i t s average magnitude. Equation (6.8) has 1 0 been previously derived ( 2 . 1 0 ) for a particular f i l t e r i n g scheme with a large number of samples i n a low SNR environment. The present derivation shows that i t i s va l i d for any SNR. Ve now use equation (6.6) to i l l u s t r a t e how varies with M for a few commonly encountered signalling waveforms. For a constant (or piecewise constant) amplitude signalling waveform, rsAS • 1, i.e. 0 dB as expected. For a half sinusoid signalling waveform, the samples are M s± - 6in-p| ( i - 0.5), 1 - 1,2,...,M. In this case, I \e±\ - cosec i - l M f l , M-1 1 M 1-1 1 I |. M > [30,1«341-1] and J, s^ 2 - ̂  M . From (6.6) r i , M - i j y - s i n 2 g j ) , M > 2 , half sii R S A S , n _ , _ 2 f * > „ . „ - 1 n u 8 0 l d . ( 6 . 9 ) The penalty T given by (6.9) is plotted as a function of M in figure 6.1. The asymptotic value of rgAS for large M i s -g- or 0.912 dB. The penalties for a f u l l sinusoid ( 6 j - sin [ ~ (i-0.5)]) and 8 raised cosine (B± - [l-cos-jp (i-0.5)]/2) signalling waveform are given respectively by 1 , M - 1 or 2 i2 lSAS |- s i n 2 (g) , M - 4,6,8... £ t l f c o s U / M ) ) 2 • M " 3 > 5 > 7 - • f u l 1 B i D U S O l d ( 6 ' 1 0 ) i n d J l , M - 1, 2 [•§ , M > 3 , ra i S A S I 4 . raised cosine . (6.11)  94 2 The asymptotic values of Tĝg as given by (6.10) and (6.11) are -g— or 3 0.912 dB and -r- or 1.76 dB. A plot of T for a f u l l sinusoid s i g n a l l i n g 2 SAS waveform is shown in figure 6.2. In all these examples, the asymptotic value of rg A g is reached rapidly. The magnitude of the difference |rSAS(s»M) - rSAg(6»")| is less than 0.1 dB for M > 11. 6.3. The Binary Partial Decision (BPD) Detector Loss The BPD detector forms the test statistic [ 2,3 ] °™* X  b* where the partial decision random vsriable D̂  is defined by if •gn(v1) » sgn(s1) # 0 if •gn(v1) - 0 or •gn(s1) - 0 (6.12) otherwise • The BPD detector chooses m - 0 if DBpD > 0 and m « 1 if DBpD < 0. Some caution ahould be exercised (section 4.5) in choosing m if DBpD ™ 0. This detector may be thought of as the special case of the WPD detector for which the weights a l l equal one (û  - 1). It can be easily Implemented using a counter which is incremented or decremented by 1 depending on the polarties of v̂  and ŝ. 2.0 rSAS< * M > (dB) 1.0 M (SAMPLES/BIT) Figure 6.2. The penalty T^i s,M ) aa a function of the number of bit samples for a sinusoid. VO The probability of error Is the probability that a majority of the transmitted samples are received vlth their polarities reversed, i . e . (6.13) where P̂  - Pr {exactly 1 polarity errors In the M samples}. By equating the right hand side of (6.13) with P~_(e) as in D M F section 5.4 the penalties rBTjr.( a-M̂ - ) can be numerically evaluated* A Plots of r__ n(«,»,0 as a function of for the raised cosine and half sinusoid signalling waveforms are shown in figures 6.3 and 6.4. The values of sample size illustrated are M « 1 - 5, 10 and 11 for both waveforms. For high SNR, i t follows from (5.8) that M W > !T < 6 ' U ) sum of the I -j I smallest terms i n i8^2}^.! where Px"] denotes the smallest integer > x . Equation (6.14) holds for an arbitrary signalling waveform. For large values of M, one has 20 l o g 1 ( ) ( A/o ) (dB) Figure 6.3. The penalty r B p o ( J»»M,~ ) as a function of the signal-to-noise ratio for a raised cosine signalling waveform. Seven values of sample size M are illustrated. The solid triangles indicate points for which p B p n ( e ) • 10~ 7. -50 -25 0 25 50 20 l o g l Q ( A/o ) (dB) Figure 6.U. The penalty r B p D ( l . M . j ) as a function of the signal-to-noise r a t i o for a half sinusoid s i g n a l l i n g waveform. Seven values of sample s i z e M are i l l u s t r a t e d . Oo T J n B 2 ( t ) d t J 8 2 ( t ) d t V PBPD( £•"»" > " T~: (6.15) T where V Is a union of Intervals In [0,T] with a total width of which minimizes / s 2 ( t ) dt. W For a raised cosine signalling waveform, (6.15) becomes r e t , , , fffri-co.»)]>dt _ 6 , ^ ( r a i s e d cosine,-,-) ^j- 2 / 0 [ j d - c o s t ) ] dt or 11.22 dB. (6.16) For a half sinusoid waveform, we have £ 8 i n 2 t d t 2, rBpD(half sinusoid,-,-) ^ - % _ 2 or 7.41 dB . (6.17) 2 L s l n ^ dt In the case of a piecewise constant signalling waveform, (6.14) { 2, M even 2 2_ M i n agreement with (5.15) M+l * The asymptotic values for low SNR's w i l l now be considered. In 100 Appendix K, It is shown that for an arbitrary signalling waveform, rBPD( -£'M*° > " rWPD( -i'M»° } * rSAS( 1»M ) ( 6 - 1 8 ) Equation (6.18) can be interpreted as follows: suppose the penalties are measured in dB; then for low SNR values, the penalty Incurred by the BPD detector Is the sum of the penalty Incurred by the SAS detector and the penalty Incurred by the WPD (or BPD) detector operating with a plecewlse constant signalling waveform ( as given by (5.13) ). Strictly speaking, this relation is only valid for - 0. It is however, nearly exact over a wide range of signal-to-noiBe ratio because of the flat nature of the penalty curves in that range. From (6.18) and the fact that lim r w p D ( _1»M»°) " ̂ 2 (section 2 . 3 ) , it follows that for large M, Wi'"»° > " 1 rSAs<.£»-> • ( 6 - 1 9 ) This result was previously derived for a particular filtering scheme in section 2.A. 6 . 4 . Discussion of Results The penalties associated with the use of the SAS and BPD detectors have been analyzed. The loss incurred by the SAS detector depends on the samples of the signalling waveform used but is Independent of the signal-to-noise ratio. In contrast, the losses associated with the BPD 101 detector (and the WPD detectors) depend on the signal-to-noise ratio as well as on the 6 i g n a l samples. While the BPD detector i s easier to Implement than the WPD detectors analyzed in chapter V, the penalty of the BPD detector was more sensitive to the choice of the signalling waveform and in 6ome cases was much greater than the penalty of the WPD detectors. Also shown in figure 6.3 are the points corresponding to p g p D ( e ) " 1 0 ~ 7 , J t can be seen from figure 6.3 that for the raised cosine waveform, the penalty using a BPD detector with M - 4 and p B p D ( e ) " 1 0 ~ 7 i s about 14.5 dB. The corresponding penalty for the optimum WPD detector i s only ~ 2.8 dB. 102 VII CONCLUSION 7.1. Summary of Results In this section, the highlights of the thesis research are summarized. Three suboptimum detectors which find application In practical digital systems have been described. The penalties of each have been analysed and identified. The relationships between the losses of the systems have been derived. Previous work In this area was mainly numerical and example specific. The present work treats the topic theoretically and gives some results that are fairly general and can be applied to systems other than those examined i n the examples. The effects of dependence among the signal samples on the penalties have been examined. It has been shown that, in some cases, the penalties can be reduced by processing more, dependent, samples. It has been found that the amount of loss recoverable depends on the prefilter characteristic and the sampling rate. The optimal detector which bases its decision on a number of hard-limited samples has been presented. This detector is optimal in the sense that it minimizes the bit error probability. The detector is simple and inexpensive since it is essentially a one-bit analog-to- digital converter (ABC) and does not require an automatic gain control (ACC). This result is general in that it applies to arbitrary SNR values, arbitrary numbers of samples, and most practical noise environment models. The generalisation of the optimum hard-limiting detector for binary signals to higher quantisation and signalling levels has also been derived. That i s , the minimum b i t error p r o b a b i l i t y detector for M-ary s i g n a l l i n g with received s i g n a l samples quantized to an a r b i t r a r y number of l e v e l s has been found. Again, these r e s u l t s have been obtained using t h e o r e t i c a l a n a l y s i s whereas previous r e l a t e d work, i n many cases, has been numerical and example s p e c i f i c . The fundamental l o s s due to h a r d - l i m i t i n g i n Gaussian noise has been investigated i n depth. This l o s s i s measured i n the s i g n a l d e t e c t a b i l i t y sense. That i s , the l o s s i s expressed as the increase i n SNR required to maintain a target value of error p r o b a b i l i t y . Much at t e n t i o n has been paid to the l o s s due to h a r d - l i m i t i n g i n the past. The r e s u l t most often quoted i s that the l o s s i s TT/2 = 1.96 dB. This r e s u l t a p p l i e s to an i n f i n i t e number of independent samples and a vanishingly small signal-to-noise r a t i o . The present work shows that the l o s s i s a function of the signal-to-noise r a t i o and of the number of samples. Independent samples are assumed i n our work and the r e s u l t s agree with the previous r e s u l t when the number of samples i s i n f i n i t e and the SNR i s vanishingly small. The r e s u l t s are important because they apply to r e a l world system conditions. The well known r e s u l t of 1.96 dB as i t a p p l i e s to an i n f i n i t e number of Independent samples does not represent a p h y s i c a l l y r e a l i s t i c system. The work done here which deals with a f i n i t e number of independent samples i s representative of r e a l systems. In a d d i t i o n to the i n v e s t i g a t i o n of the fundamental l o s s due to h a r d - l i m i t i n g , the losses incurred by some ad hoc schemes that hard- l i m i t the received samples have been examined. Some of the r e s u l t s of t h i s work are of considerable i n t e r e s t for p r a c t i c a l design. I t i s shown, f or example, that one common ad hoc procedure has very large losses at high SNRs and i s therefore unsuitable for a p p l i c a t i o n i n a strong si g n a l environment. 7.2. Suggestions for Further Research There are a number of issues a r i s i n g from the th e s i s work that provide i n t e r e s t i n g topics for further research. Some of these are presented and b r i e f l y discussed i n t h i s s e c t i o n . The e f f e c t s of dependence among the si g n a l samples on the detect- or penalties was investigated for large time-bandwidth product conditions and low SNRs. The general i z a t i o n of these r e s u l t s to a r b i t r a r y time- bandwidth products and a r b i t r a r y SNRs has not been treated i n t h i s t h e s i s The optimum detector for M-ary s i g n a l l i n g with an a r b i t r a r y , given, quantizer was derived. The optimum quantizer thresholds that minimize the b i t error p r o b a b i l i t y were not s p e c i f i e d . A rel a t e d question i s the s e n s i t i v i t y of the detector performance to v a r i a t i o n s i n threshold s e t t i n g s . The performance of the optimum detector for M-ary s i g n a l l i n g may be evaluated. Hereunder, one can consider d i f f e r e n t quantization l e v e l s and modulation formats. The losses of the suboptimum detectors under bandlimited condition with appreciable ISI are of i n t e r e s t . Performance evaluation i n these cases i s probably best done by the a p p l i c a t i o n of t i g h t bounds for the error p r o b a b i l i t i e s . \ 105 APPENDIX A ID this appendix, we show that as M + » , the probability of error for the WPD detector is He) - Q [/TIT)- ( A , 1 ) o Recall that Q( o) can be represented by the infinite series 115,16] 3 5 . , . v n 2n+l 2 ^ 6 4 0 (2n+l)2nn! Thus, from (2.15), for large values of M (small signal-to-noise r a t i o ) , o M As M - * - , the mean of D ^ , i.e J (l-2pi)|si|. Is given by (see page 14) O 0 o M , and the vsriance of D ^ , i.e. J Ap1(l-p1)|a1| , ia given by § /V(t)dt - ̂ (A .5 ) But P(e) • Q [ • variance of 0 ( / 3 )• 106 APPENDIX B It is to be shown that the probability of error for the BPD detector is O Using (A.3), it can be determined that as M-w, the mean of Dgpj}» 1«e- M I (l-2pi), is given by i - 1 4"M t i - t ^ l A . <B-2> / ^ / o | . ( t ) | - t and its variance, i.e. J 4p (1-p^. 1B given by M. Hence i-1 mean of DBpp P(e) - Q ( J • variance of DBpD Q ( 107 APPENDIX C In this appendix the calculation of P(e) for the SAS, BPD and WPD detectors for small values of M is briefly described. Let us denote the noiseless received waveform by f(t). Thus f(t) results from passing the transmitted waveform s(t) through the lowpass filter. Then, for the SAS detector, M P(e) - Q ( 1 , 1 ) (C.l) Mitt /2T o where f , i-l,2,3,...,M denotes the 1th sample of f(t). The calculation of P(e) for the BPD and WPD detector Is computationally more involved. Let p̂  denote the probability that the channel causes a reversal In the polarity of f̂ , I.e., , , f i ' , PA - Q (-7- ) CC2) n where o - / MN /2T . Also define q. • 1 - p.. n o 1 > 1 Suppose A is some subset of D - {1,2,3,...,M}. Then the probability that the samples {f.}. . have their polarities reversed by the noise and the 1 X EA remaining samples (f } retain their original polarities is given by 1 ieAc PA" C • 9±)i * c \ ) • (C'3) * ieA 1 ieAc ̂ Hence, P(e) • [ P. where the sum Is taken over all subsets A which would A A lead to a wrong decision. This brute-force method of calculating P(e) is 108 time consuming. A more efficient way of evaluating P(e) for the BPD detector can be obtained by noting that i t i s just the probability that a majority of the samples have their polarities reversed, i.e., IM/2J P(e) ~ I P (C.4) n-0 n ' M where P Pr {exactly n of the M samples are correct}. In (C.4) i t i s n ,M assumed for simplicity that M i s odd. If M i s even, then M/2 j P ( e ) " J b P n , M " * ? M / 2 » M * In any case, P u represents the generalized binomial distribution and n,H can be recursively evaluated [17] using P » q P , . + p P , n,m Tn n-1, m-1 m n,m-l P. . » 1, P. - p, p....p , and P • 0 i f n > m. (C.5) 0,0 ' 0,m r l K 2 rm* n,m A P P E N D I X D It is to be shown that as M-**> E[D^DJ] - 5 ^ - 2/n arcsin { r j j - i ) } One has that EtDjDj] » E[sgn(v is 1)sgn(v js j)] - Pr(n i> - \ B ± \ , n̂ > + P j n ^ -IsJ, -|8j|) - P r( n i> -IsJ, Dj< - | 8 j l ) - P(nA< - | B 1 | , N J > - | S J | ) , Let 1 -(x^px^+x., 2) , B-,(C, P. P) - / / — — exp{ ; M * 2 d x l denote the bivariate Gaussian distribution [15]. Then (D.2) becomes E I D ^ j ] - 2[B 1(-|s i|/o n, - l - j l / o ^ r n ( j - i ) ) + B 1 ( | . 1 | / a n , l-jl/v r n ( j-i))] - 1 M N Q Y where o 2» ^ _ • and the noise i s assumed to be stationary. The two n zcl 110 s i o n a l Taylor s e r i e s , centred at the o r i g i n , for f(x,y) i s m = 0 x = 0 y - 0 (D.5) Combining (D.5) with (D.3) gives B 1(C ,8,p) i + ^ a r c s i n p - 2/2i? 2w/l-p 2* 4itA-p2* ( D . 6 ) where e(C .6)-K) as C 3, P 3, CP 2 and C 2B. Using ( D . 6 ) i n ( D . 4 ) gives for large M, ElDjDj] • 2/tt a r c s i n { r n 2CT ( | S I-|S \y< r ( j - i ) } 3 (D.7) By proceeding as i n Appendix A i t may be shown that as M+» 1 J ^ n (D.8) Combining (D.8) with (D.7) r e s u l t s In (D.l), I l l APPENDIX E In this appendix i t w i l l be shown that as M-»-«> (3.23) v j - I I |s ||s |arcain{r <J-i)} - 2 \ arcsin{r (i)} . s 1=1 j»l 3 j=l ( E . l ) One has that M M M I I |s ||s |arcsin{r (j-i)} - l{ \ |s | |s |arcsin{r (1)} i - l j=l 1 J n k=2 K K 1 n 3*i M + I l« k l|s k_ 2|arcsin{r n(2)} + .. k=3 M + I |s k l|s k_ N + 1|arcsin{r n(N-l)} + ... } k=N M M 2{arcsin{rn(l)} \ IsJIŝ -J + arcsin{rn(2)} \ |8k||sk_2| + k*2 lce3 M + arcsin{r n(N-l)} J l*kl l»k..N+1l + ••• } • (E-2> k-N Note that, for fixed N, 112 M+» k»N M-w k»N M > » k-1 i l a I J | . ( I > < - ° ^ ) | 2 . / T B 2 ( t ) d t - E • ( E . 3 ) M-K»  M k-1 M 0 where, without loss of generality, i t has been assumed that - s([i-0.5]T/M). Using the result of (E.3) with (E.2) gives ( E . l ) . APPENDIX F It 16 to be shown that when the channel noise has a Cauchy * distribution the optimum weights for high SNR'6 are given by w^-l. In a Cauchy noise environment the probability that a sample has Its polarity C D reversed i s p. - Pr (N > |s.|) • / —-2- da. Making use of the series 1 1 1 IsJ b2+o2 - j i ^ » 1-x+x2-... yields |s 1| %a* o-|s 1 It 6, 1 i 1 * r 1 i 1 •» For large signal-to-noise r a t i o s , — - — » 1 and • I n (— - — J . Assume that s(t) * C s(t) where s(t) i s a waveform with unit amplitude and C i s a constant. Then, * *|C||sJ to* - M b 1) " *»UC|)+ togli-J) . For large values of |C| and hence large SNR's, u>* - JUi(|C|). Since the optimum decision rule i s not changed by scaling the weights, one may use * <i>4 « 1 for large SNR's. 114 APPENDIX G In this appendix i t i s shown that the optimal weights for the case of a raised cosine pulse i n Cauchy noise sampled 3 times are 4 r |s 1| i f SNR < 15.1 dB U i " 1 1 L l i f SNR > 15.1 dB . Note that there are 8 possible combinations of the 3 received signal * * , 6ample6 and that tii± - u>3. Therefore, the sample vectors _v • [+,+,+), (+,+,-) and (-,+,+) w i l l be assigned message m-0 and the vectors_v - (-,-,-), (-,-,+) and (+,-,-) wi l l be assigned m-1. This i s true for any signal-to-nolse ratio and any choice of weights provided that Wj-u^. According to the |s^| weights, the detector chooses m-0 for_v-(-,+,-) and m=l for _v»(+,-,+). But for p^ specified by the Cauchy distribution and < 15.1 1-Pi 1~P3 1-P2 dB, inf 1 + inf J < in[ ]• Thus, the optimum weights are equivalent P i P3 P2 to the |6 1| weights i n this SNR region. When ̂  > 15.1 dB, 1-Pj l - p 3 l - p 2 A Jtnf ] + Jtnf 1 > Jtn[ 1 and m - 0,1 are chosen corresponding to Pi P3 ?2 v-(+,-,+), (-,•+,-) respectively by both the optimum and unity weights detectors. APPENDIX H In thi6 appendix i t i s shown that the penalty for the general WPD detector at high SNR values i s given by (5-8) From (5.7), f o r high SNR values, the p r o b a b i l i t y of erro r becomes W e ) - r 1 I ( B ) 1 1 P i w r u B C {1,2,...M} ieB (H.l) where 1 , i f I > I wf ieB ieB 1(B) - / i , if I u>x" I u± ieB ieB 0 , i f I u± < I w± . ieB ieB" Also, for large — , / 2 * rOTn( S.M^)-! l 8 i ' "WPDV (H.2) Hence, n P j -w^M4> l i o J 2 4 j B 8 i 2 e n / 2 W W J , M A > i |. | ieB (H.3) From (H.3), i t can be seen that fypr/e) 8 6 given by (H.l) w i l l be dominated (for high SNR) by the set (or sets) B for which 1(B) > 0 and £ s 2 i s ieB 1 * * A r > minimum. Let B denote one such set, that i s , B C U - {1,2,...,M} and I(B*) > 0 and I s 2 < J s 2 for a l l B C U for which 1(B) > 0. Since * 1 1 ieB IeB r y p D ( ±t*£a ) > 1, i t follows from (H.l) that as incresses ieB Recall that / T i - i -t-0i 1,1 V e / — 8 — \ , for large - , so that 117 Equating (H.M with (H.5) y i e l d s M 2 * I E B APPENDIX I It Is to be 6 h o v n that the penalty for the WPD detector with a piecewise constant amplitude signalling waveform and low SNR is given by (5.13). We consider f i r s t the d e r i v a t i o n f o r odd values of M. A A D i f f e r e n t i a t i n g (5.10) with respect t° — and then l e t t i n g — = 0 y i e l d s M 2 2M-2 M I (JK21-M) M+l L 1 * ~2 , M odd (1.1) One ha6 I M(?) - M2 M _ 1 , M odd . . M+l ±ms — (1.3) 119 Using ( 1 .2) and ( 1 .3) In ( I - l ) o n e obtains 2 M — 2 In a similar way, i t can be shown that 2 2 M W i ' M ' ° > " / M v 2 • M e V C n * U , 5 ) It might be noted from (I.U) and ( 1 .5) that 120 APPENDIX J It i s to be shovn that for a large number M of samples the penalty for the optimum WTD detector i s upper bounded by (5.16). Let f X j } j . i be M Independent random variables defined by 1-Pj An(—-—) with probability p^ 1 1 i-Pj -Inf-p—-) with probability ( l - p 1 ) r A —v A where p± - Q(/ B , H , - ) - |s 1|) . Then * M P (e) - Pr{ X X > 0} . (J.l) i - l One can use the Chernoff bound 129: Eq« (5.4.15)] to upperbound the right hand side of (J.l) M XX P* (e) < n E[e *] (J.2) m e u i - l where X Is a positive number which can be chosen to optimize the bound. For the present problem, the best value to use Is X - -| • Then (J.2) becomes W e ) < " 2 / ' P 1 < 1 - P 1 > S • < J ' 3 ) 1 2 1 An upper bound T ™ D ( l.M,^ ) Is obtained by equating the right hand side of (J . 3 ) to P_..(e), namely Q ( " a ^ J l S l 2 ) " i - 1 2 / p i ( l ~ P i V where p A £ Q (/ O-S' M4 >i I I 5- For l a r g e M , 2 M , k - i e|r i «i2 4 / l 8 .2 ) > C A* ! Q ( ± M ^ 2 ) i - i * ° Using (J.5) i n (J.4) yields for large M, (J.5) £)2 jg s2(t)dt - - /J |*Q(/C^»-4^4 l8(t> l> [I-Q(/0±.--t>4 .•<*>!>]}* • (J-6) Letting / r J ^ D ( )'4" y. ^ c a n rewrite (J.6) a s -Y 2 s 2 ( t ) d t JOB ( „ A ^ J ° (J.7) WPD i ' * O fT • - ' - . i . r . • - v I * 1 1 . . Ji ln{4Q(y|s(t)|)[l-Q(y|B(t)|)]}dt UB A A Equation (J . 7 ) can be used to plot Tmj){ l . - r ^ ) as a function of - 122 For a plecevise constant amplitude waveform the upper bound Is also a lower bound* This i s now shown. One has (5.10) M I (?) PV-P)*1"1 M+l 2 i - , M odd (J.8) 1 - ' even where p - Q ( / 1 , * , ^ ) (-p • Therefore A ^ .A. M I 12 (V )2 ( ^ ) [ P ( 1 - P ) ] 2 t ^ f . " o d d WPD (e) >< 1 ( H ) [Pd-P>] , M even (J . 9 ) Since ( 4 0 - M+l M-1 , 2 2 »M . ^ n ( 2 + "j) > l+i w h e n M 1 8 °* d a n d »M ( M ) " 2 (MMX) > - 2 r w h e n M 1 8 e v e n ( J * 9 ) * l e l d 6 M M+T (T%) 1 / 2 [*P(1-P)? . M O D D W 6 > > M 2M j; [4p( 1-p >? , M even • (J.10) 123 Using (J.3) (with P A - p for a l l i ) , (J.5) and (J.10) one obtains for large M £ ) 2 « - Xn { 4Q(/r T O( 1.-4)4 > * ( J . l l ) [ i - Q c / r ^ c i.-4 )'4>] ^ • Letting / FypjjC ) -5 - y » ( J . l l ) ««y rewritten as r ( 1 ) rz 2. — . (J.12) W - " o > An{4Q(y) [l-Q(y) ]} A A Equation (J.12) i s used to plot hm*~a >  a s a f u n c t i o n o f • 124 APPENDIX K In this appendix, i t i s proved that W •6-,M'° } " W ^ • M , ° } * rSAS ( > ' (K.l) From equations (5.3) and (6.13), W B.M,- ) is Implicitly defined by « ( / ! - J - J i - 1 1 M M IU.1 J , M odd M+l 3 " 2 M even (K.2) where P M - Pr {exactly j (polarity) errors i n the M samples} M M k^- l k 2-k x+l H p n (l-p k) # W l * 1 A e t k l » k 2 V keU-{k l,....k j} • n d P. » Q(/r B p D( B,*£A) |BAI 4 > • Differentiating with respect to - and lBPD then letting ̂  - 0 i n ( K . 2 ) , we obtain 125 1-1 M M I I J - - ^ k r l k 2=k x+l le{k 1 (...,k^} keU-jki k^} , M odd M rif-i / r ( . m o f * K ^ " X ̂ 1 tj-* " ^BPIT -* * ' , M+l 2 J i - l M I (K.3) (K.4) Similarly, for M even, one haB / ? * , 2 - ^ f " 1 / r B P D ( i.M ,o) X I {2CJ) - ff)} X i - l 1 * * ™ , M+2 3 1 3 i - l 1 (K.5) From (K.4) and (K.5) one ha6 M W > - CM I " i 2 i - l M , M—1,2,3,.•. (X IsJ)2 i - l 1 ( K . 6 ) for some constant C^. In particular, (K .7) Using ( 6 . 6 ) and (K .7) i n (K . 6 ) , we have W ±">° 5 " W I»M'° > ' rSAS ( ±'* > REFERENCES T.C. Tozer and J. R o l l e r S t r o m , "Penalties of hard decision in signal detection," Electron. Lett.. vol. 16, pp.199-200, Feb. 1980. V.M. Milutinovic, "Suboptimum detection procedure based on the weighting of partial decisions," Electron. Lett., vol. 16, pp.237-238, Mar. 1980. V.M. Milutinovic, "Comparison of three suboptimum detection procedures," Electron. Lett., vol. 16, pp.681-683, Aug. 1980. G. B. Lockhart, "Implementation of di g i t a l matched f i l t e r s in data receivers," Electron. Lett., vol. 10, pp.311-312, July 1974. CM. Chie, "Performance analysis of d i g i t a l i n t e g r a t e - a n d - d u m p f i l t e r s , " IEEE Trans. Commun., vol. COM-30, pp.1979-1983, Aug. 1982. H. Chang, "Presampling f i l t e r i n g , sampling and quantization effects on the d i g i t a l matched f i l t e r performance," in Proc. International Telemetering Conference, San Diego,CA, USA, Sept. 28-30, 1982, pp.889-915. N.C. Beaulieu and C. Leung, "On the performance of three suboptimum detection schemes for binary signalling," IEEE Trans. Commun.. vol. C0M-33, pp.241-245, Mar. 1985. N.C. Beaulieu and C. Leung, "Optimal detection of hard-limited data signals in different noise environments," IEEE Trans. Commun., vol. C0M-34, pp.619-622, J u n e 1986. 127 [9] N.C. Beaulieu, "Penalties of Sample-and-Sum and Weighted Partial Decision Detectors in Gaussian Noise," under review. [10] N.C. Beaulieu and C. Leung, "The Optimal Hard-Limiting Detector for Data Signals i n Different Noise Environments," i n Proc. IEEE ICC, Toronto, Canada, June 23-25, 1986, pp.32.6.1-32.6.5. [11] M. Schwartz and L. Shaw, Signal Processing: Discrete Spectral Analysis, Detection and Estimation. New York: McGraw-Hill, 1975. [12] R.W. Stroh, "An experimental microprocessor-implemented 4800 bit/s limited distance voice band PSK modem," IEEE Trans. Commun., vol. COM-26, pp.507-512, May 1978. [13] V.C. Hamacher, "Analysis o f a simplified sampled signal detector," Queen's university Research Report 65-1, Mar. 1965. [14] M. Flsz, Probability Theory and Mathematical Sta t i s t i c s. New York: J. Wiley, 1963, pp.202-211. [15] J. Patel and C. Read, Handbook of the Normal Distribution. New York: Marcel Dekker, 1982, p.50. [16] M. Abramowitz and I.A. Stegun, Handbook o f Mathematical Functions. New York: Dover, 1972, pp.931-932. [17] G.P. Wadsworth and J.G. Bryan, Applications o f Probability and Random Variables. New York: McGraw-Hill, 1974, pp.89-91. [18] C. Cherry, Pulses and Transients In Communication Circuits. London: Dover, 1950. [19] A.I. Zverev, Handbook o f F i l t e r Synthesis. New York: J . Wiley, 1967, pp.67-71. 128 [20] w. Rudin, Principles of Mathematical Analysis. New York: McGraw- H i l l , 1964. [21] R.V. Churchill and J.W. Brown, Fourier Series and Boundary Value Problems. New York: McGraw-Hill, 1978, p.84. [22] R.E. Ziemer and W.H. Tranter, Systems, Modulation, and Noise. Boston: Houghton M i f f l i n , 1976. [23] M. Rosenblatt, "A Central limit theorem and a strong mixing condition," Proc. Nat. Acad. Sci., vol. 42, pp.42-47, 1956. [24] R. Deutsch, Nonlinear Transformations of Random Processes. Englewood C l i f f s , N.J. : Prentice-Hall, 1962. [25] M. Kanefsky, "Detection of weak signals with polarity coincidence arrays," IEEE Trans. Inform. Theory, vol. IT-12, pp.260-268, Apr. 1966. [26] M.J. Levin, "Generation of a sampled gaussian time series having a specified correlation function," IRE Trans. Inform. Theory, vol. IT-6, pp.545-548, Dec. 1960. [27] J.H. Miller and J.B. Thomas, "Detectors for discrete-time signals in non-gaussian noise," IEEE Trans. Inform. Theory, vol. IT-18, pp. 241-250, Mar. 1972. [28] H. Chernoff, "A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations," Annals Math. Stat., vol. 23, pp. 493-507, 1952. [29] "R.G. Gallager, Information Theory and Reliable Communication. New York: J. Wiley, 1968. 129 [30] i . s . Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products. New York: Academic Press, 1980. [31] J.M. Wozencraft and I.M. Jacobs, P r i n c i p l e s of Communication Engineering. New York: J . Wiley, 1968. [32] N.C. Beaulieu and C. Leung, "A comparison of three suboptimum detectors for binary s i g n a l l i n g , " i n Proc. IEEE ICC, Chicago, USA, June 23-26, 1985, pp.18.A.1-18.4.6. [33] N.C. Beaulieu and C. Leung, "On ha r d - l i m i t i n g i n sampled binary data systems," North American Radio Science Meeting (URSI), Vancouver, Canada, June 17-21, 1985, p.330. [34] N.C. Beaulieu, "Penalties of Weighted P a r t i a l Decision Detectors i n Gaussian Noise," i n Proc. IEEE International Montech Conference on Antennas and Communications, Montreal, Canada, Sept. 29-0ct. 1, 1986. [35] V. M i l u t i n o v i c , "Performance comparison of two suboptimum detection procedures i n r e a l environment," IEE Proc.. v o l . 131, Pt. F, pp.341- 344, July 1984. [3 6] V.M. M i l u t i n o v i c , "Generalised WPD procedure f o r microprocessor- based s i g n a l detection," IEE Proc., v o l . 132, Pt. F, pp.27-35, Feb. 1985. [37] P.M. Schultheiss and F.B. Tuteur, "Optimum and Suboptimum Detection of Di r e c t i o n a l Gaussian Signals i n an Isotropic Gaussian Noise F i e l d Part I I : Degradation of D e t e c t a b i l i t y Due to Cl i p p i n g , " IEEE Trans, on M i l i t a r y E l e c t r o n i c s , vol.MIL-9, pp.208-211, July-Oct. 1965. GLOSSARY binary antipodal s i g n a l s two s i g n a l s , each of which i s the negative of the other binary p a r t i a l d e c i s i o n a d e c i s i o n based on a s i n g l e s i g n a l sample which has two possible outcomes d i g i t a l matched f i l t e r the optimum detector which bases i t s d e c i s i o n on a number of independent s i g n a l samples hard d e c i s i o n a d e c i s i o n which has two possible outcomes h a r d - l i m i t i n g a transformation that assigns one value to a l l p o s i t i v e arguments and a second value to a l l negative arguments M-ary s i g n a l l i n g the transmitter sends one of M signals depending on the message sequence maximum a p o s t e r i o r i p r o b a b i l i t y (MAP) r u l e the receiver chooses as i t s estimate of the transmitted s i g n a l that s i g n a l which i s most l i k e l y given the received s i g n a l penalty the d e t e r i o r a t i o n i n performance of a suboptimum detector measured r e l a t i v e to an optimum detector; the increase i n signal-to-noise r a t i o required by a suboptimum detector i n order to maintain the same error p r o b a b i l i t y as an optimum detector sample-and-sum detector the received s i g n a l i s sampled and the samples are summed s i g n a l l i n g element one of the s i g n a l waveforms sent by the transmitter weighted p a r t i a l d e c i s i o n a d e c i s i o n based on a s i n g l e s i g n a l sample which may have one of several outcomes, the larger the outcome the more heavily the decision i s weighted PUBLICATIONS N.C. Beaulieu and C. Leung, "Optimal Detection of Hard-Limited Data Signals i n D i f f e r e n t Noise Environments", IEEE Trans. Commun., v o l . COM-34, no.6, pp.619-622, June 1986. N.C. Beaulieu and C. Leung, "On the Performance of Three Suboptimum Detection Schemes f o r Binary S i g n a l l i n g " , IEEE Trans. Commun., vol.COM-33, no.3, pp.241-245, March 1985. N.C. Beaulieu, "Comment on 'Calculating Binomial P r o b a b i l i t i e s When the T r i a l P r o b a b i l i t i e s are Unequal'", J . of S t a t i s t i c a l Computation and Simulation, vol.20, no.4, pp.327-328, 1985. E.V. J u l l , N.C. Beaulieu and D.C.W. Hui, " P e r f e c t l y Blazed Triangular Groove R e f l e c t i o n Gratings", J . of the Opt i c a l Society of America, vol.1, no.2, Feb. 1984. N.C. Beaulieu, "Penalties of Sample-and-Sum and Weighted P a r t i a l Decision Detectors i n Gaussian Noise", submitted. N.C. Beaulieu, "Penalties of Weighted P a r t i a l Decision Detectors i n Gaussian Noise", IEEE International Montech Conference on Antennas and Communications, Montreal, Sept. 1986. N.C. Beaulieu and C. Leung, "The Optimum Hard-Limiting Detector f o r Data Signals i n D i f f e r e n t Noise Environments", IEEE International Conference on Communications, Toronto, pp.32.6.1-32.6.5, June 1986. PUBLICATIONS N.C. Beaulieu and C. Leung, "A Comparison of Three Suboptimum Detector fo r Binary S i g n a l l i n g " , IEEE International Conference on Communication Chicago, pp.18.4.1-18.4.6, June 1985. N.C. Beaulieu and C. Leung, "On Hard-Limiting i n Sampled Binary Data Systems", North American Radio Science Meeting (URSI), Vancouver, Canada, June 1985. E.V. J u l l , N.C. Beaulieu and D.C.W. Hui, "Dual Blazed Triangular Groove R e f l e c t i o n Gratings", IEEE Antennas and Propagation Society International Symposium, Houston, Texas, May 1983. E.V. J u l l and N.C. Beaulieu, "An Unusual R e f l e c t i o n Grating Behaviour Suitable f o r E f f i c i e n t Frequency Scanning", IEEE Antennas and Propagation Society International Symposium, Quebec, June 1980.

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