OPTIMAL WEIGHTED PARTIAL DECISION COMBINING FOR FADING C H A N N E L DIVERSITY by A L A N D O U G L A S K O T B.A. Sc., University of British Columbia, 1981 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E STUDIES Department of Electrical Engineering We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F BRITISH C O L U M B I A January 1987 © A l a n Douglas Kot, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of E U f V ^ c a l FvNry^eeA^ The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date DE-6(3/81) i i Abstract A diversity combining scheme is examined that utilizes a demodulator's hard decisions i n conjunction w i th knowledge of each decision's reliabil i ty. A maximum-likel ihood bi t decision is made, based on these partial decisions from the demod-ulator and on measurements of the state of the fading channel. The technique is sub-optimal since hard decisions are processed, but it may find applicat ion in low cost receiver design. The technique is opt imal i n the sense that a m i n i m u m probabi l i ty of bit error is achieved, given a set of par t ia l decisions and knowledge of their rel iabil i ty. Performance analysis for the case of non-coherent frequency shift keying on a slow Rayleigh fading channel w i t h additive white Gaussian noise includes the derivation of a tight upper bound on the probabil i ty of bi t error, and estimates of the asymptotic performance relative to standard diversity schemes such as majori ty-voting, selection diversity, square-law, and maximal ratio combining. These results are supported by simulat ion results for bi t and packet error rates i n an example system. W i t h five independent bit repeats and a B E R of 1 0 " s , the receiver is about 3 d B more efficient than majority-voting, and about 1 d B more efficient than selection diversity. The gain in efficiency, relative to the standard par t ia l decision combination schemes, increases w i t h the number of repeats. The degradation i n performance i n a pract ical receiver implementat ion is ad-dressed, and i t is demonstrated that near ideal performance may be obtained w i th only a few rel iabi l i ty weights quantized to a small number of levels. Furthermore, this performance is maintained over a wide range of average signal to noise ratio wi thout having to adapt the rel iabil i ty weighti,. W h e n the rel iabil i ty estimate is corrupted by additive white Gaussian noise, it is demonstrated that simple low-iii pass filtering of the signal strength estimate is sufficient to obtain near ideal perfor-mance. The performance is degraded in the presence of cochannel interference, but for a moderate level of interference the performance is demonstrated to be superior to majority-voting or selection diversity. Other results include a method to estimate the optimal quantization thresholds, and a method to obtain the probability of error of selection diversity receivers employing signal to noise ratio measurement quantization. The selection diversity analysis is applicable to the more general case of Rician fading. iv Table of Contents A b s t r a c t i i L i s t of Figures v i L i s t of Tables v i i i Acknowledgements ix 1 I n t r o d u c t i o n 1 1.1 Background 1 1.2 O p t i m a l Diversi ty Combin ing 4 1.3 Thesis Organizat ion 6 2 I d e a l i z e d WPD R e c e i v e r 8 2.1 W P D Receiver Concept 8 2.2 O p t i m a l Weighting of Pa r t i a l Decisions 9 2.3 Discussion 13 2.4 Analysis of W P D Receiver Performance 15 2.4.1 Methods for an Ana ly t i c a l Solution 15 2.4.2 Bounds on the W P D B i t E r ro r Rate 17 2.5 Compar ison of W P D and Tradi t ional Diversi ty Schemes 20 2.5.1 Asympto t i c B E R Characteristics 23 2.5.2 Efficient Use of Branch Information 24 2.5.3 O p t i m a l Number of Diversi ty Branches 27 2.6 Simulat ion 33 2.6.1 Simulat ion M o d e l 33 2.6.2 Simulat ion Descript ion 39 2.6.3 Simulat ion Parameters 41 2.6.4 Simulat ion Results 43 2.7 E r ro r Rate Approximat ions 47 TABLE OF CONTENTS v 3 Some Issues in Pract ica l W P D Receiver Design 53 3.1 SNR and Weight Quantization 53 3.1.1 B E R Evaluation Method 54 3.1.2 Threshold Optimization 57 Capacity Method 57 Chernoff Bound Method 62 3.1.3 Performance With Quantized SNR 64 3.1.4 Performance with Quantized Weights 73 3.2 SNR Estimation 76 3.2.1 Receiver Model 76 3.2.2 Simulation Method, A W G N 80 3.2.3 Signal Strength Estimation 83 3.2.4 Simulation Method, Cochannel Interference 88 3.2.5 Simulation Results 93 4 Conclusions 105 References 108 A p p e n d i x A 112 A p p e n d i x B 115 A p p e n d i x C 122 A p p e n d i x D 125 v i List of Figures 1.1 Example of Rayleigh Fading 3 2.1 Weighted Par t i a l Decision Diversi ty Receiver 12 2.2 Square-Law N C F S K Diversi ty Receiver 21 2.3 M a x i m a l - R a t i o N C F S K Diversi ty Receiver 22 2.4 B i t E r ro r Ra te vs. Number of Diversi ty Branches 26 2.5 Chernoff B o u n d Exponent Functions 30 2.6 B E R vs. Number of Diversi ty Branches, F i x e d Eh/N0 32 2.7 A M P S L a n d to Mobi l e Cont ro l Channel Format 34 2.8 Normal ized Autocorre la t ion Funct ion of the Received Envelope . . . 42 2.9 B i t E r ro r Rates for 3 and 5 Repeats 45 2.10 Packet E r ro r Rates for 5 Repeats 46 2.11 W P D B i t Er ro r Rate B o u n d and Simula t ion Results 48 2.12 Uncorrected Packet E r ro r Rate 51 2.13 Single B i t E r ro r Corrected Packet E r ro r Rate 52 3.1 S N R Quant izat ion 54 3.2 Discrete Channel M o d e l Paths Corresponding to a Repeated ' 1 ' . . . 55 3.3 Discrete Memoryless Channel M o d e l 58 3.4 Discrete Channel M o d e l Capaci ty 61 3.5 Compar ison of Single Threshold Values 65 3.6 Compar ison of Er ro r Rates using a Single Threshold 66 3.7 B E R w i t h One Threshold 67 3.8 B E R w i t h Three Thresholds 67 3.9 Convergence of W P D B E R 69 3.10 Opt imized and F i x e d Threshold Performance 70 3.11 Convergence of Selection Diversi ty B E R 72 3.12 Weight Quant izat ion Characteristic 73 LIST OF FIGURES v i i 3.13 Quant ized Weight B E R 75 3.14 N C F S K Receiver, Bandpass F i l te r Implementation 78 3.15 Bandpass F i l t e r N C F S K Receiver, Block Diagram 78 3.16 Frequency Response of Signal Strength Es t imator F i l t e r 87 3.17 Tracking Behaviour of the Signal Strength Est imator 87 3.18 Phasor Representation of the IF Fi l te r Envelope Detector Output . . 89 3.19 W P D B i t E r ro r Rate w i t h S N R Es t imat ion 94 3.20 W P D B i t Er ro r Rate w i t h S N R Es t imat ion , SIR=15 d B 96 3.21 W P D Packet Er ro r Rate w i th S N R Es t imat ion 98 3.22 W P D Packet Er ro r Rate w i th S N R Est imat ion , S I R = 15 d B . . . . 99 3.23 Selection Diversi ty B E R w i t h S N R Es t imat ion 101 3.24 Selection Diversi ty B E R w i t h S N R Es t imat ion , S I R = 15 d B 102 3.25 Selection Diversi ty P E R wi th S N R Es t imat ion 103 3.26 Selection Diversi ty P E R w i t h S N R Es t imat ion , S I R = 15 d B 104 A . l A Linear B o u n d on x1/2 113 A . 2 Ratios of T w o Bounds on the Moment Generating Funct ion to the Numer ica l ly Computed Result 114 D . l Quant ized S N R Probabi l i ty Density Funct ion 126 D.2 Selection Diversi ty Quantized S N R Cumula t ive Dis t r ibu t ion Func-t ion 127 v i i i List of Tables 2.1 Asympto t i c Loss of Square-Law, Selection, and W P D Combining Relat ive to M a x i m a l - R a t i o Combin ing 25 2.2 Es t imated and Exac t Values for the O p t i m u m Number of Branches . 31 ix Acknowledgements I would like to sincerely thank my supervisor, D r . C . Leung, for his constant availabil i ty and support. I also wish to thank Glenayre Electronics L t d . for generously making available their computing facilities, and for financial support i n the form of a Glenayre Elec-tronics L t d . Fellowship. Add i t iona l financial support from N S E R C grant A-1731 and a N S E R C Postgraduate Scholarship is gratefully acknowledged. 1 Chapter 1 Introduction 1.1 Background Methods for achieving reliable and efficient mobile digi ta l communications have received considerable attention in the technical literature. M u c h of this theoret-ical and empir ical knowledge is applied in pract ical engineering designs to arrive at a best system design. Th i s best solution is a compromise between many fac-tors, including various performance cri teria, hardware cost, reliabil i ty, design ef-fort, and compat ibi l i ty w i th present and future systems. A s part of the effort to opt imize the system design, error rate performance is often sacrificed by implement-ing theoretically sub-optimal demodulators. Th i s thesis introduces and analyzes a sub-opt imum digi tal receiver that may be added to the designer's repertoire of techniques for mobile digi ta l communications. In urban or suburban environments, digi ta l communications over V H F / U H F mobile radio channels is severely impaired by mul t ipa th induced fading [1]. Loca l buildings and other objects act as scatterers that can provide mult iple propagation CHAPTER 1. INTRODUCTION 2 paths to the receiver. Unfortunately, each path has its associated attenuation and delay so that the resulting sum of these signals at the receiver will exhibit the effects of destructive and constructive interference. As the mobile receiver moves through this interference pattern the magnitude and phase of the resultant R F signal wil l fluctuate1. For urban areas the received signal energy is often modeled as the resultant of the superposition of a large number of scattered signals arriving from equally likely directions, with no line of sight (LOS) component [2]. In this model the received signal amplitude is Rayleigh distributed. A more general model incorporates a static LOS component in addition to an infinite number of scatterers and leads to a received signal amplitude that is Rician distributed [3]. The Rician distribution reduces to a Rayleigh distribution as the LOS component is decreased to zero. An additional variation in the received signal strength is due to changes in the nature of the propagation paths. Significant variations in the large scale geography, e.g. hills, can shadow the receiver from the transmitter. With this shadowing effect the logarithm of the received signal amplitude is usually modeled by a normal distribution and the rate of change of signal strength is slow relative to Rayleigh fading [4]. We will concentrate on the performance of receivers over small regions in urban centers. The appropriate model therefore is a Rayleigh distributed received signal amplitude. The typical behaviour of a Rayleigh fading channel over time is shown 1 Even if the receiver does not move it is possible to have a time variation of the received signal due to moving reflectors. For example, moving trucks or buses may reflect signals towards a stationary receiver [5]. CHAPTER 1. INTRODUCTION 3 0 1 2 3 4 5 d = METERS 0 0.2 0.4 0.6 0.8 1.01" SECONDS Figure 1.1: Example of Rayleigh Fading. Carrier frequency 850 MHz. (Figure taken from [4].) in Fig. 1.1. Fades occur at a rate that increases [4] with the Doppler frequency fo = v/X (1.1) where v is the vehicle velocity and A is the signal wavelength 2 . Thus the fade rate increases with the vehicle speed and signal frequency. Diversity techniques are often used to combat fading by combining information received from different branches. A branch is the virtual transmitter to receiver propagation path formed by the superposition of the multitude of individual paths. If the branches fade independently it is unlikely that they wil l all experience a deep fade simultaneously, and it is thus hoped that a more reliable decision may be made. 2 T h e D o p p l e r frequi ncy is the m a x i m u m Doppler frequency shift experienced by any ind iv idua l ray as it arrives at the vehicle. CHAPTER 1. INTRODUCTION 4 Diversi ty schemes may be classified according to the source of the branch in -formation. For example, space, time, and frequency diversity obtain mult iple ver-sions of the signal v i a spatially separated antennas, repeated signal elements, and transmission over mult iple frequency bands respectively. Ideally, the fading on the branches would be statistically independent, which would be true for sufficiently separated antennas, t ime samples, or frequency bands, so that the correlation be-tween branches is negligible. The effectiveness of a diversity system is reduced when the branches are correlated, but substantial gains are s t i l l achievable w i t h consid-erable correlation. For example, at a target B E R of 10~ s a two branch square-law receiver w i t h a power correlation coefficient of 0.6 achieves 85% of the d B gain provided by two completely independent branches 3 . Effective use of the branch information is determined by the combining and detection methods used. Tradi t ional methods of diversity combining, maximal-ratio, equal gain, selection, and square-law combining are discussed and analyzed i n [1,6,7,8,9,10]. 1.2 Optimal Diversity Combining In general there are different optimal combining methods, each corresponding to the part icular noise statistics, the amount of channel state information available, and the modula t ion/demodula t ion methods employed. For fading in additive white Gaussian noise ( A W G N ) the best possible combinat ion of the branch signals is obtained w i t h maximal-rat io combining. This opt imal linear combining method 3This dB gain is with respect to single branch reception of non-coherent frequency shift keying. These results are obtained using Figure 2 of [7] and (2.28). CHAPTER 1. INTRODUCTION 5 achieves the highest possible instantaneous S N R and also the lowest B E R . However, it requires the perfect measurement and correction of relative phase errors of each branch, so that the R F signal may be added coherently, and further requires that each branch be weighted by the ratio of its received signal ampli tude to noise power. In the absence of amplitude and phase measurements for the respective branches, the appropriate op t imum receiver for non coherent frequency shift keying ( N C F S K ) i n slow fading 4 uses a square-law combiner [9]. The relative performance of these schemes w i l l be discussed later. Another possible approach is to util ize a set of hard decisions derived from the branches 6 . A commonly used method for t ime diversity is majority vot ing, where each bi t is repeated an odd number, L , of times. The receiver makes a decision on each bit repeat, and then decides i n favor of the majority of these L par t ia l decisions. If channel state measurement information is available, then another pos-sible approach is to use selection diversity, where the only decision retained is that corresponding to the branch wi th the largest signal to noise ratio. Another opt ion is erasure, where the par t ia l decisions from the poorest S N R branches are ignored, and the remaining par t ia l decisions are considered equally important [4,11]. How-ever, neither of the latter two schemes use all of the available par t ia l decisions and one might expect some improvement if a l l of the par t ia l decisions were com-bined intelligently. Th i s thesis introduces and analyzes the optimal combination scheme for par t ia l decisions on a fading channel. The method consists of combining the par t ia l decisions according to their rel iabil i ty to y ie ld a m a x i m u m likelihood 4 The term slow fading refers to the case in which the received amplitude and phase are almost constant over the bit period. The condition of slow fading is assumed throughout this thesis, unless stated otherwise. 5 We shall refer to a hard decision made on a particular branch as a partial decision. CHAPTER 1. INTRODUCTION 6 decision. While the proposed technique may not be expected to perform as well as maximal-ratio or square-law combining the technique may be expected to yield an improvement over commonly used methods of partial decision combining. 1.3 Thesis Organization The remainder of this thesis is organized as follows, with the principal results of each chapter listed below. Chap t e r 2 introduces and analyzes the optimal weighted partial decision (WPD) receiver with the assumption that the knowledge of the channel state is perfect. Specific results obtained for N C F S K are; • Some useful bounds on the bit error rate are developed and used to compare the W P D receiver with various traditional combining schemes. • The optimal number of diversity branches given a fixed energy per bit is estimated. • Simulation results are presented to provide examples of the W P D receiver performance and to illustrate the accuracy with which the B E R is predicted by an upper bound derived earlier. Chap t e r 3 demonstrates the robustness of the W P D receiver when various degradations are encountered in a practical receiver design. Results include; CHAPTER 1. INTRODUCTION 7 • A n efficient method for opt imizing the W P D receiver thresholds when the S N R is quantized. • Examples of the degradation in B E R performance when the S N R and weights are quantized. • A n expression for the B E R of selection diversity when the S N R is quantized. • Examples of the performance degradation expected when the channel state estimation is corrupted by A W G N or cochannel interference. C h a p t e r 4 contains a summary of the main results presented i n the thesis, as well as a brief discussion on some related topics. 8 Chapter 2 Idealized W P D Receiver 2.1 W P D Receiver Concept A s discussed in the introduct ion, the basic concept of the W P D receiver is to weight all of the par t ia l decisions according to their reliabili ty. Whi l e we have not as yet mathematical ly defined reliabili ty, the intuit ive basis for the combination scheme is straightforward. For example, on a Rayleigh fading channel the probabil i ty that a bi t decision w i l l be wrong during a deep fade w i l l be v i r tua l ly 1/2. Clear ly there is l i t t le value i n u t i l iz ing this part icular decision. F r o m an information theoretic view we can model the fading channel as a t ime varying binary symmetric channel ( B S C ) , and i f we consider a sub-channel formed by considering only the deeply faded signal, then the information carrying capacity of this sub-channel is v i r tua l ly zero 1 . O n the other hand, consider a s i tuat ion in which the received signal is quite h igh, say 20 d B above the noise. Then the probabil i ty that a bi t decision is correct w i l l be nearly one, and the decision should be considered completely reliable. This 1This division of the fading channel into several sub-channels of varying capacity will be made mathematically explicit in section 3.1.2.1. CHAPTER 2. IDEALIZED WPD RECEIVER 9 case corresponds to a sub-channel w i th a capacity of v i r tua l ly one information bit per channel b i t , so that no redundancy is required to arrive at an error free decision. For the cont inuum of channel conditions encountered on a typical fading channel our intuit ive model of the W P D receiver would then weight bi t decisions by appropriate confidence, or reliabili ty, factors. A t the outset of this thesis work the expected gain due to a rel iabi l i ty weighting of the par t ia l decisions was investigated using some ad-hoc weighting schemes. Some of the possible candidates for the weighting functions included; • a weight function linear w i th the B S C crossover probabili ty, • a weight function equal to the received S N R , • a weight function equal to the capacity of the B S C . The performance of the capacity weighting scheme was estimated v i a simulat ion and the results indicated that the basic concept of rel iabil i ty weighting showed some promise. Rather than use an ad-hoc scheme, the question of how to optimally weight the par t ia l decisions is discussed in the next section. 2.2 Optimal Weighting of Partial Decisions Consider a set of L par t ia l decisions, { d , } ^ , made on a message bi t , m G {0,1}, where 1 i f the ith decision is a '1 ' 1 otherwise. (2.1) CHAPTER 2. IDEALIZED WPD RECEIVER 10 The maximum a posteriori probability (MAP) decision rule is to choose rh = 1 if Pr ({a\}f=1 | m = l ) Pr(m = l ) > Pr ({<*,}f=1 | m = o) Pr{m = 0) (2.2) where Pr(m — k) |t=o,i is the probability of sending m = k. This M A P decision rule is optimum in the sense of minimizing the probability of bit error. If the mes-sages are equally likely, then the M A P decision rule corresponds to the maximum likelihood (ML) decision rule, P r ({d,}f = 1 | m = l ) > Pr ({<f,}f=1 | m = o) . (2.3) It is shown in [12] that if the L partial decisions, {di}f=1, are independent, then the M L decision statistic is the weighted sum L DL = X>,4, (2.4) i=i where the weights, tw,-, are given by Wi = c ln ^ p,^1) , c any positive constant (2.5) with pi being the probability that the ITH decision is incorrect. The optimal decision rule is to decide '0' was sent if < 0. If DL > 0 then '1' is chosen. The optimal decision statistic (2.4) and weights (2.5) were derived in [12] for application in the processing of multiple hard limited samples, i.e. partial decisions, per bit. The principle is that if a signalling waveform is not rectangular, then the partial decisions should not be processed with equal weights. For example, partial decisions made on portions of the signal waveform that are of low amplitude, e.g. the beginning of a raised cosine pulse, are intuitively less reliable than partial decisions based on the high amplitude portions of the signal. Knowledge of the CHAPTER 2. IDEALIZED WPD RECEIVER 11 signall ing waveform and the pdf of the noise can then be used to calculate the op t imal weights i n (2.5). In our applicat ion we wish to know what the opt imal decision rule is for part ial decisions made on a number of bi t repeats undergoing fading. Th i s problem is closely related to that of [12]. We assume that the noise samples corrupt ing the ind iv idua l bi t transmissions are statistically independent. Each par t ia l decision made by the demodulator w i l l have a part icular probabil i ty of error depending on the channel conditions for that bi t repeat. Then for a set of L bit repeats, the L b i t decisions constitute a set of par t ia l decisions on a part icular message bi t , as i n [12], except that the probabil i ty of error for each par t ia l decision is no longer a quantity that is known a p r i o r i from the signalling waveform and the pdf of the noise. Rather , the probabil i ty of error, p,-, for each par t ia l decision, di, is to be determined from the state of the fading channel. Thus , for a particular set of L par t ia l decisions, {di}^, and the set of corresponding demodulator b i t error probabili t ies, {pi}i=i , determined from the fading channel measurements, the m a x i m u m likelihood decision statistic and weights are (2.4) and (2.5). O u r problem then is to measure the state of the fading channel so that an accurate evaluation of the demodulator error probabili ty, pi, and i n tu rn the weight, Wi, may be made. Since the B E R is directly related to the received S N R , it would be natural to measure the received signal level during the fading process. The S N R may then be estimated from the ratio of the measured received signal power to the noise power 2 . 2 If the receiver noise is dominated by internally generated front-end thermal noise, then a laboratory bench measurement of the intermediate frequency (IF) filter output power with no input signal will be a good estimate of the received noise power in actual use. As will be discussed later, external noise will adversely affect the weight determination. CHAPTER 2. IDEALIZED WPD RECEIVER 12 L Branch Signals. DEMODULATOR B l Decision SIGNAL STRENGTH WEIGHT EVALUATION MEASUREMENT Figure 2.1: Weighted Partial Decision Diversity Receiver. The W P D receiver structure is shown in figure 2.1. Rather than explicitly calculate the weights from the received SNR the receiver could make use of a lookup table indexed by the received signal level. A sufficiently fast received signal level measurement is readily available in some receivers by taking the output of the already existing average received power indicator prior to low pass filtering [13]. Practical issues involving the accuracy required of the weight determination procedure wil l be discussed in Chapter 3. CHAPTER 2. IDEALIZED WPD RECEIVER 13 2.3 Discussion Major i ty-vot ing , selection of the strongest signal, and erasure of the weaker signals were discussed earlier as combination methods that ut i l ize par t ia l decisions. It is interesting to view them as special cases of rel iabil i ty weighting. Major i ty-vot ing s imply assigns equal weights to al l of the part ial decisions. W i t h unit weights, (2.4) implements majority-voting as an up /down counter. Selection diversity assigns a non-zero weight only to the branch w i t h the strongest instantaneous S N R . Thus the single most reliable decision is used. Final ly , the erasure method ignores those branches w i t h a sufficiently low S N R and treats a l l the others as equally reliable. Since the t radi t ional methods of majority-voting, selection, and erasure are in fact non-optimal weighting schemes, we can therefore predict equal or improved performance using the op t imum W P D receiver. For the special case of two branches, the combination of two weighted par t ia l decisions w i l l result in the decision being made in favor of the par t ia l decision w i t h the higher weight. Since this higher weight corresponds to the branch w i t h the higher S N R , we see that, for L = 2, a W P D combiner is equivalent to selection diversity. Th i s equivalence does not require that the W P D receiver use opt imal weights. A n y rel iabil i ty weighting scheme that has weights monotonical ly increas-ing w i t h S N R w i l l be equivalent to selection diversity, for L = 2. In most applications the vehicle speed, or fading rate, is not constant. The choice of a method for combatt ing the effects of fading should be influenced by the dis t r ibut ion of fading rates that w i l l be encountered. We can make some observations on the performance of these various systems as the fade rate is varied. CHAPTER 2. IDEALIZED WPD RECEIVER 14 At high Doppler frequencies the received signal level wil l be almost independent from bit repeat to bit repeat. This condition is convenient mathematically and is usually assumed for the performance evaluation of diversity schemes. Expressions for the B E R are sometimes obtainable in this case and these will be reviewed in the next section. As the fade rate is lowered the bit repeats will no longer be independent and the effective gain of the diversity system is decreased. Analytical solutions for this case of correlated branches are rare [7,8]. As the fade rate is decreased still further, so that for a set of repeats the signal level is constant, the B E R will be determined by averaging the error rate obtained for this fixed SNR over the fading SNR distribution. Note that as this very slow fading rate is approached the weights for a set of bits received in the WPD combiner will be practically equal. Thus the W P D receiver performance wil l be equivalent to majority-voting at these very slow fade rates. Again, this result is independent of the optimality of the weights used. An alternative view of the majority-voting scheme at very slow fade rates is that it is optimal. Corresponding observations may be made for selection diversity at very slow fade rates. In this case the receiver wil l choose one of L equally reliable decisions. Clearly this is not the best use of equally reliable decisions since many equally good decisions have been ignored. CHAPTER 2. IDEALIZED WPD RECEIVER 15 2.4 Analysis of WPD Receiver Performance Thi s section discusses the estimation of the W P D receiver B E R when N C F S K is used on a Rayleigh fading channel corrupted by additive white Gausssian noise. 2.4.1 Methods for an Analytical Solution A n exact expression for the W P D receiver B E R has not yet been obtained. Below we briefly outline some attempts to determine the exact B E R , but stop short of a complicated series solution i n favour of a tight upper bound derived in the next section. For a Rayleigh distr ibuted received ampli tude the S N R at the receiver w i l l be exponentially distr ibuted w i th mean 7 0 > as [H] The receiver S N R is defined as T = ES/NQ, where E§ is the received signal energy i n a bi t per iod, and NQ/2 is the two-sided noise power spectral density. A n ideal b inary N C F S K receiver, w i th S N R 7, w i l l make an error w i th probabil i ty [9] Assume that a ' 1 ' is t ransmitted a number, L , of times. T h e n for each possible value of S N R , 7, the weighted par t ia l decision variable, X = to.d,-, has two values, i.e., /r(-y) = 1/70 c - ^ 0 , 7 > 0. (2.6) p.(7) = 1/2 e->l\ (2.7) ,wi th pdf value of f^l) ,wi th pdf value of fie{i) (2.8) CHAPTER 2. IDEALIZED WPD RECEIVER 16 where A« = /r(Tf)[l-P.(-Y)] (2-9) Ae = Ml)p.{l)' (2-10) Then fx{x) is obtained from a transformation of (2.9) and (2.10) using (2.8), i.e. [14] fx(x) = I A.[T(*)]|V(*)I >* < 0 fell) where 7(1) is the inverse function of (2.8). Performing the transformation in (2.11) yields, / 2 1 + 2/TO e(l+2/-7o)s 70 (i + e*)2+2^ '* < ° fx{x) = <{ 2 l + 2 / 7 Q e - 2 x / 7 0 (2.12) (1 + . - ) ' 70 . - « \ 2+2/^0 , X > 0 The probability of bit error is P . W d = Pr{DL<0) = /° /„ L(y)<fy (2.13) J-00 where //) t(y) is the pdf of the final weighted decision variable, D^, as denned in (2.4). If the bit repeats are separated by at least the reciprocal of twice the Doppler frequency, the weighted partial decision variables may be considered to be independent [1] and, fDL(x) = fx{x)*fx{z)*-.-*fx(x\ (2.14) L where * denotes convolution. However, computation of Pe,Wpd via direct convolution is unwieldy for L > 2, and one may attempt to use the moment generating function of X, Gx[s) = E[e~"] = j~j-tvfx{y)dy. (2.15) CHAPTER 2. IDEALIZED WPD RECEIVER 17 For L independent repeats, the moment generating function of the final weighted decision variable, DL, is Gx{s). Taking the inverse transform of Gx(s) and substi-tuting this in (2.13) yields Pe,v,pd = —.\ / e"GLx{s)dsdx . (2.16) J-oo J-joo It can be easily shown that Gx(s) = 1 + ^ 1 0 [Bi (1 + 2/70 + s, 1 - s) + B J ( 2 / T O - * , 2 + «)] (2.17) where B0(a, b) is the incomplete Beta function [15,16]. The direct evaluation of (2.16) appears intractable due to the nature of the incomplete Beta function. A l -ternatively, one may return to the derivation of the optimal decision rule [12] and attempt to find the pdf of the the product of the L random variables ( i ^ y (,18) using the Mell in transform [17,18,19]. Unfortunately, the expression for the Mell in transform is also in the form of an incomplete Beta function. A series approxima-tion approach could be taken by expressing the incomplete Beta function (2.17) as a hypergeometric series, [15], but the large number of terms arising from raising a truncated hypergeometric series to the Lth power would best be handled by an algebraic manipulation program such as REDUCE, or MACSYMA [20,21]. Rather than use a complicated series approximation for the WPD B E R we will develop a tight upper bound. 2.4.2 Bounds on the W P D Bit Error Rate We will find two upper bounds on the WPD bit error rate. The first, based on the Chernoff bound, is an easy to evaluate expression, but unfortunately rather CHAPTER 2. IDEALIZED WPD RECEIVER 18 loose. The second bound is a somewhat more awkward expression to evaluate, but it wil l be shown later to be very tight and can thus provide an accurate means of computing the W P D bit error rate analytically. The procedure to find a Chernoff bound is to first find s = s* that minimizes the moment generating function defined by (2.15). Then i%pd < [Gx{s*)]L. (2.19) In our case Gx(s) = f°° e-»fx(y)dy J—oo = / ; 17TZm dy. (2.20) 7o Jo (1 + e -y ) 2 + 2 h o V ; Inspection of (2.20) reveals that the integral exists only for -2 / ^ 0 < s <1 + 2/f 0, and also that the integral is symmetric with respect to s — 1/2. Thus the minimum value of (2.20) occurs at s* = 1/2. Substituting s* and v = e~v into (2.20) we obtain 22+2/iQ /-i dv ( 5 j ~ 7o io ^ / ' - ^ ( l + BJHVfo- V-21> Unfortunately, the integral in (2.21) is not expressible in terms of elementary functions. In Appendix A it is shown that a good upper bound on (2.21) is given by 4 n+5l0ywL 4 + 7o V ,4 + 37o/ (2.22) We may find a much tighter upper bound on the WPD bit error rate as follows. Since the weights given by (2.5) are optimum, the use of any other weighting scheme will result in some performance loss. The amount of the loss incurred will depend CHAPTER 2. IDEALIZED WPD RECEIVER 19 on the sensitivity of the WPD receiver to the weight accuracy. It is demonstrated later that the W P D receiver is quite insensitive to the weight accuracy, and we can thus hope to find a good upper bound on the WPD bit error rate if we can find a good approximation to the optimal weighting function and if the approximation leads to a tractable formulation. We find a reasonable approximation to the optimal weighting function by con-sidering its behaviour as the SNR is varied. The optimum weighting function for N C F S K is obtained by substituting (2.7) into (2.5), w = ln(2e l / 2 - 1). (2.23) For large signal to noise ratios the optimal weights approach w = ln2 + 7/2 « 7/2. (2.24) The behaviour of the optimal weights at small signal to noise ratios may be found by first re-writing (2.23) as tu = ln[l + (2e'T/2 - 2)]. (2.25) Using series expansions for ln( l + z) and t1^ in (2.25), and retaining only the terms up to the first powers of x it is easily shown that, for small SNR, Wi » 7. (2.26) We see that for both high and low SNR the optimal weighting function is pro-portional to the SNR, 7. It seems reasonable then to approximate the optimal weighting function by a linear function of SNR and then attempt to obtain an analytical solution. The tightness of the bound can then be checked by comparing its prediction of B E R with that obtained by simulation. CHAPTER 2. IDEALIZED WPD RECEIVER 20 Recall that a scaling of the weighting function will not affect its performance, so we will take u\- = 7,- and and attempt to carry out the exact B E R analysis. In Appendix B it is shown that the solution is e,wpd — Pt,wpd \w=t — (2 + T b ) L f c i 2L- k-L (-1)^2*^-1-2 \4 + lo) L-k L-k-i (2.27) Equation (2.27) is valid for L > 2, and it is exact for L = 2. In Section 2.7 the (2.27) wil l be demonstrated to yield an accurate estimate of the W P D bit error rate. 2.5 Comparison of WPD and Traditional Diver-sity Schemes In this section we shall make some general comparisons for NCFSK diversity re-ception using the W P D receiver versus the traditional partial decision combining schemes of majority-voting and selection diversity. These schemes wil l also be compared with the analog square-law and maximal-ratio receivers. For reference, the block diagrams of the WPD , square-law, and maximal-ratio receivers are shown in Figures 2.1, 2.2, and 2.3 respectively. Block diagrams for the selection diversity and majority-voting receivers are not shown separately since they may be considered as special cases of the WPD receiver. First, we note that the B E R of an ideal NCFSK receiver over the Rayleigh CHAPTER 2. IDEALIZED WPD RECEIVER 21 L Branch Signals. FILTER MATCHED TO fj SQUARE-LAW ENVELOPE DETECTOR FILTER MATCHED TO f 2 SQUARE-LAW ENVELOPE DETECTOR B l Decision Figure 2.2: Square-Law NCFSK Diversity Receiver fading channel is P = E[p^)} = £p.h)frb)*, = ^ . (2.28) The B E R for majority-voting on an odd number,L, of independent bit receptions is simply the probability that more than half of the partial decisions are in error. With an even number of repeats, there is the possibility of a tie. We have, Pe,mv — ' ' ( L/2 ) P l " ^ ~ P ' " ' + 1 ? , ( * ) " ' L , L odd. even. (2.29) CHAPTER 2. IDEALIZED WPD RECEIVER 22 L Branch Signals. SIGNAL AMPLITUDE MEASUREMENT CO-PHASING AND SUMMING SIGNAL PHASE MEASUREMENT DEMODULATOR BI Decision Figure 2.3: Maximal-Ratio NCFSK Diversity Receiver The B E R for selection diversity is [7,9] Pc,.d = - L = — . (2.30) 11(70 + 2*) 2lI(* + Tb/2) The analog techniques of square-law and maximal-ratio combining have bit error rates [7,9] P^i = P L L £ ( L + kk~1)(l-P)L-k (2.81) Pt<mr = 2L~lPL. (2.32) CHAPTER 2. IDEALIZED WPD RECEIVER 23 2.5.1 Asymptotic B E R Characteristics We consider the behaviour of these combination techniques for large signal to noise ratios. In this case the dependence of B E R on 7 0 is 3 ?\L/2 ) P i w H I To" 1(^ /2)!] , 7o » -L ,L odd , 7o > L , L even > £ - l Pc,tql p •*• e,mr L! ,7o > i ( 2 L - 1 ) ! 1 To L ! ( L - 1 ) ! 7 0 L 2 i - i ,70 > L 7oL , 7 o » ^ (2.33) (2.34) (2.35) (2.36) Thus w i t h selection, square-law and maximal-rat io diversity the B E R decreases inversely w i t h the Lth power of 7 0 , while for majority-voting w i t h L odd the exponent of 70 is —1/2 ( L + l ) . For majority-voting w i t h L even, the exponent of 7o is — L/2. We expect that the W P D receiver B E R should also decrease inversely w i t h the Lth power of 70. T h i s is confirmed by evaluating (2.22) or (2.27) for large 7o. Equa t ion (2.22) gives s imply wpd < 10 (2.37) A t high signal to noise ratios the asymptotic increase i n S N R required to achieve the same error rate for most of these systems may be evaluated. The loss incurred by using square-law or selection diversity receivers instead of maximal-rat io com-3 The form of Pe,,,j in (2.35) is obtained from the leading term in an alternate expression for (2.31) derived in [7].' CHAPTER 2. IDEALIZED WPD RECEIVER 24 binat ion is obtained by equating (2.35) and (2.34) to (2.36), which yields, 1»ql (2L-1)! (2.38) 7mr [L\{L-l)\2L~\ and, for selection diversity, TT = M*' (2-39) We may also evaluate (2.27) as 70 —• °°> to obtain a slightly pessimistic estimate of the loss of the W P D receiver relative to maximal-rat io combinat ion. Th i s loss estimate is i i The results of evaluating (2.39), (2.38) and (2.40) for several L are summarized i n Table 2.1. It can be seen that the W P D receiver suffers less than 1.1 d B loss relative to the square-law receiver for L < 15. Selection diversity suffers a greater loss as L is increased, and is 5.6 d B worse than square-law combining w i t h L = 15. 2.5.2 Efficient Use of Branch Information The B E R of the various combination schemes may also be compared as a function of the number of branches to see how effectively the branch information is used. The bound of (2.27) and the bit error rates as given by (2.29) through (2.32) CHAPTER 2. IDEALIZED WPD RECEIVER 25 Number of Branches L Loss [dB Square-Law Selection WPD (bound) 2 0.9 1.5 1.5 3 1.3 2.6 2.1 4 1.6 3.5 2.5 5 1.8 4.2 2.7 6 1.9 4.8 2.9 7 2.0 5.3 3.0 8 2.1 5.8 3.15 9 2.2 6.2 3.23 10 2.26 6.6 3.30 11 2.31 6.9 3.36 12 2.35 7.2 3.41 13 2.39 7.5 3.45 14 2.42 7.8 3.49 15 2.45 8.1 3.53 Table 2.1: Asymptotic Loss of Square-Law, Selection, and WPD Combining Rela-tive to Maximal-Ratio Combining are plotted in Figure 2.4 4 . It can be seen that significant gains for the WPD receiver are indicated by (2.27) as L is increased. Selection diversity will eventually offer no further improvement as L increases, while the BER for majority-voting and the WPD receiver continue to decrease. The BER of majority-voting decreases in a stepwise fashion with L. Voting on an even number of branches offers exactly zero improvement over using one less branch5. The WPD receiver makes the most efficient use of the branch 4 The formulas of (2.27), (2.29), and (2.31), are awkward to compute directly for large L since the factorial terms in the binomial coefficient become very large. One may avoid numerical over-flow by computing the binomial coefficient as Ni^Lif)i = exp{ln[Af!] - ln[N\] - ln[(M - N))}} = «P{MIL»i *] - Mn{Ll j] - InlJlJ™ k}} = explXJlx ln[t] - £ ? = 1 ln[y] - ln[k)}. 5This arises from the properties of the binomial distribution, and a proof may be found in [23], for example. CHAPTER 2. IDEALIZED WPD RECEIVER 26 1 5 10 15 20 25 30 35 40 Number of Diversity Branches Figure 2.4: Bit Error Rate vs. Number of Diversity Branches. information of any of the partial decision schemes. As one might expect, the analog combination schemes are more efficient than the partial decision combiners. In particular, it can be shown [22] that a L branch square-law combiner achieves the same probability of error as a 2L — 1 branch majority-voting combiner. CHAPTER 2. IDEALIZED WPD RECEIVER 2.5.3 Optimal Number of Diversity Branches 27 In the previous section we considered the improvement in B E R obtained by the various combiners when an increasing number of diversity branches is used. This compared their performance as more and more energy was invested in the trans-mission of a single message bit. Consider instead that the total amount of energy transmitted per message bit is fixed. Then, as an additional branch is added the av-erage power per repetition will be decreased (assuming a fixed repetition duration), but the additional branch increases the likelihood of avoiding fades. For the partial decision combiners we note that on one hand, each partial decision wil l become less reliable as more branches are added, but on the other hand we have more decisions on which to decide the final outcome. These two tendencies have opposite effects on the B E R and one would expect that an optimum number of branches exists to achieve a best compromise. The square-law combiner can also be shown [11] to have an optimum number of branches, since it employs non-coherent combination of the branch signals. This is in contrast to maximal-ratio combining which makes the best possible use of the branch signals via coherent combining. It can be shown [9] that in maximal-ratio combining an optimum number of repeats does not exist since additional branches never decrease performance. Previous results exist for the optimal number of diversity branches for selection diversity [24] and square-law receivers [11,25]. We will incorporate these results in the discussion to follow and we will estimate the optimal number of diversity branches for the WPD and majority-voting receivers. For a fixed transmitted energy per message bit and L branches with independent CHAPTER 2. IDEALIZED WPD RECEIVER 28 Rayle igh fading, the ratio of the average received energy of a message bi t to the noise density is E b = L 7 o . (2.41) N0 Recal l that the Chernoff bound of (2.19) is normally in the form Pr( bi t error ) < Gx{s*) (2.42) Th i s may be re-written as - L i n Pr{ bit error) < c L G * ( S * ) . < N0Eb [Gx{s*) < e Eb_l_ Wo 7o i n where we have defined Sfro) = — In lo (2.43) (2.44) (2.45) (2.46) (2.47) We w i l l refer to (2.47) as the Chernoff bound exponent function 6 . The form of the Chernoff bound i n (2.46) shows that for a given Eh/No, the lowest B E R indicated by the bound occurs for the m a x i m u m of the Chernoff bound exponent function. Le t 7J denote the S N R at which this m a x i m u m is attained, then our estimate of the opt imal number of branches, L * , is obtained from substi tuting 7^ into (2.41); r* 1 ^» L « — —. 1qN0 (2.48) 6 In [ll] a similar form of the Chernoff bound is referred to as an efficiency function and is used to estimate the optimum number of branches for the square-law receiver. CHAPTER 2. IDEALIZED WPD RECEIVER 29 For the W P D receiver the Chernoff bound is given by (2.22) and we have 9wPd{lo) = — In (4 + 7o) /4 + 37Q 4 H + 570 (2.49) For majori ty-voting, we can obtain a Chernoff bound by considering the majority-vot ing receiver to act as an up /down counter. It counts up w i t h probabil i ty P, and down w i t h probabil i ty 1 — P. The moment generating function is then Gmv(s) = E[etx] = P e " ' + (1 - P ) e \ The value of s that minimizes (2.50) is easily found and yields 2 Gmv(s*) = 2y/P(l-P) = —— yjl + 70 and 9mv{lo) = — In 2 + 7o 2 + 7o 2v/r+~ 7o, For the square-law receiver, we have from [ l l ] that 4 ( 1 + 7o) G.qi{s*) = 4 P ( 1 - P ) = so that 9aqi{lo) = — I n (2 + 7o)2 (2 + 7o)2^ 7o [4(1 + 70) = 2gm u(7o). (2.50) (2.51) (2.52) (2.53) (2.54) (2.55) The exponent functions, (2.49) and (2.54), are plot ted in Figure 2.5. Note that bo th are m a x i m u m at 7 0 « 5 d B « 3. Thus our estimate of the op t imum number of branches for these schemes is CHAPTER 2. IDEALIZED WPD RECEIVER 30 • i i i i i i - i — i — i — i — p i i i i i S Q U A R E - L A W W P D ' • • I i i i i 1 I I I I I I I I 1 La -10 0 5 10 Signal to Noise Ratio (dB) 15 20 Figure 2.5: Chernoff Bound Exponent Functions CHAPTER 2. IDEALIZED WPD RECEIVER 31 E„/NQ [dB] Major i ty -Vot ing Selection Square-Law W P D Es t . Exac t Es t . Exac t Es t . Exac t Es t . (Bound) 10 3 1 2 2 3 4 3 3 12 5 3 3 3 5 6 5 5 14 8 7 4 4 8 9 8 8 16 13 11 4 4 13 14 13 13 18 21 19 6 6 21 21 21 20 Table 2.2: Es t imated and Exac t Values for the O p t i m u m Number of Branches For selection diversity it has been shown [24] that the op t imum number of branches is well estimated by L* « ^Eb/2N0 . (2.57) To see how accurate these guidelines are in predicting the op t imum number of branches one can search for the op t imum L w i t h a few values of E^/No. The results of this search using the exact B E R expressions for the majori ty-voting, selection, and square-law receivers, and using the bound of (2.27) for the W P D receiver are summarized i n Table 2.2. It can be seen that the predicted op t imum numbers of branches are reasonably close to the exact result, w i t h the exception of majority-vot ing at lower S N R . The loss i n using the predicted op t imum number of branches is very smal l as demonstrated by the plot of B E R as a function of L in Figure 2.6. Note that as Ei/N0 is decreased, there w i l l be a point for each of the imperfect combinat ion schemes (i.e. a l l but maximal-ratio) where the the loss associated wi th imperfect combinat ion overcomes any gain inherent in having mult iple samples of the fading signal. In other words, the opt imal number of branches for the imperfect combinat ion schemes w i l l be one as Et/No becomes smal l . Th i s is demonstrated i n Figure 2.6 for majority-voting w i th E^/Nq = 10 d B . CHAPTER 2. IDEALIZED WPD RECEIVER 32 T i 1 i i i i | i r T i | i i i i | i i i r SELECTION WPD (Bound) SQUARE-LAW i i • * • • ' i i i i I i i i i 1 1 1 1 1——I 5 10 15 20 25 Number of Branches Figure 2.6: BER vs. Number of Diversity Branches, Fixed Eb/N0. CHAPTER 2. IDEALIZED WPD RECEIVER 33 Fina l ly , we emphasize that the maximal-rat io receiver has no op t imum number of branches since it perfectly combines a l l of the branch information. It can be shown [9] that as the number of branches approaches infinity, w i t h a fixed energy per bi t of Et/No, the effects of Rayleigh fading are completely mit igated, i.e. l i m Pe>mr = 1/2 e"'^. (2.58) Id—*oo Thus the B E R for maximal-rat io combining w i th an infinite number of Rayleigh faded branches is the same as the B E R for an unfaded signal w i t h the same bi t energy. 2.6 Simulation To obtain an estimate of the W P D receiver performance in an example system a Mon te Car lo type simulat ion program was developed. Th i s section outlines the simulat ion method and presents some results for the bi t and packet error rates. The simulat ion results to be presented are for a simple model of the Advanced M o b i l e Phone Service ( A M P S ) land to mobile control channel [26,27]. Th i s cellular system was selected since it is a well known modern system, and since it uses majority vot ing to help combat fading. 2.6.1 Simulation Model The A M P S system employs F S K modulat ion and detection of a 10 kbps Manchester encoded data stream. In the land to mobile control channel five repeats of a forty bi t packet are sent, w i t h an effective forty bi t gap between block repeats to decrease CHAPTER 2. IDEALIZED WPD RECEIVER 34 10BITS 11 BITS 40 BITS 40 LAST WORD B 6 BIT SYNC WORD SYNC WORD A, (40. 28) BCH i i i WORD B, i i i • T • t 1 r t • t 1 40 WORD A 2 WORD B 2 _-l l • • WORD A 3 40 10 40 WORD B 5 • i i BIT SYNC WORD SYNC NEXT WORD A, i i i } • t t 1 } t t t 1 NOTES: f « POINT OF BUSY-IDLE BIT INSERTION (AFTER EACH 10 MESSAGE BITS AND AFTER BIT AND WORD SYNC) A, - i T H OF FIVE REPEATS OF WORD FROM MESSAGE STREAM A B; » i ™ OF FIVE REPEATS OF WORD FROM MESSAGE STREAM B INFORMATION CONTENT OF WORDS PAGES CHANNEL DESIGNATIONS OVERHEAD WORDS FILLER TEXT Figure 2.7: A M P S Land to Mobile Control Channel Format. (Figure taken from [39].) their dependence, as shown in Figure 2.7. Upon reception of the packet repeats the mobile decoder performs majority voting on the individual bits and then corrects one bit error per packet. The error correction utilizes a (40,28) B C H code, which is a shortened version of the (63,51) B C H code 7. To efficiently simulate the W P D receiver we shall use the ideal N C F S K bit error probability model given in (2.7). This wil l not accurately predict the error rate to be expected in a practical receiver for several reasons, which will be outlined below. However, it provides a simple and reasonable approximation to the expected error 7 ' , ihe (40,28) code has a m i n i m u m distance of 5, which implies a capabi l i ty for double error cor-rect ion . However the decoder is configured to correct only one error, to achieve a lower probabi l i ty of undetected packet error. CHAPTER 2. IDEALIZED WPD RECEIVER 35 rate, and in fact is used in the performance specifications for the A M P S system [13]. The idealized N C F S K model impl ic i t ly assumes that a receiver w i t h matched filters is used. The receiver block diagram is equivalent to the square-law receiver i n Figure 2.2 w i t h L = 1. Some of the factors contr ibuting to a deviation of the actual cellular system performance from our simulat ion results are: D e m o d u l a t o r I m p l e m e n t a t i o n M e t h o d . Prac t ica l N C F S K demodulators i n cellular receivers usually employ a l imi ter /d iscr iminator followed by an inte-grate and dump circuit . Performance analysis of l imi ter /d iscr iminator based receivers attempts to incorporate the effect of click noise [28,29,30]. The gen-eration of a cl ick, or spike, may be understood by representing the received signal plus noise on a phasor diagram. The discriminator 's function is to differentiate the received phase, and a click w i l l be generated if the resultant signal plus noise phasor rapidly encircles the origin. For example, if the noise happens to cause the received phase to completely encircle the origin in the positive direction, then the phase change w i l l be 2n and a positive spike w i l l be generated of area 2ir. A t high S N R , the probabil i ty of the noise overcom-ing the signal and resulting i n an encirclement of the origin is very low. A s the S N R is lowered the signal approaches the noise level and the l ikel ihood of click occurrences increases. Other receiver implementations might conceivably uti l ize matched filters or bandpass niters instead of the l imi ter /d iscr iminator scheme. In this case there would inevitably be some cross-talk between the filters, as well as loss of signal power due to bandl imi t ing by the receiver intermediate frequency CHAPTER 2. IDEALIZED WPD RECEIVER 36 (IF) filter. The effects of these factors on the B E R is analyzed in [31]. I n t e r s y m b o l I n t e r f e r e n c e . The transmitted pulses are filtered prior to trans-mission i n order to control adjacent channel interference and are also shaped by the receiver IF filter. The resulting intersymbol interference (ISI) may increase the B E R [32,33,34]. A s the IF bandwidth is decreased there w i l l be a higher level of ISI which w i l l tend to decrease performance. However, this reduction i n IF bandwidth also admits less noise to the receiver. Thus , for a given bi t rate and frequency deviation there w i l l be an opt imal I F fil-ter bandwidth for a specified receiver structure. Several investigations into the relationship between IF filter bandwidth , frequency deviat ion, and the resulting B E R have been made, including [32,33,34]. M a n c h e s t e r E n c o d e d D a t a . The cellular signalling format uses Manchester encoded bits as inputs to the F M modulator . Th i s facilitates bit synchro-nizat ion, since each bit has a transi t ion. It also has no D C component, so that implementation v i a an integrator followed by a phase modulator w i t h l imi ted range is pract ical . O f course the mul t ip l icat ion of the data sequence by a square wave increases the occupied bandwidth of the signal [35], but i n cellular systems relatively wide channels, 30 k H z , are available and spec-t ra l efficiency is instead obtained v i a frequency re-use. The ISI w i l l be i n -creased from the use of Manchester encoding but this is t raded off for the convenient synchronization and implementation features. The performance of Manchester encoded data and non- return to zero ( N R Z ) data through l imiter discriminator systems is investigated in [33,34]. I n t e r f e r e n c e . Cel lular systems operate in co-channel and adjacent channel in -CHAPTER 2. IDEALIZED WPD RECEIVER 37 terference generated by neighbouring cells. The performance of cellular sig-nal l ing is very dependent on the cell frequency plan , cell site antenna design, and control of transmitter power [36,37]. If a l l of the transceivers increase their power levels in an effort to overcome the A W G N , the level of interference w i l l also increase and may dominate the system performance. A form of interference generated by neighbouring vehicles is ignit ion noise, which is impulsive in nature. The contribution of igni t ion noise towards the B E R may be significant in some systems [38]. The contr ibut ion of igni t ion noise towards the B E R in the A M P S system is considered to be low [39], since the arr ival rate of the impulses is low compared to the bi t rate. The effects of interference w i l l be discussed further i n Section 3.2. M i s c e l l a n e o u s P r o p a g a t i o n P h e n o m e n a . A s discussed in Section 1.1, we have assumed that the propagation is described by Clarke's multi-scatterer model , so that the received amplitude is Rayleigh distr ibuted. There w i l l be, of course, a variety of fading environments encountered i n practice, including the presence of a direct L O S component and shadowing of the signal. A l s o , fading has been assumed to be fiat across the frequency band. For U H F frequencies i n dense urban areas the coherence bandwidth is of the order of 40 k H z [4] 8 . Fading on frequencies separated less than 40 k H z w i l l then be largely correlated, so that the assumption of flat fading in the A M P S 30 k H z bandwidth should be reasonable. We have also assumed the condit ion of slow fading, i.e. that the signal ampli -tude and phase are effectively constant during a b i t . The t ime varying phase 8 Here the coherence bandwidth is taken as the bandwidth within which fading has a 0.9 or greater correlation. CHAPTER 2. IDEALIZED WPD RECEIVER 38 leads to a s m a l l 9 but irreducible B E R as the S N R is increased [ l ] . Th i s is referred to as random FM, since the variat ion of received phase over the bi t period is equivalent to an undesired random frequency modulat ion. M i s c e l l a n e o u s R e c e i v e r I m p l e m e n t a t i o n L o s s e s . In addi t ion to the items mentioned above there w i l l be addit ional degradation due to various receiver implementation losses such as imperfect bit synchronization, inaccuracies i n channel measurement information, and quantization of variables i n a digi ta l implementation. Some of these factors w i l l be addressed i n Chapter 3. In summary, there are many practical aspects of receiver implementation that w i l l alter the B E R performance from the ideal. The exact amount of degradation w i l l depend on the part icular receiver implementation, but we note from the A M P S specification [13] that it is entirely feasible to have less than 3 d B degradation from the ideal N C F S K bit error rate. This includes a l l of the effects listed above except for the interference and propagation phenomena. In Chapter 3 a more comprehensive simulat ion is developed that relaxes some of our assumptions of ideality. Another benefit of the more detailed simulat ion is that it simulates the random variables internal to the demodulator, so that bi t and packet error rates for the square-law receiver may be obtained in addi t ion to the majori ty-voting, selection, and W P D receivers. 9At 850 MHz with a 10 kbps bit rate and a 14 kHz peak to peak frequency deviation the BER limit is of the order of 10~5 for vehicle speeds below 70 mph [4]. CHAPTER 2. IDEALIZED WPD RECEIVER 39 2.6.2 Simulation Description The computer s imulat ion program utilizes routines developed in [40] to generate an exponentially distr ibuted random variable that represents the squared envelope of a sine wave under-going Rayleigh fading. The fading routines are based on the fading simulat ion program discussed i n [41]. Briefly, two pseudo-random Gaussian number sequences are generated that represent the Fourier transform of the in -phase and quadrature components of a complex Gaussian random process. The autocorrelation function of the complex valued received amplitude for a part icular vehicle speed, derived in [2], yields the power spectral density function of the received signal as The Gaussian random number sequences are weighted to achieve this spectral shape, and then inverse fast Fourier transformed to y ie ld a sequence of t ime do-m a i n samples of the complex amplitudes. F ina l ly , summing the squares of the in-phase and quadrature components yields the desired exponentially distr ibuted sequence 1 0 . For each bi t the fading sequence value is used to represent the received S N R , and is then used i n (2.7) to y ie ld p,-. The weight u>,- is then obtained from (2.5). A bi t error is injected if the output of a uniformly distr ibuted pseudo-random number generator, w i t h range (0,1), is less than p,. The program maintains various counters and registers to simulate and monitor 1 1 I / |< fD- (2.59) 1 0The phase of the received signal could be obtained from the ratio of the quadrature to in-phase components, but this is not required here since we are simulating a non-coherent receiver. CHAPTER 2. IDEALIZED WPD RECEIVER 40 the W P D , majority-voting, and selection diversity receivers. The various bi t and packet error events are counted by M separate sets of error counters that mainta in data from M different fading sequences. Th i s provides M independent tr ials, from which we may compute M sample means that estimate any part icular error rate. If we have counted a sufficient number of error events then the sample means should be approximately normally distr ibuted according to the central l imi t theorem. We can then compute a confidence interval using the ^-distribution [42]. Denot ing the mth sample mean for an actual error rate p as pm, then our final estimate p is the mean of {pi,P2> • • • >PM}J and an (1 — a) • 100% confidence interval about p is of the dis t r ibut ion, and s 2 is the sample variance. The program tests periodically to see i f a specified confidence interval has been attained and stops execution i f this cri terion has been met. Since some of the simulations at higher signal to noise ratios w i l l run for a considerable t ime the program periodically sends intermediate results to a file for prel iminary viewing. The program also periodical ly updates a simulation state file that contains the present values of the random number generator seeds and al l of the error event counters. Th i s s imulat ion state file is used to gracefully continue the simulation in the event of a computer system crash, rather than having to re-start from scratch. s (2.60) y/M where t<± is the argument of the t dis tr ibut ion that leaves an area of a / 2 i n the t a i l chapter 2. idealized wpd receiver 41 2.6.3 Simulation Parameters A l l simulation results were obtained using a Doppler frequency, fD, of 49 Hz and a bit rate of 8192 bits per second (bps). This bit rate was chosen since the fading simulation is constrained to use bit rates that are a power of two 1 1 , and 8192 bps is the closest power of two bit rate to the 10 kbps data rate used by AMPS . The fade rate was chosen using the guideline that the bit repeats will be approximately independent for separations, r, of at least the reciprocal of twice the Doppler frequency [l]. For cellular signalling with bit repeats separated by 80 bits at 10 kbps, or 8 msec, the Doppler frequency should be greater than h > h = o * 6 0 H z - ( 2 - 6 1 ) ZT Z ' O msec At 850 MHz this Doppler frequency is equivalent to a vehicle speed of 47 mph. For our simulation we must scale the Doppler frequency since we are simulating 8192 bps, rather than 10 kbps. The equivalent fD for 8192 bps is then 49 Hz. This Doppler frequency is the minimum according to our guideline to achieve indepen-dent bit repeats, and we will use it to see if there is any appreciable deviation of the simulation results from that predicted by (2.29) and (2.30), and to compare the W P D simulation results with the bound of (2.27). The degree of correlation between the fading envelopes of two bit repeats can be found from the normalized autocorrelation function of the received envelope, P ( T ) « J^(2nfDr) (2.62) where JQ(X) is the zero order Bessel function of the first kind and r is the time separation. Equation (2.62) is obtained from combining equations (16) and (17) 1 1 This constraint is due to the fact that the fading envelope generator uses a FFT to produce one second of fading envelope data. CHAPTER 2. IDEALIZED WPD RECEIVER 42 (Doppler Frequency) X (Time Separation) Figure 2.8: Normalized Autocorrelation Function of the Received Envelope in [2] and is plotted in Figure 2.8. For our bit separation of at r = 8 msec with 10 kbps, and fu = 60 Hz (2.62) yields a correlation coefficient of 0.1. Given this small correlation and the previous observation that square-law combining suffers only minor degradation with correlated branches [7], it seems reasonable to expect that the simulation results for the other combining methods will be fairly close to their predicted error rates. This will be confirmed in the next section. CHAPTER 2. IDEALIZED WPD RECEIVER 43 2.6.4 Simulation Results The B E R performance of the majority-voting, selection, and W P D receivers as determined by simulation 1 2 is shown in Figure 2.9. Also shown are the bit error rates predicted for the standard partial decision combining schemes, as given by (2.29), and (2.30). The B E R obtained from the simulations for majority-voting, selection diversity, and square-law combining are only marginally higher than predicted. This confirms that at /o-r = 0.48 the bit repeats are nearly independent, and we should not expect any significant decrease in B E R if the bit separation is further increased. The results for three and five repeats are shown. As mentioned earlier, an L branch square-law combiner achieves the same error rate as a 2L — 1 branch majority-voting receiver [22], so that square-law combining with L = 3 is equivalent to majority-voting with L = 5. With a nominal AMPS signal to noise ratio of 15 dB, the A M P S format of majority-voting on 5 repeats achieves a B E R of 2.5 x 1 0 - 4 . For this target BER, the WPD receiver is over 3.5 dB more efficient. As shown earlier, this gain will increase indefinitely in an ideal system as the SNR is increased. Relative to selec-tion diversity, the WPD receiver shows a gain of more than 1 dB. As the SNR is increased the asymptotic gain of the WPD receiver relative to selection diversity wil l be about 1.4 dB as shown in Table 2.1. With three repeats, the W P D receiver is approximately 4 dB more efficient than majority-voting and only slightly better 1 2 The BER curve for the selection and square-law receivers with L = 5 was obtained via the more general simulation method explained in Chapter 3. This was done since these receivers were not incorporated into the simple simulation method used in this chapter. All of the relevant simulation parameters such as the packet format, Doppler frequency, etc. are identical in both of the simulation runs. The error bars for all simulation results in this thesis indicate 95 % confidence intervals. CHAPTER 2. IDEALIZED WPD RECEIVER 44 than selection diversity. This is consistent with our earlier observation that W P D and selection diversity are equivalent for L = 2, and the gain offered by the W P D method increases with L. Finally, we note that for 3 or 5 repeats, the W P D receiver is less than 1 dB poorer than square-law combining. The packet error rates (PER) for five repeats are shown in Figure 2.10. For a P E R of 10~3, the W P D strategy is over 3 dB more efficient than majority-voting. Here, even with no error correction, the W P D and selection diversity receivers outperform majority-voting with single bit error correction, and the W P D receiver has more than a 1 dB advantage over selection diversity. As discussed in Section 2.3, a more complete performance comparison could av-erage the error rate over the Doppler frequency distribution. We have demonstrated that with a few independent bit repeats the performance gain over majority-voting is significant, and the gain relative to selection diversity is more modest. At very low Doppler frequencies, the error rates of the WPD and majority-voting receivers wil l be equivalent, while selection diversity wil l offer no gain relative to a single bit transmission. Thus, if there is a wide range of expected Doppler frequencies, using the W P D receiver will provide better performance than either of the majority voting or selection diversity schemes. CHAPTER 2. IDEALIZED WPD RECEIVER 45 Figure 2.9: Bit Error Rates for 3 and 5 Repeats CHAPTER 2. IDEALIZED WPD RECEIVER 46 10~5 ' — 1 — 1 — • — • — i 1 • 1 i i 1 1 ' 1 1 1 1 1 * ' 0 5 10 15 20 Signal to Noisa Ratio (dB) Figure 2.10: Packet Error Rates for 5 Repeats CHAPTER 2. IDEALIZED WPD RECEIVER 47 2.7 Error Rate Approximations In this section we demonstrate that the upper bound given by (2.27) is quite tight. We also demonstrate that some simple bounds on the packet error rate are in good agreement w i t h the simulation results. Three estimates of the W P D bit error rate are shown in Figure 2.11. The fad-ing s imulat ion results from the previous section have been re-plotted, along w i t h the bound (2.27), and another simulat ion that used completely independent bi t repeats. This latter simulation generated independent samples of an exponentially distr ibuted random variable by summing the squares of two independent Gaussian random number genera tors 1 3 . A s discussed i n the previous section, the fading s imulat ion w i t h fD • T = 0.48 yielded bi t error rates that are slightly higher than predicted for the standard diversity schemes w i t h completely independent bi t re-peats. Thus we can expect the fading simulation to y ie ld a slightly higher B E R for the W P D receiver as wel l . The independent bi t repeat s imulat ion plotted in Figure 2.11 confirms this. Note that the B E R predicted by the upper bound of (2.27) is pract ical ly identical to the independent bi t repeat s imulat ion resul ts 1 4 and can thus be used to accurately estimate the W P D bit error rate. Th i s does not show that 2.27 is accurate at higher signal to noise ratios or for large values of L , but , at the typica l values of S N R and L used here the accuracy is very good. There is no reason to expect a dramatic loss of accuracy as the S N R or L is increased. 1 3 The Gaussian random number generation program is documented in [43]. "The independent bit repeat simulation plot in Figure 2.11 extends only to 12 dB. The 95% confidence intervals are indicated by error bars, which extend only about one line thickness, at worst. CHAPTER 2. IDEALIZED WPD RECEIVER 48 i i i 5 Repeats 10"1 10* -OC 10 3 o Ul #•* m 10" 10 3 -Fading Simulation Upper Bound Independent Bit Repeat Simulation 10"1 • ' i I. -i L i i i 10 Signal to Noise Ratio (dB) 15 20 Figure 2.11: WPD Bit Error Rate Bound and Simulation Results CHAPTER 2. IDEALIZED WPD RECEIVER 49 On the contrary, there will be a fixed dB loss as the SNR approaches infinity since both the bound and the actual WPD error rate have the same asymptotic slope. For five repeats, this fixed dB loss appears to be negligible. Next, we demonstrate the effectiveness of approximating the packet error rate by some simple bounds. The Rayleigh fading channel is usually considered to be bursty, i.e. bit errors tend to cluster together rather than being randomly spaced. Since a packet in error may often contain a number of errors, relatively few packets will absorb a specified number of bit errors, compared to the number of packets that would be required to absorb the same number of independently occurring bit errors. For a bursty channel then, it would seem reasonable to form an upper bound on the uncorrected PER by assuming that bit errors occur independently, resulting in more packet errors for a given BER, i.e. P( packet error ) = P(> 0 bit errors in packet) < 1 - {l-Pe)N (2.63) where Pe = P{ bit error after L repeats), and N is the packet length. One may form a bound on the t-error corrected PER by considering that, at worst, the errors that occur are grouped into exactly t + 1 errors per packet. Over a long observation period of say M bits, there will be MPt bits in error. At worst, this will result in MPe/(t + 1) packet errors. The bound on the PER is then Equations (2.63) and (2.64) are plotted in Figures 2.12 and 2.13 along with the simulation data. For the standard diversity schemes the value of Pt used is taken from the analytical BER expressions. For the WPD receiver Pe is determined from the bound, (2.27). P(> t bit errors in packet ) < MPe/(t + 1) M/N t + 1 NPe (2.64) CHAPTER 2. IDEALIZED WPD RECEIVER 50 W i t h no error correction, Figure 2.12 shows that, for packet error rates below 1 0 _ 1 , bo th the independent bit error model (2.63) or the bound (2.64) are wi th in 1 d B of the s imulat ion results. A s the S N R is increased, the accuracy of both bounds improves. The accuracy of the P E R prediction by (2.63) or (2.64) corresponds to single bit errors being the dominant error mechanism. Th i s is due in part to diversity decreasing the effective fade durat ion [9], so that the probabil i ty of single bi t errors can dominate even at moderate Doppler frequencies. If the channel, including the effect of diversity, is such that the sum of the probabili t ies of t + 2, t + 3 , . . . , N errors per packet are significant compared to the probabil i ty of t + 1 errors per packet, then the bound of (2.64) w i l l not be very t ight. The looser bound of (2.64) w i th one bi t error correction is shown i n Figure 2.13. Below a P E R of about 1 0 - 1 the bound is about l d B off the square-law, W P D , and selection diversity curves, while for majority-voting i t is about 2 d B off. One would expect that the P E R bound (2.64) w i l l be looser as t is increased. CHAPTER 2. IDEALIZED WPD RECEIVER 51 Signal to Noise Ratio (dB) Figure 2.12: Uncorrected Packet Error Rate CHAPTER 2. IDEALIZED WPD RECEIVER 52 \ I i i i i I i i i i 1 i i 1 1 1 1 1 1 1 1 0 5 10 15 20 Signal to Noise Ratio (dB) Figure 2.13: Single Bit Error Corrected Packet Error Rate 53 Chapter 3 Some Issues in Practical WPD Receiver Design In this chapter we demonstrate the robustness of the W P D receiver when several of the assumptions discussed in the previous chapter are removed. The perfor-mance of the W P D receiver is assessed when various parameters are quantized, and when the signal strength measurement is imperfect. A l s o , a relatively efficient method is presented to determine the opt imal quantization thresholds. F ina l ly , it is demonstrated that a good signal strength estimate is obtainable by simple low pass filtering of the IF filter output envelope. 3.1 SNR and Weight Quantization Let the received S N R be quantized into N + 1 regions by TY thresholds, { T j t } ^ , as shown i n Figure 3.1. These regions w i l l give rise to TY + 1 different reliabili ty weights. We w i l l consider the effect of the S N R quantization shown i n Figure 3.1 and the effect of the quantization of the corresponding weights separately. Tha t is, we first assume that the weights are represented exactly, then consider quantization of these weights. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 54 .00 P 2 • • • *N+1 T T2 ... Figure 3.1: SNR Quantization 3.1.1 B E R Evaluation Method The probability of bit error for the WPD receiver when a perfect SNR measurement is quantized may be evaluated as follows. The random variables resulting from the thresholding of the SNR can be represented by a discrete channel model. If we assume that a '1' is sent L times, and assume that the repetitions are independent, then all of the possible combinations of the receiver random variables may be repre-sented by the tree shown in Figure 3.2. At each repeat, the receiver makes a partial decision from one of the N + 1 reliability regions, resulting in 2{N + 1) possible values. The probability that the SNR falls in the kth SNR region is represented by the branch labeled P*, and stemming from this branci are the probabilities of then making a correct decision or an error, Pe\k and Pe\t respectively. With L repeats, CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 55 th L REPEAT I/, U. M-1 U. M Figure 3.2: Discrete Channel Model Paths Corresponding to a Repeated '1' there is a total of M = [2(7Y + 1)]L sequences of random variables represented by the M final nodes in Figure 3.2. Let tuO denote the optimal weight for the kth SNR region. Since a bit received in the kth SNR region has a probability of error equal to P e | t , we have from (2.5) that 0) (3.1) CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 56 where (3.3) and rTk - T t _ i -Tb Pk = fr{l)di = e io - e TO . (3.4) JTk-i Define a set of variables, U = {UJ}JLV where the subscript j indicates one of the M possible paths in the tree of Figure 3.2. The value of Uj represents the value of the final decision variable for that part icular path, and is computed by summing for the jih path. The probabil i ty of uy occurring, p ( u ; ) , is given by the product of the probabilities of the branches associated w i t h the j*h path . The B E R is evaluated by summing the probabilities of a l l the paths that lead to an error, Pe = E P(«/) + I E PM . (3-5) ut-<0 * uy=0 where the second te rm accounts for the fact that half of any ties w i l l result in an error. T h i s procedure w i l l later be used to compute the probabil i ty of bit error for the W P D receiver w i t h unquantized and quantized weights, but before doing so the issue of how to select the thresholds is discussed. 1 Pk JTk.y (3.2) Pk 2 + lo CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 57 3.1.2 Threshold Optimization Thi s section discusses various methods that may be used to select the S N R quan-t iza t ion thresholds shown in Figure 3.1. We seek a set of thresholds {Tk}%=1 that minimizes the probabil i ty of bit error. A direct, but inefficient, approach would be to optimize the B E R by using a non-linear function opt imizat ion routine and the B E R evaluation a lgor i thm described i n Section 3.1.1. In an alternate method, which we shall refer to as the capacity method, thresholds are chosen so as to maximize the discrete memoryless channel ( D M C ) capacity. Th i s method has been used previously i n different contexts in [44,23]. We w i l l apply the direct opt imizat ion and capacity methods here, and we w i l l also introduce a method that chooses thresholds that minimize a Chernoff bound. 3.1.2.1 Capaci ty M e t h o d Recal l that Figure 3.2 summarizes the possible events that w i l l occur i n a W P D receiver for L independent repeats of the message bi t m = 1. The D M C model for one repeat may be obtained from Figure 3.2 by re-drawing the paths arising from one repeat to also show the paths for m = 0, and collapsing the transi t ion probabili t ies Pe\kPk and Pc\tPk to form Pet and Pet respectively. The result is shown i n Figure 3.3. We w i l l show that the capacity of such a channel is simply the weighted average of the capacities of the sub-channels formed by the S N R quantizat ion regions. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 58 Figure 3.3: Discrete Memoryless Channel M o d e l CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 59 The capacity of the channel in Figure 3.3 is C = max [I(X;Y)\ (3.6) where {P(x = t)} is the set of input probability assignments, and I(X;Y) is the average mutual information between the sets of inputs and outputs, X and Y. For a symmetric D M C 1 , as we have here, capacity is attained for equally likely inputs [45], Furthermore, with the maximizing input probability assignment the capacity will be equal to the mutual information between any one of the inputs2 and the set of outputs, i.e. [45, pg. 91] C = I(x = i ; Y) (3.7) = y^Pij | 0 log2 \ information bits/channel bits . From Figure 3.3 we note that P(j = l) = l / 2 P e l + 1/2P.X = l /2[P e l + P e l] = 1/2PX (3.8) P(j=2) = l /2[P e l + P c l] = l /2Pi or in general, P(j' = 2A;-l) = P(j = 2k) = l /2P f c ,fc = 1,2,...,7Y + 1. (3.9) The capacity may be written as W+i) pf.- I { \ c = E pti 10 i°g2 pfi) (3-10) = E xFor a formal definition of symmetry in a DMC, see [45, pg.94]. 2Provided that the input chosen has a non-zero probability assignment. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 60 Choosing t = 1, and noting from Figure 3.4 that P(2k — 1 | * = 1) = Pek, and P(2k | i = 1) = Pc*, we have C = N f \ P e k l o g 2 2 - ^ + P e t l o g 2 ^ | Jb=l L Fk Ft J N+1 = E P * [ P c | J t l o g 2 2P c | f c + P e | t log 2 2P e | f c ] k=l N+1 = £ Pk [{Pc\k + P.|*) l 0 g 2 2 + ( l - P,|t) log 2 ( l - P,|t) + P.|fc log 2 Pe| t] = E ft i - M^iO (3 -n) *=i N+1 = E PkCBsc{Pc\k)- (3.12) Jb=l In (3.11) /i(-) is the binary entropy function and CBsc{') is the capacity of a BSC. The final equation (3.12) shows that the capacity of the channel is the weighted average of the capacities of the N + 1 sub-channels formed by thresholding. Note that the form of this result does not depend on the actual SNR distr ibution 3 . These results are consistent with the intuitive argument presented in Section 2.1 that views independent samples from a fading channel as coming from sub-channels of different capacities. The D M C capacity is plotted in Figure 3.4 for a few values of N. The capacity rapidly approaches its limiting value as N is increased. This behaviour indicates that most of the gain is achievable, in principle, with a small number of thresholds. 3For example, the capacity formula will be similar when the fading is Rician distributed, except for the actual values of the Pit and Pe|fc-CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 61 20 Signal to Noise Ratio (dB) Figure 3.4: Discrete Channel M o d e l Capaci ty Selecting the thresholds {Tt}k=i by maximiz ing the capacity (3.12) may be effi-ciently done numerically, using a non-linear function opt imizat ion p r o g r a m 4 . One can take advantage of some reasonably efficient opt imizat ion routines by providing the function (3.12) and its first par t ia l derivatives w i t h respect to {Tk}t=v Section 3.1.3 discusses how well the capacity method predicts the opt imal thresholds. 4The non-linear function optimization program used here was LINOPT, which is documented in [46]. Alternatively, one might consider finding the thresholds by setting the partial derivatives of (3.12) to zero, and attempt to solve the resulting set of N simultaneous non-linear equations. This approach would have involved the development of routines to solve the simultaneous non-linear equations. Instead, the approach used employs an available routine for optimizing a single non-linear function of several variables that is efficient and reliable. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 62 3.1.2.2 Chernoff Bound Method In this section we introduce the Chernoff bound as a performance criterion for optimization of W P D receiver thresholds. The method is not restricted to the modulation, noise, path gain, and quantization characteristics used here 5. We begin by deriving the Chernoff bound for the W P D receiver. We represent the probability of occurrence of discrete random variables in the W P D receiver by a channel transition matrix, and require that the transition matrix be symmetric. This is satisfied, by definition, in channels that are symmetric with respect to the input symbols. The general form of a binary input, Q—ary output, channel transition matrix is T P l l Pl2 '" PlQ P21 P22 P2Q (3.13) If T is symmetric, then from the definition [45] it may be partitioned as '( pn pn \ ( P12 P22 \ ( Pit P2k \ ( Pl(M-l) ft(M-l) \ ( PlM \ V P21 Pll J \ P22 Pl2 ) ' \ P2k Pit ) " ' [ P2(Af-l) Pl(M-l) )\PIM) (3.14) The second subscript indicates the partition number, k = 1,2,... , M , which cor-respond to a reliability class. Note that the final partition in (3.14) is not a 2 x 2 matrix. This would arise if the output j = Q was equally likely given that either a '1 ' or a '0' was sent. This corresponds to an erasure, since the occurrence of this output offers no indication as to what was sent. 6For example, the Chernoff bound approach may also be applied to binary antipodal signalling on a non-fading channel with AWGN. This proposal was subsequently implemented in [47]. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 63 It is shown i n Append ix C that if the transit ion mat r ix does not contain any erasure par t i t ions 6 then the Chernoff bound on the probabil i ty of bi t error is M L (3.15) L i = l where P u and P 2 * are the transi t ion probabilities of the kth par t i t ion of T. The value of (3.15) is that it concisely gives the Chernoff bound i n terms of the t ransi t ion mat r ix elements. Thus it may be applied, for example, whether the Q outputs are derived from a fading or static channel, or whether the Q outputs are derived from a signal corrupted by Gaussian or non-Gaussian noise. O u r strategy here is to choose the receiver thresholds {Tk}k=i so that (3.15) is min imized , or equivalently, N+l m i n zZsJPikPik- (3.16) tT*>t=i k=i Equa t ion (3.16) may be efficiently solved numerically by a non-linear function opt imiza t ion program. It is straightforward to substitute the equations for P e t and Pek into (3.16), and to compute the first par t ia l derivatives of (3.16) w i t h respect to {Tj fc}^ to provide to the numerical opt imizat ion p r o g r a m 7 . The performance of the Chernoff bound method for choosing the thresholds w i l l be discussed next. 6Appendix C also presents a slight modification of (3.15) for the case when an erasure partition is present. This would be useful, for example, in optimizing the WPD receiver thresholds for binary antipodal signalling with an even number of thresholds. 7The numerical optimization routine used here was the same as used for the capacity method, namely LINOPT [46]. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 64 3.1.3 Performance With Quantized SNR In this section we uti l ize the a lgori thm of Section 3.1.1 to compare the B E R using op t imal thresholds w i t h the error rates using the thresholds found by the capacity and Chernoff bound methods. Figure 3.5 shows the value of the threshold i n a single threshold receiver as determined by direct op t imiza t ion 8 , the capacity method, and the Chernoff bound method. The op t imum threshold increases slightly w i t h the number of repeats, and generally the Chernoff bound threshold is quite close, especially as the number of thresholds is increased. T h e capacity threshold appears to differ significantly from the op t imum as the S N R is increased. However, one must evaluate the resulting B E R using the various thresholds to make a meaningful comparison. A plot of B E R vs. S N R is shown in Figure 3.6 for 3, 5, and 7 repeats. It can be seen that the capacity method performs quite wel l , but not as well as the Chernoff bound method which gives results very close to the op t imum. The previous comparisons have been for a receiver employing a single threshold. Compar ison of Figures 3.7 and 3.8 reveals that as the number of thresholds is increased, the sensitivity of B E R to the accuracy of the thresholds decreases. Th i s is what we would expect as the number of thresholds is increased, since even randomly selected thresholds would eventually quantize the received S N R sufficiently that the B E R would approach that of an unquantized channel. 8 Direct minimization of the BER evaluation algorithm was done using the routine NLPQL, which is documented in [46]. The thresholds were verified to be optimum within 0.1 <JB by per-turbing the thresholds obtained from the numerical optimization, and checking that the resulting BER was higher. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 65 Signal to Noise Ratio (dB) Figure 3.5: Compar ison of Single Threshold Values CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 66 i i i i I i i i i I— i i i i | i i . . | . . . • Signal to Noise Ratio (dB) Figure 3.6: Comparison of Error Rates using a Single Threshold CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 67 Signal to Noise Ratio (dB) Figure 3.8: B E R w i t h Three Thresholds CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 68 F r o m the previous discussion and B E R plots, it is apparent that as the number of thresholds is increased, we obtain diminishing returns i n terms of increased performance. Th i s is summarized i n Figure 3.9, where the B E R for an increasing number of opt imal thresholds is shown, along w i t h the bound on the W P D error rate (2.27) which is an accurate estimate of the B E R for an infinite number of thresholds. It can be seen that the B E R rapidly approaches its l im i t , and most of the gain attainable is reached w i t h a few thresholds. Th i s is in agreement w i t h the discussion i n Section 3.1.2.1 on the capacity of the quantized channel. Note that the thresholds used in the preceding B E R curves were opt imized for each value of average S N R . The question arises as to how much degradation is incurred when opt imizat ion is not carried out for each S N R . Figure 3.10 shows the B E R for bo th a continuously opt imized threshold and a threshold opt imized only at an S N R of 15 d B . The degradation is very slow as the S N R deviates from the 15 d B opt imizat ion point , even for this most sensitive case of a single threshold. Thus , even if the dis t r ibut ion of the average S N R is quite wide, only a slight performance degradation should result from fixing the single threshold at its nominal design l e v e l 9 . The degradation is even less when a few more thresholds are used, as w i l l be demonstrated in the next section. 9 As discussed in the introduction, the logarithm of the received signal amplitude averaged over a sufficient number of wavelengths so that the effect of Rayleigh fading is removed, is well modeled by a normal distribution. The standard deviation of this shadowing distribution was measured to be 8 to 12 dB in some large cities [l]. Thus the range of the average SNR may in fact be quite wide. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 69 Signal to Noise Ratio (dB) Figure 3.9: Convergence of W P D B E R CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 70 • i i i I i i i i i i i i i [ i i i • i i i i 5 Repeats 7 « ! • • « • * 1 ' 1 • I I I I I I 1 1 1 1 0 5 10 15 20 25 Signal to Noise Ratio (dB) Figure 3.10: Optimized and Fixed Threshold Performance CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 71 It is interesting to compare the performance of the W P D and selection diversity receivers with SNR quantization. It is possible to obtain an expression for the B E R of selection diversity in this case. It is shown in Appendix D that with TY SNR thresholds, the selection diversity B E R is N+l JV- = £ [(4k + Pkf - Af] Pe\k (3.17) k=l where AM = *E Pi (3.18) and Pt,Pe\k are defined by (3.4) and (3.3). Equation (3.17) is plotted in Figure 3.11 for a few values of TY, along with (2.30) which is the selection diversity B E R using an unquantized SNR distribution. The thresholds used are optimum, and were found by direct minimization of (3.17) using the same routine, NLPQL [46], as was used for the WPD BER algorithm. Only a few thresholds are needed to achieve almost all of the attainable gain, although the convergence to the TY —• oo result is not as rapid as for the WPD receiver (see Figure 3.9 ). CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 72 CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 73 1 2 B -3 - I [--2 - I | 1 ! 1 1 1 B 1 1" 1 2 3 • • * 2 - 2 2-1 W Figure 3.12: Weight Quantization Characteristic 3.1.4 Performance with Quantized Weights In this section we demonstrate the degradation incurred when the WPD receiver weights are quantized. Let there be B bits available for weight quantization, pro-viding 2B quantization levels. Scaling of the optimal weight values does not affect the resulting performance, so that one may normalize the weights prior to quan-tization. Since the largest weight, iw,^,^, has the greatest influence on the final decision variable, we choose to scale the weights such that i u m o s is quantized to 2B. The quantization characteristic is shown in Figure 3.12. There may be superior weight quantization schemes, but our intention is simply to demonstrate that a small number of quantization levels can be sufficient. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 74 The B E R of the WPD receiver with quantized weights is easily evaluated using the algorithm of Section 3.1.1. Figure 3.13 shows the results of quantization with a few values of B, and also the B E R for unquantized weights. Note that the quantized weights are based on fixed SNR thresholds that are optimal at an average SNR of 15 dB. Also shown is the best performance possible, achieved using unquantized weights that are optimized at each average SNR. In the example shown in Figure 3.13, we see that four bits for quantization is sufficient to obtain almost exactly the same performance as unquantized weights. With three bits for quantization the degradation is less than 0.5 dB with respect to the unquantized weights, for a B E R greater than 10~7. Note that the loss due to the use of fixed weights, instead of optimizing the thresholds and weights at each average SNR, is very small. Comparison of the three threshold example of Figure 3.13 with the one threshold case of Figure 3.10 shows that with three thresholds the degradation incurred due to the use of fixed weights has decreased. This is consistent with the earlier discussion on the reduced sensitivity of the B E R to weight accuracy as the number of thresholds is increased. The examples presented in this and the previous section demonstrate that with a reasonably small number of bits for weight quantization and a few fixed weights, the B E R performance is effectively the same as the ideal case with no SNR or weight quantization, over a very wide range of average SNR. It is reasonable to expect that the number of weights and quantization levels required to attain near ideal performance will not increase dramatically as the number of repeats is increased. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 75 Figure 3.13: Quantized Weight BER . CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 76 3.2 SNR Estimation In this section we investigate the degradation in WPD receiver performance due to imperfect knowledge of the channel state. The effects of an imperfect signal strength measurement due to noise and cochannel interference are shown. Simple low-pass filtering of the signal strength estimate is shown to improve the perfor-mance of both the WPD and selection diversity schemes, if cochannel interference is not dominant. The performance of the various receivers was estimated using computer simu-lations. By using a suitable receiver model the random variables internal to the receiver need be computed only once per bit. This approach yields an efficient simulation procedure relative to a multiple sample per bit model. The receiver model, simulation methods, and simulation results are discussed in the following sections. 3.2.1 Receiver Model The development of the receiver model was motivated by the fact that practical received signal strength indicator circuits estimate the amount of power passing through the IF filter. We will assume a dual bandpass filter implementation of a N C F S K receiver, as shown in Figure 3.14. A pair of idealized bandpass filters cor-responding to each of the two signal frequencies are followed by envelope detectors and bit rate sampling circuits. The envelope samples may be compared to form a partial decision, or squared and summed over a number of repeats to implement a square-law receiver. The estimate of the received signal strength is formed by CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 77 sampling the envelope of the IF filter output. Figure 3.15 shows an equivalent receiver block diagram to that of Figure 3.14. We assume that the IF and detection filters have ideal 'brickwall' passbands, and that the detection filters have a bandwidth, B/2, half that of the IF filter. The assumptions made for the bandpass filter model are the same as those for the matched filter model in Section 2.6.1. Briefly, it is assumed that the signal energy of a cosine burst influences only the output of its respective filter, and that all of the transmitted signal energy is received. Intersymbol interference and imperfect symbol synchronization effects are neglected. Also, we make the usual assumption of slow fading. The assumption of no filter crosstalk and reception of all the signal energy wil l only be true if almost all of the spectral energy of the frequency burst is contained within the detection filter bandwidth. The usual rule of thumb is that the occupied bandwidth of a pulse is the reciprocal of the pulse duration. Since we wish to estimate the performance degradation due to the use of an imperfect signal strength measurement, and not the degradation due to a band-limited signal, we will assume that the conditions of no filter crosstalk or signal loss are valid. This wil l be approximately true for R < B/2. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 78 X IFFtar BANDPASS FLTEH BWB.OPE DETECTOR BANDPASS FLTEF BWH-OPE 'a DETECTOR DETECTOR Figure 3.14: NCFSK Receiver, Bandpass Filter Implementation Figure 3.15: Bandpass Filter NCFSK Receiver, Block Diagram CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 79 The relationship between the B E R performance of the bandpass and matched filter receivers is evident from the following expressions, pe,ipf = l/2e-p>'2N (3.19) pe,mf = 1/2 e - * / " * (3.20) where Pa is the signal power and N is the total noise power at the output of one of the ideal bandpass filters [9]. The two B E R expressions are identical in form since in each case, the receiver decides in favour of the larger of a Rayleigh versus a Rician random variable. In each case, the filter with no signal input outputs a zero mean Gaussian random process 1 0 which is envelope detected and sampled, resulting in a Rayleigh random variable. Also, in each receiver, the filter with the signal input outputs a non-zero mean Gaussian random process which is enveloped detected and sampled, resulting in a Rician random variable. The relationship between the two exponents in (3.20) and (3.19) is simply, P, _ Eb/T _ R Eb N N0B/2 B/2 N0 { ' ' where T is the bit duration. Equation (3.21) implies that the two receivers would have the same error rate if R — B/2. However, as discussed above, this would only be approximately true since at R = B/2 there would be some crosstalk and signal loss. The random variables computed by the simulation program during each bit are derived in the next section. 1 0This is due to the fact that a Gaussian random process through a linear filter results in a Gaussian random process [11]. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 80 3.2.2 Simulation Method, A W G N Assume that a cosine burst of frequency fx is transmitted, and that the channel fading attenuates the signal energy to Es. The received signal is x{t) = sj^f- cos(27r/!* + 9) + w(t) ,0 < t < T, (3.22) where 9 is a random phase angle and w(t) is A W G N of power spectral density iVo/2. The random phase 9 is uniformly distributed over [0, 2TT) since Rayleigh fading is assumed. The outputs of the bandpass filters are Vi(t) = yf^ cos(27r/it + 9) + nel(t) cos^ T r/ i* ) - n t l sin(27r fxt) (3.23) and y 2(t) = n c 2(t) cos(2jr/2t) - na2(t) sin(27r/2t) (3.24) where we have used the narrowband representation of the noise at the filter outputs, i.e. n c l ,n , i , ra c 2 , and nt2 are all independent Gaussian random processes with zero mean and variance N = NQB/2 [48]. Expanding (3.23) yields yi(0 = \ J W cos9COS(2TTht) - y j ^ - sm9sm{2nht) +nei (t) cos(2nfit) - n n sin(27r/i« ) (3.25) = [yf^ cos9 +n c l(t)]cos(27r / 10 - [y/^ sm9 + nal(t)] sin(2jr/i*). (3.26) The output of an en relope detector is the magnitude of the vector sum of the in-phase and quadrature components. Thus, from (3.26) the samples of the envelope CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 81 detectors at time t = T are Ri(t) \t=T= { [ y ^ cos*+ n e l ( T ) ] 2 + [^fsme + ntl(T)]2y/2 (3.27) and |,=r= [n*c2(T) + n]2(T)]1/2. (3.28) Using appropriate transformations of these random variables it can be shown that Ri{T) is Rician distributed, and R2{T) is Rayleigh distributed [11]. The output of the IF filter is J/3(<) = yi{t) + y2{t) = [ ^ f cos0 + n c l(r)] cos(27r/1< ) - [ ^ f sin6 + nal{t)] sm(2irfxt ) +ne2{t) cos(27r/2t ) - n, 2 sin(27r/2t ) (3.29) = y ci(0 cos(27r/it) - ytl{t) sin(2^-/it) +ne2(t) cos(27r/2t) - n,2(r) sin(27r/ 2f). (3.30) In the last equation we have introduced yci(t) and yti[t) to denote the in-phase and quadrature components, with respect to fx, of the received signal plus noise. Writing fx = ///? — A / , /2 = fip + Af , and expanding then collecting all the terms that change slowly with respect to the carrier frequency, we obtain ys(0 = [y*{t) cos 2?rAft + ytl{t)s'm2irAft +ne2(t) cos 2n Aft — nt2(t)sin2nAft]cos27rfjpt + [ycl(t)sin2irAft - y 4 l(t)cos2TTA/* —nc2(t) sin27rA/t — n i 2 ( i ) cos2^A/* ]sin27r//jri . (3.31) CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 82 The output of the IF filter's envelope detector, Ri(t), is then the magnitude of the vector sum of the in-phase and quadrature terms in square brackets in (3.31). For signal frequencies that are centered on the bandpass niters, i.e. A / = B/4, as shown in Figure 3.15, and choosing R = B/2, the IF envelope is Rz(t) \t=T= {[yci(T) + nc2{T)}2 + [ytl(T) + n.2(T)}2}1/2 . (3.32) The evaluation of yci[T) and y*i(T) may be simplified if we ignore the random phase variable 6. That is we set 6 = 0 for each bit, which wil l not influence the pdf at the output of the envelope detector since the envelope detector output is independent of the carrier phase, assuming that fi » 1/T. This results in ya(T) = yfif + ncl(T) (3.33) y,i{T) = nal{T) (3.34) Thus, for each bit, the simulation requires the Rayleigh distributed received am-plitude, \j2Es/T, and four independent Gaussian random variables,^! (T), n , i(T), nC2(T), n, 2(T). From these Ri{T),R2{T), and R3{T) are computed via (3.27), (3.28), and (3.32). These envelope detector outputs can be compared to determine a partial decision, or squared and summed over a number of repeats to simulate a square law receiver. For the partial decision receivers, Rz(T) can be used to provide an estimate of the received signal strength, as discussed in the next section. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 83 3.2.3 Signal Strength Estimation The accuracy of using the envelope of the IF filter output as a signal strength estimate is determined by several factors. Since R${T) is the sampled envelope of the sum of the signal component plus noise, the signal strength estimate wil l be poor at low SNR. Also, since the envelope detector output is a non-linear function of the input signal, it will have a non-zero mean even with no signal input 1 1 . Also, any additional interference will contribute to the magnitude of the received signal strength estimate, so that the estimate wil l be high if the signal to interference ratio (SIR) is low. Once the signal strength estimate has been obtained, the receiver could use a lookup table to obtain the corresponding weight. From the results of the previous section on quantization we know that only a few fixed weights quantized to a small number of levels are required. The calculation of the weights would normally be done during the receiver design so that the weight accuracy is dependent on a priori knowledge of the characteristics of the expected noise. If the received noise varies from its design model, due to variability in manufacturing, installation, or interference, then we can expect some degradation in performance. We shall consider separately the effects of AWGN and cochannel interference on the WPD receiver performance. When the signal strength estimate is derived from the envelope of the fading signal plus AWGN, we note that the estimator need only track frequencies up to the 1 1 It is possible to compensate for the non-linear transfer characteristic of the envelope detector by using a non-linear; warping function for correction. We shall not consider such compensation here, but will simply observe the resulting performance without correction of the non-linearity. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 84 order of the Doppler frequency. A low-pass filter following the envelope detector could be used to remove a considerable portion of the IF bandwidth noise, since the IF bandwidth is considerably greater than the Doppler frequency. We model this filtering by digitally filtering the sampled output of the envelope detector, since the simulation only has access to the receiver random variables once per bit. The effectiveness of the filtering in a real implementation wil l be better than that shown here, since the simulation processes aliased samples of the noise. This is because the sampling frequency, R, is less than twice the noise bandwidth at the output of the envelope detector. The filter bandwidth must be chosen wide enough so that it can reliably track the signal fades, and the phase characteristic should not delay the tracked signal significantly. As a rule of thumb, we note that the the spectrum of an envelope detected Rayleigh faded signal extends to 2/p [2], so that the filter bandwidth should be at least twice the maximum expected Doppler frequency. In practice, we would be using a non-ideal filter, and the bandwidth of the filter should be chosen even greater than 2fo to accommodate the fading signal, and to minimize undesirable delay introduced by the filter. One might consider compensating for the filter delay by delaying the weighting of the received data, but this adds to the complexity of the receiver. We will use a very simple low-pass filter with no delay compensation to see how well a basic implementation performs. The digital filter used is a simple, first order, infinite impulse response filter, y(n) = a y(n — 1) + x(n) (3.35) and was chosen since it represents the least sophisticated filtering strategy that one might use in practice. The input x(n) is taken to be the logarithm of the CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 85 sampled envelope detector output, since logarithmic signal strength estimators are common in F M receiver integrated c ircuits 1 2 . We would expect that due to the logarithmic function, the bandwidth of the signal component would be wider than at the output of the envelope detector. This can be seen by considering that during a typical deep fade, which is short in duration, the argument of the log function wil l be quite small. This results in a large negative output of a short duration and thus of high frequency content. We can therefore expect that the low-pass filter cutoff frequency will have to be wider than the simple 2fry guideline for an envelope detector. The Fourier transform of the low-pass filter impulse response is 2T(e**) = H(z) U , „ = l a 1 e _ i h , . (3.36) The value of the constant, a, that yields a specified 3 dB cutoff frequency, fcot is o = [2 - cos(27r/ c o/T)] - \J[2 - cos(27r/ c o/r)] 2 - 1. (3.37) For land mobile radio communications at UHF, we expect a maximum Doppler frequency of about 100 Hz. This corresponds to 130 kmh at 850 MHz. Our guideline for the low-pass filter cutoff frequency would have us set feo > 2fo = 200 Hz, assuming that we had an ideal filter. For a trial, we will choose fe0 = 4fu = 400 Hz since we are using a logarithmic signal strength estimator, and since we are using a low pass filter with non-ideal gain and phase characteristics. The value of a for this cutoff frequency is 0.74 at our simulation bit-rate of 8192 bps. The frequency response shown in Figure 3.16 shows the poor low-pass characteristic of this filter. The tracking performance of the filter can be seen in Figure 3.17, where 1 2 For example, the Plessey SL6652 IC. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 86 a sample of a Rayleigh faded signal is plotted versus time, along with the low-pass filter output. There is a delay of the order of 5 bits and the amplitude fluctuations of the signal are not precisely followed. Despite these shortcomings, we will use this filter to observe whether it can improve the W P D receiver performance. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 87 03 T3 co O N "5 E k. o z -10 -15 -20 y(n)-0.74y(n-1) + x(n) Cutoff frequency - 400 Hz Bit rate - 8192 bps _l I I I I I I I L I i i i I i i i i i i i—i—i—L _ I _ J I L _ 1000 2000 Frequency (Hz) 3000 4000 Figure 3.16: Frequency Response of Signal Strength Estimator Filter. 10 CO DopDN iter frequency = 100 Hz. 'it rate - 8192 bps. Bit Number Figure 3.17: Tracking Behaviour of the Signal Strength Estimator CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 88 3 . 2 . 4 Simulation Method, Cochannel Interference We now consider the case where the signal strength estimate is derived from the envelope of the faded signal plus cochannel FSK interference and AWGN . The cochannel interference is assumed to be due to a number of interferers, and the received level of each is taken to be Rayleigh distributed. These conditions have been used to model the cochannel interference level for the A M P S cellular system [49]. If any number of cochannel interferers are Rayleigh faded, then the received level of interference is also Rayleigh [49]. Unfortunately, the spectrum of the interference will be same as that of the signal, so that low-pass filtering of the IF filter envelope detector output will not be of benefit in an interference dominated system. We will develop a simulation procedure that uses a pessimistic model for the signal strength estimate, by assuming that the envelope detector output is not filtered. As shown in Section 3.2.2 the IF filter envelope detector output can be written as the vector sum of the in-phase and quadrature components of the signal and narrowband noise components, and we extend this to include additional components arising from the interference. The Rayleigh distributed interference level is equivalent to sampling the magnitude of a complex Gaussian random process. Thus we may represent the signal, cochannel interference, and the A W G N by the phasor diagram in Figure 3.18, where the in-phase and quadrature components of each vector is a Gaussian random variable. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 89 Q Figure 3.18: Phasor Representation of the IF Filter Envelope Detector Output. The IF filter envelope detector output sample, R^T), is the vector sum of the faded signal and the narrowband representations of the A W G N and cochannel interfer-ence. The signal and A W G N components are discussed in Section 3.2.2. The fading cochannel interference has narrowband in-phase and quadrature components, nd(t) and nti(t), that are independent Gaussian random processes. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 90 The faded signal level and the A W G N noise components are generated as de-scribed in Section 3.2.2. If the bit repeats are separated by several fades then the envelope detector samples w i l l be independent, and the fading interference lev-els for a set of bi t repeats w i l l be accurately modeled using independent complex Gaussian noise samples. The S IR is Es/Ei, where Es and Ej are the average bi t per iod energies of the signal and interference, respectively. The narrowband model of the interference is i[t) = nei(t)cos(2irfIFt) - n „ ( t ) sin(27r fIFt). (3.38) The variances of the Gaussian components, n c t ( t ) and nM-(t) are found from ET = E \j* i\t)dt = f * « ( ' ) ] + ^E[ni(t)} (3-39) so that E[nl(t)\ = EMt)] = f. (3.40) Next , we consider the modeling of the demodulator performance i n cochannel F S K interference. W i t h fading interferers, there w i l l be a varying number of inter-ferers that contribute to a bit error. In [52] the performance of an ideal N C F S K demodulator is derived for a non-fading signal in the presence of any number of equal energy, non-fading, cochannel interferers. The exact B E R formulas are com-plex to evaluate and as such are impract ical for our simulat ion. However, as the number of cochannel interferers is increased the B E R approaches that obtained by assuming that the cochannel interference is equivalent to narrowband Gaussian noise of the same energy. Th i s assumption is pessimistic when there are a small number of interferers, and the static SIR and S N R values are high [52,53]. Al so , for high S IR and S N R , the assumption of equal strength interferers is pessimistic CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 91 [54]. At lower values of SNR and SIR, the assumption of narrowband Gaussian noise is slightly optimistic if there are a very small number of interferers, e.g. one or two [53,52]. However, in our case the instantaneous interference level is the superposition of a number of fading signals, and it is more probable that a number of interferers contribute to any significant interference level. Also, since there are a number of interferers, it is likely that some interference energy wil l be present in both of the detection filters. These considerations indicate that errors due ex-clusively to a single tone are not likely to be the most common. Thus, with more than one interferer usually causing errors, the modeling of the interference by an equivalent amount of narrowband Gaussian noise may be reasonably good. Our modeling strategy, then, is to simulate the effect of the fading interference on the demodulator by using the instantaneous level of interference to adjust the variance of additional AWGN. The interference energy for a particular bit is where n c,(T) and n,;(T) were shown earlier to have variance Ej/T. The division of A W G N between the detection filters does not affect the error rate with equally likely data bits in an ideal NCFSK receiver [52]. For our simulation we must divide the noise equally between the filters since we are only simulating the reception of a '1'. The variances of the Gaussian components in the narrowband representation of the equivalent interference noise for the filter at f\ are found from The variances of ncn, n„i, will be equal, and will also be equal to the variances of (3.41) CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 92 the equivalent noise components for the filter at ji, n^, re«t2> so that using (3.42) and (3.41) one obtains E[n*M{T)] = E[n*M{T)] = E\n\i2{T)) = E[n]i2(T)} = \ [»» (T) + n^T)] . (3.43) The simulation results for both no cochannel interference and an unfiltered sig-nal strength estimate with cochannel interference are discussed in the next section. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 93 3.2.5 Simulation Results The B E R of the W P D receiver using five repeats with no cochannel interference is shown in Figure 3.19. Also shown are the B E R curves for the majority-voting and square-law receivers as determined by simulation and as predicted by (2.29) and (2.31). The Doppler frequency, bit-rate and packet format are the same as used for the simulation discussed in Section 2.6.4, i.e. fa = 49 Hz, R — 8192 bps, and the packets are forty bits long with single packet length gaps between repeats. Four curves are shown for the WPD receiver which correspond to the following methods of obtaining the SNR estimate: E x a c t . Exact SNR measurement. S igna l plus noise. The unfiltered envelope detector output is used to estimate the received signal strength. F i l t e red s ignal plus noise. The logarithm of the envelope detector output is low-pass filtered and used to estimate the received signal strength. F i l t e red s ignal , no noise. The logarithm of the signal without noise is low-pass filtered and used to estimate the received signal strength. This will reveal the degradation of the B E R due to the distortion of the signal by the filter. Inspection of Figure 3.19 shows the separate contributions to the B E R degrada-tion by the noise and low-pass filter characteristics. At a target error rate of 1 0 - 3 the W P D receiver suffers a one dB loss using the signal plus noise estimate instead of the ideal SNR estimate. With the low-pass filter, the filtered signal plus noise curve is very close to the filtered signal with no noise curve. This indicates that the low-pass filter has removed almost all of the effect of the noise. However, as the SNR is increased the filtered signal plus noise B E R curve shows an increasing CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 94 i r r 6 I • ' • • 1 • ' • • 1 i i i i I 1 1 1 1— 0 5 10 15 2 0 Signal to Noise Ratio (dB) Figure 3.19: WPD Bit Error Rate with SNR Estimation CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 95 loss relative to the ideal SNR measurement. This is due to the distortion of the received signal level by the filter. At high SNR, errors occur mainly during deep fades, which are of short duration, and the delay introduced by our simple filter becomes significant. At very high SNR, the filter distortion wil l limit the BER, and better performance would be obtained without low-pass filtering. Despite this imperfect filtering, the low-pass filter does improve the B E R over the range of SNR shown. The filter design involves a trade-off between good noise attenuation and the effects of filter distortion at high SNR. A compromise filter design can be sought to obtain a B E R characteristic close to that of the ideal SNR measurement, over a certain operating range of average SNR. Figure 3.20 shows the performance of the WPD receiver when interference is present. We have taken the SIR to be 15 dB as an approximation to the cochannel interference expected in cellular systems. In the AMPS system the design objective is to achieve a SIR of 17 dB or greater over 90% of the coverage area in order to maintain good voice quality [36]. We have used a pessimistic value of SIR for our example. Returning to Figure 3.20, we see that compared to the case of no interference shown in Figure 3.19, the interference significantly affects the BER . Two curves for the W P D receiver are shown, with the labels describing the SNR estimation method as follows: S igna l . The SNR estimate is formed from the ratio of a perfect signal strength measurement to the value of the known noise PSD, NQ. S igna l p lus noise and interference. The SNR estimate is formed from the ra-tio of the unfiltered e: ivelope detector output to the known noise PSD, N0. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 96 l 1 1 1 1 1 1 —7 1 1 1 > 1 ' r 5 Repeats Signal to Noise Ratio (dB) Figure 3.20: WPD Bit Error Rate with SNR Estimation, SIR=15 dB CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 97 For a target B E R of IO - 3 , comparing the s ignal p lus noise and interference curve in Figure 3.20 with the s ignal p lus noise curve in Figure 3.19, i.e. no lowpass filtering of the envelope detector output, we see that despite the interference the W P D receiver has slightly increased its gain over majority-voting. Unfortunately, if the interference level is increased, then the signal strength estimate may become so inaccurate that the WPD receiver may perform worse than majority-voting. However, as discussed earlier, the SIR value used here is a pessimistic approxi-mation to that expected. If lowpass filtering of the envelope detector output is used, then we can expect the W P D performance to approach the s ignal curve in Figure 3.20 for low SNR, where the performance is dominated by AWGN . As the SNR is increased, the s ignal plus noise and interference curve wil l represent the expected performance since filtering the envelope detector output wil l not mit-igate the effects of cochannel interference. However, for the range of SNR shown in Figure 3.20 we note that the B E R curves are not yet approaching their limiting value set by the interference. Thus, there is considerable influence due to A W G N in this region, and lowpass filtering should increase the gains over that reported here. Without filtering, the gain of the W P D receiver over majority-voting is over 2 dB at a B E R of IO - 3 . Also, as the SNR is increased, the W P D receiver may achieve a lower B E R limit than majority-voting, depending on the level of interference. Similar gains in performance relative to majority-voting using the P E R as a criterion may be observed in Figures 3.21 and 3.22, for the cases of no interference and SIR = 15 dB respectively. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 98 i n - 6 I — • i i i I i i i i 1 1 1 1 1 1 1 > 1 1— , U - 5 10 15 20 Signal to Noise Ratio (dB) Figure 3.21: WPD Packet Error Rate with SNR Estimation CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 99 Signal to Noise Ratio (dB) Figure 3.22: WPD Packet Error Rate with SNR Estimation, SIR = 15 dB CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 100 It is interesting to compare the performance of selection diversity to the W P D receiver when both utilize the same signal strength estimation schemes. Figures 3.23 through 3.26 show the performance of selection diversity with and without interference. Below an SNR of about 12 dB in AWGN, the effect of the noise on the unfiltered signal strength estimate has degraded the selection diversity receiver B E R so that it is higher than majority-voting. With interference, the selection diversity B E R relative to majority-voting is further increased. The filtered signal strength estimate will improve the B E R and P ER for lower SNR, as in the WPD receiver. However, the selection diversity receiver appears to be more sensitive to SNR estimation error than the WPD receiver. This can be seen by comparing Figures 3.23 and 3.19. The loss in efficiency at a B E R of 10~3 by using an unfiltered signal strength estimate is about 50% greater for the selection diversity receiver. The greater sensitivity of the selection diversity receiver is also evident when a filtered signal strength estimate is used, or whether B E R or P E R is used as the performance measure. Compared to selection diversity, the W P D receiver with an unfiltered signal strength estimate offers an improvement of about 3 dB, at a B E R of 1 0 - 3 and an SIR of 15 dB. The improvement relative to selection diversity and majority-voting will increase with the number of repeats. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN101 Figure 3.23: Selection Diversity B E R with SNR Estimation CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN 102 5 Repeats Signal to Noise Ratio (dB) Figure 3.24: Selection Diversity B E R with SNR Estimation, SIR = 15 dB. CHAPTER 3. SOME ISSUES IN PRACTICAL WPD RECEIVER DESIGN103 Signal to Noise Ratio (dB) Figure 3.25: Selection Diversity P ER with SNR Estimation Figure 3.26: Selection Diversity P E R with SNR Estimation, SIR = 15 dB. 105 Chapter 4 Conclusions In this thesis we have examined the technique of weighted partial decision combin-ing for diversity improvement on fading channels. The motivation for using partial decisions rather than a completely soft decision process is that it may be preferable in some applications due to constraints on receiver cost. In Chapter 2, the WPD receiver performance for the case of N C F S K demod-ulation on a slow Rayleigh fading channel was analyzed. A tight upper bound on the B E R was derived, as were estimates of the asymptotic performance of the scheme relative to majority-voting, selection diversity, square-law and maximal ratio combining. The WPD receiver performance was shown to be equal to, or better than, majority voting or selection diversity. For low Doppler frequencies the WPD receiver is equivalent to majority-voting, while selection diversity is less efficient, offering no gain relative to a single transmission. With five independent bit repeats and a target B E R of 10" 3 , the WPD receiver is about 3 dB more effi-cient than majority voting, and about 1 dB more efficient than selection diversity. This gain increases with the number of repeats, '."he W P D receiver is less than CHAPTER 4. CONCLUSIONS 106 1.1 dB poorer than the square-law receiver, up to about 15 repeats, while at this number of repeats selection diversity is about 6 dB poorer than the square-law receiver. Simulation results for packet error rates show similar gains over the stan-dard partial decision combiners. A simple guideline was developed to estimate the optimal number of independent branches, given a fixed energy per bit. The optimal number of branches for the majority-voting, WPD, and square-law receivers is approximately one third of the value of the available energy per bit. The performance degradation due to various receiver implementation effects was considered in Chapter 3. It was demonstrated that near optimal WPD performance is achievable with only a few reliability weights quantized to a small number of levels, and without threshold optimization as the average SNR varies. This level of performance can be nearly maintained despite an AWGN corrupted received signal strength estimate, by using a simple low pass filter to smooth the IF filter envelope detector output. It was also demonstrated that the selection diversity receiver is slightly more sensitive to signal strength estimator errors. When the signal strength estimate is corrupted by cochannel interference, the performance degradation may be considerable, but for a moderate level of interference the WPD receiver maintains some performance advantage over majority-voting and selection diversity. The results indicate that near ideal WPD receiver performance in AWGN can be obtained with a simple implementation. Since many land-mobile radio receivers already employ a received signal strength measurement circuit, implementation of the WPD receiver requires only that the signal strength measurement be low-pass filtered, sampled, and used to index a small lookup table of reliability weights which CHAPTER 4. CONCLUSIONS 107 are summed to form the final decision variable. The WPD receiver may be appro-priate when the receiver design is constrained to use a hard decision demodulator, and the system is power limited with a modest level of cochannel interference. In such cases, one might consider the W P D receiver when a fixed number of bit repeats are used, or when a variable number of transmissions are employed with memory A R Q (automatic repeat request). Also developed in Chapter 3 was a method to estimate the optimal quantization thresholds for the W P D receiver. The method minimizes a Chernoff bound on the probability of bit error, and is applicable to other signalling, channel fading, and noise characteristics. Also, an expression for the probability of bit error for a se-lection diversity receiver with quantization was derived. The analysis is applicable for the more general case of a Rician faded signal level. Finally, it is emphasized that this thesis has concentrated on simple repetition diversity, because of its practicality in low cost receivers. However, it is possible to extend the decision rule to apply to coded diversity reception on slowly fading channels, so that the receiver forms a maximum-likelihood packet decision. 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COM-22, pp. 1948-1949, Dec. 1974. 112 Appendix A In this appendix it is shown that 2 2 + 2 /T° r l dv 4 (4 + 5 i Q \ 1 / 2 [ S ] ~ 7o Jo »i/»-»/io(l+ < 4 + 7 o V 4 + 37oJ 1 j so that we can obtain (2.22), 4 + 7o V 4 + 3 7 o / Let 1 Jo u l/2-2/To(l + v)2+2/T,o ( A > 2 ) . f* * (A.3) 7o w 5 / 2 Vl + w/ v ' By making the substitution t = u/(l + u) we can obtain J = [1,*r1'i+tt'">{l-t)1'idt = Bi ( l / 2 + 2/7o,3/2). (A.4) JO 3 One might evaluate J using tables or numerical integration routines for the incom-plete Beta function, but we can find a good upper bound on Bi(l/2 + 2/70,3/2) that is more readily evaluated. Substituting x = 1 — t, gives 1= f1 x^il-xY^'^dx. (A.5) J l / 2 APPENDIX A 113 Figure A . l : A Linear Bound on x 1 / 2 . We may upper bound J by bounding the function x 1 / 2 over the region x = (1/2,1) by a linear approximation to x 1 / 2 , as shown in Figure A . l . Thus / < ^(xo) = fl {mx + b){l-x)-1'2+2''">dx. (A.6) Jl/2 The line y i = mx + 6 is chosen to be tangent to y = x 1 / 2 at the point xo, and m and 6 are easily found in terms of x 0 . If one minimizes (A.6) with respect to the tangent point x 0 , it is found that the minimum value of 7i(x 0) occurs at XQ = (4 + 57o)/(8 + 670)- The resulting bound on J is / < Ji(xS) = ( l /2 + 2/7o)21/2+2/'"> 4 + 57o1 1/2 8 + 670 J (A.7) which, after multiplication by 2 2 + 2 / ' w /7o , is equivalent to (A. l ) . The ratio Ii/I, is plotted as a function of SNR in Figure A.2, where / was APPENDIX A 114 1.21 i i i i | i i i i | i i i i | i i i i | i i i i | i i i i | i i i i Signal to Noise Ratio (dB) Figure A.2: Ratios of Two Bounds on the Moment Generating Function to the Numerically Computed Result. evaluated numerically 1. Also shown is the ratio h/I, where I2 = I + JT/2 was found in [51] and is asymptotic to J as 70 - • 00. Over the range of SNR from -5 dB to 30 dB, the bound Ix is less than 0.5% greater than the exact value J . Although the bound I2 is asymptotic to I, it is not tighter than Ix unless the SNR is greater than about 37 dB. For more typical signal to noise ratios Ix is much tighter. *The FORTRAN routines BETA and RIBETA (50] were used to evaluate the incomplete Beta function. 115 Appendix B In this appendix we derive equation (2.27), which gives the W P D receiver B E R when weights proportional to the received SNR are used. We assume that an ideal N C F S K receiver is operating in A W G N with a Rayleigh faded signal amplitude. Assume that a '1 ' is transmitted L times, and that L partial decisions, {di}^, are made, where di is defined in (2.1). With the weights equal to the received SNR, it wil l be convenient to define the weighted partial decision variable as Y = —^idi. Note that a positive value of Y corresponds to a partial decision error. Let the sum of L independent samples of Y be denoted by YL, and denote the pdf of YT. by fyL{y)- The probability of bit error is For each possible value of SNR, 7, the weighted partial decision variable, Y, has two values, (B.l) —1 1 ,with pdf value of fnc(l) ,with pdf value of fie{l) (B.2) where M i ) fv(l)[l-Pe(l)) (B.3) M i ) fr{l)Pe{l)- (BA) APPENDIX B 116 The pdf of Y is f (v\ — I / r ( -y ) [ i -p e ( -y ) ] ,y<o ( B 5 ) M y ) ~ [ h(y)Pe(y) , y > o - ( B ' 5 ) The moment generating function of Y is Gy(s) = E[e—] = / _~c - 'Vr (y )r fy (B.6) = / o ° ° /r(y) [(1 - Pe(y)) e4" + p a ( y ) e - » ] dy (B.7) [ ( l - l / 2 e - " / 2 ) «*» + l ^ e ^ / V " ] dy (B.8) 7o Jo i r i 2 i + 2-fo L5 + ! / 2 + l/lo ~ s + 1/fo ~ s + V 2 + 1ho Thi s can be put in the form (B.9) where G y ( 3 ) = J , (B.10) Y K ) 27o5 + l/2 + l/7o V ' = -2/70.+ 1/2(1/2+ 1/70)- { B n ) K } (-s + l/7o)(-s +I/2 + I/70) V ) Note that i2(s) incorporates al l of the right half plane ( R H P ) poles of Gy(s). The moment generating function of YL is GYL = GY(s), (B.12) which can be wri t ten as GYl = C(s) + Z(s) (B.13) where £(s) is the par t ia l fraction expansion of GyL(s) corresponding to its left half plane ( L H P ) poles, and Z(s) is the par t ia l fraction expansion corresponding to its R H P poles. Note that to obtain fyL for y > 0 one needs to find the inverse APPENDIX B 117 Laplace transform of £(s). The probability of error could then be obtained via the integration in (B. l ) . We may avoid the Laplace transform inversion and subsequent integration by noting that P..wPdU=1 = r frM*V (B.14) Jo = llm Te-^fyMdy (B.15) »-»0 JO = l i m £ ( 3 ) . (B.16) Thus, the probability of error may be found by evaluating the L H P portion of GYL{S) at s = 0. GYM = , J w : , . . , (B- 1 7) In our case, using (B.10) in (B.12) gives 1 RL(s) (27o) L ( 5 + I / 2 + I / 7 0 ) The LHP partial fraction expansion of GyL(s) is e M = ( ^ 5 ( . + 1 / * % - , , ) ' ( B 1 8 ) where - - ( rh j i^^w] !.=-<«• ( B 1 9 ) In order to evaluate the ak we will find it convenient to split up R(s) into its partial fractions, -2/7o3 + (1/2 + I / 7 0 X I + 2 / 7 0 ) R(s) = ( - 3 + l / 7 o ) ( - 3 + l / 2 + l / 7 o ) 1 + 4 / 7 0 1 + 2 / 7 0 -s + I / 7 0 - 5 + 1/2 + I / 7 0 ' We now have L-j (B.20) u , t s v W v - s + i / W v - " + 1 / 2 + 1 / 7 0 J v APPENDIX B 118 Consider the nth derivative of RL(s), where we have denned bjn as dr b i n = df ( 1 + 4 / 7 0 y / - ( 1 + 2 / 7 0 ) y ~ y V-5 + l /7oJ V-5 + I /2 + I/70J For j = 0 we have ban = d 5 n fc - ( 1 + 2/70) 3 + 1/2 + I/70, ( L - l ) ! (-« + 1/2 + 1/n,) and for j = L, bm — dT d3n ( 1 + 4/70 V V-S + I/70J (l + 4/7o) L(L + n - l ) ! ( L - l ) ! (_s + i / 7 0 ) ^ ' For the terms, bji,bj2,..., * n e form of 6yn is where and V-S + 1/70; -(l+2/70) (B.22) (B.23) (B.24) (B.25) (B.26) (B.27) (B.28) (B.29) (B.30) -s + 1/2 + I/70 Using Leibnitz's theorem for the differentiation of a product of two functions [15], dT dx1 -[uv] = dTv ( n \ du dn~lv ( n\d2u dr~2v dTu Udxn + [ 1 ) dxdxn~x + { 2 ) dx2dxn~2+'"+ dxnV • = V ( n \ d n ~ i u d i v (B.31) APPENDIX B 119 we have, for n = 1,2,..., L — 1, bin = v ( n \ 17 1 + 4 ^ ° VI 17- - ( 1 + 2/70) •3 + 1/2 + I/7. SI* J(1 + 4 / 7 o ) ( i - i ) i (-, + W-• (B.32) • ( - 1 ) ^ ( 1 + 2 / 7 o ) ^ ( L " y + n " , ' " 1 ) ! ( L - y - i ) i ( - 5 + 1/2 + 1 / 7 0 ) ^ + -- .(B.33) We require the (L — k)th derivative of RL(s) evaluated at the LHP pole, which gives n=L—k «=-(l/2+l/-7o) = ZZ bi(L-k) U=-(l/2+l/-ro) i=0 (B.34) = bo(L-k) |«=-(l/2+l/io) + |a=-(l/2+l/<ro) L - l ^ + X j ( y ) 6i(i-fc) l*=-(i/2+i/70) • ( B - 3 5 ) Using (B.16) with (B.18), and (B.19), gives ^ 1 . = , - ( 2 7 0 )^^(1 / 2 + 1 / 7 0 ) * ^ Using (B.36), (B.35), (B.33), (B.25), and (B.27), yields -k)\ {dsL~k ^ L ( s ) ]L=-(i /2+i/<ro)} (B.36) e,wpd \w=i 1 A 1 1 f (2L - k - 1)! (27o) L (1/2 + l/7o) f c (L - k)l \ ( L - l ) ! (-1)* + (1 + 4/70)* + & - ^ ( > ) g ( £ 7 f c ) ( £ - ^ o r 1 ) ! ( £ - y + " 1 ) ! ( 1 + 4 / 7 0 V (1/2 + 2/70) L-k+j-i (1 + 2/7o)'} ( l + 2/7o) L - i T ( l /2 + 2/7o) 2 L - \ I ~{L-j-i)\ (B.37) APPENDLX B 120 After some algebra, the preceding equation can be simplified to (B.38) which is (2.27). L-k As a check on the above derivation of (2.27), one can use the form of GY{s) given by (B.9) and write GY(S) = ± [ A { a ) + B{s)\ (B.39) *7o where M*) = wo 1 , 7 ( B - 4 0 ) v ' s + 1/2 + 1/70 and Then * M = ^TTJT0 ~ + (B-41) ° r M = ^ t { * ) A H ° ) B L ~ k { s ) - { b a 2 ) By finding the LHP portion of this form of GyL (s), and setting s to 0, one can obtain an alternate form of (2.27), APPENDIX B 121 e,wpd | tu=7 (2 + 7o)L (2 + L-k-l E m=l (- l )^-**'- 1 , + 2 L " * s (*) s {(£ * - i 1 ) ( m + ^ ' - 0 ( L ' * ^ + ' " 0 ( ^ ) ( a 4 3 ) £ — & ^ 2 2 m + J b - , ' - L t=0 L - k - k—j+m—i ' Equation (B.43) and (2.27) were checked to give identical numerical results over a broad range of SNR and for several values of L. Note that the kth term in (B.43) is the probability that an error is caused by L — k incorrect partial decisions outweighing k correct partial decisions. 122 Appendix C In this appendix the Chernoff bound on the probability of bit error for a binary W P D receiver employing quantization is shown to be given by (3.15), Pe < [Gx(e')}L = r M L k=l where and P 2t are the transition probabilities in the kth partition of a symmetric channel transition matrix that has no erasure partitions. A Chernoff bound for the case of an erasure partition wil l also be given. As discussed in section 3.1.2.2, a symmetric channel transition matrix, T, may be partitioned as '( Pll P2l \ ( Pu P22 \ ( Pit P2k \ ( P l(Af- l) PHM-1) \ ( PlM \ . V p n Pn J V P22 Pu J \P2t Pit ) " ' { P 2 (M-I) P i (M- i) J\PiM J Let j = 1,2,..., Q, index the columns of T and let » = (0,1) index the rows, with » = 0 corresponding to the message bit m = 0. Assume that '1 ' is transmitted and that L independent outputs {./j}^ are obtained from the channel. A maximum-likelihood receiver will decide that a '1 ' was sent if P [{*}£* I f = 1] > P [b\}f=11»= 0 ] . (ci) Since we have assumed that the samples are independent, then the M L decision APPENDIX C 123 rule reduces to ftpWIi) > f[PUi\o) 1=1 1=1 L 1=1 PI* 11) > I PUi\o)\ > 0. (C.2) (C.3) (C.4) We seek to bound the probability of error by a Chernoff bound, i.e. (C.5) where Gx{s*) is the minimized moment generating function for the random variable X, where X takes on the values Xj = ln J = 1,2,. ..,<?. (C.6) L^(J I o). Since we have assumed that a '1 ' has been transmitted, x}- occurs with probability P{j | 1). To find s* we write Gx(s) = E[e—] = £ P ( y | l ) e - " > j=l - a ln I = izpui^-'puioy. (a?) This sum over the Q columns of T may be re-written as a sum over the M partitions of T, resulting in M-l Gx(s) = E[Pu'Pik + Pik'Pit]+PiM (C.8) *=i Equation (C.8) is symmetric with respect to s = 1/2, and since the moment gener-ating function is a convex function of 3 [45], Gx{s) is minimized by 3 = s* = 1/2. APPENDIX C 124 Thus and M - l Gx(s<) = £ 2P 1 1/ JP 2 1/ 2 + P, 1M (C.9) Pt < [GX(S*)} M-l 2 £ \JPlkP2k + PlM fc=l (C.10) Equation (C.10) is in a form for the case of an erasure partition. If the erasure partition is absent the channel transition matrix will consist of M 2 x 2 partitions, and (C.10) simplifies to Pe < [Gx{s*)\L = • M : k=l which is (3.15). 125 Appendix D Here we derive an expression for the probability of error of a selection diversity receiver employing a quantized SNR measurement. Let the pdf of the received SNR be denoted by fr{l)- The received SNR is quantized using N thresholds {Tk}^=1, as shown in Figure 3.1. Let the pdf of the quantized SNR be denoted as /x(x) , where x £ {x i , x2,..., XJV+I}. A representative /x(x) is shown in Figure D. l . A selection diversity receiver operating on L independent samples of the quan-tized SNR will choose the sample with SNR, x*, that is the largest of those available. The probability that x* is equal to, or less than, some value y is equal to the proba-bility that all of the samples are at the same level, y, or less. Since we have assumed that each branch fades independently the cumulative distribution function of X* is Fx>{y) = P{X1<y,X2<y , . . . , XL<y) (D.l) L (D.2) L (D.3) = [Fx(y)]L (D.4) / APPENDIX D 126 fx(x) 1 x i x , x , 1 2 3 PN +1 X X N N+1 Figure D. l: Quantized SNR Probability Density Function where from (D.3) to (D.4) we have assumed that the Xi are identically distributed. A representative Fx-{y) is shown in Figure D.2. The pdf of X', fx>{y), is the derivative of Fx-[y). When the selected SNR is in the ktH SNR region, the receiver will make an error with probability Pk JTk-i where Pk is the probability that the SNR is in region k in Figure 3.1, and pe{l) is the probability of receiver error at SNR 7. The probability of error is then the sum of the conditional probabilities given by (D.5), weighted by the pdf of X*, i.e. Pe,.el = PlLPe\l+[(Pl + P2)L-PlL]Pe\2 + ~-N + 1 r r ,1 N+1 E APPENDIX D 127 Fx.(x) ( t»+ P2> N+1 Figure D.2: Selection Diversity Quantized SNR Cumulative Distribution Function where we have denned Ak = £ P « . (D.8) «=i Equations (D.7) and (D.8) are identical to (3.17) and (3.18) in Section 3.1.3. Note that equation (D.7) is not restricted to any particular SNR distribution, 1 so that it may be used, for example, with a Rician SNR distribution. As the number of SNR quantization regions is increased, the B E R given by (D.7) wil l approach the B E R for a selection diversity receiver without SNR quantization. As demonstrated in Section 3.1.3, with 3 repeats, about 4 thresholds are required to closely approximate the unquantized selection diversity BER . 1If the Pfc and Pe|jt cannot be obtained in closed form, then one may evaluate these probabilities by numerical integration.
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Optimal weighted partial decision combining for fading channel diversity Kot, Alan Douglas 1987
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Title | Optimal weighted partial decision combining for fading channel diversity |
Creator |
Kot, Alan Douglas |
Publisher | University of British Columbia |
Date Issued | 1987 |
Description | A diversity combining scheme is examined that utilizes a demodulator's hard decisions in conjunction with knowledge of each decision's reliability. A maximum-likelihood bit decision is made, based on these partial decisions from the demodulator and on measurements of the state of the fading channel. The technique is sub-optimal since hard decisions are processed, but it may find application in low cost receiver design. The technique is optimal in the sense that a minimum probability of bit error is achieved, given a set of partial decisions and knowledge of their reliability. Performance analysis for the case of non-coherent frequency shift keying on a slow Rayleigh fading channel with additive white Gaussian noise includes the derivation of a tight upper bound on the probability of bit error, and estimates of the asymptotic performance relative to standard diversity schemes such as majority-voting, selection diversity, square-law, and maximal ratio combining. These results are supported by simulation results for bit and packet error rates in an example system. With five independent bit repeats and a BER of 10⁻³, the receiver is about 3 dB more efficient than majority-voting, and about 1 dB more efficient than selection diversity. The gain in efficiency, relative to the standard partial decision combination schemes, increases with the number of repeats. The degradation in performance in a practical receiver implementation is addressed, and it is demonstrated that near ideal performance may be obtained with only a few reliability weights quantized to a small number of levels. Furthermore, this performance is maintained over a wide range of average signal to noise ratio without having to adapt the reliability weights. When the reliability estimate is corrupted by additive white Gaussian noise, it is demonstrated that simple low- pass filtering of the signal strength estimate is sufficient to obtain near ideal performance. The performance is degraded in the presence of cochannel interference, but for a moderate level of interference the performance is demonstrated to be superior to majority-voting or selection diversity. Other results include a method to estimate the optimal quantization thresholds, and a method to obtain the probability of error of selection diversity receivers employing signal to noise ratio measurement quantization. The selection diversity analysis is applicable to the more general case of Rician fading. |
Subject |
Radio -- Transmitters and transmission -- Fading |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-07-20 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0097126 |
URI | http://hdl.handle.net/2429/26710 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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