PERFORMANCE ANALYSIS OF VITERBI D E C O D I N G IN R A Y L E I G H F A D I N G W I T H CHANNEL ESTIMATION ERRORS by Dingyi L i n B . S c , Zhongshan University, Guangzhou, C h i n a , 1998 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 THE REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF APPLIED SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES (Electrical and Computer Engineering) T H E UNIVERSITY OF BRITISH C O L U M B I A A p r i l 2006 © Dingyi L i n , 2006 Previous studies on the effect of channel estimation errors on the bit error rate ( B E R ) performance of V i t e r b i decoding ( V D ) concern various types of fading channels w i t h additive white Gaussian noise ( A W G N ) , modulation and interleaving schemes. Pairwise error probabilities ( P E P ) have been derived using Laplace transform. Studies of V D on fading channels w i t h impulsive noise and perfect channel estimation are also available i n the literature. In this thesis, the B E R performance of unquantized V D w i t h B P S K is analyzed for a frequency-nonselective slow Rayleigh fading channel w i t h A W G N and Gaussian distributed channel estimation errors. Closed-form expressions for the P E P are derived. Upper bound and lower bound on the B E R are obtained. It is shown that channel estimation errors have the same effect on B E R as channel noise. Computer simulation results show that the upper bound is fairly tight. In practice, the channel might be estimated using pilot symbols, together w i t h various interpolation filters. It is shown that the channel estimation error variances are usually unequal for different data symbol positions. T h e B E R performance of V D optimized for unequal estimation error variances is compared w i t h that of V D optimized for equal variances using computer simulation. The B E R performance of V D in Rayleigh fading and impulsive noise w i t h channel estimation error is also studied. The optimal metrics are derived for V D i n Rayleigh fading w i t h Gaussian channel estimation errors, for L a p l a c i a n noise and Gaussian mixture noise. The B E R performances of V D ' s for various scenarios are compared. Contents Abstract ii Contents iii List of Figures v List of Symbols ix List of Abbreviations xii Acknowledgements xiii Chapter 1 Introduction 1 1.1 M o t i v a t i o n and G o a l 2 1.2 Thesis Outline 3 Chapter 2 Review of Related Works 4 2.1 Convolutional Codes and the V i t e r b i Decoder 4 2.2 Fading Channel Models 5 2.3 Pilot Symbol Assisted M o d u l a t i o n 7 2.4 Impulsive Noise 13 2.5 Review of Related Results 14 Chapter 3 V D in Rayleigh Fading with A W G N 17 3.1 System M o d e l 17 3.2 Variances of E s t i m a t i o n Errors i n a Frame 20 3.3 Performance of V D optimized for unequal estimation error variances . 24 3.4 Upper and Lower Bounds 28 Chapter 4 V D in Rayleigh Fading with Impulsive Noise 47 4.1 Laplacian Noise 47 4.2 Gaussian M i x t u r e Noise 54 Chapter 5 Conclusion 59 Bibliography Appendix A p RelRi] 61 Derivation of the C o n d i t i o n a l pdf's (Re[ri]\E£[Hi] = Re[hi], Si = Sj) andp (Im[ri]\Im[Hi] lmlRi] = Im[hi], Si = Si) 65 Appendix B Derivation of O p t i m a l B i t M e t r i c for Laplace Noise Appendix C Derivation of O p t i m a l B i t M e t r i c for M i x t u r e Gaussian Noise . . 68 72 List of Figures 2.1 Pilot symbols i n P S A M 7 2.2 Filter coefficients of Gaussian interpolation for M = 100 9 2.3 Relative positions of the pilot symbols and the interpolated frame i n Gaussian interpolation 10 2.4 Filter coefficients of 4-tap ideal low-pass filter for M — 100 11 2.5 Relative positions of the pilot symbols and the interpolated frame i n 4-tap ideal low-pass filter 2.6 Gaussian, Laplace and Gaussian mixture pdf's. A l l pdf's have a variance of 1 2.7 12 14 Gaussian, Laplace and Gaussian mixture pdf's on L o g scale. A l l pdf's have a variance of 1 15 3.1 System model 18 3.2 Discrete-time representation of the system model 19 3.3 Variances of estimation error for linear interpolation at different data symbol positions 3.4 Variances of estimation error for Gaussian interpolation at different data symbol positions 3.5 21 22 Variances of estimation error for 10-tap Sine filter at different data symbol positions 23 3.6 Variances of estimation error for 10-tap Wiener filter at different data symbol positions 3.7 Comparison of B E R of V D ' s w i t h unequal variance metric and equal variance metric for convolutional code (33,25,37) 3.8 24 27 Upper and lower bounds on the B E R of unquantized V D for convolutional code (133,171) as a function of S N R for different values of ESR 3.9 35 Upper and lower bounds on the B E R of unquantized V D for convolutional code (133,171) as a function of S N R for E S R = -co d B and E S R = -10 d B 36 3.10 Upper and lower bounds on the B E R of unquantized V D for convolutional code (133,171) as a function of E S R for different values of SNR 37 3.11 Upper and lower bounds on the B E R of unquantized V D for convolutional code (33,25,37) as a function of S N R for different values of ESR 38 3.12 Upper and lower bounds on the B E R of unquantized V D for convolutional code (33,25,37) as a function of E S R for different values of SNR 39 3.13 Upper and lower bounds on the B E R of unquantized V D for convolutional code (133,171) for E S R = 1 / S N R 39 3.14 Upper and lower bounds on the B E R of unquantized V D for convolutional code (33,25,37) for E S R = 1 / S N R 40 3.15 B E R of unquantized V D for convolutional code (133,171) as a function of S N R 41 3.16 B E R of unquantized V D for convolutional code (133,171) as a function of E S R '42 3.17 Approximate B E R of unquantized V D for convolutional code (133,171) as a function of S N R for E S R = - 8 d B 43 3.18 B E R of unquantized V D for convolutional code (33,25,37) as a function of S N R 43 3.19 B E R of unquantized V D for convolutional code (33,25,37) as a function of E S R 44 3.20 Region for (upperbound - simulation)/simulation < 0.1 for convolutional code (133,171) 45 3.21 Region for (upperbound - simulation)/simulation < 0.1 for convolutional code (33,25,37) 4.1 46 Histogram of the channel estimation error i n Rayleigh fading and L a p l a cian noise w i t h 10-tap Wiener filter, compared w i t h Gaussian histogram w i t h the same variance 4.2 B E R performance of V D o p t , VD 48 E u ciid and V D s q u a r e d Euclid, i n Rayleigh fading w i t h Laplacian noise and Gaussian channel estimation error, as a function of S N R 4.3 49 B E R performance of V D o p t , VD E u c i i d and V D s q u a r e d Euclid, i n Rayleigh fading w i t h Laplacian noise and Gaussian channel estimation error, as a function of E S R 4.4 B E R performance of 50 VD s q u a r e d Euclid i n Rayleigh fading w i t h Laplacian noise and A W G N 4.5 B E R performance of and A W G N 51 VDEuciid i n Rayleigh fading w i t h Laplacian noise 52 4.6 B E R performance of V D o p t i n Rayleigh fading w i t h L a p l a c i a n noise and A W G N 4.7 53 B E R of V D i n Rayleigh fading w i t h Gaussian mixture noise and Gaussian channel estimation error, for e = 0.05, crj/a^ = 21 and crj/a^ = 181, as a function of S N R 4.8 55 B E R of V D i n Rayleigh fading w i t h Gaussian mixture noise and Gaussian channel estimation error, for e = 0.05, cr]/a^ = 21 and o^ja^ = 181, as a function of E S R 4.9 56 B E R of V D ' s w i t h optimal metrics i n Rayleigh fading w i t h Gaussian mixture noise, Laplacian noise and A W G N , and Gaussian channel estimation error, as a function of S N R 57 4.10 B E R of V D ' s w i t h optimal metrics i n Rayleigh fading w i t h Gaussian mixture noise, Laplacian noise and A W G N , and Gaussian channel estimation error, as a function of S N R 58 List of Abbreviations AWGN A d d i t i v e W h i t e Gaussian Noise BER B i t Error Rate BSC B i n a r y Symmetric Channel ESR Error-to-Signal R a t i o FIR Finite Impulse Response i.i.d. Independent identically distributed ISI Inter-Symbol-Interference ML Maximum-likelihood pdf Probability Density Function PSAM P i l o t Symbol Assisted M o d u l a t i o n PEP Pairwise Error Probability RP R a n d o m Process rv R a n d o m Variable SNR Signal-to-Noise R a t i o TCM Trellis-Coded M o d u l a t i o n VA Viterbi Algorithm VD V i t e r b i Decoder or Decoding List of Symbols ddj The number of codewords of weight d that correspond to information sequences of weight j bd T h e total number of nonzero information bits associates w i t h codewords of weight d c Complex channel gain sequence Cj Complex channel gain i?e[cj] R e a l part of q Im[ci] Imaginary part of q dfree Free distance ti Channel estimation error Re[ei] Real part of Im[ei] Imaginary part of f M a x i m u m Doppler frequency f(t) Unit-energy basis function h Estimated channel gain sequence hi Estimated channel gain Re[hi] R e a l part of hi Im[hi] Imaginary part of hi M Frame length M{ri\ hi, Si) B i t metric d M ( r | h, s) P a t h metric rii Channel noise Re[rii} Real part of rii Im[ni] Imaginary part of n(t) Channel noise Pi Channel estimation at pilot symbol position P;, B i t error rate P T h e probability that a weight d path accumulates a higher metric t h a n d the all-zero path q t Coefficient of interpolation filter Ti Received Signal i?e[rj] Real part of Ti Imfri] Imaginary part of r , ri(t) Received signal T Symbol duration s Channel symbol sequence Si Channel symbol si(t) Transmitted signal x Information bit sequence Xk Information bit y Codeword sequence Hi Encoded bit z Decoded bit sequence Zk Decoded bit o% Variance of i?e[cj] or Irn[ci] a\ Variance of -Re[e;] or Im[ei] Variance of Re[hj\ or Im[hi] Variance of impulsive noise Variance of i?e[nj] or Im[rii] Variance of Re[ri\ or Im[ri] Variance of background Gaussian Variance of Laplace distribution Impulsive index Acknowledgements I am grateful to my supervisor Prof. C y r i l Leung, who has given me constant guidance, invaluable advice, and the fundamental knowledge i n the field, and who has set up an example of being attentive and pursuing academic excellence. I also appreciate the efforts of many professors who have been reviewing my thesis, including Prof. L u t z Lampe and Prof. Hussein Alnuweiri. I would like to express my heartfelt gratitude to my parents, Shiquan L i n and Z h i a n L i u , parents in-law, Haoran W u and Haoying S h i , for their patient and constant support, physically and mentally; and to my wife, L i y u n W u , who has been always on my side, providing love, care and encouragement. Finally, to a l l my friends and relatives who care about me, thank you! T h i s work was partially supported by the N a t u r a l Sciences and Engineering Research C o u n c i l Grant OGP0001731. DINGYI THE UNIVERSITY April 2006 OF BRITISH COLUMBIA (CHRIS) L I N Chapter 1 Introduction D u r i n g the past decade, much of the growth i n the telecommunications industry has been i n the wireless sector. For cellular wireless communications, the t h i r d generation (3G) standard has been developed and systems are currently being deployed around the world. The 3 G systems are capable of supporting circuit-switched and packet d a t a at 144 kbps for high mobility (vehicular) traffic, 384 kbps for pedestrian traffic and 2 M b p s for indoor traffic [1]. T h i s data rate enables many new applications, e.g., personal applications that combine entertainment and information, multimedia message services, mobile access to intranet and extranet, etc. For fixed broadband wireless access, I E E E 802.16 standards have been developed to offer an alternative to cable network access, e.g., optical fibre, cable modem and digital subscriber line ( D S L ) . T h e standards cover the medium access control and physical layers for the frequency range 2-11 G H z and 10-60 G H z . O n the commercial side, W i M A X (world interoperability for microwave access) technical working groups have been established to develop a set of system profiles, protocol implementation conformance statements, etc., to handle some of the shortcomings of the I E E E 802.16 standards [2]. W i M A X systems are able to cover a large geographical area up to a radius of 50 k m and to deliver a large bandwidth up to 72 M b p s to end-users. For wireless L A N , I E E E 802.11a/b/g standards compliant devices are widely used and support data rates up to 54 M b p s . More recently, the I E E E 8 0 2 . l l n standard is being developed and some of its proposals include speeds up to 540 M b p s [3]. In many of these wireless communication systems, convolutional codes and V i t e r b i decoders ( V D ) are used to deliver reliable communications. For example, they have been incorporated i n the I E E E 802.11a [4] and I E E E 802.16 [5] standards, and 3 G P P (3rd Generation Partnership Project) specification [6] for channel coding and decoding. 1.1 Motivation and Goal In order for the V D to efficiently recover the transmitted signal i n a fading environment, the complex fading channel gain has to be estimated. Depending on the design and implementation, there exist various sources of error that can make the estimated channel gain different from the actual channel gain. For systems that use pilot symbols, the estimate of the complex channel gain for each data symbol is produced by interpolating the received pilot symbols. In digital implementations, quantiza- tion introduces errors i n the channel gain estimation. T h e r m a l noise is inevitable i n any hardware implementation. It is thus important to evaluate the effect of channel estimation errors on the performance of V D . In previous related papers on V D i n fading w i t h channel estimation errors, it was assumed that the channel estimation errors had equal variances. It was shown in [7], which was on uncoded pilot-symbol-assisted modulation ( P S A M ) w i t h Wiener filter, that the channel estimation error variances were unequal. It is interesting to compare the performance of V D w i t h optimal metric for unequal variances and that w i t h optimal metric for equal variances in unequal variances environment. A l t h o u g h additive white Gaussian noise ( A W G N ) is commonly assumed i n com- munications system studies, it has been found that i n many situations [8-15], the noise exhibits an impulsive nature. T h i s motivates an investigation of the performance of V D i n impulsive noise. T h e objectives of this thesis are (1) to analyze the impact of channel estimation errors on the B E R performance of V D , (2) to compare the B E R performance of V D w i t h optimal metric for unequal variances and that w i t h optimal metric for equal variances i n unequal variances environment, (3) to develop an expression that links the channel estimation error and the B E R performance of V D , (4) to derive optimal metrics for V D in impulsive noise w i t h channel estimation errors. 1.2 Thesis Outline T h e thesis is organized as follows. In Chapter 2, the V D , fading channel, pilot symbol assisted modulation ( P S A M ) and impulsive noise models are introduced, and existing results are reviewed. In Chapter 3, the system model used i n the thesis is introduced, and the variance of the estimation error i n P S A M is examined. Closed-form expressions for upper bounding and lower bounding the B E R of V D i n Rayleigh fading w i t h B P S K and A W G N are then obtained and compared w i t h simulation results. A B E R comparison of the V D w i t h optimal metric for unequal variances and that w i t h optimal metric for equal variances is provided using simulation results. In Chapter 4, optimal metrics are derived for V D in Rayleigh fading w i t h Gaussian channel estimation errors, for Laplace noise and Gaussian mixture noise. B E R results are obtained using simulation. In Chapter 5, the main findings of the thesis are summarized and directions for further research are presented. Chapter 2 Review of Related Works 2.1 Convolutional Codes and the Viterbi Decoder Convolutional codes were invented in 1955 [16]. They differ from block codes i n that they transform the whole information sequence, regardless of its length, into one single codeword rather than segmenting the sequence into blocks of fixed length. There are three main types of decoding algorithms for convolutional codes, namely sequential decoding, feedback decoding and V i t e r b i decoding. The V i t e r b i algorithm ( V A ) is the only one which provides maximum-likelihood ( M L ) decoding. T h e V A makes use of the trellis structure of convolutional codes to decode the received codewords. O n every reception of branch symbols, the V A computes the branch metrics and adds them to the current survivors' metrics; it then compares the metrics of paths entering the same node and selects the path w i t h lowest metric as the survivor. Finally, the path w i t h the lowest metric (highest probability) is the decoded path and the corresponding codeword is the decoded codeword. A detailed discussion of convolutional codes and the V D can be found in [17, chapters 11 and 12]. 2.2 Fading Channel Models W h e n a signal is transmitted in a wireless mobile environment w i t h time-varying m u l t i p a t h propagation, the pass-band received signal can be modelled as a superposition of multiple attenuated and delayed copies of the transmitted signal [18, chapter-14]: = x(t) ^ a ( t ) s [ t - r {t)} (2.1) n n n where s(t) r (t) n is the transmitted signal, a (t) n is the attenuation of the n t h path and is the delay of the n t h path. W h e n these copies of the transmitted signal add constructively, the received signal strength is high; when they add destructively, the received signal strength is low. The fluctuations i n the amplitude of the received signal are termed signal fading. For some channels, such as the tropospheric scatter channel, where the pass-band received signal can be viewed as consisting of a continuum of m u l t i p a t h components, the pass-band received signal can be expressed in an integral form: oo - r)dr a(r;t)s(t / (2.2) •CO where a(r; t) is the time-varying band-pass channel impulse response. The equivalent low-pass received signal can be expressed as CO (2.3) a{r-t)e-^^ {t-r)dT Sl / -co where si(t) is the equivalent low-pass transmitted signal. T h u s the equivalent low-pass channel impulse response is c(r;t) = a(r;t)e- j27TfcT (2.4) . T h e coherence bandwidth and the coherence time are two important parameters that characterize the fading channel. The coherence bandwidth ( A / ) c is defined as the b a n d w i d t h over which the frequency correlation function is above a specific value [19], say 0.5. W h e n the coherence bandwidth ( A / ) is much greater than the signal c bandwidth, the channel is said to be frequency-nonselective: a l l frequency components of the signal are altered by the channel i n the same way. T h e equivalent low-pass received signal can then be expressed i n a product form, instead of a convolution [18]: n(t) where C(0;t) = (2.5) C(0;t) (t) Sl represents the time-varying transfer function C(f;t) can be expressed i n a complex form C(0;t) — r(t)e^^ at / = 0 which where r(t) is the amplitude of C(0;t). T h e coherence time ( A i ) is defined as the time over which the time correlation c function is above a specific value [19], say 0.5. W h e n the coherence time ( A t ) c is much greater t h a n one signal interval, the channel is said to be slowly fading, and can be regarded as constant over a signal interval. T h e n the equivalent r{t)e^^ low-pass received signal can be further simplified as n(t) = re^siit), (2.6) 0<t<T. Three distributions are commonly used to model the envelope of the channel impulse response c(r;t), namely Rayleigh, R i c i a n and Nakagami-m distributions. If C(0;£) is modelled as a complex-valued Gaussian random process ( R P ) w i t h zeromean, its envelope R — |C(0;£)| at any instant t has a Rayleigh distribution: p (r) = R - e- ' °\ 2 r2 2 r>0 (2.7) where cr is the variance of the real or imaginary part of C(0;£). 2 If C(0;t) is a nonzero-mean complex-valued Gaussian R P w i t h an expected a m - plitude of 5, and the variance of the real or imaginary part of C ( 0 ; t) is cr , the envelope 2 R of C ( 0 ; t) has a R i c i a n distribution: PR(T) = y e^ 2 2+s2)/ ^Io(^), 6 r>0 (2.8) where IQ(X) represents the zeroth order modified Bessel function of the first kind. The envelope R is also commonly modelled by the Nakagami-m distribution: '«< > - r where 0 = E(R ), 2 F m — ^ _ ^, E 2 a R Q " and T(x) ^ ' ^ - r s 0 represents the G a m m a function. parameter m is usually larger than or equal to 1/2 <'> 2 9 The [20]. More detailed discussions of fading channel models can be found in [18, chapter 14] and [21, chapter 1]. 2.3 Pilot Symbol Assisted Modulation Pilot symbol assisted modulation ( P S A M ) is one of the techniques that can be used to estimate the channel gain. Compared to the pilot tone assisted modulation ( P T A M ) , P S A M is less complex in implementation, occupies less bandwidth and has lower peak-to-average power ratio [22]. frame of M symbols next frame 1 1 data symbols pilot symbols Figure 2.1: Pilot symbols in P S A M A t the transmitter, pilot symbols are periodically inserted into the data stream as illustrated i n F i g . 2.1. We assume that the pilot symbol is the first symbol i n a frame. The pilot sequence is usually pseudo random and is known to the receiver. The receiver then extracts the channel information at the pilot positions, and calculates the channel gains at the data positions by interpolation. There are several kinds of interpolation filters [7,23,24]. We assume that one pilot symbol is inserted for every M — 1 data symbols, thus M can be considered as the length of a frame. Let pi denote the estimated channel gain at the pilot position of the zth frame and q^j) denote the zth-tap filter coefficient that is used to calculate the estimated channel gain for the jth data symbol i n a frame. We also assume that the interpolation filter uses the L nearest pilot symbols. T h e channel gain estimate at the jth data symbol of the L ^ - J t h frame is L (2.10) i=i where [ x\ denotes the largest integer < x. Zeroth-order Interpolation W h e n the fading process is changing very slowly and the channel gain is assumed to be constant i n a frame, zeroth-order interpolation [23] can be used. The zeroth-order interpolation filter has only one tap (L = 1) and (2.11) Linear Interpolation Linear interpolation [23,25] is another simple interpolation method. Its filter has only two taps where (2.12) (2.13) 1.2 I 1 1 1 I 1 1 _ 1 1 q2 q3 0.8 V. - 0.6 o N 0.4 \ S 0.2 s s \ \ s 0 -0.2 -0.4 1 1 1 1 1 1 i 1 1 10 20 30 40 50 60 70 80 90 100 Samples Figure 2.2: Filter coefficients of Gaussian interpolation for M = 100 Gaussian Interpolation T h e Gaussian interpolation filter [23] has three taps, where 3 M (2.14) (2.15) M = i2 - ( T 7 + l) "M 3 M (2.16) T h e values of qi, q and q are plotted i n F i g . 2.2 for M = 100. T h e relative positions 2 3 of the pilot symbols pi, p and P3 are plotted i n F i g . 2.3. 2 interpolated frame f Figure 2.3: Relative positions of the pilot symbols and the interpolated frame i n Gaussian interpolation Ideal Low-pass Filter A n ideal low-pass filter can be implemented i n time domain w i t h a truncated Sine function [24] where qiU) = Sine{i-l^±±\-jj) l<i<L (2.17) and Sinc(x) = sin(iTx) TtX (2.18) or i n frequency domain w i t h fast Fourier transform ( F F T ) [26]. T h e filter coefficients of a 4-tap ideal low-pass filter are plotted in F i g . 2.4 for M = 100. T h e relative positions of the pilot symbols pi, p , PJ, and p^ are plotted i n F i g . 2.5. 2 Wiener Filter A l t h o u g h the ideal low-pass filter is commonly used in interpolating the deterministic signals, it can not minimize the mean squared estimation error i n estimating the stochastic process. In order to minimize the mean squared error, a Wiener filter can be used [7]. T h e Wiener filter provides a solution to the following problem [27, chapter 5]. Let u> = [u>(0) ui(l) • • -u(L — 1)] denote the complex coefficients of an finite impulse response ( F I R ) filter, u = [u(0) u(l) • • • u(L — 1)] denote the input to the filter where u(n) are samples of a wide sense stationary process, v(L — 1) denote the filter output Figure 2.4: Filter coefficients of 4-tap ideal low-pass filter for M = 100 at time L — 1, L-l u(L-i) = (2.19) £y(i)u(i), i=0 d(L — 1) denote the desired signal at time L — 1, and e(L-l) (2.20) = d(L - 1) - v{L - 1) denote the estimation error. T h e coefficients of the F I R filter are chosen to minimize the mean squared estimation error E{ee*}. It was shown i n [27, chapter 5] that the filter coefficients are pR" (2.21) 1 where p denotes the cross-correlation vector of d(L — l) and vector u, and R 1 denotes the inverse of the covariance matrix of u. T h e m i n i m u m mean squared estimation interpolated frame t P 4 Figure 2.5: Relative positions of the pilot symbols and the interpolated frame i n 4-tap ideal low-pass filter error is E{ee*} = E{dd*} - (2.22) E{vv*}. W h e n the mean squared estimation error is minimized, the estimation error e is uncorrelated w i t h every filter input u(z) and the filter output v(L — 1). In the case of interpolation, the input to the filter is a vector of channel estimates at the L nearest pilot positions. T h e n for every data symbol position, a set of optimized filter coefficients is obtained through (2.21), and the filter output is the i n terpolated value at that data symbol position, which has the m i n i m u m mean squared estimation error. Limit on Frame Length Let T denote the symbol duration and M denote the length of a frame. Assuming that one pilot symbol is transmitted per frame, then the pilot symbol rate (or channel sampling rate) is Let fa denote the m a x i m u m Doppler frequency, then the power spectrum of the fading channel has a bandwidth of 2f d [21, chapter 1]. According to the sampling theorem, the channel sampling rate must be at least 2/^, so that the length of a frame must not exceed ^j^f [7]. 2.4 Impulsive Noise Impulsive noise has been investigated by many researchers for characterizing m a n made R F noise, low-frequency atmospheric noise a n d underwater acoustic noise [8, chapter 3]. A s opposed to Gaussian noise, the probability density function (pdf) of impulsive noise tends to have heavier tails, which means that large deviation from the mean is more likely. Some commonly used impulsive noise models, e.g., generalized Gaussian noise, generalized Cauchy noise, mixture noise and M i d d l e t o n class A noise, are described i n [8]. In this thesis, we focus on Laplacian noise and Gaussian mixture noise. Laplacian noise has been suggested to be a fairly accurate noise model at extremely low frequency [9], and discussed for signal detection i n [10] and V i t e r b i decoding i n [14]. Laplacian noise is a special case of generalized Gaussian noise, a n d the pdf of a zero mean Laplace random variable (rv) w i t h variance 2f5 is 2 1 Inl (2-23) PN(U) = ^ P ( - j ) - M i x t u r e noise is another k i n d of widely used impulsive noise model. Its p d f is p (n) N = {l-e)r {n) (2.24) + ei(n), 1 where the impulsive index e is a small positive constant, r)(n) is a Gaussian pdf representing the background Gaussian noise, and i(n) is some other p d f w i t h a heavier tail that represents the impulsive noise. W h e n i(n) is also a Gaussian function, we have Gaussian mixture noise w i t h pdf 1 n 1 ) - _ _ x p ( - — ) + e ^— 2 p {n) N = (1 - € e n exp(-—^) 2 (2.25) where a is the variance of the background Gaussian noise and a] is the variance of 2 the impulsive noise. T h e ratio CF /CF 2 2 is usually i n the range (20,10000) [8]. T h e total variance of a Gaussian mixture r v is (1 — e)a + eaj. It is also noted that a smaller 2 e implies a heavier t a i l i n pdf. Gaussian mixture noise has been used as the noise model for multiuser detection i n [28] and for signal detection i n [29]. T h e pdf's of the Gaussian, Laplace and mixture Gaussian rv's w i t h zero mean and variance 1 are plotted i n F i g . 2.6 and F i g . 2.7. To better compare the tails of three distributions, F i g . 2.7 is plotted on log scale. Figure 2.6: Gaussian, Laplace and Gaussian mixture pdf's. A l l pdf's have a variance of 1. 2.5 Review of Related Results T h e B E R performance of V D on the binary symmetric channel ( B S C ) , A W G N channel and general memoryless channels is studied i n [30], which provides upper bounds on the B E R using the generating function approach. A B E R lower bound for the V D is derived i n [17, chapter 12] from the genie-aided approach which enables the decoder Figure 2.7: Gaussian, Laplace and Gaussian mixture pdf's on L o g scale. A l l pdf's have a variance of 1. to choose between the correct codeword and the codeword that is at distance df ree away from the correct codeword. M a n y researchers have studied the V D i n various practical systems. In [31] and [32] the V D was examined for trellis-coded modulation ( T C M ) i n uncorrelated Rayleigh fading and A W G N . The optimal metric was derived for imperfect channel estimation, which i n the case of P S K can be shown to be identical to that for perfect channel estimation. The pairwise error probability ( P E P ) was derived by the method of Laplace transform, and provided i n terms of residues. T h i s method was extended to uncorrelated R i c i a n fading i n [33]. A similar method was used i n [34] to study the V D for finite-depth interleaved convolutional codes i n Rayleigh fading. The P E P for B P S K was given i n terms of residues w i t h interpolation filter coefficients and interleaving depth as parameters. In [35] the V D was studied for convolutional codes i n spatially and temporally correlated Rayleigh fading D S - C D M A systems w i t h channel estimation error. A n upper bound on B E R was derived for perfect channel estimation by means of upper bounding the P E P , and an approximate B E R was provided for imperfect channel estimation. In [36] and [37] the V D was considered for convolutional codes i n uncorrelated R i c i a n block fading and Rayleigh block fading, where the channel gain is constant over a frame. T h e P E P and the union bound on B E R were provided for b o t h perfect and imperfect channel estimation. In [38] the V D was studied for T C M i n correlated Rayleigh fading w i t h imperfect phase reference and an approximate P E P was given. In [7] the uncoded P S A M w i t h Wiener filter was studied. It was shown that the variances of the estimation error were unequal. In all these studies, unequal variances of the channel estimation error have not been considered i n V D , and the final results for P E P were not i n closed form. W h i l e A W G N was assumed i n the above papers, the following ones were using impulsive noise models i n the study of V D . T h e V D for combating inter-symbolinterference (ISI) was studied i n [11] and [12], for Cauchy noise w i t h unknown parameters and for a-stable noise respectively. In [11] per-survivor-processing was used to estimated the noise parameters; while i n [12] a penalty function was incorporated i n the metric. O n the other hand, an approximation to the log likelihood function was used as the metric for the V D i n simplified class A noise i n [13]. O p t i m a l metrics for the V D i n Laplacian noise, Cauchy noise and Logistics noise were derived in [14]. Moreover, the B E R performance of V D i n direct-sequence spread-spectrum multiple-access system i n R i c i a n fading w i t h perfect channel estimation is discussed in [15], including Gaussian mixture noise and inter-user interference. To the best of our knowledge, the V D w i t h imperfect channel estimation i n impulsive noise has not been studied so far. Chapter 3 V D in Rayleigh Fading with AWGN 3.1 System Model T h e system model used i n this thesis is shown i n F i g . 3.1. For clarity, we use uppercase letters to denote rv's and the corresponding lowercase letters to denote their sample values. T h e information bit sequence x = ( x , £i, • • •) is first convolutionally 0 encoded. T h e n the encoded bit sequence y = (yoiZ/i, • • •) is mapped into channel symbol sequence s = (so,si, • • •)• We assume that s* = +1 is transmitted if yi = 0 and Si = —1 is transmitted if ^ = 1. D u r i n g the ith signal interval, Si is multiplied by a unit-energy basis function f(t) si(t) = and the resulting waveform, Si(t), where Si-f(t), 0<t<T (3.1) is transmitted over the channel. We assume that the channel is a frequency-nonselective, slow Rayleigh fading BPSK E n c o d e d B i t to Channel S y m b o l Mapper Convolutional Encoder Modulator Si fit) Channel G a i n Estimator hi Viterbi Demodulator Decoder Figure 3.1: System model channel. T h e received signal is r (t) t = Ci • si(t) + n(t) = Ci-Si-f(t)+n(t), 0<t<T (3.2) where Cj denotes the complex channel gain during the ith. interval and is a sample of a zero-mean complex-valued Gaussian r v C ; w i t h a real or imaginary part variance of OQ\ the real and imaginary parts of C j are assumed to be independent; n(t) denotes the low-pass A W G N and is a sample function of a zero-mean complex-valued Gaussian R P w i t h a real or imaginary part variance of a%. T h e received signal is input to a correlation demodulator, whose output is rT U ri(t)f*(t)dt = f Jo • Si • f(t)f*(t)dt+ [ n(t)r(t)dt T Jo where /*(£) denotes the complex conjugate of fit). (3.3) Letting rT rii - n(t)f*(t)dt (3.4) E n c o d e d Bit to Channel S y m b o l Mapper Convolutional Encoder Discrete-time Fading Channel Channel G a i n Estimator hi Zk Viterbi Decoder Figure 3.2: Discrete-time representation of the system model (3.3) becomes (3.5) — 7*2 ' Si ~\~ Tli where nj is a sample of a zero-mean complex-valued Gaussian rv, JVj, w i t h a real or imaginary part variance of a\. The rv N is independent of d t or Si. T h e real and imaginary parts of Ni are assumed to be independent. A discrete-time model of the system is shown i n F i g . 3.2. A channel gain estimator provides an estimation of Cj for the V D . T h e estimated channel gain hi is assumed to have the form hi where the channel estimation error (3.6) Ci is a sample of a zero-mean complex-valued Gaussian rv Ei, w i t h a real or imaginary part variance of o\ . The real and imaginary parts of Ei are independent. For simplicity, we assume that Ei is independent of C,. A l t h o u g h the estimated channel gain and the channel estimation error are affected by A W G N at the pilot symbol position, they are independent of iV, at the data symbol position. T h e V D then uses the estimated channel gain sequence h = (ho, hi,...) to decode the received sequence r = (rn,7"i> • • •)• T h e decoded bit sequence is denoted by z = (z ,z ,...). 0 l For convenience, we define S N R and E S R as S N R = (JQ/G 2 N and E S R = O\I<J' 2 C [39]. 3.2 Variances of Estimation Errors in a Frame Depending on the type of interpolation used, the variances of the estimation error are usually different at different data symbol positions. We assume that the received symbol at the pilot symbol position has the form i n (3.5), and the channel gain p; at the pilot symbol position is obtained by Pi = = — Si Ci + ^. Si (3.7) For linear interpolation as defined i n (2.12) and (2.13), the interpolated channel gain is M,) - (i--L)(c + =I)4(«> S) + There are two sources of channel estimation error: A W G N and the fading process. W h e n the A W G N is the dominant source of the channel estimation error, the estimation error is eU) « < » 3J and its variance 4, = {( -^) 1 2 + (^) }^' 2 j = l - M - l . (3.10) For Gaussian interpolation, the interpolated channel gain is W ) = ~ C l + (1 - j^)C + ^)C + 2 _ j_\VI + n - AiJL + ^ ) 2 M* M ( } S l + { + M> s 1 } 2 + -?-)— 2 M* + 3 { + (3 ID M s' } 1 3 j the estimation error is <1) - - ^ - M ^ - ^ - ^ - ^ M ^ + M ^ ' ( 3 - 1 2 ) and its variance Real Part Imaginary Part 40 60 80 100 Samples Figure 3.3: Variances of estimation error for linear interpolation at different data symbol positions Simulations are performed to measure the variance of the real and imaginary parts of the estimation error at different data symbol positions. T h e results for 0.3 Samples Figure 3.4: Variances of estimation error for Gaussian interpolation at different data symbol positions linear interpolation are plotted i n F i g . 3.3. The frame length M is 100 symbols, the fT d product values are 0.001, 0.002, 0.003, 0.004 and 0.005. T h e noise variance a\ is 0.158, corresponding to a channel gain variance UQ = 0.5 and S N R = 5 d B . A reference curve corresponding to (3.10) is also plotted. T h e figure shows that the variances of the estimation error are not constant. W h e n the frame length M is much smaller t h a n jJZr' e , § ' ' ^ 0 f° r fd^ = 0.001, the variances are very close to those predicted by (3.10). A similar plot for Gaussian interpolation is shown in F i g . 3.4. The reference curve is obtained using (3.13). A plot for a 10-tap Sine filter is shown in F i g . 3.5. It can be seen that for M < 2jjri the curves are much flatter than those for linear and Gaussian interpolation, and the variances of estimation error are very close to the variance of the A W G N . Real Part Imaginary Part 0.24 0.22 fdT=0.005 O 0.18 .§ 0.16 40 Samples 60 100 Figure 3.5: Variances of estimation error for 10-tap Sine filter at different data symbol positions T h e curves change negligibly as fdT changes from 0.001 to 0.004. Variances for a 10-tap Wiener filter are plotted i n F i g . 3.6. We notice that when M < Tjjrjy, the variances are essentially equal for all d a t a symbol positions. T h e Wiener filter provides a substantial performance improvement over the Sine filter i n terms of the estimation error variance, especially when M <C jf^f- For example, when fdT = 0.01, the estimation error variance of the 10-tap Wiener filter is about 6 d B lower t h a n that of the 10-tap Sine filter. W h e n fdT = 0.04, the difference is smaller. T h e estimation error variance of the 10-tap Wiener filter is about 1.7 d B lower than that of the 10-tap Sine filter. 20 40 60 80 100 Samples Figure 3.6: Variances of estimation error for 10-tap Wiener filter at different data symbol positions 3.3 Performance of V D optimized for unequal estimation error variances We assume that the channel gains for consecutive received symbols are uncorrelated and that the V D performs unquantized decoding. T h e uncorrelated condition can be achieved by using channel-symbol interleaving [17, section 10.4.4]. Let i?e[c*] and Im[ci] denote the real and imaginary parts of the channel gain a; Re[ci\ and ira[cj] are samples of independent identically distributed (i.i.d.) zero-mean Gaussian rv's w i t h variance OQ; Z2e[Ci] and Im[Ci] are independent. Let i?e[nj] and Im[rii] denote the real and imaginary parts of the A W G N n*; i?e[n;] and Im[rii] are samples of i.i.d. zero-mean Gaussian rv's w i t h variance a%; Re[Ni] and Im[Ni] are independent. Let Re[ri] and Im[ri) denote the real and imaginary parts of r j . From (3.5) we have Thus given Re[ri] = Re[ci\ • Si + Re[ni] (3.14) Im[ri] = Im[ci] • Si + Irn[rii\. (3.15) Re[ri] and Im\ri] are samples of i.i.d. zero-mean Gaussian rv's w i t h variance a\ where 4 = 4 + 4; (3.16) Re[Ri] and Im[Ri] are independent. T h e real and imaginary parts of the channel estimation error, i.e., Re[ei] and Im[ei] are samples of i.i.d. zero-mean Gaussian rv's w i t h variance a%.. F r o m (3.6) we have Re[hi] = Re[ci] - Re[ei] (3.17) Im[hi] = Im[ci] — Im[ei}. (3.18) Thus Re[hi] and Im[hi] are samples of i.i.d. zero-mean Gaussian rv's w i t h variance ajf. where 4, = 4 + 4; (3-19) Re[H~i] and Im[Hi] are independent. Given Re[hi], Im[hi] and samples of i.i.d. Gaussian rv's w i t h means, m [ ]\ i }=Re[hi} Re Ri variance, 4e[i? ]|*e|// ]=Ke[/ ] 1 i ll a n d Re Hi 4m[fl ]|/m[/f ]=/ [h ]» i i m i a n d Sj, i?e[rj] and im[rj] are and m [ ]\ [H \=:im[hi}, lm Ri Im i conditional pdf's as derived in A p p e n d i x A . T h e joint conditional pdf is PReiR }im{R }(R4 i\^ l i} i = i r I Re[Hi] = Re[hi], Im[Hi] Im r Pite[/y (fle[ri]|J?e[#i] = i?e[/i*], •Pjm[fli](^[rj]|/m[ffi] = = Im[h^,Si = Si) $ = s^) im[/i*], Si = s ). t (3.20) We assume that a codeword of length L is transmitted. L e t h denote the es- timated channel gain sequence, s denote the transmitted sequence corresponding to the codeword and r denote the received sequence. T h e V D selects a sequence s that maximizes the value of p (r| H = h, S = s). T h e negative log likelihood function is R l o g p ( r | H = h , S = s) R L-l = l°gPRciR ]i iR ]( [ i\i i exp m i Re r / m N I Re[Hi] = Re[hi], Im[Hi] = Im[hi], S = Si) t - (Re[r ] - SiRelhi}^ t / (2(0% - ^) i=0 / m [ r j - sjmm^j exp L-l / (2(0% - ^ ^Re[ri] - SiRe[hi]^f- ) + ( Imfri] - \ Silm[hi]^f- = £ - log i=0 ) (3.21) 1 Thus the optimal bit metric for unequal variances is 2 M(ri,hi,Si) = Re[ri] - SiRelhil^j + ^Irn[ri} - -log Silm[hi]^f- m (3.22) — It can be simplified as M(r hi,Si) u = (-Re[ri]Re[hi\si - Im[ri]Im[hi]si). (3.23) For equal variances, the optimal bit metric for the V D i n Rayleigh fading w i t h channel estimation error and A W G N , for P S K and Q A M is derived i n [31] as M{ri,hi,Si) = + [lm[ri] \Re\n} - RelhiYs^j - Im[hi]§i^J .(3.24) B y further removing the common terms, it is simplified as M(ri,hi,Si) = -Re[ri]Re[hi]si - Im[ri]Im[hi]si. (3.25) A n d the p a t h metric is L-l M(r,h,s) ^2M(ri,hi,Si). = (3.26) i=0 T h e equal variances metric is independent of the actual estimation error variance value, and is the same as that for perfect channel estimation. W e also refer to (3.24) as squared Euclidean distance metric. + 10 Unequal variances metric Equal Variances metric 10 10 LU CD 10" 10 10 -6 SNR (dB) Figure 3.7: Comparison of B E R of V D ' s w i t h unequal variance metric and equal variance metric for convolutional code (33,25,37) Simulation was performed to compare the B E R of the V D w i t h optimal metric for unequal variances ( V D variances ( V D e q u a u n e q u a i ) and that of the V D w i t h optimal metric for equal i ) . Here we use a convolutional code whose generator sequence i n octal form is (33,25,37) [17]. T h i s is a rate | code w i t h a free distance df ree of 12 [40] and is used in 3 G P P systems [6] .The implementation of the simulation is based on the MATLAB® example [41, function vitdec] which is capable of convolutional encoding and V i t e r b i decoding i n an A W G N channel. T h e ability to decode i n frequencynonselective slow fading channel is added to the V D . T h e simulation program is also built as a standalone binary file to save MATLAB® license usage and to accelerate program execution. We assume that o\. is changing according to (3.10) which corresponds to the 2-tap linear interpolation, and adjacent samples of are independent. be achieved by interleaving. F i g . 3.7 shows the simulation results for V D VD e q u a T h i s can u n e q u a i and ] . It can be seen that the two V D ' s have more or less the same B E R perfor- mance. T h i s finding agrees w i t h that in [7]. 3.4 Upper and Lower Bounds Derivation of Expressions The generating function (weight enumerator) of a convolutional code [17, chapter 11] has the form oo T(X,Y) = oo Y,Y, <i ad d=l XdYi (- ) 3 27 i=l where a^j is the number of codewords of weight d that correspond to information sequences of weight j. B y taking the partial derivative of the generating function w.r.t. Y and setting Y to 1, we have dT(X, Y) = dY X > * y=i (3-28) d d=l where oo &d = (3-29) ^J-a-dj is the t o t a l number of nonzero information bits associated w i t h codewords of weight d. W i t h o u t loss of generality, we assume that the all-zero codeword is transmitted. A n upper bound on the B E R of the V D for a (n, k) convolutional code is obtained in [30] as -. Pb < oo T Y\ b P 4 i d k where dj ree (3.30) d is the free distance of the convolutional code, {b } are the coefficients d from (3.29), P is the P E P that a weight d path accumulates a higher metric than d the all-zero path. A lower bound on the B E R of the V D is obtained i n [17, chapter 12], p b > (3.31) \Pd . Sree We assume that the variances of the channel estimation error are equal over a frame, a n d use the metric i n (3.24), so that d-1 P d = d-1 P{J2 ( i> i'Si M R H = - 1 i=0 ) > J 2 M ( R " H i ^ i = 1 ^ i=0 d-1 = P{J2(-MPi}Re[Hi) ~ ImiRijImlHi}) i=0 d-1 > ^(ReiRilReiHi] + Jm^/m^])} i=0 d-1 = PiY^iReiRilRelHi] i=0 + Im[Ri]Im[Hi\) < 0}. (3.32) A l t h o u g h a numerical evaluation of P is given i n [31] as a sum of residues of d the Laplace transform of a cumulative pdf, the effect of the channel estimation error is not clear. Here we derive a closed form expression for Pd, by first deriving the conditional probability P j ^ 0 R e [ ^ ] # e [ # i ] + /m[fli]/m[tfi]) then averaging over {Re[Hi}} < 0\{Re[Hi] = Reiki}}, {Im[Hi\ = J m ^ ] } J , and {Im[Hi\}, as follows. Let d-l u = d ^{Re[hif (3.33) + Im[hif). i=0 Since {i?e[/ij]} and {Im[hi}} are samples of i.i.d. Gaussian rvs w i t h zero-mean and variance a\ as derived i n (3.19), u is a sample of a chi-square-distributed rv w i t h 2d d degrees of freedom. T h e pdf of the corresponding rv, Ud, is p "> d ) ^2^r(d) '" = u l e "" / 2 ^' U d ~ ° - ( 3 - 3 4 ) Given Re[hj\, Im[hi] and S j , Re[ri\ and i r a [ r j ] are samples of Gaussian rvs w i t h means, m^R^iuiH^R^hi] and o'| [/e ]|/ [f/ ] / [/ ]) m i m i = m lj and mi [ ]\ m a n d and variance, , Im[Hi]=Im[hi] v Re[Ri]lRe[Hi]=Re[hi] derived i n A p p e n d i x A . A s s u m i n g that all-zero codeword s is transmitted, Si = 1, then m a Ri Re[Ri] \ Re[Hi]==Re[hi m [H ]|/ [// ]=/ [/ ], crL[K ]|Ke[// ]=fle[/i ] ], /m i m i m li i i i /m[fl ]|/m[H ]=/m[h ] become <7 i i i 2 ^He[R ]|He[// ]=He[/i ] = i ? e [ / l i ] - j i i i ^/m[K ]|/m[H ]=/m[/i ] = i i i /mf/lj]— (3.35) (3.36) 4 He[K ]|fie[/f ]=fie[^] Cr i i = /m[R ]|/m[// ]=/m[ft ] = <T i i i a f i ~~ ~2~ ~ ~2~- (3.37) (3.38) Let Re[gi] = Im[gi] = (3.39) Re[ri]Re[hi] (3.40) Im[ri]Im[hi\. Thus, given Re[hi], Re[gi] is a sample of a Gaussian r v w i t h mean = mRe[Gi]\Re[Hi]=Re[hi] R&[hi) • "T.«e[H]|fle[H]=fle[/i] j i i (3.41) •H and variance 0'iie[Gi]|/te[ifi]=fle[/ii] — Re[hi] • Ofl [/e ]|fl [// ]=.Re[fc e i e i ; (3.42) i i > similarly, given Im[hi], Im[gi] is a sample of a Gaussian rv w i t h mean ^/m[Gi]|/m[Hi]=fle[/ii] = Im[hi] • m/ [ ]|/ [// ] = Im[hi] Tn fii m i =/m [ ] hi 2C (3.43) U and variance ~ <Tlm[Gi]\Im[Hi]=Iin[hi] = I™\hif • 0'] [R \\I [H ]=Jm[h m i i i (3.44) Im[hi] [o* 2 m R 'H Let d-1 w = d '^(Re[r ]Re[h \ i + i Im[ri]Im[hi}). (3.45) t=0 Thus given {i?e[/ii]} and {Im[hi]}, % = 0, ...,d - 1, w is a sample of a Gaussian rv d w i t h mean 2 m wd ?f " d-1 J2(Re[hi] 2 + Im[hi i=0 (3.46) and variance d-l = °ia ri-5)E(*]'+ w') /m 'H j=0 4 ( a 2| - - °C^ K (3.47) T h e conditional probability is then d-l P{J^(/?e[i2 ]i2e[//' ] + Im[Ri]Im[Hi]) i < 0 i i=0 | {Re[Hi] = Reiki}}, {Im[Hi] = = P{W 1 2 Imfa]}} < 0 | {Re[Hi\ = Reiki}}, {Im[Hi\ = J m [ ^ ] } } d / 1 m —= V\/2 cr eric wd wd = 2 4 2 J ( 3 4 8 ) where erfc(-) denotes the complementary error function. We then average (3.48) over U i n (3.34). B y using [42, (8.250) and (8.253)], the d complementary error function can be expressed as an infinite sum: P, - r - erfc [ / P d ~ Jo 2 7o \2 e r f 2 C U d 6 ^2-r(d)7 i 0 0 V*F£f fe=i V 1 i V 2 ( a ^ - a ^ ) j ^ F ^ f af2^(d) 2 ^ 4 ^ /•oo l _ r <#2«T(d) 7 0 - u - e- ^du d l H m (2fc-l)(fc-l)! H d d 0 ( d a * W ( ^ \*{?\O* -O%<J* )] } u 4 12(^-4^)/ ^ fc-1/2 n (2fc-l)(*-l) B y using [42, (3.351)], the integrals i n (3.49) can be simplified as 1 < p - 1 r5f2%5) "( oo 1 1 ) ! ( 2 4 ) d , V^tC \ (°> H 2 1 4 N - °h°l) J fe-1/2 ., (2fc-l)(fc-l)! (3.50) -(k + d - h (2a ) - / . 2 ajf2 T{dy 4 2' d H k+d 1 2 B y using the properties of the G a m m a function i n [42, (8.339), (8.338) and (8.331)] and letting k' = k — 1,( (3.50) can be further simplified as: y/i T(d) V a%a% - a' 2 oo /• 4 N c k—1 (fc + d - ^ ) ! (2fc- l ) ( f c - 1 ) P 2 i r a + d) / 2 ^ 4 r ( d ) V 4 4 - <4 fc')ra + + dd+ + k') fc') fr -~ r(l( i ++ k')T^ rr((§j )) 'r(|)r(i+ 1 2 r ( f + A-) r ( ± + d) ^ T(d) V 4 4 2'2 ' 4 -'2'o%a%-o* c r a + d) 2 A r ( d ) V ( ^ + ^ ) ( ^+ 4 ) - ^ 2'2 ' 2 ' ( 4 + 4 ) ( 4+4 ) - 4 r ( i + d) / • r(d) 2 1 1 1 4 4 - 4 J 1 1 4 -- 4 T f n 1 V ( 4 / 4 + 4 / 4 x 4 / 4 + 4 / 4 ) - 3 \ 2' 2 + d ; 4 / 4 -1 2 ( 4 / 4 + 4 / 4 ) ( 4 / 4 + 4 / 4 ) ; 4 / 4 1 _ r ( ^ + d) 2 0F-T(d) f1 1 ' F {2' 2 y (1 + ESR)(1 + l / S N R ) - 1 3 + d ; —1 1 2 (1 + ESR)(1 + l / S N R ) - 1 J ' ! ( 3 " 5 1 ) where F{-} denotes the hypergeometric function [43, (15.1.1)]. Substituting Pd into (3.30) and (3.31), we obtain an upper bound and a lower b o u n d on the B E R of the VD. It might be noted that Pd i n (3.32) is the probability that a sum of products of correlated zero mean Gaussian rv's is less t h a n zero. One solution is given in [18, A p p e n d i x B] for the probability that a sum of products of correlated nonzero mean Gaussian rv's is less t h a n zero. However, if applied to zero mean Gaussian rv's, the expression has a zero denominator. A small modification yields the following result. d-l / \ / \ k / \ 2d-l where vi y/(crh + a )((rh + <7 E)-crh 2 N 2 ^/(1 + 1 / S N R ) ( 1 + E S R ) + 1 (3.53) 7(1 + 1 / S N R ) ( 1 + E S R ) - 1 It might also be noted that the metric i n (3.26) is a weighted sum of the received signals, which bears similarity to the decision variable i n maximum-ratio-combining ( M R C ) . A n expression for the P E P for M R C w i t h B P S K is given i n [44] as where 7 (o* + o%){a* + ol) - of = c ( 3 ' 5 5 ) A l l three expressions for P , i.e., (3.51), (3.52) and (3.54), show that the channel d estimation error and A W G N have exactly the same effect on the B E R of the V D . It is thus equally important to improve channel gain estimation accuracy as S N R . SNR (dB) Figure 3.8: Upper and lower bounds on the B E R of unquantized V D for convolutional code (133,171) as a function of S N R for different values of E S R Numerical Results T h e upper and lower bounds are plotted for two convolutional codes (133,171) and (33,25,37). T h e first is a rate \ code w i t h a free distance dj ree of 10 [40] and is used in I E E E 802.11a [4] and 802.16 [5] standards; the second was introduced previously in section 3.3. T h e coefficients {b } d i n (3.28) are known [45]. For our purposes, they are obtained using the MATLAB® function distspec, up to d = 25. For d > 25, the bdPd term becomes very small and can therefore be neglected from the upper bound and lower bound calculations. Since the hypergeometric function is not available i n M A T L A B ® , the value of Pd is calculated using Maple®. Computer simulations were also used to validate the upper and lower bounds. F i g . 3.8 shows the upper and lower bounds on the B E R of the unquantized V D for the convolutional code (133,171) as a function of S N R , for different values of E S R . V 1 i ' I l •^-^A ESR = -10 dB ^ _ 7 Q ESR = -°° dB \ upper b o u n d _ x. \. X X — — lower b o u n d 0 i 5 10 15 20 ^ \ 25 SNR (dB) Figure 3.9: Upper and lower bounds on the B E R of unquantized V D for convolutional code (133,171) as a function of S N R for E S R = -oo d B and E S R = -10 d B T h e B E R performance of the V D degrades rapidly w i t h increasing E S R , especially when E S R is large. For example, for a target B E R of 1 0 - 2 and using the upper bounds, there is about 1.7 d B S N R degradation for E S R = -6 d B relative to E S R = -10 d B , while there is about 8.5 d B S N R degradation for E S R = -2 d B relative to E S R = -6 d B . We also notice the B E R floors of the V D when channel estimation errors exist. T h e error floors increase w i t h increasing E S R . Moreover, the gap between the upper and lower bounds increases slightly w i t h increasing E S R . F i g . 3.9 shows the upper and lower bounds on the B E R of the V D with perfect channel estimation ( E S R = -oo d B ) and E S R = -10 d B for the convolutional code (133,171) as a function of S N R . It can be seen that, when there is no channel estimation error, the B E R of the V D does not have an error floor w i t h increasing SNR. F i g . 3.10 shows the upper bounds and the lower bounds on the B E R of the V D for -25 -20 -15 ESR (dB) -10 -5 0 Figure 3.10: Upper and lower bounds on the B E R of unquantized V D for convolutional code (133,171) as a function of E S R for different values of S N R the convolutional code (133,171) as a function of E S R , for different values of S N R . It can be seen that the B E R of the V D degrades rapidly w i t h increasing E S R , especially when S N R is small. C o m p a r i n g with F i g . 3.8, we notice that these two graphs are exactly the same except that one is the mirror image of the other, as implied by the expression of P4. F i g . 3.11 and F i g . 3.12 show the upper and lower bounds on the B E R of the unquantized V D for the convolutional code (33,25,37). T h e bounds are similar to those i n F i g . 3.8 and F i g . 3.10, except that the bounds for the convolutional code (33,25,37) are lower than those for (133,171). C o m p a r i n g their upper bounds i n F i g . 3.10 and F i g . 3.12, for the same E S R of -25 d B and same B E R of 10~ , the 6 unquantized V D requires a S N R of 6 d B for the convolutional code (133,171) while only 4 d B is needed for the convolutional code (33,25,37). that the convolutional code (33,25,37) has a larger df ree T h e reason for this is t h a n the convolutional code SNR (dB) Figure 3.11: Upper and lower bounds on the B E R of unquantized V D for convolutional code (33,25,37) as a function of S N R for different values of E S R (133,171). F i g . 3.13 and F i g . 3.14 show the upper and lower bounds for E S R = 1 / S N R , which corresponds to P S A M w i t h Sine filter, for the convolutional codes (133,171) and (33,25,37) respectively. It can be observed that i n this case, there is no B E R floor. Figure 3.13: Upper and lower bounds on the B E R of unquantized V D for convolutional code (133,171) for E S R = l / S N R SNR (dB) Figure 3.14: Upper and lower bounds on the B E R of unquantized V D for convolutional code (33,25,37) for E S R = 1 / S N R tr m 10" — upper bound — lower bound + simulation 10 10 SNR (dB) 15 20 25 Figure 3.15: B E R of unquantized V D for convolutional code (133,171) as a function of S N R Computer simulations are used to validate the upper and lower bounds. F i g . 3.15 and F i g . 3.16 show the computer simulation results for the convolutional code (133,171) compared w i t h the theoretical bounds. T h e bounds agree w i t h the simulation results very well. A s S N R increases a n d E S R decreases, i.e., the variances a 2 N and a\ de- crease, the upper bounds are very close to the simulation B E R results. A n approximate B E R of the V D is provided i n [17, chapter 12] as Ph « bd. (3.56) Pa, • B y setting S N R to oo i n (3.51), the B E R floor is thus P b r = h d bd u free free 2 P d r Sree |SNR=oo b T{\+df ) dfre ^•^(d / r e e 1 1 • F )^/ESR ' 1 2 ' 2 ' ^ 3 ree e e ; -1 1 2 ESR] (3.57) ; T h e approximate B E R i n (3.56) for the convolutional code (133,171) and E S R = -8 d B is plotted i n F i g . 3.17, along w i t h the upper bound, the lower bound and the -25 -20 -15 -10 -5 0 ESR (dB) Figure 3.16: B E R of unquantized V D for convolutional code (133,171) as a function of E S R simulation results. It can be seen that the B E R estimate given by (3.56) is very good at high S N R ' s . F i g . 3.18 and F i g . 3.19 show the simulation results, along w i t h the upper bounds, lower bounds and approximate B E R ' s for the convolutional code (33,25,37). B o t h graphs show very good correspondence of the upper bounds and approximate B E R w i t h the simulation results. In order to assess the tightness of the upper bounds on the B E R of the unquantized V D for convolutional code (33,25,37), the simulation points where (upperbound - simulation)/simulation < 0.1 are plotted in F i g . 3.20 and F i g . 3.21 as asterisks. It can be observed that, for the convolutional code (133,171), when E S R < -8 d B and S N R > 8 d B , the upper bounds are w i t h i n 10% of the simulation B E R ; for the convolutional code (33,25,37), when E S R < -4 d B and S N R > 4 d B , the upper bounds are w i t h i n 10% of the simulation B E R . SNR (dB) Figure 3.17: Approximate B E R of unquantized V D for convolutional code (133,171) as a function of S N R for E S R = -8 d B 25 SNR (dB) Figure 3.18: B E R of unquantized V D for convolutional code (33,25,37) as a function of S N R -15 ESR (dB) -10 Figure 3.19: B E R of unquantized V D for convolutional code (33,25,37) as a function of E S R CO T3, DC CD LU -15 10 15 SNR 20 30 (dB) Figure 3.20: Region for (upperbound - simulation)/simulation < 0.1 for convolutional code (133,171) CO tr C/) LU 10 15 SNR 20 (dB) Figure 3.21: Region for (upperbound - simulation)/simulation < 0.1 for convolutional code (33,25,37) V D in Rayleigh Fading with Impulsive Noise In this chapter, the V D i n Rayleigh fading w i t h channel estimation errors is studied, for Laplacian noise and Gaussian mixture noise. F i r s t , the optimal metrics are derived, and then their B E R performances are studied using simulation. 4.1 Laplacian Noise In this section, the B E R performance of V D i n Rayleigh fading w i t h Laplacian noise and channel estimation error is studied. The system model is the same as i n F i g . 3.2, except that the real and imaginary parts of the noise rv iV* now have independent Laplacian distributions defined i n (2.23). Since the estimated channel gain is a sum of rv's as i n (2.10), according to Central L i m i t theorem, its pdf approaches a Gaussian pdf as the interpolation filter grows longer. To facilitate our analysis, the channel estimation error E{ is thus assumed to be Gaussian distributed and has the same variance for a l l values of i. A histogram of the channel estimation error i n Rayleigh fading and Laplacian noise w i t h a 10-tap Wiener filter is plotted i n F i g . 4.1, and T 1 1 1 I I — — -2 -1.5 -1 -0.5 0 0.5 I Gaussian histogram actual histogram 1 1.5 2 Figure 4.1: Histogram of the channel estimation error i n Rayleigh fading and L a p l a cian noise w i t h 10-tap Wiener filter, compared w i t h Gaussian histogram w i t h the same variance compared w i t h the histogram of a Gaussian rv w i t h the same variance. It can be seen that the Gaussian distribution provides a good approximation. T h e optimal bit metric for Laplacian noise without fading is given i n [14] as M(vi, hi, Si) = \Re[ri] - s*| + \Im[ri] - s \, t (4.1) whereas the optimal metric for fading w i t h perfect channel estimation is hi, Si) = \Re[ri] - Re[ci]si\ + \Im{ri} - Irn[ci]si\. M{n, (4.2) We refer to (4.2) as Euclidean distance metric. T h e optimal bit metric for Rayleigh fading w i t h channel estimation error is derived i n A p p e n d i x B as M{r u h u Si) = f(Re[n], Re[hi], s ) + f(Im[ ], t ri Im[hi], Si), (4.3) 4 6 SNR (dB) Figure 4.2: B E R performance of V D , V D i and V D d Euclid, i n Rayleigh fading w i t h Laplacian noise and Gaussian channel estimation error, as a function of SNR o p t E u c i d s q u a r e where f(ri,hi,Si) , f -r a% f - log <^ exp{ i 2h s Pal + i 2 i ) 2 0(* E + 4) 2 •erfc( Si(3a ac\/(J E •erfc{ rj0a% + rif3a 2 c 2 E + - hiSi(5a a% 2 c Simulations were performed for three V D ' s (1) V D (4.3), (2) V D E u c + o\o , 2 o p t c (4.4) w i t h the optimal metric i n i i d w i t h Euclidean distance metric i n (4.2) and (3) V D s q u a r e c i Euclid w i t h squared Euclidean distance metric i n (3.24), for the convolutional code (33,25,37). -6 -4 ESR (dB) Figure 4.3: B E R performance of V D , V D c i i d and V D d Euclid, i n Rayleigh fading w i t h Laplacian noise and Gaussian channel estimation error, as a function of ESR o p t T h e results in F i g . 4.2 show that V D Eu o p t s q u a r e has the best performance A t low E S R , the B E R of VDEuciid approaches that of V D VD q S Uar e d Euclid is about 1 d B worse t h a n that of V D o p t o p t as expected. ; the performance of at target B E R of 10~ . T h i s 6 can be explained by examining the optimal metric and the Euclidean distance metric. 4 6 SNR (dB) Figure 4.4: B E R performance of V D noise and A W G N s q u a r e c l E u c l i d i n Rayleigh fading w i t h Laplacian Letting a\ = 0, Pi(Re[ri], Re[hi], §i) — \og{exp(2Re[hi]si) • erfc(oo) + exp(—2Re[hi]si) • erfc(—oo)}, Re[ri] - Re[hi]si < 0 = < — log{exp(2Re[hi]si) • er/c(—oo) + exp(—2Re[hi]si) • erfc(oo)}, Re[ri] - Re[hi]si > 0 2Re[hi]si - log 2, Rein] - Re[hi]§i < 0 (4.5) - log 2, Re[r^ - Re[hi]si > 0 -2Re[hi]si p {Im[ri}, 2 Irn[hi\, s ) t 2Im[hi]si — log 2, Im[ri] — Im[hi]si < 0 —2Im[hi]si — log 2, Im[ri] — Im[hi]si > 0 (4.6) 10° Euclid, dist. metric in Lap. noise Euclid, dist. metric in AWGN rr LU m 10 N -7 4 2 0 6 10 8 SNR (dB) Figure 4.5: B E R performance of V D AWGN E u c i i d i n Rayleigh fading w i t h L a p l a c i a n noise and Substituting (4.5) a n d (4.6) into (4.3), the metric only differs from the Euclidean distance metric by a constant. So that when channel estimation error is small, the optimal metric is approaching the Euclidean distance metric. A t high E S R , where the channel estimation error dominates, the performance of VD o p t and V D s q u a r e c j Euclid are almost the same, while the performance of VDEuciid is slightly worse. F i g . 4.3 shows the B E R performance of the V D ' s as a function of E S R . A t low S N R , VDEuciid almost has the same performance as that of V D VD s q u a r e d Euclid and V D o p t o p t ; at high S N R , have the same performance. It is also interesting to show how V D s q u a r e d Euclid, V D s q u a r e d Euclid and V D form i n A W G N compared w i t h L a p l a c i a n noise. The B E R performance of V D o p t s q u a r e perd Euclid optimal metric for Lap. noise in J 0 2 4 I L 6 8 10 SNR (dB) Figure 4.6: B E R performance of V D AWGN o p t i n Rayleigh fading w i t h Laplacian noise and is plotted i n F i g 4.4. It is shown that at low E S R , i.e., when Laplacian noise dominates, the performance of V D s q u a r e c j Euclid is noticeably degraded by Laplacian noise; while at high E S R , their performances are similar. T h e B E R performances of V D and V D o p t E u c ! i d are plotted i n F i g . 4.5 and F i g . 4.6. It can be seen that there is perfor- mance difference i n A W G N and Laplacian noise, for b o t h VDEuciid and V D o p t . The difference at low E S R is larger than that i n high E S R . However, the difference of VDEuciid and V D o p t at low E S R is much smaller than that of V D s q u a r e d Euclid- 4.2 Gaussian Mixture Noise In this section, the B E R performance of V D i n Rayleigh fading w i t h Gaussian mixture noise and Gaussian channel estimation errors is analyzed. T h e system model is the same as in F i g . 3.2, except that the pdf's of the real and imaginary parts of the noise rv Ni are independent Gaussian mixture distributed as defined i n (2.25). T h e optimal bit metric is derived i n A p p e n d i x C as M{r u h §i) h = - \ogp (Re[ri\ | Re[Hi] = Re[h }, S = §i) - logPim[Ri}{Im[ri\ | Im[Hi] = Im[hi], Si = Re[Ri] t t s) t (4.7) where PRe[Ri}{Re[ i) I Re[Hi] = Re[hi],Si = §i) r / ( ( a £ + a? - - (Rein] - s.ReM^rf exp ^2^(4 + ff a «E+ C a (ite[r<] - s ^ h i } ^ ) exp ^ ) ) 2 2 / 1 ( ( a £ + a] 2 ^ ) (4.8) +e- Pim{R ](I™[r ] i i exp | Im[Hi] = Im[hi],Si - (lm[n] - « ] ^ ) = 5) 4 J / ( % ° h + ^ " ^ ) ) (1-c)- exp - {lm[ri\ - silrnih,]^) 2 J (2(a% + aj - ^ ) +eSimulations were performed for different Gaussian mixture noise parameters. (4.9) The convolutional code (33,25,37) was used. Samples of the Gaussian mixture noise are generated according to the empirical transform method [46, section 7.2.2.2]. T h e B E R of the V D w i t h optimal metric in Rayleigh fading w i t h Gaussian m i x ture noise and Gaussian channel estimation error is plotted i n F i g . 4.7, for e = 0.05, 15 SNR (dB) Figure 4.7: B E R of V D in Rayleigh fading w i t h Gaussian mixture noise and Gaussian channel estimation error, for e = 0.05, a]/a = 21 and aj/a = 181, as a function of SNR 2 _ 2i a n d af/a 2 2 — 181. The pdf's of the Gaussian mixture noise are plotted in F i g . 2.7. It is shown in F i g . 4.7 that at low S N R , the V D w i t h aj/a 2 = 181 has better performance, and the performance difference increases as E S R decreases. T h i s is because at low S N R , the threshold over which a noise sample can cause error is low, say, 1 i n F i g . 2.7, and the probability of a noise sample has magnitude larger t h a n 1 is lower w i t h tf/tf = 181 t h a n w i t h a]/a 2 = 21. A t high S N R and high E S R , the V D ' s i n the two environments have the same performance. T h i s is because the effect of high channel estimation error masks the effect of the Gaussian mixture noise, even though the pdf of the noise rv w i t h crf/a aj/tf 2 — 181 has a heavier tail t h a n that w i t h = 21. For high S N R and low E S R , we expect that the V D w i t h a]/a* w i l l have better performance than that i n a]/a 2 = 181 = 21. Unfortunately, the B E R is too 10" c f /CT =181 2 o f / (^=21 -6 -4 ESR in dB Figure 4.8: B E R of V D i n Rayleigh fading w i t h Gaussian mixture noise and Gaussian channel estimation error, for e = 0.05, crj/cr , = 21 and a]/a — 181, as a function of ESR 2 2 low to be verified by simulation. F i g . 4.8 shows the B E R of the V D as a function of E S R . A s expected, the channel estimation error causes the B E R performance of the V D to degrade. It is also interesting to compare the performance of the V D ' s i n Gaussian mixture noise, Laplacian noise and A W G N . T h e results w i t h optimal metrics are plotted i n F i g . 4.9. Similar observation to those for F i g . 4.7 can be made. For E S R = 0 , the V D in Gaussian mixture noise has better performance at low S N R , because of its lower probability of having large magnitude noise; while at high S N R , a l l three V D ' s have the same performance, because the channel estimation error dominates. W h e n E S R is low, the V D i n Gaussian mixture noise still has better performance at low S N R ; while at high S N R , the V D i n Gaussian mixture noise has worse performance because Figure 4.9: B E R of V D ' s w i t h optimal metrics in Rayleigh fading w i t h Gaussian mixture noise, Laplacian noise and A W G N , and Gaussian channel estimation error, as a function of S N R of the heavy t a i l of the Gaussian mixture pdf. T h e B E R of the V D ' s as a function of E S R are plotted i n F i g . 4.10. A s E S R increases, the B E R decreases, as expected. <r LU -6 -4 ESR (dB) Figure 4.10: B E R of V D ' s w i t h optimal metrics i n Rayleigh fading w i t h Gaussian mixture noise, Laplacian noise and A W G N , and Gaussian channel estimation error, as a function of S N R Conclusion In previous related research, the P E P of V D on fading channels w i t h channel gain estimation errors and A W G N was either obtained approximately, or using Laplace transform. It was generally assumed that the channel estimation errors associated w i t h different data symbols had equal variances. Channel estimation errors have not been considered i n the study of V D i n fading and impulsive noise. In this thesis, the impact of Gaussian distributed channel estimation errors on the B E R of the unquantized V D i n frequency-nonselective slow Rayleigh fading was stud- ied, for A W G N , Laplacian noise and Gaussian mixture noise. T h e main contributions are summarized as follows. • T h e B E R performances of two V D ' s (1) V D u n e q u a i w i t h optimal metric for u n - equal channel estimation error variances and (2) V D e q U a i w i t h optimal metric for equal channel estimation error variances were compared, in an unequal variances environment. It is shown that when the frame length M is less than the V D e q u a i has more or less the same performance as V D u n e q U jj^f, ai- • Closed-form expressions of the P E P were derived for V D w i t h B P S K in Rayleigh fading w i t h A W G N and Gaussian channel estimation errors. Upper and lower bounds on the B E R are then obtained. It is found that channel estimation errors affect the B E R performance of V D i n a way identical to A W G N . Computer simulation results for two commonly used convolutional codes (133,171) and (33,25,37) show close agreement w i t h the upper bounds. T h e optimal metric was derived for V D in Rayleigh fading w i t h Laplacian noise and Gaussian channel estimation errors. T h e B E R performances of the V D ' s w i t h optimal metric ( V D o p t ) , Euclidean distance metric ( V D Euclidean distance metric ( V D s q u a r e Euc i i d ) and squared d Euclid) i n Rayleigh fading and Laplacian noise were compared, using simulation. It is shown that when Laplacian noise is dominant, the B E R of V D o p t and VDEuciid are almost the same; when channel estimation errors dominate, the B E R of V D VD o p t , VD E u c o p t and V D s q u a r e d Eu ciid are similar. i , VD u red Euclid were compared for Laplacian noise versus A W G N i d sq a It is found that the performance difference between the two noise models i n creases as the channel estimation error decreases. T h e performance is typically smallest for V D o p t and largest for V D s q u a r e difference d Euclid- The optimal metric is derived for V D i n Rayleigh fading w i t h Gaussian mixture noise and Gaussian channel estimation errors. T h e B E R performance of the V D w i t h optimal metric is obtained for different noise parameter values. For fixed S N R and E S R values, the B E R decreases as the noise becomes more impulsive. A m o n g related topics for future study are the following: V D i n which quantization is applied. Extension of the thesis work to other modulation, channel fading, and channel estimation error models. Bibliography [1] " 3 G Standardization Process," Nov. 2005. h t t p : / / w w w . 3 g a m e r i c a s . o r g / E n g l i s h / Technology _Center/standardization_process.cfm. [2] "Technical Information," Nov. 2005. http://www.wimaxforum.org/technology. 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B a t e m a n , "Performance of M P S K trellis-coded modulation in rayleigh fading w i t h an imperfect phase reference," IEE Proceedings-I munications, Com- Speech and Vision, vol. 139, pp. 329-335, June 1992. [39] D . G u a n d C. Leung, "Performance analysis of transmit diversity scheme w i t h imperfect channel estimation," Electronics Letters, vol. 39, pp. 402-403, Feb. 2003. [40] K . J . Larsen, "Short convolutional codes w i t h m a x i m a l free distance for rates 1/2, 1/3, a n d 1/4," IEEE Transactions on Information Theory, vol. 19, pp. 371-372, M a y 1973. [41] T h e M a t h W o r k s , Inc., MATLAB Help Version 7.0.1, 2004. [42] I. S. Gradshteyn and I. M . R y z h i k , Table of Integrals, Series, and Products. Academic Press, 5th ed., 1994. [43] M . A b r a m o w i t z a n d I. A . Stegun, Handbook Formulas, Graphs, and Mathematical of Mathematical Functions With Tables. U n i t e d States Department of C o m - merce, National Bureau of Standards, 1964. [44] J . W u , C . X i a o , a n d N . C . Beaulieu, " O p t i m a l diversity combining based on noisy channel estimation," i n 2004 IEEE nications, International Conference on Commu- vol. 1, pp. 214-218, June 2004. [45] J . C o n a n , " T h e weight spectra of some short low-rate convolutional codes," Transactions on Communications, vol. 32, pp. 1050-1053, Sept. 1984. [46] M . C. Jeruchim, P. B a l a b a n , a n d K . S. Shanmugan, Simulation nication Systems: Modeling, Publishers, 2nd ed., 2000. IEEE Methodology, and Techniques. of Commu- Kluwer Academic Derivation of the Conditional pdf's PRe[R^ ^^\ \. ^ Re Re = H Re \ ii Si = Si) h and P i m [ R l P m V i ] \ I m [ H i \ = Im[hi], Si = si) In this appendix, we derive the conditional p d f s p^ [/j ](i?e[rj] | Re[Hi] = Re[hi\, Si = e Si) and PimiRi](I [ i\ m r I I [Hi\ m = Im[hi],Si i = s,), which are used i n obtaining the expression for M(r{\ hi, s,) i n section 3.3 and Pa i n section 3.4. T h e covariance, C, of Re[Ri) and Re[Hi], given Si = Si, is C = EiiReiRj-EiRelRiWiRelH^-EiRelHi}})} = E {{Re[Ci] = E{Re[Ci] Si = E{Re[Ci] Si} = cr si Si 2 2 where E{-} 2 + Re[Ni]){Re[Ci] + Re[Ci]Re[Ni] - Re[Ei})} - Re[Ei]Re[Ci]si + E{Re[Ci}Re[Ni}} - Re[Ei\Re[Ni}} - E{Re[Ei}Re[Ci] } Sl - E{Re[Ei]Re[Ni]} (A.l) c denotes the expectation operation. T h e correlation coefficient, pi, of Re[Ri) a n d Re[Hi], given Si = is Si, A = Pi O'RdHi - ^ L . (A.2) O~RO- HI Thus we have PRe[Ri}(Re[ri] I Re[Hi) = Re[hi],Si = PRe[Ri]Re[Hi](Re[ri], Re[hi] | Si = Si) PRelHi](Re[hi]) exp fle[rj] _ 2 Q » Jfe[rj]fle[/tj] , fle[fc 2(1-P?) y/2«(l - p ) ^ 2 exp VMi exp - - ( i ? e N - ^ g ^ ) v ^ i exp ^ exp - (feh] - - 2 P P\)°\ / ( 2 ( l - p ) 4 ) ? K • Si 2 / (2(1 - fie[A,] f ^ ) 1 (2(o% 2 ( ^ ) 2 K ) |-)) (A.3) Given = s,, i?e[#j] = i?e[/ij], i?e[i?j] is a Gaussian R V w i t h mean ^e[« ]|He[W ]=fle[/i ] = Sj.Re[/ii]-y- (A.4) Re[-R]|fte[/f]=.Re[/ii] = aR (A.5) i i < and variance <J i i 'Hi T h e conditional pdf, mean a n d variance of Im[Rj\ can be similarly obtained as: I [ i\ Plm[Ri]( [ i] Im r exp Im Im[ ] ri = Im[hi],Si = Si) H sjm[hi}^j /(2(4-^) (A.6) r mim.[R )\Im\H }=Im[h ) i i i Im[Ri]\Im[H ]=Im[h a i i = = SiIm[hi]—£- aR „2 a Hi • (A.7) (A.8) Derivation of Optimal Bit Metric for Laplace Noise T h e optimal bit metric is derived for the V D i n Rayleigh fading with Laplace noise and channel estimation error. In the following derivation, Re[] is dropped for convenient, and all upper case and lower case letters represent the real parts of their corresponding R V ' s and samples. T h e pdf of Hi conditioned on d = Q is PHi(hi | Ci = —exp( Ci) = \ 2 % ), (B.l) the pdf of R conditioned on d = Cj and Si = Sj is t PRM I d = a,Si = Si) = jp P(- * ex ^"")' r (-) B 2 the pdf of Ci is 1 P^ici) =-7=^-exp{--\). c 2 (B.3) Given Ci = Cj, Ri and Hi are independent. Thus the joint pdf of Ri and Hj conditioned on Si = Si is PRi,Hi{ri,hi | Si = Si) oo / I Ci = Ci, Si = §i) • p {hi PRiin I Ci = Ci) • p (ci)dci Hi Ci -oo oo ^ ._^ ——> __| P exp( , + J-oo E e X r I 4nPa a £ w P r , - c ^ _ (a - hj) 2a| P [ e x v ( - C i S i ~ r - i 2o\ + T J T - ^ e _ 2 J-oo c ^ )dQ 1 W*E*c V ^/l ( - 4irP<J <Tc Jn/si 2a E ( C i " 2o* ^ - 2a a 2 c 2 2a C )a€i -$-)dc- (3 E i - A ) + c A E c 2a 2 E p - |) + ( - A + ^))dc, °E P 2a E P (B.4) Letting A = E>\ = ~^t~ '% and C\ = -£jr + ^ , the first integral of + (B.4) becomes TilSi exp(-Ac = - BiCi - 2 / Ci)dCi •00 expi-d + %) exp(-(VAa + ^=) )dci. 2 (B.5) L e t t i n g \[A~Ci + ^= = U, dcj = ^ d £ ; , (B.5) becomes 2Ar + B a,; i where erfc(-) ^ 1 eM-C^^erM- -^^!) (B.6) 2 2vC4 4A 2SiVC4 y denotes the complementary error function. Letting B = + % and C = t t V - %, the second integral of (B.4) becomes 2 2 ^E P E P / e x p ( - A c - S C j - C )dCi J ri/Si * , B\ 2Ari + B Sj^ exp(-C + -—)erfc{— —). 2^4 4A 2siVA 2 2 2 n 2 t ; ( 2 (B.7) Thus PRi,Hi(ri,hi | Si = §i) 1 - 2rj0<j% - 2nl3a -h p 2 f 4 / 3 ^ v ^ | T ^ 2 ~- P\ eX " 2P (a V Si^E^cy/^s -/i / crfcC^^ + + c sa a ^ 2 2 c / c - 2hiS (3o 2 c 2 x 002/-2 + c , _22 N sa a . 2 2 E 2 c ) 2f5\o\ + o ) c 2 E +a ) E 2 ex T i f 3 ( j 2 c 2 i _ 22 -\ + erg. / ? + 2ri0a% - 2n(3a 2 •- P( 2 y y 4/5v 2T\/a| + a + 2hCsjPo 2 c 0 / 322 / _2 2 c ~ + g < g g) g (B 8) g Using (3.19), the conditional probability PRi(ri = | /fj = h Si = Si) it PRi,Hi(ri,hi | Si = §i) PHM - 2 r j / ? a | - 2rj/3gg. + 2hjSj(ia J_ 4 / ? ^ 2 2 Ti c a ~ E r i / 9 g g + hi§i Sif3o- vcV E E a 2 7 j ^ c + H \ c^ a a a + c a +2n/?a| - 27-i/frg; - 2hiSi/3a 1 2 7 2/? (ai + cT ) f (~ ^ ' CT + + •§ ct|ct . 2 c A(3 2 c + sa a 2 2 E W\o\ + o ) eXP[ 2 Si0<TE<7cy/ E a + 2 c } c c CT Following the analysis i n section 3.3, the optimal bit metric is hi, §i) M(r , { = - logp (Re[ri] - logp { ](Im[ri] Re[Ri] Im Ri | Re[Hi] = Re[hi], Si = Si) | Im[Hi] = Im[hi\, Si = s ), { (B.10) and by removing a l l common terms, it is simplified as M(r h h u §i) = f(Re[n], Re[hi],Si) + f(Im[n], Im[hi\, s ), { (B.ll) where f(ri,hi,Si) , ,-r a f i •er/c( E + 2h s Poi 2 c — , Si(3a (Tcy'cr 2hCsil5u 2 , 2^ ) E ritr| ex v , „n„1 2 E E + °c 2 c + P\ crfcf i + ^ _u.z.R„2 ~ , + a ^2^2 ^ g) I a (B 12) Derivation of Optimal Bit Metric for Mixture Gaussian Noise T h e o p t i m a l bit metric of the V D i n Rayleigh fading w i t h mixture Gaussian noise and channel estimation error is derived. In the following derivation, Re[ ] is dropped for convenient, and all upper case and lower case letters represent the real parts of their corresponding R V ' s and samples. T h e pdf of Hi conditioned on Ci = Ci is PHiihi | Ci = Ci) = —exp(-^ Cl ), ^ (C.l) the pdf of Ri conditioned on Q = q and Si = ii is | Ci = Ci,Si = Si) PRiin ! - / £ V27ra v e (n-CiSi) 2 exp( — 2cr (n ) + V27ra/ exp( 7 2 c^) - — 2a] 2 2 ), (C.2) the pdf of Ci is 1 PCM) = nr- c P(-7T2-)- EX 2 ( - ) C 3 Given d = Cj, Ri and Hi are independent. Thus the joint pdf of Ri and Hi conditioned on Si = §i is PRi,Hi(ri,hi | Si = Si) oo PRiiXi / -f{ -oo oo oo oo J —oo 1 | 1 - Ci = e 2ixo~ Ci, , fa - P( ) + 2 1 exp(- 2a\ —oo v •ea;p( +e TTO 2a» >^?7 ,2 1 _exp(-^)dc a) • p {ci)dci Ci 2 tf n = Ci§i) ex 1 |Q Hi k 2TT v / S, = §i) • p (hi 27T<7/ 2ol ezp( z-5 ) 2a) )dci ) « r f - ^ > 1 l 2a| 24 27TCT/ (C.4) It can be seen that the first integral is actually a joint p d f of two correlated Gaussian R V ' s , which has been derived i n A p p e n d i x A , only requires replacing 0 N w i t h a . T h i s is also true for the second integral, only requires replacing aff w i t h a). 2 B y modifying (A.3), the conditional pdf is PRi{n I Hi = hi, Si = Si) r exp 2 - ^ ^ ) 2 / ( 2 ( 4 + 4 - ^ ) ) (1-6)- exp +e- S ^^) /( (^ 2 2 + ^ -^)) 2 (C.5) Following the analysis i n section 3.3, the optimal bit metric is M(ri,hi,Si) logpjfc[^](.Re[rj] | Re[Hi) = #e[^], 5, = - - logp (Im[ri] Im[Ri] I Jm[#i] = Im[hi], = Sj) S = t Sj). (C.6)
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Performance analysis of Viterbi decoding in Rayleigh fading with channel estimation errors Lin, Dingyi 2006
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Title | Performance analysis of Viterbi decoding in Rayleigh fading with channel estimation errors |
Creator |
Lin, Dingyi |
Date Issued | 2006 |
Description | Previous studies on the effect of channel estimation errors on the bit error rate (BER) performance of Viterbi decoding (VD) concern various types of fading channels with additive white Gaussian noise (AWGN), modulation and interleaving schemes. Pairwise error probabilities (PEP) have been derived using Laplace transform. Studies of VD on fading channels with impulsive noise and perfect channel estimation are also available in the literature. In this thesis, the BER performance of unquantized VD with BPSK is analyzed for a frequency-nonselective slow Rayleigh fading channel with AWGN and Gaussian distributed channel estimation errors. Closed-form expressions for the PEP are derived. Upper bound and lower bound on the BER are obtained. It is shown that channel estimation errors have the same effect on BER as channel noise. Computer simulation results show that the upper bound is fairly tight. In practice, the channel might be estimated using pilot symbols, together with various interpolation filters. It is shown that the channel estimation error variances are usually unequal for different data symbol positions. The BER performance of VD optimized for unequal estimation error variances is compared with that of VD optimized for equal variances using computer simulation. The BER performance of VD in Rayleigh fading and impulsive noise with channel estimation error is also studied. The optimal metrics are derived for VD in Rayleigh fading with Gaussian channel estimation errors, for Laplacian noise and Gaussian mixture noise. The BER performances of VD’s for various scenarios are compared. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-01-16 |
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. |
DOI | 10.14288/1.0065429 |
URI | http://hdl.handle.net/2429/18293 |
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 |
Graduation Date | 2006-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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