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UBC Theses and Dissertations

Performance analysis of Viterbi decoding in Rayleigh fading with channel estimation errors Lin, Dingyi 2006

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P E R F O R M A N C E ANALYSIS OF V I T E R B I 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 C H A N N E L E S T I M A T I O N E R R O R S by Dingyi L i n B . S c , Zhongshan University, Guangzhou, China , 1998 A T H E S I S S U B M I T T E D I N 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 S T U D I E S (Electrical and Computer Engineering) T H E U N I V E R S I T Y O F B R I T I S H 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 terb i decoding (VD) concern various types of fading channels with 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 with impulsive noise and perfect channel estimation are also available in the literature. In this thesis, the B E R performance of unquantized V D wi th B P S K is analyzed for a frequency-nonselective slow Rayleigh fading channel with A W G N and Gaussian distributed channel estimation errors. Closed-form expressions for the P E P are de-rived. 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 with various interpolation filters. It is shown that the channel estimation error variances are usually unequal for different data symbol positions. The B E R performance of V D optimized for unequal estimation error variances is compared with 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 with channel estimation error is also studied. The optimal metrics are derived for V D in Rayleigh fading with Gaussian channel estimation errors, for Laplacian 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 Motivat ion and Goal 2 1.2 Thesis Outline 3 Chapter 2 Review of Related Works 4 2.1 Convolutional Codes and the Vi terb i Decoder 4 2.2 Fading Channel Models 5 2.3 Pi lot Symbol Assisted Modulat ion 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 Model 17 3.2 Variances of Est imation Errors in 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 Mixture Noise 54 Chapter 5 Conclusion 59 Bibliography 61 Appendix A Derivation of the Conditional pdf's pRelRi](Re[ri]\E£[Hi] = Re[hi], Si = Sj) andplmlRi](Im[ri]\Im[Hi] = Im[hi], Si = Si) 65 Appendix B Derivation of Optimal Bit Metric for Laplace Noise . . 68 Appendix C Derivation of Optimal Bit Metric for Mixture Gaussian Noise 72 List of Figures 2.1 Pi lot symbols in P S A M 7 2.2 Fi lter coefficients of Gaussian interpolation for M = 100 9 2.3 Relative positions of the pilot symbols and the interpolated frame in Gaussian interpolation 10 2.4 Fi lter coefficients of 4-tap ideal low-pass filter for M — 100 11 2.5 Relative positions of the pilot symbols and the interpolated frame in 4-tap ideal low-pass filter 12 2.6 Gaussian, Laplace and Gaussian mixture pdf's. A l l pdf's have a vari -ance of 1 14 2.7 Gaussian, Laplace and Gaussian mixture pdf's on Log 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 21 3.4 Variances of estimation error for Gaussian interpolation at different data symbol positions 22 3.5 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 24 3.7 Comparison of B E R of V D ' s with unequal variance metric and equal variance metric for convolutional code (33,25,37) 27 3.8 Upper and lower bounds on the B E R of unquantized V D for convo-lutional code (133,171) as a function of S N R for different values of E S R 35 3.9 Upper and lower bounds on the B E R of unquantized V D for convo-lutional 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 convo-lutional code (133,171) as a function of E S R for different values of S N R 37 3.11 Upper and lower bounds on the B E R of unquantized V D for convo-lutional code (33,25,37) as a function of S N R for different values of E S R 38 3.12 Upper and lower bounds on the B E R of unquantized V D for convo-lutional code (33,25,37) as a function of E S R for different values of S N R 39 3.13 Upper and lower bounds on the B E R of unquantized V D for convolu-tional 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 convolu-tional 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 convolu-tional code (133,171) 45 3.21 Region for (upperbound - simulation)/simulation < 0.1 for convolu-tional code (33,25,37) 46 4.1 Histogram of the channel estimation error in Rayleigh fading and Lapla -cian noise with 10-tap Wiener filter, compared wi th Gaussian his-togram with the same variance 48 4.2 B E R performance of V D o p t , V D E u c i i d and V D s q u a r e d Euclid, in Rayleigh fading with Laplacian noise and Gaussian channel estimation error, as a function of S N R 49 4.3 B E R performance of V D o p t , V D E u c i i d and V D s q u a r e d Euclid, in Rayleigh fading with Laplacian noise and Gaussian channel estimation error, as a function of E S R 50 4.4 B E R performance of V D s q u a r e d Euclid in Rayleigh fading with Laplacian noise and A W G N 51 4.5 B E R performance of V D E u c i i d in Rayleigh fading with Laplacian noise and A W G N 52 4.6 B E R performance of V D o p t in Rayleigh fading wi th Laplacian noise and A W G N 53 4.7 B E R of V D in Rayleigh fading with 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 55 4.8 B E R of V D in Rayleigh fading with 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 56 4.9 B E R of V D ' s with optimal metrics in Rayleigh fading with Gaussian mixture noise, Laplacian noise and A W G N , and Gaussian channel es-t imation error, as a function of S N R 57 4.10 B E R of V D ' s with optimal metrics in Rayleigh fading with Gaussian mixture noise, Laplacian noise and A W G N , and Gaussian channel es-t imation error, as a function of S N R 58 List of Abbreviations A W G N Addit ive Whi te Gaussian Noise B E R B i t Error Rate B S C Binary Symmetric Channel E S R Error-to-Signal Ratio F I R Finite Impulse Response i . i .d . Independent identically distributed ISI Inter-Symbol-Interference M L Maximum-l ikel ihood pdf Probabil ity Density Function P S A M Pi lot Symbol Assisted Modulat ion P E P Pairwise Error Probabil ity R P Random Process rv Random Variable S N R Signal-to-Noise Ratio T C M Trellis-Coded Modulat ion V A Vi terb i Algor i thm V D Vi terb i Decoder or Decoding List of Symbols ddj The number of codewords of weight d that correspond to information sequences of weight j bd The total number of nonzero information bits associates with codewords of weight d c Complex channel gain sequence Cj Complex channel gain i?e[cj] Real 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 fd 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] Real part of hi Im[hi] Imaginary part of hi M Frame length M{ri\ hi, Si) B i t metric M (r| h, s) Pa th 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 Pd The probability that a weight d path accumulates a higher metric than the all-zero path qt 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 in 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. Lutz Lampe and Prof. Hussein Alnuweiri . I would like to express my heartfelt gratitude to my parents, Shiquan L i n and Zhian L i u , parents in-law, Haoran W u and Haoying Shi , 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 al l my friends and relatives who care about me, thank you! This work was partially supported by the Natura l Sciences and Engineering Re-search Counci l Grant OGP0001731. D I N G Y I ( C H R I S ) L I N THE UNIVERSITY OF BRITISH COLUMBIA April 2006 Chapter 1 Introduction During the past decade, much of the growth in the telecommunications industry has been in the wireless sector. For cellular wireless communications, the th ird 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 data at 144 kbps for high mobility (vehicular) traffic, 384 kbps for pedestrian traffic and 2 Mbps for indoor traffic [1]. This 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 (DSL) . The 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 Mbps 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 Mbps. 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 Mbps [3]. In many of these wireless communication systems, convolutional codes and Vi terb i decoders (VD) are used to deliver reliable communications. For example, they have been incorporated in 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 decod-ing. 1.1 Motivation and Goal In order for the V D to efficiently recover the transmitted signal in a fading environ-ment, 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 sym-bols, the estimate of the complex channel gain for each data symbol is produced by interpolating the received pilot symbols. In digital implementations, quantiza-t ion introduces errors in the channel gain estimation. Thermal noise is inevitable in 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 in fading with 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 ) with Wiener filter, that the channel estimation error variances were unequal. It is interesting to compare the performance of V D with optimal metric for unequal variances and that with optimal metric for equal variances in unequal variances environment. Although additive white Gaussian noise ( A W G N ) is commonly assumed in com-munications system studies, it has been found that in many situations [8-15], the noise exhibits an impulsive nature. This motivates an investigation of the performance of V D in impulsive noise. The 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 with optimal metric for unequal variances and that wi th optimal metric for equal variances in 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 with channel estimation errors. 1.2 Thesis Outline The 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 in the thesis is introduced, and the variance of the estimation error in P S A M is examined. Closed-form expres-sions for upper bounding and lower bounding the B E R of V D in Rayleigh fading with B P S K and A W G N are then obtained and compared with simulation results. A B E R comparison of the V D with optimal metric for unequal variances and that with optimal metric for equal variances is provided using simulation results. In Chapter 4, optimal metrics are derived for V D in Rayleigh fading with Gaussian channel estima-tion 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 in 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 Vi terb i decoding. The Vi terb i algorithm (VA) is the only one which provides maximum-likelihood ( M L ) decoding. The 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 with lowest metric as the survivor. Finally , the path with 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 with time-varying mul -t ipath 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 n ( t ) s [ t - rn{t)} (2.1) n where s(t) is the transmitted signal, an(t) is the attenuation of the n t h path and rn(t) is the delay of the n t h path. When 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 in 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 mult ipath components, the pass-band received signal can be expressed in an integral form: /oo a(r;t)s(t - r ) d r (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 a{r-t)e-^^Sl{t-r)dT (2.3) -co where si(t) is the equivalent low-pass transmitted signal. Thus the equivalent low-pass channel impulse response is c (r ; t ) = a(r;t)e-j27TfcT. (2.4) The coherence bandwidth and the coherence time are two important parameters that characterize the fading channel. The coherence bandwidth ( A / ) c is defined as the bandwidth over which the frequency correlation function is above a specific value [19], say 0.5. When the coherence bandwidth ( A / ) c is much greater than the signal bandwidth, the channel is said to be frequency-nonselective: al l frequency components of the signal are altered by the channel in the same way. The equivalent low-pass received signal can then be expressed in a product form, instead of a convolution [18]: n(t) = C(0;t)Sl(t) (2.5) where C(0;t) represents the time-varying transfer function C(f;t) at / = 0 which can be expressed in a complex form C(0;t) — r(t)e^^ where r(t) is the amplitude of C (0 ; t ) . The coherence time ( A i ) c is defined as the time over which the time correlation function is above a specific value [19], say 0.5. When the coherence time ( A t ) c is much greater than one signal interval, the channel is said to be slowly fading, and r{t)e^^ can be regarded as constant over a signal interval. Then the equivalent low-pass received signal can be further simplified as n(t) = re^siit), 0<t<T. (2.6) Three distributions are commonly used to model the envelope of the channel impulse response c(r;t), namely Rayleigh, Ric ian and Nakagami-m distributions. If C(0;£) is modelled as a complex-valued Gaussian random process (RP) with zero-mean, its envelope R — |C(0;£)| at any instant t has a Rayleigh distribution: pR(r) = -2e-r2'2°\ r > 0 (2.7) where cr2 is the variance of the real or imaginary part of C(0;£). If C(0;t) is a nonzero-mean complex-valued Gaussian R P with an expected am-plitude of 5, and the variance of the real or imaginary part of C (0 ; t) is cr2, the envelope R of C (0 ; t) has a Ric ian distribution: PR(T) = y2e^2+s2)/^Io(^), r>0 (2.8) 6 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 > - F R Q " ^ ' ^ - r s 0 <2'9> where 0 = E(R2), m — E^2_a^, and T(x) represents the G a m m a function. The parameter m is usually larger than or equal to 1/2 [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 1 next frame 1 data symbols pilot symbols Figure 2.1: Pi lot symbols in P S A M A t the transmitter, pilot symbols are periodically inserted into the data stream as illustrated in F ig . 2.1. We assume that the pilot symbol is the first symbol in 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 in a frame. We also assume that the interpolation filter uses the L nearest pilot symbols. The channel gain estimate at the jth data symbol of the L^-Jth frame is where [ x\ denotes the largest integer < x. Zeroth-order Interpolation When the fading process is changing very slowly and the channel gain is assumed to be constant in a frame, zeroth-order interpolation [23] can be used. The zeroth-order interpolation filter has only one tap (L = 1) and Linear Interpolation Linear interpolation [23,25] is another simple interpolation method. Its filter has only two taps where L (2.10) i = i (2.11) (2.12) (2.13) 1.2 0.8 0.6 o 0.4 0.2 0 -0.2 -0 .4 I 1 1 1 I 1 1 1 1 _ q2 q3 V. -N \ S s s \ \ s 1 1 1 1 1 1 i 1 1 10 20 30 40 50 60 70 80 90 100 Samples Figure 2.2: Fi lter coefficients of Gaussian interpolation for M = 100 Gaussian Interpolation The Gaussian interpolation filter [23] has three taps, where 3 M M = i - ( T 7 + l ) 3 2 "M M (2.14) (2.15) (2.16) The values of qi, q2 and q3 are plotted in F ig . 2.2 for M = 100. The relative positions of the pilot symbols pi, p 2 and P3 are plotted in F ig . 2.3. interpolated frame f Figure 2.3: Relative positions of the pilot symbols and the interpolated frame in Gaussian interpolation Ideal Low-pass Filter A n ideal low-pass filter can be implemented in time domain with 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 in frequency domain with fast Fourier transform ( F F T ) [26]. The filter coefficients of a 4-tap ideal low-pass filter are plotted in F ig . 2.4 for M = 100. The relative positions of the pilot symbols pi, p2, PJ, and p^ are plotted in F ig . 2.5. Wiener Filter Although the ideal low-pass filter is commonly used in interpolating the deterministic signals, it can not minimize the mean squared estimation error in estimating the stochastic process. In order to minimize the mean squared error, a Wiener filter can be used [7]. The 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 (FIR) 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: Fi lter coefficients of 4-tap ideal low-pass filter for M = 100 at time L — 1, L-l u ( L - i ) = £ y ( i ) u ( i ) , (2.19) i=0 d(L — 1) denote the desired signal at time L — 1, and e(L-l) = d(L - 1) - v{L - 1) (2.20) denote the estimation error. The coefficients of the F I R filter are chosen to minimize the mean squared estimation error E{ee*}. It was shown in [27, chapter 5] that the filter coefficients are p R " 1 (2.21) 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. The minimum mean squared estimation t interpolated frame P4 Figure 2.5: Relative positions of the pilot symbols and the interpolated frame in 4-tap ideal low-pass filter error is E{ee*} = E{dd*} - E{vv*}. (2.22) W h e n the mean squared estimation error is minimized, the estimation error e is uncorrelated with 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 esti-mates at the L nearest pilot positions. Then for every data symbol position, a set of optimized filter coefficients is obtained through (2.21), and the filter output is the in -terpolated value at that data symbol position, which has the minimum 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 maximum Doppler frequency, then the power spectrum of the fading channel has a bandwidth of 2fd [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 man-made R F noise, low-frequency atmospheric noise and underwater acoustic noise [8, chapter 3]. As 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 Middleton class A noise, are described in [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 ex-tremely low frequency [9], and discussed for signal detection in [10] and Vi terb i de-coding in [14]. Laplacian noise is a special case of generalized Gaussian noise, and the pdf of a zero mean Laplace random variable (rv) with variance 2f52 is 1 Inl PN(U) = ^ P ( - j ) - (2-23) Mixture noise is another k ind of widely used impulsive noise model. Its pdf is pN(n) = {l-e)r1{n) + ei(n), (2.24) 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 pdf with a heavier tai l that represents the impulsive noise. W h e n i(n) is also a Gaussian function, we have Gaussian mixture noise with pdf 1 n2 1 n2 pN{n) = (1 - € ) - _ _ e x p ( - — ) + e ^— exp(-—^) (2.25) where a2 is the variance of the background Gaussian noise and a] is the variance of the impulsive noise. The ratio CF2/CF2 is usually in the range (20,10000) [8]. The total variance of a Gaussian mixture rv is (1 — e)a2 + eaj. It is also noted that a smaller e implies a heavier ta i l in pdf. Gaussian mixture noise has been used as the noise model for multiuser detection in [28] and for signal detection in [29]. The pdf's of the Gaussian, Laplace and mixture Gaussian rv's with zero mean and variance 1 are plotted in F ig . 2.6 and F ig . 2.7. To better compare the tails of three distributions, F ig . 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 The B E R performance of V D on the binary symmetric channel (BSC) , A W G N channel and general memoryless channels is studied in [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 in [17, chapter 12] from the genie-aided approach which enables the decoder Figure 2.7: Gaussian, Laplace and Gaussian mixture pdf's on Log scale. A l l pdf's have a variance of 1. to choose between the correct codeword and the codeword that is at distance dfree away from the correct codeword. Many researchers have studied the V D in various practical systems. In [31] and [32] the V D was examined for trellis-coded modulation ( T C M ) in uncorrelated Rayleigh fading and A W G N . The optimal metric was derived for imperfect channel estimation, which in 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 in terms of residues. This method was extended to uncorrelated Ric ian fading in [33]. A similar method was used in [34] to study the V D for finite-depth interleaved convolutional codes in Rayleigh fading. The P E P for B P S K was given in terms of residues with interpolation filter coefficients and inter-leaving depth as parameters. In [35] the V D was studied for convolutional codes in spatially and temporally correlated Rayleigh fading D S - C D M A systems with 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 convolu-tional codes in uncorrelated Ric ian block fading and Rayleigh block fading, where the channel gain is constant over a frame. The P E P and the union bound on B E R were provided for both perfect and imperfect channel estimation. In [38] the V D was studied for T C M in correlated Rayleigh fading with imperfect phase reference and an approximate P E P was given. In [7] the uncoded P S A M with Wiener filter was studied. It was shown that the variances of the estimation error were unequal. In al l these studies, unequal variances of the channel estimation error have not been considered in V D , and the final results for P E P were not in closed form. Whi le A W G N was assumed in the above papers, the following ones were using impulsive noise models in the study of V D . The V D for combating inter-symbol-interference (ISI) was studied in [11] and [12], for Cauchy noise with unknown para-meters and for a-stable noise respectively. In [11] per-survivor-processing was used to estimated the noise parameters; while in [12] a penalty function was incorporated in the metric. O n the other hand, an approximation to the log likelihood function was used as the metric for the V D in simplified class A noise in [13]. Opt imal met-rics for the V D in Laplacian noise, Cauchy noise and Logistics noise were derived in [14]. Moreover, the B E R performance of V D in direct-sequence spread-spectrum multiple-access system in Ric ian fading with perfect channel estimation is discussed in [15], including Gaussian mixture noise and inter-user interference. To the best of our knowledge, the V D with imperfect channel estimation in impulsive noise has not been studied so far. Chapter 3 V D in Rayleigh Fading with A W G N 3.1 System Model The system model used in this thesis is shown in F ig . 3.1. For clarity, we use up-percase letters to denote rv's and the corresponding lowercase letters to denote their sample values. The information bit sequence x = (x 0 , £i , • • •) is first convolutionally encoded. Then 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. During the ith signal interval, Si is multiplied by a unit-energy basis function f(t) and the resulting waveform, Si(t), where si(t) = 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 Convolutional Encoder B P S K Encoded Bit to Channel Symbol Mapper Si fit) Channel Gain Estimator hi Viterbi Decoder Modulator Demodulator Figure 3.1: System model channel. The received signal is rt(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 rv C; with 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 with a real or imaginary part variance of a%. The received signal is input to a correlation demodulator, whose output is rT U = ri(t)f*(t)dt f • Si • f(t)f*(t)dt+ [Tn(t)r(t)dt Jo Jo (3.3) where /*(£) denotes the complex conjugate of fit). Lett ing rT rii - n(t)f*(t)dt (3.4) Zk Convolutional Encoder Encoded Bit to Channel Symbol Mapper Channel Gain Estimator hi Viterbi Decoder Discrete-time Fading Channel Figure 3.2: Discrete-time representation of the system model (3.3) becomes 7*2 — ' Si ~\~ Tli (3.5) where nj is a sample of a zero-mean complex-valued Gaussian rv, JVj, with a real or imaginary part variance of a\. The rv Nt is independent of d or Si. The real and imaginary parts of Ni are assumed to be independent. A discrete-time model of the system is shown in F ig . 3.2. A channel gain estimator provides an estimation of Cj for the V D . The estimated channel gain hi is assumed to have the form hi Ci (3.6) where the channel estimation error is a sample of a zero-mean complex-valued Gaussian rv Ei, wi th 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,. Al though 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. The V D then uses the estimated channel gain sequence h = (ho, hi,...) to decode the received sequence r = (rn,7"i> • • •)• The decoded bit sequence is denoted by z = (z0,zl,...). For convenience, we define S N R and E S R as S N R = (JQ/G2N and E S R = O\I<J'2C [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 in (3.5), and the channel gain p; at the pilot symbol position is obtained by Pi = — Si = Ci + ^. (3.7) Si For linear interpolation as defined in (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 estima-tion 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^)C2 + + ^ ) C 3 AiJL _ j_\VI + n - + + -?-)— (3 ID + 2 ( M * M } S l + { 1 M>} s2 + 2{ M* + M} s3' 1 j the estimation error is <1) - - ^ - M ^ - ^ - ^ - ^ M ^ + M ^ ' ( 3 - 1 2 ) and its variance Real Part Imaginary Part 40 60 80 Samples 100 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. The results for 0.3 Samples Figure 3.4: Variances of estimation error for Gaussian interpolation at different data symbol positions linear interpolation are plotted in Fig . 3.3. The frame length M is 100 symbols, the fdT product values are 0.001, 0.002, 0.003, 0.004 and 0.005. The 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. The figure shows that the variances of the estimation error are not constant. When the frame length M is much smaller than 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 ig . 3.4. The reference curve is obtained using (3.13). A plot for a 10-tap Sine filter is shown in F ig . 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 . 0.24 0.22 O 0.18 .§ 0.16 Real Part Imaginary Part fdT=0.005 40 60 Samples 100 Figure 3.5: Variances of estimation error for 10-tap Sine filter at different data symbol positions The curves change negligibly as fdT changes from 0.001 to 0.004. Variances for a 10-tap Wiener filter are plotted in F ig . 3.6. We notice that when M < Tjjrjy, the variances are essentially equal for all data symbol positions. The Wiener filter provides a substantial performance improvement over the Sine filter in 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 than that of the 10-tap Sine filter. When fdT = 0.04, the difference is smaller. The 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 Samples 80 100 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 es-timation error variances We assume that the channel gains for consecutive received symbols are uncorrelated and that the V D performs unquantized decoding. The 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 with 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 with 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 Re[ri] = Re[ci\ • Si + Re[ni] (3.14) Im[ri] = Im[ci] • Si + Irn[rii\. (3.15) Thus given Re[ri] and Im\ri] are samples of i . i .d . zero-mean Gaussian rv's wi th variance a\ where 4 = 4 + 4 ; (3.16) Re[Ri] and Im[Ri] are independent. The 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 with variance a%.. From (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 with variance ajf. where 4 , = 4 + 4 ; (3-19) Re[H~i] and Im[Hi] are independent. Given Re[hi], Im[hi] and S j , i?e[r j ] and i m [ r j ] are samples of i . i .d . Gaussian rv's with means, mRe[Ri]\ReiHi}=Re[hi} and mlm[Ri]\Im[Hi\=:im[hi}, variance, 4e[i?1]|*e|//i]=Ke[/ll] a n d 4m[fli]|/m[/fi]=/m[hi]» a n d conditional pdf's as derived in Appendix A . The joint conditional pdf is PReiRi}im{Ri}(R4ri\^ Imlri} I Re[Hi] = Re[hi], Im[Hi] = Im[h^,Si = Si) = Pite[/y (fle[ri]|J?e[#i] = i?e[ / i*] , $ = s^ ) •Pjm[fli](^[rj]|/m[ffi] = i m [ / i * ] , Si = st). (3.20) We assume that a codeword of length L is transmitted. Let h denote the es-timated channel gain sequence, s denote the transmitted sequence corresponding to the codeword and r denote the received sequence. The V D selects a sequence s that maximizes the value of p R(r| H = h, S = s). The negative log likelihood function is logp R (r| H = h,S = s) L - l = l°gPRciRi]imiRi](Re[ri\i / m N I Re[Hi] = Re[hi], Im[Hi] = Im[hi], St = Si) - (Re[rt] - SiRelhi}^ / (2(0% - ^) exp i=0 exp /m[rj - sjmm^j / (2(0% -L - l = £ ^ ^Re[ri] - SiRe[hi]^f- ) + ( Imfri] - Silm[hi]^f-i=0 \ - log 1 ) (3.21) Thus the optimal bit metric for unequal variances is 2 M(ri,hi,Si) = Re[ri] - SiRelhil^j + ^Irn[ri} - Silm[hi]^f-- m - l o g — It can be simplified as (3.22) M(ruhi,Si) = (-Re[ri]Re[hi\si - Im[ri]Im[hi]si). (3.23) For equal variances, the optimal bit metric for the V D in Rayleigh fading with channel estimation error and A W G N , for P S K and Q A M is derived in [31] as M{ri,hi,Si) = \Re\n} - RelhiYs^j + [lm[ri] - 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 path metric is L-l M ( r ,h , s ) = ^2M(ri,hi,Si). (3.26) i=0 The equal variances metric is independent of the actual estimation error variance value, and is the same as that for perfect channel estimation. We also refer to (3.24) as squared Euclidean distance metric. LU CD 10 10 10 10" 10 10 -6 Unequal variances metric + Equal Variances metric SNR (dB) Figure 3.7: Comparison of B E R of V D ' s with 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 with optimal metric for unequal variances ( V D u n e q u a i ) and that of the V D with optimal metric for equal variances ( V D e q u a i ) . Here we use a convolutional code whose generator sequence in octal form is (33,25,37) [17]. This is a rate | code with a free distance dfree 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 Vi terb i decoding in an A W G N channel. The ability to decode in frequency-nonselective slow fading channel is added to the V D . The 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. This can be achieved by interleaving. F ig . 3.7 shows the simulation results for V D u n e q u a i and V D e q u a ] . It can be seen that the two V D ' s have more or less the same B E R perfor-mance. This finding agrees with 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 oo T(X,Y) = Y,Y,ad<iXdYi (3-27) d=l 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 where y=i = X > * d (3-28) d=l oo &d = ^J-a-dj (3-29) is the total number of nonzero information bits associated with codewords of weight d. Without 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 -. oo Pb < T Y\ bdPd (3.30) k 4 i where djree is the free distance of the convolutional code, {bd} are the coefficients from (3.29), Pd is the P E P that a weight d path accumulates a higher metric than the all-zero path. A lower bound on the B E R of the V D is obtained in [17, chapter 12], pb > \PdSree. (3.31) We assume that the variances of the channel estimation error are equal over a frame, and use the metric in (3.24), so that d-1 d-1 Pd = P{J2M(Ri>Hi'Si = - 1 ) > J 2 M ( R " H i ^ i = 1^ i=0 i=0 d-1 = P{J2(-MPi}Re[Hi) ~ ImiRijImlHi}) i=0 d-1 > ^(ReiRilReiHi] + J m ^ / m ^ ] ) } i=0 d-1 = PiY^iReiRilRelHi] + Im[Ri]Im[Hi\) < 0}. (3.32) i=0 Although a numerical evaluation of Pd is given in [31] as a sum of residues of 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^0Re[^]#e [# i ] + /m[f l i ] /m[tf i ] ) < 0\{Re[Hi] = Reiki}}, {Im[Hi\ = J m ^ ] } J , then averaging over {Re[Hi}} and {Im[Hi\}, as follows. Let d-l ud = ^{Re[hif + Im[hif). (3.33) i=0 Since { i?e[/ij]} and {Im[hi}} are samples of i . i .d . Gaussian rvs with zero-mean and variance a\ as derived in (3.19), ud is a sample of a chi-square-distributed rv with 2d degrees of freedom. The 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 ra [r j ] are samples of Gaussian rvs with means, m^R^iuiH^R^hi] and mim[Ri]\Im[Hi]=Im[hi], and variance, vRe[Ri]lRe[Hi]=Re[hi] and o'|m[/ei]|/m[f/i]=/m[/lj]) a s derived in Appendix A . Assuming that all-zero codeword is transmitted, Si = 1, then mRe[Ri]\Re[Hi]==Re[hi], m / m[H i]|/m[// i]=/m[/ l i], crL[Ki]|Ke[//i]=fle[/ii] a n d <7/m[fli]|/m[Hi]=/m[hi] become 2 ^He[Ri]|He[//i]=He[/ii] = i?e[ / l i ] - j - (3.35) ^/m[Ki]|/m[Hi]=/m[/ii] = / m f / l j ] — (3.36) 4 CrHe[Ki]|fie[/fi]=fie[^] = a f i ~~ ~2~ (3.37) <T/m[Ri]|/m[//i]=/m[fti] = ~ ~2~- (3.38) Let Re[gi] = Re[ri]Re[hi] Im[gi] = Im[ri]Im[hi\. Thus, given Re[hi], Re[gi] is a sample of a Gaussian rv with mean (3.39) (3.40) mRe[Gi]\Re[Hi]=Re[hi] = R&[hi) • "T.«e[Hj]|fle[Hi]=fle[/ii] •H and variance 0'iie[Gi]|/te[ifi]=fle[/ii] — Re[hi] • Ofle[/ei]|fle[//i]=.Re[fc; i i > similarly, given Im[hi], Im[gi] is a sample of a Gaussian rv with mean ^/m[Gi]|/m[Hi]=fle[/ii] = Im[hi] • m/ T n[ f i i]|/m[// i] = / m[ h i] = Im[hi] 2UC and variance (3.41) (3.42) (3.43) Let d-1 <Tlm[Gi]\Im[Hi]=Iin[hi] ~ I™\hif • 0']m[Ri\\Im[Hi]=Jm[hi = Im[hi]2[o*R 'H wd = '^(Re[ri]Re[hi\ + Im[ri]Im[hi}). t=0 (3.44) (3.45) Thus given {i?e[/ii]} and {Im[hi]}, % = 0, ...,d - 1, wd is a sample of a Gaussian rv with mean 2 d-1 mwd ?f J2(Re[hi]2 + Im[hi " i=0 (3.46) and variance d - l °ia = ri-5)E(*]'+/mw') 'H 4 2 °C j=0 ( a | - - ^ K (3.47) The conditional probability is then d - l P{J^(/?e[i2 i]i2e[// ' i] + Im[Ri]Im[Hi]) < 0 i=0 | {Re[Hi] = Reiki}}, {Im[Hi] = Imfa]}} = P{Wd < 0 | {Re[Hi\ = Reiki}}, {Im[Hi\ = Jm[^]}} 1 / 1 mwd eric —= 2 V\/2 cr w d = 2 4 2 J ( 3 - 4 8 ) where erfc(-) denotes the complementary error function. We then average (3.48) over Ud in (3.34). B y using [42, (8.250) and (8.253)], the complementary error function can be expressed as an infinite sum: P , - r - erfc [ 4 / ^ 1 - ud-le-u^dud P d ~ Jo 2 e r f C i V 2 ( a ^ - a ^ ) j a * W ( ^ 7o \ 2 ^ F ^ f } \*{?\O*H-O%<J*H)] ( 2 f c - l ) ( f c - l ) ! a f 2 ^ ( d ) U d 6 m d 2 ^ 2 - r ( d ) 7 0 d i 0 0 l _ r <#2«T(d) 7 0 2 ( 4 ^ fc-1/2 n V*F£fV ^ 1 2 ( ^ - 4 ^ ) / (2fc - l ) (* - l ) fe=i /•oo B y using [42, (3.351)], the integrals in (3.49) can be simplified as 1 1 p< - r 5 f 2 % 5 ) ( " - 1 ) ! ( 2 4 ) d 1 oo , 4 N fe-1/2 ., V^tC \2(°>4H - °h°l) J ( 2 f c - l ) ( f c - l ) ! 1 -(k + d - h (2a2H)k+d-1/2. (3.50) ajf2dT{dy 2' B y using the properties of the G a m m a function in [42, (8.339), (8.338) and (8.331)] and letting k' = k — 1,( (3.50) can be further simplified as: 2 y/i T(d) V a%a% - a'c oo /• 4 N k—1 (fc + d - ^ ) ! (2fc- l ) ( f c - 1)P 2 i r a + d) / 4 2 ^ r (d) V 4 4 - <4 r ( § ) ~ r ( i + fc')ra + d + fc') r - 4 1 j  - T(l  k')T^    k') f -  1 ' r ( | ) r ( i + r ( f + A-) 1 4 4 - 4 J 1 r (± + d) 2 ^ T(d) V 4 4 - 4 2 ' 2 ' -'2'o%a%-o*c 1 r a + d) 2 A r ( d ) V ( ^ + ^ ) ( ^ + 4 ) - ^ 2 ' 2 ' 2 ' ( 4 + 4 ) ( 4 + 4 ) - 4 1 r ( i + d) / 1 2 • r (d) V ( 4 / 4 + 4 / 4 x 4 / 4 + 4 / 4 ) - 4 / 4 f n 3 - 1 \ 2' 2 + d ; 2 ; ( 4 / 4 + 4 / 4 ) ( 4 / 4 + 4 / 4 ) - 4 / 4 1 _ r ( ^ + d) 2 0 F - T ( d ) y (1 + ESR)(1 + l / S N R ) - 1 f1 1 3 —1 1 ' F {2' 2 + d ; 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 bound on the B E R of the V D . It might be noted that Pd in (3.32) is the probability that a sum of products of correlated zero mean Gaussian rv's is less than zero. One solution is given in [18, Appendix B] for the probability that a sum of products of correlated nonzero mean Gaussian rv's is less than 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 + a2N)((rh + <72E)-crh ^/(1 + 1/SNR)(1 + E S R ) + 1 (3.53) 7(1 + 1 /SNR)(1 + E S R ) - 1 It might also be noted that the metric in (3.26) is a weighted sum of the received signals, which bears similarity to the decision variable in maximum-ratio-combining ( M R C ) . A n expression for the P E P for M R C with B P S K is given in [44] as where 7 = (o*c + o%){a* + ol) - of ( 3 ' 5 5 ) A l l three expressions for Pd, i.e., (3.51), (3.52) and (3.54), show that the channel 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 The upper and lower bounds are plotted for two convolutional codes (133,171) and (33,25,37). The first is a rate \ code with a free distance djree 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. The coefficients {bd} in (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 in MATLAB®, the value of Pd is calculated using Maple®. Computer simulations were also used to validate the upper and lower bounds. F ig . 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 bound \ x. X \ . — — lower bound X ^ \ i 0 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 The B E R performance of the V D degrades rapidly with 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. The error floors increase with increasing E S R . Moreover, the gap between the upper and lower bounds increases slightly with increasing E S R . F ig . 3.9 shows the upper and lower bounds on the B E R of the V D with per-fect channel estimation ( E S R = -oo dB) 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 with increasing S N R . F ig . 3.10 shows the upper bounds and the lower bounds on the B E R of the V D for -25 -20 -15 -10 -5 0 ESR (dB) Figure 3.10: Upper and lower bounds on the B E R of unquantized V D for convolu-tional 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 with increasing E S R , especially when S N R is small. Comparing with F ig . 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. Fig . 3.11 and F ig . 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). The bounds are similar to those in F ig . 3.8 and F ig . 3.10, except that the bounds for the convolutional code (33,25,37) are lower than those for (133,171). Comparing their upper bounds in F ig . 3.10 and F ig . 3.12, for the same E S R of -25 d B and same B E R of 10~ 6 , the 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). The reason for this is that the convolutional code (33,25,37) has a larger dfree than the convolutional code SNR (dB) Figure 3.11: Upper and lower bounds on the B E R of unquantized V D for convolu-tional code (33,25,37) as a function of S N R for different values of E S R (133,171). F ig . 3.13 and F ig . 3.14 show the upper and lower bounds for E S R = 1 /SNR, which corresponds to P S A M with Sine filter, for the convolutional codes (133,171) and (33,25,37) respectively. It can be observed that in 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 convolu-tional 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 convolu-tional code (33,25,37) for E S R = 1 / S N R tr m 10" 10 — upper bound — lower bound + simulation 10 15 SNR (dB) 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 ig . 3.15 and Fig . 3.16 show the computer simulation results for the convolutional code (133,171) compared with the theoretical bounds. The bounds agree with the simulation results very well. A s S N R increases and E S R decreases, i.e., the variances a2N 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 in [17, chapter 12] as Ph « bd. Pa, • B y setting S N R to oo in (3.51), the B E R floor is thus (3.56) P = h P rb udfree rdSree |SNR=oo bdfree bdfreT{\+dfree) • F 1 1 3 - 1 1 (3.57) 2 ^ • ^ ( d / r e e ) ^ / E S R ' 1 2 ' 2 ' ^ e e ; 2 ; E S R ] The approximate B E R in (3.56) for the convolutional code (133,171) and E S R = -8 d B is plotted in F ig . 3.17, along with 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 SNR ' s . F ig . 3.18 and F ig . 3.19 show the simulation results, along with the upper bounds, lower bounds and approximate B E R ' s for the convolutional code (33,25,37). Both graphs show very good correspondence of the upper bounds and approximate B E R with the simulation results. In order to assess the tightness of the upper bounds on the B E R of the unquan-tized V D for convolutional code (33,25,37), the simulation points where (upperbound - simulation)/simulation < 0.1 are plotted in F ig . 3.20 and F ig . 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 within 10% of the simulation B E R ; for the con-volutional code (33,25,37), when E S R < -4 d B and S N R > 4 d B , the upper bounds are within 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 -10 ESR (dB) 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 20 SNR (dB) 30 Figure 3.20: Region for (upperbound - simulation)/simulation < 0.1 for convolutional code (133,171) CO tr C/) LU 10 15 20 SNR (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 in Rayleigh fading with channel estimation errors is stud-ied, for Laplacian noise and Gaussian mixture noise. First , 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 in Rayleigh fading with Laplacian noise and channel estimation error is studied. The system model is the same as in F ig . 3.2, except that the real and imaginary parts of the noise rv iV* now have independent Laplacian distributions defined in (2.23). Since the estimated channel gain is a sum of rv's as in (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 al l values of i. A histogram of the channel estimation error in Rayleigh fading and Laplacian noise with a 10-tap Wiener filter is plotted in F ig . 4.1, and T 1 1 1 I I I — — Gaussian histogram actual histogram - 2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Figure 4.1: Histogram of the channel estimation error in Rayleigh fading and Lapla -cian noise with 10-tap Wiener filter, compared with Gaussian histogram with the same variance compared with the histogram of a Gaussian rv with the same variance. It can be seen that the Gaussian distribution provides a good approximation. The optimal bit metric for Laplacian noise without fading is given in [14] as M(vi, hi, Si) = \Re[ri] - s*| + \Im[ri] - st\, (4.1) whereas the optimal metric for fading with perfect channel estimation is M{n, hi, Si) = \Re[ri] - Re[ci]si\ + \Im{ri} - Irn[ci]si\. (4.2) We refer to (4.2) as Euclidean distance metric. The optimal bit metric for Rayleigh fading with channel estimation error is de-rived in Appendix B as M{ru hu Si) = f(Re[n], Re[hi], st) + f(Im[ri], Im[hi], Si), (4.3) 4 6 SNR (dB) Figure 4.2: B E R performance of V D o p t , V D E u c i i d and V D s q u a r e d Euclid, in Rayleigh fading wi th Laplacian noise and Gaussian channel estimation error, as a function of S N R where f(ri,hi,Si) , f f-ria% + 2hisiPal - log <^  exp{ 2 2 ) •erfc( 0(*2E + 4) Si(3aEac\/(J2E + a% •erfc{ rj0a% + rif3a2c - hiSi(5a2c + o\o2c, (4.4) Simulations were performed for three V D ' s (1) V D o p t wi th the optimal metric in (4.3), (2) V D E u c i i d wi th Euclidean distance metric in (4.2) and (3) V D s q u a r e c i Euclid with squared Euclidean distance metric in (3.24), for the convolutional code (33,25,37). -6 - 4 ESR (dB) Figure 4.3: B E R performance of V D o p t , V D E u c i i d and V D s q u a r e d Euclid, in Rayleigh fading with Laplacian noise and Gaussian channel estimation error, as a function of E S R The results in F ig . 4.2 show that V D o p t has the best performance as expected. A t low E S R , the B E R of VDEuciid approaches that of V D o p t ; the performance of VD S q U a r ed Euclid is about 1 d B worse than that of V D o p t at target B E R of 10~ 6 . This 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 s q u a r e c l E u c l i d in Rayleigh fading with Laplacian noise and A W G N Lett ing 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 -2Re[hi]si - log 2, Re[r^ - Re[hi]si > 0 p2{Im[ri}, Irn[hi\, st) 2Im[hi]si — log 2, Im[ri] — Im[hi]si < 0 —2Im[hi]si — log 2, Im[ri] — Im[hi]si > 0 (4.5) (4.6) 10° Euclid, dist. metric in Lap. noise Euclid, dist. metric in AWGN rr LU m N 10 - 7 0 2 4 6 8 10 SNR (dB) Figure 4.5: B E R performance of V D E u c i i d in Rayleigh fading with Laplacian noise and Substituting (4.5) and (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 V D 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 ig . 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 o p t ; at high S N R , V D s q u a r e d Euclid and V D o p t 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 o p t per-form in A W G N compared with Laplacian noise. The B E R performance of V D s q u a r e d Euclid A W G N optimal metric for Lap. noise in J I L 0 2 4 6 8 10 SNR (dB) Figure 4.6: B E R performance of V D o p t in Rayleigh fading with Laplacian noise and A W G N is plotted in F i g 4.4. It is shown that at low E S R , i.e., when Laplacian noise domi-nates, 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. The B E R performances of V D E u c ! i d and V D o p t are plotted in F ig . 4.5 and F ig . 4.6. It can be seen that there is perfor-mance difference in A W G N and Laplacian noise, for both VDEuciid and V D o p t . The difference at low E S R is larger than that in 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 in Rayleigh fading with Gaussian mixture noise and Gaussian channel estimation errors is analyzed. The system model is the same as in F ig . 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 in (2.25). The optimal bit metric is derived in Appendix C as where M{ru hh §i) = - \ogpRe[Ri](Re[ri\ | Re[Hi] = Re[ht}, St = §i) - logPim[Ri}{Im[ri\ | Im[Hi] = Im[hi], Si = st) PRe[Ri}{Re[ri) I Re[Hi] = Re[hi],Si = §i) exp - (Rein] - s.ReM^rf / ( 2 (a£ + a? - ^ ) ) ^2^(4 + ffa «E+aC1 exp + e -(ite[r<] - s ^ h i } ^ ) 2 / ( 2 (a£ + a] - ^ ) Pim{Ri](I™[ri] | Im[Hi] = Im[hi],Si = 54) exp ( 1 - c ) -- (lm[n] - « ] ^ ) J / ( % ° h + ^ " ^ ) ) exp +e-- {lm[ri\ - silrnih,]^)2 J (2(a% + aj - ^ ) (4.7) (4.8) (4.9) Simulations were performed for different Gaussian mixture noise parameters. 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]. The B E R of the V D with optimal metric in Rayleigh fading with Gaussian mix-ture noise and Gaussian channel estimation error is plotted in F ig . 4.7, for e = 0.05, 15 SNR (dB) Figure 4.7: B E R of V D in Rayleigh fading with Gaussian mixture noise and Gaussian channel estimation error, for e = 0.05, a]/a2 = 21 and aj/a2 = 181, as a function of S N R _ 2 i a n d af/a2 — 181. The pdf's of the Gaussian mixture noise are plotted in F ig . 2.7. It is shown in F ig . 4.7 that at low S N R , the V D with aj/a2 = 181 has better performance, and the performance difference increases as E S R decreases. This is because at low S N R , the threshold over which a noise sample can cause error is low, say, 1 in F ig . 2.7, and the probability of a noise sample has magnitude larger than 1 is lower wi th tf/tf = 181 than wi th a]/a2 = 21. A t high S N R and high E S R , the V D ' s in the two environments have the same performance. This 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 with crf/a2 — 181 has a heavier ta i l than that wi th aj/tf = 21. For high S N R and low E S R , we expect that the V D with a]/a* = 181 wi l l have better performance than that in a]/a2 = 21. Unfortunately, the B E R is too 10" c f /CT2=181 of / (^=21 - 6 - 4 ESR in dB Figure 4.8: B E R of V D in Rayleigh fading with Gaussian mixture noise and Gaussian channel estimation error, for e = 0.05, crj/cr2, = 21 and a]/a2 — 181, as a function of E S R low to be verified by simulation. F ig . 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 in Gaussian mixture noise, Laplacian noise and A W G N . The results with optimal metrics are plotted in F ig . 4.9. Similar observation to those for F ig . 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 , al 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 in Gaussian mixture noise sti l l has better performance at low S N R ; while at high S N R , the V D in Gaussian mixture noise has worse performance because Figure 4.9: B E R of V D ' s with optimal metrics in Rayleigh fading with 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 ta i l of the Gaussian mixture pdf. The B E R of the V D ' s as a function of E S R are plotted in F ig . 4.10. As 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 with optimal metrics in Rayleigh fading with 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 with 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 with different data symbols had equal variances. Channel estimation errors have not been considered in the study of V D in 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 in frequency-nonselective slow Rayleigh fading was stud-ied, for A W G N , Laplacian noise and Gaussian mixture noise. The main contributions are summarized as follows. • The B E R performances of two V D ' s (1) V D u n e q u a i wi th optimal metric for un-equal channel estimation error variances and (2) V D e q U a i with optimal metric for equal channel estimation error variances were compared, in an unequal vari -ances environment. It is shown that when the frame length M is less than jj^f, the V D e q u a i has more or less the same performance as V D u n e q U a i -• Closed-form expressions of the P E P were derived for V D with B P S K in Rayleigh fading with 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 in 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 with the upper bounds. The optimal metric was derived for V D in Rayleigh fading with Laplacian noise and Gaussian channel estimation errors. The B E R performances of the V D ' s with optimal metric ( V D o p t ) , Euclidean distance metric ( V D E u c i i d ) and squared Euclidean distance metric ( V D s q u a r e d Euclid) in 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 o p t and V D s q u a r e d E u ciid are similar. V D o p t , V D E u c i i d , VD s q u a red Euclid were compared for Laplacian noise versus A W G N It is found that the performance difference between the two noise models i n -creases as the channel estimation error decreases. The performance difference is typically smallest for V D o p t and largest for V D s q u a r e d Euclid-The optimal metric is derived for V D in Rayleigh fading with Gaussian mixture noise and Gaussian channel estimation errors. The B E R performance of the V D with 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. Among related topics for future study are the following: V D in which quantization is applied. Extension of the thesis work to other modulation, channel fading, and channel estimation error models. Bibliography [1] "3G Standardization Process," Nov. 2005. http : / /www.3gamericas .org /Engl ish/ Technology _Center/standardization_process.cfm. [2] "Technical Information," Nov. 2005. http: / /www.wimaxforum.org/technology. [3] "Glossary of terms," Nov. 2005. http : / /www.wi- f i .org /OpenSect ion/ glossary, asp. 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Derivation of the Conditional pdf's PRe[R^Re^^\Re\.H^ = Re\hii Si = Si) 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^e[/j i](i?e[rj] | Re[Hi] = Re[hi\, Si = Si) and PimiRi](Im[ri\ I Im[Hi\ = Im[hi],Si = s,), which are used in obtaining the expression for M(r{\ hi, s,) in section 3.3 and Pa in section 3.4. The covariance, C, of Re[Ri) and Re[Hi], given Si = Si, is C = EiiReiRj-EiRelRiWiRelH^-EiRelHi}})} = E {{Re[Ci]Si + Re[Ni]){Re[Ci] - Re[Ei})} = E{Re[Ci]2Si + Re[Ci]Re[Ni] - Re[Ei]Re[Ci]si - Re[Ei\Re[Ni}} = E{Re[Ci]2Si} + E{Re[Ci}Re[Ni}} - E{Re[Ei}Re[Ci]Sl} - E{Re[Ei]Re[Ni]} = cr2csi ( A . l ) where E{-} denotes the expectation operation. The correlation coefficient, pi, of Re[Ri) and Re[Hi], given Si = Si, is A Pi = Thus we have O'RdHi - ^ L . (A.2) O~RO-HI PRe[Ri}(Re[ri] I Re[Hi) = Re[hi],Si = PRe[Ri]Re[Hi](Re[ri], Re[hi] | Si = Si) PRelHi](Re[hi]) exp 2(1 -P?) fle[rj]2 _ Q » Jfe[rj]fle[/tj] , fle[fc exp y/2«(l - p 2 ) ^ VMi - P\)°\ exp - ( i ? e N - ^ g ^ ) 2 / ( 2 ( l - p 2 ) 4 ) exp v ^ i - P ? K ^ • / (2(1 - ( ^ ) 2 K ) exp - ( f e h ] - Sifie[A,] f ^ ) 2 1 (2(o% - |-)) Given = s,, i?e[#j] = i?e[/ij], i?e[i?j] is a Gaussian R V with mean ^e[«i]|He[Wi]=fle[/i<] = Sj .Re[ / ii ] -y-and variance <JRe[-Ri]|fte[/fi]=.Re[/ii] = a R 'Hi (A.3) (A.4) (A.5) The conditional pdf, mean and variance of Im[Rj\ can be similarly obtained as: Plm[Ri](Im[ri] I Im[Hi\ = Im[hi],Si = Si) exp Im[ri] - sjm[hi}^j / ( 2 ( 4 - ^ ) (A.6) rmim.[Ri)\Im\Hi}=Im[hi) = SiIm[hi]—£- (A.7) aIm[Ri]\Im[Hi]=Im[hi = a R „2 • a Hi (A.8) Derivation of Optimal Bit Metric for Laplace Noise The optimal bit metric is derived for the V D in 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 RV ' s and samples. The pdf of Hi conditioned on d = Q is PHi(hi | Ci = Ci) = —exp( \ 2 % ), (B . l ) the pdf of Rt conditioned on d = Cj and Si = Sj is PRM I d = a,Si = Si) = jpexP(-r* ^"")' (B-2) the pdf of Ci is 1 c 2 P^ici) =-7=^-exp{--\). (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 PRiin I Ci = Ci, Si = §i) • pHi{hi I Ci = Ci) • pCi(ci)dci -oo ——> oo ^ P V ^ / l ^ w £ ._^ e x p (__| ) d Q 1 , r , - c ^ _ (a - hj)2 _ W*E*c J-oo e X P [ P 2 a | 2o*c)a€i + I r e x v ( - C i S i ~ r i - ( C i " ^ - -$-)dc-4nPaEac J-oo 2o\ 2a2c a2E (3 2a2E p + T J T - e ^ ( - i - A ) + c A - | ) + ( - A + ^ ) ) d c , 4irP<JE<Tc Jn/si 2aE 2aC °E P 2aE P (B.4) Lett ing A = + E>\ = ~^t~ '% and C\ = -£jr + ^, the first integral of (B.4) becomes /TilSi exp(-Ac2 - BiCi - Ci)dCi •00 = expi-d + %) exp(-(VAa + ^=)2)dci. (B.5) Lett ing \[A~Ci + ^= = U, dcj = ^ d £ ; , (B.5) becomes 2Ari + B1a,; - ^ eM-C^^erM-2-^^!) (B.6) 2vC4 4 A y 2SiVC4 where erfc(-) denotes the complementary error function. Lett ing B2 = + % and C 2 = t tV - %, the second integral of (B.4) becomes ^ E P E P / e x p ( - A c 2 - S 2 C j - C2)dCi J ri/Si * , n B\ t (2Ari + B2Sj^ exp(-C2 + -—)erfc{— — ) . (B.7) 2 ^ 4 4 A ; 2siVA Thus PRi,Hi(ri,hi | Si = §i) 1 f-h2p2 - 2rj0<j% - 2nl3a2c + 2hCsjPo2c + s2a2Ea2c^ ~-eXP\ 0 / 3 2 / _ 2 i _ 2 -\ / 4 / 3 ^ v ^ | T ^ " V 2P2(a2E + a2c) Si^E^cy/^s + erg. - / i 2 / ? 2 + 2ri0a% - 2n(3a2c - 2hiSx(3o2c + s2a2Ea2c. •-exP( 0 0 2 / - 2 , _ 2 N ) 4/5v / 2T\/a| + a2c y y 2f5\o\ + o2c) crfcC^^ + T i f 3 ( j 2 c ~ + g < g g g g ) (B 8) Using (3.19), the conditional probability PRi(ri | /fj = hit Si = Si) = PRi,Hi(ri,hi | Si = §i) PHM J _ -2rj/?a| - 2rj/3gg. + 2hjSj(ia2c + •§2ct|ct27 . 4 / ? ^ 2/?2(ai + cT27) j CTfc(~Ti^a'E ~ r i / 9 g g + hi§i^ac + Ha\ac^ Sif3o-EvcVaE + ac 1 +2n/?a| - 27-i/frg; - 2hiSi/3a2c + s2a2Ea2c  + A(3eXP[ W\o\ + o2c) } Si0<TE<7cy/aE + CTc Following the analysis in section 3.3, the optimal bit metric is M(r{, hi, §i) = - logpRe[Ri](Re[ri] | Re[Hi] = Re[hi], Si = Si) - logpIm{Ri](Im[ri] | Im[Hi] = Im[hi\, Si = s{), (B.10) and by removing all common terms, it is simplified as M(rh hu §i) = f(Re[n], Re[hi],Si) + f(Im[n], Im[hi\, s{), ( B . l l ) where f(ri,hi,Si) , f ,-riaE + 2hisiPo-2c • e r / c ( — , Si(3aE(Tcy'crE + °c r i t r | - 2hCsil5u2c +exP\ 2 , 2^ ) v2 , „n„1 _u.z.R„2 , ^ 2 ^ 2 crfcf E + ^ ~ + a ^ a g ) I (B 12) Derivation of Optimal Bit Metric for Mixture Gaussian Noise The optimal bit metric of the V D in Rayleigh fading with 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 RV ' s and samples. The pdf of Hi conditioned on Ci = Ci is PHiihi | Ci = Ci) = —exp(-^ C l ^ ), (C . l ) the pdf of Ri conditioned on Q = q and Si = ii is PRiin | Ci = Ci,Si = Si) ! - £ / (n-CiSi)2 e (n - c ^ ) 2 exp( —7 ) + exp( —2 ), (C.2) V27rav 2cr2 V27ra/ 2a] the pdf of Ci is 1 c 2 PCM) = nr- EXP(-7T2-)- ( 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 | Ci = Ci, S, = §i) • pHi(hi | Q = a) • pCi{ci)dci -oo oo oo -f{ J —oo k 1 - e , fa - Ci§i)2 exP( ) + oo 1 2ixo~n 2TT 1 2tf 1 27T<7/ ezp( z - 5 ) 2a) 2a\ 1 exp(-2ol )dci •ea;p( TTO ) —oo v >^?7 ,2 2a» 1 « r f - ^ > v / _ e x p ( - ^ ) d c l +e 27TCT/ 2 4 2a| (C.4) It can be seen that the first integral is actually a joint pdf of two correlated Gaussian RV ' s , which has been derived in Appendix A , only requires replacing 0 N with a2. This is also true for the second integral, only requires replacing aff wi th a). B y modifying (A.3), the conditional pdf is PRi{n I Hi = hi, Si = Si) exp ( 1 - 6 ) -r 2 - ^ ^ ) 2 / ( 2 ( 4 + 4 - ^ ) ) exp +e-S ^ ^ ) 2 / ( 2 ( ^ + ^ 2 - ^ ) ) Following the analysis in section 3.3, the optimal bit metric is (C.5) M(ri,hi,Si) = - logpjfc[^](.Re[rj] | Re[Hi) = #e[^], 5, = Sj) - logpIm[Ri](Im[ri] I Jm[#i] = Im[hi], St = S j ) . (C.6) 

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