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Performance degradation of a transmit diversity scheme due to correlated fading Zheng, Tao 2005

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PERFORMANCE DEGRADATION OF A TRANSMIT DIVERSITY SCHEME DUE TO CORRELATED FADING by TAO ZHENG B. Eng., Shanghai Jiaotong University, 1993 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES ELECTRICAL ENGINEERING THE UNIVERSITY OF BRITISH COLUMBIA Feburary 2005 © Tao Zheng, 2005 Abstract The error performance of the Alamouti simple transmit diversity (STD) scheme in the presence of time-selectivity and channel estimation errors has been previously studied. In this thesis, results are obtained for two other scenarios: (1) non time-selective, spatially correlated Rayleigh fading with imperfect channel estimation and (2) time-selective, spatially correlated Rayleigh fading with perfect channel estimation. Exact expressions for the conditional bit error rates given the estimated channel gains are derived and approximations for average bit error rates over correlated Rayleigh fading are obtained using matrix transformations. It is found that STD performance generally degrades with increase in channel estimation errors, decrease in temporal correlation and increase in spatial correlation. The degradation is greatest with channel estimation errors, then time-selectivity and thirdly with spatial correlation. iii Table of Contents Abstract ii Table of Contents iii List of Figures v List of Acronyms and Symbols vi Acknowledgement x 1 Introduction 1 1.1 STD Scheme 4 1.2 Generalized STD Expression 6 2 STD in Non Time-selective, Spatially Correlated Fading with Imperfect Channel Estimation 9 2.1 System Model 9 2.2 Performance Analysis Based on Estimated Channel Gains 12 2.3 Approximation of Average BER Performance 17 2.4 Numerical Results 28 3 STD in Time-selective, Spatially Correlated Fading with Perfect Channel Estimation 33 3.1 System Model 33 3.2 Performance Analysis 35 3.3 Average Performance Approximation 38 3.4 Numerical Results 44 4 Conclusion 48 4.1 Main Thesis Contributions 48 4.2 Topics for Further Study 49 Appendix A Derivation of the Means and Variances of Random Variables in (2.7) 50 Appendix B Derivation of the Variances and Correlation Coefficients of Matrix Transformed Spatially Correlated Fading 54 Appendix C Derivation of the Means and Variances of Random Variables in (2.55)57 Appendix D Derivation of the Correlation Coefficient Between Time-selective, Spatially Correlated Channel Gains 60 Appendix E Linear Transformation of Jointly Gaussian Random Variables 62 Appendix F Derivation of the Mean and Variances of Random Variables in (3.13) 66 Appendix G Derivation of the Variances and Correlation Coefficients of Matrix Transformed Time-selective, Spatially Correlated Fading 68 Appendix H Distribution of Hermitian Quadratic Form 71 Appendix I Derivation of Channel Estimation Correlation Coefficient as a Function of SNR 76 Bibliography 77 List of Figures Fig. 1.1 STD scheme 4 Fig. 1.2 STD in time-selective fading with imperfect channel estimation 6 Fig. 2.1 Comparison of approximate analytic BER to simulation results 28 Fig. 2.2 Comparison of approximate analytic BER to results in [ 1 2 ] 29 Fig. 2.3 Simulated BER curves as a function of average SNR for different (pe,ps) values 30 Fig. 2.4 Approximate analytic BER curves as a function of average SNR 32 Fig. 3.1 Comparison of approximate analytic BER to simulation results 44 Fig. 3.2 Simulated BER curve as a function of average SNR for different (p,,ps) values 45 Fig. 3.3 Simulated BER curve as a function of average SNR for different (p,,ps) values (zoomed) 46 List of Acronyms and Symbols Acronyms BS BER BPSK DF ML MRC MS OTD pdf SNR STD STS STTD ZF Base Station Bit Error Rate Binary Phase Shift Keying Decision Feedback Maximum Likelihood Maximal Ratio Combining Mobile Station Orthogonal Transmit Diversity Probability Density Function Signal to Noise Ratio Simple Transmit Diversity Space-Time Spreading Space-Time Transmit Diversity Zero Forcing vii Symbols X Wavelength s0, sl Transmitted bits r0, rT Received signals at time 0 and time T G, Channel gain from transmit antenna i to the receive antenna g, Sample of G, G;J Channel gain from transmit antenna i to the receive antenna at time j gu Sample of G0 Hi Estimated channel gain from transmit antenna i to the receive antenna hj Sample of Hi Hjj Estimated channel gain from transmit antenna i to the receive antenna at time j hy Sample of H{j Z, Channel estimation error on channel i z. Sample of Z, n0, nT Samples of noise at time 0 and T s0, ?, Output of STD combiner a2G Variance of the channel gain o2H Variance of estimated channel gain o\ Variance of channel estimation error a2D Variance of G, and G 2 given hx and h2 (j2w Variance of GlT and G 2 0 given gi0 and g1T <72N Variance of noise N pe Correlation coefficient of G, and Ht ps Spatial correlation coefficient pt ... Time-selective correlation coefficient pd . Correlation coefficient of G, and G 2 given h{ and h2 pw Correlation coefficient of G i r and G 2 0 given g10 and g Re(?,) Decision variable for decoding of si Pe Conditional error probability Pf Average error probability C G Covariance matrix of G C H Covariance matrix of H C Covariance matrix of (H, G) X H Vector (#,, H2) pH)iH2(hx,h2) Joint pdf of Hx, H2 X Vector (#,, H2, G,, G 2 ) PH^H^GS^^S^gi) Jointpdfof /Y,, /7 2 , G, and X T Transpose of X Aj Eigenvalues of C H fM(ju) pdf of M T Transformation matrix Acknowledgement I would like to express my sincere thanks to my thesis supervisor, Dr. Cyril Leung, for his attentive guidance, constructive suggestions and continuous encouragement. His valuable comments and encouragement were important to the completion of this work. I would also like to thank my parents, Mr. Yongfa Zheng and Mrs. Jun Hu, for their respect and understanding of my decision to return to school and for their constant love and encouragement during these two years. They have also played an important role in the completion of this work. This work was partially supported by NSERC Grant OGP0001731. 1 1 Introduction The first and second generation (1G and 2G) cellular systems have enabled wireless voice communications. However, the data services in 1G and 2G systems are limited mostly to text messaging. Besides voice service, 3G systems are supposed to support higher data rates and make it possible to offer enhanced services such as web browsing, transmission of high quality images and videos, etc. Due to the nature of the voice and text messaging services, the requirements for downlink and uplink capacities are similar. However, the battery life of mobile stations (MS) restricts the maximum power at which a MS can transmit and results in a poor uplink than downlink. As a result, many enhancements have been introduced on the base station (BS) side, such as increasing the BS receiver sensitivity or exploiting receive diversity at the BS to improve the uplink. In 3G systems, the services are more data-centric than voice-centric. Most of the services, such as web browsing, picture downloading and video downloading, require more downlink capacity than uplink capacity. However, in 3G systems, the uplink data throughput is higher than that of the downlink [ 1 ]. The data throughput on the uplink in macro cells is typically 1,040 kbps whereas that on the downlink is only 660 kbps. Therefore, improving the downlink capacity becomes more important in 3G systems. Because of the power, size, weight and cost limitations on the MS side, improving receiver sensitivity or implementing receive diversity to improve the downlink may not be practical. 2 However, on the BS side, since receive diversity has already been widely deployed, there are usually two receive antennas installed on the BS side. We can achieve transmit diversity on the downlink by duplexing the downlink transmission to the receive antennas. Since a BS can serve hundreds to thousands of MSs, the use of transmit diversity at the BS is a more cost-effective solution to improving downlink qualify. Much research work including techniques such as time diversity, frequency diversity, polarization diversity" [ 2 ], space-time coding [3], orthogonal transmit diversity [4], time switched transmit diversity [ 5 ], selective transmit diversity [ 5 , 6 ] and transmit adaptive array [ 4 ] has been carried out in order to achieve high-speed and reliable data transmission using transmit diversity. Some of these technologies have been proposed for 3G evolutions [ 7 ]. Previous works on transmit diversity can be classified into two categories: open loop diversity and closed loop diversity." Closed loop transmit diversity relies on feedback information from the MS while open loop transmit diversity does not use feedback information. Generally, the performance of closed loop transmit diversity is better, as the channel state information can be used to calculate optimal transmit weights, which makes it possible to maximize the desired received signal power at the desired MS and minimize the interference to other MSs. However, closed loop diversity requires the MS to send back channel information and this requires extra signalling overhead. Open loop diversity does not have this requirement. It is a "one size fits all" approach. 3 The advantages of this kind of diversity are two-fold: signalling overhead is lower and the MS receiver complexity is relatively low. Some of the open loop transmit diversity techniques, such as orthogonal transmit diversity (OTD), space-time spreading (STS) and space-time transmit diversity (STTD), have already been adopted in 3G standards. STTD has been included in the 3GPP standard [ 8 ] while the other two methods, OTD and STS, are part of the 3GPP2 standard [ 9 ]. STS is a variation of STTD. In STTD, the symbols are transmitted over two time slots using a single Walsh code; whereas in STS, the symbols are transmitted over a single time slot using two Walsh codes [ 1 0 ] . In STTD, the symbols are transmitted using the simple transmit diversity (STD) scheme proposed by Alamouti [ 1 1]. STD is well-known for its simplicity in decoding. It has been the subject of many studies with some focused on the BER performance of STD in different channel conditions. These studies include the performance of STD with imperfect channel estimation, STD in time-selective Rayleigh-fading channels and STD in spatially correlated Rayleigh fading [ 1 2 , 1 3 ] . They show that the performance of STD generally degrades as channel estimation errors, time-selectivity and spatial correlation increase. In [ 1 3 ], different detection strategies, such as maximum-likelihood (ML), decision-feedback (DF) and zero-forcing (ZF), are used to assess the BER performance assuming perfect channel estimation. The results show that the ML detector significantly outperforms the other two detectors. 4 In this thesis, we analyze the performance degradation of STD with the receiver structure in [ 1 1 ] in different channel conditions. 1.1 STD Scheme Transmit Antenna 1 Noise Channel Estimator h, Combiner Decision Device . w w Transmit Antenna 2 Fig. 1.1 S T D scheme In the STD scheme, two information bits s0 and s] are sent simultaneously by transmit antennas 1 and 2 in two consecutive bit periods. It is assumed that the bandwidth of the signal is narrow compared to the channel coherence bandwidth and the channel coherence time is much larger than T, so that the channels can be considered as non frequency-selective and non time-selective [ 1 4 ] . In the first bit period, s0 is sent by 8x 82 V + n0 * — * * * Jr. .82 ~8i_ nT transmit antenna 1 and s{ is sent by antenna 2; in the second bit period, - s* and s*0 are sent by transmit antennas 1 and 2 respectively. The signals r0 and rT at the receive antenna in these two bit periods can be expressed as (i.i). where g,, g 2 are samples of the channel gains from the two transmit antennas to the receive antenna and n0 and nT are samples of thermal noise and interference at the receive antenna at time 0 and time T. After both signals are received, the receiver combines the received signals using the estimated channel gains. It is assumed in [ 1 1 ] that the channel estimation is perfect, i.e., h\ = g, and h2 - g2. Then the combiner can generate two combined signals s0 and ?, as 8 \ 8 2 ° (1.2) .82 -8ijlrT. Substituting (1.1) into (1.2), we obtain — ^ - I |2 I |2 \8i\ +\82\ 0 0 'so' + 81 82 n0 |2 1 |2 * * 8i\ +\82\ _ .82-81. (1.3) The combined signals ?n and are then sent to a maximum likelihood detector to recover the original bits s0 and s{. It is shown in [ 1 1 ] that with perfect channel estimation, STD has the same BER performance as 2-branch MRC for a fixed value of the radiated power per transmit antenna. 1.2 Generalized STD Expression In the original STD scheme, several assumptions are made: (1) the fading channels from the two transmit antennas to the receive antenna are spatially uncorrected; (2) each channel is frequency flat and non time-selective; (3) the channel estimator provides perfect channel estimations. To better reflect conditions in a real system, the.following model changes are introduced: Transmit Antenna 1 Channel Estimator h i : " : -Combiner Decision Device V . , ! ' ^ : : * Transmit Antenna 2 Fig. 1.2 S T D in time-selective fading with imperfect channel estimation Due to time-selectivity, G> and G 2 may be different at time 0 and time T. To reflect this, they are denoted by G 1 0 , G 2 0 , GlT and G2T respectively. Due to spatial correlation, G I 0 and G 2 0 , GlT and G2T may not be 7 independent. The corresponding estimated channel gains are denoted by Hl0, H20, H1T and H2T, Following the STD scheme, we can rewrite (1.1) and (1.2) as 8io 820 V + "n0~ * • — * * * Jr. _8lT nT ~?o~ A o h2T ' _n20 -h * JT. (1.4) (1.5) By substituting (1.4) into (1.5), we get the general expression for the signals at the output of the combiner as _ nio8io n2r82T nw820 n2TglT n20810 ~ niT82T n2o820 ~*~ nvr8\T + h w h2T h*20-hlT (1.6) Depending on various conditions, (1.6) can be simplified to different forms which can be used for signal detection and system performance evaluation. In the case of non time-selective and perfect channel estimation, we can rewrite (1.6) as 0 2 2 Si + # 2 0 2 2 81 + 82 po + * 81 82 n0 * u. 8*2 nT (1.3) which is the same expression as in [ 1 1 ]. The BER for spatially uncorrelated channels is given in [ 1 1 ] and the BER for spatially correlated channels is given in [ 1 2 ]. In the case of time-selective, spatially uncorrelated fading with imperfect channel 8 estimation, the expression of the signals from the combiner is the same as (1.6). In this case, the actual and estimated channel gains of channel 1 are independent of those of channel 2. Although the inter-channel interference term (h*wg2Q -h2Tg*lT)sl and (^20c?io - n\T§*2T)so a r e non-zero in contrast to (1.3), each of the four product terms within the parentheses are products of two independent random variables. So, the BER performance can be evaluated exactly as in [ 1 2 ]. For STD in non time-selective, spatially correlated fading with imperfect channel estimation and STD in time-selective, spatially correlated fading with/without channel estimation errors, the BERs are more difficult to obtain because the terms within the parentheses are products of two dependent random variables. In this thesis, we study the performance for two cases: (1) STD in non time-selective and spatially correlated fading with channel estimation errors; (2) STD in time-selective and spatially correlated fading with perfect channel estimation. 9 2 STD in Non Time-selective, Spatially Correlated Fading with Imperfect Channel Estimation This chapter evaluates the performance of STD with BPSK modulation in non time-selective, spatially correlated Rayleigh fading with imperfect channel estimation. The variance of any complex Gaussian random variable, i.e., X, will be defined as the variance of either its real or imaginary component, denoted as o \ in this thesis. 2.1 System Model In the original STD scheme, the channel gains from two transmit antennas are assumed independent. Theoretically, the channels are spatially independent if the antenna spacing is greater than A/2. However, in reality, an antenna spacing of 50A and 100/1 are necessary at the BS [1 5 ]. If the antennas are allocated too close to each other, the channels can no longer be considered independent. In this section, we discuss the performance of STD when the channels are spatially correlated. Similar to the original STD scheme, we use two transmit antennas and one receive antenna, but here the two transmit antennas are very close to each other. We denote the gains of the two diversity paths as Gj and G 2 , which both are zero mean complex Gaussian random variables with variance aG . Gx and G 2 are spatially correlated with correlation coefficient ps. As to [ 1 6 ], ps is defined as 10 p, = (2.1) 4E[\ GX I2 ]£[| G 2 | 2] Consequently, the covariance matrix of GX and G 2 , CG can be expressed as ol Ps°l (2.2) In the original STD scheme, the channel gains are known. However, in this model, the channel gains are unknown. They have to be estimated by channel estimator from the received signals. We denote the estimations of the channel gain G, and G 2 as H{ and H2, where Hl and H2 are zero mean complex Gaussian random variables. Each pair of Hi and G, are correlated. Following [ 1 7 ], we define Ht as ht = g,+z,, where Z, is the channel estimation error. Z(. is a zero mean independent complex Gaussian random variable with variance o\ and independent of G , , i.e., It can be shown that the variance of/7, is rj 2 = o 2G + 6\. The correlation coefficient between G,. and Hi is defined as pe, where E[Z,. Z* ] = E[G,. z ; ] = E[G,. Z] ] = 0 ( i , j =1, 2 ) (2.3) Pe = £[G,.(G,.+Z,.)'] V(2cr2)(2tT2) (2.4) Therefore 11 o2H=o2Glp2e (2.5) and a\ = (1/'p2 -Y)02G The covariance of Hl and H2 is expressed as E[ Hl Hi ]=E[(Gl+Zl)(G2 + Z2)*] = 2ps a2G Now we can get the covariance matrix of Hi and H2 as C H = .Ps°C PsVG Pi (2.6) (2.7) (2.8) Similarly, we can prove E[G, H*2]=E[Gl(G2 + Z2)*] = 2ps <j2G E[G 2 H'l]=E[G2(G1+Z1)'] = 2ps o2G (2.9) (2.10) Consequently, we can write the covariance matrix of HY, H2, G[ and G 2 as C = p] PsOl ol PsOl Pi PsOl ol °l PsOl ol PsOl PsOl PsOl ol (2.11) If we rewrite (1.16) according to this model, we have 12 KS\+Ks\ h*g2-h2gl KSx-Kg'i h2g2+hxg\ K k2 n0 + h*2-hx * (2.12) 2.2 Performance Analysis Based on Estimated Channel Gains Since H{ and H2 are zero mean complex joint Gaussian random variables, based on their covariance matrix (2.8) we can write the joint pdf of H, and H2 as [ 1 8 ] l- exp(-±xlc-JxH) 2;r(detCH) / 2 2 (2.13) where X 7 , = [H{ H2] is the transpose of X H , a 2x1 column vector of random variables HX and H 2 . Similarly, we can write the joint pdf of / / , , H 2 , G, and G 2 as 1 PH,,H2,G,,G2(h\>h2'8l'82^ ~ (2.14) (2;r)2(detC) / 2 where X T = [//, H2 GX G 2 ] is the transpose of X , a 4x1 column vector of random variables H X , H 2 , Gj and G 2 . Now that we know the joint pdf PH],H2,G„G2(ni'n2'8\'82) M & PH^HSK'K)> W E C A N write the joint pdf of G, and G 2 given HX = / i , and 7/2 = h2 as [ 1 9] (hl,h2,gl,g2) PG,G2\H,,HS8gi82 ^ 1 ^ 2 ) — PH],H2 (Kh) 13 27ta2Dyll-l :exp (81 ~ m i ) ~2Pd(8i-"Q(g2 -m 2 ) + (g 2 -m 2 Y 2c72Dd-p/) (2.15) where \-p]pl (2.16) p'e[ps(\-pie)K+(\-P:P:)h2] l-plpt (2.17) 2 a-p;)(i-psve2) <7D = 2 „4 (2.18) Pd = P,<\-P:) 2 „ 2 i-p;p (2.19) This joint pdf is in the form of a bivariate Gaussian pdf given by [ 1 8 ]. Hence, given Hx = hx and H2 = h2, Gx and G2 are correlated complex Gaussian random variables with means m, and m2, variance <Jp and correlation coefficient pd. Following the conditioned pdf, we can express gx, g2 by hx, h2 as g, = mx + dx = ahx + bh2 + dx g2 = m2 + d2 = bhx + ah2 + d2 (2.20) 2 2 2 2 2 where a = ^e ^s^e , b = ^e^s „ ."' and D,; D7 are zero mean correlated complex Gaussian random variables with variance cr^  and correlation coefficient pd. According to STD scheme, the signals received at time 0 and time Tcan be expressed as: 14 r0 = gls0 + g2si+no ( 2 2 1 ) where n0 and nT are samples of channel noises, which are zero mean independent complex Gaussian random variables with variance o2N . After receiving r 0 and rT, the signal can be decoded based on the value of s0 and sx, where 5 0 =Vo+Vr ( 2 2 2 ) When the real part of s0 is greater than 1, s0 = 1 is selected; otherwise, s0 =-1 is selected. The same decision rule applies to the decoding of B y using (2.19) ~ (2.22), we can write sQ as ~ | [2 I |2 * s0 = (as0 + bsx)\hx\ + (as0-bsx)\h2\ +2bs0hih2 + h*(dxs0 + d2sx) + h2{d*2sQ - dx*sx) + hxn0 + h2nT (2.23) Since s0 and sx are either +1 or -1 with equal probability, the chances of sx = s0 and sx = —s0 are equal. Therefore, we can calculate the B E R of S T D as Pe = \(Pe,sr-s0 + Pe,sr-s0) (2-24) When s0 = sx, the decision variable from (2.23) can be expressed as Re(? 0 ) = [(a + b)\hx\2 +(a-b)\h2\2 +2bRe(h*h2)]s0 • + Re[hx(dx + d2)]s0 + Re[h2(d*2 -d\)]s0 15 + Re[hx n 0 ] + Re[h2nT ] (2.25) When hx and h2 are given, it is shown in Appendix A that Rc[h* (Dx + D2 )]s0, Re[h2 (D*2 - D*x )]s0, Re[/i*Af0] and Re[/z2A^] are zero mean independent Gaussian random variables with variances 2(1 +pd)Gl\hx\2 , 2(1-pd)al\h2\2 , G2N|/z,|2 and c r 2 | / i 2 | 2 respectively. Thus, Re(?0) is the sum of [(a + b ) ^ 2 +(a-b)\h2\2 + 2bRe(hfh2)]s0 and a zero mean independent Gaussian random variable with variance [2(1 + pd )G2D + G2n ]\hx | 2 + [2(1 - pd )G2D + G2n ]\h2\2. When s0 = sx = 1, there is an error if the decision variable is less than 0. Thus, we can express the error probability as = Q f (a + b)\hx\2+(a-b)\h2\2+2bRe(hxh2) ^ ^[2(1 +Pd)G2D +G2N)\hl\2 +[2(1-pd)G2D +G2N ]\h2\2 (2.26) When s0 = sx = - 1 , and if the decision variable is greater than 0, there is an error. So, the error probability can be expressed as f -[(a + b)\h§+(a-b)\h2^+2bRe(hlh2)\ ^ ^[2(1 + P d )G2D +G2N}\h$+ [2(1 - P i )G2D + G%\h§ = Q (a + b)\hx\ + (a-b)\h2\ + 2bRe(hxh2) ^[2(1 +Pd)G2D +G2N ]\hx\2 +[2(\-Pd)G2D +G2N ]\h2\2 (2.27) This is the same as (2.26). Similarly, when sx =- s0, we can get the decision variable as Re(?0) = [(a-b)\hx\2+(a + b)\h2\2+2bRe(h*lh2)]s0 + Re[hx (dx -d2)]s0 + Re[h2(d2 + dx )]s0 16 + Re[hx n0 ] + Re[h2nT ] (2.28) and the corresponding error probability as P = Q f I 12 I ,2 . ^ (a-b)\hx\ +(a + b)\h2\ + 2bRe(hxh2) j[2(l-pd)cr2D +CJ2n ]\hx\2 +[2(\ +Pd)(j2D + <72N ]\h2\2 (2.29) Therefore, given estimated channel gains hx and h2, we can write the error probability as P = Qi (a + fc)|/i,|2 +(a-Z?)|^2|2 +26Re(/z,*/z2) [^2(1 + )<x2 + C T 2 | 2 + [2(1 - pd )a2D + G2N ]\h2 +Q (a - b)\hx |2 + (a + 6)|/i 21 2 + 2b Re(hxh2) [^2(1 - P D ) C J 2 d + a2N ]\hx\2 + [2(1 + pd )a2D + CJ2n ]\h2 (2.30) By the same method, we can derive the conditional error probability for sx. It is exactly the same expression as (2.30). Hence, for given hx and h2, the conditional error probability of STD in non time-selective, spatially correlated fading with channel estimation errors can be expressed as (2.30). Whenever we collect a pair of estimated channel gains from the channel estimator, we can calculate the error probability by (2.30). In case of ps = 0, no spatially correlated fading, we can have a = p2, b = 0, pd =0 and o2D = (1 - p])o2G. So (2.30) reduces to 17 P. = Q Pe \2O2D+CJ2N (2.31) v It is the same result as in [ 1 2 ] for STD in non time-selective, spatially uncorrelated fading with imperfect channel estimation. In case of pe =1, with perfect channel estimation, we can have a = I, b = 0, pd = 0 and a2D =0. Then (2.30) reduces to which is the same result as in [ 1 2 ] for STD in non time-selective, spatially correlated fading with perfect channel estimation.. We can see from (2.31) and (2.32) that, given estimated channel gains hx and h2, the introducing of the channel correlation will not affect the BER performance; however, the introducing of the channel estimation error will degrade the performance. 2.3 A p p r o x i m a t i o n of A v e r a g e B E R P e r f o r m a n c e Normally, given estimated channel gains Hx = hx and H2 = h2, if we know the error (2.32) probability Pe and the joint pdf of Hx and H2, p (hx,h2)[2 0 ], we can evaluate the average error probability over the fading channels as (2.33) 18 Since the error probability expression Pe in (2.30) has the terms with | hf \, \ h\ | and Re(/i,*/i2) inside the Q-function and Hl and H2 are jointly Gaussian, this makes it difficult to calculate the overall BER performance. To simplify the expression, we use the transformation technique discussed in [ 2 1 ] to convert the two correlated Rayleigh fading channels into two independent Rayleigh fading channels, then use the new channels to evaluate the performance of the model. Since we consider only BPSK modulation here, we can rewrite (1.1) as 82 + n0 Jr. .82 'Si. nT (2.34) and write the estimated channel gains as V Si +*i h2 .82+z2_ (2.35) According to [ 2 1 ], if we define the transformation matrix T as V2 41 2 2 Jl '2 2 . (2.36) then apply it to (2.34) and (2.35) 42 42 2 2 fl 2 2 . 42 42 2 2 Jl fl 2 2 J 81 82 + 42 42 2 2 _42 42 2 2 . 42 42 2 2 Jl fL 2 2 ' 42 42~ 8i +zx h2 1 2 Jl fL . 2 2 . .82 + z2. (2.37) (2.38) After simplifying (2.37) and (2.38) to the same form as (2.34) and (2.35), we can write 19 V 83 84 V + n3 zzz / 4 . _#4 " ~ ^ 3 . (2.39) h3 ~83+z3 h2 _84+z4_ (2.40) where fading channels 3 and 4 are new channels generated from the matrix transformation. Accordingly, R3, R4, G 3 , G 4 , N3, N4, H3, H4, Z 3 and Z 4 are new received signals, new channel gains, new noises, new estimated channel gains and new channel estimation errors. We can express them as r3 = ^ (>"0+>V) 83 = ^(81+82) 84 2~^2 8\) n3 = — (n0+nT) 42 , x h3 = ^ ( h 1 + h 2 ) K = : ^ ( h 2 - n i ) Z3 = ^-(Zi+Z2) Z4 2~(^ 2 Zj) (2.41) (2.42) (2.43) (2.44) (2.45) (2.46) (2.47) (2.48) (2.49) (2.50) Since Gx, G 2 , A/ 0 , NT, Hx, H2, Zx and Z 2 are all zero mean Complex Gaussian 20 random variables, it is clear that the sums of these random variables, G 3 , G 4 , N3, N4, H3, H4, Z 3 and Z 4 are also zero mean complex Gaussian random variables. It is shown in Appendix B that they are statistically independent with variances of (1 + ps )<7G , (l-ps)crG, a2N, <J2n, (-\ + ps)a2G, (\-Ps)a2G, a\ and o\ respectively. It is P e Pe also shown in Appendix B that the correlation coefficient between new channel gains G, and new estimated channel gains Hi are 1 + Pe3=Pe,- (2-51) V + PsPl Pe< = P e , T ] — ^ (2-52) Pe Because the new channels are independent, the new channel gains G, can be expressed exclusively by its channel estimations Hi. They can be written as 83 = P ^ h 3 + d3 (2.53) 84 = plK + d4 (2.54) where D 3 and D4 are zero mean independent complex Gaussian random variables with variances '~o2m = (1 -p 2 3)<7c 3 and a2D4. =. (l-p24)aG4. Both are independent of H3 and H4. Now we have converted two correlated channels G, and G 2 into two independent channels G 3 and G 4 . We can use the same performance evaluation method to evaluate the performance of these new channels. By using the method for getting (2.23), we can 21 prove the new combined signal ?0 is + hl(d3s0 + d4s{) + h4(d*4s0 -d*3sx) + h*3n3 +h4n*4 (2.55) Then, as shown in Appendix C, the decision variable Re(?0) is the sum of pl3s0\h3\2 + pl4s0\h4\2 ±(p2e4 - p2e3)sx • Re(h*3h4) and a zero mean Gaussian random variable with variance (\h3\2 + | / * 4 | 2 ) ( C T 2 3 +<J2D4 + <72n)±2R&(h3h*4)(a2D4 -<J2D3). We can write the conditional BER as P. = Q P r i N +PIA\K\ + ( P 2 4 - P 2 3 ) - R e ( ^ > 4 ) h32 + h4 2)((72D3 +<T2D4+<T2N) + 2Re(h3h*4)(cr2D4 -a2D3) +Q f 2 2 ^ P r f N +PeM - ( P 2 4 - P 2 3 ) - R e ( ^ > 4 ) • + h4 )(cx2D3 + a2D4+a2N)-2Re(h3h*4)(a2D4 -a2D3) f Q L V f +Q Pert + PcSt + (PeA ~ Pa )V4 COS(03 -94) 4(r2 + r2)(a2D3 +CT2D4+CT2N) + 2r3r4 cos(03 - d 4 ) ( ( 7 2 D 4 -o\3) P*rl + Pl^l ~ (P24 ~ P « 3 V 4 C 0 S ( ^ 3 ~ ^ 4 ) v W + rl ) ( ^ D 3 + °D4 + °2N ) - 2r3r4 cos(03 - B4 )(o2D4 - CJ2D3 ) (2.56) Here we rewrite the estimated channel gain as hi = riexp(jdi) = rjcos(8i)+jrt 5111(61), i = 3, 4, where rt is Rayleigh distributed and 9i is uniformly distributed in (0, 2n). Because H3 and H4 are independent, we can write the pdf separately as 22 2(7 ^ 3 ( ^ 3 ) = —he xP 2al (2.57) 1 In (2.58) where 1 ak =(— + Ps)°c Pe ak = ( - T - A ) ^ c Pe (2.59) Then we can write the average BER as Pf = JT JT \T \l" P< PRJr3)pR/rJp0/63)p9f9Jd03d83dr3dr4 (2.60) As 83 and 84 are independent, we use 8 = 63-64 to reduce the average BER as p/=rrrp«y«3(r'^(r4^^ ( m ) Although we eliminate one integral in (2.61) and H3 and H4 are independent compared to Ht and H2 in (2.33), we still have to perform a triple integral in (2.61). It would be better if we can eliminate all integrals and get a closed-form expression for Pf. Although the method described in [ 1 6 ] can diagonalize a Hermitian quadratic form in complex Gaussian variables and then get the corresponding pdf, it is shown in Appendix H that a closed-form expression is difficult to obtain. 23 We noticed that the STD scheme gives the best performance when two channels are spatially independent. As a result, in a real system, all efforts will be made to minimize the spatial correlation between channels to achieve the best performance. In that case, the performance of STD over small spatial correlation is more significant than that of big correlation. In the case of small spatial correlation, that is, when ps « 1 , we have found that p e 3 and P e 4 are very close to pe. Although the actual condition for peJ and pe4 to equal pe is ps = 0 or pe =1, the difference between pej and pe is very small when ps « 1 or pe » 0. Therefore, to simplify the analysis, when the spatial correlation is very small, i.e., P s « l or the channel estimation is almost perfect, i.e., P e » 0, we use approximations to complete the analysis. Therefore, we rewrite (2.51) and (2.52) as (2.62) (2.63) and, (2.53) and (2.54) as (2.64) g 4 ~ PeK+d4 (2.65) Consequently, the variances of / / , , H2, D 3 and D4 can be simplified to 24 2 _ ° G 3 U H 3 ~ 7 (2.66) CT 2 = u HA 1 (2.67) (2.68) (2.69) By substituting (2.62) and (2.63) to (2.55), we can rewrite (2.55) as ? 0 ~ / ^ e - ^ O C|^ 312 +|^4|2) + ^3*(^ 3'S'o + ^4^1 ) + / l 4 (^4*^0 ~ ^3 ^ ) + ^3 "3 + ^ 4«I (2.70) Now the decision variable Re(?0) is the sum of p2(|/*3|2 + | /J 4 | 2 )S 0 and a zero mean Gaussian random variable with variance (|/*3|2 +|/z4|2)(2<72 + C T 2 ) + 4 p s ( l - p2e)<JG Re(h3h*4) ~ (\h3\2 +|/i4|2)(2cr2j +cr 2). Thus, we can write the conditional error probability given h3 and h4 as Pe = Q ^ 2 / 1 , |2 I, | 2 . ^ ^ 3 | 2 + | / I 4 | 2 ) ( 2 C T 2 + C T 2 ) = Q \P^H+\K\ ) 2aD + aN = Q(^2K(\h3\2+\h4\2) (2.71) where 25 K = 2(2<j2D+o2) 2[2(l-p 2 )cr 2 +CT^] (2.72) ju = K(\h3\2+\h4\2) (2.73) We know that after approximation, H3 and H4 become zero mean independent complex Gaussian random variables with variances 2a2G31p2 and 2oG41p2e . We can write the covariance matrix of H3 and H4 as C H34 ^ G 3 P\ 0 ^ G 4 Pi (2.74) If the pdf of M = K(\H3 2 + H4 2) is fM (//), then from [ 1 6 ] , its Laplace transform can be written as P(S) = ff—L_ (2.75) where r,. = 2KAt, At are the eigenvalues of (2.74) as * 3 = (1 + P S P 2 ) C T 2 (2.76) ^4 = (2.77) From [ 2 2 ] fM(ju) can be written as /«G") =Y,d J exp(s jfi), a>0 (2.78) 26 where dj are the poles and Sj are the residues of (2.75). Then the average error probability can be calculated as Pf = Q(^2pT)fM(p)dp (2.79) By using [2 3, 2 4 ], we can reduce (2.79) to P f 2(r 3-r 4) l + n i + r„ (2.80) where T 3 = 2 ^ = p2(l + psp2)cj2. 2(\-p))o2G+o2N (2.81) T4 = 2KA4 p2(l-psp2)<j2 2(\-p2e)e2G+o2N (2.82) This is the approximate BER for STD with spatially correlated fading and channel estimation error when ps « 1 or pe » 0 . In the case of pe =1, (2.81) and (2.82) reduce to (1 + P J C T 2 (2.83) (2.84) The result is the same as in [ 1 2 ] for STD in non time-selective, spatially correlated fading with perfect channel estimation. 27 In the case of ps = 0, (2.81) and (2.82) reduce to r = r 1 3 1 4 2 2 2l\-p))o2G+ol (2.85) We can write fM(ju) as [ 1 6 ] fM(ju) = ^ e x p ( - ^ ) A 3 1 3 (2.86) Then (2.79) can be reduced to P<=-•\2 2 + l r 3 i + r , (2.87) This result is exactly the same as shown in [ 1 2 ] for STD in non time-selective, spatially uncorrelated fading with imperfect channel estimation. 28 2.4 Numerical Results 10" Al • A A A 1 A A A A A 1 A A A A A 7 v -. fr. j^-l V V V V " \? y Jf. ^ % —r—^ $ % ^ _ —%— —#— ^ —~^  -01 LU 10' e — — e - — e « — — e -4> p=0.99 p=0.1 Pe=0.99 p=0.7 p=0.99 p=0.9 p=0.99 p=1.0 P =0.9 Ps=0.4 P,f0.9 p=1.0 Pe=OJ ps=0.3 p=0.7 p-1.0 P=0-5 p=0.3 P=0.5 p=1.0 P=0.3 .pM.O Pg=0.1 ps=10 — < H — 4 - A A 10 15 20 Average S N R (dB) 25 30 Fig. 2.1 Comparison of approximate analytic B E R to simulation results (approximation: solid lines; simulation: dotted lines) The approximate and simulated BER curves are plotted as a function of the average SNR, defined as the ratio of the variance of the channel gain to the variance of the additive Gaussian noise, i.e.,• aGJ(72N , for different (pe, ps) values.. As expected from the analysis in Section 2.3, the approximate BER agrees very well with the values from simulation when ps is close to 0 and pe is close to 1. Moreover, the approximate and simulation results are close for all (pe, ps) values plotted. The largest percentage error is about 6% and occurs for (pe = 0.9, ps = 1). For SNR values greater than about 25 29 dB, the approximate and simulated values agree very closely for any (pe, ps) value. .3:10" 0 ; „ A . ...1. •- C\ ' . A1 :::'iiit:u!,..iii.,:,,i; •' :;' A ^ ' ; ~ .. A ' ' i , ! i ' i i ! A . . ' ! L i ; i ' . A • ,-; f . i-- % A. ; .:• \J • - Z. V" ' . . . V — V . - ~\> ... . , . . C\... " " f t . ,-. v . . KT * — — * -" , " V " " — — 10' pe=0-9 ps=10 P=0.7 p=0.3 p=0.7 p=1.0 p=0 5 Ps=0.3 Pe=0 5 p=1.0 Pe=°3 Ps=10 p=0.1 p=1.0 0.5 1 1.5 2 Average SNR (dB) 2.5 Fig. 2.2 Comparison of approximate analytic B E R to results in [ 1 2 ] (approximation: solid lines; simulation: dotted lines; result from [ 1 2 ] : dashed lines) Compared to the BER expression in [ 1 2 ], (2.80) gives the same result when ps = 0 or pe = 1. For other values of (pe, ps), it is shown in Fig. 2.2 that the method discussed in Section 2.3 gives more accurate results. 3 0 10 £ 1 0 CO a: t LU ; ; | 1 0 ' 4 10 ~* * #• -# *- * $ * *--• - X - . = 1.0 = 1.0 """ Pr=0-5 : - e - =m -10 mm .•"'•X--' /3 =0.99 r ft mm: ,. ....<J.. =0.99 Ps'05 !..0.. =0.99 Prr08 - * • • =0.99 p?r10 - X - - P< =0.9 =0.9 ! Ps'05 , - e - p =0.9 Ps=0-8 -*•- Pi =0.9 Ps=10 Pi =0.0 10 15 Average SNR (dB) 20 25 30 Fig. 2.3Simulated B E R curves as a function of average S N R for different (pe, ps) values In the case of spatially correlated fading with channel estimation error, the BER increases as ps increases from 0 to 1 and as pe decreases from 1 to .0.. The BER degradation for pe = 1 - A with ps fixed is larger than for ps = A with pe fixed. As shown in Fig. 2.3, for pe = 1, ps = 0 and a target BER of 10"3, there is about 0.6 dB degradation when ps increases to 0.5 and about 2.1 dB degradation when ps increases to 0.8. The degradation is about 3.2 dB when pe decreases from 1.0 to 0.99; if pe continues to decrease, i.e., to 0.9, the target BER cannot be attained. We can also see from Fig. 2.3 that for each pe value, there is a performance floor which occurs at ps = 0. 31 This is the best BER performance STD can achieve for a given pe value. The exact expression for this BER floor is given by (2.87). It can be-observed from Fig. 2.3 that the spatial correlation influences the BER performance more as the channel estimation error increases. For a target BER of 10 3 and pe =1, the degradation is about 0.6 dB and 2.1 dB when ps changes from 0 to 0.5 and 0.8 respectively. When pe = 0.99, the degradations increase to 1.3 dB and 6.9 dB respectively. The worst BER performance of STD occurs as pe approaches 0. In such a case, the channel estimations become random and the BER approaches 0.5. In the analysis above, for each BER curve, we assume that the channel estimation correlation coefficient pe is fixed. However, the changes of SNR will affect the accuracy of the channel estimation. The influence of SNR to pe varies with different channel estimation models. In a simple model described in Appendix I, we can see that P e = . * -. By using this result together with the analytic results from Section 2.3, V SNR we can plot the BER curves for this model as a function of SNR where pe changes with SNR. 32 10 P S = 0 . . . v . Ps-o:8 . : . 0 Ps=1 o 10 15 20 Average SNR (dB) Fig. 2.4 Approximate analytic B E R curves as a function of average S N R (p = -\=: clotted lines; p„ = , 1 : solid lines) V2 I 1 1 + -SNR Fig. 2.4 shows with the increase of SNR form 0 dB, pe = increases from 1 + -SNR V2 and the performance is continuously improving. The channel estimation becomes perfect when the SNR increases to infinity. 33 3 S T D in Time-selective, Spatially Correlated Fading with Perfect Channel Estimation In this chapter, we analyze the performance of STD in time-selective, spatially correlated fading channels with perfect channel estimation. 3.1 System Model We combine the time-selective fading from [ 2 5 ] together with the spatially correlated fading mentioned before, but without channel estimation error in this model. For each spatial channel, the channel gain is constant over one symbol duration but can be changed in the successive symbol period. We denote the channel gains from two transmit antennas to the receive antenna as G 1 0 , GiT, G 2 0 and G2T. They are zero mean complex Gaussian random variables with the same variance <JG. G 1 0 and G i r , and G 2 0 and G2T are spatially correlated with the correlation coefficient ps. Meanwhile, G 1 0 and G1T , and G 2 0 and G2T are correlated with the time-selective correlation coefficient pt. Appendix D shows that the correlation coefficient between G 1 0 and G2T , and between G 2 0 and GlT are ps p, . Thus, we can express the 2x2 covariance matrix of G 1 0 , G 2 0 , GlT and G2T as 34 < Ps°G P,°l PsPPl Ps°G °2G PsP^l P,°l PsP,02G Ol Ps°2G PsPt°l PPl Ps°G (3.1) We already know the covariance matrix of G 1 0 and G 2 0 is Ps°l PS°G (3.2) By using the linear transformation [1 8 ] of G 1 0 , G 2 0 together with two new complex Gaussian random variables, we can, express GiT and G2T by G 1 0 and G 2 0 . Appendix E shows that we can write them as 8IT = P,8w+V\ ' (3-3) 82T = P,8z0+v2 (3.4) where V, and V2 are zero mean correlated complex Gaussian random variables with variance = (l-p?)crG and correlation coefficient ps. Following the STD scheme, the received signals at time 0 and time T can be written as r0 = 8 WS0 + <?20^ 1 ~*~ n0 (3.5) (3.6) Then the combined signals s0 and ?, can be got from •^ o = 8\oro + 8irRT (3-7) "^ 1 — c?20r0 8\TRT (3.8) 35 Based on the values of the combined signals, we can use maximum-likelihood to decode the information bits s0 and s,. If the real part of ?0 is greater than 0, .s0 = 1 will be chosen; otherwise, s0 = -1 will be selected. Similarly, the same decoding rule applies to recover s,. 3.2 Performance Analysis When there is no channel estimation error, the estimated channel gain. Htj is equal to the channel gain Gtj. Therefore, we can simplify (1.6) as (3.9) Now we consider the case for sQ first. From Appendix E, we can represent G 2 0 and GlT by G 1 0 and G2T as 1 I 2 1 I 2 * * r- - l * - ~n0~ * so = | < ? i o | * + |<?2r| * 81082O 1 I 2 — 8\T82T 1 1 2 + 810 t * ?2T Jl. _<?10<?20 — 8\T&2T |<?2o| + \8\T\ 8IT _ nT 8IT =a8w +b82f+w\ 820 =bgw+ag2T+w2 (3.10) where a = p, b = ps l - p 2 p 2 1-P, 2 1 - P 2 P 2 (3.11) Wx and W2 are zero mean correlated complex Gaussian random variables with variance < = (1-P, 2) ( ——^J-r \jr and correlation coefficient 0 = -p<p,. i-Psp?; 36 By substituting (3.10), s0 in (3.9) can be expressed as ^ .1 2 2, , * * \ * * ^0 = ( | £ l O <?2T )S0 + (<?10<?20 ~ 8\T§2T)S\ + 8\0n0 + §2TNT = (|Slof +|«?2r| 2) J0 +^(klo| 2 -|«?27-|2)*1 +(S 10W2 ~82TWl)S\ + 8wH0 +82THT (3.12) When sl = s0, the decision variable Re(?0) can be written as R e ( ? o ) = [(l + *0|<?io|2 +Q--b)\82r\2]so +Re(g*0w2 -g2Tw*)s0 + Re(g*0n0 + g2TnT) (3.13) Appendix F shows that Re(g*0W2 - g2TWx), Re(g*0A/0) and Re(g2TNj) are zero I 12 I |2 mean independent Gaussian random variables with variances [g 1 0 + g2r -2pw Re(g 1 0g 2 r)]<7 2 , |g 1 0| 2O"^ and I g ^ V 2 , respectively. Thus Re(?0) is the sum of [(l + £>)|g10|2 + (1 —^)|g2r|2],so a n < ^ a n z e r 0 m e a n Gaussian random variable with variance (|<?iof +|<?2r | 2 ) (° 'w +<?l)-2PwRe(g\o82T)(7l • Given the channel gains gl0 and g2T, the conditional error probability can be written as P = Q (l + % 1 0 | 2 + ( l - % 27" V ^ k i o T + | « ? 2 7 - | 2 X<^ + < J l ) - 2 P w R e (c? io< ? 2 r ) ° ' i (3.14) Similarly, when s'x = --s0, the conditional error probability can be written as = Q a-b)\gw\2+(\+b)\g 2T\ V ( k i o | 2 +\82T\2)(PI +&l)-2Pw Re(<?io<?2r)^ (3.15) 37 Because the chances of sx = s0 and s{ = —s0 are equal, given gl0 and g2T, the conditional error bit of s0 can be expressed as Pe,s0 = \ Q\ (l + ^)|g 1 0 | 2+(l-fc)|g 2 r | 2 +-Q 2 * V'(|«ioT +\82T\2)(°1 +o2N)-2pwRe(gl0glT)(7] (l-b)\gl0\2+(l + b)\g2Tf (3.16) J(\8io\2 +\82T\2)(°W +°2N)-2PW ^(8w81T)^W Similarly, given giT and g10, we can prove that the conditional error probability of st is P.* = -Q i 2 (i+b)\g20\2+a-b)\8lT a-b)\g20\2+(i+b)\g IT <J(\8TD\2 +\8IT\2)(.°W +°2N)-2PW Re(82o8lT)^ (3.17) When ps = 0, we have a = pt, b = 0, p w = 0and c 2 = (1 - p,2 )oG. Thus, we can reduce (3.16)to P = Q e,s0 *Z 8w 2 + 82T 2 (1 -p2)o 2 +a2 G N (3.18) This is the same result shown in [ 1 2 ] for STD in time:selective, spatially uncorrected fading with perfect channel estimation. When p,= 1, we have a = 1, b = 0, pw = -ps and 0^=0. Then we can reduce (3.16) 38 to P = Q 2 2 + 82T (3.19) which is the same result shown in [ 1 2 ] for STD in non time-selective, spatially correlated fading with perfect channel estimation. From (3.18) and (3.19) we can see that given the channel gains glQ and g2T, introducing the spatial correlation will not increase the BER; however, introducing the time-selective correlation will degrade the BER performance. V 810 820 V + n0 _<?2r ~ 8IT _ nT 3.3 Average Performance Approximation In case of STD in time-selective, spatially correlated fading with perfect channel estimation, with BPSK modulation, (1.4) can be rewritten as (3.21) As in section 2.3, if we use the transformation technique to simplify the analysis, we can apply the transformation matrix T in (2.36) to both sides of (3.21) and get the new received signals as r/r 1 r 0 0 T c 1 r„ 1 (3.22) #30 840 V + " " 3 " / 4 _ _8AT ~ 83T _ . n 4 . where channel 3 and channel 4 are new channels generated from the matrix transformation 39 of channel 1 and channel 2. Correspondingly, R3, R4, G 3 0 , G 4 0 , G3T, G4T, N3 and A 4^ are random variables for new received signals, new channel gains and new channel noises after the transformation. The samples of the new channel noises are the same as in (2.45) and (2.46). The samples of new channel gains are As in Section 2.3, we can prove that all these new random variables are zero mean complex Gaussian random variables. The new channel noises N3 and N4 are statistically independent with the same variance o 2N . Appendix G shows that the gains of channel 3 and channel 4 are statistically independent. Therefore, the covariance of each channel can be expressed as <?30 <?40 _ V2 <?10+<?27- <?20 8 IT SAT 83T J \_82T~810 8IT + 820 (3.23) (3.24) (3.25) where (3.26) °GA = (I-PSP<)02G (3.27) "'3 1 , 1 + (3.28) 40 P,A = j 1 - ^ ( 3 - 2 9 ) Hence the channel gains can be written as SIT = A 3 < ? 3 0 + v 3 ( 3 - 3 ° ) 840 = P,484T+V4 (3-31) where V3 and V4 are zero mean independent complex Gaussian random variables with variances <*V3 = 0--p,3)^G3 = : G (3-32) l + A A 0V 4 = ( I - P M ) ^ = • ° G (3-33) L~P,P, where V3 and V4 are independent of G 3 0 and GAT. If we substitute (2.30) and (2.31) into (3.9), we get r** I 12 1 12 * * * * ^0 = (|<?30| "Hi) 47" I ^ 0 + (<?30<? 40 _ 8 37 8 4T^S] + 8 30N3 +8 4TN4 = (|g3of + k 4 r | 2 K + (A 4 - A 3 ) < ? 3 0 c?4r^ + ( < ? 3 0 v4 -<?4r v3 )* i +<?3o" 3 +c§4r"I (3-34) As in Section 2.3, if we look at the case of small spatial correlation, we can assume that pl3 and pt4 are very close to pt.. The condition for this assumption is that ps «1 and p t » p s . By using pl3 ~ pl4 ~ p,, we can rewrite (3.30) ~ (3.34) as S 3r = A S 3 o + v 3 (3-35) 41 8 40 ** P,84T+V4 (3.36) (3.37) ? o = (\8w\2 ^84T\2)sv+(8wVA-gATvl)sx+gl0n3+g4TnA (3.38) Because V3 , V4 , A/3 and A/4 are independent, given g 3 0 and g 4 r ' , the BER expressions from (3.38) are the same for all combinations of the BPSK signals sQ and sx. They can be expressed as P. = Q ( | |2 | ,2 ^ |<?3o| "*"|c?47'| yj(CTv+CT2N)(\g30\2 +\gAT\2) = Q 830 84T ov +<yN Q^lK^g^ +\g4T\2)^j (3.39) where K = 1 1 2 (o - , z +^) 2 [ ( l - p , 2 ) o - 2 + ^ ] (3.40) 0 = ^ ( | c ? 3 0 r + k 4 7 - | 2 ) (3.41) We know that G 3 0 and G4T are zero mean independent Gaussian random variables with variances (1 + psp, )crG and (1 - pspt )aG . Thus, we can write the covariance matrix of 42 G 3 0 and G 4 r as 0 0 {\-psp,)a2G (3.42) Its eigenvalues are A3 = (l + psp,)a2c K = (l-PsP<)v2c (3.43) (3.44) By using the same method in section 2.3,. given ps «1 and' pt » ps, we can obtain the approximation of the average BER of STD in time-selective, spatially correlated fading with perfect channel estimation as P f 2(r 3-r 4) L V i+n l -i+r„ (3.45) where I* =2**3 = (l + PsPt)G2c (\-p2)a2G +cr, (3.46) r 4 = 2 ^ 4 = ^ ( 1 - f ; ^ {\-p2)a2G+a2N (3.47) When p = \, (3.46) and (3.47) reduce to r 3 = <X + P,Xrl (3.48) r 4 <X-P,)<rG (3.49) 43 The result is the same as in [ 1 2 ] for STD in non time-selective, spatially correlated fading with perfect channel estimation. When ps = 0, (3.46) and (3.47) reduce to r = r = 1 3 1 4 (\-p2)a2G+cr2N (3.50) As in Section 2.3, we can obtain the exact BER expression as 1- ^ l + r, 2 + l r 3 l + r, (3.51) which is the same as shown in [ 1 2 ] for STD in time-selective, spatially uncorrelated fading with perfect channel estimation. 44 3.4 Numerical Results Average SNR (dB) Fig. 3.1 Comparison of approximate analytic B E R to simulation results (Approximation: Solid Lines; Simulation: Dotted Lines) The approximate and simulated BER curves are plotted in Fig. 3.1 as a function of the average SNR for different (p t, ps) values. As expected from the analysis in Section 3.3, the approximate BER agrees very well with the values from simulation when ps is close to 0 and pt is much greater than ps. Moreover, the approximate and simulation results are close for p < —. 45 15 20 Average SNR (dB) Fig. 3.2 Simulated B E R curve as a function of average S N R for different (p,, ps) values In the case of time-selective, spatially correlated fading with perfect channel estimation, for a fixed ps, the BER increases as p, decreases from 1 to 0.1 When ps is small, for a fixed pt, the BER increases with ps. The BER degradation for pt = 1 - A when ps is fixed is larger than the degradation for ps = A when pt is fixed. As shown in Fig. 3.2, for a target BER of 10"3 for pt = 1, there is about 1.3 dB degradation when ps changes from 0 to 0.7 and about 10 dB degradation when ps changes to 1. For the 1 Although pt changes from 1 to 0, when pt = 0, the channel changes so fast that the gains are independent in two consecutive symbol periods. Thus, when pt is small, the assumption that the channel is constant over one symbol interval is not valid. 46 same target BER for ps = 0, when p, changes from 1 to 0.99, there is about 1.3 dB degradation; when pt changes to 0, the degradation is more than 25 dB. Average SNR (dB) Fig. 3.3Simulated B E R curve as a function of average S N R for different (p,,ps) values (zoomed) When ps becomes bigger, for instance, ps is equivalent to pt or even bigger, from the simulated BER curve, we observe that the BER decreases with ps. However, this occurs only when the average SNR exceeds certain thresholds. As shown in Fig. 3.3, for p, = 0.9, when the SNR is small, the BER for ps =1 is bigger than ps = 0.9; when the SNR increases to about 14.5 dB, the BER for ps = 1 is the same as ps = 0.9. As the SNR continues increasing, the BER for ps = 1 becomes smaller than the BER for ps = 47 0.9 and eventually, becomes smaller than all other smaller ps cases. If we look at the extreme case, i.e., ps = 1, we can write (3.9) as Ji In (3.52), when SNR increases to infinity, the only interference is the inter-channel interference. Compared to (3.9), the signal to noise ratio changes from (|G | 2 + L I 2) 2 (\g \2+\g I 2) 2 . -^P—— to 1 0 2 — — , which is always equal or greater than 1. This I <?io<?2o ~8\T82T I (|<?io| _ |<?i7-| ) 2 explains why for high SNR values, STD performance improves as ps increases. From the simulation results, we also observe that the SNR threshold for performance reversal decreases as pt decreases. For instance, for ps = 1, the SNR threshold for performance reversal is about 14.5 dB when pt = 0.9; however, when pt =0.1, the threshold decreases to about 2.5 dB. k i o f + k i r f | S i o f - | S i r | 2 I I2 I I2 I I2 I I2 <?io| — |<?i7- |c?io| <?ir + * 8 IT * * J l . .810 ' 8 IT _ (3.52) 48 4 Conclusion 4.1 Main Thesis Contributions This thesis presents a performance study of STD in non time-selective, spatially correlated fading with imperfect channel estimation and STD in time-selective, spatially correlated fading with perfect channel estimation. • In the case of STD in non time-selective, spatially correlated fading with imperfect channel estimation, the error probability conditioned on the estimated channel gain is derived. A simple, approximate expression for the average BER over Rayleigh fading is given. A comparison with simulation results show that the approximate is quite accurate over a wide range of (pe, ps) values. The results also show that the channel estimation error has a bigger impact on STD performance than spatial correlation. • In the case of STD in time-selective, spatially correlated fading with perfect channel estimation, the error probability conditioned on the channel gain is derived. A simple, approximate expression for the average BER over Rayleigh fading is given for (ps « 1 and pt » ps). It is found that time-selectivity has a bigger impact on STD performance than spatial correlation. ' 49 • From the results of STD in time-selective, spatially uncorrected fading with channel estimation error in [ 1 2 ], it was found that the channel estimation error has a bigger impact on STD performance than time-selectivity. Combining this with our results, it can be deduced that STD performance is affected primarily by channel estimation errors, secondly by time-selectivity and thirdly by spatial correlation. 4.2 Topics for Further Study • It would be useful to extend the derivation of the conditional error probability of STD to the general time-selective, spatially correlated fading with imperfect channel estimation scenario. Although it is possible to build the system model by introducing channel estimation error in Section 3.1 and using the same method to derive the conditional BER, the derivation of the joint pdf of G 1 0 , G 2 0 , GlT and G2T given Hl0 , H20, HlT and H2T requires a Gaussian distribution that involves 8x8 and 4 x 4 covariance matrices to be solved and is thus awkward to deal with. • A new combing scheme to cancel the inter-channel interference from the temporal and spatial correlation. • A n average BER expression over correlated Rayleigh fading and generalized fading, e.g. Ricean, Nakagami, etc.. 50 Appendix A Derivation of the Means and Variances of Random Variables in (2.7) In Chapter 2, the decision variable in (2.7) is expressed as Re(? 0 )= [(a + b)^2 +(a-b)\h2\2 +2bR6(h*h2)]s0 + Re[h*(dx + d2)]s0 + Re[h2(d*2 -d*)]s0 + Re[h*1n0] + Re[h2nT] (2.7) When . hx and h2 are given, the first term in (2.7) [(a + fr)|/z,|2 +(a-b)\h2\2 +2bRe(h*h2)]s0 is determinate. For the rest, it is a sum of four random variables Re[h*(D, + D2)]s0 , Re[h2(D*2 -D*)]s0 , Re[/z*A 0^] and Re[h2N*T]. It is shown below that the means of ti[(D, + D2), h2(D*2-D*), h*N0 and h2N*T are zero. E[h*(Dl+D2)] = h*lE[(Dl+D2)] = hl\E[D[] + E[D2])=0 (A.l) E[h2(D*2 - D\)] = h2E[{D*2 - D;)] = 0 • ' (A.2) E[h;N0] = h;E[N0] = 0 (A.3) E[h2N*T]=h2E[NT] = 0 (A.4) Thus, the mean of Re[h*(D, + D2)]s0, Re[h2(D*2 -D*)]s0, Re[h*N0] and Re[h2N*T] are zero too. 51 Next we prove that these random variables are independent of each other. As we know, N0 and NT are statistically independent of any random variables, when h{ and h2 are fixed, Re^'TV,,] and Re[h2Nj] are independent and they are also independent of Re[h* (D{ + D2 )]s0 and Re[/z2 (D*2 - D* )]s0 . The variances of Ret^A/J and Re[/z2A^] are E{Re2[h;N0]}=\h?\(J2N (A.5) ^ { R e ^ V V ^ H ^ K (A.6) Now we need to prove that Re[/i* (Dx + D2 )]s0 and Re[h2 (D*2 - D*x )]S0 are independent. Or that Re[/i,* (D, + D2)] and Re[h2 (D2 - D*x)] are independent, as s0 is either+1 or - 1 . Because D and Z)2 are zero mean correlated complex Gaussian random variables, we can express d{ and d2 as dx = +]yx (A.l) d2 = x2 +}y2 (A.8) where , y,, x2 and y 2 are samples of zero mean correlated real Gaussian random variables with variances <J2D and correlation coefficient pd . The variances and covariances can be expressed as E[X2] = E[Y2] = a2D (A.9) 52 E[XtYj] = 0 (1,7 = 1,2) (A.10) E[XxX2] = E[YxY2) = Pdcr2D (A.l l) Then we have dx +d2 = (xx +x2 ) + j(yx +y2) (A. 12) d2-dx=(x2-xx) + j(y2-yx) (A.13) Similarly, if we denote /i, by its real and imaginary parts as ht — U, + jwt, we can have Reft* (dx +d2)] = ux (xx + x2) + w, (yx + y2) (A. 14) Re[h2(d2 -dx)*] = u2(x2 -xx) + w2(y2 - yx) (A. 15) The covariance of Reft* (Dx + D2)] and Reft (D*2 - Dx)] can be expressed as £[Reft*(D, + D 2 ))Reft(D 2 -D,)*)] = £ { [ M l ( X 1 + X 2 ) + w 1 (F 1 +F 2 ) ] [ M 2 (X 2 -X 1 ) + w 2(F 2-y i)]} = 0 (A.16) The result shows that Reft* (Dx + D2)] and Reft (D 2 - £)*)] are independent. The variances of Re[hx(Dx + D2)] and Reft(D 2 -£>*)] can be expressed as £{Re 2 f t*(D 1 + D 2 ) ]} = E[uf(Xx +X2 + 2X XX 2) + w2 (Y2 + Y2 + 2YXY2) + 2uxwx(X x +X2)(Yl + Y2)] = 2(1 + pd)(w2 + w2)a2D = 2(1 + pd)\ h2 I <To (A.17) £{Re 2[/z 2(D 2-D 1)*]} • - - " 53 = E[u\(X2 + X2 - 2 X 2 X X ) + w\(Y22 + Y2 -2Y2YX) + 2u2wx(X2 -X , ) (Y 2 -F,)] = 2(\-pd)(u22+w22)a2D = 2(l-pd)\h2\a2D (AAS) Thus, we have proved that the decision variable Re(?0) is a Gaussian random variable with mean [(a + b)|/i,|2 +(a-b)\h2f +2bRe(h*h2)]s0 and variance [2(1 + pd) \ h2 \ a2D + 2 ( l -p , ) | / * 2 2 |<x 2 ] s0 + (\ hi\ + \h2 \)a2N. 54 Appendix B Derivation of the Variances and Correlation Coefficients of Matrix Transformed Spatially Correlated Fading Originally, G, and G 2 are zero mean correlated complex Gaussian random variables with variance oG and spatial correlation coefficient ps. After matrix transformation, two new zero mean complex Gaussian random variables G 3 and G 4 are generated. From (2.41) and (2.42) we know that • V2 : g3 = — (8i +82) (2.41) 8, = —^82 ~8i) (2.42) We can calculate the covariance of G 3 and G 4 as £[G 3 G 4 ] = E ^ ( G 1 + G 2 ) ^ ( G 2 * - G ; ) ±E\G2\ 2 -\GX\2 +Gfi*2-GlG2] = 0 (B.l) The result shows that G 3 and G 4 are uncorrected and statically independent. We can calculate their variances as < = ^E[G3G;] =X-E ^ ( G , + G 2 ) ^ ( G ; + G 2 * ) (B.2) 55 < = \E[GAG;] =X-E ±-(G2-GX)^-(G2-GX) = (\-ps)<72G (B.3) The original channel noises N0 and NT are zero mean independent complex Gaussian random variables with variance o2N. Based on (2.43) (2.44) V2 « 3 = ^ - K + « 7 - ) (2.43) V2 n4= — (nT -nQ) (2.44) we can calculate the covariance of N3 and N4 as £[/VX] = E = 0 (B.4) They are also uncorrelated and statically independent. Their variances are o2N = -E[N3N;] = -E A j 2 - 2 V2 V2 . • = 0- (B.5) < = \E[NX] = \E V2 V2 . (B.6) Same way, we can prove that Z 3 and Z 4 are zero mean independent complex Gaussian random variables with variance <rz . It can be shown that G, and Z, are independent. EiGtf] = E V2 V2 . ^ - ( G 2 ± G , ) ^ - ( Z 2 ± Z 1 ) = 0 i = 3,4 (B.7) 56 From (2.38) we know that Hi is the sum of G, and Z,, / = 3,4 h3 A. J4+Z4. (2.38) Therefore, the variance of Ht is the sum of the variances of G, and Z ( . That is, J _ - ^ = iiaaL.,2 (B.8) 2 2 0> +cr7 = ( 1 - A ) ^ + l~AP E -2 „ 2 ° C (B.9) The covariance of G, and i / , can be expressed as E[G3H;] = £ [ G 3 ( G ; + Z 3 * ) ] = 2ol (B.10) £[G4H4*] = 2<rl (B.ll) Now we can obtain the correlation coefficients of G. and //, as g[G 3/y 3] _ 1+A A 3 - 1 - P M " ^ (B.12) PE4 = A > J r - ^ 7 (B.13) P - A P 2 57 Appendix C Derivation of the Means and Variances of Random Variables in (2.55) In Chapter 2, the combined signal s0 in (2.55) is expressed as + h*3{d3s0 + d4si) + h4(d*4s0 -d*3sx) + h*3n3 + h4n4 (2.55) The correspondent decision variable is Re(?0). When hx and h2 are given, the first three terms of R e ^ ) are determinate, as p23s^hj\2 + p24s0\h4\2+(p24-p^s^-Reihlh^. The rest part is variable; it is the real part of the sum of four random variables, expressed as Re[hl(D3s0 + D4sx) + h4(D*4s0 -D 3*5,) + h3N3 +h4N*4]. When sx = s0, the variable part can be expressed as Re[/i*(D3 + D4)s0 + h4(D*4 -D*3)s0 + h*3N3 +h4N*4] = Re[(/i3* - hi )D3]s0 + Re[(/z3* + ti4 )D4 ]s0 + Re(/z3*yV3) + Re(h4N*4) (C.l) Because D3, D4, N3 and N4 are zero mean independent complex Gaussian random variables, it is obvious that the mean of the variable part in (C.l) is zero and Re[(/z3 -h*4)D3]s0 , Re[(/*3 +h*4)D4]s0 , Re(/z3;V3) and Re(h4N*4) are statistically independent. As in Appendix A, we write the samples of and D4 as d3 = x3 + j y 3 (C.2) 58 d4 = xA +jy 4 (C.3) where x3, y3, x4, y4 are samples of zero mean independent real Gaussian random variables with variances cr2D3 and a2D4 respectively. If we substitute ht by its real and imaginary parts as ht = w,. + jwt, we have Re[(/** - h\ )d3 ] = Re-fl^ - u4)- j(w3 - w4 )Jx3 + jy3)} = (w 3 - u4 )x3 + (w3 - w4 )y3 (C.4) Re[(h3*+hl)d4] = Re^(u3+u4)-j(w3+w4)Jx4 + jy4)} • • = (u3 +u4)x4 +(w3 + w4)y4 (C.5) The variances of Re[(/i3* -h*4)D3]sQ and Re[(h*y +h*4)D4]s0 are E{Re[(h*3 -h*4)D3]s0}2 = E[(u3 -u4)2X2 +{w3 - w4)2Y2 + 2(u3 -u4)(w3 -w4)X3Y3] = (u2 + w3 +u24 +w24 - 2u3u4 - 2w3w4)CJ2D3 + \h4\2-2Re(h3h*4)]a2D3 (C.6) E{Rc[(h*3 + h*4)D4]s0}2 = E[(u3+u4)2X24+(w3 + w4)2Y2+2(u3+u4)(w3+w4)X4Y4] 2 2 2 2 2 = (u3 + w3 + u4 + w4 + 2u3u4 + 2w3w4)(TD4 = [\h3\2 +\h4\2 .+ 2Re(h3h*4)](T2D4 (C.l) The variances of Re [^A^] and Re[h4N*4] have already been shown in Appendix A, as .2 I „ 2 , I .2 I ^ 2 h2 | <72N and \h2 \a2N respectively. 59 So when sx = s0, the decision variable Re(s0) is a Gaussian random variable with mean [pUh3\2 + pl4\h4\Z+(p2e4 -p2e3)Re(hlh4)]s0 and variance (N2 +M2)(°'D3 + 0 " D 4 + 2Rt(h3h*4){a2D4 -<J2m). By using the same method, we can prove that when s{ = —s0, the decision variable Re(?0) is a Gaussian random variable with mean [p e 2 3|/i 3| 2 + p 2 4 |^ 4 | 2 -( P ^ - P ^ R ^ X ) ] ^ a n d variance (\h3\2 + \h4\2){a2D3 +<J2D4 +<J2N )-2Rz{h3h*4)(o2D4-(J2D3). 60 Appendix D Derivation of the Correlation Coefficient Between Time-selective, Spatially Correlated Channel Gains In this model, the channel gain G 1 0 is correlated with both G 2 0 and G 1 7 . . When setting up the model, we first generate G 1 0 , then use G 1 0 to generate G 2 0 and G i r . Therefore the expression of g20 and glT given gw can be written as 820 = Ps8w+u0 (D- 1) 8 IT = P,8w+Vx (D.2) where U0 and Vl are zero mean independent complex Gaussian random variables with variances o2v = (1 - p] )aG and crv - (1 - p] )crG. U0 and V{ are independent of G w Then the covariance of G 2 0 and G, r is given by E[G20G;T] = E(pSGi0+u0)(p,G;0+v;) = 2psp,G2G (D.3) Thus the correlation coefficients of G 2 0 and G L T can be expressed as E[G20G;T] _ 2psp,a2G = PSP, (D.4) • A/£[|G 2 0! 2]^[|G 1 /.| 2] ' . 42°c • 2al Similarly, if we start with G 2 0 , we can also get the correlation coefficient of G 1 0 and 61 G2T as pspt. 62 Appendix E Linear Transformation of Jointly Gaussian Random Variables From [1 8 ] we know that a linear transformation of a set of jointly Gaussian random variables results in another set of jointly Gaussian random variables. If we have a set of jointly Gaussian random variables, denoted as X, which is a n x l column vector with nx 1 mean vector mx and nxn covariance matrix Cx, by using Y = AX (E.l) where A is a nxn non-singular matrix, we can transform X into a new set of jointly Gaussian random variables Y, which is a n x l column vector with n x l mean vector my and nxn covariance matrix Cy. Correspondingly, the transformation of the mean vector and the covariance matrix can be done by mv = Amx (E.2) C Y = AC X A T (E.3) Where A T denotes the transpose of A. Frist transformation case: In case of representing G i r and G2T by G l 0 and G 2 0 , a set of jointly Gaussian random variables is defined as x T = [£io 820 £1 « 2 1 (E-4) 63 where E, and E 2 are zero mean independent Gaussian random variables with variance <7G . Ej and E 2 are independent of Gw and G 2 0 . By using the linear transformation, we can transform X into a new set of jointly Gaussian random variables ^ — [#10 82O 8\T §27"] (E.5) with the desired means and variances. Because the means of the random variables in our model are all zero, we only need to look after the transformation of covariance matrix in (E.3). From the previous definition, we can write the covariance matrixes of X , Y as (E.6) C Y = Ps<rl 0 0 " ps<yl ol 0 0 0 0 ol 0 0 0 0 < ol PsOl . P,ol PsP,0l ol PsP,0l " P.OI p,o2c PsP,0l ol PsOl PsPVl P.OI PsOl ol Using Maple®[ 2 6 ], we can find a A that complies with (E.3)~(E.5). A = 1 0 0 1 0 0 0 0 0 P l 0 VTPF 0 p, P s ^ p J V(l-A2Xl-A2) If we use (E.8) to rewrite (E.l), we can get 8IT = P,8w+^-Pl (E.7) (E.8) (E.9) 64 SIT = PISIQ +Ps^~P?£\ PsX17A*)e2 (E.10) If we replace the two zero mean independent complex Gaussian random variables E, and E 2 by two correlated complex Gaussian random variables V, and V2, we can rewrite the expression as 8 IT = P,8w+V\ (3-3) SIT = P , S 2 0 + V 2 (3-4) where V, and V2 are zero mean complex Gaussian random variables with variance cr2 = (1- pf)crc and correlation coefficient ps. They are independent of G 1 0 and G 2 0 . Second transformation case: When representing GlT and G 2 0 by G 1 0 and G2T, we write X, Y, Cx and Cy as (E.H) Y T = [g 1 0 SIT Sir £20 ] ( E - 1 2 ) cx = io SIT «1 £ 2 ] IO I  <?ir &PsPt^G 0 0 P,P,°l ° l 0 0 0 0 0 0 0 0 (E.13) < PsP,02G P,<r2G Ps°G P.PPl 02G Ps°G Pt°2G P.OI ps<yc K ° l PsP<°2G Ps02G- Pl°G PsPl°G . <• (E.14) and can get a A as 65 A = 0 1 0 0 0 0 1 0 0 a b c 0 b a d e 0 0 / ( l -p 2 ) ( l -p 2 ) 0 0 • 0 V r i r » (E.15) Thus, we can express GlT and G20 as = Agio + ^ 2 T + « 1 (E.16) (E.17) Similarly, if we replace E, and E 2 by a new pair of correlated Gaussian random variables W, and W2, we can rewrite (E.16) and (E.17) as S i r = agw+bg2T+wx (3.10) §20 = bgw+ag2T+w2 (3.11) where TV, and W2 are zero mean correlated complex Gaussian random variables with variance <7^ = c2aG and correlation coefficient pw = -pspt, 66 Appendix F Derivation of the Mean and Variances of Random Variables in (3.13) In (3.13) we need to evaluate the sum of four random variables, Re(g*l0W2), Re(g2TWx), Re(g*wN0) and Re(g2TNj). We know that W,, W2, N0 and NT are zero mean Gaussian random variables. Therefore, their sum in (3.13) is also zero mean. From Appendix A, we know that Re(g*0iV0) and Re(g2TNj) are zero mean independent Gaussian random variables with variances \gw\ CTN and \g2T\ aN respectively... They are independent of any other random variables. For W, and W2, we know that they are correlated. Therefore, we analyze them as one term Re(g*wW2 - g2TWx)We express the samples of -W and W2 as Wl. = x \ +J v i (F-l) w2 = x2 +jy 2 (F.2) where x{, yt, x2 and y2 are samples of zero mean real Gaussian random variables with variance cr^ and correlation coefficient pw, i.e., E[Xf] = E[Yi2] = a2w (F.3) E[X.Y.] = 0 i, y = l,2 (F.4) E[XlX2] = E[YlY2] = pwcr2w (F.5) If we substitute gi} by its real and imaginary parts as g.. = utJ +.jvtj, we have 67 Re(g*0W2 -g2TW*) = (ul0x2 - K 2 r X j ) + (v 1 0y 2 -v2Tyl) (F.6) Then the variance of Re(g*i0W2 - g2TW*) is £ [ R e 2 ( g l > 2 - g 2 X ) ] = £[(u 1 0X 2 - u 2 r X , ) 2 +(v10F2 - v 2 r y , ) 2 +2(w 1 0X 2 -MjrX^Cv^Kj -v 2 r F,)] = ( M 2 0 + u\T -2uwu2Tpw)al + (vf0 + v\T - 2 v w v 2 T p w ) a l 68 Appendix G Derivation of the Variances and Correlation Coefficients of Matrix Transformed Time-selective, Spatially Correlated Fading Before transformation, G 1 0 , G 2 0 , G i r and G2T are zero mean complex Gaussian random variables with variance o2G. Their covariance matrix is expressed in (3.1) as PS°2G P,°l P,<*1 PsPt°G P,02G P,°l P,P,°l _2 Ps<?G PsP^G P,<7G Ps°l (3.1) psp,<yG After matrix transformation, four new zero mean complex Gaussian random variables G 3 0 , G 4 0 , G 3 T and G 4 r are created by (3.23) £ 3 0 840 _SAT 83T. 2 810 + 82T 8ir ~ 8w 820 8\T 8\T + 820 (3.23) The variance of G 3 0 can be written as ° G 3 0 = - ^ b 3 o G 3 o ] = —E 2 (G10 + G2T ) ( G 1 0 + G2T ) (G.1) Similarly, the variances of G 4 0 , G3T and GAT can be proved as o2GA, (T^  and cr^ respectively, where alGA = (l-psp,)<JG. 69 The covariances of G 3 0 and G 4 n , G 3 0 and G 3 T can be written as £[G 3 0 G 4 0 ] = £ V2 V2 . — (G 1 0 + G 2 r ) — (G 2 0 - G i r ) = E[GWG20 -GlTG2T -GwGlT + G20G2T] =0 (G.2) E[G30G3T] = E V2 V2 . — (G 1 0 + G 2 r ) — ( G i r + G 2 0 ) = 77 ^ b io^o + GlTG2T + Gl0GlT + G20G2T ] (G.3) The correlation coefficient between G 3 0 and G 3 r is P* = £T_G30G3r] P,+Ps ^E[\Gi0\2]E[\G,T\2] 1 + PsP, (G.4) Similarly, we can prove the covariance and the correlation coefficient between G 4 0 and GAT as E[GA0G;T] = 2(p,-ps)cTG (G5) P,4 = Pt-Ps \-P",P, (G.6) Because the rest of the covariances between new channels are 0, the new covariance matrix can be written as 70 0 (P,+Ps)°G 0 0 0 (P,-P,)°G (P,+Ps)°G 0 (l + PsPt)a2 0 0 (P,-P,)°c 0 Since channel 3 and channel 4 are uncorrelated, we can write (G.7) separately as {\ + PsP,)G2c (P,+Ps)°2C SP,+Ps)vl (l + PsPl)CT2G_ P,1°G} (1-PSP,)°G (Pt-PS)<7G A 4 < SP,-Ps)cl (J-PsP<)°2G_ (JQ (G.8) (G.9) 71 Appendix H Distribution of Hermitian Quadratic Form Using (2.23), we can write the decision variable for s0 as • ' ' / = Re(s0) = (asQ + bsx)\hx\ + (as0-bsx)\h2\ + bsQ(hxh2 + hxh2) + (h*dl + hxdx)s0 + (h*d2 + hxd2)sx + (h2d*2 + h2d2)sQ - (h2dx + h*2dx )sx + (h* + hx )n0 + (h2 + h2 )n*T = ZT* F Z (H.1) where Z = [HX H2 Dx D2 N0 NT]T, a 6x1 column matrix of six jointly distributed complex Gaussian variables, with covariance matrix R 1 2 2 Pe and R = 2 1 2 Ps°G Pe 0 0 0 0 F = 0 0 0 0 0 0 0 0 0 0 0 0 ol p d ° l 0 0 Pd°2D ol 0 0 0 0 ol 0 0 0 0 ol as0 + bsx bs0 1 2 0 1 2 1 1 — s 2 1 — s 2 bs0 as0 - bsx 1 2 0 1 s \ 2 1 0 0 1 2 1 1 2 1 2 0 0 0 0 1 . 2 0 0 0 0 0 0 0 (H.2) (H.3) 72 It is clear that both F and R are Hermitian. Thus a unitary 6x6 matrix, U, "V2 V2 U 2 V2 2 0 0 0 0 2 V2 2 0 0 0 0 0 0 V2 2 V2 2 0 0 0 0 _V2 2 V2 2 0 0 0 0 0 0 2 V2 0 0 0 0 _V2 2 V2 (H.4) can be formed with the 6 eigenvectors of R as its columns, such that U T * U = I U T * R U = A U A U T * = R (H.5) (H.6) (H.7) where I is the identity matrix and A is a diagonal matrix with the six eigenvalues of R. A = „ 2 ° C 0 0 0 0 0 0 0 0 0 0 P.p] a2 •) G 0 0 0 0 Pe 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (H.8) There is an infinity of matrices that allow a factorization of A in the form (H.9) 73 One such factorization is the "square-root" matrix. T = Or 0 0 0 0 0 PsPe 0 0 0 0 0 4X + Pd0D 0 0 0 0 0 0 0 0 0 0 0 0 0 ON 0 0 0 0 (H.10) Thus, Z can be transformed into W, a new set of Gaussian random variables, in which the random variables are statistically independent with covariance matrix I. W = ¥ 1 U T Z = Hx+H2 V Pe H2 -Hx ^-psp2e<yG ^ 1 D. + D, 1 D2-Dx 1 N0 + NT 1 NT-NQ (H.ll) The inverse of (H.ll) is Z = U* *FW (H.12) The quadratic form of (H.l) becomes / = WT* (TT* U T F If ¥ ) W = W'1' T W (H.13) where 74 T = ¥ T * U T F U * ¥ = a p 7 s £ 0 P <r V 9 0 I r V 0 0 0 0 s 9 0 0 0 0 £ 0 0 0 0 0 0 i 0 0 0 0 a -P l + P s P e - c r 2 ( a + b)s0 Pe .2 „4 PsPe 7 2 <*GbSl (H.14) (H.15) (H.16) 7 = 4(\ + PSP2E)(\ + PD) 2pe 4(\+PsP2e)d-Pd) 2Pe (TGCJDS0 aGaDsx £ = ^+~PS 2Pe °G°N (H.17) (H.18) (H.19) 1 P'2P< a2G(a-b)s0 (H.20) 77 = j a - P s P 2 ) ( i + P d ) 2pe aGaDsl (H.21) 0 = 2P, (H.22) i = V i - p , y 2p e oGoN (H.23) Since T is also Hermitian, it can be diagonalized in a form T = SOS T * (H.24) where S is a unitary matrix of orthonormalized eigenvectors of T, and O is the diagonal 75 matrix of its eigenvalues, (j>i. Thus, one can introduce the transformation X = ST* W (H.25) in terms of which the quadratic form is diagonal, / = X T * O X = X^kT- (H-26) and the covariance matrix of X is still I. Since the Hermitian quadratic form f here is in a zero mean complex Gaussian process, the characteristic function of/, defined as a Fourier transform on its pdf, is Gf(%) = ^—i (H.27) f * det(I-2/-£R*F) with its pdf given as the inverse PFU) = ^ -r e x P W £ / ) Gf(t)dt (H.28) 2n J-°° We can derive Gf(£) here, but due to its complexity, it is difficult to obtain the pdf by inverse Fourier transform. 76 Appendix I Derivation of Channel Estimation Correlation Coefficient as a Function of SNR In a simple model, with a transmitted signal s, i.e., a B P S K signal, the received signal r can be represented by r = gs + n CIA) where G is the channel gain, which is a zero mean complex Gaussian random variable with variance aG and N is the additive Gaussian noise, which is also a zero mean complex Gaussian random variable with variance <72N . If we use pilot symbols, we can obtain an estimate of g as 8=- = 8+- (1-2) s s Thus G is a zero mean complex Gaussian random variable with variance CT26=CTG+CT2N. (1.3) The correlation coefficient pe between G and G can then be obtained as E[GG*] _ E[(G + N)-G*] _ E[\G\2] Pe = V £ [ | G | 2 ] - £ [ | G | 2 ] T]E[\G\2]-E[\G\2] JE[\G\2]-E[\G\2] 2a2 'G V2(c7 2+<7 2)-2a 2 f a l l a l Vl + l / S N R (1.4) where SNR = <72G I <72N, 77 Bibliography [1] H. Holma and A. Toskala, WCDMA for UMTS, Radio Access For Third Generation Mobile Communications, John Wiley & Sons, 2000. [2] W. C. Jakes, Microwave Mobile Communications, IEEE Press, 1974. [3] V. Tarokh, N. Seshadri, and A. R. Calderbank, "Space-time codes for high data rate wireless communication: Performance criterion and code construction," IEEE Trans, on Inform. Theory, pp. 744-765, Mar. 1998. [4] K. Rohani, M . Harrison, and K. Kuchi, "A comparison of base station transmit diversity methods for third generation cellular standards," in Proc. IEEE 49th Vehicular Technology Conf, vol. 1, pp. 351-355, 1999. [5] M. Raitola, A. Hottinen, and R. Wichman, "Transmission diversity in wideband CDMA," in Proc. IEEE 49th Vehicular Technology Conf, vol. 2, pp. 1545-1549, 1999. [6] A. Hottinen and R. Wichman, "Transmit diversity by antenna selection in CDMA downlink," Proceedings of IEEE 5th International Symposium on Spread Spectrum Techniques Applications, pp. 767-770,1998. [7] R. T. Derryberry, S. D. Gray, D. M Ionescu, G. Mandyam and B. Raghothaman, "Transmit Diversity in 3G CDMA Systems," IEEE Communications Magazine, pp. 68-75, April 2002. 78 [8] 3GPP, "Physical channels and mapping of transport channels onto physical channels (FDD)," 3GPP TS25.211, version 6.3.0, December 2004. [9] 3GPP2, "Physical Layer Standard for cdma2000 Spread Spectrum Systems -Revision C," 3GPP2 C.S0002-C, version 2.0, July 2004. [10] C. D. Frank, "MMSE Reception of DS-CDMA with Open-Loop Transmit Diversity," Second International Conference on 3G Mobile Communication Technologies, Conf. Publ. No. 477, pp. 156-160, March 2001. [11] S. M. Alamouti, "A Simple Transmit Diversity Technique for Wireless Communications," IEEE Journal on Selected Areas in Communications, vol. 16, no. 8, pp. 1451-1458, October 1998. [12] D. Gu, "Performance Analysis of a Transmit Diversity Scheme in Correlated Fading with Imperfect Channel Estimation," MASc. Thesis, Department of Electrical & Computer Engineering, UBC, March 2003. [13] A. Vielmon, Y. L i and J. R. Barry, "Performance of Transmit Diversity over Time-Varying Rayleigh-Fading Channels," IEEE Global Telecommunications Conference, 2001, vol. 5, pp. 3242-3246, November 2001. [14] T. S. Rappaport, Wireless Communications Principles and Practice, Prentice Hall, 1996. 79 [15] W. C. Y. Lee, Mobile Communications Design Fundamentals, 2n edition, Wiley, 1993. [16] M . Schwartz, W. R. Bennett and S. Stein, Communication Systems and Techniques, McGraw-Hill, 1966. [17] X. Feng and C. Leung, "Performance Sensitivity Comparison of Two Diversity Schemes," Electronics Letters, vol. 36, no. 91, pp. 838-839, April 2000. [18] J. G. Proakis, Digital Communications, McGraw Hill, 4 t h edition, 2001. [19] A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, 2 n d edition, 1984. [20] W. C. Y. Lee, Mobile Communications Engineering, McGraw-Hill, 1982. [21] L. Fang, G. Bi and A. C. Kot, "New Method of Performance Analysis for Diversity Reception with Correlated Rayleigh-fading Signals," IEEE Transactions on Vehicular Technology, vol. 49, no. 5, pp. 1807-1812, September 2000. [22] J. N. Pierce and S. Stein, "Multiple Diversity with Non-independent Fading," Proc. IRE, vol. 48, pp. 89-104, January 1960. [23] I. S. Gradshteyn and I. M . Ryzhik, Talbe of Integrals, Series, and Products, Academic Press Inc., 1994. 80 [24] M . Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, 1972. [25] Z. Lie, X. Ma and G. B. Giantess, "Space-time Coding and Kalman Filtering for Diversity Transmission Through Time-selective Fading Channels," Proc. MILCOM, Los Angeles, CA, pp. 382-386, October 2000. [26] J. S. Robertson, Engineering Mathematics with Maple, McGraw-Hill, 1996. 

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