UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Performance analysis of a transmit diversity scheme in correlated fading with imperfect channel estimation Gu, Donghui Charles 2003

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_2003-0141.pdf [ 2.24MB ]
Metadata
JSON: 831-1.0090897.json
JSON-LD: 831-1.0090897-ld.json
RDF/XML (Pretty): 831-1.0090897-rdf.xml
RDF/JSON: 831-1.0090897-rdf.json
Turtle: 831-1.0090897-turtle.txt
N-Triples: 831-1.0090897-rdf-ntriples.txt
Original Record: 831-1.0090897-source.json
Full Text
831-1.0090897-fulltext.txt
Citation
831-1.0090897.ris

Full Text

P E R F O R M A N C E A N A L Y S I S O F A T R A N S M I T D I V E R S I T Y S C H E M E I N C O R R E L A T E D F A D I N G W I T H I M P E R F E C T C H A N N E L E S T I M A T I O N by DONGHUI CHARLES GU B. E. (Industrial and Electrical Automation), Shanghai Jiaotong University, China, 1991 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 2003 © Donghui Charles Gu, 2003 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of bLc{rlc( Gf^^W The University of British Columbia Vancouver, Canada Date DE-6 (2/88) Abstract Exact closed-form expressions are derived for the bit error rate of Simple Transmit Diversity (STD) with 2 transmit and M receive antennas in time-selective, spatially independent Rayleigh fading with imperfect channel estimation and in non time-selective, spatially correlated Rayleigh fading with imperfect channel estimation. The performance analysis is also presented. For spatially independent fading, it is found that for the same values of the channel gain time correlation coefficient pt and the channel gain estimation error correlation coefficient p e , the error performance in non time-selective fading with imperfect channel estimation is worse than in time-selective fading with perfect channel estimation. The BER floor resulting from channel estimation errors and time-selective fading is determined. For the same values of pt and p e , say p, the error floor limits are approached at lower signal to noise ratio (SNR) values for (pt = 1, p e = p) than for (pt = p, p e = 1 ) . The effects of channel estimation errors on error performance of STD and Maximum Ratio Combining (MRC) were compared and it was shown that for large values of signal to noise and estimation error to noise ratios, STD suffers a 3 dB loss compared to MRC in non-time selective, spatially independent fading. i i Table of Contents Abstract ii Table of Contents iii List of Figures v Acknowledgment vi Chapter 1 Introduction 1 1.1 Space Diversity 1 1.2 Simple Transmit Diversity 2 1.3 Thesis Overview 3 Chapter 2 Background and Related Works 5 2.1 Statistical Models for Fading Channels 6 2.2 Maximal Ratio Combining (MRC) 6 2.3 Simple Transmit Diversity (STD) 9 2.3.1 STD with Two Transmit and One Receive Antenna 9 2.3.2 STD with Two Transmit and M Receive Antennas 10 2.4 Maximum Ratio Transmission (MRT) 12 Chapter 3 Performance Analysis of STD with Correlated Fading and Channel Estimation 16 3.1 2M-Branch STD in Time-selective, Spatially Independent Fading with Imperfect Channel Estimation 16 3.1.1 System Model 17 3.1.2 BER Analysis 19 3.2 2M-Branch STD with QPSK Modulation ....24 iii 3.3 2M-Branch STD in Non Time-selective, Spatially Correlated Fading with Imperfect Channel Estimation 25 3.3.1 System Model 25 Chapter 4 Numerical Results 32 Chapter 5 Conclusion 44 5.1 Main Thesis Contributions 44 5.2 Topics for Future Study 45 Glossary 49 Appendix A. Derivation of the Correlation Coefficient Between the Estimated Channel Gain and the Actual Channel Gain 49 Appendix B. Derivation of the Means and Variances of the Random Variables in Equation (3.8) v-' 51 Appendix C. Derivation of the Correlation Coefficient between the Estimated Channel Gains 57 Appendix D. Derivation of Means and Variances of Random Variables in Equation (3.29) 58 Bibliography 62 iv List of Figures Figure 2.1 MRC with M receive antennas 7 Figure 2.2 STD scheme with one receive antenna 9 Figure 2.3 The STD scheme with M receivers 11 Figure 2.4 MRT with N transmit and M receive antennas 13 Figure 3.1 The STD scheme with M receivers in time-selective, spatially independent fading with imperfect channel estimation 17 Figure 3.2 The STD scheme with M receivers in non time-selective, spatially correlated fading with imperfect channel estimation 26 Figure 4.1 BER of two-branch STD as a function of SNR for different values of when 33 Figure 4.2 BER of two-branch STD as a function of SNR for different values of when 34 Figure 4.3 BER of two-branch STD as a function of SNR for different values of and 35 Figure 4.4 BER for MRC and STD in non time-selective Rayleigh fading with diversity order of two as a function of SNR for ESR = dB, dB, dB and dB ....37 Figure 4.5 BER for MRC and STD in non time-selective, spatially independent Rayleigh fading with diversity order of two as a function of ESR for SNR = 4 dB, 10 dB and 15 dB 38 Figure 4.6 BER for STD and MRC with diversity order of four as a function of SNR for different values of and ! 39 Figure 4.7 BER of two-branch STD as a function of SNR for different values of and 40 Figure 4.8 BER of two-branch STD and MRC as a function of SNR for different values of and 41 Figure 4.9 BER of two-branch STD as a function of ESR for two different values of and two different SNR values 43 V Acknowledgment I would like to express my sincere gratitude to my thesis supervisor, Dr. Cyril Leung, for his guidance and encouragement. Without his valuable comments and great help, I can not complete my thesis. Of special mention is my friend, Mr. Vaibhav Dinesh, who spent a lot of time to review my thesis and gave me valuable comments. Special thanks are also due to my friends and fellow students. I would like to thank Ms. Xiaoyan Sarina Feng, Mr. Xinrong Wang, Mr. Cyril Iskander, Mr. Kaiduan Xie and other people in the Communications Group for their helpful suggestions and much support. Thanks also go to the members of my thesis examination committee for their time and efforts. Finally, I wish to express my gratitude to my wife, Ms. Jun Hou, my parents, Mr. Shigeng Gu and Ms. Xiuweng Huang, my parents-in -law, Mr. Xiaokui Hou and Ms. Fengying Ou, my sister Ms. Dongyun Gu, my brother Mr. Dongbiao Gu and their families, for their endless love and continuous support. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) under grant OGP0001731. vi 1 Chapter 1 Introduction The primary objective of this thesis is to study the effectiveness of space diversity techniques in improving the performance of wireless communication systems such as cellular systems. Currently cellular systems enjoy widespread use around the world since the introduction of the Advanced Mobile Phone Service (AMPS) in the United States in 1983 [1]. Currently the number of cellular phone subscribers worldwide is 1.3 billion [2]. With the introduction of new applications in 3G systems and beyond, it is anticipated that much research and development activity will be needed. Mobile radio or indoor wireless communication channels commonly suffer from signal fading which can cause severe performance degradation [3], [4]. The adverse effects of fading can be mitigated by employing diversity techniques which exploit the randomness in signal propaga-tion to establish independent (or at least highly uncorrelated) signal paths for communication so as to reduce the probability that all the signal paths will fade simultaneously. There are a number of diversity techniques which can provide significant link improvement. Depending on the propagation mechanisms, these may include [3]: space diversity, frequency diversity and time diversity. 1.1 Space Diversity Space diversity, also known as antenna diversity, is one of the most popular forms of diversity used in wireless communications. It is conceptually simple and relatively easy to implement. The method is based upon the principle of using two or more antennas at the base station or at the mobile terminal to provide diversity. Conventional cellular communication Chapter 1 Introduction 2 systems consist of an elevated base station antenna and an antenna at the mobile unit which is close to the ground. The existence of a direct path between the transmitter and the receiver is not guaranteed and the possibility of a number of scatters in the vicinity of the mobile unit suggests Rayleigh fading in the wireless channel. If there are more than one antenna at the base station or the mobile unit, to achieve decorrelation among the received signals, separation between antennas on the order of several tens of wavelengths is required at the base station and at least one half wavelength at the mobile unit [1]. Space diversity techniques include Selection Diversity (SD), Maxima] Ratio Combining (MRC) and Equal Gain Combining (EGC) [5]. Among these three diversity techniques, MRC is theoretically the optimum diversity combin-ing method for branch signals [6]. It provides the highest average output signal-to-noise (SNR) and the lowest probability of occurrence of deep fades. All the branch information is used to improve the overall receiver performance by cophasing the signals from different branches, weighting them according to their individual SNR's, and then summing the cophased and weighted signals. It is well-known that the output SNR is equal to the sum of the individual SNRs [3]. 1.2 Simple Transmit Diversity The classical MRC technique uses multiple antennas at the receiver to obtain the optimum performance. However, in cellular communications systems, this approach is not desirable for the mobile handsets because of cost, size and power considerations. Recently, a technique known as simple transmit diversity (STD) was proposed by Alamouti [7]. This technique employs two transmit antennas and one receive antenna to achieve the same diversity order as MRC. Two signals are simultaneously transmitted from the two antennas during a given symbol period and a Chapter 1 Introduction 3 transformed version of the signal pair is transmitted during the next symbol period. The technique can also be used for space-frequency coding. The proposed scheme was shown to have the same error performance in non time-selective channels as MRC when perfect channel estimation is available at the receiver. STD can be easily generalized to two transmit antennas and M receive antennas to provide a diversity order of 2M. It does not require any feedback from the receiver to the transmitter and involves small computation complexity. As a result, STD has now been incorporated in third generation cellular communication systems [8], [9]. It is thus important to understand its performance under non-ideal conditions. With imperfect channel estimation, STD was shown in [10] to have a poorer performance than MRC. Bit error rate (BER) curves for STD in Rayleigh fading with imperfect channel estimation were obtained by computer simulation. In [11], the performance of STD in time-selective Rayleigh fading channels was investigated with perfect channel estimation and an approximate BER expression was obtained. In this thesis, closed-form expressions are derived for the BER of STD in time-selective, spatially independent Rayleigh fading with imperfect channel estimation and in non time-selective, spatially correlated Rayleigh fading with imperfect channel estimation. This not only obviates the need for time-consuming simulations but also provides greater insight into the effects of channel estimation errors, time-selectivity and spatially correlated fading. 1.3 Thesis Overview In Chapter 2, some background and previous related studies are reviewed. The MRC combining technique is described. The STD and MRT scheme [12] are also discussed. Chapter 1 Introduction 4 In Chapter 3, exact, closed form expressions are derived for the BER of 2 transmit and M receive antennas STD with time-selective spatially independent Rayleigh fading with imperfect channel estimation and with non time-selective spatially correlated Rayleigh fading with imperfect channel estimation. BER expressions for time-selective spatially independent fading channels with perfect channel estimation or non time-selective spatially independent fading channels with imperfect channel estimation are obtained as special cases. Both binary phase shift keying (BPSK) and quadrature phase shift keying (QPSK) modulation methods are considered. In Chapter 4, numerical results are provided to illustrate the performance of STD in variety of channel model. In particular, the BER performances of STD and MRC are compared in the presence of imperfect channel estimation and non time-selective fading. In Chapter 5, the main contributions and conclusions of this thesis are summarized and some suggestions for future work are given. Chapter 2 Background and Related Works 5 Chapter 2 Background and Related Works The classical maximal ratio combining (MRC) technique has been shown to be optimum if the channel state is know perfectly [13]. This technique gives the best statistical reduction of fading for any known linear diversity combiner [1]. However, mobile station (MS) receiver diversity may not be desirable for wireless systems because of cost, size and power considerations. Recently, a simple but effective technique, the simple transmit diversity (STD) was proposed in [7]. For the same level of radiated power per transmit antenna, it was shown in [7] that STD in non time-selective Rayleigh faded channels has the same error performance as MRC when perfect channel estimation is available at the receiver. The STD technique can be generalized to two transmit antennas and M receive antennas to provide a diversity order of TM. Maximum ratio transmission (MRT) was proposed in [12] to allow a generalization to an arbitrary number of transmit antennas. However, it requires feedback from the receiver to the transmitter so that the latter can estimate the channel. In a slow fading channel where the fading channel is treated constant over the symbol duration, assume that the phase shift can be estimated from the received signal without error, coherent detection can be used [13]. Coherent combining systems do not suffer degradation from phase transients [14], therefore, coherent detection is considered more desirable when a large number of diversity branches are employed and it is used for the diversity schemes in this study. In this chapter, we describe the fading channel model to be used and briefly review MRC, STD and MRT schemes. Chapter 2 Background and Related Works 6 2.1 Statistical Models for Fading Channels When there are a large number of scatterers in the channel that contribute to the signal at the receiver, the central limit theorem suggests that the channel gain can be modeled as a complex Gaussian random process. If the process has zero-mean, the envelope of the channel response at any time instant has a Rayleigh distribution and the phase is uniformly distributed in the interval (0,27t) [13]. The complex channel gain corresponding to the kth branch is denoted by 2 gk = akexp(jdk), k = 1,2, . . . M , where is Rayleigh distributed with variance 2oK and 0£ is uniformly distributed in (0, 2rc). We can express the channel gain as gk - xk + jyk, where xk = a^ cosB .^ and yk = a s^inO,,., are samples of independent zero-mean Gaussian 2 2 2 random variables (r.v.'s) with variance ax = oy = cK. 2.2 Maximal Ratio Combining (MRC) Figure 2.1 shows the baseband representation for MRC with diversity order of M. The received signal form each of the M diversity branches are co-phased and weighted to maximized the received SNR. The received signal on the /th branch corresponding to the transmission of signal s0 can be written as rk,MRC = Sks0 + nk, k = 1...M, (2.1) where gk is the independent channel gain and nk is an independent complex Gaussian r.v. representing the noise and interference at the receiver. For clarity, we will use uppercase letters to Chapter 2 Background and Related Works 7 Noise -e-Channel estimator Noise Channel estimator 5 0 , MRC 8M=aMe Transmitter Noise -e-Channel estimator Figure 2.1 MRC with M receive antennas denote r.v.'s and corresponding lowercase letters to denote their sample values. The receiver combining rule for M branch MRC is as follows: M ~SQ,MRC = X 8k rk, MRC k = 1 M = X 8k(8kso + nk) (2-2) k = 1 M M S a ^ o + S 8k* nk • Jfc =1 k=\ where s0 M R C is the combined signal at the output and * denotes the complex conjugate. The theoretical analysis of the error performance for MRC was discussed in [6] and it was shown that the instantaneous received SNR y, at the output of the diversity combiner, is the sum of the SNR's on the individual branches, i.e. Chapter 2 Background and Related Works M = k = 1 k=\zaN M f. * = 1 2(7^ (2.3) t 2 where we define the received SNR on the individual branches as yk = — - . cj is the energy of 5, 2a N and aN is the variance of the real or imaginary component of Nk. It is assumed that the average 2 2 received energy gain for each diversity channel is equal, i.e. aK = a . The average SNR per branch is then % 2 To = 2 a >N The probability density function (pdf) of y is given by [13] (2.4) M -p(y) = lo(M-\)\ e~y/y\y>0, (2.5) The BER is obtained by averaging over the fading channel statistics (2.5), i.e. re,MRC ~ \P{y)Q(JTy)dy (l-u) (2.6) Chapter 2 Background and Related Works 9 where u = 2.3 Simple Transmit Diversity (STD) The STD scheme with one receive antenna is first reviewed, followed by M receive antennas case. 2.3.1 STD with Two Transmit and One Receive Antenna Figure 2.2 shows the baseband representation of the STD scheme with one receive antenna. transmit antenna 2 Figure 2.2 STD scheme with one receive antenna In this scheme, independent and equiprobable data bits are transmitted from each transmit antenna at symbol rate 1/T. In the first symbol period, sQ and s] are sent from antenna 1 and antenna 2 respectively. In the second symbol period, -s* is transmitted from antenna 1 and SQ from antenna 2, where * denotes the complex conjugate operation. The delay spreads are small compared to T and the coherence times are much larger than T, so that the channels are treated as Chapter 2 Background and Related Works 1 0 frequency flat and non time-selective, i.e. the received signal in the first and second bit period can be expressed as r \ = S\so + 82s\ +ni (2.7) r2 = S \ s i + 82s0+n2 where n, and n2 are samples of independent Gaussian r.v.'s representing noise and interference at the receiver at successive intervals. The decoding of s0 and sl is based on s 0 s t d and s l s l d respectively where The resulting signals, s0 s t d and s t d , are then sent to the maximal likelihood detector. The combined signals in (2.8) are equivalent to those obtained from two-branch MRC except for the phase rotations on the noise component which do not degrade the effective SNR. Thus assuming perfect channel estimation in non time-selective fading, STD scheme provides the same error performance as 2-branch MRC for a fixed value of the radiated power per transmit antenna. 2.3.2 STD with Two Transmit and M Receive Antennas Figure 2.3 shows the baseband representation of the STD scheme with two transmit and M receive antennas. s, 0,std \,std 8 * r 0 + 8 2 r \ (a 2 + a22)s0 + gfnQ + g2n* 8*r0 - g2r\ (a]+a2)sl + gfn0-g2n* (2.8) Chapter 2 Background and Related Works 11 tx antenna 2 Figure 2.3 The STD scheme with M receivers The encoding and transmission sequence of the information symbols is identical to the case of a single receiver. The channel gains from transmit antenna j to receive antenna k is denoted by gjk,j = \,2,k= 1,2, ...M. The received signals at the /th receive antenna are: rk = 8\ksQ + g2ks\+nk (2.9) rk,T = -8iksf +82kso+nk,T k = 1,2, . . . M where rk is the signal received in the first symbol period and rk T is the signal received in the second symbol period. The output signals are expressed as Chapter 2 Background and Related Works 12 M M ~so,std = X rk+ X 82kr%: k = 1 k = 1 , std M M M X ( a i * + 04)*o+ X Stk nk+ X # 2 * 4 , r fe= 1 *= 1 k=\ M M X # * 2 * rk - X S l t ' l . r = 1 * = 1 X (a l f t + + x ^ * 2 ^ - X ^ u n l r k = 1 Jt= 1 k = 1 (2.10) • Equation (2.10) shows that the error performance of STD with two transmit and M receive antennas is equal to that of 2M branch MRC. 2.4 Maximum Ratio Transmission (MRT) The MRT scheme [12] was proposed to make use of an arbitrary number of transmit antennas. The channel gain matrix can be represented by #11 ••• S\M ! \ ! SNI • • • 8NM (2.11) where gjk,j = \,2...,N,k = 1,2..., M represents the channel gain from transmit antennay to receive antenna k. As shown in Figure 2.4, the symbol to be transmitted, s, is weighted by a transmit weighting vector V = [v l s v2...vA,] with V = 1-{GW)H (2.12) Chapter 2 Background and Related Works 13 > S MRT Figure 2.4 MRT with N transmit and M receive antennas where f i s a M x l receive weighting vector and a = \GW\. [. ] denotes the Hermitian operation, i.e. complex conjugate transpose. The received signal vector is given by 1 = S-{GW)HG + n (2.13) T T where n = [nx...nM] and [. ] denotes the transpose operation, n-is an independent Gaussian r.v representing noise and interference. The estimate of the signal is given by W = S-(GW)HGW + nTW = as + n W (2.14) The overall output SNR can be written as Chapter 2 Background and Related Works 14 7 MRT ~ a t\ a % 2WWHc>2N M 2 I K 2 2 (2.15) In [12], it is assumed that w, = w2 = ...= w w , and (wpw*) = *\* - i- 1 N J = 7 .g * > pi°qi (2.16) where p and q take on values in the set {1,2, N}. For (TV x 1) MRT, i.e. N transmit and one receive antennas, the weighting function at the receiver, w,, is set to unity for convenience. From (2.15), the resulting output SNR can be written as yN x 1, MRT ~ 2 e aNx\C> 2a (2.17) N where aNx , = \G\ = f N 2 v V/ = i J . Equation (2.17) is the same as (2.3), the output from MRC combiner. Since the pdf of the output SNR for both N x 1 MRT and 1 x N MRC are the same, they have the same error performance. For (1 x M) MRT with one transmit and M receive antennas, the output SNR can be written as Chapter 2 Background and Related Works 15 £ f M "\2 Y - _1_ ' 1 x M, MRT ~ 2Mr>NKk=l j (2.18) which is the same as the output SNR from equal gain combiner (EGC) [3]. However, the constraint that = |w2| = •••= |wM| in [12] results in degraded perfor-mance and a new scheme, maximum ratio transmission and combining (MRTC) was proposed recently in [15]. It is shown that MRTC can offer significant gains over MRT by using optimum transmit and receive weights. (N x 1) MRTC has the same error performance as MRC with same diversity order. Chapter 3 Performance Analysis of STD with Correlated Fading and Channel Estimation 16 Chapter 3 Performance Analysis of STD with Correlated Fading and Channel Estimation STD was shown to have the same error performance in non time-selective Rayleigh fading as MRC when perfect channel knowledge is available at the receiver. However, with imperfect channel estimation, STD has a poorer performance [10]. BER curves for STD in Rayleigh fading with imperfect channel estimation were obtained using computer simulation in [10]. The perfor-mance of STD in time-selective Rayleigh fading was investigated in [11] assuming perfect channel knowledge. An approximate expression for the BER was obtained. In this chapter, exact closed-form expressions are derived for the BER of STD with two transmit and M receive antennas in time-selective, spatially independent Rayleigh fading with imperfect channel estimation and in non time-selective, spatially correlated Rayleigh fading with imperfect channel estimation. BER expressions for time-selective spatially independent Rayleigh fading with perfect channel estimation or non time-selective spatially independent Rayleigh fading with imperfect channel estimation are obtained as special cases. BPSK and quadrature phase shift keying (QPSK) modulation methods are considered. 3.1 2M-Branch STD in Time-selective, Spatially Independent Fading with Imperfect Channel Estimation In this section, we present the performance analysis of STD with two transmit and M receive antennas in time-selective, spatially independent Rayleigh fading with imperfect channel estima-tion using BPSK modulation. Chapter 3 Performance Analysis of STD with Correlated Fading and Channel Estimation 17 3.1.1 System Model Figure 3.1 shows the baseband representation of the STD scheme with M receive antennas in tx antenna 2 Figure 3.1 The STD scheme with M receivers in time-selective, spatially independent fading with imperfect channel estimation. time-selective, spatially independent fading with imperfect channel estimation. Independent and equiprobable data bits, each of duration T, are transmitted. With BPSK modulation, the transmit-ted signals sQ and from the two transmit antennas are either +1 or -1. It is assumed that the bandwidth of the signal is narrow compared to the channel coherence bandwidth so that the channel can be considered as non frequency-selective [1]. We use the time-selective fading model in [11] in which the channel gain is constant over a symbol duration but can change in successive symbol periods. The channel gains from transmit antennas 1 and 2 to receive antennas i, i = 1 M at time 0 and time T are denoted by r.v.'s GU 0, G2I Q, GU T, G2I T . In STD, the received signals at time 0 and time T at receive antenna / can be written as: Chapter 3 Performance Analysis of STD with Correlated Fading and Channel Estimation 18 ri = Su,0s0 + 82i,0sl+ni,0 (3.1) ri,T = -Su,Ts\+S2i,TsQ + ni,T 1 = h---,M where gu t , g2i ?2> ni t » h> ^  e T} a r e outcomes of independent complex Gaussian distributed r.v.'s with zero means, i.e. the channel gains are spatially independent. The variances 2 2 of the corresponding r.v.'s G^ t and Ni t are denoted by oG and o~N respectively. In this thesis, we define the variance of a complex r.v. as the variance of either its real or imaginary component. G 7 o and G ; i T , j = 1, 2, are correlated with correlation coefficient p, which is defined by [3] as E[GJit0GJi>T*] P, = jE[\GjJ2]E[\GJitl\2] ( 3 2 ) E[Re(Gj- 0)Re(Gjitr)] + E[Im(GjU0)/m(Gy,- T)] 2 ' 2rj, G where E[. ] denotes the expected value, Re(G- t) and Im(G - t) are the real and imaginary components of Gji t . In STD, the decoding of sQ and is based on M M ~s0,s,d = Z ^ ' > 0 rt + Y.h2i,Tr*T , = 1 ' ' = 1 (3.3) M M v ' ~Sl,std = ^Zh%Ori - Y*h\i.Tr*T i = l i=1 where h-{ t is the estimate for g •• ,. If the real part, Re(sk STD), of ~sk S T D , k = 0, 1 , is greater than 0, sk = + 1 is chosen; otherwise sk =-1 is chosen. Following [10], we write Chapter 3 Performance Analysis of STD with Correlated Fading and Channel Estimation 19 hjt t = g • • t + Zji t where the channel estimation error, Zjt t , is a sample of a zero mean, variance 2 a z complex Gaussian r.v which is independent of G-,- t . H •• f is thus a zero mean complex Gaussian r.v. with variance <5H = <JG + a z . It is shown in Appendix A that the correlation coeffi-cient of Gj^, and H}i , is p e = oG/cH. 3.1.2 B E R Analysis Due to the symmetry in the STD scheme, the BER for the two transmitted signals are equal and we need only consider one of the signals, say s0. We will make use of the following result for joint Gaussian r.v.'s [16]. For the two jointly Gaussian r.v.'s X and Y with zero means, i.e. E(X) = E(Y) - 0, assuming X = x' , then E{Y\X = x'} = -^x' °* , (3.4) °Y\X = X' = a y V ] - P 2 2 2 where <5X, aY are the variances of X and Y respectively and p is the correlation coefficient of X and Y. Since G ; - t and / / •• t are jointly Gaussian, hence the channel fading gain G ; 7 t conditioned on 2 . Hji { = hji t is a complex Gaussian r.v. with mean pe nhjt t where p e n = — = p e and vari-2 2 ance (1 - pe)oc . Thus, we can write g •• f as Chapter 3 Performance Analysis of STD with Correlated Fading and Channel Estimation 20 8\i,0 = P e , n h U 0 + dU u ' A ( 3 - 5 ) S2i,T = Pe,nh2i,T + d2i 2 2 2 where du and d2i are samples of zero mean, variance aD = (1 - p e ) a c , complex Gaussian r.v.'s, Dj- is independent of Hu 0 and D2i is independent of H2i T. Similarly, given gu 0 and g2i T , we can express gu T and g2i Q as S\i,T = Pr«l«,o + V l«-82i,o = P / % r + v2/ ( 3 - 6 ) 2 2 2 where v h and v2i are samples of zero mean, variance ov = (1 - p , ) o G , complex Gaussian r.v.'s, Vu is independent of G 1 ;- 0 and V2i is independent of G2i T . Using (3.1) - (3.6), sQ s t d can be written as M M ~sQ,std = X Pe,n(| / lh-,o|2 + |^2 ( )7-| 2) 50+ I W i,0 <*1« + >*2«. 7" <*2*«^0 i=l (=1 M + X 0 V2i ~ h2i, T v\1 + h^i, 0 Ptd2i ~ h2i, T Pt dTJs\ (3-7) i = 1 M M + X hU 0 n i , 0 + X A 2 i , T n t T • i=\ i = l -Since S ^ J Q and s} = -s0, each with probability ^, we can calculate the BER for STD as Pe,std 2^e's]-so+ ^e'si = -s() ' For the case = s0, from (3.7) we can write the decision variable for s0 S T D as Chapter 3 Performance Analysis of STD with Correlated Fading and Channel Estimation 21 M M ReCsQ,std)= ^Pe,n(\h\i,o\2 + \h2i,T\2)S0  + R e \ I [ f c f i , 0 (m\i +  v 2 ^ \ s 0 /= 1 M + Re\ £ ih2itT(m2i-vu)*) \sQ + Re M X ^ ' . 0 ni,0 U= 1 + Re M X^., Tn*,T (3.8) where m 1 ( = dli + ptd2i and m2i = (d2i - ptdu). It is shown in Appendix B that Re{hfl0Mu), Re(h2-TM*), Re(h^0 V2i) , Re(h2-TV*), Re(hfltQNl0) and 2 2 i . 2 Re(h2i T N*T ) are independent, zero mean Gaussian r.v.'s with variances (1 + pt )oD\hu 0 , 2 2 2 2 2 2 2 2 2 2 2 (1+p,)rjD|/i 2 l- r| , oK|/?1 ;-0| , ov\h2iT\ , CF^|^if,0| a n d  aN\h2i,T\ respectively. Thus, M 2 2 7?e(50 ^) is the sum of ^ p e 0[|^h o| + 1^2/ r| a n o - a n independent, zero mean Gaussian i = 1 r.v. with variance [(1 + p2)o2D + a2/ + o^Kj/zj^ 0 [ z + \h2i T\A) • The BER is given by i= 1 r M "I 2 \ Xp*,«(h«-.o|2 + N , H 2 ) •J = 1 1 X [ ( ] + P?)°? ) + 4 + 0w](|;ih-,o|2 + | / ? 2 i ,r | 2 ) (3.9) V 1 i = l where M a = X^IVo|. + N,;r| )» (3.10) Chapter 3 Performance Analysis of STD with Correlated Fading and Channel Estimation 22 with K = p] / { 2 [ ( l + p 2 ) a 2 + G 2 + c 2 ] } . (3.11) Similarly, it can been shown that the BER for s] = -s0 is also given by (3.9). For non time-selective fading with imperfect channel estimation, (3.10) reduces to 2 M Pe,n nts 2 2 — 2[2aD + o>],-= 1 Since Hu 0 and H2i T are independent and identically distributed zero-mean complex M 2 2 Gaussian r.v.'s, A - ^ K(]HU 0| + \H2i r | ) has a chi-square distribution with AM degrees of i = 1 freedom and its pdf is given by [13] 2M 2 M - 1 f . . (2M) fl -{2Ma)/\lA fA(a) = 1-JIi e ,a>0, (3.13) U / M ( 2 M - 1 ) ! where \La = 4MKo2H =2Mp2eo2G/[(\ +pf)(l -p])<5c + {\ -p2)G2G + o2N] 2 2 2 2 2 2 2 2 2 ( 3 J 4 ) = 2 M a J / [ ( l + p 2 ) a z a G + a 2 7 (a 2 / + a N ) - i - a z ( a 2 / + aA,)] . The overall BER for STD with BPSK modulation on a Rayleigh fading channel can then be obtained by averaging over the fading channel statistics (3.13), i.e. Chapter 3 Performance Analysis of STD with Correlated Fading and Channel Estimation 23 Pf, STD = J / A ( f l ) 2 ( ^ ) ^ a (3.15) The integral of (3.15) can be simplified as [13] Pf, STD (l-u) 2M2M-UaM-\+i i = 0 V i "1 "" 2<1 + ") (3.16) where u = . For the special case of STD with two transmit and one receive A/2M + ^ antenna, the overall BER is given by (3.16) with M = 1 , i.e. Pf.STD- 4 -\2f 2 + u, 2 + 2 + ut (3.17) 2 2 For given values of M, pe, pt and aN, \iA increases monotonically with a c . The limiting value of \iA as a c —> °° is \iA 2Mp e 2 The BER thus has an error floor expression 2 - P , 2 - P 2 P 2 given by replacing ]XA by [iAi/nax in (3.16), i.e. u = P 2 f2-p2p2' For a non time-selective Rayleigh fading channel, p;= \,cv = 0 and (3.14) reduces to V>A,n,s = 2 M a J / [ 2 o 2 r j G + a 2 a ^ + a 2 a ^ ] (3.18) 2 2 2 With perfect channel estimation, pe = 1, aD= 0, aH= aG and (3.14) reduces to Chapter 3 Performance Analysis of STD with Correlated Fading and Channel Estimation VA,Pce =2Mc52c/(a2v + a2N) 24 (3.19) With both non time-selective Rayleigh fading and perfect channel estimation, (3.14) reduces to the result in [7], i.e. V-A.nts/pcs =2M02G/a2N . (3.20) The corresponding BER is given by replacing \iA by \xAillts, \iAi p c s and \iAiltts/pcs in (3.16). For comparison purposes, we note that the BER of MRC with M receive antennas and imperfect channel estimation is given by [17] Pe, MRC (3.21) where hi is the estimated channel gain for the i th receive antenna. A comparison of (3.12) for the non time-selective fading case and (3.21) shows that, for the same diversity order, MRC has a 2 2 smaller BER and for aD » aN, STD is 3 dB worse than MRC. 3.2 2M-Branch STD with QPSK Modulation The performance of 2M-branch STD with QPSK modulation is now considered. To calculate the symbol error rate (SER), we note that coherent demodulation ideally results in the two demodulated signals being separated at the outputs of the quadrature mixers at the receiver [18]. Chapter 3 Performance Analysis of STD with Correlated Fading and Channel Estimation 25 Thus a coherent QPSK system can be considered equivalent to two parallel independent coherent BPSK systems [13]. Hence using (3.16), the SER is given by [13] 2 Pe, QPSK = 1 - ( 1 - Pf, STD) = ^Pf^sTD~Pf,STD • 3.3 2M-Branch STD in Non Time-selective, Spatially Correlated Fading with Imperfect Channel Estimation In this section, we investigate the performance of 2M-branch STD with BPSK modulation in non time-selective, spatially correlated Rayleigh fading with imperfect channel estimation. 3.3.1 System Model Figure 3.2 shows the complex baseband representation of the STD scheme with M receive anten-nas in non time-selective, spatially correlated fading with imperfect channel estimation. Indepen-dent and equiprobable data bits, each of duration T, are transmitted. With BPSK modulation, the transmitted signals sQ and s, from the two transmit antennas are either +1 or -1 . 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 channel can be considered as non fre-quency-selective and non time selective [1]. The gains of the 2M diversity paths, denoted by 2 G J J , G 1 2 , G 1 M , G 2 1 , G2M are zero mean, variance oc correlated complex Gaussian r.v.'s. The 2M x 2M covariance matrix, C , of these r.v.'s is assumed to be of the form [19]: Chapter 3 Performance Analysis of STD with Correlated Fading and Channel Estimation 26 tx antenna 2 Figure 3.2 The STD scheme with M receivers in non time-selective, spatially correlated fading with imperfect channel estimation. 2 ° C <4P, 0 0 o"cP* 2 ° G 1 0 2 * ° G P * 2 ' 0 0 o <4P* 2 In STD, the received signals at time 0 and time T at receive antenna i can be written as [7] [10]: ri = Slis0 + 82isl+ni,0 ri,T = -8usi+82iso + ni,T 1 = U~-,M where gu and g2i are path gain samples and «• t , te {0, T} is a sample of a zero mean, 2 variance aN independent complex Gaussian r.v. which represents the channel noise. In STD, the decoding of s0 and is based on Chapter 3 Performance Analysis of STD with Correlated Fading and Channel Estimation 27 M M ~s0,std = I>f; ri + Y.h2ir*T i=\ (=1 M M ~s\,std = ri~ H,h\ir*,T (3.25) i = 1 i = 1 where h-t is the estimate for g ••. If the real part, ^ (^5^ STD), of 5^  k = 0, 1, is greater than 0, sk = + 1 is chosen; otherwise sk = -1 is chosen. Following [10], we write hj{ =gjj + Zjt where 2 the channel estimation error, z , is a sample of a zero mean, variance a z complex Gaussian r.v. which is independent of G-t-. //•,• is thus a zero mean complex Gaussian r.v. with variance 2 2 2 <5H = a G + o z . It is shown in Appendix C that the correlation coefficient of Hu and H2i is ph = — ps. The 2Mx 2M covariance matrix, Ch, of Hn, Hn, H]M, H2], / / 2 M can be expressed as Ch = °ff <*HPh 0 0 C7 G p | 0 0 °HPh 0 acp*, oH (3.26) It is shown in Appendix A that G-(- and / / •• are correlated with correlation coefficient p g = <5g/GH . Using (3.4), we can write g-• as Chapter 3 Performance Analysis of STD with Correlated Fading and Channel Estimation 28 S\i = Pe,nhU + d\i Sli = Pe,nh2i + d2i (3.27) 2 2 2 2 2 2 where p e n=pe = <5Q./<5h , du and d2i are samples of zero mean, variance <sD = (1 - p e ) o c , complex Gaussian r.v.'s, Du is independent of Hx • and D2i is independent of H2i. It is shown in 2 Appendix D that Du and D2i are correlated with correlation coefficient p^ = (1 - p e)p . Using (3.24), (3.25) and (3.27), sQ s t d can be written as M M ~S0,std = yLPe,n(\hu\2 + \h2l\2)s0+ £ W« dU + h2id2i)s0 i = 1 M M M (3.28) + £.(/!*,• d2i-h2i du)sx + Y,hu n , 0 + Y,h2in%T i = 1 Since s^ = s0 o r si = ~s0' e a c n w i t n probability - , we can calculate the BER for STD as Pe,std ^Pe,s} = sQ^~Pe,= -Sf) ' For the case st = s0, we can write the decision variable for s0 s t d as M r M ReCs0,std> ^Pe.n(\hu\2 + \h2i\2)s0 + Re\llWi (du + d2i)]\sQ M + Re\Yi[h2i(d2i-dur}\s0 + Re + Re i = • M M X h2in* i, T (3.29) It is shown in Appendix D that Re(h*u (Du + D2i)), Re(h2i(D2i-D, •)*), Re(h*u Nii0) and Chapter 3 Performance Analysis of STD with Correlated Fading and Channel Estimation 29 2 . .2 Re(h2iN*iT ) are independent, zero mean Gaussian r.v.'s with variances [2(1 + pd)oD] ftjJ , 2 i 12 2 i 12 2 i 12 [[2( 1 + prf)rj£)]]\h2i\ , cyJ/iJ and oN\h2i\ respectively. Thus, Re(sQ std) is the sum of M X P e + |^ 2rj an^ a n independent, zero mean Gaussian r.v. with variance i = i M X [[2( 1 + p^)a2 ] + a2v](|/z1;|2 + \h2\2). The BER is given by i = 1 f r M -i IP..n(|fclJ2 + N 2 ) 2 N S= Q s0 -«' = 1 A X[[2(l+p,)a23] + a2,](|/z1(|2 +N 2) (3.30) U i = l ) = 2(7^) , where M Zp2n(M2 + NI2) 2{[2(l+p^)a2] + a^ } (3.31) M i = 1 with 2 K = — . (3.32) 2{[2(\+pd)oZD) + CJ2N} Similarly, it can be shown that the BER for 5 , = -sQ is also given by (3.30). Chapter 3 Performance Analysis of STD with Correlated Fading and Channel Estimation 30 Since Hu and H2i are correlated and identically distributed zero-mean complex Gaussian M r.v.'s, if the pdf of A = ^ £(|//h|2 + |#2(|2) 1S f A ^ ' t n e n LLS Laplace transform, P(s), can be i = 1 written as [3] 2M 1 a A i + sr, k=\ 1 where Tl = 2KXl and Xl are the eigenvalues of (3.26) and are given by [19] (3.33) - 2 2p;icos 2M+ 1 , / = 1,2, . . . ,2M (3.34) Then f A(a) is given by [19] 2 M / A O ) = ^dpexp(spa)> (3-35) P= 1 where .y are the poles of P(s) and dp are the corresponding residues of P(s). The overall BER is given by P/.STD = \f A(a)Q42ada (3.36) By using [20], [21], (3.36) reduces to 2 M ^ , Chapter 3 Performance Analysis of STD with Correlated Fading and Channel Estimation 31 For the special case of STD with two transmit and one receive antennas, the overall BER is given by (3.36) with M = 1, i.e. where Pf.sTD- 2 ( r 1 - r 2 ) I -l +r i +r n (3.38) T, = 2tfoJ(l+pA) = r 2 = 2 * o £ ( i - p A ) = [2(1 +pd)a2D] + a2N [2(1 + pd)o2D] + o2N (3.39) Chapter 4 Numerical Results 32 Chapter 4 Numerical Results Numerical results based on the analytic results derived in Chapter 3 are presented in this chapter. For convenience, we define the signal-to-noise ratio (SNR) as the ratio of the variance, 2 2 2 2 OG , of the channel gain, to the variance, oN, of the additive Gaussian noise, i.e. aG/aN and the 2 2 2 estimation error-to-signal ratio (ESR) as <3z/o~G, where G z is the variance of the channel estimation error. A fixed value of ESR corresponds to a fixed value of pe since p] =1/(1 +ESR). The theoretical BER of two branch STD in non time-selective Rayleigh fading with imperfect channel estimation is given by substituting (3.18) into (3.17). The corresponding curves are plotted in Figure 4.1 as a function of SNR for different correlation coefficient values, pe , between the estimated channel gain and actual channel gain. It can be seen that the error performance degrades rapidly as pe decreases from 1. The performance difference with perfect channel estimation increases with SNR. For a target BER of 10 , there is about 3.5 dB degradation for p e = 0.99 relative to perfect channel estimation, i.e. p e = 1 . For p e < 1 , the BER curve exhibits an error floor with a value obtained by substituting uA m a x = -—e—^ in (3.17). It can be seen that the error floor limit is approached for lower SNR values as pe decreases. When pe = 0, the estimated channel phase is completely random and hence the BER is 0.5. The BER of two branch STD in time-selective Rayleigh fading with perfect channel estima-Chapter 4 Numerical Results 3 3 10 6 6 $ 10" CD H r i - 3 QC o LU S 10"4 10 10" 10 — Pe = 1 - X - Pe = 0.99 - 3 - Pe = 0.9 Pe = 0.5 -e- Pe = 0.2 -*- Pe = 0 > £ fc> &> E> > £ E> > fc> E> > j> 10 15 SNR, dB 20 25 30 Figure 4.1 BER of two-branch STD as a function of SNR for different values of pe when pt = 1. tion as given by substituting (3.19) into (3.17) is plotted in Figure 4.2 as a function of SNR for different correlation coefficient, p ;, between the channel gains at time 0 and time T. The error performance degrades rapidly as pt decreases from 1. The performance difference with non time-selective fading increases with SNR. For a target BER of 10 , there is about 4dB degradation for pt = 0.9 relative to pt = 0.99 and there is only about 0.4 dB degradation for pt = 0.99 relative Chapter 4 Numerical Results 3 4 10" £ 10" t o rr i_ O k . UJ m 10~4k 10 -7 Pt = 1 -*- P, = 0.99 Pt = 0.9 Pt = 0.5 -e- Pt = 0.2 -*- Pt = 0 10 t t t 8 = j t ^ — « — < — « — « — < — ^ 15 SNR, dB 20 * * 25 30 Figure 4.2 BER of two-branch STD as a function of SNR for different values of p ; when pe = 1 . to non time-selective fading case, i.e. pt = 1. For small values of pt, 0<pt< 0.2 , the perfor-mance is almost identical. The BER of two branch STD in time-selective Rayleigh fading with imperfect channel Chapter 4 Numerical Results 35 10 1 0 ^ rr o i_ LU m 10~ 4 - 5 10 10" 10 -7 -* * * *-1 1 1 - © e e e e e -- < - - 4 - -- e e -->• - i>-- e e -- O - - £>--<3- -X. -x - x X -X X --e--X X < < Pt = i.P e = i X • Pt = 0.99, p =1 • < • Pt = 1,p =0.99 • •>• P. = 0.99, p =0.99 - X - Pt = 0.9, p =1 -<- Pt = 1,p =0.9 ->- P . = 0.9, p =0.9 - e - Pt = 0 , p e = 1 -*- P. = 1 ' P e = ° 10 15 S N R , dB 20 25 30 Figure 4.3 BER of two-branch STD as a function of SNR for different values of pt andp e. estimation as given by substituting (3.14) into (3.17) is plotted as a function of SNR for several pt and pe values in Figure 4.3. For the same pt and pe values, the performance in non time-selective fading with imperfect channel estimation (p, = 1, 0 < pe < 1) is worse than in time-selective fading with perfect channel estimation (0 < pt < 1, p = 1). For a target BER of 10 , Chapter 4 Numerical Results 36 there is about 1 dB degradation for (pt = 0.99, pe = 1), about 3 dB degradation for (pt = 1, p = 0.99) and about 7 dB degradation for (p; = 0.99, pg = 0.99) relative to non time-selective fading with perfect channel estimation i.e. (p, = \, pe = 1). For pt < 1 or pe < 1 , the 2 p 2 BER curve exhibits an error floor with a value obtained by substituting \iA 2-p2-pe2p2 in (3.17). Figure 4.4 shows the BER curves for STD and MRC in non time-selective Rayleigh fading with diversity order of two as a function of SNR for four different values of ESR. It can be seen that the BER for STD or MRC degrades quiet rapidly with increase in ESR. STD is more sensitive _4 to channel estimation error than MRC. For a target BER of 10 , an ESR of -20 dB results in an SNR loss of about 2 dB for MRC and about 6 dB for STD relative to the perfect channel estima-tion case. For ESR > 0, the BER curve exhibits an error floor with a value given by substituting UA,MAX = i n(3- 1 7)-Figure 4.5 shows the BER curves for STD and MRC in non time-selective Rayleigh fading with a diversity order of two as a function of ESR for three different SNR values. The BER differ-2 2 ence between STD and MRC increases with SNR and ESR. For aD » cN, or equivalently SNR » 1 + I/ESR, STD is 3 dB worse than MRC as expected from (3.12) and (3.21). The theoretical BER performance of STD and MRC with diversity order of 4 in time-selective Rayleigh fading with imperfect channel estimation, as given by (3.16) and (3.21), is plotted in Figure 4.6 as a function of SNR for different values of p and p.. As in the case of a Chapter 4 Numerical Results 37 15 S N R , d B Figure 4.4 BER for MRC and STD in non time-selective Rayleigh fading with diversity order of two as a function of SNR for ESR = -5 dB, -10 dB, -20 dB and - o o dB. diversity order of two, the error performance of STD degrades rapidly as pt and pe decrease from 1 and the error performance of MRC degrades only as pe decrease from 1. For values of p ; or pe less than 1, each curve exhibits an error floor. As expected, STD is more sensitive to channel _4 estimation error than MRC. For a target BER of 10 , the error performance of MRC with Chapter 4 Numerical Results 38 Figure 4.5 BER for MRC and STD in non time-selective, spatially independent Rayleigh fading with diversity order of two as a function of ESR for SNR = 4 dB, 10 dB and 15 dB. p e = 0.99 and STD with (pf = 0.99, p = 1) are almost identical with about 0.6 dB degradation relative to the curve for non time-selective fading with perfect channel estimation i.e. (p, = 1, pe = 1). However, there is about 0.6 dB degradation for STD with (pf = 1, p = 0.99) and about 1.2 dB degradation for STD (p, = 0.99, p = 0.99) relative to MRC with p e = 0.99 . Chapter 4 Numerical Results 39 10" 10" * * * * *- • 10 - 4 £ 1 0 CO CC o i— U J m 10" -6 10 •10 10 10 - 1 2 •14 •* * * - - • * - * *• -0- -o- - -e-•4- --o- - e • - > • -- 4 - - < -• - * • - • - * • -o- - e - -o- - $ -p- - ->- -"X— --x - - X- - - * X -3 — f r -•4. •4 -.. STD:p e=1,p=1 + MRC:p =1 X • STD:p e=1,p=0.99 - B - MRC, p =0.99 • < • STD:p e=0.99, p=1 - • » • STD: p6=0.99, p =0.99 —X- STD:p e=1,p=0.9 -0- MRC:p =0.9 <- STD:p e=0.9, p=1 ->-STD:p e=0.9, p=0.9 -o- S T D : p ° = 1 , p = 0 S T D : p ° = 0 , p = 1 * MRC: p =0 4 -• 4 --0-•4-- o — e -* X * • 4 -> t> •<•• ••<• 10 15 SNR, dB 20 25 30 Figure 4.6 BER for STD and MRC with diversity order of four as a function of SNR for different values of p, and pe . The theoretical BER performance of two branch STD in spatially correlated Rayleigh fading with imperfect channel estimation, as given by (3.38), is plotted in Figure 4.7 as a function of SNR for different values of ps and pe . It can be seen that the error performance degrades as ps increases from 0 and as p decreases from 1. For p = 1 and a target BER of 10 , there is Chapter 4 Numerical Results 40 15 SNR, dB Figure 4.7 BER of two-branch STD as a function of SNR for different values of and pe. about 1 dB degradation for ps = 0.5 and about 2 dB degradation for p^  = 0.8 relative to the spatially independent case, i.e. ps = 0. For pe = 0.99, there is about 4.6 dB degradation for ps - 0.5 and about 10 dB degradation for ps = 0.8 relative to spatially independent fading with perfect channel estimation. It can also be seen that, in spatially correlated fading with imperfect channel estimation, the error floor limit is approached because of the estimation error while there Chapter 4 Numerical Results 41 is no error floor for STD in spatially correlated fading with perfect channel estimation. Figure 4.8 shows the BER curves for STD and MRC in spatially correlated Rayleigh fading with diversity order of two as a function of SNR for three different values of pe. It can be seen that the BER for STD or MRC degrades quiet rapidly with decrease in pe and increase in ps. Figure 4.8 BER of two-branch STD and MRC as a function of SNR for different values of ps and pe. Chapter 4 Numerical Results 42 With perfect channel estimation, i.e. pe = 1, the error performance of STD and MRC in spatially correlated fading is identical; with pe = 0, the estimated channel phase is completely random and the BER is 0.5 regardless of the value of . With channel estimation error, the error performance of STD is worse than that of MRC in both spatially independent and correlated fading. For pe = 0.8 and a target BER of 10"1, there is 2.3 dB degradation in independent fading (ps - 0) and 3.2 dB degradation in correlated fading with p^  = 0.6 for STD relative to MRC. Figure 4.9 shows the BER curves for STD and MRC in spatially correlated Rayleigh fading with a diversity order of two as a function of ESR for two different SNR values. The BER differ-ence between STD and MRC increases with SNR and ESR. Chapter 4 Numerical Results Figure 4.9 BER of two-branch STD as a function of ESR for two different values of p^  and two different SNR values Chapter 5 Conclusion 44 Chapter 5 Conclusion 5.1 Main Thesis Contributions • In this thesis, exact closed-form expressions for the bit error rate of the simple transmit diversity scheme (STD) [7] in time-selective, spatially independent Rayleigh fading with imperfect channel estimation and in non time-selective, spatially correlated Rayleigh fading with imperfect channel estimation are derived. For spatially independent fading, it is found that for the same values of the channel gain time correlation coefficient p, and the channel gain estimation error correlation coefficient pe, the error performance in non time-selective fading with imperfect channel estimation is worse than in time-selective _3 fading with perfect channel estimation. For a target BER of 10 , there is about 1 dB degradation for (pf = 0.99, p e = 1) and about 3 dB degradation for (pt =1, pg = 0.99) relative to non time-selective fading with perfect channel estimation i.e. (p, = 1, pe = 1). • An expression for the BER floor resulting from channel estimation errors and time-selective fading is determined. For the same values of pt and pe, say p , the error floor limits are approached at lower SNR values for (pt= 1, pe = p) than for (p, = p, pe = 1). • The effects of channel estimation errors on error performance of STD and MRC were compared and it was shown that for large values of signal to noise and estimation error to noise ratios, STD suffers a 3 dB loss compared to MRC in non-time selective, spatially independent fading. Chapter 5 Conclusion 45 5.2 Topics for Future Study • This thesis investigated STD in a frequency flat Rayleigh fading channel. It would be useful to analyze the error performance of STD in frequency selective channels with Rican fading. • The BER result for time selective fading is based on the assumption that the channel gains are spatially independent. It would be interesting to study the effects of spatial correlation on performance in time selective fading. Glossary Acronyms AMPS Advanced Mobile Phone Service BER Bit Error Rate BPSK Binary Phase Shift Keying dB decibel EGC Equal Gain Combining ESR Estimation Error to Signal Ratio MRC Maximal Ratio Combining MRT Maximum Ratio Transmission MRTC Maximum ratio transmission and combining MS Mobile station pdf Probability of density function QPSK Quadrature Phase Shift Keying SD Selection Diversity SER Symbol Error Rate SNR Signal to Noise Ratio STD Simple Transmit Diversity 47 Symbols N M GJU Sk ak 9 * xk yk C rk, MRC s0> s] nk 2 oN s0, MRC J Yk Yo p(y) Pe, MRC T s0, STD ' i, T Number of transmit antennas Number of receive antennas The channel gain from transmit antenna j to receive antenna / at time t The sample of G - t The sample of the channel gain corresponding to /cth branch Amplitude of gk Phase of gk The real part of gk The imaginary part of gk Variance of Xk Variance of Yk Variance of G J ( t Signal energy Received signal on branch k in MRC scheme Transmitted signal noise and interference on branch K Variance of the real (or imaginary) component of N\ Signal output from MRC combiner Output SNR from combiner SNR on /th branch Average SNR per branch pdf of y BER of MRC Symbol duration STD Signal output from STD combiner Received signal on branch i in the first symbol period Received signal on branch i in the second symbol period Symbols 48 G V W H R S MRT a H, Jl,t H z ]l,t 2 P * Pe Pe,n Re (s0> S T D Pe,std Pf, STD Pe, QPSK c Ps Ph h Channel coefficient matrix for MRT Transmit weighting vector for MRT Receive weighting vector for MRT Hermitian operation Transpose operation Noise vector for MRT Signal output from MRT combiner A normalization factor Estimated channel gain for G t Variance of H • • , Channel estimation error for G ji,t Variance of Z;(7) Correlation coefficient of Gp 0 and Gjt T Correlation coefficient of G 0 and H^ t Normalized correlation coefficient equal to pe ) The real component of (50 S T D ) BER for STD The overall BER for STD SER for QPSK The covariance matrix of the channel gains Spatially correlation coefficient of the channel gains Spatially correlation coefficient of the estimated channel gains Eigenvalues of C 49 Appendix A . Derivation of the Correlation Coefficient Between the Esti-mated Channel Gain and the Actual Channel Gain Here we derive the correlation coefficient, pe, between the channel gain, G t and the estimated channel gain Hji r In the following, we will leave out the index t and / for brevity. Following [10], we can express the estimated channel gain H} and H2 as (A.l) h2 = 82 + z2 where g • and z • are samples of independent zero mean complex Gaussian r.v.'s. We express Gj as 81 = x \ + jy\ (A.2) 82 = x2 + n 2 where xx, yx, x2, y2 a r e samples of independent zero mean Gaussian r.v.'s. The variances of Xj, Y], X 2 , Y2 can be expressed as We express Z • as E(X\) = E(Y\) = <J2G (A.3) E{X22) = E(Y22) = a2G Z = u +jW (A.4) z2 = u2 + jw2 where ux, wv u2, w2 are samples of independent zero mean Gaussian r.v.'s. The variances of Appendix A. 50 Up Wv U2, W2 can be expressed as E{U\) = E(w\) = a z E(U\) = E(W22) = a z (A.5) Thus hx = xl + ul + j(yl + wl) h2 = x2 + u2 + j(y2 + w2) (A.6) (A.7) The correlation coefficient pe is given by [3] E[GjHn P = = E^Xj + JYj^Xj + Uj) ~ KYj + Wj™ JE(X2 + Y2)E[{Xj + Uj)2 + •( T • + Wjf] _ E[X2 + XjUj + Y2 + YJWJ + jjUjYj - WJXJ)] JE(X) + f^EiX2 + r j 2 + 2x ; . r j 7 . + y 2 + W 2 + 2YjWj] Since XJ,YJ,UJ,WJ are independent zero mean Gaussian r.v.'s, E(XJYJ) = E(XjWj) = E(XjUj) = E(YjWj) = E(UjYj) = E(WJXJ) = 0 . From 3.1.1, we have c2Hj = a2Gj + G2z. (A.8) Using (A.5), (A.7) and (A.8), pe can be expressed as Pe = °c/aHr (A-9) 51 Appendix B . Derivation of the Means and Variances of the Random Vari-ables in Equation (3.8) Here we derive the means and variances of Re(hfiQ Mu), Re(h2iTM2*), Re(hUo V2i) , Re(h2iTV*), Re(hfit0 Nii0) and Re{h2i< T /V* T ) as given in (3.8). Hjit,je (1,2), z = 1...M, te (0, T) can be denoted by h j i t = a..(e = u j i t + jw j i t where u„ = a.. cos6„- , and w-- = a.. ,sin0„- ,. First of all, we prove that Du, D7i, Vu Jl,t ji,t Jht Jl,t jitt ]l,t r I I ' Z l ' II and V2i are independent with each other. From (3.1) and (3.2), we have 8li,0 = Pe,nh]i0 + dU (B.l) 8li,T = Pe,nh2i,T  + d2i 8\i,T = P/Sw,o + vw 82i,0 = Pt82iT + v2l ( B - 2 ) 2 where g u t and g 2 i t , tx, t2e (0, T), each with variance a G , are samples of zero mean complex Gaussian r.v.'s which are independent with each other; du and d2i, each with variance 2 2 2 oD = (1 - p e )o G , are samples of zero mean complex Gaussian r.v.'s which are independent of 2 2 2 hu 0 and h2i T , vu and v2i, each with variances ov = (1 - p;)<3G, are samples of zero mean complex Gaussian r.v.'s which are independent of gu 0 and g2i T . t =gji t + t where 2 Zji t is a sample of a zero mean, variance a z , complex Gaussian r.v which is independent of Appendix B. Derivation of the Means and Variances of the Random Variables in Equation (3.8) 52 Gp t . Zjt o is independent of Zjt Tl_ Using (B.l), the covariance of Gu 0 and H2i j can be expressed as [3] E ( G \ i , o H % T ) = E ( G U 0 G % T + GXUQZ\IT ) = E(penHuoH%T +DUH%T) Since GU T is independent of GU ^; Zjt t i is independent of G;-- ^  and Zjt 0 is independent of Zjit T , we have E { G I U Q G % T ) = E ( G X U Q Z % T ) = E(HU0H%T) = 0, (B.4) thus using (B.3) and (3.3), we have E(DUH%T) = 0. (B.5) Similarly, it can been shown that E(D$iHUt0) = 0. (B.6) Using (B.l) and (B.2), the covariance of GU 0 and G2i T can be expressed as E(GU>0G%T) = E(plnHU0H%T +pe>nH*2iT Du + pe> „ D%, Hx,. 0 + DUD^ ) (B.l) Using (B.4), (B.5) and (B.6), thus we have E(DuD*2i) = 0 (B.8) Since Du and D2i are zero mean complex Gaussian r.v.'s, they are uncorrelated and statistically Appendix B. Derivation of the Means and Variances of the Random Variables in Equation (3.8) 53 independent [23]. Similarly we can show Du and Vki, Vu and Vki, l,ke (\,2),l^k, are independent with each other. In the following, we will show that Dji is independent of V ••. Using (B.2), the covariance of HXi 0 and GXi T can be expressed as E{HUQG^T) = E(ptHuoG^i>0 +HuoV]i) (B.9) Since E(H]1,0G*U,T) = E(G\i,0G*U,T + Zli,0G*i,T ) = E(GUi0G^itT) (B.10) = 2P,°G E(HUQG%0 ) = 2oG (B.ll) thus using (B.9) and (B.10), we have E ( f f h . 0 V T i ) = 0 (B.12) Since DXi is independent of HXi 0 , using (B.l), the covariance of Gu 0 and DXi can be expressed as E{GUQD*U) = E(PenHUQD*u +DUD*U) (B.13) = 2 c 2 , Appendix B. Derivation of the Means and Variances of the Random Variables in Equation (3.8) 54 Using (B.l) and (B.2), the covariance of GXi 0 and GXi T can be expressed as E(GU0G% T) = E(penPtHuoG*Ut0 + p ^ t f , . > Q +ppuG*Ui0 + D,,. V*u ) (B.14) Using (B.10) - (B.13), we have E{DuVXi) = 0 (B.15) Du and Vu are uncorrelated and statistically independent. Similarly we can show D2j and V2j are also independent with each other. Thus Du, D2i, Vu and V2i are independent with each other. In the following, we derive the means and variances for i?e(/zf(>0 Mu), Re(h2i T M2*) , ReWi,0 V2i) - Re(h2i,T VJ) , Re(hfitQ N-Q) and Re(h2iiTN%T) in (3.8). From (3.8), we have *nu = *u + 9r*» ( B 1 6 ) m2i = d2i-PtdM Since Du and D2i are independent, zero mean complex Gaussian r.v.'s, we assume D » = X » + J Y " CB.,7) dn = x2« + tin where ;c1(-, t-» JC2/»3'2* a r e s a m p l e s of independent, zero mean Gaussian r.v.'s. Thus Appendix B. Derivation of the Means and Variances of the Random Variables in Equation (3.8) 55 mu = xu + ptx2i + j(yu + p,y2i) m2i = X2i-Ptx]i + J(y2i-P,y\i) (B.18) and Re(h1[ii0 Mu) = uUiQ(xu + p,x2i) + w U t 0 ( y u + pty2i) Re{h2iTM*) = u 2 i T { x 2 i - p t x u ) + w 2 i T { y 2 i - p t y u ) (b.19) Since the means of Xu, Yu,X2i,Y2i , E(XU) = E(X2i) = E(YU) = E{Y2i) = 0, the means of Mu, M2i, Re(lr\i 0 Mx •) and Re(h2i T M2*) are also equal to zero. From (3.5), The variances of Xu, Yu,X2i,Y2i can be expressed as E(X2U) = E(Y2U) = cj2D = ( l - p e 2 ) G 2 E(X22) = E(Y22i) = a 2 = ( l - p 2 ) o G (B.20) and the variances of Re(hfi Q Mxi) and Re(h2iTM2*) can be expressed as E{Re[(hfito Mu)]2} =E[u2Ut0(x2u + p)x22i + 2ptxux2i) +w2- 0(y 2 (. + p)y22i + 2ptyuy2i) + 2 Mii,o wii,o( J cii + P^2i)(3 ' i I - + P,y2(-)] (B.21) £{Re[( / i 2 - TM2*{)] } = E[u2iT(x2i + ptxu-2ptxux2A + w 2 i T ( y 2 i + ptyu-2ptyuy2i) + 2u2it Tw2i> T(x2i - ptxu){y2i - p,yu) ] Since XXi, Yu,X2i,Y2i are independent zero mean Gaussian r.v.'s, E{XjjYu) = 0, le {1, 2}, l^j. using (B.20), the variances of Re(/if / 0 M 1 ( ) and Re(/z2(- -pMfi) can be expressed as Appendix B . Derivation of the Means and Variances of the Random Variables in Equation (3.8) 56 E{Re[(hf10 Mu))2} = a2U0(a2D + p2a2D) E{Re[(h2i jMfi)}2} = a22i T(a2D + p2o2D) (B.22) and the covariance of Re(hUt 0 MXi) and Re(h2iTM2*i) can be expressed as E[(Rohfit0Mu )Ro(h2- TM*)] = 0 (B.23) It indicates that Re(hfi)0Mu ) and Re(h2j T M2*() are uncorrected and statistically indepen-2 dent.The variance of M, G M , where M - Re(/zf,- 0 Mu) + Re(h2i T M 2 *), can be derived as o2M = E{{Re[h0*(0)M0]}2] + E{{Re[hl(T)Mf]}2} (B.24) Since Du, D2i, Vu, V2i, Nt 0 and Nt T are zero mean independent Gaussian r.v.'s with 2 2 2 variances aD, av and aN, M, Re(/ifI ;0 V2i) , Re(h2i T V,*) , Re(/rf ( ;0 /V- 0) and Re(^2; r ^ t r ) a r e a l s o z e r o mean independent Gaussian r.v.'s. The corresponding variances are [a 2,- 0 +a2,-r](l + p 2 ) ^ , a 2 a]iQ , a 2 a 2 ( . r , o2N a2u 0 and a^a 2 , .^ . 57 Appendix C. Derivation of the Correlation Coefficient between the Estimated Channel Gains From 3.3.1, we assume that Gu and G2i are correlated with correlation coefficient . The estimated channel gain, HXi and H2i are zero mean complex Gaussian r.v.'s with variances 2 2 2 csH = <5G + G z where the channel estimation error, Zu and Z2i, are independent zero means 2 complex Gaussian r.v's with variances az . The correlation coefficient, ph, of Hu and H2i, is given by [3] Since E[GU G2*] = p^c^, also Zu and Z2i are independent Gaussian r.v.'s, E[GuZ2i*] = E[G2iZ2i*] = E[G2i*Zu] = E[GU*ZU] = E[ZuZ2i*] = 0, and hence the Ph = E[HuH*2i] jE[\Hu\2]E[\H2l\2] E[(Gu + Zu)(G2i + Z2i)*) 2oi (Cl) correlation coefficient ph can be expressed as 2 Ph = (C.2) 58 Appendix D. Derivation of Means and Variances of Random Variables in Equation (3.29) Here we derive the variances of Re(h*U(DU + D2i)), Re(h2i(D2i- Du)*), 7?e(fr*i,-iV- 0) and Re{h2iN*iT) as given in (3.29). h^, j = 1, 2, i = 1, M can be denoted by hi; - a.e J' = ui;+ jw,-,: where = a. .cos0„ and w •• - a . . s in0„. First of all, we prove J' Jl J' J J< Jl Jl Jl Jl Jl Jl ' r 2 that Du, D2i are correlated with correlation coefficient pd - (1 - pe)ps. From (3.27), we have Su = P2ehu + d\i Sli = P2eh2i + d2i (D.l) 2 2 2 2 2 2 where pe - <3Q/OH , du and d2i are samples of zero mean, variance aD =(1 - pe)oG complex Gaussian r.v.'s which are independent of Hu and H2i. The complex Gaussian r.v.'s Gu and G2i are correlated with the correlation coefficient p , i.e. E(GuG%i ) P, = 2 • ( D - 2 ) 2a G Using (D.l), the covariance of Gu and H2i can be expressed as E(GuHli) = E(GU G% +GuZli) = E(p2eHuH^ +DuH*2i) (D.3) since Appendix D. Derivation of Means and Variances of Random Variables in Equation (3.29) 59 E(GuZ%i) = 0, (DA) E(GUH%1 ) = E(HuH\i ) = 2pso2G, (D.5) Thus using (D.2) - (D.5), we have E(DuH%i) = 2 ( l - p 2 ) P j a c . (D.6) Similarly, the covariance of HXi and D2i can be expressed as E(HuD%i) = 2(\-p2e)psa2G. (D.7) Using (D.l), the covariance of GXi and G2i can be expressed as E{GuG\i) = E(p4eHuH^ +p2eDuH*2i +p2HuD*2i +DuD*2i) (D.8) Using (D.2), (D.5) - (D.7), we have E(DuD*2i) = 2(\-p2e)2psa2G (D.9) The correlation coefficient of DXi and D2l, pd, can be expressed as EWUD2*] Pd = jE[\Du\2]E[\D2i\2) = 2 ( l - P e 2 ) V G (D-1Q) 20-p e V c = ( l - P e 2 ) P , • Appendix D. Derivation of Means and Variances of Random Variables in Equation (3.29) 60 In the following, we derive the variances of Re(h^t (Du + D2i)), Re(h2i(D2i -Du)*) and prove that i?e(/zf(- (Dx• + D2i)) and Re'h2i(D2i -DXi)*) are independent. Since D u and D2i are correlated, zero mean complex Gaussian r.v.'s, we can express the samples of Du and D2i, du and d2i as d\i = Xu + Jy\i (D . l l ) d2i = x2i + jy2i 2 where xu, yx i,x2i,y2i are samples of zero mean correlated Gaussian r.v.'s with variances aD, i.e. Then we have EiX^] = £ [ T 2 . ] = oj, (D.12) E[XuYki] = 0 / = 1,2; k = 1,2 (D.13) E[XuX2i] = E[YuY2i] = pda2D (D.H) d u +  d2i = x u +  x2i + J(y\i + y2i) d2i~du =  x 2 i - x \ i + j(y2i-yn) (D.15) and Re(h^ (du + d2i)) = uu(xli + x2i) + wu(y]i + y2i) Re(h2i(d2i-du)*) = u 2 i ( x 2 i - x u ) + w 2 l ( y 2 i - y u ) The covariance of Re(hfi (DXi + D2i)), Re(h2i(D2i - Du)*) can be expressed as (D.16) Appendix D. Derivation of Means and Variances of Random Variables in Equation (3.29) 6 1 E[Re(hu (Du + D2[))Re(h2i(D2i-Du)*)] = E[uu(Xu + X2i) + wu(Yu + Y2i)][u2i(X2i-Xu) + w2i(Y2[-Yu)] . Using (D.12), (D.l3) and (D.14), (D.l7) can be reduced to E[Re(hfi (Du + D2i))Re(h2i(D2i-Du)*)] = 0 (D.18) This shows that Re{h^ (Du + D2i)) and Re(h2i(D2i - D, •)*) are uncorrelated and statistically independent with each other. The variances of Re(hfi (Du + D2i)), Re(h2i{D2i - D} •)*) can be written as E{Re[hti (Du + D2i)f} = E[u2u(X2u + X22i + 2XuX2i) + W2u(Y2u + Y22[ + 2YuY2i) + 2uuwu(Xu + X2i)(Yu+Y2i)] E{Re[h2l(D2i-Du)*]2} = E[u22i(X22i + X2r2XuX2l) + w22i(Y22[ + Y2r2YuY2l) + 2u2lw2i(X2l-Xu)(Y2l-Yu)] Using (D. 12), (D. 13) and (D. 14), (D. 19) can be reduced to E{Re[hu (Du + D2i))2} = 2(\+pd)aua2D (D.20) E{Re[h2i(D2i-Du)*]2} = 2(\+pd)a22ic2D 62 Bibliography [I] T. S. Rappaport, Wireless Communications Principles and Practice, Upper Saddle River, N.J.: Prentice Hall PTR 1996. [2] http://www.cellular.co.za, "Latest Global, Handset, Base Station, & Regional Cellular Statistics", Jan. 2003. [3] M. Schwartz, W. R. Bennett and S. Stein, Communication Systems and Techniques, New York: McGraw-Hill, 1966. [4] A. S. Acampora and J. H. Winters, "Systems applications for wireless indoor communications," IEEE Communications Magazine, vol. 25, no. 8, pp. 11-19, Aug. 1987. [5] Jakes, W. C , "A comparison of specific space diversity techniques for reduction of fast fading in UHF mobile radio systems," IEEE Transactions on Vehicular Technology, vol. VT-20, no. 4, pp. 81-93, November 1971. [6] D. G. Brennan, "Linear diversity combining techniques," Proc. IRE, vol. 47, pp. 1075-1101, June 1959. [7] S. M. Alamouti, "A simple transmit diversity technique for wireless communications," IEEE Journal on Select Areas in Communications, vol. 16, no. 8, pp. 1451-1458, October 1998. [8] 3GPP, "Physical channels and mapping of transport channels onto physical channels (FDD)," 3GPP TS25.211 version 3.12.0, Sept. 2002. [9] A. Hottinen, K. Kuchi and O. Tirkkonen, "A space-time coding concept for a multi-element transmitter," Proc. 2001 Canadian Workshop on Information Theory, June 2001, Vancouver BC, ISBN: 0-9697216-1-7. [10] X. Feng and C. Leung "Performance sensitivity comparison of two diversity schemes," IEE Electronics letters, vol. 36, no. 91, 27th April 2000 pp. 838-839. [II] Z. Lie, X. Ma and G. B. Giantess "Space-time coding and Kalman filtering for diversity transmissions through time-selective fading channels", Proc. MILCOM, Oct. 2000, Los Angeles, CA, pp.382-386. [12] T. K. Y. Lo, "Maximum ratio transmission," IEEE Transactions on Communications, vol. 47, no. 10, pp. 1458-1461, October 1999. [13] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 3rd ed., 1995. [14] W. C. Jakes, Ed., Microwave Mobile Communications, New York: Wiley, 1974. Bibliography 63 [15] C. Leung and X. Feng "Jointly optimal transmit and receive diversity," IEE Electronics letters, vol. 38, no. 12, 6th June 2000 pp. 594-596. [16] A. Papoulis, Probability, Random Variables and Stochastic Processes, New York: McGraw-Hill,1965. [17] P. Polydorou and P. Ho, "Error performance of MPSK with diversity combining in non-uniform Rayleigh fading and non-ideal channel estimation," Proc. IEEE Veh. Technol. Conf., 2000, Tokyo, Japan, pp. 627-631. [18] R. E. Ziemer and W. H. Tranter, Principles of Communications, 3rd edition, Boston: Houghton Mifflin, 1990. [19] J. N. Pierce and S. Stein, "Multiple diversity with nonindependent fading," Proc. IRE, vol. 48, pp.89-104,Jan. 1960. [20] I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, San Diego California: Academic Press Inc., 1994. [21] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover, 1972. [22] W. C. Y. Lee, Mobile Communications Engineering. New York: McGraw-Hill, 1982. [23] P. Z. Peebles, Probability, Random Variables, and Random Signal Principles. New York: McGraw-Hill, 1993. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0090897/manifest

Comment

Related Items