P E R F O R M A N C E A N A L Y S I S O F S O M E N E W A N D E X I S T I N G T R A N S M I T A N D R E C E I V E A N T E N N A D I V E R S I T Y S C H E M E S by XIAOYAN SARINA FENG A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F E L E C T R I C A L A N D C O M P U T E R E N G I N E E R I N G We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 2000 © Xiaoyan Sarina Feng, 2000 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 shal) not be allowed without my written permission. The University of British Columbia Vancouver, Canada Date Q-ct. H , l^e>0v DE-6 (2/88) ll Abstract A well-known method for symbol detection involving signals received over a number of independent diversity branches perturbed by additive white Gaussian noise is Maximum Ratio Combining ( M R C ) . Two other diversity schemes, Simple Transmit Diversity (STD) and Maximum Ratio Transmission (MRT), have recently been proposed. The performances of these three schemes are compared. The STD scheme, which is applicable only to two transmit antennas, is shown to have the same error performance as M R C with the same diversity order when perfect channel estimates are available. The MRT scheme with ./V transmit and one receive antennas is shown to have the same performance as M R C with N receive antennas, but M R T with one transmit and N > 1 receive antennas or MRT with N > 1 transmit and M > 1 receive antennas do not perform as well as M R C . The performance degradations of M R C , STD and MRT due to errors in estimating the channel parameters are analyzed and compared. It is found that STD is significantly more suscep-tible than M R C and MRT to errors in the channel estimates. An improved scheme for MRT (IMRT) and a new optimal maximum ratio transmission and combining scheme (MRTC) are proposed. In the IMRT scheme, the same transmit weighting functions are used as in MRT. At the receiver, M R C combining rules are used to choose the weights for the received signals. The MRTC scheme maximizes the SNR using optimal transmit and receive weighting factors. Simulation results indicate that both IMRT and M R T C provide significant performance improvements over MRT. iii Table of Contents Abstract ii Table of Contents iii List of Tables v Acknowledgment vi Chapter 1 Introduction 1 1.1 Frequency Diversity 3 1.2 Time Diversity 3 1.3 Space Diversity 4 1.3.1 Selection Diversity 5 1.3.2 Feedback Diversity 6 1.3.3 Equal Gain Diversity 6 1.3.4 Maximal Ratio Combining 7 Chapter 2 Performance Comparison with Perfect Channel Estimation 8 2.1 Review of Maximal Ratio Combining (MRC) 9 2.2 Review of Simple Transmit Diversity (STD) 12 2.2.1 Two-Branch STD 12 2.2.2 STD With Two Transmit and Receive Antennas 14 2.3 Review of Maximum Ratio Transmission (MRT) 15 2.4 Performance Comparison with Perfect Channel Estimates 18 Chapter 3 Performance with Imperfect Channel Estimation 23 3.1 B P S K Modulation 23 3.1.1 M R C 24 IV 3.1.2 STD 25 3.1.3 M R T 26 3.1.4 Channel Model 27 3.1.5 Numerical Results with BPSK modulation 28 3.2 QPSK Modulation 32 3.2.1 M R C 33 3.2.2 STD 35 3.2.3 M R T 36 3.2.4 Numerical Results with QPSK modulation 36 Chapter 4 Two New Diversity Schemes 39 4.1 Improved MRT (IMRT) Scheme 39 4.2 New Maximum Ratio Transmission and Combining (MRTC) Scheme 41 4.3 Numerical Results 45 Chapter 5 Conclusion 48 5.1 Main Thesis Contributions 48 5.2 Topics for Further Study 49 Glossary 51 Bibliography 56 Appendix A. Means and Variances of MRC and STD Decision Variables 59 Appendix B. Comparison of (3.5) and (3.9) 62 V List of Tables Figure 1.1 Principles of selection diversity from [2] 5 Figure 1.2 Basic form of feedback diversity 6 Figure 2.1 M R C with M receive antennas 10 Figure 2.2 The two-branch STD scheme with one receiver 13 Figure 2.3 STD with two receive antennas 14 Figure 2.4 MRT with N transmit and M receive antennas 16 Figure 2.5 Comparison of theoretical and simulation results 19 Figure 2.6 BER's of M R C and MRT schemes with diversity order of two 20 Figure 2.7 SER's of M R C , STD and MRT with diversity order of four. 21 Figure 3.1 BER's of M R C and STD schemes as a function of SNR for a randomly chosen set of parameter values 30 Figure 3.2 BER's of M R C and STD schemes as a function of ESR for SNR = 4 and lOdB 31 Figure 3.3 BER's of M R C and STD schemes as a function of SNR for four different ESR values 32 Figure 3.4 SER's of M R C and STD schemes as a function of SNR for ESR = -20 and - lOdB 37 Figure 4.1 IMRT with two transmit and two receive antennas 40 Figure 4.2 MRTC with two transmit and two receive antennas 42 Figure 4.3 SER performances of 4-branch M R C , MRT, IMRT and MRTC 46 Figure B . l Illustrating (B.5) 55 VI Acknowledgment I would like to express my sincere thanks and deep gratitude to my thesis advisor, Dr. Cyril Leung, for his guidance and encouragement. His critical reviews and many constructive suggestions were very essential to the completion of this work. This work was partially supported by NSERC Grant OGP0001731. Friends and fellow students have certainly made my studies here a memorable and interesting one. I would like to thank Mr. Kelvin Ho, Mr. Cyril Iskander, Mr. Lawrence Chen, Mr. Peter Chong, Mr. Shailesh Sheoran, Mr. Dave Martin and other people in the Communication Group for their support and help. Finally I would like to thank my parents, Mr. Xixian Feng and Mrs. Shuhua Liu, my sister, Ms. Xiaol i Feng and her family, for their constant love and encouragement. They, too, have played a significant role in the completion of this work. Chapter 1 Introduction 1 Chapter 1 Introduction In recent years there has been an explosive growth in the mobile communication market worldwide. This demand is expected to grow unabated over the next decade as new services are offered and new markets developed. Multipath fading is a major obstacle to the efficient and reliable transmission of data over many radio channels. Possible solutions to this problem are to increase the transmission power, antenna size, or antenna height [1]. These solutions may not be compatible with the need for portability and reduced energy consumption. Another standard technique which can be used to mitigate the effects of fading is diversity. The motivation behind diversity techniques is that we provide several independent paths, hence there are hopefully always some paths that may have strong signals so as to reduce the probability that all the signal components will fade simultaneously. Depending on the propagation mechanism, there are several effective diversity techniques. Independent transmission paths suitable for the diversity method could be obtained by using different frequencies, different transmission times or spatially separated antennas. In most scatter-ing environments, space diversity (antenna diversity) is a practical, effective and widely applied technique for reducing the effects of multipath fading [2]. A l l the discussion for diversity has been premised on the assumption that the fading processes among the diversity branches are mutually statistically independent. The absence of correlation between the branches is an important desired feature for diversity techniques, since it would not help the receiver to have additional copies of the signal if the copies are all equally poor [3]. In practice, there will be some cases in which this cannot be achieved, for example, insufficient antenna spacing (due to siting limitation in spaced antenna diversity). A number of papers have appeared in this subject area [4], [5], [6], [7]. Bit Chapter 1 Introduction 2 error rate (BER) performance of space diversity systems with channel correlation was studied in [4]. In [5], numerical results demonstrate the impact of the different correlation coefficients in combating fading and reducing co-channel interference (CCI). In [6], a maximum likelihood sequence estimation receiver structure was derived for the case of correlated diversity sources. A brief introduction to alternative diversity techniques is given in the latter part of this chapter. In Chapter 2, related studies found in the literature are reviewed. A classical approach, maximal ratio combining (MRC), involves the use of multiple antennas at the receiver. The signals received at the various antennas are weighted such that the signal-to-noise ratio (SNR) of their sum is maximized. The major problem with using M R C is the cost, size, and power of the remote units. Simple transmit diversity (STD) is a simple but effective scheme proposed by Alamouti [8]. In STD, a pair of symbols is transmitted using two antennas during the first time unit, and a transformed version of the pair is transmitted during the second time unit to obtain a MRC-like diversity. Two transmit and M receive antennas are used for the generalization of the STD scheme so as to provide a diversity of the order of 2M. A scheme called maximum ratio transmission (MRT) is suggested by Lo [9]. MRT can be generalized to any number of antennas for both transmission and reception. In Chapter 3, bit error rate (BER) performances for M R C , STD and MRT are compared. The B E R degradations due to imperfect channel estimates are analyzed. Both B P S K and QPSK modulation methods are considered. The analysis starts with the simplest case, two branch diversity, and then is generalized to multiple branches. Chapter 1 Introduction 3 In Chapter 4, an improved M R T (IMRT) scheme and a new optimal maximum ratio transmission and combining scheme (MRTC) are introduced. Symbol error rate (SER) perfor-mances of these two new schemes are presented and compared with those of M R C and MRT. A summary of the main results of this work and some suggestions for possible future work appear in Chapter 5. 1.1 Frequency Diversity In frequency diversity, the information bearing signals is transmitted on more than one carrier frequency. If the frequency separation is larger than the coherence bandwidth, independent fading variations can be assumed [10]. In [11], coherence bandwidth is defined as a statistical measure of the range of frequencies over which the channel can be considered "flat". Those spectral components passed through the "flat" channel are subject to an approximately equal gain and a linear phase shift. For the mobile radio case with a coherence bandwidth on the order of 500kHz, it has been measured that the separation between the branches has to be at least 1 ~ 2MHz [2]. The advantages of the frequency diversity over the space diversity is the reduction of the number of antennas. However, to achieve M branch diversity, the bandwidth and transmit-ting power required will be M times larger. Therefore, this technique is not commonly used for land mobile communication systems in which spectrum efficiency and power savings are important issues. 1.2 Time Diversity In time diversity, the information bearing signal is repeatedly transmitted so that the multiple repetitions of the signal undergo nearly independent fading, thereby providing diversity. Chapter 1 Introduction 4 It has been shown in [12], [13] that the required time slot interval is at least as great as the recipro-cal of the fading bandwidth, or 0.5/fd, in the mobile radio case to obtain diversity branch signals, where fd is the maximum Doppler frequency. Time diversity is effective for C D M A systems, where the multipath channel provides redundancy in the transmitted message. However, it is less effective when the channel is slowly varying because a very long time slot interval is necessary to obtain sufficient diversity gain. Moreover, when the mobile is stationary, we may obtain no diversity gain at all. 1.3 Space Diversity Space Diversity is another effective approach to combat multipath fading. It has histori-cally been the most commonly used form of diversity in mobile radio link base station [14]. Sufficiently spaced antennas are an attractive means of obtaining this diversity advantage since they do not incur bandwidth expansion. The method is based upon the principles of using two or more attennas in order to receive uncorrected signals. Conventional cellular radio systems consist of elevated base station antennas and mobile antennas 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 scatterers in the vicinity of the mobile suggests a Rayleigh fading signal. At the mobiles, an antenna spacing greater than X/2 is sufficient to achieve very low fading correlation between branches [10], [14], [15], whereas 50X, and 100A, are necessary at the base station [16]. Space diversity can be implemented at either the mobile terminal or base station, or both, depend-ing on the particular combining technique used and degree of the signal enhancement required. In [17], the research results show that theoretically, space diversity (with optimum combining) can substantially increase the capacity of most interference limited wireless communication systems. Chapter 1 Introduction The following combining techniques have been considered in the literature. • Selection Diversity • Feedback Diversity • Equal Gain Diversity • Maximal Ratio Combining 1.3.1 Selection Diversity Referring to Figure 1.1, M receivers are used to achieve M branch diversity. The diversity branch having the highest instantaneous signal to noise ratio (SNR) is connected to the output [2]. In practice, a selection diversity system cannot function on a truly instantaneous basis. It must be designed so that the internal time constants of the selection circuitry are shorter than the recipro-cal of the signal fading rate [11]. The drawback of this scheme is its sub-optimal performance since it does not use all of the branch channel information. Transmitter V V R R R receiver LOGIC T output best one of the M receivers Figure 1.1 Principles of selection diversity from [2]. Chapter 1 Introduction 1.3.2 Feedback Diversity 6 V Y control Comparator mean signal level Instantaneous signal level Receiver 1 Figure 1.2 Basic form of feedback diversity. As shown in Figure 1.2, the feedback diversity scheme is similar to selection diversity. A feedback link is provided to switch between transmitting antennas at the remote station within a limited amount of time delay. Instead of always using the best of M signals, the M signals are scanned in a fixed sequence until one is found which is above a predetermined threshold. This signal is then demodulated until it falls below a certain threshold and the scanning process is initiated again. This scheme avoids excessive switching when both antennas are in simultaneous fades [11]. The performance of this scheme is not as good as those obtained by other methods but it requires only one receiver. 1.3.3 Equal Gain Diversity In equal gain combining, signals from all the branches are coherently combined using the same weighting factor. The signals received over the diversity channels are co-phased and added up. The performance of this method is marginally inferior to that of maximal ratio combining and superior to that of selection diversity. Chapter 1 Introduction 7 1.3.4 Maximal Ratio Combining It was stated that in selection diversity only one of the diversity branch signals is used for demodulation. In contrast, in M R C , all the branch information is used to improve the overall performance. The signals from the received antenna elements are co-phased and weighted so as to maximize the overall SNR [1]. Owing to the recent development of the pilot signal-aided scheme as well as DSP technologies, the maximum combining schemes can be implemented with a simple hardware configuration [18], which is especially effective for base station to implement three branch diversity or more. Chapter 2 Performance Comparison with Perfect Channel Estimation 8 Chapter 2 Performance Comparison with Perfect Channel Estimation The maximal ratio combining (MRC) approach uses a maximal-ratio combining receiver to process the signals received at multiple antennas. The signals from the received antenna elements are cophased and weighted according to their individual signal voltage to noise power ratios. The realization of this combiner is based on the assumption that the channel state is known perfectly [19]. In some applications, e.g. the third generation cellular communication systems currently under development, such a receive diversity scheme may not be desirable for the mobile handsets because of cost, size and power considerations. A simple transmit diversity (STD) scheme [8], which uses two transmit antennas and M receive antennas, was proposed. There is no feedback required from the receiver to the transmitter. It was shown that, for a fixed level of radiated power per transmit antenna, this STD scheme has the same BER as M R C with the same diversity order. The STD scheme can be generalized so as to include any number of receive antennas, but cannot be easily extended to more than two transmit antennas. The maximum ratio transmission (MRT) scheme [9] can be applied to any number, N, of transmit antennas and any number, M, of receive antennas, even though feedback is required from the receiver so that the transmitter can estimate the channel. For convenience, we refer to this a s a ( i V x M ) MRT scheme. Those aspects of the M R C , STD and MRT schemes which are necessary in this study are briefly reviewed in this section. Following [8], a complex baseband representation of the systems Chapter 2 Performance Comparison with Perfect Channel Estimation 9 is used. The channel diversity branch from antenna n at the transmitter to antenna m at the receiver is denoted by hnm = anme nm, in = 1...N, m= I...M), where anm is the amplitude gain of the diversity branch and 0 is the phase distortion introduced by the channel. Systems that coherently combine independent signals from spatially separated antennas have better carrier statistics and less random F M than selection diversity systems [2]. Coherent combining systems do not suffer degradation from phase transients that are inherent in antenna-switching systems. It has been shown in [12] that absolute phase coherent detection gives the best theoretical B E R performance for BPSK and QPSK with a given number of diversity branches under flat Rayleigh fading conditions. Therefore, Coherent detection is considered to be more desirable when a large number of diversity branches are employed, and is used for those schemes reviewed in this section. Noncoherent detection is also used in practice. In this study, we use coherent detection. 2.1 Review of Maximal Ratio Combining (MRC) Figure 2.1 shows the baseband representation of the M R C scheme with a diversity order of M. The signal received on branch i corresponding to the transmission of a signal s0 is r\i,MRC = hlis0 + n i > i = l - M , (2.1) where {«•, i = I...M} are outcomes of independent complex Gaussian random variables (r.v.) representing noise and interference. Through the thesis, we use uppercase letters to denote r.v.'s and the corresponding Chapter 2 Performance Comparison with Perfect Channel Estimation 10 lowercase letters to denote their samples. The combined signal sQ M R C is then defined as channel 1 ^ur^u estimator " 2 , channel estimator \h, u r~\nn >0 channel ft-iMf-tym estimator *\jy Receiver >) H - K * — H •Kx—H *S0,MRC Figure 2.1 M R C with M receive antennas. M ~s0,MRC ~ ^^\ir\i,MRC i= 1 M = yLh*i(huso+ni) i= 1 f M \ M Vi = l J i = 1 (2.2) where ^ • denotes the complex conjugate of hu. The theoretical analysis of the error perfor-mance for a binary digital communications system with diversity has been discussed in [1][19] assuming Rayleigh faded diversity channels. The instantaneous output signal to noise ratio (SNR) from the combiner was shown to be the sum of the instantaneous SNR's on the individual branches, Chapter 2 Performance Comparison with Perfect Channel Estimation 11 M Y = £ Y I , - . ( 2 - 3 ) I = 1 With coherent binary PSK, the output of the maximal ratio combiner can be expressed as a single decision variable in the form UMRC = Re(So,MRc), ( 2 . 4 ) where Re(So, MRC) denotes the real part of So, MRC • The data bits to be transmitted are assumed to be independent and equally likely to be 0 or 1. The B E R is the probability of that UMRC is less than zero, i.e. P(l) = 2 ( V 2 Y ) , (2.5) where Y = 2 Vi = 1 J (2.6) 2 where £ is the signal energy, and <5N is the variance of the real or imaginary components of Nt. 2 The noise variance o"yy has been assumed to be identical for all branches. It is assumed that the average energy gain of each diversity channel is the same, i.e. E(a1{) = x. The average SNR per branch is then written as Chapter 2 Performance Comparison with Perfect Channel Estimation 12 (2.7) 'N The p.d.f of r , / r ( y ) , is derived in [1][19] as 1 M - l f M - 1 Y -Y/YO / r m " ( M - l ) ! M e i o (2.8) The B E R can be evaluated by averaging the conditional error probability given in (2.5) over the fading channel statistics in (2.8), i.e. = J j > ( Y ) / r ( Y W Y (2.9) The integral of (2.9) can be simplified as indicated in [20] M -Pe = 2 d - ) M '"^ \(M-\+i i = 0 V i (2.10) where u = l + Y o (2.11) 2.2 Review of Simple Transmit Diversity (STD) The system with two transmit and one receive antennas is introduced in this section. The generalization to a system with diversity order of 2M can be achieved by using the combiner for Chapter 2 Performance Comparison with Perfect Channel Estimation 13 one receive antenna and adding the combined signals from M receive antennas. 2.2.1 Two-Branch STD In the STD scheme, two bits represented by s0 and s-^ are sent simultaneously during two consecutive bit periods. In the first bit period, sQ is sent from antenna A and s1 is sent from * * antenna B . In the second bit period, the signals sent from antennas A and B are -Sj and s0 respectively. s0 -Sj* noise Figure 2.2 The two-branch STD scheme with one receiver. Figure 2.2 shows the baseband representation of the STD scheme with a diversity order of two. Assuming that the channel fading does not change significantly during two consecutive bit periods, the received signals are r0,STD - ^\lsQ + ^ l ^ l + n l * * r\,STD = ~ h \ \ s \ + ^2150 + n2 (2.12) Chapter 2 Performance Comparison with Perfect Channel Estimation 14 where nx and n2 are complex random variables representing receiver thermal noise and interfer-ence. (2.13) It is proposed in [8] that the decoding of s0 and s] be based on sQ S T D and J, S T D respec-tively, where _ * * SO,STD = hur0STD + h 2 l r l S T D 2 2 * * = ( a n + a 2 1) SQ + hnn^ + h21n2 _ * * S1,STD = ^ 2 1 r 0 , STD ~ h\\r\, STD 2 2 * * = (OCJJ + a 2 1) 5j - hlln] + h2]n2 • Those combined signals are then sent to the decision device. The STD scheme yields the same BER as M R C for a fixed value of the power radiated per transmit antenna assuming that the channel gains hu and h2X can he perfectly estimated by the receiver. 2.2.2 STD With Two Transmit and M Receive Antennas Two transmit and M receive antennas can be used to yield a diversity order of IM. The case of two transmit and two receive antennas is shown in Figure 2.3. The encoding and transmission sequence of the information symbols are identical to the case of one receiver. The received signals at the two receive antennas are: R\,STD = ^\lS0 + H2\sl + " l * * R2,STD = -hus] + h2ls0 +n2 - u u ( 2 , 1 4 ) R3,STD ~ " ^ O + "22^1 + N3 * * R4, STD = ~^\2sl + ^22S0 + N4> Chapter 2 Performance Comparison with Perfect Channel Estimation 15 noise channel w estimator hji h21 ' i r S0,STD —• combiner w decision device w k i \ S1,STD h22 channel w estimator Figure 2.3 STD with two receive antennas. where rl S T D and r3 S T D are the signals received in the first symbol period, r2 5 r z ) a n d r4 S T D are the received signals corresponding to the second symbol period. The output signals from the combiner are achieved as: ^0, STD = ^ l l r l , S 7 D + ^21 r 2, STZ) + ^12 r 3, S7T) + ^22 r 4, STD 2 2 2 2 * * * * = (ocjj + oc 1 2 + oc21 + oc2 2) 5 0 + hntix + / i 2 1 « 2 + ^i2 n3 + h22nA _ * * * * ^l,STD = n2\r\,STD~ nl\r2,STD+^22r3, STD ~^l2r4, STD 2 2 2 2 * * * * = ( a n + a 1 2 + a 2 1 + a 2 2 ) .Sj - h l l n 1 + h2Xn2-hun4 + h22n3 . (2.15) Equation (2.15) shows that those two combined signals are the summation of the combined signals from each receive antenna. The combiner with two transmit and M receive antennas can be built by using the combiner for each receive antenna, and then adding the combined signals from all the receive antennas to obtain a diversity order of 2M. Chapter 2 Performance Comparison with Perfect Channel Estimation 16 2.3 Review of Maximum Ratio Transmission (MRT) The MRT scheme [9] can be applied to a system with N transmit and M receive antennas. The channel can be represented by a channel coefficient matrix, H= hu ... hlM hNl ... hNM (2.16) where hnm, n = 1, 2, N,m = 1, 2, M represents the channel from antenna n at the transmitter to the antenna m at the receiver. Figure 2.4 MRT with N transmit and M receive antennas. As shown in Figure 2.4, the source signal s is weighted by a ( N x 1) transmit weighting vector V before transmission, where V is defined as Chapter 2 Performance Comparison with Perfect Channel Estimation 17 V = -a{HW)H. (2.17) where a = \HW\, which is the length of the vector HW, is a normalization factor to ensure that the transmission power is normalized to an average value, and W is a (M x 1) receive weighting vector. The superscript H in (2.17) denotes the Hermitian operation, i.e. complex conjugate transposed. The transmitted signal vector can be expressed as st = -(HW)H, (2.18) _ a and the received signal vector as r = -{HW)HH + n, (2.19) a where n = [nl, nM] is the noise vector. The received vector components are then weighted and sent to the combiner. The estimate of the signal is given by J = -(HW)HHW + nW MRT A \ -> - — ( 2 2 Q ) = as + nW The overall output SNR can be written as WW " , , Z h i i= 1 2 where y 0 M R T = • To maximize the y in (2.21), It is concluded in [9] that W should satisfy Chapter 2 Performance Comparison with Perfect Channel Estimation 18 the condition \w w2\ = • • • = \WM\ > A N D N X hPiKt i = 1 (2.22) N i= 1 where p and q take on values in the set { 1, 2, ..., Af} . 2.4 Performance Comparison with Perfect Channel Estimates The BER's of M R C , STD and MRT with diversity orders of two and four are compared in this section. It is assumed that each channel undergoes independent slow fading, whose amplitude is Rayleigh distributed. The interferences and noises are modeled as additive complex white Gaussian random variables. The complex channel gains hnm, n = 1, 2, N, m = 1, 2, M can be perfectly estimated by the receiver and/or the transmitter. The theoretical B E R performance of M R C is given by (2.10), and plotted in Figure 2.5. The simulation results for two-branch M R C and STD are also shown and agree very closely with the theoretical M R C curve. Figure 2.6 shows BER's for two-branch M R C and MRT with uncoded coherent BPSK. It can be seen that the diversity schemes provide substantial improvement over the no diversity case. The improvement increases with SNR. For BER= 10 , there is an improvement of about 8 dB. M R C and STD are about 0.5 dB more efficient than M R T ( N x M = 1 x 2) at BER= 10"2. From Figure 2.5 and Figure 2.6, we can see that STD and (N x M = 2 x 1) MRT have the same perfor-Chapter 2 Performance Comparison with Perfect Channel Estimation 19 10 10 -4—» o PQ 10 10 -i 1 1 1 r STD(NxM=2xl) simulalion results o MRC(NxM= 1 x2) simulation results + theoretical refills SNR (dB) 10 12 14 Figure 2.5 Comparison of theoretical and simulation results, mances as (N x M = 1 x 2 ) M R C . This is proved at the end of this section. It is interesting to note that the BER performance of (N x M = 1 x 2 ) MRT is worse than that of (N x M = 2 x 1) MRT. The symbol error rate (SER) curves for diversity order four M R C , STD and MRT with Q P S K modulation are plotted in Figure 2.7. It can be seen that M R C ( / V x M = 1 x 4 ) , Chapter 2 Performance Comparison with Perfect Channel Estimation 20 i r I i n " I ' ' i I i I I I I I i I i I I I I I 1 1 1 1 1 1 1 1 1 1 0 2 4 6 8 10 12 14 SNR (dB) Figure 2.6 BER's of M R C and MRT schemes with diversity order of two. MRT (TV x M = 4 x 1), and STD(N x M = 2 x 2 ) have the same performances for a given value of the average power radiated per transmit antenna as shown below. For MRT with diversity order of four, the SER performances are given for three different cases (i.e. (N x M = 1 x 4 ) , (N x M = 2 x 2) and ( J V x M = 4 x l ) ) . Comparing these cases, it can be seen that MRT with only transmit diversity ( i V x M = 4 x l ) gives the same SER performance as M R C , and that Chapter 2 Performance Comparison with Perfect Channel Estimation 21 in" I i i i I i i i I L — i 1 1 1 1 1 1 1 1 0 2 4 6 8 SNR (dB) Figure 2.7 SER's of MRC, STD and MRT with diversity order of four. MRT with both the transmit and receive diversity (N x M = 2 x 2) gives the worst SER perfor-mance. The performance of M R T wi th (NxM = 1 x 4 ) is i n f e r i o r to that of (N x M = 4 x 1) MRT and superior to that of (N x M = 2 x 2). For SER= 10" 3, there is a 0.9 dB degradation for M R T ( N x M = 1 x 4 ) and 1.3dB for M R T ( N x M = 2x2) compared to Chapter 2 Performance Comparison with Perfect Channel Estimation 22 M R T ( i V x M = 4 x 1). From Figure 2.6 and Figure 2.7, we notice that for (A^x 1) MRT, the error probability is the same as that of M R C with the same diversity order. For (TV x 1) MRT, the weighting function, W j , at the receiver is set to 1 for convenience. Then, we have from (2.21) l = \ a , (2.23) M where a2 = \H\2 = ocfj. The overall SNR y can then be written as i = l I ( M ' ^ °JVV« = 1 J (2.24) which is the same as (2.6), the output SNR from M R C combiner. From (2.9), we can see that Pe is a function of fr(y). Since the output SNR p.d.f. for both (1 x N) M R C and ( i V x l ) MRT are the same, they have the same error performance. Chapter 3 Performance with Imperfect Channel Estimation 23 Chapter 3 Performance with Imperfect Channel Estimation It is of practical importance to study the performance degradations which result from the use of imperfect channel estimates. The BER's of M R C , STD and MRT with channel estimation errors are now analyzed. Two modulation methods B P S K and QPSK are considered in the analysis. We begin with the performance analysis for those three schemes. The channel model to be used in the analysis is then described. Performance results are compared in the last section. A two-branch diversity model is discussed in this section. The estimated complex channel gains are expressed as Km = hnm + znm,n,m= 1,2 , (3.1) where znm = $nme^"m represents the estimation error for the diversity branch from the n th transmit antenna to the m th receive antenna. 3.1 BPSK Modulation We first derive the B E R for given values of hnm, n, m= 1, 2 and znm, n, m= 1,2. The results can then be used to obtain the B E R when hnm and znm are drawn according to any arbitrary probability distribution. It can be shown in [21] that the maximum likelihood decision rule, which minimizes the BER, is equivalent to choosing sQ = 1 if Re(s0) > 0 and choosing sQ = -1 otherwise, where denotes the combined signal. Chapter 3 Performance with Imperfect Channel Estimation 3.1.1 MRC 24 The decision r.v. for M R C is denoted by UMRC = RC(SQ M R C ) . For given values of sQ, h n , h l 2 , z u and z 1 2 , we have from (2.2) and (3.1) that - * * ^0,MRC = ( ^ l l + Z l l ) r0, MRC + (^ 12 + Zu) r\,MRC 2 2 * * = ( « H + "12)^ 0 + ^ l l ^ l l ^ O + 1^2^ 12-^ 0 ( 3 , 2 ) + (hn+zn)Nl + (hn + z n ) N 1 . After simplification, the mean and variance of the decision r.v. can be written as E(UMRC) = [(a?, + a i 2 ) +oCi iP 1 1 cos(0 n -4> H ) and ° u m c = [ a 2 1 + (3 2 1 +2a 1 1 (3 1 1 cos (e 1 1 - ( | ) n ) + a 2 2 + p 2 2 + 2a 1 2 (3 1 2 cos(9 1 2 - ( l ) 1 2 ) ]a^ Since UMRC has a Gaussian distribution, the BER is given by where (3.3) (3.4) a = (a2n + a 2 2 ) + a 1 1 p 1 1 c o s ( 0 1 1 -tyn) + a 1 2 p 1 2 c o s ( e ] 2 - ( t ) 1 2 ) , (3.6) Chapter 3 Performance with Imperfect Channel Estimation 25 1 f° - y 2 / 2 and <2(JC) = - = e dy. A detailed derivation of (3.3) and (3.4) is given in Appendix A . 3.1.2 STD Recall that in the STD scheme, two bits sQ a n d * i are simultaneously transmitted. By symmetry, the BER's for both bits are equal so that we only need to consider the B E R for one of the bits, say bit s0. Denote the corresponding decision r.v. by USTD = Re(S0i STD). For given values of s0, sx, hn , h2l, zu and z2l, we have from (2.13) and (3.1) that So, STD = (^11+^n) rQ,STD + (^21 + z 2 1 ) r 1, STD = (<A\ +U2\)SQ + ZuhUs0 + z2\h2\sQ + z n h 2 \ s \ (3-7) * * - z2lhnsx+{hu+zn) Nl + (h2l+z2l) N2 . After simplification (see details in Appendix A.) , it can be shown that the mean of the decision r.v. can be written as E(USTD)= [ ( a 2 j + a 2 1 ) + a 1 1 ( 3 1 1 c o s ( e 1 1 - ( j ) n ) + a 2 1 p 2 1 c o s ( 0 2 1 - 4 i 2 1 ) ] 5 o (3.5) + [ a 2 1 p n c o s ( 0 2 1 — <t> 11) — a 1 1 p 2 1 c o s ( 0 1 1 -(t)^ )]^ !. 2 The variance of USTD can be shown to be the same as the variance, ^uMRC' °f ^'MRC a s g i v e n m (3.4). Since USTD has a Gaussian distribution, the BER for STD is given by (3.9) where a is given by (3.6) and Chapter 3 Performance with Imperfect Channel Estimation 26 b = a 2 1 (3 1 1 cos(6 2 1 - ( ) ) 1 1 ) -a 1 1 (3 2 1 cos(0 1 1 -<|>21). (3.10) 3.1.3 MRT It was shown in Section 2.4 that the B E R performance for the (7V x M = 1 x 2 ) case is worse than that for the {N x M = 2 x 1) case. Furthermore, the (TV x M = 2 x 1) case has the same performance as two-branch M R C and STD, when perfect channel estimates are available. The (7V x M = 2 x 1) case is considered in the following theoretical analysis to compare the system sensitivity performance with M R C and STD. From the description of M R T ( N x M = 2 x l ) in Section 2.3, the weighting functions at the transmitter can be written as V i = v 2 = hu * L2\ The weighting function at the receiver can be normalized to 1 in this case. When the channel estimates are erroneous, the estimated v., i = 1,2 can be expressed as (fci i+Zn) v, = J\hn+zn\2 + \h2l+z * (h2l +z 2 1 ) 2 211 (3.12) V2 = ^ n + z n f + l ^ i + ^ i l 2 From Section 2.3, the combined signal r.v. can then be written as Chapter 3 Performance with Imperfect Channel Estimation 27 So, MRT = v l s 0 h u + v 2 s Q h 2 l + /Y j \hu\2 + hnzu + \h2\\2 + h2lz2\ (3- 1 3) *l\hu+zn\2 + \h2l+z2l\ The corresponding decision r.v. is denoted as UMRT = Re(Soi MRT) • For given values of s0, hn , h2l, zu and z 2 1 , we have from (3.13) that =vrr N h l ^ + l ^ l ^ + ^ ^ l l ^ l l + / ?2l4) E(UMRT) = 1 2 2 *0 # 1 1 + Z l l | +1^ 21+2211 _ [(«?i +«2i) + « i i ! 3 n c o s ( 9 1 1 - 0 1 1 ) + a 2 1 (3 2 1 cos(9 2 1 - ( | ) 2 1 ) ]5 o *lan + P i i + 2 a 1 1 P 1 1 c o s ( e , 1 -<(>„) + a 2 1 + p 2 1 + 2a 2 ] (3 2 1 cos(0 2 1 -<|>21) 2 2 and G f ; = G N . With (3.4) and (3.6), the mean o f UMRT can be expressed as (3.14) E(UMRT) = • (3.15) The BER for MRT is then given by Pe,MRT = Q[^-\ (3-16) 3.1.4 Channel Model As described in Section 2.4, a Rayleigh faded channel model is assumed. The channel gains h , n= 1 ...N, m= 1 ...M are then modeled as complex Gaussian r.v.'s. The fading Chapter 3 Performance with Imperfect Channel Estimation 28 processes among the (N x M) diversity channels are assumed to be independent. At the receiver, an ideal coherent detector is assumed. The signal in each branch is corrupted by an additive zero-mean white Gaussian noise process. 3.1.5 Numerical Results with BPSK modulation From (3.5) and (3.16), it can be seen that (N x M = 1 x 2 ) M R C and {NxM = 2x\) M R T have identical channel error sensitivity performances. For two-branch STD, using the symmetry of the Q(.) function, it can be shown from (3.5) and (3.9) that PetsTD-^e,MRC' STD has a higher BER than M R C and M R T ( N x M = 2 x 1), if a is positive. Details are given in Appendix B . From (3.6), a is positive for $ik<aik,i = 1,2, k = 1,2, since c o s ( . ) > - l . Typically, the magnitude of the estimation error will be small compared to that of the channel gain and the STD will have a higher BER. For presentation of the numerical results, we consider two cases. For case 1, the channel gains and estimation errors are fixed with values chosen at random other than to ensure $ik < aik, i = \,2,k = 1, 2. The BER's for M R C and STD can be calculated using (3.5) and (3.9) respectively. For the second case, a Rayleigh faded channel is used and the channel estimation errors zik, i = 1,2, k = 1,2 are modeled as samples of independent complex Gaussian r.v.'s. The estimate hnm, n = 1,2, m = 1,2, gain amplitude and phase shift of the n —> m th channel can be derived either from the transmission of a pilot signal or from demodulation of the information bearing signals received in previous signaling intervals [19]. The BER's for M R C and STD can Chapter 3 Performance with Imperfect Channel Estimation 29 then be obtained by averaging (3.5) and (3.9) respectively over the probability distributions for aik, Bilc, $ik, and §ik, i = \,2,k= 1,2. These samples are al l independent. The samples aik,$ik,i = \,2,k= 1,2 are Ray le igh distr ibuted and Qik,fyik,i = 1,2, k = 1,2 are uniformly distributed in [0, 2n]. 2 The variances of the real and imaginary components of Hik and Zik are denoted by <3H and a z . The signal-to-noise ratio (SNR) and the estimation error-to-signal ratio (ESR) are 2 2 2 2 defined as (EBGH)/<5N and GZ/GH respectively, where Eb is the bit energy. Since \Hik\ has 2 2 zero mean, here we have oH = E[\Hik\ ]. Figure 3.1 shows the BER as a function of SNR for Case 1. For the parameter values _3 shown in the figure caption, it can be seen that at a BER of 10 , STD is worse than M R C by about 1.5 dB. The results for Case 2 are shown in Figure 3.2 and Figure 3.3. When perfect channel estimates are available, M R C and STD have the same BER, given by (2.10). For two branch diversity 1 3 / SNR If SNR Y / 2 r-i . + 7 T - T ^ • ( 3 - 1 7 ) e, perfect 2 4^1+ SNR 4 U + SNR) In Figure 3.2, the BER's of the M R C and STD schemes are plotted as a function of ESR for SNR = 4 and 10 dB . From (3.17), the B E R values of SNR = 4 and 10 dB are 1.69 x 10"2 Chapter 3 Performance with Imperfect Channel Estimation 30 \ \ \ I A " I i i i I i i i l i i i I i i i 1 1 1 1 — d > 0 2 4 6 8 10 SNR (dB) Figure 3.1 BER's of M R C and STD schemes as a function of SNR for a randomly chosen set of parameter values: hQ = - 0.43 - 1.66j, hx = 0.13 + 0 . 2 9 z 0 = - 0.811 + 0.842j, z, = 0.841 - 0.027j. and 1.6 x 10 respectively. It can be seen that the B E R increases quite rapidly with ESR for ESR > -10 dB . For SNR = 10 dB and BER = 10~2, there is a 2 dB degradation for STD compared to M R C . Chapter 3 Performance with Imperfect Channel Estimation 31 10 10 -|—i—i—i—|—i—i—i—|—i—r -1—|—i—i—i—|—i—i—i—f , -"'MRC4 STD BER=1.6xl0' 3 with perfect channel est. -1 I I 1 I I I I I I L -20 -18 -16 -14 -12 -10 ESR (dB) Figure 3.2 BER's of M R C and STD schemes as a function of ESR for SNR = 4 and 10 dB . Figure 3.3 shows the BER's of the M R C , STD and MRT (TV x Af = 1 x 2 ) schemes as a function of SNR for four different values of ESR. The curve for MRT(iV x M = 2 x 1) with perfect estimate appears in Figure 2.6 and has been omitted from Figure 3.3 to reduce cluttering. At low ESR values, estimation error tend to be small and there is little difference in the BER's Chapter 3 Performance with Imperfect Channel Estimation 32 between M R C and STD. At ESR = -10 dB and BER= 10~2, there is a 3.5 dB degradation for STD and 0.6 dB for M R T ( N x M = 1 x 2 ) compared to M R C . It can be seen that the difference increases fairly rapidly with SNR. From Figure 2.6, we can see that with perfect channel estimation, STD has better B E R performance than MRT(7V x M = 1 x 2 ) ; however, with imperfect channel estimation, MRT may have a lower B E R than STD. For ESR •= -20 dB , the B E R performance for STD is still better than that for MRT, but the difference decreases with SNR. It can be seen that for SNR > 10 dB , the B E R for STD is almost the same as that for MRT. For ESR = -10 dB , STD has a better B E R performance than M R T only for SNR < 2 dB . We concluded that STD is more sensitive to channel estimation errors than M R C and MRT. It is interesting to note that M R C and MRT show similar sensitivity to channel estimation errors. 3.2 QPSK Modulation The performance sensitivity to channel error estimates for M R C , STD and M R T with QPSK modulation is now considered. We first derive the symbol error rate (SER) for given values of znm, n= \...N,m= 1 ...M. The results are then used to obtain SER when znm is drawn according to any arbitrary probability distribution. Gray coding is used in this study. For a given symbol error, the most probable number of bit errors is one, subject to the mapping constraint [22]. In the analysis below, we make use of the fact that coherent demodulation ideally results in the two messages being separated at the outputs of the quadrature mixers [23]. Thus, the transmit-ted signal for a QPSK system can be viewed as two binary PSK signals. Chapter 3 Performance with Imperfect Channel Estimation 33 T 1 1 1 1 1 1 1 1 1 1 j 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r Figure 3.3 BER's of MRC and STD schemes as a function of SNR for four different ESR values: (i) ESR = - o o dB, i.e. perfect estimate, MRC and STD (ii) ESR = -20 dB (Hi) ESR = -10 dB (iv)ESR = -3 dB 3.2.1 M R C For given values of sQ, hn , h n z n and z 1 2 , the combined signal can be written as in Chapter 3 Performance with Imperfect Channel Estimation (3.2). After simplification, the means and variances of the decision r.v.'s can be written as 34 E[Re(S0,MRc)] = [(oc n + a 2 2 ) +a 1 1P 1 1cos(0 1 1-(t) 1 1)+a 1 2p 1 2cos(e 1 2-(t) 1 2)]5 o 2 2 (3.18) E[Im(S0tMRC)] = [ ( a n + a 1 2 ) + a 1 1 p 1 1 s i n ( e 1 1 - ( t ) 1 1 )+a 1 2 P 1 2 s in ( e ] 2 -4 ) 1 2 ) ]5 0 , where Re(So, MRC) denotes the real part of So, MRC a n a " Im(Sot MRC) denotes the imaginary part of S0, MRC and °MRC = [ a 2 i + p 2 i + 2 a n p 1 1 c o s ( 0 1 1 - ( t ) 1 1 ) + a 2 2 + p 2 2 + 2 a 1 2 p 1 2 c o s ( 0 1 2 - 4 ) 1 2 ) ] a 2 v , (3.19) 2 which is the same as Gn in (3.4). Since both Re(sMRC) and Im(~sMRC) are Gaussian distributed, the BER ' s of the two symbol bits are given by PMRC, b «• eGr-) P MRC,bit! = Q{~Z J' (3.20) where c = a n + a22 + 7 2 a i i P ] 1 c o s ( 0 n - § n + 7 t / 4 ) + ^ a i 2 p 1 2 c o s ( 0 1 2 - < j ) 1 2 + n/4) 2 2 ( 3 - 2 1 ) d = a n + a ] 2 + 72anp ] 1 s i n ( 0 1 1 -(t>n + 7 t / 4 ) + ^ a i 2 p 1 2 s i n ( 0 1 2 - ( | ) 1 2 + 7i /4) . The SER can then be derived as Chapter 3 Performance with Imperfect Channel Estimation 35 PMRC, QPSK ~ 1 - ( 1 - PMRC, bitO( 1 ~ P MRC, bitl) • (3.22) 3.2.2 STD For given values of s0, s, , hu , h2i, z n and z 2 1 , we have the combined signal r.v. S T D as in (3.7). After simplification, the means for the decision symbol bits can be expressed as E[Re(S0!sTD)]= [(an +o t2 i ) +oc 1 1p 1 1cos(0 1 1-4) 1 1) + a 2 1 | 3 2 1 c o s ( e 2 1 - ( | ) 2 1 ) ] 5 o +[a 2 1(3 1 ]cos(0 2 ]-(|) 1 1)- a n (3 2 1 cos(0 1 1 - ( | ) 2 1 ) ]5 1 E[Im(S0tSTD)] = [(a 2 ! + a 2 1 ) + a 1 1 P 1 1 s i n ( 0 1 1 -<t)11)+a21(321sin(021 -4> 2 1 ) ]5 0 +[a 2 1 p H s in(0 2 1 - ( ) ) „ ) - a 1 1 p 2 1 s in(0 1 1 - ( ] ) 2 1 ) ] 5 1 . 2 2 The variance O^TD ^ ^ e s^me as GMRQ in (3.19). The BER s of the two symbol bits are given as 4 ^MRCJ 4 ^MRCJ 4 \-°MRCS * ^°MRC 4 y^MRcJ 4 y°MRCJ 4 ^°MRCS 4 V ° M R C 2 (3.24) where el = V2a 2 i P n cos(0 2 1 - § n + 7t/4)-A/2anp 2 1cos((|) 2 1 - 0 U + K / 4 ) e2 = 72a 2 ip ] ] cos(0 2 1 - ( | ) n -7 t / 4)-72anP 2 1 cos ( ( l )2 1 - 0 ^ -n/4) 1 1sin(0 2 1 - ^ n + 7t /4)-72anP 2 1 sin((|)2 1 - 0 n +K/4) f2 = ^ a 2 i p ] 1 s i n ( 0 2 1 - ( ( ) , ] - 7 t / 4 ) - ^ a n P 2 1 s i n ( ( | ) 2 1 - 0 i i - 7 t / 4 ) c and d are given in (3.21). The SER can then be written as Chapter 3 Performance with Imperfect Channel Estimation 36 PSTD, QPSK = 1 - ( 1 - PSTD, bitO^ ~ P STD, bill) • (3.26) 3.2.3 MRT The outcome of the combiner has been given in (3.13). The means of i\e(So, MRT) a r*d Itn(So, MRT) can be written as [(a2j + a2!) + a 1 1 (3 1 1cos(e 1 1 -$u) + a 2 1 p 2 1cos(0 2 1 - <)) 2 1)]J 0 E[Re(S0yMRT)]= - 2 2 Jan + P n + 2a 1 1 p 1 1cos (0 1 1 -$n)+a2l + p 2 1 + 2a 2 1P 2 1cos(0 2 1-4> 2 1) [(a2j + a 2 1 ) +a 1 1P 1 1sin(6 1 1 -<^ n) + a 2 1 p 2 1 sin (0 2 1 - ( ) ) 2 1 ) ] 5 0 £[/m(5 0,Afi?r)] = , 2 2 : f n + Pn + 2a 1 1 p 1 1cos (e 1 1 -(j)u)+a21 + p 2 1 + 2a 2 1 p 2 1 cos (6 2 1 - <|>21) (3.27) 2 The variance for those two decision bits are identical to oN. Thus, the BER's for those two symbol bits can be derived as PMRT,bit\ - 2^ P MRT, bit! ~ Q MRC d °MRC (3.28) where c and d are as given in (3.22). The SER can then be obtained as PMRT, QPSK = 1 ~ ( 1 - PMRT, 1 ~ PMRT, bill) (3.29) 3.2.4 Numerical Results with QPSK modulation We can see from (3.20) and (3.28) that MRT ( i V x M = 2 x l ) has the same B E R and Chapter 3 Performance with Imperfect Channel Estimation 37 10 t3 Pi l-O i-w 1 -: >^ 10 00 —i 1 r _L J I I L_ 4 6 S7V/J (dB) 10 Figure 3.4 SER's of M R C and STD schemes as a function of SNR for £Sfl = -20 and -10 dB . SER performance as M R C (NxM = 1 x 2 ) . From (3.20) and (3.24), using the same procedure derived in Appendix B., it can be shown that PSTD, bill -PMRC, bit\ PSTD, bit! - PMRC, bit! , (3.30) Chapter 3 Performance with Imperfect Channel Estimation 38 if c and d are positive. The SER's for M R C and STD with imperfect channel estimates are shown in Figure 3.4, using the same channel model as in Section 3.1.4. The SER performance for STD is worse than that for M R C , as in the case of BPSK. Figure 3.4 shows that the SER difference between the two schemes increases with SNR and ESR. Chapter 4 Two New Diversity Schemes 39 C h a p t e r 4 T w o N e w D i v e r s i t y S c h e m e s A number of diversity combining schemes have been devised to exploit the uncorrelated fading exhibited by separate antennas in a space diversity array. The Maximal Ratio Combining (MRC) scheme is known to be optimal in the sense that it yields the best statistical reduction of fading of any linear diversity combiner [14]; however, the M R C technique has so far been exclusively for receiving applications. The Maximum Ratio Transmission (MRT) scheme is proposed for a system using both transmit and receive diversity [9]. In the M R T scheme, the weighting factors wt, i = 1 . . .M, in Figure 2.4, are set to have equal absolute values, i.e. |w,| = |w 2 | = •••= | w M | ; the absolute values are set to 1 for simplicity. However, it should be noted that this constraint affects the maximum SNR. An improved M R T scheme (IMRT) is proposed in Section 4.1. In the IMRT scheme, the receiver weights the signals on different branches according to M R C , but uses the same transmit weighting factors as in Figure 2.4. A new optimal maximum ratio transmission and combining (MRTC) scheme is then proposed in Section 4.2, which maximizes the SNR using optimal transmit weighting function and a MRC-like receive weighting function. Here we use a (NxM = 2x2) model to illustrate its operation. The SER performance of the new scheme together with that of the IMRT is discussed in Section 4.3. 4.1 Improved MRT (IMRT) Scheme In MRT, the received signals on all diversity branches are equally weighted. In this section, an improved MRT (IMRT) scheme is proposed in which the signals on each branch are weighted by a factor proportional to the received signal amplitude. In the IMRT scheme, the two received signals in a (N x M = 2 x 2) MRT scheme can be Chapter 4 Two New Diversity Schemes viewed as two inputs for another MRC combiner. The system model is shown in Figure 4.1. 40 SIMRT : w2g2 Figure 4.1 IMRT with two transmit and two receive antennas The weighting factors kt,i =1 ,2 are the products of the weighting factors w•, i = 1,2 from MRT and gi, i = 1, 2. The values of gt ,i= 1,2 are set according to the M R C rules. From (2.17), the transmit weighting factors are 1 * v l = SW\hU+W2h\l) 1 * v 2 = -(wlh2l+w2h22) , (4.1) 2 2 2 2 2 where a = | /z u | + | /J 1 2 | +|^2i| + l^nl + 2 ^11^12 + ^21^22 , and Wj = w2 = 1 The combined signal is then given as Chapter 4 Two New Diversity Schemes 41 ~SIMRT = (sv1hn+sv2h2l+nl)kl + (svlhl2 + sv2h22 + n2)k2 = (svxhxl + sv2h2l + nx)wxgx + (svxhl2 + sv2h22 + n2)w2, s, = ; ^ i i | 2 + h i | 2 + + s-(\hn\2 + \h22\2 + n\\n\2 + ^21^22 n\\n\2 + n2\n22 )g2 + W282n2-(4.2) According to the combining rule in M R C , the optimum weight for each branch has a magnitude proportional to the signal magnitude and inverse to the branch noise power level [1], here we choose si = ( h i i 2 + h i i 2 + 2 2 #2 = (\hn\ +\h22\ + nlln\2 + ^21^22 ^11^12 + ^21^22! )/a )/a. (4.3) The combined signal can then be written as ^IMRT- ~ ( | ^ l l | + | ^ 2 l | + a ^11^12 + ^21^22 + ^ ( | ^ 1 2 | 2 + \h22^ + 1^11^12 + h2\h2^) + k2n2> a ) + kxnx 2 (4.4) where kx = wxgx, k2 = w2g2. The IMRT is actually a combination of a MRT and a MRC-like receiver. 4.2 New Maximum Ratio Transmission and Combining (MRTC) Scheme It has been shown in Section 4.1 that the IMRT scheme can improve the output SNR by using a MRC-l ike weighting function at the M R T receiver. However, the improvement is not optimal for maximizing SNR when multiple transmit and multiple receive antennas are used. In this section, we introduce a new maximum ratio transmission and combining scheme (MRTC) Chapter 4 Two New Diversity Schemes with both optimal transmit and receive weighting functions. 42 : Wi!• \i2 SMRTC Figure 4.2 MRTC with two transmit and two receive antennas As shown in Figure 4.2, = 1,2 are the transmit weighting factors, and /,-,/= 1,2 are the receive weighting factors. The two received signal components can be written as rx = {t\hu + t2h2X)s r2 - (*1^12 + h^22)s-(4.5) The weighting functions, /,-,/= 1, 2 at the receiver, are chosen according to the M R C scheme, i.e. lx = (tlhn+t2h21) * l2 = (t\hn + t2h22) (4.6) The combined signal can then be written as Chapter 4 Two New Diversity Schemes 43 ~SMRTC = l\(r\ + n0 + l2(r2 + n2) • (4J) 2 Assuming that each receiving branch has the same average noise power oRN, which is the 2 2 sum of the powers, oN, in real and imaginary components, the total noise power aNT is simply the weighted sum of the noise in each branch [11]. Thus, 2 i= 1 The SNR, y, at the output of the combiner can be written as y = r2/o2NT, (4.9) where r = /,(txhn + t2h2l)s + l2(txhX2 + t2h22)s• Using Schwarz Inequality [24], y is 2 maximized when /• = r/GRN, which leads to ( 2 2 2 f X ri/aRN 2 2 2 Y = Hr - = S " T - = 5 > f . < 4 1 0 > The SNR at the output of the diversity combiner is simply the sum of the SNR's for each receiving branch, i.e. SNR = SNRX + SNR2. From equation (4.5), we have SNRX = \txhix + t2h2l\ ^/GRN 2 2 ( 4 , 1 1 ) SNR2 = \txhn + t2h22\ ^/o Chapter 4 Two New Diversity Schemes 44 where C, is the signal energy. Maximizing SNR is then equivalent to maximizing 2 2 K= \txhu +t2hlx\ +\tlhl2 + t2h22\ (4.12) i . 2 I .2 with respect to tx and t2 subject to the transmit power constraint, i.e. \tx\ + \t2\ < 1 . We use the same complex representation for hnm = Ctnme nn, n, m=l, 2 as in previous chapters. The transmit weights t{, i = 1,2 can be represented by t• = \|/,-e7*', / = 1,2. With the 2 2 2 2 constraint \tx\ + \t2\ =1, i.e. + \ j / 2 = 1, A' can be expressed as K = y(9ii + <l>i) , „ ;'(e2i + <t>2) 2 + 7(6,2 + <t>l) 7(922 + <t>2) ¥ l t t i 2 « + ¥ 2 « 2 2 e = \|/2a21+y2a2]+2\|/1\|/2a11a21cos(011+(|)1-e21-())2) 2 2 2 2 + V|/ja 1 2 + \|/2oc22 + 2\|/ 1\)/ 2a 1 2a 22cos(0 ]2 + <t>i - ©22~ ^2) = \\f2(a2u + a22) + y22(a221 + «L) + V i ¥ 2 2 , where Q = 2 a n a 2 i c o s ( 0 n + (j), - 0 2 1 -^>2) + 2ai 2 a 2 2cos (0 1 2 + <|>i-622_ ^2) • S ince V|/2 = Jl-\\f\, equation (4.13) can be re-written as K = \ i / 2 ( a 2 1 + a 2 2 ) + ( l - x | / 2 ) ( a 2 1 + a22) + 2 v | / i A / 1 - V ? - ( 4 - 1 4 ) For given values of hnm = anme "m, n, m=l, 2, we wish to maximize K w.r.t. , (j)] and <|)2. To find the stationary points of \\fx, §x and (J>2, we set the partial derivation , and to Chapter 4 Two New Diversity Schemes 45 zero [25]. From = 0 and = 0, we have the same equation tan(d> -<)>) = a n a 2 i s i n ( 9 n - 9 2 i ) + a i 2 a 2 2 s i n ( 9 i 2 - e 2 2 ) = D ( 4 1 5 ) 1 2 - a j j a ^ s i n t O i , - 0 2 1 ) - a 1 2 a 2 2 s i n ( 0 1 2 - 0 2 2 ) Thus, the optimal solution for K depends only on the difference A between (j)j and <j)2. From (4.15), in the interval (-K, it), there are two roots for A , which is (atanD) and either (atanD + n) or (atanZ) - n). From = 0, we have 3Vi ¥ 2 = l ± V l - 4 / ( 4 + C 2 ) ; ( 4 1 6 ) 2 2 2 2 2 where C = 2 ( a n + a 1 2 - ot21 - a 2 2 ) / 2 . Since 0 < \\fx > 1, K will obtain its maximum value at one of the four possible points \ 0, (1 ± Vl - 4 / ( 4 + C 2 ) ) / 2 , 1 i . Since 2 depends on A , for each A , there are four possible y's. In our simulation, the eight possible points are compared to select the proper weighting functions which maximize the function K. 4.3 Numerical Results The channel, interference and noise models are the same as those described in Section 3.1.4. QPSK is used in the simulation. It is assumed that perfect knowledge of channel fading coefficients are available to both transmitting and receiving stations. A (N x M = 2 x 2) diversity Chapter 4 Two New Diversity Schemes 46 model is used in this section. The symbol error rate (SER) curves are shown in Figure 4.3. It can be seen that IMRT and 0 2 4 6 8 SNR (dB) Figure 4.3 SER performances of 4-branch M R C , MRT, IMRT and MRTC MRTC offer significant improvements over MRT. At a SER of 10 3 , IMRT and M R T C are about 0.5dB and 0.6dB more efficient than M R T ( / V x M = 2 x 2 ) . With optimal transmit weighting Chapter 4 Two New Diversity Schemes 47 function, M R T C exhibits a slightly better SER performance than IMRT. Among the schemes shown in Figure 4.3, M R C (N x M = 1 x 4 ) has the best performance. Chapter 5 Conclusion 48 C h a p t e r 5 C o n c l u s i o n 5.1 Main Thesis Contributions In this thesis, the effects of channel estimation errors on error probability performance of M R C , STD and MRT were studied. An improved scheme for MRT and a new optimal maximum ratio transmission and combining scheme were proposed. • The M R C scheme yields the best statistical reductions of fading for any linear diversi-ty combiner. However, a major problem with using the receive diversity approach in a cellular communication system is the cost, size and power of the mobile units. The STD and MRT schemes allow implementations of diversity without requiring multiple antennas at the receiver. This is attractive for small mobile handsets. The error proba-bility performances of the three diversity schemes were compared in this thesis. The STD scheme has previously been shown to have the same BER as the M R C scheme with the same diversity order when perfect channel estimates are available. For MRT, when multiple antennas are used at the transmitter and only one antenna at the receiv-er, i.e. (N> 1,M = 1), the BER performance is shown to be the same as that of M R C with the same diversity order. When multiple antennas are used at the receiver, the BER performance is much worse than that for the (N > 1, M = 1) case. For a 4th or-_3 der diversity scheme, at BER=10 , there is a 0.9 dB degradation for MRT(7V x Af = 1 x 4 ) and 1.3 dB for MRT(7V x M = 2 x 2) compared to M R T ( / V x M = 4 x 1). Chapter 5 Conclusion 49 • The performance sensitivity of M R C , STD and MRT to channel estimation errors was studied. It was shown that (N > 1, M = 1) MRT has the same BER as M R C , whereas STD can have a significantly worse performance than M R C when the channel cannot be estimated accurately. With BPSK, at a BER of 10~3 , STD is worse than M R C by about 1.5 dB. The BER and SER degradations for STD increase rapidly with ESR and SNR relative to M R C . • Two new schemes, IMRT and MRTC, which yield improved error performances com-pared to the MRT scheme were proposed. In IMRT, the same transmit weighting func-tion as in MRT is used. Unlike MRT, M R C combining is applied at the receiver. This results in a substantial error performance improvement over MRT. The MRTC scheme is an optimal maximum ratio transmission and combining scheme. It maximizes the SNR using both optimal transmit and receive weighting factors. For a (NxM = 2 x 2 ) system, at BER=10~ 3, IMRT and MRTC are about 0.5dB and 0.6 dB more efficient than MRT(/V x M = 2 x 2). With an optimal transmit weighting function, MRTC was shown to have a slightly better SER performance than IMRT. 5.2 Topics for Further Study • For (N> \,M> 1) MRT, further theoretical analysis of the error performances due to channel estimation errors would be useful. • The sensitivity to channel estimation errors for the IMRT and MRTC schemes needs to be investigated. Chapter 5 Conclusion 50 • The results of this thesis is based on the assumption that channel fadings are indepen-dent. It would be interesting to study the effects of correlated fades on performance with and without channel estimation errors. Glossary 51 Glossary Acronyms B E R - Bit Error Rate BPSK - Binary Phase Shift Keying CCI - Co - Channel Interference C D M A - Code Division Multiple Access dB - decibel ESR - Estimation Error to Signal Ratio IMRT - Improved Maximum Ratio Transmission M R C - Maximal Ratio Combining MRT - Maximum Ratio Transmission MRTC - Maximal Ratio Transmission and Combining QPSK - Quadrature Phase Shift Keying SER - Symbol Error Rate SNR - Signal to Noise Ratio STD - Simple Transmit Diversity Glossary 52 Notations fd - Maximum Doppler frequency X - Wave length N - Number of transmit antennas M - Number of receive antennas hnm - Diversity branch gain from transmit antenna n to receive antenna m anm - Amplitude of hnm dnm - Phase of hnm r\i MRC ' Received signal on branch 1 —> i in M R C scheme SQ, 5J - Transmitted signals ni - noise and interference on branch i s0 m r c - Signal output from M R C combiner y - output SNR from combiner j u - SNR on 1 - M branch Glossary 53 UMRC ' Decision r.v. for M R C °UMKC - V A R I A N C E 0 F UMRC C, - Signal energy 2 aN - Variance of the real (or imaginary) components of 7V(-y 0 - Average SNR per branch /r(Y) -p.d.fof r PMRC, QPSK ' SER for M R C Pe - BER r • S T D - Received signal on branch / in STD scheme •*o STD ' 1^ STD ' Signal outputs from STD combiner H - Channel coefficient matrix V - Transmit weighting vector W - Receive weighting vector " - Hermitian operation Glossary 54 st - Transmitted signal vector n - Noise vector sMRT - Signal output from MRT combiner a - A normalization factor hnm - Estimated complex branch gain znm - Estimation error for the n x m branch (3 n w j - Amplitude of zn §nm - P h a S e 0 f USTD ' Decision r.v. in STD scheme UMRT ' Decision r.v. in MRT scheme Re(S0, MRC) - Real part of S0, MRC Im(S0, MRC) - Imaginary part of 50, MRC g • - Receive weighting function in the IMRT scheme ~SIMRT ' Signal output form IMRT combiner Glossary 55 ~SMRTC ' Signal output from MRTC combiner ti - Transmit weighting factor in the MRTC scheme /• - Receive weighting factor in the MRTC scheme °RN ' Average noise power in each receive branch in the MRTC scheme <5NT - Total noise power in the MRTC scheme Bibliography 56 Bibliography [1] M . Schwarz, W. R. Bennett and S. Stein, Communication Systems and Techniques, New York: McGraw-Hill, 1966. [2] W. C. Jakes, Ed., Microwave Mobile Communications, New York: Wiley, 1974. [3] B . Sklar, "Rayleigh fading channels in mobile digital communication systems, i i . Mitigation", IEEE Communications Magazine, 35(7):, pp. 102-109, September 1997. [4] J. R. Abeysinghe and J. A . Roberts, "Bit error rate performance of antenna diversity systems with channel correlation", IEEE Global Telecommunications Conference, GLOBECOM'95, vol.3, no.3, pp. 2022 - 2026, Nov. 1995. [5] J. Cui, A . U . H . Sheikh and D. D. Falconer, "BER analysis of optimum combining and maxi-mal ratio combining with channel correlation for dual antenna systems", IEEE 47th Vehicu-lar Technology Conference, vol. 3, no. 1, pp. 150-154, May 1997. [6] B. D. Hart and D. R Taylor, " M L S E for correlation diversity source and unknown time-vary-ing frequency-selective Rayleigh fading channels", IEEE Transaction on Communications, vol. 46, no. 2, pp. 169-172, Feb. 1998. [7] J. Cui and A. U . H . Sheikh, "Outage probability of radio cellular systems using maximal ratio combining in the presence of multiple interferers", IEEE 48th Vehicular Technology Confer-ence, vol. 3, no. 2, pp. 731-735, May 1998. [8] 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, Oct.1998. [9] T. K. Y. Lo, "Maximum Ratio Transmission", IEEE Trans, on Communications, vol. 47, no. 10, pp. 1458-1461, Oct. 1999. [10] J. F. Lemieux, M . S. Tanany and H. M . Hafez, "Experimental evaluation of space/frequency/ polarization diversity in the indoor wireless channel", IEEE Transactions on Vehicular Tech-nology, vol. 40, no. 3, pp. 569-574, August 1991. [11] T. S. Rappaport, Wireless Communications Principles and Practice, Upper Saddle River, N.J.: Prentice Hall PTR, 1996. Bibliography 57 [12] S. Sampei, Application of Digital Wireless Technologies to Global Wireless Communications, Upper saddle River, N.J.: Prentice Hall PTR, 1997. [13] B. Sklar, "Rayleigh fading channels in mobile digital communication systems, i . Character-ization", IEEE Communications Magazine, 35(7):, pp. 90-100, September 1997. [14] A . Paulraj, "Diversity methods" in chapter 12, The Handbook on Mobile Communications, J.Gibson, editor, vol. 12, pp. 166-176, CRC Press, 1995. [15] A. Wittneben, "Base station modulation diversity for digital simulcast", IEEE 41st Vehicular Technology Conference, St. Louis, M O , pp. 848-853, May 1991. [16] W. C. Y. Lee, Mobile Communications Design Fundamentals, 2nd edition, New York: Wiley, 1993. [17] J. H. Winters, J. Salz, and R. D. Gitlin, "The capacity of wireless communication systems can be substantially increased by the use of antenna diversity", 1st International Conference on Universal Personal Communications, ICUPC '92 Proceedings, pp. 02.01/1 - 02.01/5, 29 Sept. - 1 Oct. 1992. [18] T. Sunaga and S. Sampei, "Performance of multi-level Q A M with maximal ratio combining space diversity for land mobile communications", IEEE Transactions on Vehicular Technol-ogy, vol. 42, no. 3, pp. 294-301, August 1993. [19] J. G. Proakis, Digital Communications, 3rd edition, New York: McGraw-Hill, 1995. [20] A. Kegel, W. Hollemans and R. Prasad, "Performance analysis of interference and noise lim-ited cellular land mobile radio", IEEE 41st Vehicular Technology Conference, St. Louis, M O , pp. 817-821, May 1991. [21] S. Haykin, Communication Systems, 3rd edition, New York: John Wiley & Sons, 1994. [22] S. B . Wicker, Error Control Systems for Digital Communication and Storage, Upper Saddle River, N.J.: Prentice Hall PTR, 1995. [23] R. E. Ziemer and W.H.Tranter, Principles of Communications, 3rd edition, Boston: Houghton Mifflin, 1990. [24] L . W. Couch II, Digital and Analog Communication Systems, 4th edition, New York: Mac-millan, 1993. Bibliography 58 [25] E. Kreyszig, Advanced Engineering Mathematics, 7th edition, New York: John Wiley & Sons, 1993. Appendix A. Means and Variances of MRC and STD Decision Variables 59 A p p e n d i x A . M e a n s a n d V a r i a n c e s o f M R C a n d S T D D e c i -s i o n V a r i a b l e s Here we derive the means of the decision r.v.'s for MRC and STD, as given in (3.3) and (3.8) respectively, and the variance which is given in (3.4). For MRC, using hnm = anme nm and znm = $nme^nm, n,m = 1,2 , (3.3) can be written as •>0, MRC = («H + a 1 2 ) * 0 + a l l e P\\e s0 + a l 2 e P\2e s0 (A.l) +(hu+zn) Nx + (hn + zn) N2 . The decision r.v. UMRC can then be expressed as U MRC = Re(S0,MRc) = (a2n + a 2 2 ) 5 o + a 1 1 (3 1 1 cos(0 1 1 -(t) 1 1 )5 o + a 1 2 p 1 2 c o s ( e 1 2 - ( l ) 1 2 ) ^ o (A.2) + Re[(/z n + z n )*/Vj] + Re[(hl2 + z12)*N2]. Since the channel noise is zero mean Gaussian, the mean of UMRC can be derived as E(UMRC)= t ( a i i + «i2) + « i iPnCos (0 1 1 -<j>n) + a12fJ,2cos(812-<|>12)].s0, (A.3) which is the same as (3.3). The variance of UMRC can be expressed as Appendix A. Means and Variances of MRC and STD Decision Variables 60 = E[Re2(c\Nx) + Re2{C2N*2)] , where we assume Cx = (hn + z n ), C 2 = (h2l + z 2 1 ) , and the noise processes in the diversity branches are assumed to be statistically independent with equal powers. Let Nl = X1+jYl ; C 1 = Ux+jVx N2 = X2 + jY2;C2= U2 + jV2, where U{ = a 1 1 c o s 0 1 ] + PjjCos^jj U2 = ot 2 1 cos0 2 1 + P 2 i c o s ( t ) 2 i Vx = oc n s i n 0 n + PjjSin^n V2 = a 2 1 s in0 2 1 + p21 sin<l)21, we can then write (A.5) (A.6) G 2 U m r c = E[(XlUl + YlVl)2 + (X2U2+Y2V2)2] = U\E{X\) + V]E{Y2X) + U\E{X\) + V22E{Y22) = oJ,(f72 + y 2 + r/2 + v2) (A.7) = °^ aii + Pii +2a 1 1p 1 1cos(0 1 1 -(()„) + a 1 2 + p22 + 2a 1 2p 1 2cos(0 1 2 - <j)12) ] where E{x])= E{Y\)= E(X\)= E{Y\)= G 2 N . Appendix A. Means and Variances of MRC and STD Decision Variables 61 F o r S T D , the comb ined s ignal g iven in (3.7) can be writ ten as 7, ,2 2 N fiun -J<t>n _ ^ ' e 2 i Q ^21 So, STD = ( « H +CL2l)sQ + ane P n g $ 0 + a 2 1 g P2\e S0 + a 2 1 e " ' e 2 ' P 1 1 e ^ n s l - a u e ^ u $ 2 \ e ^ 2 l s \ (A-8) +(hn+zn) Nl + (h2l+z21) N2 . The dec is ion r.v., USTD, is U STD = Re(So,STD) = («11 + a 2 i ) ^ 0 + a l l P l l c o s ( e i l - 4 > n ) ^ O + a 2 l P 2 1 c o s ( 0 2 1 - ^ 2 1 ) ^ 0 (A.9) + a 2 1 p n c o s ( 9 2 1 — 4>n>*i - a n P 2 i c o s ( e n - ( t ) 2 i ) 5 i + Rc[(hu+zu)*Nl]+Re[(h2]+z20 N*] . The mean of USTD can then be der ived as E(USTD)= [ ( a 2 j + a 2 1 ) + a 1 1 p 1 1 c o s ( 0 1 1 - ( ( ) „ ) + a 2 1 p 2 ] c o s ( 0 2 1 - ( t ) 2 1 ) ] 5 o + [ a 2 1 p n c o s ( 0 2 1 -tyu) - a 1 1 p 2 1 c o s ( 0 1 1 -$21)]sx, wh ich is the same as (3.8). T h e der ivat ion o f the var iance o f USTD is the same as that for UMRC. Appendix B. Comparison of (3.5) and (3.9) 62 A p p e n d i x B . C o m p a r i s o n o f (3.5) a n d (3.9) Here we compare the BER's of M R C and STD, as given by (3.5) and (3.9) respectively, using the symmetry property of the Q(.) function, which is defined as Q(x) = - L f Y ^ V (B.l) From (3.5) and (3.9), to prove that Pe S T D > Pe M R C when a is positive is equivalent to proving that U MRC MRC MRC The inequality (B.2) can be further written as Q(c)<±[Q(c + d) + Q(c-d)],c>0, (B.3) where c = —-—, d =—-— . Assuming that c, d > 0, i.e. a, b > 0, we have U MRC U MRC Q(c)-Q(c + d) = ^Lf+ e~y/2dy J2KJC Q(c-d)-Q(c) = -Lf e~y/2dy. J2KJc-d (B.4) Due to the symmetry property of the Q(.) function, we can see from Figure B . l that Figure B. l Illustrating (B.5)
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Performance analysis of some new and existing transmit and receive antenna diversity schemes Feng, Xiaoyan Sarina 2000
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Title | Performance analysis of some new and existing transmit and receive antenna diversity schemes |
Creator |
Feng, Xiaoyan Sarina |
Date Issued | 2000 |
Description | A well-known method for symbol detection involving signals received over a number of independent diversity branches perturbed by additive white Gaussian noise is Maximum Ratio Combining (MRC). Two other diversity schemes, Simple Transmit Diversity (STD) and Maximum Ratio Transmission (MRT), have recently been proposed. The performances of these three schemes are compared. The STD scheme, which is applicable only to two transmit antennas, is shown to have the same error performance as MRC with the same diversity order when perfect channel estimates are available. The MRT scheme with N transmit and one receive antennas is shown to have the same performance as MRC with N receive antennas, but MRT with one transmit and N > 1 receive antennas or MRT with N > 1 transmit and M > 1 receive antennas do not perform as well as MRC. The performance degradations of MRC, STD and MRT due to errors in estimating the channel parameters are analyzed and compared. It is found that STD is significantly more susceptible than MRC and MRT to errors in the channel estimates. An improved scheme for MRT (IMRT) and a new optimal maximum ratio transmission and combining scheme (MRTC) are proposed. In the IMRT scheme, the same transmit weighting functions are used as in MRT. At the receiver, MRC combining rules are used to choose the weights for the received signals. The MRTC scheme maximizes the SNR using optimal transmit and receive weighting factors. Simulation results indicate that both IMRT and MRTC provide significant performance improvements over MRT. |
Extent | 2390078 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-07-10 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0065168 |
URI | http://hdl.handle.net/2429/10609 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2000-11 |
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UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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