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Transmit beamforming for frequency-selective channels with suboptimum equalization Liang, Yang Wen 2006

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T R A N S M I T B E A M F O R M I N G F O R F R E Q U E N C Y - S E L E C T I V E C H A N N E L S W I T H S U B O P T I M U M E Q U A L I Z A T I O N by L I A N G , Y A N G W E N B . E N G . , McMaster University, 2004 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S (Electrical and Computer Engineering) T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A August 2006 © Liang, Yang Wen, 2006 Abstract In this thesis, we propose beamforming schemes for frequency-selective chan-nels wi th decision-feedback equalization ( D F E ) or linear equalization ( L E ) at the receiver and wi th , respectively, perfect and quantized channel state in -formation (CSI) at the transmitter. For beamforming wi th perfect CSI and infinite impulse response (HR) beamforming filters ( B F F s ) we derive a closed-form expression for the opt imum B F F s . We also provide two efficient numerical methods for recursive calculation of the opt imum finite impulse response (FIR) B F F s wi th perfect CSI . For beamforming wi th quantized CSI and finite-rate feedback channel, we propose a global vector quantization ( G V Q ) algorithm for codebook design. This algorithm is deterministic and independent of ini t ia l conditions and does not impose any constraints on the number of transmit and receive antennas, the antenna correlation, or the fading statistics. Our simu-lation results for typical G S M 1 [1] and E D G E 2 [2, 3, 4] channels show that in general short F I R B F F s are sufficient to closely approach the performance of I IR B F F s even in severely frequency-selective channels. Furthermore, f inite-rate feedback beamforming wi th only a few feedback bits achieves significant performance gains over single-antenna transmission, transmit antenna selec-tion, and optimized delay diversity [5] in frequency-selective fading. 1 G S M : Global System for Mobile Communication. 2 E D G E : Enhanced Data Rates for G S M Evolution. Table of Contents A b s t r a c t i i T a b l e o f C o n t e n t s i i i L i s t o f F i g u r e s v i L i s t o f A b b r e v i a t i o n s a n d S y m b o l s ix A c k n o w l e d g m e n t s x i i 1 I n t r o d u c t i o n 1 1.1 Background and Mot iva t ion 1 1.2 Contributions 2 1.3 Thesis Organization 5 2 T r a n s m i s s i o n S y s t e m 6 2.1 Channel M o d e l 6 2.1.1 Frequency-nonselective Channel 7 2.1.2 Frequency-selective Channel 8 2.2 Correlat ion of C I R Coefficients . . 10 2.3 Equivalent Baseband System M o d e l 12 2.3.1 Transmit Pulse Shaping 14 2.3.2 Receiver Processing 15 2.4 Equivalent Discre te-Time System M o d e l 16 2.4.1 Discrete-t ime Channel M o d e l 16 2.4.2 Equivalent S I M O M o d e l 18 i i i 2.4.3 Feedback Channel 19 3 Bearnforming with Perfect CSI and IIR Filters 20 3.1 I IR Bearnforming wi th Decision-Feedback Equal izat ion 21 3.1.1 Opt imiza t ion Problem 21 3.1.2 O p t i m u m I IR B F F s 23 3.2 I IR Bearnforming wi th Linear Equal izat ion 29 3.2.1 Opt imiza t ion Problem 30 3.2.2 Op t imum I IR B F F s 32 4 Bearnforming with Perfect CSI and F I R Filters 39 4.1 F I R Bearnforming wi th Perfect CSI and D F E 39 4.1.1 O p t i m u m F I R B F F s . . . . 39 4.1.2 Calculat ion of the O p t i m u m F I R B F F s 41 4.2 F I R Bearnforming wi th Perfect CSI and L E 44 4.2.1 O p t i m u m F I R B F F s 44 4.2.2 Calculat ion of the O p t i m u m F I R B F F s 45 5 F I R Bearnforming with Quantized CSI 47 5.1 Vector Quantizat ion - Preliminaries 47 5.2 Mean Quantizat ion Error ( M Q E ) and Distor t ion Measure . . . . 48 5.3 L B G Algor i thm 49 5.4 Globa l Vector Quantizat ion ( G V Q ) Algor i thm 51 6 Simulation and Numerical Results 54 6.1 Bearnforming wi th Perfect CSI 55 6.1.1 Decision-Feedback Equal izat ion 55 6.1.2 Linear Equal izat ion 58 6.2 F in i t e -Ra te Feedback Bearnforming 59 iv 7 Conclusions and Future Work 66 7.1 Conclusions 66 7.2 Recommendations for Future Work 67 Bibliography 69 v List of Figures 2.1 Frequency selective M I M O channel 9 2.2 Block diagram of the continuous-time overall transmission sys-tem. b[k] are estimated symbols at the receiver 13 2.3 Block diagram of the discrete-time overall transmission system. b[k] are estimated symbols at the receiver 17 3.1 Water F i l l i ng for one realization of the E Q channel wi th L = 7, NT = 3, NR = 1, equal antenna correlation p = 0.5, and 1 0 1 o g 1 0 ( £ s / / V 0 ) = 10 d B 27 3.2 Water F i l l i ng for one realization of the E Q channel wi th L = 7, NT = 3, NR — 1, equal antenna correlation p = 0.5, and 1 0 1 o g 1 0 ( £ s / / V 0 ) = 2 0 d B 28 3.3 Quasi Water F i l l i ng for one realization of the E Q channel w i th L = 7, NT = 3, NR = 1, equal antenna correlation p — 0.5, and 1 0 1 o g 1 0 ( £ s / / V 0 ) = 1 0 d B 36 3.4 Quasi Water F i l l i ng for one realization of the E Q channel w i th L = 7, NT = 3, NR = 1, equal antenna correlation p = 0.5, and 101og 1 0 (£ s /7Vo) = 2 0 d B . 37 6.1 S N R of M M S E - D F E vs. iteration i of the proposed G A and M P M for one realization of the E Q channel wi th L — 7, NT = 3, NR — 1, equal antenna correlation p = 0.5, Lg = 1, and 1 0 1 o g 1 0 ( £ y / V o ) = 1 0 d B 55 vi 6.2 Average S N R of D F E for beamforming ( B F ) wi th perfect CSI and different B F F s . E Q channel wi th L = 7, NT = 3, NR = 1, and equal antenna correlation p = 0.5. Results for single-antenna transmission are also included 56 6.3 Average S N R of D F E for beamforming (BF) wi th perfect CSI and different B F F s . T U channel wi th L = 5, NT = 3, NR = 1, and equal antenna correlation p = 0.5. Results for single-antenna transmission are also included 57 6.4 Average S N R of L E for beamforming ( B F ) wi th F I R and I IR filters. Transmission over T U channel wi th L = 5, NT = 3, NR = 1, and equal antenna correlation p = 0.5. The result for single-antenna transmission is also included 58 6.5 B E R of M M S E - D F E vs. number of feedback bits B per chan-nel update. B P S K transmission over E Q channel wi th L = 7, NT — 3, NR = 1, equal antenna correlation p = 0.5, and 101og 1 0 (£6/Yvo) = 1 0 d B - T h e B E R i s obtained from E q . (5.2). . 60 6.6 B E R of M M S E - D F E vs. number of feedback bits B per chan-nel update. 8 - P S K transmission over T U channel wi th L = 5, NT = 3, NR = 1, equal antenna correlation p = 0.5, and 101og 1 0(£7 6 /rWn) = 10 d B . The B E R is obtained from E q . (5.2). . 61 6.7 Simulated B E R of M M S E - D F E for finite-rate feedback beam-forming wi th B F F s of length LG = 1. B P S K transmission over E Q channel wi th L = 7, NT — 3, NR = 1, and equal antenna correlation p = 0.5. Results for single-antenna transmission (NT = 1, NR = 1), antenna selection, and beamforming wi th perfect CSI are also included 62 v i i 6.8 Simulated B E R of M M S E - D F E for finite-rate feedback beam-forming wi th B F F s of length LG = 3. B P S K transmission over E Q channel wi th L — 7, NT = 3, NR = 1, and equal antenna correlation p = 0.5. Results for single-antenna transmission (NT = 1, NR = 1), O D D [5], and beamforming wi th perfect CSI are also included 63 6.9 Simulated B E R of M M S E - D F E for finite-rate feedback beam-forming wi th B F F s of length LG = 1. 8 - P S K transmission over T U channel wi th L = 5, NT = 3, NR = 1, and equal antenna correlation p — 0.5. Results for single-antenna transmission (NT = 1, NR = 1), antenna selection, and beamforming wi th perfect CSI are also included 64 v i i i List of Abbreviations and Symbols A c r o n y m s 4G The fourth generation wireless communications technology A W G N Addi t ive white Gaussian noise B E R B i t error rate B F F s Bearnforming filters B T Time-bandwidth C C Centroid condition C I R Channel impulse response CSI Channel state information D D Delay diversity D F E Decision-feedback equalization E D G E Enhanced data rates for G S M evolution E Q Equalizer test F B F Feedback filter F F F Feed-forward filter F I R Fini te impulse response G M S K Gaussian minimum shift keying G S M Global system for mobile communication G V Q Global Vector Quantization H T H i l l y terrain I IR Infinite impulse response ISI Intersymbol interference L B G Linde-Buzo-Gray algorithm (or generalized L l o y d algorithm) L E Linear equalization L O S Line-of-sight L S S E Least sum of squared errors M I M O Mul t ip le - inpu t multiple-output M L S E Maximum-l ikel ihood sequence estimation M M S E M i n i m u m mean-square error M Q E Mean quantization error M S E Mean-square error N N C Nearest neighborhood condition O D D Opt imized delay diversity O F D M Orthogonal frequency division multiplexing pdf Probabi l i ty density function P E P Pairwise error probabili ty P S K Phase shift keying Q A M Quadrature amplitude modulation R A Rura l area S D D Standard delay diversity S I M O Single-input multiple-output SISO Single-input single-output S N R Signal-to-noise ratio S R C Square-root raised cosine T U Typ ica l urban area W i M a x Worldwide Interoperability for Microwave Access W L A N Wireless Loca l Area Network W M F Whitened matched filter Z F Zero forcing X Operators and Notation | • | Absolute value of a complex number • ® • Convolution 5(-) Dirac delta function £ {•} Expectat ion det (•) Determinant tr(-) Trace [•]* Complex conjugate M a t r i x or vector transposition M a t r i x or vector hermitian transposition [x]+ max(x ,0 ) 0 m A l l - ze ro column vector of length m Im Identity matr ix wi th dimension m x m xi Acknowledgments Undoubtedly, the completion of this thesis came from the support of many individuals. Firs t and foremost, I would like to extend my sincere thanks to my supervisor Professor Robert Schober for his invaluable guidance and for giving me immense insight into my research. Professor Schober provided much of the ini t ia l motivation for pursuing this investigation and also provided priceless feedback that has improved this work in nearly every aspect. Wi thout him, this would never have been completed. I would also like to thank my family who have always encouraged me in my quest for higher education and especially my wife Y i f a n for her persistent support. Final ly, I would like to extend my thanks to the colleagues at the Depart-ment of Electr ical and Computer Engineering, U B C , for creating a stimulating and a friendly environment at work, and especially Simon Y i u for his detailed explanation on the simulation program that I use for this work. YANG W E N LIANG The University of British Columbia Vancouver, Canada August 2006 x i i Chapter 1 Introduction The following section provides an overview of the background information and motivation for this work in detail. We review the related work that has been proposed by other researchers in this field in this section as well. The contributions of this work are briefly summarized in the second section of this chapter, and the concluding section outlines the organization of this thesis. 1.1 Background and Motivation In recent years, the application of multiple antennas in wireless communication systems has received considerable interest from academia and industry [6, 7, 8]. In particular, beamforming was shown to be a simple yet efficient technique for exploiting the benefits of multiple transmit antennas, cf. e.g. [9] and references therein. Beamforming generally requires channel state information (CSI) at the transmitter. Since perfect CSI may not be available at the transmitter in prac-t ical systems, recent research in this field has focused on beamforming wi th imperfect CSI , cf. e.g. [10, 11, 12]. F rom a practical point of view the assump-tion of quantized CSI is particularly interesting. In this case, the transmitter selects the beamforming vector from a pre-designed codebook based on infor-mation received from the receiver over a finite-rate feedback channel [11, 12]. 1 Depending on the adopted performance measure and the statistical properties of the underlying channel, a closed-form solution for the opt imum bearnform-ing vector codebook may or may not exist, cf. [11, 12, 13]. For the latter case vector quantization algorithms have been proposed for codebook construction. In particular, L i n d e - B u z o - G r a y ( L B G ) type algorithms [14] (also referred to as generalized L loyd- type algorithms) have been adopted in [13] and [15], re-spectively. However, the L B G algorithm is a local search procedure and its performance heavily depends on the starting conditions [16]. Most of the existing literature on transmit bearnforming wi th perfect or im-perfect CSI has assumed frequency-nonselective fading. A notable exception is [17] where it was shown that bearnforming wi th infinite impulse response (IIR) filters is asymptotically capacity achieving in strongly correlated frequency-selective mult iple- input multiple-output ( M I M O ) fading channels. In addi-t ion, in [17] jointly opt imum I IR bearnforming and equalization filters were derived for various optimization criteria. For systems employing orthogonal frequency-division multiplexing ( O F D M ) to cope wi th the frequency selectiv-ity of the channel, effective bearnforming techniques for the imperfect CSI case were proposed in [18, 19]. 1.2 C o n t r i b u t i o n s In this thesis, we consider transmit bearnforming wi th , respectively, perfect and quantized CSI for frequency-selective fading channels which are typically encountered in high-rate transmission. We focus on single-carrier transmission and the developed bearnforming schemes may be used to e.g. upgrade existing communication systems such as the Globa l System for Mobi le Communications ( G S M ) and the Enhanced Da ta Rates for G S M Evolut ion ( E D G E ) system. Note that the multi-carrier based techniques in [18, 19] are not applicable in this case. Due to the intersymbol interference (ISI) caused by the frequency selectivity of the channel, equalization is necessary at the receiver and the 2 opt imum beamformer depends on the equalizer used. Here, we adopt decision-feedback equalization ( D F E ) and linear equalization ( L E ) because of their low complexity, good performance, and practical relevance [20, 21, 22].' Al though the main emphasis of this paper is on the practically relevant case of finite impulse response (FIR) beamforming, for the sake of completeness and since there are many interesting parallels and differences between the F I R and the I IR cases, we also consider I IR beamforming. The main contributions of the present research work are as follows: • For perfect CSI we derive a closed-form solution for the opt imum I IR beamforming filters ( B F F s ) maximizing the signal-to-noise ratio (SNR) for receivers wi th D F E or L E . Interestingly, although our derivation of the opt imum IIR B F F s is much simpler than that presented in [17] as we do not perform a joint optimization of the B F F s and the D F E or L E filters, our final result is identical to that given in [17]. More importantly, our approach can be readily extended to F I R B F F s which does not seem to be easily possible for the approach taken in [17]. • We show that, similar to the opt imum I IR B F F s , the opt imum F I R B F F s wi th perfect CSI are the solution to a nonlinear eigenvalue problem. However, in the F I R case a closed-form solution to this problem does not seem to exist, and we provide two efficient numerical methods for calculation of the optimum F I R B F F s . • We propose a practical finite-rate feedback beamforming scheme for frequency-selective channels. The beamforming vector codebook design is based on the opt imum F I R B F F s for perfect CSI . In particular, ex-ploit ing the findings in [23], we propose a global vector quantization ( G V Q ) algorithm for codebook design which performs a deterministic global search. The G V Q algorithm does not depend on starting condi-tions and employs the L B G algorithm as a local search procedure. • Our simulation results show that short F I R B F F s can closely approach the performance of the optimum I IR B F F s . In fact, for quantized CSI 3 with small codebook sizes B F F s of length one (i.e., beamforming weights) are preferable. If the channel is severely frequency selective, longer B F F s become beneficial as the codebook size increases. • For typical G S M / E D G E channel profiles beamforming wi th finite-rate feedback enables large performance gains compared to single-antenna transmission, transmit antenna selection, and optimized delay diversity [5]. The results of our work are summarized in the following papers: • Y . Liang, R . Schober, and W . Gerstacker. Transmit beamforming for frequency-selective channels wi th decision-feedback equalization. Sub-mitted to IEEE Transactions on Wireless Communications, M a y 2006. • Y . Liang, R . Schober, and W . Gerstacker. F I R beamforming for frequency-selective channels wi th linear equalization. Submitted to IEEE Commu-nication Letters, M a y 2006. • Y . Liang, R . Schober, and W . Gerstacker. Transmit beamforming wi th finite-rate feedback for frequency-selective channels. Accepted for pre-sentation at the IEEE Global Telecommunications Conference (GLOBE-COM), San Francisco, USA, December 2006. • Y . Liang, R . Schober, and W . Gerstacker. Transmit beamforming for frequency-selective channels. Accepted for presentation at the IEEE Ve-hicular Technology Conference (VTC), Montreal, Canada, September 2006. • Y . Liang . Transmit beamforming wi th linear equalization. Accepted for presentation at the First Canadian Summer School on Communications and Information Theory, Banff, Canada, August 2006. 4 1.3 Thesis Organization To explain the above findings in detail, the thesis is organized as follows. In Chapter 2, we describe the adopted correlated M I M O frequency-selective Rayleigh fading model, and the G S M and E D G E power delay profiles. The adopted overall communications system model is also presented in this chapter. The optimization of I IR and F I R B F F s wi th perfect CSI is discussed in Chap-ters 3 and 4, respectively. In Chapter 5, basic vector quantization concepts are introduced, finite-rate feedback bearnforming is discussed, and the proposed G V Q algorithm is presented. Simulation results are provided in Chapter 6, and some conclusions and recommendations for future work are given in Chapter 7. 5 Chapter 2 Transmission System In this chapter, the overall transmission system consisting of signal mapper, bearnforming filters ( B F F s ) , pulse shaping filters, correlated M I M O channel, receiver input filters, and equalizer wi l l be discussed. It w i l l be first shown that the correlated M I M O channel wi th NT transmit antennas and NR receive antennas can be modeled by matrices wi th dimension NR X NT- We wi l l then show that the overall channel model, continuous in time, can be obtained by convolving the correlated M I M O channel wi th the pulse shaping filters, and the receiver input filters. Furthermore, an overall discrete-time channel model is obtained by sampling and truncating the continuous-time channel impulse response (CIR) . Final ly , an equivalent channel model containing the combined effect of the overall discrete-time channel and B F F s is derived. 2.1 Channel Model In order to design transmit bearnforming schemes, an established knowledge of the transmission channel and its properties are crucial. The correlated M I M O frequency-selective fading channel model is adopted in this work. In a M I M O wireless link, the data stream is broken up into separate signals and •sent over different transmit antennas. To get a proper understanding of this M I M O frequency-selective channel, we w i l l first explain M I M O frequency-6 nonselective channel. In the later section, we w i l l extend our discussion from frequency-nonselective channel to frequency-selective channel. 2.1.1 Frequency-nonselective Channel The complex baseband frequency-nonselective M I M O channel can be modeled by the following channel matr ix [24]: Hc(t) = h%(t) h%{t) h1^ hcR\t) h^2(t) ... h%«NT(t) (2.1) /i£r™'(£) is the continuous-time channel gain between transmit antenna nt, 1 < nt < NT, and receive antenna nr, 1 < nr < NR, where NT and NR are the total number of transmit and receive antennas, respectively. Furthermore, h1cJnt(t) can be modeled as a Rayleigh fading channel or a Ric i an fading channel, if there is a line-of-sight (LOS) path. In case of Rayleigh fading channels, the complex gain h^fnt (£) can be de-scribed as a continuous-time zero mean Gaussian random process h£nt{t) = h?nt(t) + jti%nt(t), (2.2) where hnjrnt(t) and h^nt(t) are the real and imaginary parts of / ^ " ' ( i ) , re-spectively [25]. The envelope of the process, (Urnt(t) = \h!£nt(t)\, is Rayleigh distributed wi th probability density function (pdf) ;|exp(-4)> o, Pdx) for x > 0 for x < 0 (2.3) where O - Q is the variance of the two quadrature channels. Since hITnt(t) and hQrnt(t) are assumed to be independent and identically distributed, the vari-ance of ^ " ' ( t ) . is equal to 1a\. 7 In case of R ic ian fading channels, / ^ " ' ( i ) has a time dependent complex mean value rh(t), and can be modeled as (2.4) where / i " r n ' ( £ ) and h^nt{t) are the real and imaginary parts of h^rnt(t), respec-tively [25]. Furthermore, hJ}Tnt{t) and hgrnt(t) are statistically independent zero mean Gaussian random processes wi th common variance a\. The envelope of the process, (nrnt(t) = \h^Tnt(t)\, is now Rice distributed wi th probability density function (pdf) where IQ(X) is the modified zero order Bessel function of the first k ind , <7Q is the variance of the two quadrature channels, and p 2 = £ 2 { / i " r " ' ( £ ) } + £ 2 { / i £ r " ! ( t ) } , where £{ •} denotes statistical expectation. Again , since h"rTlt(t) and hg'nt(t) are assumed to be independent, the variance of hQrnt(t) is equal to 2cr2. From E q . (2.5), the Rice pdf converges to the Rayleigh pdf for p —> 0. 2.1.2 Frequency-selective Channel The frequency-nonselective model described by E q . (2.1) is only valid when the signal bandwidth is much smaller than the coherence bandwidth of the channel. If the signal has a bandwidth greater than the coherence bandwidth, the transmitted signal is subjected to different gains and phase shifts across the band. In such a case, the channel is said to be frequency-selective [25]. A frequency-selective channel causes intersymbol interference (ISI). The received signal w i l l be the superposition of several transmitted signals. A frequency-selective M I M O model wi th L mult ipath components is shown in Figure 2.1. xnt(t) represents the signal transmitted by transmit antenna nt, while ynr(t) represents the signal received by receive antenna n r . 77, I = for x > 0 for x < 0 (2.5) 8 xdt) -T2 TL-1 J NT Nn NR Na yM) Figure 2.1: Frequency selective M I M O channel. 1 , . . . , L — 1, represents the delay of the mult ipath component Each matrix, Hlc(t), I — 0,... ,L — 1, has dimension NR X NT and its elements can be wri t ten as [24, 25] Hlc(t) (2.6) The matrices Hlc(t) are independent for different I, I = 0, . . . , L — 1, and their elements, ti£nt'l(t), are continuous-time random processes as defined in E q . (2.2) or E q . (2.4). The overall M I M O weight function Hc{r,t) is also a matr ix w i th dimen-sion NRx NT- It relates to the matrices Hlc(t) in the following way L - l Hc(T,t) = Y,Hlc(t)5(T-Tl), (2.7) (=0 where 5(-) is the Dirac delta function [25] and ro is equal to zero. Therefore, the matr ix elements in Eqs. (2.6) and (2.7) are related by hZnt(T,t) = ^hZ^l(t)5(T-n). (2. (=0 Adopt ing the power delay profile given in [26], we define the equivalent power delay profile of the channel for each receive-transmit antenna pair as 2 L - l 1=0 (2.9) 9 where is the variance of h\ nrnt, I •c • (t) defined as (2.10) In practice, the summation of \ a^Jnt' J w i th respect to I is normalized to 1, i.e., For G S M and E D G E system, four different power delay profiles are spec-ified [1]: rural area ( R A ) , hi l ly terrain ( H T ) , typical urban area ( T U ) and equalizer test (EQ) . For E Q , H T , and T U it is assumed that the amplitudes of al l propagation paths, h^nt'l(t), are continuous-time zero mean Gaussian random processes as described by E q . (2.2). Their envelopes are Rayleigh dis-tr ibuted wi th pdf as defined in E q . (2.3). For R A , it is assumed that the ampli-tudes of al l propagation paths are continuous-time non-zero mean Gaussian random processes as defined in Eqs. (2.4) and (2.5). The mean value is due to the L O S path between a transmit antenna and a receive antenna. This results in a Ric ian fading channel. In this work, the E Q , H T , and T U profiles are considered. Final ly , it should be noted that if NT and NR are both equal to one, the M I M O channel in E q . (2.6) reduces to a single-input single-output (SISO) frequency-selective fading channel. Furthermore, if L = 1, the channel reduces to a frequency-nonselective channel resulting in only scalar multiplicative dis-tort ion of the transmitted signals. In general, an independent and identically distributed (i.i.d.) channel model assuming rich uniform scattering w i l l not be an accurate description of rea l -world mult i-antenna channels [27], since in practice, insufficient antenna spac-ing and a lack of scattering cause the antennas to be correlated. Therefore, L-l (2.11) 1=0 2.2 Correlation of C I R Coefficients 10 spatial correlation is assumed to occur at both the transmit and receive an-tennas in this work. Under this assumption, the matr ix taps in E q . (2.6) can be writ ten as [27] Hlc{t) = R1/2Hl(t)(S1/2)H, (2-12) where Hl(t), R = R}'2{Rll2)H, and S = S1/2(S1/2)H are the channel ma-t r ix taps wi th i . i .d . entries, the receive correlation matrix, and the trans-mit correlation matrix, respectively. The superscripts 1/2 and H denote the matr ix square-root and Hermit ian transposition, respectively. Al though not completely general, this simple correlation model has been validated through recent field measurements as a sufficiently accurate representation of the fade correlations seen in actual cellular systems [27, 28]. S and R are positive definite matrices wi th dimensions NT X NT and NR x NR, respectively. From now on, we assume the M I M O model defined in E q . (2.7) to be a spatially correlated frequency-selective M I M O channel w i th matr ix taps de-scribed by E q . (2.12). For simplicity, we assume that the spatial correlation is identical for al l matr ix taps. Setups wi th up to three transmit and two receive antennas are considered in this work. Since matrices S and R have the same form, we w i l l concentrate on the transmit correlation matr ix in the following discussion. There is only one correlation factor for the two antennas case. The corre-lation matr ix S can be writ ten as P12 (2.13) Ai 1 where p\2 is the correlation factor between transmit antenna one and transmit antenna two and it is defined by p\2 = 1 , V 1 J - (2.14) V(<#1,')2(<#2,')a There are three correlation factors for the three antennas case, p\2, p23, and p\3. They represent the correlation between transmit antenna one and transmit 11 antenna two, between transmit antenna two and transmit antenna three, and between transmit antenna one and transmit antenna three, respectively. The resulting correlation matr ix is a 3 x 3 matr ix wi th elements similarly defined as in E q . (2.14) S = 1 Pl2 Pl3 Pl2 1 Pl3 P23 P23 1 (2.15) The square root of the correlation matr ix can be calculated by using Cholesky decomposition such that S1^2 and R1^2 are lower triangular ma-trices whereas (S1/2)H and (R1/2)H are upper triangular matrices [29]. S ' for the two and three transmit antennas case can be writ ten as 1 0 S1/2 = Ai Vi - (pi2)2 (2.16) and A* V i - ( P I 2 ) 2 0 0 Pl3 PJ3 Pl2Pl3 1 - (Pl3) _ (P23-P12P13)2 i-(pi2)2 (2.17) respectively. Similar results can be obtained for the receive antennas by re-placing the correlation factors in the above matrices wi th the respective receive correlation factors. For future convenience, Pt=U 2 p\, p\A (2-18) and P'12 Ph P'23 I ( 2 - i 9 ) are defined, where pt and pr are the correlation vectors for the transmit and receive antennas, respectively. 2.3 E q u i v a l e n t B a s e b a n d S y s t e m M o d e l The channel model presented in the previous section is only a part of the overall mobile communications transmission model. The other parts of the 12 model are discussed in this section. The block diagram of the continuous-time overall transmission model in complex baseband representation is shown in Figure 2.2. i[k] Figure 2.2: Block diagram of the continuous-time overall transmission system. b[k] are estimated symbols at the receiver. The i . i .d . symbols b[k] are taken from a scalar symbol alphabet A such as phase-shift keying ( P S K ) or quadrature amplitude modulat ion ( Q A M ) , and have variance cr2 = £{\b[k]\2} = 1. Since we consider G S M and E D G E in this work, the mapped symbol is either a 2 - P S K or an 8 - P S K symbol depending on whether G S M or E D G E is used. Note that G S M uses binary Gaussian M i n i m u m Shift Key ing ( G M S K ) , which can be approximated as filtered 2 -P S K . E D G E improves spectral efficiency by employing 8 - P S K modulation instead. However, other system parameters such as symbol rate and burst duration remain unchanged in order to enable a smooth transition from G S M to E D G E [2]. Before transmit pulse shaping, the modulated symbols, b[k], are first f i l -tered by the discrete-time beamforming filters ( B F F s ) , gnt[k}- The B F F s de-picted in Figure 2.2 are discrete-time filters, which can be realized as tapped delay lines. The transmit B F F impulse response coefficients of antenna nt, 13 1 < nt < NT, are denoted by gnt[k], —qi < k < qu, and their energy is normal-ized to 2 ^ = i J2k=-qi \9n,.[k}\2 — 1. For infinite impulse response (IIR) B F F s <?( —> oo and qu —> oo and for finite impulse response (FIR) B F F s qi — 0 and qu = Lg — 1, where L s is the F I R B F F length. For future reference we define the F I R B F F vector g 4 [ 5 l[0] . . . 9l[Lg - 1] 5 a [0] •.. gNT[Lg - 1]] T of length NTLg. A s a result, the filtered symbols snt[k] of antenna nt can be obtained for the modulated symbols b[k] at time k by S n t W = S n t [ f c ] . ® 6 W . (2-20) where <8> refers to convolution. 2.3.1 Transmit Pulse Shaping For transmit pulse shaping, the linearized impulse ht(t) corresponding to G M S K wi th t ime-bandwidth ( B T ) product 0.3 is employed [25]. Therefore, the trans-mit filter impulse response is given by [2, 25] ht(t) n s(t+kT), k=0 with sin (7T j p ( r )d r s(t) = { / t-AT sin ( | — 7T J g(r)d,T V o 0, 0 < t < 5T; otherwise, 0 < t < 4T; 4T < t < 8T; otherwise, (2.21) (2.22) where T = 3.69//S is the symbol duration. The impulse g(t) of duration AT is given by 9(t) = _1_ 2T Q 2TT • 0.3 t 5T T^/W) 3T (2.23) Q 2TT • 0.3 TJW2) 0 < * < 4T, 14 where Q(-) denotes the complementary Gaussian error integral [25], +00 Q(t) = ~j=l e-^2dr. (2.24) t B y this choice of the transmit filter for 2 - P S K and 8 - P S K symbols, the trans-mit spectra of E D G E and G S M are approximately equal, and the spectral masks of G S M are fulfilled [2]. 2.3.2 Receiver Processing The continuous-time signals are transmitted over the correlated M I M O chan-nel HC(T, i) discussed in the last section. A t the receiver, the continuous-time received signal at antenna n r is impaired by additive white Gaussian noise ( A W G N ) nnr(t). The choice of the receiver input filter, hr(t), is up to the receiver designer. We assume a filter wi th square-root Nyquist frequency re-sponse. Th is allows us to model the channel noise after sampling as a spatially and temporally white discrete-time Gaussian random process, which w i l l be discussed in Section 2.4. Two filters which have a square-root Nyquist frequency response are the whitened matched filter ( W M F ) [25], which belongs to the class of opt imum re-ceiver input filters [30], and the square-root raised cosine (SRC) filter [25, 26]. We use a fixed filter in this work, namely the S R C receive filter wi th roll-off factor 0.3 [26]. Th is filter offers a similar performance as the opt imum W M F . However, the implementation of the S R C filter is much simpler because, in contrast to the W M F , it does not have to be adapted to a particular chan-nel impulse response [2]. The discrete-time received signals are obtained by sampling the output of the receiver input filters at times t — kT. Final ly , the receiver, assumed to have perfect knowledge of the overall channel impulse re-sponse (CIR) , performs equalization of the received signals and estimates the transmit symbols. In other words, ISI can be mitigated by employing an equal-izer at the receiver side. Maximum-l ike l ihood sequence estimation ( M L S E ) , 15 decision-feedback equalization ( D F E ) , and linear equalization ( L E ) are some of the equalization methods which are commonly used in practice. We adopt D F E and L E at the receiver in this work due to their low complexity, good performance, and practical relevance. It should be noted that the B F F s gnt [k], the pulse shaping filters ht(t), and the receiver input filters hr(t), introduce additional ISI to the M I M O channel. In addition, the pulse shaping filters and receiver input filters introduce tem-poral correlation to the channel. 2.4 E q u i v a l e n t D i s c r e t e - T i m e S y s t e m M o d e l The overall channel model discussed in the previous section is in continuous-time and contains different blocks including the pulse shaping filters ht(t), the physical channel H(r,t), and the receiver input filters hr(t). It is convenient to derive an equivalent discrete-time model containing the combined effects of all these blocks. In this section, we w i l l show how the discrete-time model can be obtained. Let us consider again a M I M O system wi th NT transmit and NR receive antennas. The block diagram of the discrete-time overall transmission system in complex baseband representation is shown in Figure 2.3. 2.4.1 Discrete-time Channel Model In this work, block fading is assumed. Tha t is, the wireless channel coefficients hcrnt'l(t) defined in E q . (2.6) are approximately constant during one burst but vary from burst to burst. In other words, the coefficients h1(\rnt'l(t) are time-invariant wi th in each burst. This assumption is valid for small-to-moderate burst lengths and low vehicle speeds. W i t h this assumption, the time depen-16 b[k] Si [k] tolk] ryw.,.[fc] ni >-& "2 •k] nN„[k] 6[fc] Equalizer Feedback Channel Figure 2.3: Block diagram of the discrete-time overall transmission system. b[k] are estimated symbols at the receiver. dence of / i ^ r " t , ( ( t ) can be dropped and E q . (2.6) reduces to Hln = 1,21,/ 121 ,22,1 h c ,NRl,l hNR2,l h1™ h™T<1 (2.25) Now, the continuous-time overall C I R can be obtained from hnrnXt) = h{t)®hn(Jnt{t)®hr{t), where L-l (2.26) (2.27) 1=0 One can also obtain the above equation from E q . (2.8). Since the channel is t ime-invariant, t in E q . (2.8) is fixed and can be dropped from the equation. Therefore, the only variable left is r . Replacing r wi th t yields hr}jnt(t). In principle, the overall C I R is of infinite length. However, in practice, it can be sampled and truncated to L consecutive taps which exhibit maximum energy [31]. Therefore, the sampled and truncated overall C I R can be writ ten as hr, hnrnt(lT + *o)i 1 = 0 L - l , (2.28) 17 where to is a small time delay, and T is the symbol duration. L and t 0 are chosen so that only a negligible amount of power is disregarded. 2.4.2 Equivalent S I M O Model W i t h this discrete-time channel model, the T-spaced, sampled version of the received signal at receive antenna nr, 1 < nr < NR, can be modeled as NT L-l rnr [k] = ]P E hnrn<- WS™' [k~l]+ nnT [k] nt = l 1=0 NT = E hntnT[k]®snt[k]+nnT[k],. (2.29) nt = l where snt[k] is defined in E q . (2.20) and nnr[k] denotes additive (spatially and temporally) white Gaussian noise ( A W G N ) wi th variance a\ = £ {\nnr[k]\2} — N0, where N0 denotes the single-sided power spectral density of the underlying continuous-time passband noise process. Note that nnr[k] — nnr(kT + t0) in E q . (2.29) is spatially and temporally white because the S R C receive filter autocorrelation function fulfills the first Nyquist criterion [25]. Likewise, E q . (2.29) can be writ ten as NT rnT [fc] = YI h n ^ M ® blk] 0 9nt [k] + nnT [k] nt=l = h2[k}®b[k)+nnr[k} L+Lg-2 = E h2[l}b{k-l]+nnr[k], (2-30) (=0 where [k] is the equivalent C I R wi th length Leq = L + N — 1 corresponding to receive antenna nr and is defined as NT KqM = J2hnrnt[k}®9nt[k}- (2.31) nt = l E q . (2.30) shows that a M I M O system wi th bearnforming can be modeled as an equivalent single-input multiple-output (SIMO) system. Therefore, at the receiver the same equalization, channel estimation, and channel tracking 18 techniques as for single-antenna transmission can be used. Here, we adopt receive-diversity zero-forcing (ZF) or minimum mean-squared error ( M M S E ) D F E [21] and L E [22] for detection and assume perfect channel estimation at the receiver. 2.4.3 Feedback Channel Furthermore, we assume that a feedback channel from the receiver to the transmitter is available, cf. Figure 2.3. In case of perfect CSI , the receiver sends the opt imum B F F s (or equivalently the C I R ) to the transmitter. If the feedback channel allows only the transmission of B bits per channel update, the receiver and the transmitter have to agree on a pre-designed B F F vector codebook Q = {g1, g2, • • •, g^} of size N = 2B, where gn is a vector of length NTL9 and g^gr„ = 1, 1 < n < AT. For a given C I R , the receiver determines the address n of the codeword ( B F F vector) gn € G, 1 <n < N, which yields the min imum bit error rate ( B E R ) . Subsequently, address n is sent to the transmitter which then utilizes g — gn for beamforming. Since the primary goal of this work is to investigate the achievable performance of beamforming wi th , respectively, perfect and quantized CSI at the transmitter, similar to [11, 12, 13] we assume that the feedback channel is error-free and has zero delay. The discussions in this chapter are valid for both G S M and E D G E systems because they use the same frequency bands, transmit pulse shaping filters, and receiver input filters [1]. It should also be noted that the model is not restricted to G S M and E D G E systems, but is applicable to any system that employs linear single-carrier modulation and has a feedback channel. 19 Chapter 3 Beamforming with Perfect CSI and IIR Filters In this section, we assume perfect CSI and I IR B F F s . For this scenario jointly opt imum B F F s and equalization filters were derived in [17, Section IV] . Thereby, the B F F s and equalization filters were optimized for a fixed overall C I R (including the B F F s , the channel, and the equalization filters), and subsequently this overall C I R was chosen or optimized. Here, we pur-sue a much simpler and more straightforward approach. In particular, we assume that the receiver employs the opt imum D F E ( Z F - D F E or M M S E -D F E ) filters or L E ( Z F - L E or M M S E - L E ) filters for given M I M O chan-nel and B F F s , and optimize the B F F s for maximizat ion of the S N R un-der this assumption. The S N R s achievable wi th I IR B F F s given in this chapter w i l l serve as upper bounds for the S N R s achievable wi th the F I R B F F s , which w i l l be discussed in Chapter 4 and 5. For convenience the fre-quency responses of the I IR B F F s Gnt(f) = F{gnt[k)} are collected in vector G(f) = [Gi(f) G2(f) ... GNT(f)]T in the following sections. 20 3.1 I I R B e a m f o r m i n g w i t h D e c i s i o n - F e e d b a c k E q u a l i z a t i o n In this section, we consider I IR beamforming wi th Z F - and M M S E - D F E at the receiver. The optimization problem is carefully stated first, then the opt imum B F F s for maximizat ion of the S N R of a given channel realization are derived under these assumptions. 3.1.1 Optimization Problem For a given channel realization and a given beamforming vector G(f), the unbi-ased S N R for D F E wi th opt imum I IR feedforward and corresponding feedback filtering is given by [20, 21] 1/2 SNR(G(/)) = 4exp< / In NR e+]rro/)i2 nr=l df}-X, (3-1) -1/2 where x = 0, f = 0 and x = 1, f = ^l/^l for Z F - and M M S E - D F E filter optimization, respectively. In E q . (3.1), the equivalent channel frequency re-sponse H%(f) 4 F{h<*[k]} is given by H%(f) = E^ = i G n t(/)iW/) w i th Hntnr(f)=F{hntnr[k}}. The opt imum B F F vector G(f) = [Gi(f) G2(f), . . . GNT{f)]T shall max-imize S N R ( G ( / ) ) subject to the transmit power constraint 1/2 j GH(f)G(f)df = l. (3.2) -1/2 A convenient approach for calculating G(f) is the classical Calculus of Var i -ations method [32]. Following this method, we model the B F F of antenna nt as Gnt(f) = Gnt(f) + £ntBnt(f), where Bnt(f) and ent denote an arbitrary function of / and a complex-valued variable, respectively. The optimization 21 problem can now be formulated in terms of its Lagrangian 1/2 L(e)=SNR(G(/))-/z J GH(f)G(f)df -1/2 ^Ei R=i E"=i (G„ t(/) + ^ B n t ( / ) ) tf„tnr(/) In -1/2 + X V 2 JV. -1/2 n ' = 1 d / (3.3) where e = \e\ £2 • • • £NT]T and / i is the Lagrange multiplier. The opt imum B F F vector G(f) has to fulfill the condition [32] 8L(e) de* e=o NT (3.4) for arbitrary Bnt(f), 1 < nt < NT- Therefore, for al l s = 1,2, ...,NT, E q . (3.4) can be writ ten as dL{e)\ de* -1/2 EnXiB:(f)H:nr(f) E^=i {Gnt(f) +entBnt(f)) Hntnr(f) Z—mr=l E?=i {Gnt(f) + entBnt(f)) Hntnr(f) 2 t-df £ = 0 * 1/2 - / x y ( G s ( / ) + e S J B a ( / ) ) B ; ( / ) d / -1/2 £ = 0 E^=i Gnt(f)HntnT(f) E^=i Gnt(f)Hntnr(f) 2 1/2 -1/2 - M y G.(f)B*a(f)df -1/2 =0, (3.5) where [•]* denotes complex conjugate. In order to achieve E q . (3.5), the fol-22 lowing condition must be satisfied for all s = 1, 2 , . . . , NT, / NR NT 2 \ \ n r = l n't=l / NR / NT N = ?ZH*snM)[Y,Gnt{f)Hntnr{f) (3.6) nr = l \nt = l or equivalently, / NR NT \ n r = l n't=l / NT / NR N = TtGnt(f)[J2H*mr(f)Hntnr(f) (3.7) nt = l \nT = l 3.1.2 Optimum IIR B F F s Combining Eqs. (3.4) and (3.7), it can be shown that vector G(f) has to fulfill S(f) G(f) = jl [GH(f)S(f)G(f) + £] G(f), (3.8) where fx is a constant and S(f) is an NT X iVr matrix given by NR S(f)=J2Hnr(f)HZ(f) nr — l NR NR E H*lnr(f)Hlnr(f) E H*lnT(f)H2nr(f) nr = l nr = l NR NR E H2nr(f)Hlnr(f) E H2nrU)H2nrU) nr = l nr=l NR E Hlnr{f)HNTnr{f) nr = l NR E H2nr(f)HNTnr(f) nr=l NR NR NR E H*NTnr(f)Hlnr(f) E HNrnr(f)H2nr(f) - E HNTnr(f)HNTnr(f) n r = l nr = l n r = l (3.9) and H n r ( / ) = # 2 „ r ( / ) • • • ^ . n r ( / ) ] T . (3.10) Eq. (3.8) is a nonlinear eigenvalue problem and G ( / ) can be expressed as G(f) = X(f)E(f) (3.11) 23 where E(f) = [Ei(f) E2(f) ••• ENT(f)}T is that un i t -norm eigenvector of S(f) which corresponds to its largest eigenvalue A m a x ( / ) , and X(f) is a scalar factor. For example, for NR = 1 we obtain E(f) = H^f)/J Hf (f)Hi(f). E q . (3.11) shows that in general the opt imum I IR B F F s can be viewed as concatenation of two filters: A filter X(f) which is common to a l l transmit antennas and a filter Ent(f) which is transmit antenna dependent. In the following, we derive filter X{f) for Z F - D F E and M M S E - D F E . 1) ZF-DFE: In this case, £ = 0 is valid. Combining Eqs. (3.8) and (3.11), we can get or equivalently | X ( / ) | = \j\fji- Furthermore, from the power constraint in E q . (3.2), we obtain S(f) XU) E(f) = ft E (f) X*{f) S(f)X(f) E(f) X(f) E{f) A m a x ( / ) XU) = AAmax(/) \X{f)\2 X{f) \xu)\2 = i (3.12) 1/2 - 1 / 2 1/2 - 1 / 2 1/2 - 1 / 2 1. (3.13) Therefore, the opt imum I IR B F F s for Z F - D F E are given by G(f) = E(f) e M / ) (3.14) 24 where <p(f) is the phase which can be chosen arbitrarily. Replacing G(f) in Eq. (3.1) by G(f) from Eq. (3.14) yields the maximum SNR for Z F - D F E 1/2 SNR(G( / ) ) = ^ e x p NR , NT E J2Gnt(f)Hntnr(f) In . n r = l n t = l GH(f)S(f)G(f) df -1/2 1/2 = ^ e x J J l n [ | X ( / ) | 2 A m a x ( / ) ] df -1/2 1/2 ^ e x p i J l n ( A m a x ( / ) ) d / 1-1/2 (3.15) 2) MMSE-DFE: For M M S E - D F E £ = c r ^ /o f holds. As in the derivation for Z F - D F E case, combining Eqs. (3.8) and (3.11), we can get S(f)X(f)E(f) = ii[EH(f)X*(f)S(f)X(f)E{f) + Z] X(f)E(f) A m a x ( / ) X(f) = ix [ A m a x ( / ) \X{f)\2 + H] X(f) \x(f)\2 = P- (3.16) Amax(/) where jj, = 1//2. Therefore, it can be shown that in this case the magnitude of the optimum X(f) IS given by l * ( / ) l = Amax(/) J (3.17) 25 where we took into account that |-X"(/)| has to be non-negative. Furthermore, from the power constraint in E q . (3.2), we obtain 1/2 J GH(f)G(f)df = l -1/2 1/2 \X(f)\2 d / = l -1/2 1/2 i f r df = 1. (3.18) 1/* I ^ m a x ( / ) -1/2 A s E q . (3.18) suggests, finding the optimum ft is a typical Water F i l l i n g prob-lem [29]. In particular, motivated by the power constraint in E q . (3.2) we define m * / 1/2 e 1 + A m a x ( / ) df. (3.19) -1/2 The opt imum / i o p t can be found by slowly increasing a very small starting value fi = /t 0 which yields P ( Ao ) > 1 unti l P(fi — / t o pt) = 1 is achieved. Figures 3.1 and 3.2 illustrate results for the Water F i l l i n g algorithm for the same channel realization of the E Q channel wi th channel length L = 7 at ES/NQ = 10 d B and ES/N0 = 20 d B , respectively, where ES is the average received energy per symbol. The communications system consists of 3 trans-mit antennas, 1 receive antenna and M M S E - D F E filter at the receiver. F rom Figures 3.1 and 3.2, the opt imum "water" level, / } o p t , is a constant wi th in the normalized frequency range, - 1 / 2 < / < 1/2. The opt imum I IR B F F s uti-lize the channel realization and allocate more power to frequencies wi th large eigenvalues, A m a x ( / ) , of matr ix S ( / ) . A s the ES/NQ increases from 10 d B to 20 d B , the "water" level p,opt drops from 1.15 to 1.01, respectively. Also , the applied power, [fiopt — £ / A o p t ( / ) ] + , becomes flatter in Figure 3.2 compared to Figure 3.1 as ES/N0 increases. 26 -O.S -0.4 0.2 0.3 0.4 O.S Figure 3.1: Water F i l l i n g for one realization of the E Q channel wi th L = 7, NT = 3, NR = 1, equal antenna correlation p = 0.5, and 10 log 1 0(.E s/ . /Vo) = 10 d B . The opt imum IIR B F F s for M M S E - D F E are given by G(f) = J Aopt E(f)e^\ (3.20) Amax(/) J where </?(/) is again the phase which can be chosen arbitrarily. The corre-27 a> 3 0.6 S 0 4 - n (water level) -0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5 Figure 3.2: Water F i l l i n g for one realization of the E Q channel wi th L = 7, NT = 3, NR = 1, equal antenna correlation p = 0.5, and 1 0 1 o g 1 0 ( £ , 5 / A r 0 ) = 20 d B . sponding maximum S N R is O", SNR(G<(f)) = ^ e x p cr ot NR , NT nr — l nt = l df}-l exp{ I In G"(/)S(/)G(/)+£ d / J> - 1 = ^ exp | j In [ | X ( / ) | 2 A m a x ( / ) + f] d / 1 - 1 -1/2 1/2 ^ exp J | In ( [ A m a x ( / ) A o P t - £ ] + + 0 d / | - 1. (3.21) 28 Regarding the B F F s and the maximum S N R s for Z F - D F E and M M S E - D F E , we make the following interesting observations. a) Al though we have optimized the B F F s for fixed D F E filters, the opt imum IIR B F F vectors given by Eqs. (3.14) and (3.20) are identical to the jointly opt imum solution given in [17]. b) A s expected, for c r 2 -> 0 (i.e., £ -> 0) the opt imum I IR B F F s for M M S E -D F E approach those for Z F - D F E . Furthermore, for arbitrary a\ and L = 1 (i.e., frequency-nonselective channel) the opt imum B F F s for Z F -D F E and M M S E - D F E , respectively, have only one non-zero tap and are identical to the well-known bearnforming weights, cf. e.g. [9]. c) In case of Z F - D F E , the opt imum B F F frequency response vector G(f) at frequency / = fo is just the dominant eigenvector of matr ix S(f) at frequency f = fo- Since S(f0) only depends on the channel frequency responses Hntnr(f) at frequency / = fo, G(fo) is independent of the Hntnr(f), f ^ fo- This is not true for M M S E - D F E , where the opt imum frequency response vector G(f) at frequency f = fo also depends on the channel frequency responses Hntnr(f) at frequencies / ^ fo, because of the constraint P{jl) = 1, cf. E q . (3.19). In fact, for M M S E - D F E X(f) may be interpreted as a power allocation filter which allocates more power to frequencies wi th large eigenvalues A m a x ( / ) . 3.2 I I R B e a r n f o r m i n g w i t h L i n e a r E q u a l i z a -t i o n In this subsection, we consider I IR bearnforming wi th Z F - and M M S E - L E at the receiver. Similar to the previous subsection's approach, the optimization problem is addressed followed by the derivation of the opt imum B F F s for max-imization of the S N R of a given channel realization under these assumptions. 29 3.2.1 Optimization Problem From [20], the unbiased S N R for L E wi th opt imum I IR feedforward filtering is (7, S N R ( G ( / ) ) = - § - X , (3.22) where x = 0 and x = 1 for zero-forcing (ZF) and min imum mean-squared error ( M M S E ) L E filter optimization, respectively. The L E error variance is 1/2 o-2e=o-2n [ 5 df, (3.23) 1/2 where £ = 0 and f = <r2/cr2 are valid for Z F - L E and M M S E - L E , respectively [22]. Equivalently, we can rewrite E q . (3.22) as / V2 , \ ^ - X- (3.24) S N R ( G ( / ) ) = 4 1 d f y_t/2 ES=iTO/)i 2+e y The equivalent channel frequency response H^(f) = ^"{^[ fc ]} is given by H%U) = E i i Gnt(f)Hninr(f) wi th tfntnr(/) 4 J F ^ J A : ] } . Aga in , the opt imum B F F vector G ( / ) = [G\(/) G 2 ( / ) , . . . GNr(f)]T shall maximize S N R ( G ( / ) ) subject to the transmit power constraint 1/2 J GH(f)G(f)df = l. (3.25) -1/2 A s in the D F E convenient approach for calculating G(f) is the classical Calculus of Variations method [32]. Following this method, we model the B F F of antenna nt as Gnt(f) = Gnt(f) + entBnt(f), where Bnt(f) and ent denote an arbitrary function of / and a complex-valued variable, respectively. The 30 optimization problem can now be formulated in terms of its Lagrangian 1/2 L(e) =SNR(G(/)) - p j GH(f)G(f) df -1/2 1/2 £ £ L i E ^ L (Gnt(f) + entBnt{f)) Hntnr(f) -1/2 df l l 2 NT -y. J El^t(/) + ^ A t ( / ) | 2 d / - 1 / 2 N ' = 1 (3.26) where £ = [ei e 2 • • • £NT}T and // is the Lagrange multiplier. The opt imum BFF vector G(f) has to fulfill the condition [32] dL(e)\ de* £ = 0 0 W t (3.27) NT for arbitrary Bnt(f), 1 < nt < NT- Therefore, for al l s — 1,2, ...,NT, Eq. (3.27) can be writ ten as dL(e) I e=oh -1/2 EnrR=lB;(f)H;nr(f) E Z i {Gnt{f)+entBnt{f)) Hntnr{f) Z ^ n r = l E " = i {Gnt(f) + entBnt(f)) Hntnr(f) 2 2 d / £=CK 1/2 y (Ga(f)+eaBa(f))B;(f)df - 1 / 2 E ^ = l G„t(f)Hntnr(f) Z - m r = l E n t = l Gnt{f)HntrlT(f) 2 2 1/2 - 1 / 2 - M y cs(f)B:(f)df - 1 / 2 d / =0, (3.28) In order to achieve Eq. (3.28), the following condition must be satisfied for al l 31 s = 1,2,..., NT, N R , N T \ 2 nr n't = l / i V R / N T (3.29) (3.30) n r = l \ n ( = l / or equivalently, / N R N T 2 \ 2 (^/) E|E^ (/)^ w(/) +e = \ nr n't=l J N T / N R • \ n t = l \ n r = l / 3.2.2 Optimum IIR BFFs Combining Eqs. (3.27) and (3.30), it can be shown that vector G(f) has to fulfill S(f)G(f) = i±[GH(f)S(f)G(f)+t\2 G(f), (3.31) where p, is a constant and S(f) is an NT X NT matr ix given by E q . (3.9). The only difference between E q . (3.31) and E q . (3.8) of the D F E case is that there is a square term in E q . (3.31) compared to E q . (3.8). This difference is resulted from changes of the objective function for optimization. Similarly, E q . (3.31) is a non-linear eigenvalue problem and G(f) can be expressed as G(f) = X(f)E(f), (3.32) where E(f) = [Ei(f) E2{f) ••• ENT(f)]T is that un i t -norm eigenvector of S(f) which corresponds to its largest eigenvalue A m a x ( / ) , and X(f) is a scalar factor. Consequently, as in the D F E case, in general the opt imum I IR B F F s can be viewed as concatenation of two filters: A filter X(f) which is common to all transmit antennas and a filter Ent(f) which is transmit antenna dependent. 1) ZF-LE: In this case, £ = 0 is valid. Combining Eqs. (3.31) and (3.32), we 32 get S(f) X(f) E(f) = A [EH(f) X*(f) S(f)X(f) E(f) XU)]2 X{f) E(f) \ m a x U ) x U ) = A M \ x U ) Y x u ) \x(f)\4 A^max(/) (3.33) or equivalently | X ( / ) | — 1/\/jlXmSiX(f)- Furthermore, from the power con-straint in E q . (3.2), we obtain 1/2 j GHU)G(f)df = l -1/2 1/2 (X*(f)EH(f)) X(f)E(f)df = l - 1 / 2 1/2 - 1 / 2 V7^max(/) d/ = l / 1/2 = f - r 1 V-l/2 y/KnaxU) df (3.34) Thereby, the opt imum I IR B F F s for Z F - L E are given by \ - i / 2 G(f) 1/2 \J A m a x ( / ) V Amax(/) df E{f) eMf), (3.35) -1/2 / where (p(f) is the phase which can be chosen arbitrarily. Unl ike the result X(f) = 1 given by E q . (3.14) for Z F - D F E , X(f) is a frequency dependent term for Z F - L E . Interestingly, X(f) allocates more power to frequencies wi th large eigenvalues A r a a x ( / ) of S(/). Thereby, X(f) for Z F - L E can be interpreted as a power allocation filter. Replacing G{f) in E q . (3.24) by G(f) from E q . (3.35) yields the corre-33 sponding maximum S N R for Z F - L E SNR(G(/ ) ) or. jrr: \_l/2 2~2n*=l 2^2n^=lGnt(f)Hntnr(f) J /1/2 _ X- 1 -1 / -/2 GH(f)S(f)G(f) ( 1/2 , : \ - 1 / ^ f # d / at df J V-l/2 1/2 \ -1 -1/2 \J A max(/) df n / 1 / 2 | / — I n I V \ / A m a x ( / ) V-l/2 -2 / (3.36) 2) MMSE-LE: For M M S E - L E f = cr 2/cr 2 holds. Combining Eqs. (3.31) and (3.32), we can get S(f) X(f) E{f) = A [ # " ( / ) X * ( / ) £ ( / ) * ( / ) + e ] 2 X{f) E{f) A m a x ( / ) * ( / ) = A [ A m a x ( / ) | X ( / ) | 2 + C ] 2 X{f) \ / A m a x ( / ) (3.37) yArnaxT/) where / i = l / v T * is a constant and we took into account that | X ( / ) | 2 has to be non-negative. Furthermore, from the power constraint in E q . (3.26), we 34 obtain 1/2 J GH(f)G(f)df = l -1/2 1/2 / -1/2 \X(f)\2 d / = l 1/2 / 1 \ / A m a x ( / ) -1/2 A4 \ A n a x ( / ) d / = 1. (3.38) A s E q . (3.38) suggests, unlike the result derived in M M S E - D F E case, finding the opt imum p is a quasi Water F i l l i n g problem [29] due to the fact that the "water" level, p/y/\max(f), is not a constant but depends on frequency / . In particular, motivated by the power constraint in E q . (3.26) we define 1/2 V A m a x ( / ) 1/2 \J A m a x ( / ) df. (3.39) The opt imum popt can be found by slowly increasing a very small starting value p. = p,0 which yields P(po) < 1 unti l P(p — popt) = 1 is achieved. Figures 3.3 and 3.4 illustrate the quasi Water F i l l i n g results for the same channel realization of the E Q channel wi th channel length L = 7 at ES/NQ = 10dB and ES/N0 = 20 d B , respectively. The simulated system consists of 3 transmit antennas, 1 receive antenna and M M S E - L E filter at the receiver. Figures 3.3 and 3.4 confirm that, unlike the "water" level in the D F E case, the opt imum "water" level, popt j\J'Amax(/), is not a constant wi th in the frequency range but a variable wi th respect to / . A s the ES/N0 increases from 10 d B to 20 d B , the "water" level varies more smoothly. Al though it is hard to justify how the opt imum IIR B F F s allocate power from these two figures, the resulting opt imum S N R , i.e. E q . (3.41) derived in the later section, at the equalizer confirms that the optimum IIR B F F s utilize the channel realization and allocate more power to frequencies wi th large eigenvalues, A m a x ( / ) , of matr ix S(/). 35 Figure 3.3: Quasi Water F i l l i n g for one realization of the E Q channel w i th L — 7 ; NT — 3, NR = 1, equal antenna correlation p = 0.5, and 10logw{Es/N0) = 10 dB. The opt imum I IR B F F s for M M S E - L E are given by 1 G(f) ^opt E{f)e ,Mf) (3.40) \ A i m u i ( / ) i where tp(f) is again the phase which can be chosen arbitrarily. Replacing G(f) in E q . (3.24) by G(f) from E q . (3.40) yields the corresponding maximum S N R 36 2.5 - (f) ^ max ' [ ^ ™ y ( f ) ) " 2 - ^ m a v ( 0 ] + 0) / \ / \ -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Figure 3.4: Quasi Water F i l l i ng for one realization of the E Q channel wi th L = 7, NT = 3, NR — 1, equal antenna correlation p = 0.5, and 10logw(Ea/N0) = 20 d B . for M M S E - L E SNR(G(/)) / 1/2 V 1 •df ^2 \ - l / 2 12n?=l X)n t=l Gnt ( V 2 ' _ 1 I ^TTj d / | - 1 J/2 GH(f)S(f)G(f) + t: ( 1/2 - 1 -1 V 1 ' / 2 y/\m**(f) ( 1/2 L •df -i \_l /2 \/Amax(/)Aopt ~ £ +•£ y d / - 1. - 1 (3.41) 37 Comparing the B F F s and the maximum S N R s for Z F - L E and M M S E - L E , we make the following interesting observations: a) For a2n - » 0 (i.e., £ -> 0) the opt imum IIR B F F s for M M S E - L E approach those for Z F - L E . Al though we have optimized the B F F s for fixed L E filters, the opt imum I IR B F F vectors given by Eqs. (3.35) and (3.40) are identical to the jointly opt imum solution given in [17]. b) In both Z F - L E and M M S E - L E cases, the opt imum B F F frequency re-sponse vector G(f) at frequency f = fo depends on the channel fre-quency responses HntTlr(f) at al l frequencies —1/2 < / < 1/2. In fact, X(f) may be interpreted as a power allocation filter which allocates more power to frequencies wi th large eigenvalues A m a x ( / ) in Z F - L E and M M S E - L E cases. c) If the underlying channel is frequency-nonselective, S(/) = S for al l / . In this case, it is easy to see that the opt imum B F F s have only one non-zero tap and are identical to the well-know beamforming weights for frequency-nonselective channels [9]. In this case, the L E structure collapses to a simple threshold detector, of course. Al though we derived promising I IR B F F s in this chapter, implementing I IR B F F s in practice is not realistic. There are two main reasons. First ly, it is well know that designing stable I IR filters for any channel realization is difficult. Secondly, the transmitter requires to know perfect CSI in order to perform IIR beamforming. Due to the fact that the feedback channel is bandwidth l imited, perfect CSI at the transmitter cannot be provided. However, the S N R s achievable wi th I IR B F F s given by Eqs. (3.15), (3.21), (3.36), and (3.41) w i l l serve as upper bounds for the S N R s achievable wi th the F I R B F F s considered in Chapter 4 and 5. 38 Chapter 4 Beamforming with Perfect CSI and F I R Filters The I IR B F F s derived in the previous chapter are not suitable for practical implementation. Therefore, in this chapter, we derive the opt imum F I R B F F s for perfect CSI using the same approach as for the I IR case in Chapter 3. We note that although F I R B F F s are adopted in this section, the D F E and L E are s t i l l assumed to employ an I IR feedforward filter. This is not a major restric-t ion as reasonably long F I R D F E and L E feedforward filters yield practically the same performance as I IR feedforward filters. 4.1 F I R B e a m f o r m i n g w i t h P e r f e c t C S I a n d D F E 4 . 1 . 1 O p t i m u m F I R B F F s Since the F I R B F F s have length Lg, the resulting equivalent overall C I R h^[k] has length L e q = L + Lg — 1. The frequency response of the equivalent channel can now be expressed as H£(f) = dH(f)Hnrg, (4.1). 39 where d(f) 4 [1 eP2^ . . . e ^ / ( ^ , - D ] 3 , ^ [-H"lnr H-2nr H NTnr\ and Hntnr is an L e q x L 9 column-circulant matrix defined as hntnr [0] hntnr [L - 1] 0 0 T J A 0 h n t n r [0] (4.2) h n t n r [L — 1] Combining Eqs. (3.1) and (4.1) the S N R of Z F - and M M S E - D F E wi th F I R B F F s is obtained as SNR( B) = 4 exp | / ln(g"B(/)s + Q df 1 - X, (4.3) with i V T L g x i V T L 3 matrix B ( / ) = E n r t i H^d{f)dH(f)HnT. The opt imum B F F vector g shall maximize SNR(g ) subject to the power constraint gHg = 1. Thus, the Lagrangian of the optimization problem can be formulated as L(g) = SNR(g) + LLgHg, (4.4) where \x denotes again the Lagrange multiplier. The opt imum vector g has to fulfill dL(g)/dg* = OJV T L„ , which leads to the nonlinear eigenvalue problem 1/2 / - 1 / 2 B{f) + £INTLg g" {B(f)+lrINTLg)g zdf 9 = 9, (4.5) where \i does not appear since E q . (4.5) already includes the constraint gHg = 1. However, in contrast to the I IR eigenvalue problem in E q . (3.8), E q . (4.5) does not seem to have a closed-form solution. To substantiate this claim, we discretize the integral in E q . (4.3) and rewrite the optimization problem in E q . (4.4) as £to) = f K ( B ( / ' ) . * - { J * 0 ' . ( « ) i = l 9H9 40 where = - 1 / 2 + (i - l)/(S - 1) and S is a large integer. Note that E q . (4.6) is scale invariant (i.e., L(g) does not depend on the magnitude of g) and the resulting solution has to be scaled to meet gHg = 1. E q . (4.6) is a product of Rayleigh quotients. Unfortunately, it is wel l -known that the maximizat ion of a product of Rayleigh quotients is a difficult mathematical problem which is not well understood and a closed-form solution is not known except for the t r iv ia l case S = 1, cf. e.g. [33, 34]. Therefore, we also do not expect to find a closed-form solution for the nonlinear eigenvalue problem in E q . (4.5). Instead, we provide two efficient numerical methods for recursive calculation of the opt imum F I R B F F vector g in the next section. 4.1.2 Calculation of the Optimum F I R B F F s For calculation of the opt imum F I R B F F s we propose two different algorithms. B o t h algorithms recursively improve an ini t ia l B F F vector g0. Since the cost function in E q . (4.3) is not concave, we cannot guarantee that the algorithms converge to the global maximum. However, adopting the ini t ial izat ion pro-cedure explained below, the solutions found by both algorithms seem to be close-to-optimum, i.e., for Lg » 1 the obtained F I R B F F s approach the per-formance of the opt imum IIR B F F s derived in Chapter 3. 1) Gradient Algorithm (GA): The first algorithm is a typical G A where in iteration i + 1 the B F F vector gi from iteration i is improved in the direction of the steepest ascent [29]. The G A is summarized as follows: 1. Let i = 0 and initialize the B F F vector wi th some g0 fulfilling g"gQ = 1. 2. Update the B F F vector 9i+i =9i + &i 1 / 2 B(f) + dINTLg g?{Btf) + ZINTLg)g. - 1 / 2 (4.7) where <5j is a suitable adaptation step size. 41 3 Normalize the B F F vector 9i+i = 9i+i (4.8) 9i+l9i+l 4. If 1 — Ig i^ffjl < e, goto Step 5, otherwise increment i —> % + 1 and goto Step 2. 5. g i + l is the desired B F F vector g. The main drawback of the G A is that its speed of convergence crit ically de-pends on the adaptation step size Si, which has to be empirically optimized. For the termination constant e in Step 4 a suitably small value should be chosen, e.g. e = 10~ 4 . Ideally the adaptation step size 5i should be op-timized to maximize the speed of convergence. Here, we follow [29] and choose.6i proportional to 1/Aj where Aj is the maximum eigenvalue of ma-tr ix j^2B(f) df j (g?B(f)gi + in iteration i. In particular, we found empirically that Si = 0.01/Aj is a good choice. 2) Modified Power Method (MPM): It is interesting to observe that if the de-nominator under the integral in E q . (4.5) was absent, g would simply be the maximum eigenvalue of matrix Jj1^2(B(f) + £1NTL9) df, which could be cal-culated efficiently using the so-called Power Me thod [29]. Mot ivated by this observation, we propose an M P M for recursive calculation of g. The corre-sponding algorithm does not involve an adaptation step size and is summarized as follows: 1. Le t i — 0 and initialize the B F F vector wi th some g0 fulfilling gQ = 1. 2. Update the B F F vector 1 / 2 B{f) + tjINTL9 9i+i = I g" (B(/) + fe,)ft df (4.9) 1 - 1 / 2 3. Normalize the B F F vector 9i+i (4.10) 42 4. If 1 — I g ^ f f J < e, goto Step 5, otherwise increment i —> i + 1 and goto Step 2. 5. g i + 1 is the desired B F F vector g. A s mentioned earlier global convergence of the G A and the M P M to the maximum S N R solution cannot be guaranteed. However, we found empirically that convergence to the opt imum or a close-to-optimum solution is achieved i f the B F F length is gradually increased. If the desired B F F length is Lg, the G A or the M P M are executed Lg times. For the z/th, 2 < v < Lg, execution of the algorithm the first v — 1 B F F coefficients of each antenna are init ialized wi th the opt imum B F F coefficients for that antenna obtained from the [y — l ) t h execution and the j / th coefficients are initialized wi th zero. For the first [y = 1) execution, the B F F vector is initialized wi th the normalized all-ones vector of length NT. If this ini t ial ization procedure is used to calculate the opt imum B F F s of length Lg, the opt imum or close-to-optimum F I R B F F s of lengths 1, 2, . . . , Lg — 1 are also obtained as a by-product . This property may be useful when comparing F I R B F F s of different length. Of course, this comes at the cost of increased computational complexity. However, computational complexity is not a major concern here, since in practice beamforming wi th perfect CSI is not possible anyway. Nevertheless, F I R beamforming wi th perfect CSI is of interest because it (a) constitutes a benchmark for beamforming wi th quantized CSI and (b) the opt imum F I R B F F s are the input to the (off-line) codebook design for beamforming wi th quantized CSI , cf. Chapter 5. Extensive simulations have shown that the F I R B F F s obtained by respec-tively the G A and the M P M wi th the proposed init ial ization procedure ap-proach the performance of the opt imum I IR B F F s as the F I R filter length Lg increases. Exemplary simulation results for both algorithms are shown and discussed in Chapter 6. 43 4.2 F I R B e a r n f o r m i n g w i t h P e r f e c t C S I a n d L E Similar to the approach in the previous section, we introduce a numerical method for calculation of the opt imum F I R B F F s for L E in this section. 4.2.1 Optimum F I R B F F s Combining Eqs. (3.24) and (4.1), the S N R for Z F - L E and M M S E - L E can be expressed as - l SNR (o) = -i 1/2 -1/2 (f)9 + Z df X, (4.11) wi th NTLg x NTLg matr ix B(f) ± ZnrU Hld(f)dH(f)Hnr. The opt imum B F F vector gopt shall maximize S N R ( g ) subject to the power constraint gHg — 1. Thus, the Lagrangian of the optimizat ion problem can be formulated as L(g) = SNR(g) + figHg, (4.12) where fj, denotes the Lagrange multiplier. The opt imum vector g has to fulfill dL(g)/dg* =-0/VTL , which leads to the non-linear eigenvalue problem 1/2 B(f) -1/2 (gHB(f)g + tY 9 = 1*9, (4.13) wi th eigenvalue p,. The difference between E q . (4.5) and E q . (4.13) is that there is a square term in the denominator in E q . (4.13). Unfortunately, E q . (4.13) does not seem to have a closed-form solution. Instead, we proposed a gradient alori thm ( G A ) for recursive calculation of the opt imum F I R B F F vector g in the next section. 44 4.2.2 Calculation of the Optimum F I R B F F s In this subsection, we propose a gradient algorithm ( G A ) for calculation of the opt imum F I R B F F s , which recursively improves an ini t ia l B F F vector g0. Since the cost function in E q . (4.12) is not concave, we cannot guarantee that the algorithm converges to a global maximum. However, adopting the init ial ization procedure similar to that for the D F E case, the solution found by the proposed G A seems to be close-to-optimum, cf. Chapter 3. The Gradient Algorithm is summarized as follows: 1. Let i — 0 and initialize the B F F vector w i th a suitable g0 fulfilling 9o9o = 1-2. Update the B F F vector 9i+i =9i + $i 1/2 B(f)df -1/2 (9?B(f)gi + ty 9i, (4.14) where <5, is a suitable adaptation step size. 3. Normalize the B F F vector: 9i+i 9i+i 9i+i9i+i (4.15) 4. If 1 — \g^+1gi\ < e, goto Step 5, otherwise increment i —> i + 1 and goto Step 2. 5. gi+l is the desired B F F vector g. For the termination constant e in Step 4 a suitably small value should be chosen, e.g. e = 10~ 4 . Ideally the adaptation step size Si should be op-timized to maximize the speed of convergence. Here, we follow [29] and choose Si proportional to 1/Aj where Aj is the maximum eigenvalue of ma-t r ix J*y2B(f) df/ (gf'B(f)gi + £ ) 2 in iteration i. In particular, we found empirically that Si = 0.01/Aj is a good choice. 45 . A s the G A for the D F E case, the global convergence of the G A for L E cannot be guaranteed, but convergence to the opt imum or a close-to-optimum solution is achieved if the B F F length is gradually increased. It is also worth mentioning that M P M is not suitable for calculating F I R B F F s for L E re-ceivers. The reason for that is s t i l l unknown, and it may be the objective for future investigation. Our numerical results in Chapter 6 show that the performance of opt imum I IR bearnforming can be closely approached wi th short F I R B F F s derived in this Chapter. 46 Chapter 5 F I R Bearnforming with Quantized CSI In this chapter, we consider F I R bearnforming wi th quantized CSI . For this purpose, we first introduce vector quantization in the context of finite-rate feedback bearnforming for frequency-selective channels. Subsequently, we dis-cuss the mean quantization error and the distortion measure adopted for quan-tizer design. Then, we adapt the basic L B G algorithm to the problem at hand and present a global vector quantization ( G V Q ) algorithm for calculation of the B F F vector codebook Q for D F E and L E . 5.1 V e c t o r Q u a n t i z a t i o n - P r e l i m i n a r i e s Clustering is an important and fundamental instrument in engineering and other scientific disciplines to solve problems such as pattern recognition, image processing, machine learning, and so on [16]. The simplest form of clustering is the parti t ioning clustering approach known as Vector Quantization [35], which aims at parti t ioning a given data set into disjoint subsets so that the objective function (distortion) representing the quantization error is minimized. In this manner, each element of the data in the same subset is represented by only one corresponding codeword. In this section, we apply general vector quantization 47 techniques to our specific F I R B F F s codebook searching problem. For convenience we define the channel vector h = [/in[0] ... hn[L — 1] 1^2(0] ... hf]TNR{L — 1]] T of length NTNRL. We assume that a training set Ti — {hi, h2, • • •, hr} of T channel vectors hn is available. Thereby, chan-nel vector hn has length NTNRL and contains the C I R coefficients of al l NTNR C I R s of the n th realization of the M I M O channel. In practice, the hn may be obtained either from measurements or by simulating the M I M O channel. Using the G A or M P M summarized in Chapter 4 we calculate the opt imum B F F vector gn for each channel hn, 1 < n < T. The resulting B F F vector training set is denoted by QT — g2, • • • i 9T)-A vector quantizer Q is a mapping of the B F F vector training set QT w i th T entries to the B F F vector codebook Q = {glt g2, . . . , gN} w i th N entries, where N <C T [14]. Therefore, the vector quantizer can be represented as a function Q : QT —* Q'. The elements gn of the codebook Q are also referred to as codewords. Once Q is determined, we can define part i t ion regions 1Zn constituted by subsets of the original training set 1ln = {g G gT\Q(g) = 9n}, l<n<N, (5.1) i.e., if g falls into lZn it is quantized to gn. 5.2 M e a n Q u a n t i z a t i o n E r r o r ( M Q E ) a n d D i s -t o r t i o n M e a s u r e In general, a vector quantizer is said to be opt imum if it minimizes the mean quantization error (MQE) for a given codebook size N. The M Q E is defined as MQE =-J2d(Q(gn),gn), (5.2) n=l where d(gm,gn) is the so-called distortion measure and denotes the distortion caused by quantizing gn € QT to gm € Q. 48 Here, our a im is to design a codebook Q which minimizes the average B E R , i.e., the average B E R is the M Q E . Therefore, the distortion measure d(gm,gn) is the B E R Pe(gm, hn) caused by gm £ Q for channel hn € fi wi th opt imum B F F vector gn 6 QT, i.e.,' d(gm,gn) = Pe(gm,hn). (5.3) In order to obtain a tractable expression for the B E R , we assume Gray map-ping, and that the residual error at the input of the equalizer, D F E or L E , decision device is a Gaussian random variable independent of the data symbol b[k]. The latter assumption is true for Z F - L E and Z F - D F E , and is a good approximation for M M S E - L E and M M S E - D F E . W i t h these assumptions the B E R of D F E (with error-free feedback) or L E can be approximated as Pe(gm,hn) = C-Q (y< i n S N R ( £ m , M / 2 ) (5.4) where Q(x) = ^ = fx°° e~' 2/ 2 dt, and both the unimportant constant C and the min imum Euclidean distance dmm depend on signal constellation A. For example, C = 2 / l o g 2 M and dmm = 2s in(7r /M) for M - a r y P S K (MPSK). SNR(<) m , hn) can be obtained from E q . (4.3) for D F E or E q . (4.11) for L E . 5.3 L B G A l g o r i t h m The L B G algorithm [14] can be used to improve a given ini t ia l codebook. This algorithm exploits two necessary conditions that an optimal vector quantizer satisfies ( L l o y d - M a x conditions): 1. Nearest Neighborhood Condition (NNC): For a given codebook Q the part i t ion regions Hn, 1 < n < N, satisfy TZn = {ge QT\d(gn,g) < d(gm,g),Vm ^ n}, (5.5) i.e., 7ln is the Voronoi region of codeword gn, 1 < n < N. 49 2. Centroid Condition (CC): For given partitions TZn, 1 < n < N, the opt imum codewords satisfy where the M Q E for region lZn and candidate codeword g is defined as where \lZn\ denotes the number of t raining B F F vectors in region lZn. The L B G algorithm applies the above two conditions in an iterative fashion. In the first part of each iteration, using the N N C the training set Qr is partitioned into N regions lZn, 1 < n < N, based on the current codebook Q.. In the second part of the iteration, the C C is used to find a new codebook based on the partitions found in part one of the iteration. We note that, in contrast to the typically used Euclidean distance distortion measure [14, 16], for the distortion measure in E q . (5.3) it is not possible to find a closed-form solution for the centroid in E q . (5.6). Therefore, M Q E n (g ) is computed for a l l training vectors g G lZn, and that training vector which minimizes MQEn(g) is chosen as the new codeword, for region 1Zn. The complexity of this operation can be considerably reduced by pre-computing and storing a l l T 2 possible d(gm,gn), 1 < TO, n < T. The L B G algorithm terminates if the reduction in the global M Q E given by E q . (5.2) from one iteration to the next iteration becomes negligible. Unfortunately, a codebook Q satisfying the N C C and the C C may be a lo-cal optimum, and therefore, the final codebook obtained by the L B G algorithm may be a local opt imum as well [35]. The most common approach to mitigate the effects of the local opt imum problem is to run the L B G algorithm for many different random ini t ia l codebooks [16]. The final codebook which yields the lowest global M Q E is then used for quantization. Recently, more systematic approaches to overcome the local opt imum problem have been reported in the neural network and pattern recognition literature [16]. Two prominent exam-ples are the enhanced LBG algorithm [36] and the adaptive incremental LBG 9n = argmin{MQE„(g)} , (5.6) geiin (5.7) 50 algorithm [37]. These algorithms t ry to optimize the codeword placement us-ing the assumption that for the globally opt imum vector quantizer the M Q E s of a l l part i t ion regions are approximately equal [38]. However, while this as-sumption is valid for large codebooks, it is questionable for codebooks wi th a small number of codewords [38]. Since for bearnforming small codebooks are both desirable and sufficient to achieve close-to-perfect-CSI performance, we do not further pursue the enhanced and the adaptive incremental L B G algorithms here. Instead, we adapt the so-called global fc-means clustering algorithm [23] to our problem, since it is particularly well suited for small codebooks [16]. 5.4 G l o b a l V e c t o r Q u a n t i z a t i o n ( G V Q ) A l g o -r i t h m The global A;-means clustering algorithm [23] is based on the assumption that the opt imum codebook wi th i codewords can be obtained by ini t ial izing the L B G algorithm wi th the optimal codebook wi th i — 1 codewords. Al though this assumption is difficult to prove theoretically, the excellent performance of the global A;-means clustering algorithm has been shown experimentally in [23]. Therefore, we adapt the global A;-means clustering algorithm to the finite-rate feedback bearnforming problem and refer to the resulting algorithm as G V Q algorithm in the following. The G V Q algorithm can be used to either design the opt imum codebook for a given number of codewords N or to find the codebook wi th the minimum number of codewords for a given target M Q E . The proposed GVQ algorithm is summarized as follows: 1. Pre-define the total number of codewords as N or pre-define the target M Q E as M Q E t a r . 2. Initialize the number of codewords wi th i = 1 . 3. Calculate the opt imum codeword g j l ] by searching the entire training 51 set GT for that gn which minimizes the M Q E in E q . (5.7), where TZi = GT-Set G[l] — and record the corresponding MQE[1] . If N = 1 or MQE[1] < M Q E t a r goto Step 7, otherwise goto Step 4. 4. Increment the iteration number i —> i + 1. 5. Execute the L B G algorithm described in Section 5.3 for al l T — (i — 1) in i t ia l codebooks given by {g\[i — 1], fl^-l]j • • • > —1]> 9n } i where 9n ^ £ T , <7„ & — 1]- Reta in the final codebook delivered by the L B G algorithm wi th min imum M Q E and record it as Q[i]. Record the corresponding M Q E [ i ] . 6. If i < N or if M Q E [ i ] > M Q E t a r goto Step 4, otherwise goto Step 7. 7. G[i] is the desired codebook. It is interesting to note that in order to find the opt imum codebook of size N, the G V Q algorithm computes al l intermediate codebooks of size 1, 2, . . . , N — 1. This property is useful when comparing the performance of codebooks of different size as they can be obtained by executing the G V Q algorithm only once. Note that the proposed G V Q algorithm is completely deterministic, which is a major advantage over related algorithms such as the enhanced and the adaptive incremental L B G algorithms [16]. O n the other hand, for applica-tions wi th large codebooks (e.g. N > 500) such as image compression the main drawback of the underlying fc-means clustering algorithm is its high complex-ity. In particular, the fc-means clustering algorithm (and therefore also the proposed G V Q algorithm) requires 0(TN) executions of the L B G algorithm. However, for finite-rate feedback bearnforming problems complexity is not a major concern as typically only a few thousand training B F F vectors are re-quired to capture the statistical behavior of the channel and codebooks wi th N < 200 are usually sufficient to achieve close-to-perfect-CSI performance. Furthermore, it is important to note that codebooks for finite-rate feedback 52 beamforming are designed off-line. Therefore, the proposed G V Q algorithm is an attractive and feasible solution. 53 Chapter 6 Simulation and Numerical Results In this chapter, we present simulation and numerical results for D F E and L E wi th transmit beamforming wi th perfect and quantized C S I , respectively. For all results shown we assume NR = 1 receive antenna and NT = 3 equally mutu-ally correlated transmit antennas wi th correlation coefficient p = 0.5. As rele-vant practical examples we consider the severely frequency-selective equalizer test (EQ) and the moderately frequency-selective typical urban ( T U ) channel profiles of the G S M / E D G E system [1], As the system model introduced in Chapter 2, for modulation we consider G M S K and 8 - P S K used in G S M and E D G E , respectively. In section 6.1, we wi l l show that significant performance gains of beamform-ing wi th perfect CSI for both D F E and L E over single-antenna transmission can be achieved. We also find out that very short F I R B F F s closely approach the performance upper bound set by the I IR B F F s . In section 6.2, we only consider M M S E - D F E wi th finite-rate feedback beamforming due to the fact that M M S E - D F E gives better performance and G V Q , in Chapter 5, can be easily adopted wi th other equalizations. 54; 6.1 B e a m f o r m i n g w i t h P e r f e c t C S I 6.1.1 Decision-Feedback Equalization In this subsection, we compare F I R and I IR beamforming assuming perfect C S I at the transmitter. However, first we illustrate the convergence behavior of the proposed G A and M P M for calculation of the F I R B F F vector g for D F E . We assume Lg = 1 and for both algorithms the B F F vector is initialized wi th g 0 = ^ [ 1 1 1 ] T as discussed in Section 4.2.2. Figure 6.1 shows the S N R of M M S E - D F E [calculated from E q . (4.3)] vs. iteration number i for one random realization of the E Q channel and 10 l o g 1 0 ( £ , s / A r 0 ) = 10 d B , where EA is the average received energy per symbol. Whi le both algorithms converge to 12.51 1 1 1 1 1 1 I I I | '0 10 20 30 40 50 60 70 80 90 100 i • Figure 6.1: S N R of M M S E - D F E vs. iteration i of the proposed G A and M P M for one realization of the E Q channel wi th L = 7, = 3, NR = 1, equal antenna correlation p = 0.5, Lg = 1, and 10 logW(ES/N0) = 10 d B . 55 the same S N R value, the convergence speed of the M P M is much higher than that of the G A . Therefore, for al l F I R bearnforming results shown in the following for D F E , the B F F s were calculated for each channel realization using the M P M wi th e = I O - 4 . The B F F vectors were initialized using the procedure proposed in Section 4.2.2. In Figures 6.2 and 6.3 we show for, respectively, the E Q and the T U chan-nel the average S N R (SNR) of Z F - D F E wi th I IR B F F s and M M S E - D F E wi th F I R and I IR B F F s . Thereby, S N R was obtained by averaging the respective 0 2 4 6 8 10 12 14 16 18 20 101og 1 0(£./yV 0)[dB] • Figure 6.2: Average S N R of D F E for bearnforming ( B F ) wi th perfect CSI and different B F F s . E Q channel wi th L = 7, NT = 3, = 1, and equal antenna correlation p = 0.5. Results for single-antenna transmission are also included. S N R s in Eqs. (3.15), (3.21), and (4.3) over 500 independent realizations of the E Q and T U channels. For comparison we also show the average S N R of Z F -D F E and M M S E - D F E for single-antenna transmission (NT = 1, NR = .1) in 56 ;l • ' 1 i i • i i : i : i : i : i : I 0 2 4 6 8 10 12 14 16 18 20 101og 1 0 (£ ? / JV 0 ) [dB] • Figure 6.3: Average S N R of D F E for beamforming ( B F ) wi th perfect CSI and different B F F s . T U channel wi th L = 5, NT = 3, NR — I, and equal antenna correlation p = 0.5. Results for single-antenna transmission are also included. Figures 6.2 and 6.3. For both channel profiles transmit beamforming wi th I IR filters leads to performance gains of 5 d B or more compared to single-antenna transmission. However, while for the moderately frequency-selective T U chan-nel an F I R B F F length of LG = 1 achieves practically the same performance as I IR beamforming, especially at high ES/N0, for the severely frequency-selective E Q channel increasing the F I R B F F length beyond LG — 1 is highly beneficial. The fact that the F I R B F F s wi th large enough LG approach the perfor-mance of I IR beamforming in Figures 6.2 and 6.3 also confirms the effectiveness of the M P M for F I R B F F computation. 57 6.1.2 Linear Equalization Figures 6.4 shows the average S N R (SNR) vs. Es/N0 of M M S E - L E wi th F I R and I IR beamforming, respectively, where Es denotes the average received en-ergy per symbol. S N R was obtained by averaging the respective S N R s over 500 independent realizations of the T U channel. For this purpose, in case of F I R beamforming, the S N R given in E q . (4.11) was used and the corresponding B F F s were calculated wi th the G A introduced in Chapter 4. For I IR beam-forming the results given by Eqs. (3.36) and (3.41) in Chapter 3 are used. A s can be observed, I IR beamforming achieves a performance gain of more than 6.5 d B for Z F - L E and 4 d B for M M S E - L E compared to single-antenna transmission. 1 1 ! : ! : ! : ! : T T : i : i : i - i : i : i . i : 1 . 1 : 1 0 2 4 6 8 10 12 14 16 18 20 lOlogm^/iVoMdB] Figure 6.4: Average S N R of L E for beamforming ( B F ) wi th F I R and I IR filters. Transmission over T U channel wi th L = 5, NT = 3, NR = 1, and equal antenna correlation p = 0.5. The result for single-antenna transmission is also included. 58 A s expected, bearnforming wi th I IR B F F s constitutes a natural perfor-mance upper bound for bearnforming wi th F I R filters. However, interestingly, for the T U channel an F I R filter length of Lg = 3 is sufficient to closely approach the performance of I IR bearnforming. This also confirms the effec-tiveness of the proposed G A . We note that for high ES/NQ even an F I R B F F length of Lg — 1 achieves a performance gain of more than 4.5 d B compared to single-antenna transmission (NT — 1, NR = 1). Addi t iona l simulations for other G S M / E D G E channel profiles have shown that in general F I R B F F lengths of Lg < 6 are sufficient to closely approach the performance of I IR bearnforming. Thereby, the F I R filter length required to approach the performance of I IR bearnforming seems to be shorter if the channel is less frequency selective. Comparing Figures 6.3 and 6.4, they both achieve performances sufficiently close to I IR bearnforming wi th short F I R B F F s . A s expected, the average S N R achieved by I IR B F F s wi th D F E is about 1.5dB higher than that achieved by I IR B F F s wi th L E . However, about more than l d B performance gain over single antenna transmission can be achieved at high Eb/N0 by I IR bearnforming wi th L E than that wi th D F E . 6 . 2 F i n i t e - R a t e F e e d b a c k B e a r n f o r m i n g In this section, we present simulation and numerical results for the bit error rate ( B E R ) of M M S E - D F E wi th finite-rate feedback bearnforming. For codebook design the proposed G V Q algorithm is applied to training sets of T — 5000 independent M I M O channel realizations generated for the E Q and the T U channel profiles, respectively, assuming an S N R of 101og 1 0(.E&//V 0) = 10 d B , where E\, = Esj log 2 M denotes the received energy per bit. Figures 6.5 and 6.6 show the B E R of M M S E - D F E for finite-rate feedback bearnforming (solid lines) as a function of the number of feedback bits B for, respectively, B P S K transmission over the E Q channel and 8 - P S K transmis-59 x 10 3.5 2.5 1.5 0.5 . Finite-Rate Feedback, L = 1 s . Perfect CSI, L =1 g . Finite-Rate Feedback, L = 2 - # - Perfect CSI, L g = 2 — A — Finite-Rate Feedback, L - A _ Perfect CSI, L = 3 0- - -O o -o-* * — ft A- -*-- o -*-0-3 B Figure 6.5: B E R of M M S E - D F E vs. number of feedback bits B per channel update. B P S K transmission over E Q channel wi th L — 7, NT = 3, NR = 1, equal antenna correlation p = 0.5, and 101og10(-E6/iVo) = 10 d B . The B E R is obtained from E q . (5.2). sion over the T U channel. The B E R is identical to the global M Q E and is obtained by evaluating E q . (5.2) for the 5000 training channels. For compari-son Figures 6.5 and 6.6 also contain the respective B E R s of M M S E - D F E for bearnforming wi th perfect CSI (dashed lines). For B = 0 the codebook has just one entry and no feedback is required. In this case, bearnforming degener-ates to delay diversity. A s can be observed from Figures 6.5 and 6.6 finite-rate feedback bearnforming approaches the performance of the perfect CSI case as B increases. For the severely frequency-selective E Q channel increasing the B F F length from LG = 1 to LG — 2 or LG = 3 results in a performance gain for both quantized and perfect CSI , cf. Figure 6.5. In contrast, as already 60 0.018 0.016 -e— Finite-Rate Feedback, L_ = 1 e - Perfect CSI, Lg = 1 _ * — Finite-Rate Feedback, L. = 2 * - Perfect CSI, L, = 2 - A — Finite-Rate Feedback, L = 3 B Figure 6.6: B E R of M M S E - D F E vs. number of feedback bits B per channel update. 8 - P S K transmission over T U channel wi th L = 5, Nr = 3, NR = 1, equal antenna correlation p = 0.5, and 10logw(Eb/N0) = 10 d B . The B E R is obtained from E q . (5.2). expected from Figure 6.3, for the moderately frequency-selective T U channel Lg — 1 is near opt imum and while small gains are possible wi th Lg = 2, 3 for perfect CSI , these gains cannot be realized wi th quantized CSI and B < 7 feedback bits, cf. Figure 6.6. Figures 6.7 and 6.8 show the simulated B E R s (averaged over 100,000 chan-nel realizations) for B P S K modulation wi th M M S E - D F E and the E Q channel assuming finite-rate feedback beamforming wi th B F F s of lengths Lg — 1 and Lg = 3, respectively. For the simulations we implemented M M S E - D F E wi th F I R feedforward filters of length 4 L e q which caused negligible performance degradation compared to I IR feedforward filters. The D F E feedback filter had 61 optimum length L e q — 1 = L + Lg — 2. Figure 6.7 shows that even finite-rate feedback beamforming with BFFs of length Lg = 1 and a small number of feedback bits can achieve substantial per-formance gains over single-antenna transmission (NT = 1, NR = 1). Further-more, finite-rate feedback beamforming with B = 1 feedback bit outperforms antenna selection which employs the codebook Q = {[1 0 0]T, [0 1 0]T, [0 0 1]T}' and requires B = 2 feedback bits. Finite-rate feedback beamforming with B — 7 bits closely approaches the performance of beamforming with perfect CSI. 10 10 Pi W PQ 10" 10 _ ^ _ N T = 1 , N R = 1 - * — Antenna Select ion — e — F in i te -Rate Feedback (0 bit) — B — F in i te -Ra te Feedback (1 bit) — * — F in i te -Rate Feedback (3 bit) — V — F in i te -Rate Feedback (5 bit) 1 — Fin i te -Rate Feedback (7 bit) Perfect C S I , L =1 101og1 0(£6/JV0)[clB] :"S>: , \ s N . \ \ \ \ X • \ \ - • 10 15 Figure 6.7: Simulated BER of MMSE-DFE for finite-rate feedback beamform-ing with BFFs of length Lg = 1. BPSK transmission over EQ channel with L = 7, NT = 3, NR = 1, and equal antenna correlation p = 0.5. Results for single-antenna transmission (NT = 1, NR = 1), antenna selection, and beamforming with perfect CSI are also included. Figure 6.8 shows that for the degenerate case of B = 0 feedback bits the proposed method with Lg = 3 achieves a slightly better performance than 62 the optimized delay diversity ( O D D ) scheme in [5]. Note that both schemes 1 0 1 o g 1 0 ( £ ; 6 / i V o ) [ d B ] » Figure 6.8: Simulated B E R of M M S E - D F E for finite-rate feedback bearnform-ing wi th B F F s of length Lg = 3. B P S K transmission over E Q channel wi th L = 7, NT = 3, NR = 1, and equal antenna correlation p = 0.5. Results for single-antenna transmission (NT — 1, NR — 1), O D D [5], and bearnforming wi th perfect CSI are also included. employ fixed transmit filters. However, the numerical methods used for filter calculation are completely different leading to small performance differences. Significant performance gains over O D D are possible even wi th few feedback bits. For example, for B E R = 10~ 4 finite-rate feedback bearnforming wi th 1, 3, and 5 feedback bits yields a performance gain of 1.0 d B , 1.8 d B , and 2.4 d B over O D D , respectively. We also observe from Figure 6.8 that finite-rate feedback bearnforming wi th Lg = 3 and B — 7 feedback bits outperforms bearnforming wi th Lg = 1 and perfect CSI . We note that the B E R s at 101og 1 0 ( .Ei ) / iVn) = 10 d B in Figures 6.7 and 6.8 are somewhat higher than the corresponding B E R s in Figure 6.5. This 63 can be attributed to the fact that the B E R in Figure 6.5 has been obtained by evaluating E q . (5.2) which does not take into account the effect of error propagation in the D F E feedback filter, whereas the simulation results shown in Figures 6.7 and 6.8 include this effect, of course. Thereby, the performance for Lg — 3 is slightly more affected by error propagation than that for Lg = 1 since Lg = 3 results in a longer overall channel requiring a longer D F E feedback filter for equalization. This explains why the small performance gain promised by Figure 6.5 when increasing Lg from 1 to 3 cannot be observed in Figures 6.7 and 6.8 for B < 7. Figure 6.9 contains the same B E R curves as Figure 6.7. However, now 2 4 6 8 10 12 14 16 18 20 101og10(£?6/iVo)[dB] • Figure 6.9: Simulated B E R of M M S E - D F E for finite-rate feedback bearnform-ing wi th B F F s of length Lg = 1. 8 - P S K transmission over T U channel wi th L = 5, NT = 3, NR = 1, and equal antenna correlation p = 0.5. Results for single-antenna transmission (NT = 1, NR = 1), antenna selection, and bearnforming wi th perfect CSI are also included. 8 - P S K transmission over the T U channel is considered instead of B P S K trans-64 mission over the E Q channel. Figure 6.9 shows that also for 8 - P S K and the T U channel the performance loss incurred by quantized CSI becomes negligi-ble for B = 7 feedback bits. Furthermore, for high Eb/N0 finite-rate feedback beamforming wi th a sufficiently large number of feedback bits can achieve a performance gain of up to 2.2 d B compared to antenna selection. 65 Chapter 7 Conclusions and Future Work This chapter concludes the thesis wi th some general comments on the beam-forming system wi th feedback capability proposed in this work, followed by a discussion on possible future work for further investigation of general beam-forming systems wi th finite-rate feedback channel. 7.1 C o n c l u s i o n s In this work, we have considered beamforming wi th perfect and quantized CSI for single-carrier transmission over frequency-selective fading channels wi th D F E and L E at the receiver. For the case of perfect CSI we have provided a simple approach for derivation of closed-form expressions for the opt imum IIR B F F s and we have developed two efficient numerical methods for calculation of the opt imum F I R B F F s for D F E and one for L E . For beamforming wi th finite-rate feedback channel we have proposed a G V Q algorithm for codebook design. The G V Q algorithm performs a deterministic global search and is therefore independent of the starting conditions. This algorithm is applicable for any number of transmit and receive antennas, arbitrary antenna correlation, and arbitrary fading statistics. Simulation results for typical G S M / E D G E channels have shown that short F I R B F F s can approach the performance of I IR B F F s . Furthermore, for finite-rate feedback beamforming wi th B F F s of 66 length Lg = 1 few feedback bits are sufficient to approach the performance of beamforming wi th perfect CSI . In severely frequency-selective channels longer B F F s can further improve performance if a sufficient number of feedback bits can be afforded. Please refer to [39, 40, 41, 42, 43] for a summary of this work. 7.2 R e c o m m e n d a t i o n s for F u t u r e W o r k We believe that the research work we initiated here on finite rate feedback beamforming for frequency-selective channels only scratch the t ip of the ice-berg and many important questions remain to be answered. We list some recommendations for future work as follows: • In the system model, we assume that the feedback channel is error-free and has zero delay. However, in practice, a zero delay feedback channel cannot be achieved, and the channel realization is time variant wi th in each channel usage. In this regard, researchers consider part ia l feed-back channel wi th realistic short delay, using statistical models to model channel variations wi th in one burst transmission [44]. It is interesting to perform investigation on imperfect feedback channel in the future work. • Furthermore, since we only discuss transmit beamforming for uncoded systems in this work, it w i l l be interesting to combined space-time coding and beamforming in the transmitter to cope wi th imperfect feedback channels and frequency-selective fading environments. However, current research in this field considers only the frequency non-selective case [45, 46]. • Also , in the B E R simulations, the assumption that the receiver has per-fect CSI is made. In practice, a least sum of squared errors (LSSE) channel estimation algorithm can be used to estimate the CSI from a known training sequences [47]. However, the impact of the channel es-t imat ion errors on the B E R performance wi th feedback channel is un-67 known. Therefore, channel estimation errors can be taken into account in the future work; Last but not least, O F D M has received considerable interests in recent years due to its easy implementation wi th current powerful digital sig-nal processor (DSP) . Due to its high data rate and frequency-selective resistance ability, O F D M is a promising technology for the fourth gener-ation (4G) wireless communications, and has been adopted in the I E E E 802.11x Wireless Loca l Area Network ( W L A N ) standard and the I E E E 802.16e Worldwide Interoperability for Microwave Access ( W i M a x ) stan-dard. It is also interesting to note that W i M a x and I E E E 8 0 2 . l l n sup-ports partial feedback for bearnforming purposes. Therefore, it w i l l be interesting to extend the present work, which considers only single carrier transmission, to O F D M for single user and multiuser cases. 68 Bibliography [1] GSM Recommendation 05.05: "Propagation Conditions", Vers. 5.3.0, Re-lease 1996. [2] W . 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