Transmit Beamforming for Frequency Selective Channels with Suboptimum Equalization

Transcription

1 Transmit Beamforming for Frequency Selective Channels with Suboptimum Equalization Yang Wen Liang Supervisor: Dr. Robert Schober Department of Electrical and Computer Engineering The University of British Columbia August 22, 2006 Yang Wen Liang () The University of British Columbia August 22, / 53

2 Outline 1 Introduction 2 System Model 3 Beamforming with Perfect CSI and IIR Filters 4 Beamforming with Perfect CSI and FIR Filters 5 FIR Beamforming with Quantized CSI 6 Simulation and Numerical Results 7 Conclusions and Future Work Yang Wen Liang () The University of British Columbia August 22, / 53

3 Outline 1 Introduction 2 System Model 3 Beamforming with Perfect CSI and IIR Filters 4 Beamforming with Perfect CSI and FIR Filters 5 FIR Beamforming with Quantized CSI 6 Simulation and Numerical Results 7 Conclusions and Future Work Yang Wen Liang () The University of British Columbia August 22, / 53

4 Background In recent years, the application of multiple antennas in wireless communication systems has received considerable interest from academia and industry, e.g. GSM, WLAN etc In particular, transmit beamforming has attracted considerable attention because of its simplicity to exploit the benefits of multiple transmit antennas Since perfect channel state information (CSI) may not be available at the transmitter in practical systems, recent research in this field has focused on the impact of noisy and/or quantized CSI. Some techniques have been adopted by WLAN: IEEE n (expected release date mid-2007) WMAN: WiMAX e WWAN: W-CDMA Yang Wen Liang () The University of British Columbia August 22, / 53

5 Related Work I Space time codes are generally referred to as open loop because they operate without the benefit of CSI at the transmitter, e.g. space time block codes (STBC) [Tarokh et al, 1999], space time trellis code (STTC) [Tarokh et al, 1998] etc Compared with traditional space time codes, beamforming ( closed loop ) systems provide the same diversity order as well as significant more array gain (power gain) at the expense of requiring perfect/quantized CSI at the transmitter From a practical point of view, the assumption of quantized CSI is particularly interesting, but beamforming with perfect CSI leads to the theoretical performance limit of the system Yang Wen Liang () The University of British Columbia August 22, / 53

6 Related Work II With finite rate feedback channel from the receiver to the transmitter, channel matrix quantization vs. beamforming vector quantization From independent works of Love and Mukkavilli, the construction of optimum beamforming code book for the i.i.d. Rayleigh flat-fading channel environment was solved and related to the problem of Grassmannian line packing [Mukkavilli et al, 2003] [Love et al, 2003] Furthermore, Linde Buzo Gray (LBG) type vector quantization algorithms have been proposed for codebook construction for correlated flat-fading cases [Xia and Giannakis, 2006] [Roh and Rao, 2006] For systems employing orthogonal frequency division multiplexing (OFDM) to cope with the frequency selectivity of the channel, effective beamforming techniques with finite rate feedback channel were proposed in [Xia et al, 2004] [Choi and Heath, 2005] Yang Wen Liang () The University of British Columbia August 22, / 53

7 Motivation Limitations Most of the existing literature assumed frequency nonselective fading. This assumption may not be realistic, especially for high rate transmission. Problem How to design beamforming filters at the transmitter for a specific equalizer at the receiver over a frequency selective fading environment with perfect/quantized CSI? Yang Wen Liang () The University of British Columbia August 22, / 53

8 Motivation Limitations Most of the existing literature assumed frequency nonselective fading. This assumption may not be realistic, especially for high rate transmission. Problem How to design beamforming filters at the transmitter for a specific equalizer at the receiver over a frequency selective fading environment with perfect/quantized CSI? Yang Wen Liang () The University of British Columbia August 22, / 53

9 Contributions For perfect CSI we derive a closed form solution for the optimum IIR beamforming filters (BFFs) maximizing the signal to noise ratio (SNR) for receivers with LE and DFE We provide two efficient numerical methods for calculation of the optimum FIR BFFs since closed form solution does not seem to exist We propose a practical finite rate feedback beamforming scheme for frequency selective channel Our simulation results show that short FIR BFFs can closely approach the performance of the optimum IIR BFFs For typical GSM/EDGE channel profiles beamforming with finite rate feedback enables large performance gains compared to single antenna transmission, transmit antenna selection, and optimized delay diversity Yang Wen Liang () The University of British Columbia August 22, / 53

10 Outline 1 Introduction 2 System Model 3 Beamforming with Perfect CSI and IIR Filters 4 Beamforming with Perfect CSI and FIR Filters 5 FIR Beamforming with Quantized CSI 6 Simulation and Numerical Results 7 Conclusions and Future Work Yang Wen Liang () The University of British Columbia August 22, / 53

11 System Model I g 1 [k] s1[k]... n 1 [k] n 2 [k] b[k] g 2 [k]. s2[k] h ntnr [k].. Equalizer ˆb[k] n NR [k] g NT [k] snt [k] Feedback Channel N T transmit antennas and N R receive antennas The i.i.d. symbols b[k] are taken from a scalar symbol alphabet A such as PSK or QAM, and have variance σ 2 b E{ b[k] 2 } = 1 Yang Wen Liang () The University of British Columbia August 22, / 53

12 System Model II g nt [k], < k <, are the transmit BFF impulse response coefficients of antenna n t. For comparison purposes, the energy of transmit BFFs is normalized to 1, i.e., È N T n t=1 È k= gnt [k] 2 = 1 Transmit signal at antenna n t s nt [k] = g nt [k] b[k] h nrn t [l], 0 l L 1, is the channel impulse response (CIR) from transmit antenna n t to receive antenna n r Define H C[k] as a discrete time correlated MIMO frequency-selective Rayleigh fading channel with L matrix taps, H[l], l = 0,..., L 1 H C[k] = L 1 l=0 H[l]δ[k l], H[l] = ¾ h 11[l] h 12[l]... h 1NT [l] h 21[l] h 22[l]... h 2NT [l] h NR 1[l] h NR 2[l]... h NR N T [l] Yang Wen Liang () The University of British Columbia August 22, / 53

13 System Model III Assumptions: H C[k] contains the combined effects of transmit pulse shaping, physical channel (EQ, TU and, HT from GSM/EDGE), and receiver input filters Block Rayleigh fading channel Antennas are spatially correlated Feedback channel is error free, zero delay infinite rate: the receiver sends the optimum BFFs (or equivalently the CIR) to the transmitter finite rate: the receiver conveys the index of the pre designed codeword (BFF vector) to the transmitter Yang Wen Liang () The University of British Columbia August 22, / 53

14 System Model III Assumptions: H C[k] contains the combined effects of transmit pulse shaping, physical channel (EQ, TU and, HT from GSM/EDGE), and receiver input filters Block Rayleigh fading channel Antennas are spatially correlated Feedback channel is error free, zero delay infinite rate: the receiver sends the optimum BFFs (or equivalently the CIR) to the transmitter finite rate: the receiver conveys the index of the pre designed codeword (BFF vector) to the transmitter Yang Wen Liang () The University of British Columbia August 22, / 53

15 System Model IV Received signal at antenna n r with equivalent CIR Observations r nr [k] = N T n t=1 h nrn t [k] s nt [k] + n nr [k] =h eq n r [k] b[k] + n nr [k] h eq n r [k] = N T n t=1 h nrn t [k] g nt [k] overall channel is a single input multiple output (SIMO) channel with CIRs h eq n r [k] with length of equivalent CIR, L eq = L + N 1 conventional equalizers (MLSE, DFE, LE) and channel estimation/tracking techniques can be used Yang Wen Liang () The University of British Columbia August 22, / 53

16 Outline 1 Introduction 2 System Model 3 Beamforming with Perfect CSI and IIR Filters 4 Beamforming with Perfect CSI and FIR Filters 5 FIR Beamforming with Quantized CSI 6 Simulation and Numerical Results 7 Conclusions and Future Work Yang Wen Liang () The University of British Columbia August 22, / 53

17 Optimization Problem I For convenience, the frequency responses of the IIR BFFs G nt (f) F{g nt [k]} are collected in vector G(f) G 1(f) G 2(f)... G NT (f) T. Recall the transmit power constraint 1/2 1/2 G H (f)g(f) df = 1 The unbiased SNR of DFE with optimum IIR feedforward filtering is [Baddour and McLane, 1999] SNR(G(f)) = σ2 b σ 2 n exp 1/2 1/2 ln ξ + N R n r=1 H eq n r (f) 2 df χ ξ = 0 and χ = 0 is valid for ZF DFE ξ = σn/σ 2 b 2 and χ = 1 is valid for MMSE DFE Define the equivalent channel frequency response Hn eq(f) r F{h eq n r [k]} Yang Wen Liang () The University of British Columbia August 22, / 53

18 Optimization Problem II Problem How to choose G(f) so that the SNR is maximized for a given channel realization subject to the transmit power constraint? Solution The classical Calculus of Variations method! Yang Wen Liang () The University of British Columbia August 22, / 53

19 Optimization Problem II Problem How to choose G(f) so that the SNR is maximized for a given channel realization subject to the transmit power constraint? Solution The classical Calculus of Variations method! Yang Wen Liang () The University of British Columbia August 22, / 53

20 Optimization Problem III Calculus of Variations Method 1 Denote the optimum BFF vector as Ḡ(f) [Ḡ 1 (f) Ḡ 2 (f),... Ḡ N T (f)] T 2 Model the BFF of antenna n t as G nt (f) = Ḡnt (f) + ε nt B nt (f), where B nt (f) and ε nt denote an arbitrary function of f and a complex valued variable, respectively 3 The optimization problem can now be formulated in terms of its Lagrangian L(ε) = SNR(G(f)) µ ε [ε 1 ε 2... ε NT ] T µ is the Lagrange multiplier 1/2 1/2 G H (f)g(f) df 4 The optimum BFF vector Ḡ(f) has to fulfill the condition L(ε) ε=0 = 0 ε NT NT for arbitrary B nt (f), 1 n t N T Yang Wen Liang () The University of British Columbia August 22, / 53

21 Optimization Problem IV After some calculations, it can be shown that vector Ḡ(f) has to fulfill S(f) Ḡ(f) = µ ḠH (f)s(f)ḡ(f) + ξ Ḡ(f) (1) µ is a constant S(f) is an N T N T matrix given by = S(f) ¾ N RÈ N RÈ N RÈ N R n r=1 H nr (f)h H n r (f) H1nr (f)h 1nr (f) nr=1 N RÈ nr=1 N RÈ H 1nr (f)h 2nr (f)... N RÈ H1nr (f)h N T nr (f) nr=1 H2nr (f)h 1nr (f) H2nr (f)h N nr=1 nr=1 nr=1 T nr (f) HN nr=1 T nr (f)h 1nr (f) N RÈ nr=1 H 2nr (f)h 2nr (f)... N RÈ H N T nr (f)h 2nr (f)... N RÈ HN nr=1 T nr (f)h N T nr (f) with H nr (f) [H 1nr (f) H 2nr (f)... H NT n r (f)] T Yang Wen Liang () The University of British Columbia August 22, / 53

22 Optimization Problem V Eq. (1) is a nonlinear eigenvalue problem and Ḡ(f) can be expressed as Ḡ(f) = X(f) E(f) e jϕ(f) X(f) is a scalar factor E(f) [E 1(f) E 2(f)... E NT (f)] T is that unit norm eigenvector of S(f) which corresponds to its largest eigenvalue λ max(f) ϕ(f) is the phase which can be chosen arbitrarily Therefore, in general the optimum IIR BFFs can be viewed as concatenation of two filters: a filter X(f) which is common to all transmit antennas a filter E nt (f) which is transmit antenna dependent Yang Wen Liang () The University of British Columbia August 22, / 53

23 IIR beamforming for ZF DFE From Eq. (1) and the power constraint, the optimum IIR BFFs for ZF-DFE are given by Ḡ(f) = E(f) e jϕ(f) ϕ(f) is the phase which can be chosen arbitrarily The corresponding maximum SNR for ZF-DFE is SNR(Ḡ(f)) = σ2 b σ 2 n exp 1/2 1/2 ln (λ max(f)) df Yang Wen Liang () The University of British Columbia August 22, / 53

24 IIR beamforming for MMSE DFE From Eq. (1) and the power constraint, we obtain 1/2 1/2 ˆµ + ξ df = 1 λ max(f) Finding the optimum ˆµ, denoted by ˆµ opt, is a typical Water Filling problem The optimum IIR BFFs for MMSE DFE are given by + ξ Ḡ(f) = ˆµ opt E(f) e jϕ(f) λ max(f) ϕ(f) is again the phase which can be chosen arbitrarily The corresponding maximum SNR for MMSE DFE is SNR(Ḡ(f)) = σ2 b σ 2 n exp 1/2 1/2 ln [λ max(f)ˆµ opt ξ] + + ξ df 1 Yang Wen Liang () The University of British Columbia August 22, / 53

25 Observations Regarding the BFFs and the maximum SNRs for DFE, we make the following interesting observations: a) For σn 2 0 (i.e., ξ 0) the optimum IIR BFFs for MMSE-DFE approach those for ZF-DFE. b) In case of ZF DFE, the optimum BFF frequency response vector Ḡ(f) at frequency f = f 0 is just the dominant eigenvector of matrix S(f) at frequency f = f 0. Since S(f 0 ) only depends on the channel frequency responses H ntnr (f) at frequency f = f 0, Ḡ(f 0) is independent of the H ntnr (f), f f 0. c) In case of MMSE-DFE, the optimum BFF frequency response vector Ḡ(f) at frequency f = f 0 depends on the channel frequency responses H ntnr (f) at all frequencies 1/2 f < 1/2. In fact, X(f) may be interpreted as a power allocation filter which allocates more power to frequencies with large eigenvalues λ max(f) in MMSE-DFE cases. d) If the underlying channel is frequency-nonselective, S(f) = S for all f. In this case, it is easy to see that the optimum BFFs have only one non-zero tap and are identical to the well-know beamforming weights for frequency-nonselective channels. In this case, the DFE structure collapses to a simple threshold detector, of course. Yang Wen Liang () The University of British Columbia August 22, / 53

26 IIR Beamforming Limitations Two reasons why implementing IIR BFFs in practice is not realistic: It is well know that designing stable IIR filters for any channel realization is difficult The transmitter requires to know perfect instantaneous CSI in order to perform IIR beamforming Yang Wen Liang () The University of British Columbia August 22, / 53

27 Outline 1 Introduction 2 System Model 3 Beamforming with Perfect CSI and IIR Filters 4 Beamforming with Perfect CSI and FIR Filters 5 FIR Beamforming with Quantized CSI 6 Simulation and Numerical Results 7 Conclusions and Future Work Yang Wen Liang () The University of British Columbia August 22, / 53

28 Optimum FIR BFFs (DFE) Let the FIR BFFs have length L g, the resulting equivalent overall CIR h eq n r [k] has length L eq = L + L g 1. The frequency response of the equivalent channel can now be expressed as Hn eq r (f) = d H (f)h nr g where d(f) [1 e j2πf... e j2πf(leq 1) ] T, H nr [H 1nr H 2nr... H NT n r ], and H ntnr is an L eq L g column circulant matrix The SNR of ZF- and MMSE DFE with FIR BFFs is obtained as SNR(g) = σ2 b σ 2 n exp 1/2 1/2 ln(g H B(f)g + ξ) df χ The optimum BFF vector ḡ shall maximize SNR(g) subject to the power constraint g H g = 1. The Lagrangian of the optimization problem leads to the nonlinear eigenvalue problem ¾ 1/2 1/2 B(f) + ξi NT L g df ḡ H B(f) + ξi NT L g ḡ ḡ = ḡ which is the same as maximization of products of Rayleigh quotients (an well-known unsolved mathematical problem) Yang Wen Liang () The University of British Columbia August 22, / 53

29 Calculation of the Optimum FIR BFFs (DFE) Gradient Algorithm (GA) 1 Let i = 0 and initialize the BFF vector with some g 0 fulfilling g H 0 g 0 = 1. 2 Update the BFF vector g i+1 = g i + δ i ¾ 1/2 1/2 g H i B(f) + ξi NT L g df B(f) + ξi NT L g gi g i where δ i is a suitable adaptation step size (we found empirically that δ i = 0.01/λ i is a good choice). 3 Normalize the BFF vector g i+1 = Õ g i+1 g H i+1 g i+1 4 If 1 g H i+1 g i < ǫ, goto Step 5, otherwise increment i i + 1 and goto Step 2. 5 g i+1 is the desired BFF vector ḡ. Yang Wen Liang () The University of British Columbia August 22, / 53

30 Calculation of the Optimum FIR BFFs (DFE) Modified Power Method (MPM) 1 Let i = 0 and initialize the BFF vector with some g 0 fulfilling g H 0 g 0 = 1. 2 Update the BFF vector g i+1 = ¾ 1/2 1/2 g H i B(f) + ξi NT L g df B(f) + ξi NT L g gi g i 3 Normalize the BFF vector g i+1 = Õ g i+1 g H i+1 g i+1 4 If 1 g H i+1 g i < ǫ, goto Step 5, otherwise increment i i + 1 and goto Step 2. 5 g i+1 is the desired BFF vector ḡ. Yang Wen Liang () The University of British Columbia August 22, / 53

31 Outline 1 Introduction 2 System Model 3 Beamforming with Perfect CSI and IIR Filters 4 Beamforming with Perfect CSI and FIR Filters 5 FIR Beamforming with Quantized CSI 6 Simulation and Numerical Results 7 Conclusions and Future Work Yang Wen Liang () The University of British Columbia August 22, / 53

32 Vector Quantization Preliminaries I Clustering is an important and fundamental instrument in engineering and other scientific disciplines to solve problems, e.g. machine learning The simplest form of clustering is the partitioning clustering approach known as Vector Quantization, which partitions a given data set into disjoint subsets so that the objective function (quantization error) is minimized For convenience we define the channel vector h [h 11[0]... h 11[L 1] h 12[0]... h NT N R [L 1]] T of length N T N RL. We assume that a training set H {h 1, h 2,..., h T } of T channel vectors h n is available Using the GA or MPM, we calculate the optimum BFF vector ḡ n for each channel h n, 1 n T. The resulting BFF vector training set is denoted by G T {ḡ 1, ḡ 2,..., ḡ T } Yang Wen Liang () The University of British Columbia August 22, / 53

33 Vector Quantization Preliminaries II A vector quantizer Q is a mapping of the BFF vector training set G T with T entries to the BFF vector codebook G {ĝ 1, ĝ 2,..., ĝ N } with N entries, where N T. Therefore, the vector quantizer can be represented as a function Q : G T G The elements ĝ n of the codebook G are also referred to as codewords Once Q is determined, we can define partition regions R n constituted by subsets of the original training set R n {ḡ G T Q(ḡ) = ĝ n }, 1 n N In general, a vector quantizer is said to be optimum if it minimizes the mean quantization error (MQE) for a given codebook size N. The MQE is defined as MQE 1 T T n=1 d(q(ḡ n ), ḡ n ) Yang Wen Liang () The University of British Columbia August 22, / 53

34 Vector Quantization Preliminaries III d(ĝ m, ḡ n ) is the so called distortion measure and denotes the distortion caused by quantizing ḡ n G T to ĝ m G To minimizes the average BER, the distortion measure d(ĝ m, ḡ n ) is the BER P e(ĝ m, h n) caused by ĝ m G for channel h n H with optimum BFF vector ḡ n G T, i.e., d(ĝ m, ḡ n ) P e(ĝ m, h n) If we assume Gray mapping and Gaussian random residual error at the decision device, the BER of DFE can be approximated as P e(ĝ m, h n) C Q Õd 2min SNR(ĝ m, hn)/2 Ê where Q(x) 1 2π x e t2 /2 dt, and both the unimportant constant C and the minimum Euclidean distance d min depend on signal constellation A Yang Wen Liang () The University of British Columbia August 22, / 53

35 LBG Algorithm The LBG algorithm can be used to improve a given initial codebook. This algorithm exploits two necessary conditions that an optimal vector quantizer satisfies (Lloyed Max conditions): Nearest Neighborhood Condition (NNC) For a given codebook G the partition regions R n, 1 n N, satisfy R n = {ḡ G T d(ĝ n, ḡ) d(ĝ m, ḡ), m n} i.e., R n is the Voronoi region of codeword ĝ n, 1 n N Centroid Condition (CC) For given partitions R n, 1 n N, the optimum codewords satisfy ĝ n = argmin ḡ R n {MQE n (ḡ)} where the MQE for region R n and candidate codeword ḡ is defined as MQE n (ḡ) 1 d(ḡ, ḡ ) R n ḡ R n where R n denotes the number of training BFF vectors in region R n Yang Wen Liang () The University of British Columbia August 22, / 53

36 LBG Algorithm The LBG algorithm can be used to improve a given initial codebook. This algorithm exploits two necessary conditions that an optimal vector quantizer satisfies (Lloyed Max conditions): Nearest Neighborhood Condition (NNC) For a given codebook G the partition regions R n, 1 n N, satisfy R n = {ḡ G T d(ĝ n, ḡ) d(ĝ m, ḡ), m n} i.e., R n is the Voronoi region of codeword ĝ n, 1 n N Centroid Condition (CC) For given partitions R n, 1 n N, the optimum codewords satisfy ĝ n = argmin ḡ R n {MQE n (ḡ)} where the MQE for region R n and candidate codeword ḡ is defined as MQE n (ḡ) 1 d(ḡ, ḡ ) R n ḡ R n where R n denotes the number of training BFF vectors in region R n Yang Wen Liang () The University of British Columbia August 22, / 53

37 LBG Algorithm The LBG algorithm can be used to improve a given initial codebook. This algorithm exploits two necessary conditions that an optimal vector quantizer satisfies (Lloyed Max conditions): Nearest Neighborhood Condition (NNC) For a given codebook G the partition regions R n, 1 n N, satisfy R n = {ḡ G T d(ĝ n, ḡ) d(ĝ m, ḡ), m n} i.e., R n is the Voronoi region of codeword ĝ n, 1 n N Centroid Condition (CC) For given partitions R n, 1 n N, the optimum codewords satisfy ĝ n = argmin ḡ R n {MQE n (ḡ)} where the MQE for region R n and candidate codeword ḡ is defined as MQE n (ḡ) 1 d(ḡ, ḡ ) R n ḡ R n where R n denotes the number of training BFF vectors in region R n Yang Wen Liang () The University of British Columbia August 22, / 53

38 LBG Algorithm Limitations Unfortunately, a codebook G satisfying the NCC and the CC may be a local optimum, and therefore, the final codebook obtained by the LBG algorithm may be a local optimum as well [Gersho and Gray, 1991] Solution 0 Brute force searching?! Yang Wen Liang () The University of British Columbia August 22, / 53

39 LBG Algorithm Limitations Unfortunately, a codebook G satisfying the NCC and the CC may be a local optimum, and therefore, the final codebook obtained by the LBG algorithm may be a local optimum as well [Gersho and Gray, 1991] Solution 0 Brute force searching?! Yang Wen Liang () The University of British Columbia August 22, / 53

40 LBG Algorithm Limitations Unfortunately, a codebook G satisfying the NCC and the CC may be a local optimum, and therefore, the final codebook obtained by the LBG algorithm may be a local optimum as well [Gersho and Gray, 1991] Solution 1 Assumption: the MQEs of all partition regions are approximately equal for the globally optimum vector quantizer [Gersho, 1979] The enhanced LBG algorithm [Patanè and Russo, 2001] and the adaptive incremental LBG algorithm [Shen and Hasegawa, 2006] This assumption is valid for large codebooks, say > 500 Yang Wen Liang () The University of British Columbia August 22, / 53

41 LBG Algorithm Limitations Unfortunately, a codebook G satisfying the NCC and the CC may be a local optimum, and therefore, the final codebook obtained by the LBG algorithm may be a local optimum as well [Gersho and Gray, 1991] Solution 2 Assumption: the optimum codebook with i codewords can be obtained by initializing the LBG algorithm with the optimal codebook with i 1 codewords. The Global Vector Quantization (GVQ) Algorithm Although this assumption is difficult to prove theoretically, the excellent performance of the global k means clustering algorithm has been shown experimentally in [Likas et al, 2003]. Due to complexity, it is suitable for small codebook, say < 500 Yang Wen Liang () The University of British Columbia August 22, / 53

42 GVQ Algorithm 1 Pre define the total number of codewords as N or pre define the target MQE as MQE tar. 2 Initialize the number of codewords with i = 1. 3 Calculate the optimum codeword ĝ 1 [1] by searching the entire training set G T for that ḡ n which minimizes the MQE, where R 1 = G T. Set G[1] = ĝ 1 [1] and record the corresponding MQE[1]. If N = 1 or MQE[1] MQE tar goto Step 7, otherwise goto Step 4. 4 Increment the iteration number i i Execute the LBG algorithm described the previous Section for all T (i 1) initial codebooks given by {ĝ 1 [i 1], ĝ 2 [i 1],..., ĝ i 1 [i 1], ḡ n }, where ḡ n G T, ḡ n G[i 1]. Retain the final codebook delivered by the LBG algorithm with minimum MQE and record it as G[i]. Record the corresponding MQE[i]. 6 If i < N or if MQE[i] > MQE tar goto Step 4, otherwise goto Step 7. 7 G[i] is the desired codebook. Yang Wen Liang () The University of British Columbia August 22, / 53

43 GVQ Algorithm The GVQ algorithm can be used to either design the optimum codebook for a given number of codewords N or to find the codebook with the minimum number of codewords for a given target MQE In order to find the optimum codebook of size N, the GVQ algorithm computes all 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 GVQ algorithm only once The proposed GVQ algorithm is completely deterministic the proposed GVQ algorithm requires O(T N) executions of the LBG algorithm. In our experiment shown later, T = 5000 and N < 200 are usually sufficient to achieve close to perfect CSI performance Since codebooks for finite rate feedback beamforming are designed off line, the proposed GVQ algorithm is an attractive and feasible solution Yang Wen Liang () The University of British Columbia August 22, / 53

44 Outline 1 Introduction 2 System Model 3 Beamforming with Perfect CSI and IIR Filters 4 Beamforming with Perfect CSI and FIR Filters 5 FIR Beamforming with Quantized CSI 6 Simulation and Numerical Results 7 Conclusions and Future Work Yang Wen Liang () The University of British Columbia August 22, / 53

45 System Model Specifications N R = 1 receive antenna and N T = 3 equally mutually correlated transmit antennas with correlation coefficient ρ = 0.5 As relevant practical examples, we consider the severely frequency selective equalizer test (EQ) channel profile of the GSM/EDGE system. Maximum channel length L = 7 BPSK transmission perfect CSI, optimum DFE at the receiver Yang Wen Liang () The University of British Columbia August 22, / 53

46 Numerical Results Beamforming with Perfect CSI Yang Wen Liang () The University of British Columbia August 22, / 53

47 20 15 ZF DFE, N T =1, N R =1 MMSE DFE, N T =1, N R =1 ZF DFE, IIR BF MMSE DFE, IIR BF MMSE DFE, FIR BF, L g =1 MMSE DFE, FIR BF, L g =2 MMSE DFE, FIR BF, L g =3 MMSE DFE, FIR BF, L g =4 MMSE DFE, FIR BF, L g =5 10 log 10 (SNR)[dB] 10 5 single antenna transmission log 10 (E s /N 0 )[db] Average SNR of DFE for beamforming (BF) with perfect CSI and different BFFs. Yang Wen Liang () The University of British Columbia August 22, / 53

48 Simulation Results Finite Rate Feedback Beamforming (Beamforming with Quantized CSI) Yang Wen Liang () The University of British Columbia August 22, / 53

49 5 x Finite Rate Feedback, L = 1 g Perfect CSI, L g = 1 Finite Rate Feedback, L = 2 g Perfect CSI, L g = 2 Finite Rate Feedback, L g = 3 Perfect CSI, L g = 3 3 BER Quantized CSI Perfect CSI B BER of MMSE DFE vs. number of feedback bits B per channel update. 10 log 10 (E b /N 0) = 10 db. Yang Wen Liang () The University of British Columbia August 22, / 53

50 single antenna antenna selection 10 3 FIR beamforming with perfect CSI, L g = 1 BER 10 4 N T =1, N R =1 Antenna Selection Finite Rate Feedback (0 bit) Finite Rate Feedback (1 bit) Finite Rate Feedback (3 bit) Finite Rate Feedback (5 bit) Finite Rate Feedback (7 bit) Perfect CSI, L g = log 10 (E b /N 0 )[db] Simulated BER of MMSE DFE for finite rate feedback beamforming with BFFs of length L g = 1. Yang Wen Liang () The University of British Columbia August 22, / 53

51 10 1 single antenna 10 2 BER FIR beamforming with perfect CSI, L g = 3 FIR beamforming with perfect CSI, L g = 1 N T =1, N R =1 ODD, L g =3 Finite Rate Feedback (0 bit) Finite Rate Feedback (1 bit) Finite Rate Feedback (3 bit) Finite Rate Feedback (5 bit) Finite Rate Feedback (7 bit) Perfect CSI, L g =1 Perfect CSI, L g = log 10 (E b /N 0 )[db] Simulated BER of MMSE DFE for finite rate feedback beamforming with BFFs of length L g = 3. *Note: ODD [Yiu et al, 2004]. Yang Wen Liang () The University of British Columbia August 22, / 53 ODD

52 Outline 1 Introduction 2 System Model 3 Beamforming with Perfect CSI and IIR Filters 4 Beamforming with Perfect CSI and FIR Filters 5 FIR Beamforming with Quantized CSI 6 Simulation and Numerical Results 7 Conclusions and Future Work Yang Wen Liang () The University of British Columbia August 22, / 53

53 Conclusions We have considered beamforming with perfect and quantized CSI for single carrier transmission over frequency selective fading channels with DFE and LE at the receiver For the case of perfect CSI Provided a simple approach for derivation of closed form expressions for the optimum IIR BFFs Developed two efficient numerical methods for calculation of the optimum FIR BFFs for DFE For beamforming with finite rate feedback channel, we proposed a GVQ algorithm for codebook design Simulation results for typical GSM/EDGE channels have shown significant performance gain of our scheme Yang Wen Liang () The University of British Columbia August 22, / 53

54 Future Work In our work, we assume that the feedback channel is error free and has zero delay. It is interesting to investigate the impact of imperfect feedback channel. We only discuss transmit beamforming for uncoded system in this work. Combination of space time coding and beamforming to cope with imperfect feedback channels and frequency selective fading environment can be a future research direction. Possible extension of the the present work to OFDM for single user and multiuser cases. Yang Wen Liang () The University of British Columbia August 22, / 53

55 Contributions The results of our work are summarized in the following papers: Y. Liang and R. Schober and W. Gerstacker, Transmit Beamforming for Frequency Selective Channels with Decision Feedback Equalization, Submitted to IEEE Trans. Wireless Commun., May Y. Liang and R. Schober and W. Gerstacker, FIR beamforming for frequency selective channels with linear equalization, Submitted to IEEE Commun. Lett., May Y. Liang and R. Schober and W. Gerstacker, Transmit Beamforming with Finite Rate Feedback for Frequency Selective Channels, Accepted for presentation at the IEEE Global Telecommunications Conference (GLOBECOM), San Francisco, USA, Apr Y. Liang and R. Schober and W. Gerstacker, Transmit Beamforming for Frequency Selective Channels, Accepted for presentation at the IEEE Vehicular Technology Conference, Montreal, Canada, Mar Y. Liang, Transmit Beamforming with Linear Equalization, Accepted for presentation at the First Canadian Summer School on Communications and Information Theory, Banff, Canada, June Yang Wen Liang () The University of British Columbia August 22, / 53

56 Thank you for your time! Questions? Yang Wen Liang () The University of British Columbia August 22, / 53

57 References I V. Tarokh, H. Jafarkhani, and A. R. Calderbank, Space time block codes from orthogonal designs, IEEE Trans. Inform. Theory, vol. 45, pp , July V. Tarokh, N. Seshadri, and A. R. Calderbank, Space time codes for high data rate wireless communication: Performance criterion and code construction, IEEE Trans. Inform. Theory, vol. 44, pp , Mar D. Love and R. Heath and T. Strohmer, Grassmannian beamforming for multiple input multiple output wireless systems, IEEE Trans. Inform. Theory, vol. 49, pp , Oct K. Mukkavilli and A. Sabharwal and E. Erkip and B. Aazhang, Beamforming with finite rate feedback in multiple antenna systems, IEEE Trans. Inform. Theory, vol. 49, pp , Oct J. Roh and B. Rao, Transmit Beamforming in Multiple Antenna Systems with Finite Rate Feedback: A VQ based Approach, IEEE Trans. Inform. Theory, vol. IT-52, pp , Mar P. Xia and G. Giannakis, Design and analysis of transmit beamforming based on limited rate feedback, IEEE Trans. Signal Processing, vol. 54, pp , May J. Choi and R. Heath, Interpolation based transmit beamforming for MIMO OFDM with limited feedback, IEEE Trans. Signal Processing, vol. 53, pp , Nov Yang Wen Liang () The University of British Columbia August 22, / 53

58 References II P. Xia and S. Zhou and G. Giannakis, Adaptive MIMO OFDM based on partial channel state information, IEEE Trans. Signal Processing, vol. 52, pp , Jan K.E. Baddour and P.J. McLane, Analysis of optimum diversity combining and decision feedback equalization in dispersive Rayleigh fading, Proceedings of IEEE International Communications Conference, pp , Vancouver, Jun S. Yiu and R. Schober and W. Gerstacker, Optimization of delay diversity for decision feedback equalization, Proceedings of IEEE Symposium on Personal, Indoor and Mobile Radio Communications, pp , Barcelona, Spain, Sep A. Gersho and R. M. Gray, Vector quantization and signal compression, Kluwer Academic Publishers, Norwell, MA, USA A. Gersho, Asymptotically Optimal Block Quantization, IEEE Trans. Inform. Theory, vol. 25, pp , Jul G. Patanè and M. Russo, The Enhanced LBG Algorithm, Neural Networks, vol 14, pp , Sep F. Shen and O. Hasegawa, An Adaptive Incremental LBG for Vector Quantization, to appear in Neural Networks, Yang Wen Liang () The University of British Columbia August 22, / 53

59 References III A. Likas and N. Vlassis and J. Verbeek, The global K means clustering algorithm, Pattern Recognition, vol. 36, pp , 2003 Yang Wen Liang () The University of British Columbia August 22, / 53