Transmit Beamforming for Frequency Selective Channels

Transmit Beamformin for Frequency Selective Channels Yan-wen Lian, Robert Schober, and Wolfan Gerstacker Dept of Elec and Comp Enineerin, University of British Columbia, Email: {yanl,rschober}@eceubcca Institute for Mobile Communication, University of Erlanen Nürnber, Email: ersta@lntde Abstract Ihis paper, we propose beamformin schemes for frequency selective channels with decision feedback equalization (DFE) at the receiver We consider both finite impulse response (FIR) and infinite impulse response (IIR) beamformin filters (BFFs) In case of IIR beamformin, we are able to derive closed form expressions for the optimum BFFs In addition, we provide an efficient numerical method for recursive calculation of the optimum FIR BFFs Simulation and numerical results for typical GSM/EDGE channels confirm the sinificant performance ains achievable with beamformin compared to sinle antenna transmission and optimized delay diversity I INTRODUCTION Iecent years, the application of multiple antennas in wireless communication systems has received considerable interest from academia and industry In particular, transmit beamformin has received much attention as a simple yet efficient technique to exploit the benefits of multiple transmit antennas, cf e [1] and references therein Beamformin enerally requires channel state information (CSI) at the transmitter Since perfect CSI may not be available at the transmitter in practical systems, recent research ihis field has focused ohe impact of noisy and/or quantized CSI, cf e [2], [3], [4] However, most of the existin literature oransmit beamformin with perfect or imperfect CSI has assumed frequency nonselective fadin In practice, this assumption may not be realistic, especially for hih rate transmission For systems employin orthoonal frequency division multiplexin (OFDM) to cope with frequency selective channels, effective beamformin techniques were proposed in [5], [6] However, these techniques cannot be applied to sinle carrier systems, which are the focus of this paper Beamformin for sinle carrier transmission is especially relevant for the uprade of existin systems such as the Global System for Mobile Communications (GSM) and the Enhanced Data Rates for GSM Evolution (EDGE) system Ihis paper, we consider transmit beamformin for frequency selective fadin channels with perfect CSI at the transmitter Due to the intersymbol interference (ISI) caused by the frequency selectivity of the channel, equalization is necessary at the receiver and the optimum beamformer depends ohe equalizer used Here, we adopt decision feedback equalization (DFE) because of its low complexity and ood performance Contribution: This paper makes the followin contributions We show that, in contrast to the flat fadin case, for frequency selective fadin channels beamformin filters (BFFs) instead of simple beamformin weihts are beneficial Thereby, we consider both finite impulse response (FIR) and infinite IR (IIR) BFFs In case of IIR BFFs, we obtain a closed form solution for the optimum BFFs which maximize the sinal to noise ratio (SNR) For FIR BFFs we provide an efficient numerical method for calculation of the optimum BFFs We show that transmit beamformin with perfect CSI results in sinificant performance ains for frequency selective channels and can also uide the desin of beamformin with imperfect CSI Oranization: In Section II, the considered system model is presented The optimization of IIR and FIR BFFs is discussed in Sections III and IV, respectively Simulatioesults are provided in Section V, and some conclusions are drawn in Section VI Notation: Ihis paper, ( ) T, ( ) H, ( ), I X, 0, E{ }, and, denote transpose, Hermitiaranspose, complex conjuate, the X X identity matrix, the all zero column vector, expectation, and discrete time convolution, respectively Furthermore, X(f) F{x[k]} is the discrete time Fourier transform of x[k] and [x] + max(x, 0) II SYSTEM MODEL Let us consider a multiple input multiple output (MIMO) system with N T transmit and N R receive antennas The block diaram of the discrete time overall transmission system in complex baseband representation is shown in Fi 1 The modulated symbols b[k] are taken from a scalar symbol alphabet A such as phase shift keyin (PSK) or quadrature amplitude modulation (QAM), and have variance σb 2 = E{ b[k] 2 } = 1 The transmit BFF impulse response coefficients of antenna, 1 N T, are denoted by nt [k], < k <, and their enery is normalized to N T k= [k] 2 = 1 The sinal transmitted over antenna at time k is iven by s nt [k] = nt [k] b[k] (1) The discrete time received sinal at receive antenna, 1 N R, can be modeled as r nr [k] = [k] s nt [k] + n nr [k], (2) where n nr [k] denotes additive (spatially and temporally) white Gaussian noise (AWGN) with variance σ 2 n = E{ n nr [k] 2 } = N 0, and N 0 denotes the sinle sided power spectral density of the underlyin continuous time passband noise process [k], 0 k L 1, denotes the overall channel impulse

1 [k] b[k] 00 11 2 [k] [k] DFE n NR [k] NT [k] s 1[k] s 2[k] s NT [k] n 1[k] n 2[k] Feedback Channel ˆb[k] Fi 1 Block diaram of the overall transmission system ˆb[k] are estimated symbols at the receiver response (CIR) betweeransmit antenna and receive antenna of lenth L In our model, [k] contains the combined effects of transmit pulse shapin, wireless channel, receive filterin, and samplin We assume a block fadin model, ie, the channel is constant for the duration of at least one data burst before it chanes to a new independent realization In eneral, the [k] are spatially and temporally correlated because of insufficient antenna spacin and transmit/receive filterin, respectively By substitutin Eq (1) into Eq (2) we obtain r nr [k] = h eq [k] b[k] + n nr [k], (3) where the equivalent CIR h eq [k] correspondin to receive antenna is defined as h eq [k] = [k] nt [k] (4) Eq (3) shows that a MIMO system with beamformin can be modeled as an equivalent sinle input multiple output (SIMO) system Therefore, at the receiver the same equalization, channel estimation, and channel trackin techniques as for sinle antenna transmission can be used III BEAMFORMING WITH IIR FILTERS We first consider IIR beamformin For convenience the frequency responses of the IIR BFFs G nt (f) F{ nt [k]} are collected in vector G(f) [G 1 (f) G 2 (f) G NT (f)] T The unbiased SNR for DFE with optimum IIR feedforward filterin is iven by [7] SNR(G(f)) = σ2 b σe 2 χ, (5) where χ = 0 and χ = 1 for zero forcin (ZF) and minimum mean squared error (MMSE) DFE filter optimization, respectively In Eq (5), the DFE error variance is iven by [8] [ ] σe 2 = σn 2 exp N R ln ξ + Hn eq r (f) 2 df, (6) where ξ = 0 and ξ = σn 2/σ2 b for ZF DFE and MMSE DFE, respectively The equivalent channel frequency response H eq (f) F{h eq [k]} is iven by H eq (f) = G nt (f)h nt (f) (7) with H nt (f) F{ [k]} The optimum BFF vector G opt (f) [G opt 1 (f) G opt 2 (f) G opt N T (f)] T shall maximize SNR(G(f)) subject to the transmit power constraint G H (f)g(f)df = 1 (8) A convenient approach for calculatin G opt (f) is the classical Calculus of Variations method [9] Followin this method, we model the BFF of antenna as G nt (f) = G opt (f) + ε nt B nt (f), where B nt (f) and ε nt denote an arbitrary function of f and a real valued variable, respectively The optimization problem can now be formulated ierms of its Laranian L(ε) = SNR(G(f)) + µ G H (f)g(f)df, (9) where ε [ε 1 ε 2 ε NT ] T and µ is the Larane multiplier The optimum BFF vector G opt (f) has to fulfill the condition [9] L(ε) ε ε=0 = 0, (10) for arbitrary B nt (f), 1 N T Combinin Eqs (9) and (10), it can be showhat vector G opt (f) has to fulfill S(f)G opt (f) = µ [G H opt(f)s(f)g opt (f) + ξ] G opt (f), (11) where µ is a constant and S(f) is an N T N T matrix iven by S(f) N R n H r (f)h H (f), H nr (f) [H 1nr (f) H 2nr (f) H NT (f)] T Eq (11) is a typical eienvalue problem and G opt (f) can be expressed as G opt (f) = X(f)E(f), (12) where E(f) [E 1 (f) E 2 (f) E NT (f)] T is that unit norm eienvector of S(f) which corresponds to its larest eienvalue, and X(f) is a scalar factor Eq (12) shows that in eneral the optimum IIR BFFs can be viewed as concatenation of two filters: A filter X(f) which is common to all transmit antennas and a filter E nt (f) which is transmit antenna dependent Ihe followin, we derive filter X(f) for ZF DFE and MMSE DFE A ZF DFE Ihis case, ξ = 0 is valid and combinin Eqs (11) and (12) results in X(f) = 1 µ (13) Furthermore, from the power constraint in Eq (8), we obtain µ = 1 Therefore, the optimum IIR BFFs for ZF DFE are iven by G opt (f) = E(f)e jϕ(f), (14)

where ϕ(f) is the phase which can be chosen arbitrarily Replacin G(f) in Eq (5) by G opt (f) from Eq (14) yields the maximum SNR for ZF DFE SNR(G opt (f)) = σ2 b σ 2 n exp ln () df (15) B MMSE DFE For MMSE DFE ξ = σn/σ 2 b 2 holds Therefore, it can be showhat ihis case the manitude of the optimum X(f) is iven by [ ] + 1 X(f) = µ ξ, (16) where we took into account that X(f) has to be non neative Findin the optimum µ is a typical Water Fillin problem [10] In particular, motivated by the power constraint in Eq (8) we define P( µ) [ ] + 1 µ ξ df (17) The optimum µ opt can be found by slowly increasin a very small startin value µ = µ 0 which yields P( µ 0 ) > 1 until P( µ = µ opt ) = 1 is achieved The optimum IIR BFFs for MMSE DFE are iven by G opt (f) = [ 1 µ opt ] + ξ E(f)e jϕ(f), (18) where ϕ(f) is aaihe phase which can be chosen arbitrarily The correspondin maximum SNR is SNR(G opt (f)) = (19) σ b 2 ( [λmax ] + (f) σn 2 exp ln ξ + ξ) df µ opt 1 Comparin the BFFs and the maximum SNRs for ZF DFE and MMSE DFE, we make the followin interestin observations 1) For σn 2 0 (ie, ξ 0) the optimum IIR BFFs for MMSE DFE approach those for ZF DFE 2) In case of ZF DFE, the optimum BFF frequency response vector G opt (f) at frequency f = f 0 is just the dominant eienvector of matrix S(f) at frequency f = f 0 Since S(f 0 ) only depends ohe channel frequency responses H nt (f) at frequency f = f 0, G opt (f 0 ) is independent of the H nt (f), f f 0 This is not true for MMSE DFE, where the optimum frequency response vector G opt (f) at frequency f = f 0 also depends ohe channel frequency responses H nt (f) at frequencies f f 0, because of the constraint P( µ) = 1, cf Eq (17) In fact, for MMSE DFE X(f) may be interpreted as a power allocation filter which allocates more power to frequencies with lare eienvalues 3) If the underlyin channel is frequency nonselective, S(f) = S for all f Ihis case, it is easy to see that the optimum BFFs have only one non zero tap and are identical to the well known beamformin weihts for frequency nonselective channels, cf e [1] Ihis case, the DFE structure collapses to a simple threshold detector, of course IV BEAMFORMING WITH FIR FILTERS Althouh the IIR BFFs derived ihe previous section achieve a hiher performance than FIR BFFs, IIR BFFs may not be feasible in practice due to their hih computational cost This motivates the investiation of FIR BFFs whose impulse responses nt [k], 0 k L 1, 1 N T, have L non zero taps As a consequence, the equivalent overall CIR h eq [k] has lenth L eq = L+L 1 Introducin the BFF vector [ T 1 T 2 T N T ] T, nt [ nt [0] nt [1] nt [L 1]] T, we can now express the frequency response of the equivalent channel as Hn eq r (f) = d H (f)h nr, (20) where d(f) [1 e j2πf e j2πf(leq 1) ] T, H nr [H 1nr H 2nr H NT ], and H nt is an L eq N T matrix defined as [0] 0 H nt [L 1] 0 0 (21) hntnr [0] 0 [L 1] Since we still assume that the DFE at the receiver side employs IIR feedforward filters, Eqs (5) and (6) remain valid Therefore, the SNR for ZF DFE and MMSE DFE can be expressed as SNR() = σ2 b σ 2 n exp 1/2 ln( H B(f) + ξ)df χ, (22) with N T L N T L matrix B(f) N R HH d(f) d H (f)h nr The optimum BFF vector opt shall maximize SNR() subject to the power constraint H = 1 Thus, the Laranian of the optimization problem can be formulated as L() = SNR() + µ H, (23) where µ denotes aaihe Larane multiplier The optimum vector opt has to fulfill L()/ = 0, which leads to the non linear eienvalue problem H opt B(f) + ξi NT L ( ) df B(f) + opt = opt, (24) ξint L opt where µ does not appear since Eq (24) already includes the constraint H opt opt = 1 However, Eq (24) does not seem to have a closed form solution In fact, this is not surprisin since

by discretizin the interal in Eq (22), it can be showhat the optimization problem in Eq (23) is equivalent to maximizin a product of Rayleih quotients L() = S i H (B(f i ) + ξi NT L ) H, (25) where f i + (i 1)/(S 1) and S is a lare inteer Note that Eq (25) is scale invariant (ie, L() does not depend ohe manitude of ) and the resultin solution has to be scaled to meet H = 1 It is well knowhat the maximization of a product of Rayleih quotients is a difficult mathematical problem which is not well understood and a closed form solution is not known except for the trivial case S = 1, cf e [11], [12] In [12] a radient method is advocated to search for the optimum vector opt However, radient methods usually suffer from a slow converence Here, we avoid a radient approach and propose a novel fast converin numerical method based on Eq (24) If the denominator under the interal in Eq (24) was absent, opt would simply be the maximum eienvalue of matrix 1/2 (B(f) + ξi NT L )df, which could be calculated efficiently usin the so called Power Method [10] Motivated by this observation, we propose a Modified Power Method (MPM) for recursive calculation of opt : 1) Let i = 0 and initialize the BFF vector with some 0 fulfillin H 0 0 = 1 2) Update the BFF vector i+1 = H i 3) Normalize the BFF vector B(f) + ξi NT L ( ) df B(f) + i (26) ξint L i i+1 = i+1 H i+1 i+1 (27) 4) If 1 H i+1 i < ɛ, oto Step 5), otherwise increment i i + 1 and oto Step 2) 5) i+1 is the desired BFF vector For the termination constant ɛ in Step 4) a suitably small value should be chosen, e ɛ = 10 4 Because of its involved nature, we are not able to prove lobal converence of the MPM alorithm to the maximum SNR However, in our simulations for ξ > 0 the alorithm always achieved hih SNR values for different initializations 0 The converence time of the alorithm depends on L For example, for L = 2 and L = 5 the alorithm typically terminated after less than 20 and 100 iterations, respectively V SIMULATION AND NUMERICAL RESULTS Ihis section, we present simulation and numerical results for the SNR and the bit error rate (BER) of DFE with transmit beamformin As relevant practical example, we consider the equalizer test (EQ) channel of the GSM system [13] As is usually done for GSM, we model Gaussian minimum shift keyin (GMSK) modulation as filtered binary PSK (BPSK) For all results we assume N T = 3 transmit and N R = 1 receive antennas and a maximum channel lenth of L = 7 The correlation coefficient between all transmit antennas is ρ = 05 Before we show SNR and BER results, we illustrate the converence behavior of the proposed MPM alorithm for calculation of the FIR BFF vector We assume BFFs of lenth L = 5 and two different initializations: (1) = 1 3 [1 0 0 0 0] T and (2) = 1 15 [1 1 1 1 1] T, 1 3 Fi 2 shows the anle distance measure H i+1 i [4], [3] vs iteration number i for one random realization of the EQ channel and 10 lo 10 (E b /N 0 ) = 10 db, where E b is the averae received enery per bit As can be observed, the alorithm converes in both cases to the steady state value 1 after less than 20 iterations For the FIR beamformin results shown in Fis 3 and 4 we calculated the BFFs for each channel realization usin the MPM alorithm with ɛ = 10 4 The BFF vector was initialized with the normalized all ones vector and the alorithm was stopped after 100 iterations if it had not terminated before In Fi 3, we show the averae SNR (SNR) vs E b /N 0 of ZF DFE with IIR BFFs and MMSE DFE with FIR and IIR BFFs SNR was obtained by averain the respective SNRs in Eqs (15), (19), and (22) over 500 independent realizations of the EQ channel For comparison we also show the averae SNR of ZF DFE and MMSE DFE for sinle antenna transmission (N T = 1, N R = 1) in Fi 3 As can be observed, even FIR beamformin with L = 1 leads to performance ains of 2 3 db compared to sinle antenna transmission These ains increase with increasin L but FIR BFFs with L = 5 achieve practically the same performance as IIR BFFs This observation is also useful for the desin of beamformin with imperfect CSI and can uide the search for suitable BFF lenths L ihat case Fi 4 shows the simulated BER (averaed over 50,000 channel realizations) of MMSE DFE with FIR BFFs of different lenths L For the simulation we implemented MMSE DFE with feedforward filter lenth 4L which causes neliible performance deradation compared to IIR feedforward filters For comparison, Fi 4 also includes simulatioesults for transmission with just one antenna (N T = 1, N R = 1) and optimized delay diversity (ODD) for DFE as proposed in [14] The ODD filters of lenth L = 3 are optimized based ohe channel statistics, ie, in contrast to the BFFs the ODD filters do not depend ohe particular realization of the channel Furthermore, we show in Fi 4 the BER of MMSE DFE with IIR BFFs obtained by numerically averain Q(2 SNR(G opt (f))) over the channel realizations, where Q(x) 1 2π x exp( t2 /2)dt denotes the Gaussian Q function and SNR(G opt (f)) is obtained from Eq (19) We note that this numerical method does not take into account the effect of error propaation ihe DFE feedback filter, whereas the simulated BER performances of beamformin with FIR filters, ODD, and sinle antenna transmission include this effect, of course This explains why the performance difference between MMSE DFE with IIR BFFs and MMSE DFE with FIR BFFs is larer in Fi 4 than expected from Fi 3 (which does not include the effect of error propaation for either scheme) However, Fi 4 also clearly shows that FIR transmit

1 10 1 simulation 095 09 085 (2) (1) 10 2 H i+1 i 08 075 BER 10 3 07 065 5 10 15 20 25 30 35 40 i Fi 2 Anle distance measure H i+1 i vs i for BPSK transmission over EQ channel with L = 7, N T = 3, N R = 1, equal antenna correlation ρ = 05, L = 5, and 10lo 10 (E b /N 0 ) = 10 db 10 4 N T IIR BF ODD, L FIR BF, L FIR BF, L =2 FIR BF, L FIR BF, L =4 numerical evaluation 10 5 0 1 2 3 4 5 6 7 8 9 10 10 lo 10 (E b /N 0 )[db] Fi 4 BER of MMSE DFE for beamformin (BF) with different filters BPSK transmission over EQ channel with L = 7, N T = 3, N R = 1, and equal antenna correlation ρ = 05 Results for ODD [14] and sinle antenna transmission are also included ZF DFE, N T 10 lo 10 (SNR)[dB] 20 15 10 5 0 MMSE DFE, N T ZF DFE, IIR BF MMSE DFE, IIR BF MMSE DFE, FIR BF, L MMSE DFE, FIR BF, L =2 MMSE DFE, FIR BF, L MMSE DFE, FIR BF, L =4 MMSE DFE, FIR BF, L =5 5 0 2 4 6 8 10 12 14 16 18 20 10 lo 10 (E b /N 0 )[db] Fi 3 Averae SNR of DFE for beamformin (BF) with different filters BPSK transmission over EQ channel with L = 7, N T = 3, N R = 1, and equal antenna correlation ρ = 05 Results for sinle antenna transmission are also included beamformin achieves a lare performance ain compared to sinle antenna transmission and ODD For example, at BER = 10 3 FIR beamformin with L = 3 achieves a ain of more than 3 db over ODD with L = 3 Of course, this ain comes at the cost of an increased complexity as beamformin requires instantaneous CSI whereas ODD only needs averae CSI VI CONCLUSION Ihis paper, we have considered beamformin with perfect CSI for sinle carrier transmission over frequency selective fadin channels with ZF DFE and MMSE DFE at the receiver We have derived closed form solutions for IIR BFFs and we have provided an efficient numerical method for calculation of FIR BFFs of arbitrary lenths L The FIR BFFs approach the performance of the IIR solution as L increases However, sinificant performance ains over sinle antenna transmission and delay diversity can be achieved already with very short FIR BFFs The derived SNRs for beamformin with perfect CSI can also serve as valuable upper bounds for the performance of beamformin with imperfect CSI REFERENCES [1] P Dihe, R Mallik, and S Jamuar Analysis of Transmit-Receive Diversity in Rayleih Fadin IEEE Trans Commun, 51:694 703, April 2003 [2] G Jonren, M Skolund, and B Ottersten Combinin Beamformin and Orthoonal Space-Time Block Codin IEEE Trans Inform Theory, 48:611 627, March 2002 [3] D Love, R Heath, and T Strohmer Grassmannian Beamformin for Multiple-Input Multiple-Output Wirelss Systems IEEE Trans Inform Theory, 49:2735 2747, October 2003 [4] K Mukkavilli, A Sabharwal, E Erkip, and B Aazhan Beamformin with Finite Rate Feedback in Multiple Antenna Systems IEEE Trans Inform Theory, 49:2562 2579, October 2003 [5] J Choi and R Heath Interpolation Based Transmit Beamformin for MIMO-OFDM with Limited Feedback IEEE Trans Sinal Processin, 53:4125 4135, November 2005 [6] P Xia, S Zhou, and G Giannakis Adaptive MIMO-OFDM Based on Partial Channel State Information IEEE Trans Sinal Processin, 52:202 213, January 2004 [7] J Cioffi, G Dudevoir, M Eyubolu, and G Forney Jr MMSE Decision- Feedback Equalizers and Codin - Part I: Equalization Results IEEE Trans Commun, 43:2582 2594, October 1995 [8] KE Baddour and PJ McLane Analysis of Optimum Diversity Combinin and Decision Feedback Equalization in Dispersive Rayleih Fadin In Proceedins of IEEE ICC, paes 21 26, June 1999 [9] R Weinstock Calculus of Variations, with Applications to Physics and Enineerin Dover, New York, 1974 [10] TK Moon and WC Stirlin Mathematical Methods and Alorithms for Sinal Processin Prentice Hall, New York, 2000 [11] R Cameron Minimizin the Product of Two Raleih Quotients Linear and Multilinear Alebra, 13:177 178, 1983 [12] R K Martin et al Unification and Evaluation of Equalization Structures and Desin Alorithms for Discrete Multitone Modulation Systems IEEE Trans Sinal Processin, 53:3880 3894, October 2005 [13] GSM Recommendation 0505: Propaation Conditions, Vers 530, Release 1996 [14] S Yiu, R Schober, and W Gerstacker Optimized Delay Diversity for Suboptimum Equalization Accepted for publication in IEEE Transactions on Vehicular Technoloy, 2006