Performance Comparisons of MIMO Techniques with Application to WCDMA Systems

Home , Beamforming, Diversity scheme, MIMO

EURASIP Journal on Applied Signal Processing 2004:5, 649–661 c 2004 Hindawi Publishing Corporation

Chuxiang Li Department of Electrical Engineering, Columbia University, New York, NY 10027, USA Email: [email protected]

Xiaodong Wang Department of Electrical Engineering, Columbia University, New York, NY 10027, USA Email: [email protected]

Received 11 December 2002; Revised 1 August 2003

Multiple-input multiple-output (MIMO) communication techniques have received great attention and gained significant development in recent years. In this paper, we analyze and compare the performances of different MIMO techniques. In particular, we compare the performance of three MIMO methods, namely, BLAST, STBC, and linear precoding/decoding. We provide both an analytical performance analysis in terms of the average receiver SNR and simulation results in terms of the BER. Moreover, the applications of MIMO techniques in WCDMA systems are also considered in this study. Specifically, a subspace tracking algorithm and a quantized feedback scheme are introduced into the system to simplify implementation of the beamforming scheme. It is seen that the BLAST scheme can achieve the best performance in the high data rate transmission scenario; the beamforming scheme has better performance than the STBC strategies in the diversity transmission scenario; and the beamforming scheme can be effectively realized in WCDMA systems employing the subspace tracking and the quantized feedback approach. Keywords and phrases: BLAST, space-time block coding, linear precoding/decoding, subspace tracking, WCDMA.

1. INTRODUCTION ing power and/or rate over multiple transmit antennas, with partially or perfectly known channel state information [7]. Multiple-input multiple-output (MIMO) communication Another family of MIMO techniques aims at reliable trans- technology has received significant recent attention due to missions in terms of achieving the full diversity promised by the rapid development of high-speed broadband wireless the multiple transmit and receive antennas. Space-time block communication systems employing multiple transmit and coding (STBC) is one of such techniques based on orthog- receive antennas [1, 2, 3]. Many MIMO techniques have been onal design that admits simple linear maximum likelihood proposed in the literature targeting at different scenarios in (ML) decoding [8, 9, 10]. The trade-off between diversity and wireless communications. The BLAST system is a layered multiplexing gain are addressed in [11, 12], which are from a space-time architecture originally proposed by Bell Labs to signal processing perspective and from an information theo- achieve high data rate wireless transmissions [4, 5, 6]. Note retic perspective, respectively. that the BLAST systems do not require the channel knowl- Some simple MIMO techniques have already been pro- edge at the transmitter end. On the other hand, for some ap- posed to be employed in the third-generation (3G) wireless plications, the channel knowledge is available at the trans- systems [13, 14]. For example, in the 3GPP WCDMA stan- mitter, at least partially. For example, an estimate of the dard, there are open-loop and closed-loop transmit diver- channel at the receiver can be fed back to the transmitter sity options [15, 16]. As more powerful MIMO techniques in both frequency division duplex (FDD) and time division emerge, they will certainly be considered as enabling tech- duplex (TDD) systems, or the channel can be estimated di- niques for future high-speed wireless systems (i.e., 4G and rectly by the transmitter during its receiving mode in TDD beyond). systems. Accordingly, several channel-dependent signal pro- The purpose of this paper is to compare the perfor- cessing schemes have been proposed for such scenarios, for mance of different MIMO techniques for the cases of two example, linear precoding/decoding [7]. The linear precod- and four transmit antennas, which are realistic scenarios ing/decoding schemes achieve performance gains by allocat- for MIMO applications. For a certain transmission rate, we 650 EURASIP Journal on Applied Signal Processing compare the performance of three MIMO schemes, namely, 2.1. BLAST versus linear precoding for high-rate BLAST, STBC, and linear precoding/decoding. Note that transmission both BLAST and STBC do not require channel knowledge Assume that there are nT transmit and nR receive antennas, at the transmitter, whereas linear precoding/decoding does. where nR ≥ nT . In this section, we assume that the MIMO For each of these cases, we provide an analytical performance system is employed to achieve the highest data rate, that is, analysis in terms of the receiver output average signal-to- nT symbols per transmission. When the channel is unknown noise ratio (SNR) as well as simulation results on their BER to the transmitter, the BLAST system can be used to achieve performance. Moreover, we also consider the application of this; whereas when the channel is known to the transmitter, these MIMO techniques in WCDMA systems with multipath the linear precoding/decoding can be used to achieve this. fading channel. In particular, when precoding is used, a subspace tracking algorithm is needed to track the eigenspace of 2.1.1. BLAST the MIMO system at the receiver and feed back this infor- n mation to the transmitter [17, 18, 19, 20]. Since the feedback In the BLAST system, at each transmission, T data sym- s s ... s s ∈ A A channel typically has a very low bandwidth [21], we contrive bols 1, 2, , nT , i ,where is some unit-energy (i.e., ffi ff E{|si|2}=1) constellation signal set (e.g., PSK, QAM), are an e cient and e ective quantized feedback approach. n The main findings of this study are as follows. transmitted simultaneously from all T antennas. The received signal can be represented by (i) In the high data rate transmission scenario, for exam-         y h h ··· h s n ple, four symbols per transmission over four transmit 1 1,1 1,2 1,nT 1 1  y   h h ··· h   s   n  antennas, the BLAST system actually achieves a bet-  2  ρ  2,1 2,2 2,nT   2   2    =       ter performance than the linear precoding/decoding  .   . . . .   .  +  . ,  .  nT  . . .. .   .   .  schemes, even though linear precoding/decoding yn hn hn ··· hn n sn nn make use of the channel state information at the trans- R R,1 R,2 R, T T R mitter. y H s n (ii) In the diversity transmission scenario, for example, (1) one symbol per transmission over two or four trans- y i mit antennas, beamforming offers better performance where i denotes the received signal at the th receive an- h i than the STBC schemes. Hence the channel knowledge tenna; i,j denotes the complex channel gain between the th j ρ at the transmitter helps when there is some degree of receive antenna and the th transmit antenna; denotes the ∼ N freedom to choose the eigen channels. total transmit SNR; and n c(0, InR ). = (iii) By employing the subspace tracking technique with an Thereceivedsignalisfirstmatchedfilteredtoobtain z H ρ/n H H Ω H efficient quantized feedback approach, the beamform- H y = T H Hs + H n.Denote H H and w H ing scheme can be effective and feasible to be employed H n, and thus, w ∼ N c(0, Ω). The matched-filter output is in WCDMA systems to realize reliable data transmis- then whitened to get sions. / ρ / u = Ω−1 2z = Ω1 2s + v˜,(2) The remainder of this paper is organized as follows. In nT Section 2, performance analysis and comparisons of differ- −1/2 ent MIMO techniques are given for the narrowband scenario. where v˜ Ω w ∼ N c(0, InR ). Based on (2), several meth- Section 3 describes the WCDMA system based on the 3GPP ods can be used to detect the symbol vector s.Forexample, standard, the channel estimation method, the algorithm of theMLdetectionruleisgivenby tracking the MIMO eigen-subspace, as well as the quantized

2 feedback approach. Simulation results and further discus- ρ = − Ω1/2 sions are given in Section 4. Section 5 contains the conclu- sˆML arg minn u s ,(3) s∈A T nT sions. which has a computational complexity exponential in the n 2. PERFORMANCE ANALYSIS AND COMPARISONS number of transmit antennas T . On the other hand, the ff OF MIMO TECHNIQUES sphere decoding algorithm o ers a near-optimal solution 3 to (2) with an expected complexity of O(nT )[22]. More- In this section, we analyze the performance of several MIMO over, a linear detector makes a symbol-by-symbol decision schemes under different transmission rate assumptions, for sˆ = Q(x), where x = Gu and Q(·) denotes the symbol slicing the cases of two and four transmit antennas. BLAST and lin- operation. Two forms of linear detectors can be used [5, 6], / ear precoding/decoding schemes are studied and compared namely, the linear zero-forcing detector, where G = Ω−1 2, 1/2 − for high-rate transmissions in Section 2.1. Section 2.2 fo- and the linear MMSE detector, where G = (Ω +(nT /ρ)I) 1. cuses on the diversity transmission scenario, where different Finally, a method based on interference cancellation with STBC strategies are investigated and compared with beam- ordering offers improved performance over the linear de- forming and some linear precoding/decoding approaches. tectors with comparable complexity [22]. Note that among MIMO Techniques Comparisons and Application to WCDMA Systems 651 the above-mentioned BLAST decoding algorithms, the lin- Remark 1. The BLAST system can be viewed as a special case ear zero-forcing detector has the worst performance. The de- of linear precoding with the transmitter filter F = ρ/nT InT . cision statistics of this method is given by And the zero-forcing BLAST detection scheme corresponds 1/2 to choosing the receiver filter G = Ω . ρ −1/2 −1/2 x = Gu = Ω u = s + Ω ˜v. (4) = nT Remark 2. An alternative precoding scheme is to choose F H ρ/nT V and G = V . Then the output of the linear decoder s It follows from (4) that the received SNR for symbol j is can be written as −1 (ρ/nT )/[Ω ]j,j , j = 1, 2, ..., nT . Hence the average received ρ ρ SNR under linear zero-forcing BLAST detection is given by = H H H H = Λs x n V H HVs + V H n n + w, (10) T T nT = ρ 1 1 . ∼ N Λ SNRBLAST-LZF n2 Ω−1 (5) where w c(0, ). Hence this scheme also transforms T j=1 j,j the MIMO channel into a set of independent channels, but with different SNRs. The received SNR for the jth symbol 2.1.2. Linear precoding and decoding is (ρ/nT )λj,whereλj is the jth eigenvalue contained by Λ. When the channel H is known to the transmitter, a linear pre- We call this method the whitening precoding. The average coder can be employed at the transmitter and a correspond- received SNR is given by ing linear decoder can be used at the receiver [7]. Specifically, n m ≤ n = s s ··· s T T Ω suppose T symbols s [ 1 2 m] are transmit- = ρ 1 λ = ρ tr( ) . m = SNRwhitening precoding n2 j n2 (11) ted per transmission, where rank(H). Then the linear T j=1 T precoder is an nT × m matrix F such that the transmitted signal is Fs.ThenR × 1 received signal vector is then Note that the whitening precoding is different from the equal-error precoding in (8). In particular, different received y = HFs + n,(6)SNRs are achieved over different subchannels for the whitening precoding, whereas the equal-error precoding provides ∼ N where n c(0, InR ). At the receiver, y is first matched fil- the same SNR over all subchannels. tered, and then an m × nT linear decoder G is applied to the matched-filter output to obtain the decision statistics 2.1.3. Comparisons We have the following result on the relative SNR perfor- H H x = GH y = GΩFs + GH n. (7) mance of the BLAST system and the two precoding schemes discussed above. The linear precoder F and decoder G are chosen to minimize a weighted combination of symbol estimation errors, that is, Proposition 1. Suppose that an nT × nR MIMO system is em- 1/2 2 minF,G E{D (s−x) },whereD is a diagonal positive def- ployed to transmit nT symbols per transmission, using either inite matrix subject to the total transmitter power constraint the BLAST system, the equal-error precoding scheme, or the tr(FFH ) ≤ ρ. The weight matrix D is such that all decoded whitening precoding scheme, then symbols have equal errors (equal error design). Denote the H H eigendecomposition of Ω as Ω = VΛV + V˜ Λ˜ V˜ ,whereΛ SNRwhitening precoding ≥ SNRBLAST-LZF and V contain the m largest eigenvalues and the correspond- (12) ≥ SNRequal-error precoding. ing eigenvectors of Ω,respectively;andΛ˜ and V˜ contain the remaining (nT −m) eigenvalues and the corresponding eigen- Proof. We first show that vectors, respectively. Denote γ = ρ/tr(Λ−1). Then the linear precoder and decoder are given by [7] SNRBLAST-LZF ≥ SNRequal-error precoding. (13)

/ F = γ1/2VΛ−1 2, Since (8) nT nT n = 1 Λ−1/2 H . 1 λ−1 = 1 Ω−1 ≥ T G − / / V j j j nT − , (14) γ 1 2 γ1 2 nT nT , / Ω 1 + j=1 j=1 j=1 1 j,j H / γ−1 γ It can be veriﬁed that GH HF = (1 ( + ))Im.Hence we have this precoding scheme transforms the MIMO channel into n a scaled identity matrix. Furthermore, the received SNRs for 1 T 1 1 ≥ n . (15) all decoded symbols are equal, given by γ, that is, n2 Ω−1 T λ−1 T j=1 j,j j=1 j ρ ρ SNR = = . (9) It follows from (5), (9), and (15) that SNR ≥ equal-error precoding tr Λ−1 tr Ω−1 BLAST-LZF SNRequal-error precoding. 652 EURASIP Journal on Applied Signal Processing

≤ We next show that SNRBLAST-LZF SNRwhitening precoding. 16 First, we have the following. 14 n×n Fact 1. Suppose that A is a positive deﬁnite matrix, then 12

1 H − 10 = Ai i − ˜ 1 i −1 , a˜i Ai a˜ , (16) A i,i 8 where A˜ i is the (n − 1) × (n − 1) matrix obtained from A 6 by removing the ith row and ith column; and a˜i is the ith Receiver SNR (dB) 4 column of A with the ith entry Ai,i removed. Note that A˜ i is a principal submatrix of A; since A is positive deﬁnite, so is, 2 −1 A˜ i,andA˜ i exists. To see (16), denote the above-mentioned 0 partitioning of the Hermitian matrix A with respect to the −2 ith column and row by A = (A˜ i, a˜i, Ai,i). In the same way, we 0 5 10 15 −1 partition its inverse B A = (B˜ i, b˜i, Bi,i). Now from the Transmitter SNR (dB) fact that AB = In, it follows that Whitening precoding A B H ˜ = B ˜ ˜ = . BLAST-LZF i,i i,i + a˜i bi 1, a˜i i,i + Aibi 0 (17) Equal-error precoding

Solving for Bi,i from (17), we obtain (16). (a)

Using (16), we have

− nT nT nT 10 1 1 H −1 = Ωi i − ω˜ Ω˜ ω˜ i ≤ Ωi i = Ω . Ω−1 , i i , tr( ) j=1 j,j j=1 j=1 (18) 10−2 It then follows from (5), (11), and (18) that SNRBLAST-LZF ≤ SNR .

whitening precoding BER

Figure 1 shows the comparisons between the BLAST and −3 the linear precoding/decoding schemes in terms of the aver- 10 age receiver SNR as well as the BER for a system with nT = 4 and nR = 6. The rate is four symbols per transmission. The SNR curves in Figure 1a are plotted according to (9), (5), 10−4 and (11). It is seen that the SNR curves conﬁrm the conclu- 0 5 10 15 sion of Proposition 1. Moreover, it is interesting to see that SNR (dB) the SNR ordering given by (12) does not translate into the corresponding BER order. This can be roughly explained as BLAST-ML i BLAST, ordered ZF-IC follows. The BER√ for the th symbol stream can be approx- BLAST-LZF imated as Q(γ SNRi), where γ is a constant determined by Equal-error precoding the modulation scheme. The average BER is then Whitening precoding

(b) nT ∼ 1 p = Q γ SNRi . (19) nT i=1 Figure 1: Comparisons of the average receiver SNR and the BER between the BLAST and the linear precoding/decoding schemes: Since Q(·) is a concave function, we have nT = 4andnR = 6; the rate is four symbols/transmission. p ≤ Q γ SNR . (20)

2.2. Space-time block coding versus beamforming Hence, the average SNR value does not directly translate into for diversity transmission the average BER. Moreover, it is seen from the Figure 1b in Figure 1 that the interference cancellation with ordering [6] In contrast to the high data rate MIMO transmission sce- BLAST detection method oﬀers a signiﬁcant performance nario discussed in Section 2.1, an alternative approach to ex- gain over the linear zero-forcing method, making the BLAST ploiting MIMO systems targets at achieving the full diver- outperform the precoding schemes by a substantial margin. sity. For example, with nT transmit antennas and nR receive MIMO Techniques Comparisons and Application to WCDMA Systems 653

H antennas, a maximum diversity order of nT nR is possible where ΩA HA HA, λ1 and λ2 are the two eigenvalues of ΩA, when the transmission rate is one symbol per transmission. and When the channel is unknown at the transmitter, this can be   n = h1,1 h1,2 achieved using STBC (for T 2); and when the channel is    h h  known at the transmitter, this can be achieved using beam-  2,1 2,2  H HA =  . .  , ΩA = HA HA. (26) forming.  . .  h h nR,1 nR,2 2.2.1. Two transmit antennas case Alamouti scheme Beamforming Beamforming can be referred to as maximum ratio weighting When nT = 2, the elegant Alamouti transmission scheme can be used to achieve full diversity transmission at one symbol [23], and it is a special case of the linear precoding/decoding discussed in Section 2.1.2,where per transmission [8]. It transmits two symbols s1 and s2 over two consecutive transmissions as follows. During the first F = ρv1, transmission, s1 and s2 are transmitted simultaneously from (27) = H antennas 1 and 2, respectively; during the second transmis- G v1 , −s∗ s∗ sion, 2 and 1 are transmitted simultaneously from transmit antennas 1 and 2, respectively. The received signals at re- and v1 is the eigenvector corresponding to the largest eigen- i value of Ω. Hence in the beamforming scheme, at each trans- ceive antenna corresponding to these two transmissions are s given by mission, the transmitter transmits v1 from all transmit antennas, where s is a data symbol. The received signal is given by yi(1) ρ s1 s2 hi,1 ni(1) = ∗ ∗ + , i=1, 2, ..., nR. yi(2) −s s hi ni(2) 2 2 1 ,2 y = HFs + n = ρHv1s + n. (28) (21) At the receiver, a decision on s is made according to sˆ = Q(u), u u = H H = Note that (21) can be rewritten as follows: where the decision statistic is given by v1 H y √ρ H Ω s H H v1 v1 + v1 H n . The received SNR in this case is yi ρ hi hi s ni (1) ,1 ,2 1 (1) λ N λ ∗ = ∗ ∗ + ∗ , 1 c(0, 1) yi(2) 2 hi −hi s2 ni(2) ,2 ,1 (22) = ρλ . yi H˜ i s ni SNRbeamforming 1 (29) i = 1, 2, ..., nR, Comparing (25)with(29), it is obvious that SNRbeamforming ≥ SNRAlamouti. Note that in this case, the SNR order indeed i.i.d. where ni ∼ N c(0, I2). Note that the channel matrix H˜ i is translates into the BER order; since in the Alamouti scheme, H 2 2 orthogonal, that is, H˜ i H˜ i = (|hi,1| + |hi,2| )I2. both symbols have the same SNR, then At each receive antenna, the received signal is matched filtered to obtain ρ pbeamforming = Q γ ρλ1 ≤ Q γ λ1 + λ2 = pAlamouti. 2 H ρ 2 2 zi = H˜ i yi = hi + hi s + wi, i = 1, 2, ..., nR, (30) 2 ,1 ,2 (23) 2.2.2. Four transmit antennas case 2 2 where wi ∼ N c(0,(|hi,1| + |hi,2| )I2). The final decision on s One symbol per transmission is then made according to sˆ = Q(z), where Q(·) denotes the It is known that rate-one orthogonal STBC only exists for symbol slicing operation, and nT = 2, that is, the Alamouti code. For the case of four transmit antennas (nT = 4), we adopt a rate-one (almost orthog- nR ρ nR nR 2 2 onal) transmission scheme with the following transmission z = zi = hi,1 + hi,2 s + wi. (24) i=1 2 i=1 i=1 matrix:   s s s s The received SNR is therefore given by  1 2 3 4  s∗ −s∗ s∗ −s∗ =  2 1 4 3  . n 2 S (31) R 2 2 s3 −s4 −s1 s2  (ρ/2) hi + hi ∗ ∗ ∗ ∗ i=1 ,1 ,2 s s −s −s = 4 3 2 1 SNRAlamouti nR 2 2 hi + hi i=1 ,1 ,2 Such a transmission scheme was proposed in [24]. Hence ρ ρ H λ1 + λ2 four symbols s1, s2, s3,ands4 are transmitted across four = tr HA HA = tr ΩA = ρ , 2 2 2 transmit antennas over four transmissions. The received sig- (25) nals at the ith receive antenna corresponding to these four 654 EURASIP Journal on Applied Signal Processing transmissions are given by When the channel state is known at the transmitter, the       optimal transmission method to achieve one symbol per y h n i(1) i,1 i(1) transmission is the beamforming scheme described by (27),   ρ     yi(2) hi,2 ni(2) (28), and (29).   = S   +   , i = 1, 2, ..., nR. (32) yi(3) 4 hi,3 ni(3) Note that the received SNR of the above block coding yi(4) hi,4 ni(4) scheme with linear zero-forcing detector is given by

Note that (32)canberewrittenas ρ n2 = · R         SNR nR −1 , (38) 4 i= Γi yi(1) hi hi hi hi s ni(1) 1 1,1    ,1 ,2 ,3 ,4   1    y ∗ ρ −h∗ h∗ −h∗ h∗  s  n ∗  i(2)  =  i,2 i,1 i,4 i,3   2  i(2)  whereas the SNR of the beamforming scheme is given by  y  −h h h −h  s + n , i(3) 4 i,3 i,4 i,1 i,2 3 i(3) = ρλ ∗ ∗ ∗ ∗ ∗ ∗ SNRbeamforming 1. yi(4) −hi −hi hi hi s ni(4) ,4 ,3 ,2 ,1 4

yi H˜ i s vi Two symbols per transmission i=1, 2, ..., nR. Now suppose that a rate of two symbols per transmission is (33) desired using four transmit antennas. When the channel is unknown at the transmitter, we can use one pair of the trans- s s T The matched-ﬁlter output at the ith receive antenna is given mit antennas to transmit s1 = [ 1 2] using the Alamouti s s T by scheme, and use the other pair to transmit s2 = [ 3 4] also using Alamouti scheme. In this way, we transmit four sym- ρ = ˜ H = Ω˜ bols over two transmissions. At the ith receive antenna, the zi Hi yi is + wi, (34) T 4 received signal yi = [yi(1) yi(2)] corresponding to the two transmissions is given by where   ρ ρ γi 0 αi 0 = ˜ ˜ i = ... n   yi Hi,1s1 + Hi,2s2,+n, 1, 2, , R, (39)  γ −α  2 2 Ω˜ = ˜ H ˜ =  0 i 0 i i Hi Hi −α γ  , (35) i 0 i 0 hi,1 hi,2 hi,3 hi,4 where H˜ i,1 = h∗ −h∗ and H˜ i,2 = h∗ −h∗ . Therefore, we 0 αi 0 γi i,2 i,1 i,4 i,3 have γ = nT |h |2 α =  h∗ h h∗ h =     i j=1 i,j , i 2 ( i,1 i,3 + i,4 i,2), and wi   y1 H˜ 1,1 H˜ 1,2 H s1 H˜ i ni ∼ N c(0, Ω˜ i). By grouping the entries of zi into two        y2  ρ  H˜ 2,1 H˜ 2,2      s2 pairs, we can write  .  =  . .    +n. (40)  .  2  . .  s3 . . . z ρ s w ˜ ˜ s4 i(1) 1 i(1) ynR HnR,1 HnR,2 = Γi + , zi(3) 4 s3 wi(3) s y H˜

zi,1 s1 wi,1 The received signal y is first matched filtered to obtain (36) zi(4) ρ s4 wi(4) = Γi + , H ρ H H zi(2) 4 s2 wi(2) z = H˜ y = H˜ Hs˜ + H˜ n. (41) 2 zi,2 s2 wi,2 Denote γ α Γ = i i ∼ N Γ =   where i −αi γi and wi, c(0, i), 1, 2. Note that nR ΓH = Γ ff ×  1 ˜ H ˜  i i. Note also that (36)aree ectively 2 2 BLAST sys-  I2 n Hj,1Hj,2  R j=  tems and they can be decoded using either linear detection or Ω˜ ˜ H ˜ = n ·  1  . H H R  nR  (42) ML detection. For example, the linear decision rule is given  1 H  nR H˜ j H˜ j,2 I2 = Q i i = n ,1 by sˆ [ i=1 G , z , ], 1, 2, where the linear detector R j= −1 1 can be either a zero-forcing detector, that is, Gi, = Γi ,oran − MMSE detector, that is, Gi = (Γi +(4/ρ)I ) 1. On the other , 2 Then the output of the whitening filter is given by u = hand, the ML detection rule is given by −1/2 1/2 Ω˜ z = ρ/2Ω˜ s + w,wherew ∼ N c(0, I4). Now we can nR ρ H ρ use any of the aforementioned BLAST decoding methods to −1 sˆ = min zi, − Γis Γi zi, − Γis decode s. ∈A2 4 4 s i=1 When the channel is known at the transmitter, linear ! " nR nR precoding/decoding can be used to transmit two symbols H ρ H = max s zi, − s Γi s , = 1, 2. per transmission. For example, the equal-error precoding ∈A2 s i=1 4 i=1 scheme is specified by (8)and(9)withm = 2. The re- (37) ceived SNR of this method is given by SNRequal-error precoding = MIMO Techniques Comparisons and Application to WCDMA Systems 655

− 10−1 10 1

10−2

10−2 10−3 BER BER 10−4 10−3

10−5

− 10 6 10−4 0 2 4 6 8 10 12 0 1 2 3 4 5 6 7 SNR (dB) SNR (dB)

Beamforming: 2 trans. ant., 3 recv. ant. Equal-error precoding Alamouti: 2 trans. ant., 3 recv. ant. Whitening precoding Beamforming: 4 trans. ant., 6 recv. ant. Rate-2 STBC, BLAST-LZF STBC (ML): 4 trans. ant., 6 recv. ant. Rate-2 STBC, BLAST-ML STBC (LZF): 4 trans. ant., 6 recv. ant. Figure 3: Comparisons of the BER performances between the linear Figure 2: Comparisons of the BER performances among the precoding/decoding strategies and the rate-two STBC: nT = 4and MIMO techniques for one symbol/transmission: beamforming ver- nR = 6; the rate is two symbols/transmission. sus Alamouti with nT = 2andnR = 3; beamforming versus rate-one STBC with nT = 4andnR = 6.

out channel knowledge requirement at the transmitter. And ρ/ λ−1 λ−1 ( 1 + 2 ). The whitening precoding method, on the this phenomenon is evident especially in the high-data rate H other hand, is specified by F = ρ/2[v1v2]andG = [v1v2] ; transmission scenario, that is, BLAST versus linear precod- n = and the average received SNR of this method is given by ing/decoding schemes with T 4. This can be explained as follows. Note that, for the linear precoding/decoding strate- SNRwhitening precoding = ρ((λ1 + λ2)/4). Note that λ1 and λ2 are the two largest eigenvalues contained in Λ. gies discussed above, the adaptive modulation is not employed, and thus, the performance gain is limited for the fixed 2.2.3. Comparisons modulation. Figure 2 shows the performance comparisons among the MIMO techniques to achieve one symbol per transmission. 3. WCDMA DOWNLINK SYSTEMS Specifically, the beamforming scheme is compared with the In this section, a WCDMA downlink system based on the Alamouti code for a system with two transmit antennas, 3GPP standard, a subspace tracking algorithm, as well as a and the beamforming scheme is compared with the rate-one quantized feedback approach are specified. In Section 3.1, STBC for a system with four transmit antennas. It is observed we describe the WCDMA system, including the structures from Figure 2 that the beamforming scheme achieves about of the transmitter and the receiver, the channelization and 2 dB gain over the Alamouti code, and similarly, the beam- scrambling codes, the frame structures of the data and the forming can achieve much better performance than the rate- pilot channels, the multipath fading channel model, as well one STBC strategy. as the channel estimation algorithm. In Section 3.2,wede- Figure 3 shows the performance comparisons between tail the subspace tracking method and the quantized feed- the linear precoding/decoding schemes and the rate-two back scheme. STBC strategy for a system with nT = 4andnR = 6toachieve two symbols per transmission. It is seen from Figure 3 that 3.1. System description the rate-two STBC achieves a better performance than the linear precoding/decoding schemes, and the performance 3.1.1. Transmitter and receiver structures gap is not so large. In particular, the rate-two STBC with The system model of the downlink WCDMA system is shown BLAST-LZF decoding has an approximate performance to in Figure 4. The left part of Figure 4 is the transmitter struc- the equal-error precoding scheme. ture. The data sequences of the users are first spread by It is observed from Figures 1 and 3 that although the unique orthogonal variable spreading factor (OVSF) codes linear precoding/decoding schemes exploit the channel (Cch,SF,1, Cch,SF,2, ...), and then, the spread chip sequences knowledge at the transmitter, they may not have perfor- of different users are multiplied by downlink scrambling mance gains compared to those MIMO techniques with- codes (Ccs,1, Ccs,2, ...). After summing up the scrambled data 656 EURASIP Journal on Applied Signal Processing

Cch,SF,1 Csc,1 Cch,SF,0 Csc,0

User X X 1 Pilot X X r Sum 11 Finger r12 . Channel tracking . . estimator r Cch,SF,2 Csc,2 Sum for pilot 1L

User X X Sum 2 r11 . Pilot X X . . . Finger r . . . 12 tracking . Decoding . for data r1L Cch,SF,0 Csc,0

Figure 4: Transmitter and receiver structures of the downlink WCDMA system. sequences from different users, the data sequences are com- 3.1.2. Multipath fading channel model bined with the pilot sequence, which is also spread and and channel estimation C C scrambled by the codes ( ch,SF,0, cs,0) for the pilot chan- Each user’s channel contains four paths, that is, L = 4. The nel sent to each antenna. The specifications of OVSF and channel multipath profile is chosen according to the 3GPP scrambling codes can be referred to [15]. The right part of specifications. That is, the relative path delays are 0, 260, 521, Figure 4 shows the receiver structure of this system with one and 781 nanoseconds, and the relative path power gains are receive antenna. We assume the number of multipaths in the 0, −3, −6, and −9 dB, respectively. L WCDMA channel is . Each receive antenna is followed by There are two channels in the system, namely, common a bank of RAKE fingers. Each finger tracks the correspond- control physical channel (CCPCH) and common pilot chan- ing multipath component for the receiver antenna and per- nel (CPICH), whose rates are variable and fixed, respectively. L forms descrambling and despreading for each of the mul- For more details, see [15]. The CPICH is transmitted from all tipath components. Such a receiver structure is similar to antennas using the same channelization and the scrambling the conventional RAKE receiver but without maximal ratio code, and the different pilot symbol sequences are adopted L combining (MRC). Hence, there are outputs for each re- on different antennas. Note that in the system, the pilot sig- L ceive antenna, and thus, each of the antenna outputs can nal can be treated as the data of a special user. In other words, be viewed as a virtual receive antenna [14]. With the received the pilot and the data of different users in the system are com- pilot signals, the downlink channel is estimated accordingly. bined with code duplexing but not time duplexing. This channel estimate is provided to the detector to perform Here we use orthogonal training sequences of length T ≥ demodulation of the received users’ signals. nT based on the Hadamard matrix to minimize the estima- It is shown in [14] that the above receiver scheme with tion error [25]. Note that, although the channel varies at the virtual antennas essentially provides an interface between symbol rate, the channel estimator assumes it is fixed over at MIMO techniques and a WCDMA system. The outputs of least nT symbol intervals. the RAKE fingers are sent to a MIMO demodulator that op- erates at the symbol rate. The equivalent symbol-rate MIMO 3.2. Subspace tracking with quantized feedback channel response matrix is given by for beamforming   3.2.1. Tracking of the channel subspace h h ... h n  1,1,1 1,1,2 1,1, T    Recall that in the beamforming and general precoding trans-  . . .. .   . . . .  mission schemes, the value of the MIMO channel H has to be  h h ... h   L L L nT  provided to the transmitter. Typically, in FDD systems, this  1, ,1 1, ,2 1, ,   . . .  H =  . . .. .  , (43) can be done by feeding back to the transmitter the estimated  . . . .  ˆ h h ... h  channel value H. However, the feedback channel usually has  nR,1,1 nR,1,2 nR,1,nT    a very low data rate. Here we propose to employ a subspace  . . . .   . . .. .  tracking algorithm, namely, projection approximation sub- h h ... h nR,L,1 nR,L,2 nR,L,nT space tracking with deflation (PASTd) [20], with quantized feedback to track the MIMO eigen channels. Figure 5 shows where hi,l,j denotes the complex channel gain between the jth the diagram of the MIMO system adopting a subspace track- transmit antenna and the lth finger of the ith receive antenna. ing and the quantized feedback approach. In particular, the Hence (43) is equivalent to a MIMO system with nT transmit receiver employs the channel estimator to obtain the esti- antennas and (nR · L) receive antennas [14]. mate of the channel Hˆ and subsequently, PASTd algorithm MIMO Techniques Comparisons and Application to WCDMA Systems 657

Feedback

Weight adjustion

Pilot Tx Array Rx Array

Subspace Data tracking

Figure 5: The MIMO linear precoding/decoding system with subspace tracking and quantized feedback schemes.

100 100

− 10−1 10 1

− 10−2 10 2 BER BER

10−3 10−3

10−4 10−4 −20 −19 −18 −17 −16 −15 −14 −13 −12 −11 −10 −20 −19 −18 −17 −16 −15 −14 −13 −12 −11 −10 Ic/Ior (dB) Ic/Ior (dB)

v = 3km/h v = 30 km/h v = 3km/h v = 30 km/h v = 10 km/h v = 35 km/h v = 10 km/h v = 35 km/h v = 15 km/h v = 40 km/h v = 15 km/h v = 40 km/h v = 20 km/h v = 120 km/h v = 20 km/h v = 120 km/h v = 25 km/h v = 300 km/h v = 25 km/h v = 300 km/h

(a) (b)

Figure 6: BER performance of beamforming under diﬀerent doppler frequencies: (a) nT = 4, nR = 1 (beamforming, perfect known channel, lossless feedback (2 frames)), (b) nT = 2, nR = 1 (perfectly known channel, lossless feedback (1 frame)).

is adopted to get F = V = [V1, ..., Vm], which contains the For the beamforming scheme, we employ the feedback principal eigenvectors of Ω = HH H. approach as follows. The average eigenvector of the channel over one frame or two frames is fed back instead of the eigen- 3.2.2. Frame-based feedback vectors of each symbol or slot duration. Note that such feed- Note that, for the uplink channel in the 3GPP standard [21], back approach assumes the downlink WCDMA channel as the bit rate is 1500 bits per second (bps), the frame rate a block fading one, and actually, it is effective and efficient is 100 frames per second (fps), and thus, there are fifteen under low doppler frequencies. Figure 6 shows the BER per- bits in each uplink frame. On the other hand, the down- formances of the MIMO system employing the beamform- link WCDMA channel is a symbol-by-symbol varied chan- ing scheme under different doppler frequencies. In Figure 6b, nel. Thereby, it is necessary to consider an effective and effi- two transmit antennas are adopted, and the average eigen- cient quantization and feed back scheme, so as to feed back F vectors over one frame duration are losslessly fed back. That to the transmitter via the band-limited uplink channel. is, the eigenvector information is precisely fed back without 658 EURASIP Journal on Applied Signal Processing

Table 1: Frame structures for quantized feedback. Case 1: two transmit antennas and one receive antenna, (5, 5) quantization : 5 bits for the absolute value component and 5 bits for the phase component of each vector element; Aij: jth bit for the absolute value of ith vector element; Pij: jth bit for the phase of ith vector element. Case 2: two transmit antennas and 1 receive antenna, (4,7) quantization. Case 3: four transmit antennas and 1 receive antenna, (3,6) quantization. Case 1 Slot 123456789101112131415

Bits A11 A12 A13 A14 A15 A21 A22 A23 A24 A25 P21 P22 P23 P24 P25

Case 2 Slot 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Bits A11 A12 A13 A14 A21 A22 A23 A24 P21 P22 P23 P24 P25 P26 P27

Case 3 Slot 123456789101112131415

Bits A11 A12 A13 A21 A22 A23 P21 P22 P23 P24 P25 P26 A31 A32 A33

Slot 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Bits P31 P32 P33 P34 P35 P36 A41 A42 A43 P41 P42 P43 P44 P45 P46

quantization. It is seen that the system achieves a good per- over, our simulations show that the (5, 5) and (4, 7) quan- formance for the speeds lower than 30 km/h, and the BER tized feedback approaches actually have very approximated curves are shown as “floors” when v is higher than 30 km/h. performances. The appearance of such “floor” is due to the severe mismatch between the precoding and the downlink channel. Similarly, Figure 6a gives the BER performances of the system employ- 4. SIMULATION RESULTS FOR WCDMA SYSTEMS ing the beamforming with four transmit antennas, where the In the simulations, we adopt one receive antenna (nR = 1), average eigenvectors over two frames are losslessly fed back. which is a realistic scenario for the WCDMA downlink re- It is seen that the BER performances degrade to “floors” for ceiver. For the multipath fading channel in the WCDMA sys- the speeds higher than 15 km/h. It is observed from (6) that tem, the number of multipath is assumed to be four (L = the frame-based feedback approach is feasible for the beam- 4), and the mobile speed is assumed to be three kilome- forming system under the low-speed cases. In particular, it is ters per hour (v = 3 km/h). QPSK is used as the modula- feasible for the system employing two transmit antennas and tion format. The performance metric is BER versus signal- four transmit antennas, under the cases of v ≤ 25 km/h and Ic/Ior Ic/Ior v ≤ to-interference-ratio ( ). is the power ratio between 10 km/h, respectively. the signal of the desired user and the interference from all 3.2.3. Quantization of the feedback other simultaneous users in the WCDMA system. Subse- quently, several cases with different transmission rates over Table 1 shows the feedback frame structures for the MIMO two and four transmit antennas are studied. system employing beamforming schemes, that is, the quantization of the elements of the eigenvector to be fed back. We consider three cases here. Case 1 and Case 2 are con- BLAST versus linear precoding trived for the beamforming system with two transmit an- Figure 7 shows the performance comparisons between the tennas. These two bit allocation strategies of one feedback BLAST and the linear precoding/decoding schemes for a rate frame are, namely, (5, 5) and (4, 7) quantized feedback, re- of four symbols per transmission over four transmit anten- spectively. In particular, (5, 5) quantized feedback allocates nas (nT = 2). In particular, the channel estimator given in 5 bits each to the absolute value and the phase component Section 3.1.2 is adopted to acquire the channel knowledge. of one eigenvector element; and (4, 7) quantized feedback al- For the linear precoding/decoding schemes, lossless feedback locates 4 bits and 7 bits to the absolute value and the phase is assumed. It is seen from Figure 7 that the BLAST scheme component of one eigenvector element, respectively. Case 3, with ML detection achieves the best BER performance over namely, (3, 6) quantized feedback, is contrived for the beam- all linear precoding/decoding schemes. Note that the reason forming system with four transmit antennas. Two feedback that precoding does not offer performance advantage here frames are allocated for the average eigenvector over two is that we require the rate for different eigen channels to be frames. Note that relatively more bits should be allocated to the same, that is, no adaptive modulation scheme is allowed. the phase component, since the error caused by quantiza- Hence we conclude that to achieve high throughput, it suf- tion is more sensitive to the preciseness of the phase com- fices to employ the BLAST architecture and the knowledge of ponents than that of the absolute value components more- the channel at the transmitter offers no advantage. MIMO Techniques Comparisons and Application to WCDMA Systems 659

10−1

10−2

BER 10−1 BER

10−3

−20 −15 −10 −5 −20 −19 −18 −17 −16 −15 −14 −13 −12 −11 −10 Ic/Ior (dB) Ic/Ior (dB)

BLAST ML Equal-error precoding BF, perfectly known channel BLAST LZF Whitening precoding BF, perfectly known channel, lossless feedback BLAST ZF BF, perfectly known channel, quantized feedback BF, subspace tracking, quantized feedback Figure 7: BER comparisons between the BLAST and the trans- Alamouti STBC, estimated channel mit precoding schemes: nT = 4andnR = 1; four QPSK symbols/transmission; v = 3km/h, L = 4. Figure 8: BER comparisons between Alamouti and beamforming with subspace tracking and quantized feedback schemes: nT = 2 and nR = 1; one QPSK symbol/transmission; v = 3km/h;(4,7) STBC versus beamforming quantized feedback.

Figure 8 gives the performance comparisons between the 0 Alamouti STBC and the beamforming schemes for a rate 10 of one symbol per transmission over two transmit antennas (nT = 2). The eﬀects of the quantized feedback approach is also shown in Figure 8. In particular, the cycled line is the 10−1 BER performance when perfect channel knowledge is available at both the transmitter and the receiver. The solid line is the performance when perfect channel knowledge is avail- 10−2 able at the receiver and the frame-based feedback without BER quantization in Section 3.2.2 is adopted. It is seen that the frame-based feedback approach only causes very trivial per- − formance degradation. The asteriated line is the performance 10 3 when perfect channel knowledge is available at the receiver and the frame-based feedback with (4, 7) quantized feedback approach in Section 3.2.3 is adopted. It is seen that the quan- 10−4 tization of the feedback only generates about 0.5dBperfor- −20 −19 −18 −17 −16 −15 −14 −13 −12 −11 −10 I /I mance loss. Moreover, the triangled line is the performance c or (dB) when the channel estimator in Section 3.1.2, the subspace BF, known channel tracking in Section 3.2.1, and the (4, 7) quantized feedback BF, known channel, lossless feedback approach are adopted. It is shown that the subspace track- BF, known channel, quantized feedback ing and the channel estimation cause about 1 to 1.5dBper- BF, PASTd, quantized feedback Rate-1 STBC, (ML) formance degradation. Finally, the squared line is the perfor- Rate-1 STBC, (LZF) mance of the Alamouti STBC, where the channel estimator is adopted at the receiver. It is observed from Figure 8 that the Figure 9: BER comparisons between rate-one STBC and beam- WCDMA system employing beamforming can have a bet- forming with subspace tracking and quantized feedback schemes: ter performance than that employing the Alamouti STBC nT = 4andnR = 1; one QPSK symbol/transmission; v = 3km/h; scheme, though the performance gain is not very evident. (3, 6) quantized feedback. Figure 9 gives the comparison between the beamforming scheme and the rate-one STBC strategy discussed in Section 2.2.2 for a rate of one symbol per transmission over feedback, and quantized feedback cases are shown. From four transmit antennas (nT = 4). Similarly, perfectly known bottom up, the ﬁrst curve is the result of the beamform- channel knowledge, estimated channel knowledge, lossless ing scheme with perfectly known channel knowledge at both 660 EURASIP Journal on Applied Signal Processing

Table 2: Summary of the performance comparisons of the MIMO techniques. (a)

High-rate transmission MIMO techniques Channel information BER performance BLAST Receiver Better Four symbols/transmission over nT = 4 Transmit precoding Transmitter/receiver Worse

(b)

Diversity transmission MIMO techniques channel Information BER performance Beamforming Transmitter/receiver Better One symbol/transmission over nT = 2 Alamouti Receiver Worse One symbol/transmission Beamforming Transmitter/receiver Better over nT = 4 Rate-one STBC Receiver Worse Two symbols/transmission Transmit precoding Transmitter/receiver Worse over nT = 4 Rate-two STBC Receiver Better

the transmitter and the receiver; the second curve is the re- ing/decoding schemes without employing adaptive modula- sult of the beamforming scheme with perfectly known chan- tion. On the other hand, the beamforming scheme achieves nel knowledge at the receiver and the frame-based feedback better performances than the STBC schemes in the diversity without quantization; the third curve is the result of the transmission scenario. Table 2 gives a summary of the per- beamforming scheme with perfectly known channel knowl- formance comparisons of the MIMO techniques in different edge at the receiver and the frame-based feedback with (3, 6) scenarios. Moreover, it is well confirmed the effectiveness and quantization; the fourth curve is the result of channel esti- feasibility of the combination of the subspace tracking algo- mator, subspace tracker, and the frame-based feedback with rithm and the quantized feedback approach for beamform- (3, 6) quantization; the top two curves are the results of the ing transmission in the MIMO WCDMA system. Finally, we rate-one STBC scheme with different detection methods. It is note that in this paper, we only consider the linear precoding observed from Figure 9 that the beamforming can achieve a scheme. Significant performance improvement is expected much better performance than the STBC for the case of four when nonlinear precoder (e.g., adaptive modulation and bit transmit antennas. loading) is employed [26, 27, 28]. Moreover, it is also well confirmed that the subspace tracking algorithm discussed in Section 3.2.1, the frame- ACKNOWLEDGMENT based feedback in Section 3.2.2, as well as the quantization approach discussed in Section 3.2.3 offer a practical way of This work was supported in part by the U.S. National Science realizing beamforming in MIMO WCDMA systems. Foundation (NSF) under Grants CCR-0225721 and CCR- 0225826. 5. CONCLUSIONS REFERENCES In this paper, we have analyzed and compared the perfor- [1] C.-N. Chuah, D. N. C. Tse, J. M. Kahn, and R. A. Valenzuela, mance of three MIMO techniques, namely, BLAST, STBC “Capacity scaling in MIMO wireless systems under correlated and linear precoding/decoding, and considered their appli- fading,” IEEE Transactions on Information Theory, vol. 48, no. cations in WCDMA downlink systems. For a certain trans- 3, pp. 637–650, 2002. mission rate, we compared the different scenarios with dif- [2] G. J. Foschini and M. J. Gans, “On limits of wireless com- ferent transmit antennas both analytically in terms of the av- munication in a fading environment when using multiple antennas,” Wireless Personal Communications,vol.6,no.3,pp. erage receiver SNR, as well as through simulations in terms 311–355, 1998. of the BER performance. To cope with the channel feedback [3] D.-S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, “Fad- in WCDMA systems for beamforming, we adopted a sub- ing correlation and its effect on the capacity of multielement space tracking method with a quantized feedback approach antenna systems,” IEEE Trans. Communications,vol.48,no.3, to make the principle eigenspace of the MIMO channel avail- pp. 502–513, 2000. able to the transmitter. [4] O. Damen, A. Chkeif, and J.-C. Belfiore, “Lattice code decoder Some instructive conclusions are drawn in this study. On for space-time codes,” IEEE Communications Letters, vol. 4, no. 5, pp. 161–163, 2000. the one hand, the optimal BLAST scheme can achieve the [5] G. J. Foschini, “Layered space-time architecture for wireless best performance in the high-rate transmission scenario, al- communication in a fading environment when using multi- though with channel knowledge available at the transmit- element antennas,” Bell Labs Tech. Journal,vol.1,no.2,pp. ters, no performance gain is achievable by the linear precod- 41–59, 1996. MIMO Techniques Comparisons and Application to WCDMA Systems 661

[6] G. D. Golden, C. J. Foschini, R. A. Valenzuela, and P. W. time spreading,” IEEE Journal on Selected Areas in Communi- Wolniansky, “Detection algorithm and initial laboratory re- cations, vol. 19, no. 1, pp. 48–60, 2001. sults using V-BLAST space-time communication architec- [24] C. B. Papadias and G. J. Foschini, “A space-time coding ture,” Electronics Letters, vol. 35, no. 1, pp. 14–16, 1999. approach for systems employing four transmit antennas,” [7] H. Sampath, P. Stoica, and A. Paulraj, “Generalized linear in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing precoder and decoder design for MIMO channels using the (ICASSP ’01), pp. 2481–2484, Salt Lake City, Utah, USA, May weighted MMSE criterion,” IEEE Transactions on Communi- 2001. cations, vol. 49, no. 12, pp. 2198–2206, 2001. [25] T. L. Marzetta, “Blast training: estimating channel char- [8] S. M. Alamouti, “A simple transmit diversity technique for acteristics for high capacity space-time wireless,” in Proc. wireless communications,” IEEE Journal on Selected Areas in 37th Annual Allerton Conference on Communication, Control, Communications, vol. 16, no. 8, pp. 1451–1458, 1998. and Computing, pp. 958–966, Monticello, Ill, USA, September [9] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time 1999. block codes from orthogonal designs,” IEEE Transactions on [26] P. S. Chow, J. M. Cioffi, and J. A. C. Bingham, “A practical dis- Information Theory, vol. 45, no. 5, pp. 1456–1467, 1999. crete multitone transceiver loading algorithm for data trans- [10] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time mission over spectrally shaped channels,” IEEE Trans. Com- block coding for wireless communications: performance re- munications, vol. 43, no. 2, pp. 773–775, 1995. sults,” IEEE Journal on Selected Areas in Communications, vol. [27] R. F. H. Fischer and J. B. Huber, “A new loading algorithm 17, no. 3, pp. 451–460, 1999. for discrete multitone transmission,” in Proc. IEEE Global [11]Z.Hong,K.Liu,R.W.Heath,andA.M.SayeedJr.,“Spa- Telecommunications Conference (GLOBECOM ’96), pp. 724– tial multiplexing in correlated fading via the virtual channel 728, London, England, November 1996. representation,” IEEE Journal on Selected Areas in Communi- [28] B. S. Krongold, K. Ramchandran, and D. L. Jones, “Com- cations, vol. 21, no. 5, pp. 856–866, 2003. putationally efficient optimal power allocation algorithms for [12] L. Zheng and D. N. C. Tse, “Diversity and multiplexing: a fun- multicarrier communication systems,” IEEE Trans. Commu- damental tradeoff in multiple-antenna channels,” IEEE Trans- nications, vol. 48, no. 1, pp. 23–27, 2000. actions on Information Theory, vol. 49, no. 5, pp. 1073–1096, 2003. [13] K. Majonen and M. J. Heikkila, “Higher data rates with space- Chuxiang Li received the B.S. and M.S. de- time block codes and puncturing in WCDMA systems,” in grees from the Department of Electronics Proc. IEEE International Symposium on Personal, Indoor, and Engineering, Tsinghua University, Beijing, Mobile Radio Communications (PIMRC ’01), pp. 36–40, San China, in 1999 and 2002, respectively. He Diego, Calif, USA, October 2001. is currently working toward the Ph.D. de- [14] D. Samardzija, P. Wolniansky, and J. Ling, “Performance eval- gree in the Department of Electrical En- uation of the VBLAST algorithm in W-CDMA systems,” in gineering, Columbia University, New York, IEEE Vehicular Technology Conference (VTC ’01), vol. 2, pp. NY. His research interests fall in the area of 723–727, Atlantic City, NJ, USA, October 2001. wireless communications and statistical sig- [15] Texas Instruments, “Space-time block coded transmit an- nal processing. tenna diversity for WCDMA,” proposal TDOC# 662/98 to ETSI SMG2 UMTS standards, December 1998. [16] Lucent Technologies, “Downlink diversity improvements Xiaodong Wang received the B.S. degree through space-time spreading,” proposal 3GPP2-C30- in electrical engineering and applied math- 19990817-014 to the CDMA-2000 standard, August 1999. ematics (with the highest honor) from [17] B. C. Banister and J. R. Zeidler, “Tracking performance of a Shanghai Jiao Tong University, Shanghai, gradient sign algorithm for transmit antenna adaptation with China, in 1992; the M.S. degree in electri- feedback,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal cal and computer engineering from Purdue Processing (ICASSP ’01), Salt Lake City, Utah, USA, May 2001. University in 1995; and the Ph.D. degree in [18] B. C. Banister and J. R. Zeidler, “Transmission subspace track- electrical engineering from Princeton Uni- ing for MIMO communications systems,” in Proc. IEEE Global versity in 1998. From July 1998 to Decem- Telecommunications Conference (GLOBECOM ’01), vol. 1, pp. ber 2001, he was an Assistant Professor in 161–165, San Antonio, Tex, USA, November 2001. the Department of Electrical Engineering, Texas A&M University. [19] W. Utschick, “Tracking of signal subspace projectors,” IEEE In January 2002, he joined the Department of Electrical Engineer- Transactions on Signal Processing, vol. 50, no. 4, pp. 769–778, ing, Columbia University, as an Assistant Professor. Dr. Wang’s re- 2002. search interests fall in the general areas of computing, signal pro- [20] X. Wang and H. V. Poor, “Blind multiuser detection: a sub- cessing, and communications. He has worked in the areas of dig- space approach,” IEEE Transactions on Information Theory, ital communications, digital signal processing, parallel and dis- vol. 44, no. 2, pp. 677–690, 1998. tributed computing, nanoelectronics, and bioinformatics, and has [21] Third Generation Partnership Project, “Tx diversity solutions published extensively in these areas. His current research inter- for multiple antennas (release 5),” proposal 3G TR25.869 V0.1.01 to Technical Specification Group Radio Access Net- ests include wireless communications, Monte-Carlo-based statisti- work, November 2001. cal signal processing, and genomic signal processing. Dr. Wang re- [22] G. J. Foschini, G. D. Golden, R. A. Valenzuela, and P. W. ceived the 1999 National Science Foundation (NSF) Career Award, Wolniansky, “Simplified processing for high spectral effi- and the 2001 IEEE Communications Society and Information The- ciency wireless communication employing multi-element ar- ory Society Joint Paper Award. He currently serves as an Associate rays,” IEEE Journal on Selected Areas in Communications, vol. Editor for the IEEE Transactions on Communications, the IEEE 17, no. 11, pp. 1841–1852, 1999. Transactions on Wireless Communications, the IEEE Transactions [23] B. Hochwald, T. L. Marzetta, and C. B. Papadias, “A trans- on Signal Processing, and the IEEE Transactions on Information mitter diversity scheme for wideband CDMA based on space- Theory.