Analysis of Space–Time Coding in Correlated Fading Channels Ahmadreza Hedayat, Student Member, IEEE, Harsh Shah, and Aria Nosratinia, Senior Member, IEEE

2882 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 6, NOVEMBER 2005 Analysis of Space–Time Coding in Correlated Fading Channels Ahmadreza Hedayat, Student Member, IEEE, Harsh Shah, and Aria Nosratinia, Senior Member, IEEE

Abstract—Antenna spacing and the properties of a scattering ces. Their analysis yields expressions that are equivalent to, environment can create correlation between channel coefficients. but different from, the ones we provide in this paper in the Temporal correlation between fading coefficients may also be special case of the quasi-static channel. In another work [7], present, because one may be unable or unwilling to fully interleave the channel symbols. This paper presents a comprehensive analy- Bölcskei et al. studied the performance of space–frequency sis of multiple-input multiple-output systems under correlated coded MIMO-orthogonal frequency-division multiplexing fading. We calculate pairwise-error-probability (PEP) expressions (OFDM) for frequency-selective Rician channels. Uysal and under quasi-static fading, fast fading, block fading, as well as ar- Georghiades [8] derived PEP expressions for transmit-antenna bitrarily temporally correlated fading, under Rayleigh and Rician correlation (with no receive correlation or temporal correla- conditions. We use the PEP expressions to calculate union bounds on the performance of trellis space–time codes, super orthogonal tion). Dogandzic´ [9] gives PEP expressions with spatial cor- space–time codes, linear-dispersion codes, and diagonal algebraic relation, but does not consider temporal correlation. space–time codes. This work augments the existing literature in the following Index Terms—Correlated channels, diversity, fading channels, ways. First, instead of focusing on Chernoff bounds, we de- MIMO signaling, pairwise error probability, space–time codes, rive exact expressions in a relatively straightforward manner. uniform error property. The exact expressions can replace Monte Carlo simulations, and thus, are helpful in large-scale simulations of wireless I. INTRODUCTION networks. Also, our analysis covers a wider range of channel conditions (e.g., joint spatio-temporal correlation), as well as N A multiple-antenna system, insufficient antenna spacing, application to a number of MIMO signaling schemes. We also I angle spread, or the lack of rich scattering may cause spatial explore the uniform error property (UEP) of codes in correlated correlation between antennas. Fading coefficients may also ex- channels and show that UEP may be lost if there is transmit-side hibit temporal correlations. This paper studies the performance antenna correlation. of multiple-input multiple-output (MIMO) radio systems in Throughout this paper, we use the following notation. Ma- correlated fading channels. We begin by calculating pairwise- trices are shown in boldface uppercase letters, e.g., A, and error-probability (PEP) expressions in a variety of scenarios, vectors are shown in boldface lowercase, e.g., x.TheHer- including quasi-static fading, fast fading, block-fading, general mitian, the determinant, and the Frobenius norm of A are Rayleigh fading, and Rician fading. We consider spatially denoted by AH, |A|, and A, respectively. The vectorizing correlated fading on the transmit or receive side (or both), as operator vec(A) stacks the columns of the matrix A in a well as combined spatially and temporally correlated fading. column vector. The Kronecker product of A and B is shown We then use the PEPs to develop union bounds for a variety of as A ⊗ B. MIMO signaling scenarios. In particular, we provide bounds for trellis space–time codes [1], superorthogonal space–time II. SYSTEM MODEL (SOSTT) codes [2], linear-dispersion (LD) codes [3], and diagonal algebraic space–time (DAST) codes [4]. We consider an MIMO system with nT transmit and nR A brief history of past work in this area is as follows. receive antennas. The binary data are encoded and modulated (n) ··· (n) T Damen et al. [5] studied, via simulations, the effect of para- by a space–time encoder. Let xn =[x1 xnT ] denote meters such as angle spread and Rician factor on the perfor- the transmitted signal vector in the nth time interval. Let mance of various space–time signaling and detection schemes. { (n)} × Hn = hji be the nR nT channel matrix at time n, where Bölcskei and Paulraj [6] studied the quasi-static channel and each entry hji is the channel gain between transmit antenna i found Chernoff bounds that in the high-SNR regime link the and receive antenna j. The channel gains are assumed to be diversity to the rank and eigenvalues of the correlation matri- circularly symmetric complex Gaussian random variables, so their magnitude exhibits a Rayleigh distribution.√ The received- signal vector at time n is given by rn = EsHnxn + zn, Manuscript received October 29, 2003; revised October 26, 2004; accepted where z is an nR × 1 independent identically distributed (i.i.d.) November 1, 2004.The editor coordinating tha review of this paper and ap- E 2 proving it for publication is M. Uysal. This paper was supported in part by the zero-mean Gaussian noise vector and Es = ( x )/nT is National Science Foundation under Grant CCR-9985171. the symbol energy. We assume that the receiver has perfect The authors are with the Department of Electrical Engineering, University of knowledge of the channel matrix and performs coherent de- Texas at Dallas, Richardson, TX 75083 USA (e-mail: [email protected]; [email protected]; [email protected]). tection. The maximum-likelihood√ (ML) metric is given by N − 2 Digital Object Identifier 10.1109/TWC.2005.858338 m(r, x)= n=1 rn EsHnxn , where N is the total

1536-1276/$20.00 © 2005 IEEE HEDAYAT et al.: ANALYSIS OF SPACE–TIME CODING IN CORRELATED FADING CHANNELS 2883 frame (codeword) length, r is the received signal in this period, analytical tools for the calculation of PEP under correlated and x is a given codeword in the same period. channel conditions. We ﬁrst consider spatially correlated antennas. Correlations can be modeled with an i.i.d. “innovations” channel H˜ in III. PERFORMANCE ANALYSIS combination with a linear system. One can think of this as The overall error rates can be approximated via the union a diagonalization of the correlated channel. The most general bound, e.g., for the frame error rate form of such a correlation model is constructed using the vectorizing operator vec(·) as follows: Pe ≤ P (x) P(x → xˆ) (1) 1 vec(H)=R 2 vec(H˜ ) x xˆ= x where R is the spatial correlation matrix. A slightly less gen- where P (x) is the probability of codeword x (assumed uni- eral, but more useful, model considers correlations on transmit form) and P (x → xˆ) is the so-called PEP, i.e., the probability and receive sides separately. Representing the transmit-side that a codeword x is decoded as xˆ. In this section, we concen- correlation matrix as R and receive-side correlation matrix trate on the development of PEP expressions under a variety of Tx as R , the correlated channel is represented as [15] correlated channel conditions. Rx

Using the methods originated by Craig [10] and widely 1 T 2 ˜ 2 applied by Simon and Alouini [11], we first calculate the H = RRxHRTx (4) conditional PEP and then integrate over channel√ gain co- → − where it can be verified that R = RTx ⊗ RRx. We use this efficients.√ An error x xˆ occurs when r EsHx > − latter spatial correlation model in our analyses. r EsHx√ˆ , or equivalently, the noise vector z must be ∗ ∗ 2 such that { EszH (xˆ − x) } >Es/2H(x − xˆ) . Denot- − ing the codeword differences by en = xn xˆn, the PEP for a A. Quasi-Static Fading With Spatial Correlation given channel realization is [11] In quasi-static fading channels, the matrix channel is as-   N sumed to be constant over the duration of a codeword, hence Es P (x → xˆ|H)=Q  H e 2 (2) Hn = H for n =1,...,N. Denote the difference of two code- 2N n n ··· 0 n=1 word matrices with ∆ =[e1 eN ]. Then, the argument of the MGF in (3) can be described in terms of RRx and RTx where N is the one-sided noise-power spectral density. Now, 0 N use the alternative integral expression of Q-function [10], [11] 2 Hnen

π n=1 2 1 x2 = H∆2 Q(x)= exp − dθ π 2 2sin θ =tr(H∆∆HHH) 0 = vec(HH)H I ⊗ ∆∆H vec(HH) and subsequently integrate over the randomness of the channel nR H 1 coefficients to obtain the unconditional PEP ˜ H H 2 ⊗ H 2 ˜ H = vec(H ) Rs InR ∆∆ Rs vec(H ) (5) π 2 N H − where Rs = RRx ⊗ RTx is the covariance matrix of vec(H ) → | 1 Es 2 P (x xˆ H)= exp 2 Hnen dθ (note the difference with the definition of R above). We wish to π 4N0 sin θ 0 n=1 calculate the MGF of the above random variable, which consists π of a positive semidefinite quadratic form involving Gaussian 2 1 E vectors vec(H˜ H). P (x → xˆ)= Φ − s dθ (3) 2 Fact [16]: A u π 4N0 sin θ Let be a Hermitian matrix and acir- 0 cularly symmetric complex Gaussian vector with mean u¯ and covariance matrix R . The MGF of the quadratic form · u where Φ( ) is the moment-generating function (MGF) of the y = uAuH is N 2 random variable n=1 Hnen . We note that alternative ∞ methods for calculating the PEP using MGF do exist, notably − −1 H sy exp suA¯ (I sRuA) u¯ the residue method [12]–[14], but Craig’s technique is easier Φy(s)= e pY (y)dy = |I − sRuA| and more versatile. 0 When channel coefficients are i.i.d., the MGF can be de- (6) composed into a product of marginal MGFs and (3) is easily calculated. In the presence of spatial or temporal correlation, where I is the identity matrix with appropriate size. however, the MGF cannot be decomposed, and thus, complica- Clearly u = vec(H˜ H)H is a zero-mean Gaussian vector with H/2 ⊗ H 1/2 tions arise. The remainder of this section is devoted to providing covariance matrix Ru = InTnR and Rs (InR ∆∆ )Rs 2884 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 6, NOVEMBER 2005 is a Hermitian matrix, hence, using the properties of the (innovations) channel and a diagonalizing argument to arrive at Kronecker product [17] a useful expression for the PEP as follows:

− − H 1 H 1 1 Φ (s)= I − s I ⊗ e e R − 2 ⊗ H 2 n nRnT nR n n s Φ(s)= InRnT sRs InR ∆∆ Rs (7) − H 1 − = I − sR ⊗ e e R − ⊗ H 1 nRnT Rx n n Tx = InRnT s InR ∆∆ Rs nR −1 −1 − ⊗ H ⊗ = (1 − sλjµn) . (13) = InRnT s InR ∆∆ (RRx RTx) (8) j=1 − − ⊗ H 1 = InRnT sRRx ∆∆ RTx The nonzero eigenvalue of e eHR is denoted µ , and λ are nT nR n n Tx n j −1 the eigenvalues of R = (1 − sλjµi) (9) Rx i=1 j=1 π 2 N nR −1 1 Esλjµn where λ are the eigenvalues of R and µ are the eigenvalues P (x → xˆ)= 1+ dθ. (14) j Rx i π 4N sin2 θ H n=1 j=1 0 of ∆∆ RTx. Therefore, the PEP in the quasi-static case is 0

π Unlike the quasi-static case, this equation shows time diversity 2 n n − 1 T R E 1 (when ∆ is nonzero). Interestingly, the overall diversity de- P (x → xˆ)= 1+ s λ µ dθ 2 j i pends only on the rank of R , and not that of R . This is due π 4N0 sin θ Rx Tx 0 i=1 j=1 to the fact that the rank of e eH is one, and hence, e eHR n n n n Tx nT nR −1 has only one nonzero eigenvalue. In fact, we will see that to Es ≤ 1+ λjµi . (10) exploit full antenna diversity, the channel must remain fixed 4N0 i=1 j=1 over nT symbol durations. Finally, the coding gain depends on the eigenvalues of both RRx and RTx. It is possible to obtain a Equation (10) is the Chernoff bound for the PEP. The special coding-gain expression in this case in a manner similar to [6]. case of the above analysis for the i.i.d. channel has been It is insightful to evaluate (14) for the special case where the reported in [8] and [18]. Also, Bölcskei and Paulraj [6] offer antennas are uncorrelated. In this case, the nonzero eigenvalue H nT | (n) − (n)|2 an equivalent expression for this Chernoff bound. Liu and of enen is equal to i=1 xi xî and we have Sayeed [19] have reported the Chernoff bound corresponding −n to (8). nT R 2 − (n) − (n) For the high-SNR approximation of the Chernoff bound, Φn(s)= 1 s xi xî . substitute (9) in (3) to get i=1

2 − Substituting in (12) and setting s = −E /4N sin θ yields a 1 E rrˆ r rˆ 1 s 0 P (x → xˆ) ≤ s (11) result similar to the one reported in [18]. 2 4N λ µ 0 i=1 j=1 j i

H C. Block Fading With Spatial Correlation where r = rank(∆∆ RTx) and rˆ = rank(RRx). In the high-SNR regime, the quality of a code is usually The block fading model is a useful approximation for a time- analyzed via the diversity order and the coding gain [1]. The varying fading channel when fading is not fast enough to be diversity order of a pair of codewords is the exponent of SNR, represented with a temporally i.i.d. process, but also not slow H i.e., rrˆ = rank(∆∆ RTx) · rank(RRx). enough to be well approximated with a quasi-static model. The block fading model has been widely used in the literature. In this model, time is divided into intervals of M symbol trans- B. Fast Fading With Spatial Correlation missions. Fading remains constant over each interval, and is statistically independent from one interval to another. Denoting We consider the case where Hn are independent across n, the length of a codeword with N, we assume for simplicity but the entries of each matrix Hn are correlated. In this case, the unconditional PEP can be calculated as follows: that a codeword length consists of an integer number of block- fading intervals, i.e., N = KM.

π Using a combination of the methods already described for 2 1 N E fast and quasi-static cases, the PEP under block fading can be → ˆ − s P (x x)= Φn 2 dθ (12) calculated as π 4N0 sin θ 0 n=1 π 2 K nT nR −1 · 2 → 1 Esλjµk,i where Φn( ) is the MGF of the random variable Hnen . P (x xˆ)= 1+ 2 dθ (15) π 4N0 sin θ Similar to the quasi-static case, we use the concept of a virtual 0 k=1 i=1 j=1 HEDAYAT et al.: ANALYSIS OF SPACE–TIME CODING IN CORRELATED FADING CHANNELS 2885

where ∆k =[ek,1 ··· ek,M ], and µk,i are the eigenvalues of corresponding to this PEP is H ∆k∆k RTx. We notice that the time diversity in this case is at most K. The rank of RRx affects the diversity directly. But −rrˆ r rˆ −1 H ≥ 1 Es 1 the rank of ∆k∆k RTx is equal to nT only if block size M P (x → xˆ) ≤ (18) 2 4N λ µ nT. Thus, in order to achieve full transmit-antenna diversity, 0 i=1 j=1 j i the channel must be constant for at least nT symbol durations. Both RRx and RTx affect the coding gain. H where the µi are the nonzero eigenvalues of ΛΛ (Rt ⊗ RTx). The high-SNR bounds can be used, as usual, for deriving design D. Joint Spatio-Temporal Correlation criteria. In the previous sections, the channel was modeled either as fast fading (interleaved) or quasi-static. In many cases, due to E. Performance Under Rician Fading the size of the codewords or other constraints, one may not be able to use interleaving. In the absence of interleaving, block The presence of a line-of-sight component creates a nonzero fading is only an approximation to a slowly fading channel. A mean component for the complex Gaussian channel, leading to more accurate model of channel variations must describe gen- Rician fading. We denote the mean value of the matrix channel eral sample-to-sample correlations. In this section, we consider as H¯ . The matrix channel can be represented as H¯ + H, where this general case. H is a circularly symmetric complex Gaussian matrix. The For calculating PEP, we are interested in codeword coordi- innovations of the channel gain coefficients is calculated in the nates where x and xˆ are not identical. Assume that among the same manner as before to give RH˜ =(1/1+K)InTnR , where N time indices, the two codewords x and xˆ are different in the K is the Rician factor. time instances {k1,...,kd}. Let the channel matrix at time ki Recall that the key to the developments for the Rayleigh case be denoted as Hi, and define H =[H1 ··· Hd]. Each Hi may was an expression for the MGF of a quadratic form, namely be spatially correlated with vec(Hi) modeled by a matrix Rs = (6). The calculations in the Rician case follow the same general RTx ⊗ RRx. We assume the statistics to be stationary (time procedure, except that the numerator of the MGF expression, invariant), therefore only one spatial correlation matrix suffices. which simplified to unity in the Rayleigh case, will be more − ⊕···⊕ Let ei = xki xˆki and let Λ = e1 ed, according to complicated in the Rician channel. For the sake of brevity, in the notation of [17], i.e., Λ is the block diagonal matrix whose the following we omit the details and give only the final PEP diagonal elements are vectors ei. The PEP for a given channel expressions. realization is H/2 ⊗ H For the quasi-static case, define Γ = Rs (InR ∆∆ ) π 1/2 2 Rs . Then, the PEP is as represented by (19) found at the 1 Es bottom of the page. P (x → xˆ|H)= exp − HΛ2 dθ π 4N sin2 θ H/2 ⊗ H 1/2 0 For the fast-fading case, let Γn = Rs (InR enen )Rs . 0 In this case, the PEP as represented by (20), found at the bottom thus, the unconditional PEP depends on the MGF of the random of the next page. variable HΛ2. Assume that the temporal correlation of the One may verify the validity of (20) in the special case of ⊗ H channel is modeled by Rt, that is, Rt(i, j) is the correlation an uncorrelated channel, by setting Γn = InR enen , which of the channel between two time instances i and j.Aftersome leads to a result that has been reported in [18]. algebra, H can be described as follows: For the spatially and temporally correlated Rician channel, using existing definitions, the PEP expression is given by (21) 1 T H 2 H˜ ⊗ 2 and (22), found at the bottom of the next page, where Γ = = RRx (Rt RTx) (16) H/2 ⊗ H 1/2 ⊗ ⊗ R (InR ΛΛ )R , and R = RRx Rt RTx. where H˜ is a (nTnR) × d matrix with i.i.d. elements. In the high-SNR regime, (22) can be written as H 1/2 H Note that vec(H )=R vec(H˜ ), where R = RRx ⊗ Rt ⊗ RTx. Using arguments similar to those in Section III-A −rrˆ → ≤ 1 Es − P (x xˆ) (23) − ⊗ H ⊗ 1 2 4N0(1 + K) Φ(s)= IdnTnR sRRx ΛΛ (Rt RTx) . (17) r rˆ 1 We can substitute this expression in (3) to obtain the appro- exp −(1 + K)H¯ 2 (24) λ µ priate PEP. High-SNR approximations of the Chernoff bound i=1 j=1 j i

−1 π E H E 2 − s ¯ H s ¯ H exp 4N sin2 θ vec(H ) Γ InRnT + 4(1+K)N sin2 θ Γ vec(H ) 1 0 0 P (x → xˆ)= dθ (19) π Es In n + 2 Γ 0 R T 4(1+K)N0 sin θ 2886 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 6, NOVEMBER 2005

where λj are the eigenvalues of RRx, and µi are the eigenvalues H 2 of ΛΛ RTx. It can be shown that H¯ = dnTnRK/(1 + K). Thus, in high SNR, the line-of-sight component offers a ﬁxed coding gain, but no diversity advantage. Note that the result of [7], when specialized to a frequency-ﬂat case, is similar to (24).

F. Insights and Design Issues In the high-SNR regime, the quality of a code is usually analyzed via the diversity order and the coding gain [1]. The diversity order of a pair of codewords is the exponent of H SNR, i.e., rrˆ = rank(ΛΛ (Rt ⊗ RTx)) · rank(RRx) in (24). Thus, there is loss in diversity if the receive-side correlation is rank deﬁcient. The transmit-side correlation appears via H r ≤ r˜, where r˜ = min(rank(ΛΛ ), rank(Rt ⊗ RTx)).IfRTx is full rank, then for the quasi-static channel r ≤ min(d, nT), and for the fast-fading channel r ≤ d, where d = rank(ΛΛH). | ⊗ H ⊗ ⊗ Fig. 1. STT code, 2-Tx, 2-Rx, fast Rayleigh fading, ρt = ρr =0.7. The coding gain depends upon (InR ΛΛ )(RRx Rt H n RTx)|= |ΛΛ | R |RRx ⊗ Rt ⊗ RTx|. Thus, the code-design criteria can be formalized as follows. as reference. However, in many cases, the symmetries in the 1) Diversity advantage: Maximize the minimum of r˜ over all code (and its corresponding trellis) leads to a condition where the error event probabilities as well as their multiplicities are codeword pairs. If RTx and Rt are full rank, maximize the minimum of rank(ΛΛH)=d. That is, the minimum independent of the transmitted codeword. This class, known as symbolwise Hamming distance over all pairs of code- UEP codes [11], includes many of the interesting and practical words should be maximized. scenarios. The past analyses have mostly relied on this property. 2) Coding advantage: Maximize the minimum |ΛΛH| over Interestingly, we found that transmit-side correlation may codeword pairs. destroy the UEP property of a code. For example, consider the eight-state QPSK space–time trellis (STT) code of [1], which is For full-rank RTx and Rt, irrespective of the channel correlation, these design criteria are the same as in [1]. We note that as ordinarily a UEP code. In Fig. 1, we show the error performance long as the receive- and transmit-side correlation matrices are over a spatially correlated channel, for the all-zero codeword full rank, there is no loss in diversity, but only in coding gain. with numeral (I), the codeword 02, 20, 02, 20, ..., with numeral A nonidentity temporal correlation matrix also introduces loss (II), and a random codeword with numeral (III). Clearly, the in coding gain. error probability depends on the transmitted codeword, thus, the UEP property has been destroyed. This phenomenon can be explained as follows. When the IV. UEP UNDER TRANSMIT-SIDE CORRELATION channel is spatially uncorrelated, the phase of the received In general, space–time signaling is nonlinear, thus it does signals are random, so the signals from the multiple transmit not immediately follow that one can use the all-zero codeword antennas may add constructively or destructively with equal

−1 π E H E 2 − s ¯ H s ¯ H N exp 4N sin2 θ vec(H ) Γn InRnT + 4(1+K)N sin2 θ Γn vec(H ) 1 0 0 P (x → xˆ)= dθ (20) π Es n=1 In n + 2 Γn 0 R T 4(1+K)N0 sin θ

−1 π E E 2 − s H¯H H s H¯H exp 4N sin2 θ vec( ) Γ IdnRnT + 4(1+K)N sin2 θ Γ vec( ) 1 0 0 P (x → xˆ)= dθ (21) π Es Idn n + 2 Γ 0 R T 4(1+K)N0 sin θ −1 − Es H¯H H Es H¯H exp 4N vec( ) Γ IdnTnR + 4(1+K)N Γ vec( ) 1 0 0 ≤ (22) 2 Es Idn n + Γ T R 4N0(1+K) HEDAYAT et al.: ANALYSIS OF SPACE–TIME CODING IN CORRELATED FADING CHANNELS 2887

V. A PPLICATION AND RESULTS In this section, we use the PEP expressions to calculate union bounds for a variety of multiple-antenna signaling and coding schemes, including the space–time trellis (STT) codes [1], SOSTT codes [2], LD codes [3], and DAST block codes [4]. The evaluation of the union bound is possible via either the transfer function or the weight enumerating function. We use the partial weight enumerating function2 of the codes for all our results. Union bounds for the quasi-static channel are often loose [21], due to the fact that each codeword experiences only a single fading coefficient. Malkämaki and Leib [21] proposed a method of limit before averaging to tighten the bounds in quasi- Fig. 2. Received constellation with transmit correlation ρt =1. static fading.3 Nevertheless, the bounds are still not as tight as the fast-fading case, a fact widely recognized and reported. In probabilities. But a positive correlation coefficient means that the presence of diversity (either temporal or spatial) the union signals add constructively more often than destructively. For ex- bounds will be tighter and limit before averaging is not needed. ample, consider the case of two transmit antennas, one receive This can happen in MIMO channels, thus some of the MIMO antenna, and QPSK modulation, where the modulation symbols quasi-static bounds are tighter than the single-input single- are denoted {0,1,2,3}. In this case, transmission vectors that output (SISO) case. send the same signal from two antennas (e.g., 00) have an In the following figures, dashed lines denote simulations and advantage over vectors that send opposite values (e.g., 02 or solid lines denote bounds. In the case of two antennas, RTx and 13). Worst case codewords are those that are made entirely of RRx are each fully defined by a single correlation coefficient: these worst case segments. Best case codewords are those made ρt for the transmit antenna, and ρr for the receive antenna. entirely of the same symbol, e.g., the all-zero codeword. A When a higher number of transmit or receive antennas are used, random codeword will contain a mixture of both, and therefore, we consider the exponential correlation structure [11] that may will fall somewhere in between.1 correspond to the case of equi-spaced antennas with decaying To further demonstrate this effect, we study the case where factor ρr. the two antennas are perfectly correlated. Assuming channel knowledge at the receiver, the equivalent receive constella- A. STT Codes tion is as shown in Fig. 2. Although the minimum distance for all codewords is the same, some codewords (e.g., 00) We present the results for STT codes in the context of the have a smaller number of nearest neighbors compared to eight-state QPSK-STT code in [1], with a spectral efficiency some others (e.g., 02). This effect is further demonstrated of 2 bits/s/Hz and 130 information symbols per codeword (260 information bits). in Fig. 3, which depicts the scaled Chernoff bound Psc = | H |−1 Consider the performance of the STT code in quasi-static InTnR +(Es/4N0)∆∆ RTx for the PEPs, as well as the corresponding multiplicities. On the left-hand side, we see that fading with four receive antennas (see Fig. 4). The results when ρ =0, the PEP multiplicities are the same for the two for the uncorrelated quasi-static channel are also included for comparison. We notice that the code has a 2-dB loss due to codewords 00 and 02. When ρ =1the multiplicities of Psc for the two codewords are different, hence the UEP does not hold. antenna correlation on the transmit and receive sides. A similar effect is seen for other nonzero values of ρ. The union bounds in Fig. 4 are relatively tight, thanks to high We note that PEP analysis may not reveal the effect of diversity. However, the union bound can be loose whenever channel correlation on UEP, because in our examples, the diversity is low, e.g., if the number of receive antennas is small. minimum distances are not affected by correlation; only the For spatially correlated cases, we present the union bound for multiplicity varies. This calls for special care in analyzing codes the worst case reference codeword, as well as for the best in the presence of transmit-side correlation. codeword. Simulations are shown for both the best codeword There are exceptions to the loss of UEP property. For ex- or random codeword cases. ample, the two-state superorthogonal code of [20] with nT =2 and BPSK modulation preserves the UEP property even in the B. SOSTT Codes presence of transmit correlation, because all trellis paths trans- Simon and Jafarkhani [20] analyzed the performance of mit space–time block (STB) codewords that are symmetric with SOSTT codes under spatially and temporally uncorrelated respect to spatial correlations; therefore, they are all affected similarly. 2Error events on the order of hundreds were used for the union bounds reported in this paper. 3Limit before averaging is relatively straight forward in uncorrelated channels. In correlated channels, limit before averaging requires the multidimen- 1In the presence of negative correlation, the role of the best case and worst sional integration of a nonlinear function that cannot be decomposed, thus, one case codewords will be reversed. must use Monte Carlo integration. 2888 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 6, NOVEMBER 2005

Fig. 3. PEP versus multiplicities for transmitted codewords 00, 00 and 02, 20, and ρt =0, i.e., RTx = I,andρt =1, codewords with lengths up to 4, SNR =10dB.

Fig. 4. STT code, 2-Tx, 4-Rx, quasi-static Rayleigh fading, ρt = ρr =0.7. Fig. 5. SOSTT code, 2-Tx, 1-Rx, fast Rayleigh fading, ρt =0.7. channel, for which they developed a specialized analysis. Un- is given by (15). This change of notation does not affect the like [20], we evaluate the performance of an SOSTT with results for the quasi-static channel. the same PEP expressions reported in Section III simply by a For demonstration purposes, we consider the two-state change of notation, which is as follows. The subscript n in the SOSTT code [2] with nT =2and BPSK modulation, which definition of xn and ∆n now stands for the nth time-interval has a spectral efficiency of 1 bit/s/Hz. Unlike the STT code in with length nT, i.e., the nth trellis section. For example, for Section V-A, this code preserves its UEP property even when nT =2,wehave RTx = I, because all trellis paths transmit STB codewords that are symmetric with respect to spatial correlations, therefore ˆ ∗ ∗ ˆ jθn − jθn − jθn jθn xn,1e xˆn,1e xn,2e +ˆxn,2e they are all affected similarly. This may not be true for all block ∆n = − ∗ − ∗ (25) xn,2 xˆn,2 xn,1 xˆn,1 space–time codes, however. The results for fast Rayleigh fading for one receive antenna are shown in Fig. 5. The numeral (I) where xn,1 and xn,2 are the two signals transmitted by the refers to the all-zero transmitted codeword, and (II) refers to a STB code in the nth trellis section, and θn is the corresponding random codeword. Simulations show no difference, thus con- phase shift, which depends on the encoder state. Using this firming UEP property. For comparison, the results for spatially modification, the PEP in each trellis section (i.e., each block) independent fast fading are presented. We see a loss of about HEDAYAT et al.: ANALYSIS OF SPACE–TIME CODING IN CORRELATED FADING CHANNELS 2889

Fig. 6. SOSTT code, 2-Tx, 4-Rx, quasi-static Rayleigh fading, ρt = Fig. 8. STB code, 4-Tx, 2-Rx, QPSK, Rayleigh fading. ρr =0.7. code has rate 1/2. Using QPSK modulation, this code provides 1-bit/s/Hz spectral efﬁciency. Fig. 8 presents the symbol error rate of this code for two receive antennas. The bounds are calculated with respect to all codewords. The loss in coding gain is about 1.5 dB when the transmit antennas are correlated with factor ρt =0.7.

D. LD Codes LD codes [3] transmit codeword matrices (in time and space) that have the following structure

Q X = (αqAq + jβqBq) q=1

where xq = αq + jβq is a signal chosen from a multitone phase-shift keying (MPSK) or M-order quadratic amplitude Fig. 7. SOSTT code, 2-Tx, 4-Rx, temporally correlated Rayleigh fading, modulation (M-QAM) modulation, and Aq and Bq are com- × fdTs =0.01. plex spreading matrices with size nT T , where T is the time span of each codeword. We analyze an LD code reported in [3], 2 dB in coding gain due to transmit-side correlation, but no with nT =2, nR =2, T =2, and Q = nT × T =4. We apply change in diversity. the block fading model, i.e., ﬁxed fading coefﬁcient over each The performance of the SOSTT code with four receive block, varying independently between blocks. For QPSK, we antennas in a quasi-static spatially correlated channel is shown have 256 codewords and a rate of 4 bits/s/Hz. The simulated in Fig. 6 and for the temporally and spatially correlated channel and analytical frame-error-rate (FER) results of this code are is presented in Fig. 7. shown in Fig. 9 for uncorrelated as well as spatially correlated antennas. When correlation exists in both the transmit and C. STB Codes receive sides, the loss in performance is about 3 dB. We note that LD codes are not UEP, thus, all codeword pairs must be Orthogonal STB codes are introduced in [22] as the general- considered for error analysis. For larger constellations (e.g., ization of the simple Alamouti signaling for nT > 2. A related 16-QAM) this may be cumbersome, and we discovered that class of block codes with rate-1, entitled quasi-orthogonal considering only adjacent constellation points often gives an codes was later produced by Jafarkhani. It is noteworthy that, acceptable approximation. for orthogonal STB codes under correlated fading, exact performance analysis (as opposed to bounds) is possible via methods presented in a companion paper [23]. For nonorthogonal block E. DAST Codes codes, however, the methods in this paper are necessary. The DAST codeword is given by For demonstration purposes, we consider the orthogonal STB H code of [22] for nT =4. The time span of this code is 8, thus the X = nT diag (x1,...,xnT ) 2890 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 6, NOVEMBER 2005

includes quasi-static and fast fading, block fading, and arbi- trarily temporally correlated fading. PEP expressions are then used to generate union bounds on the error probabilities of STT codes, SOSTT codes, and LD codes. Other space–time codes may also be analyzed with the methods presented in this paper.

REFERENCES [1] V. Tarokh, N. Seshadri, and A. Calderbank, “Space–time codes for high data rate wireless communication: Performance criteria and code construction,” IEEE Trans. Inf. Theory, vol. 44, no. 2, pp. 744–765, Mar. 1998. [2] H. Jafarkhani and N. Seshadri, “Super-orthogonal space–time trellis codes,” IEEE Trans. Inf. Theory, vol. 49, no. 4, pp. 937–950, Apr. 2003. [3] B. Hassibi and B. Hochwald, “High-rate codes that are linear in space and time,” IEEE Trans. Inf. Theory, vol. 48, no. 7, pp. 1804–1824, Jul. 2002. [4] M. O. Damen, K. Abde-Meraim, and J. C. Belfiore, “Diagonal algebraic space–time block codes,” IEEE Trans. Inf. Theory, vol. 48, no. 3, pp. 938– 952, Mar. 2002. Fig. 9. LD code, 2-Tx, 2-Rx, QPSK, Rayleigh fading. [5] M. Damen, A. Abdi, and M. Kaveh, “On the effect of correlated fading on several space–time coding and detection schemes,” in Proc. IEEE Vehicular Technology Conf., Atlantic City, NJ, 2001, pp. 13–16. [6] H. Bölcskei and A. J. Paulraj, “Performance of space–time codes in the presence of spatial fading correlation,” in Proc. Asilomar Conf. Signals, Systems and Computers, Pacific Grove, CA, Oct. 2000, pp. 687–693. [7] H. Bölcskei, M. Borgmann, and A. J. Paulraj, “Impact of the propaga- tion environment on the performance of space–frequency coded MIMO- OFDM,” IEEE J. Sel. Areas Commun., vol. 21, no. 3, pp. 427–439, Apr. 2003. [8] M. Uysal and C. Georghiades, “On the error performance analysis of space–time trellis codes,” IEEE Trans. Wireless Commun., vol. 3, no. 4, pp. 1118–1123, Jul. 2004. [9] A. Dogandzic,´ “Chernoff bounds on pairwise error probabilities of space–time codes,” IEEE Trans. Inf. Theory, vol. 49, no. 5, pp. 1327– 1336, May 2003. [10] J. W. Craig, “A new, simple, and exact result for calculating the probability of error for two-dimensional signal constellations,” in Proc. IEEE Military Communications Conf. (MILCOM), McLean, VA, Oct. 1991, pp. 571–575. [11] M. K. Simon and M. S. 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[22] V. Tarokh, H. Jafarkhani, and A. Calderbank, “Space–time block codes Aria Nosratinia (M’97–SM’04) received the B.S. from orthogonal designs,” IEEE Trans. Inf. Theory, vol. 45, no. 5, degree in electrical engineering from the University pp. 1456–1467, Jul. 1999. of Tehran, Tehran, Iran, in 1988, the M.S. degree in [23] H. Shah, A. Hedayat, and A. Nosratinia “Performance of concatenated electrical engineering from the University of Wind- channel codes and orthogonal space–time block codes,” IEEE Trans. sor, Windsor, ON, Canada, in 1991, and the Ph.D. Wireless Commun., to be published. degree in electrical and computer engineering from the University of Illinois at Urbana-Champaign, in 1996. Ahmadreza Hedayat (S’00) received the B.S.E.E. From 1995 to 1996, he was with Princeton Uni- and M.S.E.E. degrees from the University of Tehran, versity, Princeton, NJ. From 1996 to 1999, he was a Tehran, Iran, in 1994 and 1997, respectively, and Visiting Professor and Faculty Fellow at Rice Uni- the Ph.D. degree in electrical engineering from the versity, Houston, TX. Since July 1999, he has been with the faculty of the University of Texas at Dallas, in 2004. University of Texas at Dallas, where he is currently Associate Professor of From 1995 to 1998, he was a Systems and Hard- Electrical Engineering. His research interests are in the broad area of commu- ware Engineer with Pars Telephone Kar. From 1998 nication and information theory, particularly coding and signal processing for to 1999, he was a Systems and R&D Engineer the communication of multimedia signals. with Informatic Services Corporation. His research Dr. Nosratinia is Associate Editor for the IEEE TRANSACTIONS ON IMAGE interests are in wireless and digital communications PROCESSING. He was the recipient of the National Science Foundation (NSF) with emphasis on multiple-antenna systems, channel CAREER Award in 2000 and has twice received chapter awards for outstanding coding, and cross-layer schemes. service to the IEEE Signal Processing Society.

Harsh Shah was born in Ahmedabad, India, in 1978. He received the B.E. degree in electronics and communications from Gujarat University, Ahmedabad, India, in 2001, the M.S. degree in electrical engineering from University of Texas, Dallas, in 2003, and is currently pursuing the Ph.D. degree in electrical engineering at University of Texas at Dallas. He is currently interested in research on digital watermarking, channel coding, and multiple-input multiple-output (MIMO) systems. Mr. Shah was the recipient of a Kodak Graduate Fellowship for 2004.