On the performance of space-time codes over correlated ... - CiteSeerX

On the Performance of Space-Time Codes over Correlated Rayleigh Fading Channels Jibing Wang* , Marvin K. Simon† , Michael P. Fitz* and Kung Yao* *

Electrical Engineering Department, University of California, Los Angeles † Jet Propulsion Laboratory, California Institute of Technology

Abstract— We derive the exact pairwise error probability (PEP) for the space-time coding over quasi-static Rayleigh fading channels in the presence of spatial fading correlation. We show that receive correlation always degrades the PEP for all SNRs. We quantify the effect of receive correlation by employing the notion of “majorization”. We show that the stronger the receive correlation, the worse the PEP for all SNRs. For the orthogonally designed space-time blocks (ODSTBC), we analyze the effects of transmit and receive correlation on the performance in terms of PEP and symbol error probability. We show that transmit correlation also deteriorates the performance of ODSTBC. We observe a duality between the transmit correlation and receive correlation for the ODSTBC. Index Terms— Space-time Coding, spatial correlation, pairwise error probability, majorization

I. I NTRODUCTION Multiple antennas promise very high date rate for wireless communications [2][3]. An effective approach to increasing the data rate as well as the power efficiency over wireless channels consists of introducing temporal and spatial correlation into signals transmitted from different antennas (see, e.g., [1] and references therein). The design criterion of spacetime codes over Rayleigh fading channels involving rank and eigenvalues of certain matrices was pioneered in [4][5] from the quadratic forms of complex Gaussians. Later on, Tarokh, et al. [6] generalized the results to multiple receive antennas based on the Chernoff bound on the pairwise error probability (PEP). Recently, several researchers have presented tighter upper and lower bounds on PEP to gain further insights into the performance of space-time codes [13][16][17]. Most work up to this point assumes the idealistic case of independent and identically distributed (i.i.d.) channels, i.e., the spatial fading is uncorrelated. However, in reality, the individual antennas could be correlated due to insufficient antenna spacing and lack of scattering [8]-[14]. In [14], the authors studied the impact of spatial fading correlation on the performance of the space-time codes by quantifying the loss of diversity and coding gain. The results in [14], derived based on the Chernoff bound on the PEP, are therefore relevant for the case of asymptotically high SNR. In this work we derive the exact PEP for space-time coding over quasi-static Rayleigh fading channels in the presence of spatial correlation. We show that the PEP degrades in the presence of receive correlation for all SNRs. We also show that the receive correlation has This work is partially supported by NASA/Dryden grant NCC2-374 and UC CoRe grant sponsored by ST Microelectronics, Inc.

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negligible effect on the performance of space-time coding at low SNR. Further more, by employing the notion of “majorization” [20], we provide a complete study of how receive correlation degrades the performance of space-time codes. Our results show that for all level of SNRs, the more “spread out” the receive correlation matrix, the worse the performance in terms of PEP. Unlike the receive correlation, the effect of transmit correlation depends on the specific space-time code used [22]. For the orthogonally designed space-time block codes (ODSTBC) [7], we show that the transmit correlation as well as receive correlation deteriorate the performance in terms of PEP and symbol error probability (SEP). The following notations are adopted throughout this paper: scalars are denoted in lower case, vectors are column vectors unless otherwise indicated and are denoted in lower case bold, while matrices are in upper case bold. RH denotes the conjugate transpose of the matrix R, and RT and R∗ denote transpose and elementwise conjugate of R, respectively. IM is the M × M identity matrix. II. C HANNEL M ODEL Consider a MIMO link with MT transmit antennas and MR receive antennas. Assume a flat fading channel where the fading coefficient from the i-th transmit antenna to the j-th receive antenna is denoted by hj, i . Assuming linear modulation, the signal received by antenna j is given by T √
γt hj, i xi + vj ,

M

yj =

i=1

where the noise vj is modeled as independent samples of a zero-mean complex Gaussian random variable with variance 0.5 per dimension, xi is the  transmitted from antenna  signal 2 = 1. Here γt = γ/MT i normalized such that E |xi | denotes the SNR per transmit antenna, where γ is the SNR per receive antenna, regardless of the number of transmit antennas. We obtain the following matrix form of the input output relation of the MIMO channel model √ y = γt Hx + v, (1) T  is the received signal vecwhere y= y1 · · · yMR  T tor, x= x1 · · · xMT is the transmitted signal vector, T  is the i.i.d. AWGN noise vector at v= v1 · · · vMR the receiver with covariance matrix given by Rv =E vvH =

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IMR , and H is the MIMO channel matrix. We restrict ourselves to a Rayleigh fading scenario, therefore, the MR × MT elements of H are composed of circularly symmetric zero mean complex Gaussian random variables (possibly correlated). We assume that spatial fading correlation occurs both at the transmitter and the receiver. The spatial fading correlation depends on the physical geometries of the channel (see, e.g., [8]-[9]). In this paper, for analytical tractability, we employ the spatial fading correlation model in [14][10], namely, H = R1/2 Hω S1/2 ,

(2)

where Hω is an MR × MT matrix composed of i.i.d. complex Gaussian entries with zero mean unit variance, and S =  H  andH and R = R1/2 R1/2 are the transmit and S1/2 S1/2 receive correlation matrices, respectively. As noted in [15], the product-form of the assumption in (2) does not incorporate the most general case of spatial fading correlation, but could be thought of as a first order approximation of the true correlation structure [10], yielding a reasonable compromise between validity and analytical tractability. In [14], the physical parameters such as angle spread, and antenna spacing are incorporated in the spatial correlation model. Assuming a uniform linear array at the receiver, the correlation matrix R is written as [12][14]   (3) [R]m,n = ρ (n − m) ∆r , θr , σθ,r with 2   ρ s∆r , θr , σθr = e−j2πs∆r cos(θr ) e−0.5(2πs∆r sin(θr )σθ,r ) , (4) where ∆r is the relative antenna spacing, θr is the mean angle of arrival at the receiver, and σθ,r corresponds to the standard deviation of the angle spread. The transmit correlation matrix S could way, therefore, we have   be modeled the same [S]m,n = ρ (n − m) ∆t , θt , σθ,t . III. E XACT PAIRWISE E RROR P ROBABILITY (PEP) FOR C ORRELATED R AYLEIGH FADING C HANNELS For the slow Rayleigh fading channels, we employ the block-fading model used in [6] wherein the channel remains constant over N ≥ MT symbols periods and changes in an independent fashion to a new realization from block to block. The bit stream is encoded by the space-time encoder into a  codeword D = d1 d2 · · · dN of size MT × N (dn represents the data sent from the MT transmit antennas in the nth symbol period) with the individual data being taken from a finite complex constellation whose members have average energy normalized to one. The reconstructed data matrix at

 = d  N , with 2 · · · d 1 d the receiver is denoted by D  k = √γt Hdk +vk ,k = 1, 2, · · · , N. d

P (D → E) −1  π2 M R M T    γt 1 λm (D, E, S) λn (R) dθ, 1+ = π 0 n=1 m=1 4 sin2 θ (7) M

T where {λm (D, E, S)}m=1 are the set of eigenvalues of the ∗ T MR are the matrix (D − E) (D − E) ST , and {λn (R)}n=1 eigenvalues of R. The PEP in the above equation could be given in closed form using the residue method or partial-fraction expansion [13][17]. However, we keep its integral form since, as we will show later, it gives us more insight into the effect of spatial fading correlation.

IV. E FFECT OF R ECEIVE C ORRELATION ON THE PEP OF S PACE -T IME C ODES OVER C ORRELATED R AYLEIGH FADING C HANNELS In this section, we discuss the effect of spatial fading correlation on the performance of space-time codes in terms of PEP. Unlike the receive correlation, the effect of transmit correlation depends on the specific space-time code used [22]. At low SNR, transmit correlation can either improve or degrade the PEP performance [22]. For the ODSTBC, we characterize the effect of transmit correlation as we will show in a later section. In this section, we focus on the case of receive correlation only. However, the results also apply to the case where transmit correlation is included. MT = Assuming S = IMT , we have {λm (D, E, S)}m=1 MT MT {λm (D, E)}m=1 , where {λm (D, E)}m=1 is the set of eigenT ∗ values of (D − E) (D − E) . From Proposition 1, we have P (D → E) −1  π2 M R M T    1 γt = dθ. 1+ λm (D, E) λn (R) π 0 n=1 m=1 4 sin2 θ (8) At low SNR, i.e. γt  1, we have P (D → E) −1  π2  MT MR

γt 1 λm (D, E) λn (R) dθ 1+ ≈ π 0 4 sin2 θ m=1 n=1 −1  π2  1 γt MR 2 = dθ 1+ D − E F π 0 4 sin2 θ    2 γt MR D − EF 1 1− = , (9) 2 2 γt MR D − EF + 4

(5)

We assume that the channel-state information (CSI), that is, the realization of H, is perfectly known at the receiver. For maximum likelihood (ML) detection, the PEP that the receiver decides erroneously in favor of E when the codeword D is transmitted is given by [6]   γt 2 H (D−E) , (6) P (D → E) = E Q 2 0-7803-7954-3/03/$17.00 ©2003 IEEE.

N 2 2 where H (D−E) = k=1 H (dk −ek ) . By using Craig’s formular [18], we have the following proposition: [22] Proposition 1: For quasi-static Rayleigh fading channels, we have

2

where  .F is the Frobenius norm of a matrix: DF = T r DDH . Therefore, at low SNR, receive correlation has negligible effect on the performance of PEP. A similar argument is given in [10] by considering asymptotic mutual information.

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Denote the product of the nonzero r(R)eigenvalues of the receive correlation matrix R as ς = m=1 λn (R), with r (R) the rank of the matrix R. Therefore, at high SNR, the PEP depends on the receive correlation matrix only through ς, assuming r (R) is fixed [14]. However, we note that ς only characterizes the PEP performance at asymptotically high SNR. In general, larger ς does not necessarily imply better performance in terms of PEP for all SNR. We illustrate this by an example. Consider a radio link using a single transmit antenna and three receive antennas and BPSK modulation. The PEP in this case is exactly the bit error probability (BEP). Consider two receive fading correlation scenarios R1 and R2 . We assume that the eigenvalues of R1 and R2 are given by {0.1133, 0.1433, 2.7433} and {0.02, 1.49, 1.49}, respectively. Then ς1 = 0.1133 × 0.1433 × 2.7433 = 0.0445, ς2 = 0.02 × 1.49 × 1.49 = 0.0444. The bit error performance is shown in Fig.1. As we can see, although ς2 ∼ = ς1 , the bit error performance is better for a wide range of SNR when the receive fading correlation matrix is R2 . To clearly characterize the effect of spatial correlation, we employ the definition of majorization [20]. Definition 1: Let a = [ai ] ∈ Rn and b = [bi ] ∈ Rn be given. The vector b is said to majorize the vector a if   k 
 min bij : 1 ≤ i1 < i2 < · · · < ik ≤ n   j=1   k 
 ≥ min aij : 1 ≤ i1 < i2 < · · · < ik ≤ n  

Proof: Let the eigenvalues of Ri be denoted by MR . Define am (θ) = [am,n (θ)] ∈ RMR and {λn (Ri )}n=1 bm (θ) = [bm,n (θ)] ∈ RMR as am,n (θ) = 1+

γt λm (D, E) λn (R1 ) , n = 1, 2, · · · , MR , 4 sin2 θ

bm,n (θ) = 1+

γt λm (D, E) λn (R2 ) , n = 1, 2, · · · , MR . 4 sin2 θ

M R 

k
i=1

bm i ≥

k


aji , k = 1, 2, · · · , n

(10)

i=1

with equality for k = n. By definition, the real vector b majorizes a if the sum of the k smallest elements of b is greater than or equal to the sum of the k smallest elements of a for k = 1, 2, · · · , n and the sums of the elements of b and a are equal. Definition 2: The correlation matrix R1 is said to be less spread out than another correlation matrix R2 , denoted by R1 R2 , if the vector of eigenvalues of R1 majorizes the vector of eigenvalues of R2 . The notion of spread out can be taken as a measure of the strength of correlation: the more spread out, the stronger the correlation [10]. Lemma 1: [20] Let a = [ai ] ∈ Rn and b = [bi ] ∈ Rn be two given nonnegative real vectors and suppose that b majorizes a . Then n n   bi ≥ ai . i=1

i=1

am,n (θ) ≥

n=1

M R 

bm,n (θ) , m = 1, 2, · · · , MT .

n=1

We note from (8) that 1 PR1 (D → E) = π 1 PR2 (D → E) = π



π 2

0

 0

π 2

M T 

M R 

m=1

n=1

M T 

M R 

m=1

−1 am,n (θ)

dθ,

−1 bm,n (θ)

dθ.

n=1

Therefore we obtain PR1 (D → E) ≤ PR2 (D → E) . Since i.i.d. fading corresponds to the identity correlation matrix, which is the least spread out among all the correlation matries, we obtain P (D → E) ≥ Pi.i.d. (D → E) ,

j=1

for all k = 1, 2, · · · , n with equality for k = n. If we arrange the elements of a and b in increasing order aj1 ≤ aj2 ≤ · · · ≤ ajn , bm1 ≤ bm2 ≤ · · · ≤ bmn , the defining inequalities can be restated in the equivalent form

∗

T

Since R1 R2 and (D − E) (D − E) is positive semidefinite (i.e., λm (D, E) ≥ 0), we have am (θ) majorizes bm (θ). Therefore, for any m, we have

(12)

which means that for all level of SNRs, the presence of receive correlation leads to higher PEP. We have proved that the more spread out is R, i.e., the stronger the receive correlation, the worse is the performance in terms of PEP for quasi-static Rayleigh fading channels. The fact that spatial correlation degrades the performance of space-time codes is well known (see, e.g. [14]). By employing the notion of “majorization”, the above results, however, provide us with a more complete understanding of how receive correlation degrades the performance of space-time codes. Our results show that at all level of SNRs, the more “spread out” the receive correlation matrix, the worse the performance in terms of PEP. V. T HE P ERFORMANCE OF ODSTBC OVER C ORRELATED R AYLEIGH FADING C HANNELS A. PEP Analysis In this section, we discuss the performance of ODSTBC [7] in terms of PEP and SEP over correlated Rayleigh fading channels. For the ODSTBC, we have [7] ∗

T

(D − E) (D − E) = βD,E I,

Theorem 1: For MIMO quasi-static Rayleigh fading channels with receive correlation, if R1 R2 , then

where βD,E is a scaling factor depending on the codeword pair. Therefore, we have

PR1 (D → E) ≤ PR2 (D → E) .

λm (D, E, S) = βD,E λm (S) , m = 1, 2, · · · , MT ,

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(11)

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M

T where {λn (S)}n=1 are the eigenvalues of S. From Proposition 1 we obtain the PEP for the ODSTBC over the correlated quasi-static Rayleigh fading channel as

P (D → E) −1  π2 M R M T    1 γt = λ (S) λ (R) dθ. 1+ β D,E m n π 0 n=1 m=1 4 sin2 θ

For square M -QAM constellations, using the techniques in [21], we can calculate the symbol error probability PeM QAM . We can show that for M -PSK or square M -QAM constellations, stronger correlation leads to higher SEP. Theorem 3: If S1 S2 , and R1 R2 , then

An interesting observation we make here is that there is a duality between the transmit and receive correlations for ODSTBC. So if the average signal to noise ratio per transmit antenna, i.e. γt , is kept the same, then the performance of ODSTBC does not change if the role of transmitters and receivers is interchanged. This duality does not hold in general for arbitrary space-time codes, as can be seen in Proposition 1. Furthermore, we can prove the following theorem: Theorem 2: For ODSTBC over MIMO Rayleigh fading channels with joint transmit-receive correlation, denote

M P SK M P SK M P SK Pe,S ≤ Pe,S ≤ Pe,S . 1 ,R1 1 ,R2 2 ,R2

(17)

M QAM M QAM M QAM Pe,S ≤ Pe,S ≤ Pe,S . 1 ,R1 1 ,R2 2 ,R2

(18)

VI. N UMERICAL R ESULTS

In this section, we provide numerical results to investigate the effects of spatial fading correlation on the performance of ODSTBC. We use numerical integration to calculate the SEP in (16). Fig.2 shows the average bit error probability of 4 × 4 BPSK modulation for the i.i.d. (uncorrelated) case and the cases of transmit correlation only (∆t = 0.5, θt = π/2, σθ,t = 0.25), receive correlation only (∆r = 0.1, θr = π/2, σθ,r = 0.25), and joint transmit and receive correlation (∆t = PS1 ,R1 (D → E)    π2 M 0.5, θt = π/2, σθ,t = 0.25, ∆r = 0.1, θr = π/2, σθ,r = M −1 R T   1 γt 0.25), respectively. In the plot, the SNR refers to the signal λ (S ) λ (R ) dθ, = 1+ β D,E m 1 n 1 π 0 n=1 m=1 4 sin2 θ to noise ratio per receive antenna. Clearly we see that the bit error performance degrades in the presence of spatial fading PS2 ,R2 (D → E) correlation. We also notice that the performance in the case of −1 transmit correlation only is better than the performance in the  π2 M R M T    1 γt dθ.case of receive correlation only. This can be explained by the = 1+ βD,E λm (S2 ) λn (R2 ) π 0 n=1 m=1 4 sin2 θ duality. The case of receive correlation only with configuration (∆r = 0.1, θr = π/2, σθ,r = 0.25) is the same as transmit If S1 S2 , and R1 R2 , then correlation only with (∆t = 0.1, θt = π/2, σθ,t = 0.25). Due PS1 ,R1 (D → E) ≤ PS2 ,R2 (D → E) . (13) to the closer antenna spacing, the correlation is higher than the configuration (∆t = 0.5, θt = π/2, σθ,t = 0.25). We can also B. Symbol Error Probability for the ODSTBC see that the performance degradation due to spatial correlation For the ODSTBC, it turns out that we can calculate the exact is small when SNR is low. Fig.3 illustrates the SEP of 2 × 4 QPSK ODSTBC symbol error probability (SEP). The ODSTBC corresponds to with the same transmit correlation (∆t = 0.1, θt = π/2, orthogonal signaling and therefore leads to a simple receiver = 0.25), but different receive correlation given σ θ,t structure. The orthogonal design has the effect of reducing (∆r = 0.1, θr = π/2, σθ,r = 0.25), R2 by R 1 the space-time channel to a group of parallel identical AWGN (∆ = 0.1, θr = π/2, σθ,r = 0.5), R3 (∆r = 0.1, r channels, each carrying a different information symbol. Mathθ = π/2, σθ,r = 0.75). The eigenvalues of R1 , R2 r ematically, for each information symbol xi , the sufficient and R are given by {0.0000, 0.0012, 0.1173, 3.8815}, 3 statistic to detect xi is given by [19] {0.0003, 0.0171, 0.4065, 3.5761}, ! 2 yi = γt HF xi + ni , (14) {0.0029, 0.0728, 0.7341, 3.1902}, respectively. Therefore, we have R1  R2  R3 . We can clearly see that the where ni is the AWGN noise with zero mean and variance 0.5 performance of SEP improves with increasing angle spread. per dimension. For M -PSK constellations, conditioned on the channel realization H, the symbol error probability is given VII. C ONCLUSIONS by [21] " # In this work, we have derived the exact pairwise error prob−1)π  (MM 2 gpsk γt HF 1 ability of space-time codes over spatially correlated Rayleigh exp − Pe|H = dθ, (15) π 0 sin2 θ fading channels. We showed that for all SNRs, receive correlation always degrades the PEP. The stronger the correlation, where gpsk = sin2 (π/M ). We can show the SEP for M -PSK the worse the performance in terms of PEP. For ODSTBC, constellations is given by we characterize the effect of transmit correlation as well as the receive correlation on the performance of PEP and SEP. PeM P SK −1)π M    (MM We showed that the presence of transmit correlation also M −1 R T   1 gpsk γt deteriorates the performance. We also observed the duality = (S) λ (R) dθ. 1+ λ m n π 0 sin2 θ between the transmit correlation and receive correlation for n=1 m=1 (16) ODSTBC.

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Fig. 3. Symbol error probability versus SNR for QPSK ODSTBC with 2 transmit antennas and 4 receive antennas.

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