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Error Performance of MIMO-BICM with Zero-Forcing Receivers in Spatially-Correlated Rayleigh Channels
Index Terms— BICM, MIMO, zero-forcing, spatial correlation
I. I NTRODUCTION Bit-interleaved coded modulation (BICM) is a pragmatic technique which can achieve large diversity orders in fading wireless channels [1]. Recently, BICM was employed in a multiple-input multiple-output (MIMO) antenna system with maximum likelihood (ML) detection [2], and was shown to outperform space-time trellis codes in fast fading conditions. Unfortunately, the computational complexity of the ML receiver increases exponentially with the product of the number of transmit antennas and the number of bits per modulation symbol, thereby making it prohibitive in practice. A reduced complexity scheme was proposed in [3], and analyzed in [4], which employed a linear zero-forcing (ZF) receiver. It was shown in [4] that this receiver has desirable signal-space properties in i.i.d. Rayleigh fading MIMO-BICM systems, which allow near-ML performance to be achieved in many scenarios. The challenge is to investigate the performance of this practical low-complexity receiver in the more common case of spatially-correlated channels. In this letter, we derive tight analytical expressions for the coded bit error rate (BER) of MIMO-BICM with ZF receivers Manuscript received April 13, 2005; revised October 14, 2005; accepted November 4, 2005. The editor coordinating the review of this paper and approving it for publication is J. Garcia-Frias. This paper was presented in part at the Australian Communication Theory Workshop (AusCTW), Perth, Australia, Feb. 2006. M. R. McKay is with the Telecommunications Laboratory, School of Electrical and Information Engineering, University of Sydney, NSW, Australia, and also the Wireless Technologies Laboratory, ICT Centre, CSIRO, Australia. I. B. Collings is with the Wireless Technologies Laboratory, ICT Centre, CSIRO, Australia. (e-mail:
[email protected];
[email protected]). Digital Object Identifier: 00.0000/TWC.2005.000000
data
Enc
Map Int
Enc
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a
Dem
Dec Mux
Abstract— In this letter we derive tight analytical expressions for the coded bit error rate of MIMO bit-interleaved coded modulation (BICM) in spatially-correlated Rayleigh fading channels. We consider the case where low complexity zero-forcing receivers are employed. The analysis is simpler and more direct than the standard BICM error event expurgation technique, and yields efficient, accurate expressions for the error probability. Based on the analytical results, we obtain the diversity order and show that it is independent of the spatial correlation. Moreover, we show that the presence of spatial correlation induces an SNR loss with respect to i.i.d. channels. We quantify this loss, and show that it is a function of the antenna configuration and the eigenvalues of the spatial correlation matrices, and is independent of the coding and modulation parameters.
Demux
Matthew R. McKay, Student Member, IEEE and Iain B. Collings, Senior Member, IEEE
-1
Int
ZF
Map
Dem
ˆ data
Dec
Fig. 1. Structure of a MIMO-BICM system with a zero-forcing receiver and 2 × 2 antennas.
(hereafter denoted ZF-BICM) in spatially-correlated channels. The results are for Gray-labeled PSK/QAM constellations1 , and are based on the typical BICM assumption of ideal interleaving. Our analysis is simpler and more direct than the standard BICM expurgation technique (eg. as used in [1, 4]), and yields accurate, efficient BER expressions for channels with transmit and receive spatial correlation. Based on the analytical results, we obtain the diversity order and show that it is independent of the spatial correlation. Moreover, we show that the presence of spatial correlation induces an SNR loss with respect to i.i.d. channels. We quantify this loss, and show that it is a function of the antenna configuration and the eigenvalues of the spatial correlation matrices, and is independent of the coding and modulation parameters. II. S IGNAL M ODEL Consider a MIMO-BICM system with Nt transmit and Nr receive antennas (denoted Nt × Nr ). Fig. 1 shows the transmitter structure with separate modulators/encoders applied on each of the transmit layers. The data sequence is cyclically demultiplexed across the layers and each resulting stream is binary convolutionally encoded. Bit interleaving is applied within and across the layers, prior to mapping to Gray-labelled 2M -ary QAM or PSK for transmission. The received signal vector is given by √ (1) r = γ Ha + n where a ∈ C Nt ×1 is the transmitted vector with ith element ai chosen from the complex scalar constellation A with unit average power, n ∈ C Nr ×1 is the noise vector ∼ CN (0Nr ×1 , INr ), and γ is the average SNR per transmit antenna. Also, H ∈ C Nr ×Nt is the spatially-correlated channel matrix, assumed to be known perfectly at the receiver, having the (common) correlation structure 1
1
H = R 2 Hw S 2
(2)
1 We note that the analysis approach could be extended to other labelings, but is beyond the scope of this letter.
c 2005 IEEE 0000–0000/00$00.00
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where R ∈ C Nr ×N r and S ∈ C Nt ×Nt are the normalized receive and transmit correlation matrices respectively, and Hw ∈ C Nr ×Nt contains independent entries Hi,j ∼ CN (0, 1). As such, the average SNR per receive antenna is Nt γ. At the receiver, the ZF filtering step is z = Wr √ = γa + n
(3)
−1 † where W = H† H H and n = W n. The BICM log-likelihood metrics for each of the bits i (= 1, . . . , M ) corresponding to ak are calculated from zk using P a k , wk ) a ˜ ∈Ai,k p (zk |˜ Lk,i = ln P k 1 p (zk |˜ a k , wk ) a ˜ k ∈Ai,k 0 √ P a |2 |z − γ˜ exp − kkwk k2k a ˜ k ∈Ai,k 1 = ln P (4) √ |zk − γ˜ a k |2 i,k exp − 2 a ˜ k ∈A kwk k 0
i,k Ai,k 0 , A1
where are the signal subsets within the k th transmit constellation with ith bit equal to 0 and 1 respectively. These metrics are deinterleaved into the appropriate output streams, and each parallel stream is decoded using a conventional softdecision Viterbi algorithm. III. E RROR P ERFORMANCE A NALYSIS IN S PATIALLY-C ORRELATED MIMO C HANNELS
Further assuming that the all-zero codeword is transmitted, we have the following well-known result for BIOS channels [5] ! d X f (d, µ, A, γ) = Pr Li > 0 (6) i=1
where, in this ZF-BICM case P
√ |zi − γ˜ a|2 exp − kw 2 i,k k Li = ln P √ |zi − γ˜ a|2 mi ,k exp − 2 kw k a ˜∈A i,k mi ,k a ˜∈Au ¯i
(7)
ui
where mi is the bit position in the transmitted symbol’s binary label which corresponds to the ith equivalent binary-input transmission. Unfortunately Li is a complicated function for which the distribution (and hence the tail probability in (6)) is difficult to calculate. We can however evaluate the tail probability based on the moment generating function (m.g.f.) M(·) as [6] Z c+j∞ 1 ds f (d, µ, A, γ) = MPd Li (s) . (8) i=1 2πj c−j∞ s Since the likelihood ratios are i.i.d. under the assumption of ideal interleaving, we have4 Z c+j∞ 1 ds f (d, µ, A, γ) = ML (s)d (9) 2πj c−j∞ s where
√ |z− γ˜ a| 2 exp − kwk k2 ML (s) = Ez,m,u,k,wk exp s ln P √ |z− γ˜ a| 2 Throughout this paper we consider linear binary convolum,k exp − 2 a ˜ ∈Au kwk k tional codes with rate Rc = kc /nc , for which a tight BER √ s 2 P | γ(a−˜ a )+n k| union bound is given by [5] m,k exp − a ˜ ∈Au kwk k2 ¯ √ ∞ = Ea,m,u,k,nk ,wk P X | γ(a−˜ a)+nk |2 1 m,k exp − 2 WI (d)f (d, µ, A, γ) (5) Pb ≤ a ˜ ∈Au kwk k kc d=dfree (10) where WI (d) is the input weight of all error events at Averaging over m, u, k and a ∈ Am,k gives u Hamming distance d, and f (.) is the codeword pairwise error M X 1 X Nt X X probability (C-PEP) which is also a function of the labeling 1 M (s) = Im,u,a (s) (11) L map µ, the signal set A, and the SNR γ. Also, dfree is the M M 2 Nt m=1 u=0 k=1 a∈Am,k u free Hamming distance. where A. BER Union Bound
B. Evaluation of the C-PEP For the ZF-BICM system operating under the assumption of ideal interleaving, the BICM metrics which are fed into the decoder for each stream are i.i.d. Consequently, each encoderdecoder pair acts as an equivalent single-input single-output (SISO) BICM system with the same error characteristics; thereby allowing us to focus on a single arbitrary stream error probability. We also adopt the approach of [1] and force the BICM subchannels2 to behave as binary-input outputsymmetric (BIOS) channels by introducing a time-varying labeling map. Specifically, we define u as a random variable which independently selects, for every MIMO transmission, the labeling map µ or its complement µ ¯ with probability 21 3 . 2 These are the equivalent channels between the transmitted binary codeword and the corresponding BICM bit metrics. 3 Note that in the following we select µ ¯ when u = 1.
P
m,k a ˜ ∈Au ¯
Im,u,k,a (s) = √ s P | γ(a−˜ a)+nk |2 m,k exp − 2 a ˜ ∈Au kwk k ¯ √ Enk ,wk P . | γ(a−˜ a)+nk |2 exp − a ˜ ∈Am,k kwk k2 u
(12)
We note that the expectation (12) can be evaluated via numerical integration using a combination of Gauss-Laguerre and Gauss-Hermite quadratures [7], and the integral in (9) evaluated using Gauss-Chebyshev quadratures [8]. We now show that at high SNR, computing f (·) reduces to only a single numerical integration - the result of which can be written in closed form. Note that at high SNR the ratio in (12) is dominated by a single (minimum distance) 4 Note that we drop the subscript on L when we are considering the log likelihood ratio statistics at a single instant in time.
ERROR PERFORMANCE OF MIMO-BICM WITH ZERO-FORCING RECEIVERS IN SPATIALLY-CORRELATED RAYLEIGH CHANNELS
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−1 d trNt −1 (λ(R)) det INr + Nt |P M| X 1 X ! PM,i Ψt = −1 Nt k=1 i=1 γEM,i (1+tan2 ( (2t−1)π λ(R) )) 2v trNt −1 λ(R) INr + 4[S−1 ]
γEM,i (1+tan2 ( (2t−1)π ))λ(R) 2v 4[S−1 ]k,k
(20)
k,k
term in the numerator and denominator, and as such we can apply the Dominated Convergence Theorem [9]. (The Dominated Convergence Theorem has previously been used in the analysis of BICM [10, App. III] where they considered SISO systems and were deriving the asymptotic slope of the error probability.) We now have √ s a)+nk |2 | γ(a−˜ exp − 2 kwk k Im,u,k,a (s) = Enk ,wk |nk |2 exp − kwk k2
(13)
where a ˜ ∈ Aum,k is the nearest neighbour to a ∈ Am,k ¯ u . We proceed to average over the random noise as follows Im,u,k,a (s)
# √ −s| γ(a − a ˜) + nk |2 + s|nk |2 = Ewk ,nk exp kwk k2 " # √ −sγ|a − a ˜|2 − 2sRe( γ(a − a ˜)n∗k ) = Ewk ,nk exp kwk k2 " 2 Z −sγ|a − a ˜ |2 1 s γ|a − a ˜ |2 = Ewk exp exp kwk k2 π kwk k2 # √ −|nk + s γ(a − a ˜)|2 × exp dnk kwk k2 " # −γ|a − a ˜|2 s(1 − s) = Ewk exp . (14) kwk k2 "
It is now possible to express the expectation (14) in the form of an integral equation, however it appears difficult to evaluate the resulting expression in closed form. However, we can obtain a simple and accurate closed-form approximation by first noting that (14) can be written as Im,u,k,a (s) = Mγ/kwk k2 −|a − a ˜|2 s(1 − s) . (15) Using the Laplace approximation to the ratio of quadratic forms [11], a general expression for the m.g.f. Mγ/kwk k2 (·) was first obtained in [12], which considered the analysis of symbol error probabilities in uncoded ZF systems. Applying this result in (15), we have !−1 γ|a − a ˜|2 s(1 − s)λ(R) Im,u,k,a (s) ≈ det INr + [S−1 ]k,k trNt −1 (λ(R)) × h i−1 |2 s(1−s)λ(R) trNt −1 λ(R) INr + γ|a−˜a[S −1 ] k,k
(16)
where λ(·) denotes a diagonal matrix containing the eigenvalues of the matrix argument, and [·]k,k represents the k th diagonal element. Also, tr` (·) denotes the `th elementary symmetric function (e.s.f.) defined as [13, 14] tr` (X) =
` XY
λx,αi =
{α} i=1
X {α}
det Xα α
(17)
for arbitrary Hermitian positive-definite X ∈ C n×n , where the sum is over all ordered α = {α1 , . . . , α` } ⊆ {1, . . . , n}, λx,i α denotes the ith eigenvalue of X, and Xα is and ` × ` matrix formed by taking only the rows and columns indexed by α. Now, substituting (16) into (11) gives a closed-form expression for the m.g.f. Unfortunately this is quite numerically challenging to evaluate, since it requires calculation of M 2M Nt terms Im,u,a (s), each containing determinants and e.s.f.s. We can see from (16) however, that Im,u,a (s) only depends on a through the squared Euclidean distance to its nearest neighbour a ˜ in the complement subset. Since in this paper we consider Gray-labelled PSK/QAM constellations, we can further simplify (11) by exploiting multiplicities of the Euclidean distances. In Table I we present (for BPSK, QPSK, 16QAM, and 64QAM Gray-labelled constellations) the set of all distinct squared Euclidean distances EM , and the corresponding set PM containing the frequency of occurrence of each distance (this is the number of occurrences of each distance in the summations over m, u and a in (11), normalized by the total number of distances, M 2M ).
BPSK
PM
EM
{1}
{4.0}
QPSK
{1}
{2.0}
16QAM
{3/4, 1/4}
{0.4, 1.6}
64QAM
{7/12, 1/4, 1/12, 1/12}
{ 0.0952, 0.3810, 0.8571, 1.5238}
TABLE I B REAKDOWN OF DISTANCE MULTIPLICITIES BETWEEN COMPLEMENT BICM SUBSETS FOR VARIOUS G RAY- LABELED CONSTELLATIONS
All Distances
Simplified
BPSK
4
2
QPSK
16
2
16QAM
128
4
64QAM
768
8
TABLE II C OMPARISON BETWEEN NUMBER OF TERMS IN THE C-PEP EXPRESSION , BEFORE AND AFTER SIMPLIFICATION USING DISTANCE MULTIPLICITIES .
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In this way, the m.g.f. (11) can be written as ML (s) =
Nt |P M| X 1 X PM,i I˜k,EM,i (s) Nt i=1
(18)
k=1
Also, we can make use of the property [14] −1 det Skk S k,k = det (S)
(25)
where Skk corresponds to S with the k th row and column where | · | represents set cardinality, PM,i and EM,i denote removed. Combining (22)-(25) gives the ith element of PM and EM respectively, and I˜k,EM,i (s) is −d(Nr −Nt +1) given as in (16), but with the factor |a − a ˜|2 replaced by EM,i . γKS KR (26) fch (d, µ, A, γ) = In Table II we compare the number of terms in (18) and (11) 4 for Nt = 2, and for various modulation schemes. We see that where KR and KS are correlation factors corresponding to the (18) yields a significant computational advantage over (11), receive and transmit correlation matrices respectively, given by especially for high order constellations. 1 !− N −N We finally substitute (18) into (9), apply the Gaussr t +1 trNt −1 (λ(R)) ∆ KR = Chebyshev quadrature rules to the single remaining integraNr Nt −1 det (R) tion, and simplify to obtain Nt |P M| −(Nr −Nt +1) X v X det (S) 2 ∆ 1X (2t − 1) π PM,i KS = f (d, µ, A, γ) ≈
