Space-Time Linear Precoding and Iterative LMMSE Detection for ...

2011 IEEE International Symposium on Information Theory Proceedings

Space-Time Linear Precoding and Iterative LMMSE Detection for MIMO Channels without CSIT Xiaojun Yuan and Li Ping Department of Electronic Engineering, City University of Hong Kong, Hong Kong

Abstract—We propose a space-time coding scheme for efficient transmission over multiple-input multiple-output (MIMO) channels without channel state information at the transmitter (CSIT). The proposed scheme involves linear precoding (LP) at the transmitter and iterative linear minimum mean-square error (LMMSE) detection at the receiver. We develop a procedure to jointly optimize the forward-error-control (FEC) coding and LP, taking into consideration of the iterative detection process. Our analysis shows that the proposed scheme can perform close to the outage capacity of MIMO channels. I.

residual interference from the output of the LMMSE estimator. A main contribution of this paper is to develop a procedure to jointly design the FEC coding and LP, taking into account the iterative process. The proposed scheme has several distinguished advantages. It can provide performance close to the outage capacity, as demonstrated by both analysis and simulation results. It is very flexible and can be applied to an arbitrary number of antennas. It can also be implemented in a distributive way due to its random nature, and thus is well suited for ad hoc and relay applications. II. SYSTEM DESCRIPTION

INTRODUCTION

This paper is concerned with space-time coding for multiple-input multiple-output (MIMO) systems without channel state information at the transmitter (CSIT). The information-theoretic measure in this case is the outage capacity [1]. A coding technique is said to be universal if it can achieve the outage capacity. The solutions of universal codes are abundant for the multiple-input single-output (MISO) systems [2]-[5], such as those based on algebraic techniques [3][4]. Capacity-approaching quasi-random techniques [5] are also available for MISO channels. A key difference between a MISO channel and a more general MIMO channel is that the former has only one eigenmode while the latter has multiple eigenmodes. For MIMO channels, the Bell-Laboratories-Layered-Space-Time (BLAST) scheme [6] is widely studied. Theoretically, BLAST can asymptotically achieve outage capacity if universal codes designed for fading parallel channels are employed. However, the design of good universal codes in realistic channel conditions (i.e., with a finite SNR instead of an infinite SNR as in [7]) still remains an open problem. For this reason, there is a considerable gap between the simulated performance of a practical BLAST system and the outage capacity. Linear precoding (LP) [8][19] has been studied as an alternative to algebraic space-time coding. The available work is mostly on achieving diversity gain measured by the asymptotic slope of the performance curve. The related LP schemes usually perform far away from the outage capacity. One major reason is that those schemes employ sub-optimal detection methods (such as linear minimum mean-square error (LMMSE) estimation [19]). LP in general introduces inter-symbol interference (ISI). The residual interference at the output of the detector may cause significant performance loss. FEC coding is unable to resolve this problem completely if the residue interference is simply treated as noise. In this paper, we propose an LP-based space-time coding scheme for MIMO systems. We consider an iterative receiver, in which the decoder feedback is used to gradually eliminate the

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A. Channel Model A MIMO Gaussian channel with N transmit antennas and M receive antennas is generally modeled as ri = Hui + ηi (1) where ri is an N-by-1 received signal vector at the ith channel use, H is an N-by-M channel transfer matrix, ui is a transmit signal vector, ηi ∼ (0, σ2I) is an additive noise vector, and I is an identity matrix of a proper size. We assume that the channel is quasi-static. It remains unchanged in a frame consisting of K channel uses and is independent from frame to frame. The transmission power is normalized as N−1tr{Q} = 1, where Q = E[uiuiH] is the channel input covariance matrix. Perfect channel knowledge at the receiver is assumed. B. Outage Capacity For Gaussian signaling, the channel input-output mutual information per transmit antenna is given by 1 1 (2) C ( H , Q ) = log det I + 2 HQH H . N σ Let R be the transmission rate per transmit antenna. The outage capacity given an outage probability ε is defined as (3a) Coutage (ε ) = max R Q

s.t. (3b) Pr{C ( H , Q ) < R} ≤ ε . The problem in (3) is in general difficult to solve. It is conjectured in [8] that, without CSIT, independent transmission with equal power allocation is optimal, i.e., the optimal Q is given by Q = I. (4) We will assume that (4) is true throughout this paper. C. Transmitter Structure The coding scheme proposed in this paper is illustrated in Fig.1. At the transmitter side, the data is encoded by an FEC encoder (ENC), producing a codeword x. A transmission frame

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u (with the length denoted by L = K×N) is produced by applying a linear transform ΠP (referred to as a space-time linear precoder (STLP)) to x, i.e. u = ΠPx. (5) The STLP consists of two matrixes: Π and P. Π is an L×L permutation matrix and P is a block-diagonal matrix with K identical diagonal blocks, i.e., P = diag(F, F, …, F) = I⊗F, where “⊗” is the Kronecker product, and each block F is an N×N normalized DFT matrix with the (i, k)th entry given by (6) F (i, k ) = N −1/ 2 exp(− j 2π ik / N ) . We divide x into K blocks as x = {x0, x1, … xK−1}, and define vi = Fxi, for i = 0, 1, …, K−1. Correspondingly, u is equi-partitioned into K blocks: u = [u0T, u1T, ..., uK−1T]T. Each block is used for transmission over the channel in (1): ri = Hui + ηi , i = 0, 1, ..., K−1. (7) From our earlier assumption that the channel remains static for K blocks, (7) can be rewritten as r = HBD u + η (8) where r = [r0T, r1T, ..., rN−1T]T, HBD = diag(H, H, ..., H) = I⊗H (with the subscript meaning block-diagonal), and η = [η0T, η1T, ..., ηN−1T]T.

signal estimator (ESE) and the decoder (DEC). The two local operators are executed iteratively to suppress interference. The DEC follows standard soft-in soft-output decoding, and so only the ESE is discussed in detail below. The ESE estimates x following the LMMSE principle [9][10]. Let xPRI and vΙ, respectively, be the a priori mean and covariance matrix of x seen by the ESE.1 Note that xPRI = 0 and v = 1 at the first iteration, meaning that no information is available from the DEC. The well-known LMMSE estimator of x given y in (9) is given by (10a) x POST = x PRI + vH ′H R −1 ( y − H ′x PRI ) and the a posteriori MSE matrix as (10b) = vI − v2 H ′H R−1 H ′ where R is the a priori covariance matrix of y given by H (10c) R = vH ′H ′H + σ 2 I = vH BD H BD +σ 2I . Note that the last step above follows from (9b) and the fact that both P and Π are unitary matrices. The extrinsic mean xiEXT and variance viEXT of each xi are calculated by excluding the contribution of the a priori mean and variance from the a posteriori message as (cf., (34) in [10])

( viEXT ) EXT i EXT i

−1

= ( Ω (i , i ) ) − v − 1 −1

POST i

(11a)

PRI i

x x x . (11b) = − v Ω (i, i) v Finally, the ESE outputs the likelihood function p( xi | xiEXT ) for each xi, calculated by treating xiEXT as an observation from an equivalent AWGN channel with SNR ρi = 1 viEXT [10]. The complexity of the above LMMSE approach is dominated by the inversion of R. From (10c), R is a block-diagonal matrix with block size M-by-M and overall size KM-by-KM. Thus, the complexity involved in inverting R is only O(KM3).

and

III. PERFORMANCE ANALYSIS OF THE PROPOSED STLP SCHEME Fig.1. An example of the proposed transceiver structure with K = N = 4.

In this section, we analyze the performance of the STLP scheme using the signal-to-interference-plus-noise-ratio (SINR) -variance transfer chart proposed in [9].

The permutation matrix Π meets the following conditions: (i) For each index i, no two entries of vi are transmitted at a same channel use (so that the entries of vi do not interfere with each other at the receiver side); and (ii) No two entries of vi are transmitted at a same antenna (so that vi experiences all the channel fading conditions). The choices of Π satisfying conditions (i) and (ii) are plentiful. An example for K = N = 4 is illustrated in Fig.1. The precoding matrices P and Π ensure that the behavior of the iterative receiver can be characterized by single-variable transfer functions, as detailed in Section III. Combining (5)-(8), we obtain an equivalent channel as y=Hx+n (9a) H =HBDΠP. (9b) with The equivalent channel H′ introduces ISI in the received signal y. Iterative detection is used for efficient interference cancellation, as described next.

A. SINR-Variance Transfer (SVT) Chart Analysis In the SVT chart analysis, the behaviors of the ESE and DEC can be characterized by single-variable SVT functions, i.e. ρ = φ(v) for the ESE, and v = ψ(ρ) for the DEC. In the above, v is the input variance of the ESE (i.e., the output extrinsic variance of the DEC), and ρ is the output extrinsic SNR of the ESE (i.e., the input SNR of the DEC). With φ and ψ described above, the iterative process of the ESE and DEC can be tracked by a recursion of ρ and v. Let q be the iteration number. Based on (4) and (5), we have ρ ( q ) = φ (v ( q −1) ) and v ( q ) = ψ ( ρ ( q ) ) , q = 1, 2, … The recursion continues and converges to a point v* satisfying φ (v∗ ) = ψ −1 (v∗ ) and φ (v) > ψ −1 (v) , for v ∈ (v*, 1].

D. Iterative LMMSE Detection As illustrated in the lower half of Fig. 1, the iterative receiver [12] consists of two local operators, namely, the elementary

1 We assume that random interleaving is performed in constructing x so that, for a sufficiently large frame length, the inputs of the ESE/DEC are independent and identically distributed (cf., [9], [10], and [17]).

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where ψ−1(⋅) is the inverse of ψ(⋅), which exists since ψ(⋅) is monotonic (cf., [16] for the monotonicity). In particular, v* = 0 implies that the transmit signal can be perfectly recovered at the receiver side. Thus, we say that the system performs error-free decoding and the corresponding rate is achievable if v* = 0, or equivalently, φ (v) > ψ −1 (v) , for v ∈ (0, 1]. (12) B. SVT Function of the ESE We next show that the ESE outputs can be characterized by a single SINR. Let λn be the nth eigenvalue of HHH. Our main result is presented below. Proposition 1: The output SNR of the ESE is given by

1 ρ i = ρ = φ (v ) = N

N −1 n=0

1 λn + v σ2

−1

−1

1 − , v

(13)

where ρi is invariant with respect to the index i. Proof: With (9b) and HBD = I⊗H, we can rewrite (10b) as = P H H ( I ⊗ B) P = P H P B = vI − v2 H H (vHH H + σ 2 I )−1 H .

with

= H ( I ⊗ B) in a block-wise form with each block of size N-by-N. Let Di be the ith diagonal block of Φ. Every Di has the following properties: • Di is a diagonal matrix; • Di has the same diagonal entries as B has (counting multiplicity but not necessarily in the same order). The first property above is ensured by condition (i) of Π in Section II.C; and the second property by condition (ii). Together with P = I⊗F, we obtain diag{F H D0 F} 0

We can express

diag{ } =

. H

diag{F DN −1F}

0 Then

(b) v2λn 1 1 N −1 1 N −1 −1 λn tr{B} = = v− v + 2 2 N N n =0 vλn + σ N n=0 σ where step (a) follows from the fact that, for each i, (a)

Ω (i, i) =

(

diag{F H Di F} = N −1

N −1 i =0

−1

(14)

)

D(i, i) I = N −1tr{B}I ,

and step (b) from the fact that the ith eigenvalue of B is given by v − v2 λn (vλn + σ 2 )−1 . Substituting (14) into (11a) and noting ρi = 1 viEXT , we arrive at (13). C. SVT Function of the DEC Now consider the DEC module. Let C be the FEC codebook employed by the DEC. From the discussions in Section II.D and Lemma 1, the DEC decodes x based on x EXT ≡ {xiEXT } which can be modeled as observations from an equivalent AWGN channel with SNR = ρ. The output extrinsic variance of each xi is defined as [10] 1 N −1 2 v= E xi − E[ xi | x~EXT (15a) i , x ∈ C] N i =0

where x~iEXT represents the subset obtained by removing xiEXT from x EXT . Clearly, v is a function of ρ. Thus, the SVT function of the DEC can be obtained from (15a) by varying ρ. The MMSE of each xi after decoding is defined as 2 1 mmse( ρ ) = E x − E[ x | x EXT , x ∈ C ] . (15b) N Compared with (15a), the MMSE in (15b) is obtained by excluding the contribution of xiEXT in estimating xi. We next establish a relationship between ψ(ρ) and the code rate (denoted by R) by assuming that x is Gaussian distributed and the decoding output can be modeled as observations from an equivalent AWGN channel. Based on the Gaussian-message combining rule (cf., (9) in [10]), mmse(ρ) and ψ(ρ) are related as [10] mmse = (ψ ( ρ ) −1 + ρ ) . −1

(16)

From [14, Lemma 1], we can express the code rate R as ∞ ∞ 1 (17) dρ . R = mmse ⋅ dρ = −1 +ρ ψ ( ρ ) 0 0 The rate in (17), though derived based on some unjustified assumptions, is indeed approachable using superposition coded modulation (SCM), as stated in the proposition below. Denote by Ln an n-layer SCM code with rate Rn. Let ψn(ρ) be the SVT function of Ln.

Proposition 2: Assume that ψ(ρ) is first-order differentiable and monotonically decreasing in ρ, and that +∞ 1 (18) R= dρ < ∞ . −1 +ρ 0 ψ (ρ ) There exists {Ln }∞n =1 such that: (i) ψn(ρ) ≤ ψ(ρ) for any index n and ρ ≥ 0; and (ii) Rn → R as n→ ∞. The proof of Proposition 2 is omitted here due to space limitation; the details can be found in [18]. This proposition says that, for an arbitrary ψ(ρ) (satisfying some regularity conditions), we can construct an SCM code to approximately approach the rate in (18). Later, we will see that (18) plays an important role in analyzing the performance of the proposed STLP scheme. D. Curve-Matching Principle We next consider the optimization of the STLP scheme by assuming perfect CSIT. We will show that the proposed scheme is potentially information lossless. Definition 1: The transfer functions φ and ψ are matched if (19) φ (v) = ψ −1 (v) , for v ∈ (0, 1]. Note that: φ(1) > 0 since the ESE output always contains the information from the channel even if there is no information from the DEC; and that φ(0) < ∞ since the ESE cannot resolve the uncertainty introduced by the channel noise even if the messages from the DEC are perfectly reliable. Then, we can equivalently express the curve-matching condition (19) as (20a) ψ ( ρ ) = φ −1 (φ (1)) = 1 , for 0 ≤ ρ < φ(1);

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ψ ( ρ ) = φ −1 ( ρ ) ,

for φ(1) ≤ ρ < φ(0);

(20b)

ψ (ρ ) = 0 ,

for φ(0) ≤ ρ < +∞. We are now ready to present the following result.

(20c)

Proposition 3: The maximum achievable rate of the proposed STLP scheme is given by λ 1 N −1 R= log 1 + n2 , (21) N n=0 σ which is achieved when the ESE and DEC are matched. Proof: An achievable rate of the scheme is given by ( a ) φ (0)

R=

1 dρ + ( φ ( ρ )) −1 + ρ φ (1)

φ (1)

−1

0

1 dρ 1+ ρ

(b ) 0

=

1 φ ′(v)dv + log(1 + φ (1)) −1 v + φ (v ) 1

(c) 0

=

1

−

dω ( v ) ω ( v ) + 2 dv − log ω (1) ω (v ) v

= − log ω (v) −

1 N

N −1 n=0

log

1 λn + v σ2

v=0

− log ω (1)

v =1

λ 1 N −1 log 1 + n2 N n =0 σ where step (a) follows from Proposition 2 and the curve-matching condition in (20), (b) by substituting ρ = φ(v), (c) from the fact that dω (v) dv φ ′(v)dv = − + ω (v ) 2 v 2 with ω(v) defined as −1 1 1 N −1 1 λn , ω (v) = −1 = + 2 v + φ (v ) N n = 0 v σ and (d) from the fact that (d )

=

lim ω (v) v→0

N −1

∏ n=0

1 λn + v σ2

1 N

=1.

any v ∈ (0, 1] (or equivalently, (12) is not met). Thus, we say that the system is in outage if (12) is not met. We optimize ψ(ρ) to maximize the achievable rate given a target outage probability ε. This problem can be formulated using (18) as: ∞ 1 (22a) max dρ −1 ψ (ρ) +ρ 0 ψ (ρ ) s.t.

Pr {φ (v) > ψ −1 (v), for v ∈ (0,1]} ≥ 1 − ε .

(22b)

Note that the probability in (22b) is defined on the distribution of H as φ(v) in (13) is a function of the channel H. The problem in (22) is in general difficult to solve. To simplify the problem, we first discretize the probability space of H by random sampling. Let S be a set of random realizations of H. Denote by Hi the ith channel realization in S, and by φi(v) the corresponding ESE SVT function. Let Sε be a subset of S with size |Sε| ≤ ε|S|, and S1-ε be the complementary set of Sε. Then, we formulate the following problem: ∞ 1 (23a) max dρ −1 ψ ( ρ ), S1−ε ψ ( ρ ) +ρ 0 s.t. (23b) φi (v) > ψ −1 (v), for v ∈ (0,1], ∀H i ∈ S1−ε . Based on the law of large numbers, the empirical distribution of {Hi} converges to the probability distribution of H. Thus, (23) converges to (22) as |S| tends to infinity. In practice, we may choose a sufficiently large |S|, so as to ensure that (23) is a good approximation of (22). Now consider solving (23). The constraint in (23b) can be equivalently rewritten as ψ ( ρ ) < φi−1 ( ρ ), for ρ ∈ [0, ∞), ∀H i ∈ S1−ε . Together with the fact that the objective function in (23a) is monotonically increasing in ψ(⋅), the optimal ψ(⋅) for a given S1-ε can be expressed as ψˆ ( ρ ; S1−ε ) = min φi−1 ( ρ ) , for ρ ∈ [0, ∞) . (24) i: H i ∈S1−ε

Note that the rate in (21) is the channel input-output mutual information with Q = I (cf., (2)). From the Shannon’s channel coding theorem, any reliable transmission cannot exceed this rate (given Q = I), hence the proof of Proposition 3. From the above proof, we see that the proposed STLP scheme is information lossless if the ESE and DEC are matched. However, perfect curve-matching requires full CSIT. We next discuss the design of the STLP scheme with no CSIT. IV. FEC CODE OPTIMIZATION FOR NO CSIT The design of exactly matched FEC code for the proposed STLP scheme is not feasible if the CSIT contains uncertainty (since then the ESE transfer curve is not perfectly known by the transmitter). Instead, we can design a fixed FEC code that statistically best matches the distribution of the ESE transfer curves, which is our focus below. Consider an FEC code with the SVT function denoted by ψ(ρ). Let H be a random channel realization, with the corresponding SVT function φ(v) given by (13). The system cannot perform error-free decoding if φ(v) falls below ψ−1(v) for

Then, the optimization problem in (23) reduces to ∞ 1 max dρ . −1 S1−ε ˆ ψ ( ρ ; S +ρ 1−ε ) 0

(25)

The number of different choices of S1-ε is exponential in |S|. A brute-force search for the optimal solution to (25) involves high complexity even for a moderate |S|. Here, we introduce a greedy algorithm to search for a suboptimal solution. Let T be a set of channel realizations. Define ∞ 1 R(T ) = dρ . ˆ ψ ( ρ ; T ) −1 + ρ 0 The greedy algorithm works as follows. Denote by S(q) the remaining subset of S at the qth round of recursion. Initialize S(0) = S. Each S(q+1) is constructed by removing one channel realization from S(q) such that R(S(q+1)) is maximized. The recursion continues until q = ε|S|. Finally, we obtain S1-ε = S(ε|S|) and the optimized ψ ( ρ ) = ψˆ ( ρ ; S1−ε ) . V. NUMERICAL RESULTS We next demonstrate by numerical results that the above algorithm yields a good choice of ψ(ρ). Consider a block-fading

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4×4 MIMO channel. The channel coefficients are i.i.d. circularly complex Gaussian random variables; they remain unchanged for every 512 channel uses. Each transmission frame consists of 2048 symbols. The system throughput is 4 bits per channel use. The receiver performs iterative LMMSE detection. The optimized ψ(ρ) is obtained by solving (23) using the greedy algorithm with |S| = 5000 and ε = 0.01. The performance limit of the proposed STPL scheme is then obtained by calculating the outage probability of the optimized ψ(ρ). To approach the above performance limit, we consider the design of irregular LDPC codes that matches the predetermined ψ(ρ) (obtained by solving (23)), following the design method proposed in [15]. The optimized degree distributions are given by λ(x) = 0.3315x+0.1872x2+0.0215x14+0.2764x15+0.1555x76 +0.0279x77 and ρ(x) = x7. The code rate is 0.50. For comparison, we also consider the irregular LDPC code optimized for AWGN channel (with the degree distributions given by the last column of Table I in [17]). The coded bits from the LDPC encoder are Gray-mapped into QPSK symbols in transmission. Fig.3 demonstrates the simulated FER performance of the proposed STLP scheme. At FER = 10-2, the performance limit of the proposed scheme is only 0.3 dB away from the outage capacity. We see that the optimized irregular LDPC code achieves a gain of 0.8 dB over the LDPC code in [17], but there is still a gap of 1.3 dB compared with the performance limit. This gap is mainly caused by the binary constraint of the LDPC code used. We will explore the possibility of narrowing this gap in our future work.

ACKNOWLEDGEMENTS This work has been performed in the framework of the ICT project ICT-248894 WHERE2, which is partly funded by the European Union. REFERENCES [1] [2]

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[16] Fig. 3. The outage capacity and the performance of the proposed STLP scheme in the quasi-static Rayleigh-fading 4x4 MIMO channel. System throughput = 4 bits per channel use.

[17] [18] [19]

VI. CONCLUSIONS In this paper, we propose an STLP scheme for efficient transmission over MIMO channels without CSIT. We establish an analytical tool to characterize the performance limit of the proposed scheme. Numerical results show that the optimized STLP scheme can potentially perform close to the outage capacity of a MIMO channel within a marginal gap.

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