Fifth IEEE Workshop on Signal Processing Advances in Wireless Communications, Lisboa, Portugal, July 11-14, 2004
Code-aided joint channel estimation and frame synchronization for MIMO systems Henk Wymeersch
Frederik Simoens
Marc Moeneclaey
Digcom Research Group TELIN Dept. Ghent University, Sint-Pietersnieuwstraat 41, 9000 GENT, BELGIUM E-mail:
[email protected]
Digcom Research Group TELIN Dept. Ghent University, Sint-Pietersnieuwstraat 41, 9000 GENT, BELGIUM E-mail:
[email protected]
Digcom Research Group TELIN Dept. Ghent University, Sint-Pietersnieuwstraat 41, 9000 GENT, BELGIUM E-mail:
[email protected]
Abstract — This contribution deals with the problem of joint frame synchronization and channel estimation for a system using bit-interleaved coded modulation in a MIMO context. A receiver, based on the EM algorithm, is derived. This receiver iterates between detection and estimation. We illustrate how initial estimates may be obtained and how convergence problems may be avoided. Through computer simulations the performance of the proposed receiver is investigated, both in terms of frame error rate (FER) and estimation error variance. We show that exploiting code properties for estimation purposes allows to reduce the length of training sequences, and thus results in an increase in spectral efficiency.
I. Introduction The diversity resulting from the use of multiple antennas at the transmitter and/or the receiver has been advocated as a means to increase capacity and combat fading effects [1]. While a number of powerful coding techniques have been proposed for MIMO channels (such as space-time codes [2, 3] and coded modulation schemes [4]), these invariably assume ideal coherent detection, including perfect knowledge of the channel gains, propagation delay, etc. To achieve accurate synchronization without suffering the loss in spectral efficiency related to conventional dataaided (DA) estimation algorithms, so-called code-aided algorithms have recently been getting a lot of attention. These generally fall into two classes: on the one hand there exist several ad-hoc algorithms that iterate between decoding and estimation [5, 6]. On the other hand several research groups have been focusing on the EM algorithm [7] as a means of performing code-aided estimation. The impact of delay estimation errors or frame synchronization errors on such schemes applied to MIMO configurations has not yet been investigated. The latter problem will be the main focus of this paper. In this contribution we propose a new EM-based receiver that performs joint channel estimation and frame synchronization. While these problems have been tackled separately (frame synchronization in the context of Single-Input Single-Output channels [8, 9] and channel estimation for MIMO channels [4]), we here extend this research to joint
channel estimation and frame synchronization in a MIMO context. This paper is structured as follows: in section II we present the system setup. Section III describes how ML estimation may be performed by means of the EM algorithm. This algorithm is then applied to the MIMO system in section IV. Simulation results are presented in section V before conclusions are drawn in section VI. Our main conclusion is that performing code-aided synchronization and channel estimation can increase the overall spectral efficiency of the system, at the cost of additional complexity at the receiver. II. System Model We consider a block-fading MIMO system with NT transmit and NR receive antennas. The transmitter is shown in Fig. 1: a block of information bits is channelencoded, interleaved (denoted by Π in Fig. 1) and mapped onto a unit-variance M -point signaling constellation. The resulting frame, consisting of LNT constellation symbols, is then converted into NT sub-blocks of length L and sent in parallel over the NT transmit antennas using a square root cosine roll-off pulse p (t). The energy per symbol is denoted Es . The static block fading channel response matrix is denoted by H. This matrix is constant during a frame, but can change from one frame to the next. In addition, the channel introduces a propagation delay (τ ). We assume that some form of energy detection takes place to determine the arrival of a burst within Mτ symbol intervals. Finally, each received signal is independently corrupted by a complex AWGN process with spectral density N0 . The received signal vector, normalized by a factor (E s )−1 , at time t is given by:
y (t) =
H
L−1 X k=0
ak p (t − kT − τ ) + w (t)
(1)
with T m • ak = [am k ]m=1...NT , where ak is the k-th symbol transmitted at the m-the transmit antenna
• y (t) = [yn (t)]Tn=1...NR , where yn (t) is the input signal at the n-th receive antenna
ˆ kˆτ H,
update source
encoder
mapper
Π
S/P
CHANNEL
estimate of
H, kτ
detected symbols DETECTOR
Z
ˆ kˆτ ) P (ak |y, H,
Figure 1: MIMO Transmitter with channel coder and bitinterleaving
Figure 2: MIMO Receiver with EM estimation • w(t) = [wn (t)]Tn=1...NR , where wn (t) is a complex AWGN process (with power spectral density N0 /Es ) at the n-th receive antenna. We denote by A the NT × L space-time matrix of transmitted symbols: A = [a0 , a1 , . . . , aL−1 ]. The goal of the receiver is two-fold: apart from detecting the transmitted sequence A, it needs to estimate the channel matrix H and the propagation delay τ . The delay τ can be broken up into τ = kτ T + ετ , with 0 ≤ ετ < T and kτ ∈ {0, . . . , Mτ − 1}. A very accurate estimate of ετ can in many cases be obtained using the Oerder&Meyr estimator [10], independently of the transmitted symbols and the channel matrix. For that reason we will focus on joint frame synchronization (i.e., determining the discrete valued parameter kτ ) and channel estimation (i.e., determining the continuous valued matrix H), assuming that ε τ is known. III. ML estimation through the EM algorithm Assume we want to estimate a parameter vector b from an observation r in the presence of a so-called nuisance vector a. The maximum likelihood estimate of b maximizes the log-likelihood function (LLF): n o ˜ ˆ M L = arg max ln p r b (2) b ˜ b
where
Z ˜ ˜ p (a) da. = p r a, b p r b
˜ is difficult to calculate. The EM algorithm Often p r b is a method which iteratively solves (2). Defining the complete data x = [r, a], the EM algorithm breaks up in two parts: the Expectation part (Eq. 4) and the Maximization part (Eq. 5). In the case that a and b are independent:
ˆ (i+1) b
h i ˜ = Ea|r,bˆ (i) ln p r|a, b n o ˜ b ˆ (i) = arg max Q b, ˜ b
o n oo n n ˜ d, b ˆ (i) b ˜d ˜ d, b ˜c , b ˆ (i+1) b ˜ d = arg max Q b . b c c ˜c b
(6) (0) ˜ ˆ An initial estimate bc bd is required to start the pro ˆc b ˜ d the estimate obtained after cess. Let us denote by b convergence of (6). The final estimate of bd then becomes ˆ d = arg max b
˜ d ∈Bd b
IV. Code-aided estimation for MIMO systems In this section we apply the EM algorithm to the MIMO system from section II. The resulting receiver is shown in Fig. 2. It operates by iterating between detection and estimation. A. Expectation step
Denoting by y a random vector obtained by expanding y (t) onto a suitable basis, we now make use of the EM algorithm for estimating b = {H, kτ }, with the coded data symbols denoting the nuisance parameter. Defining the NR × L matrix of matched filter outputs Z (kτ ) by R +∞ [Z (kτ )]n,k = −∞ yn (t)p∗ (t − kT − kτ T − ετ )dt, it can easily be shown that
(4)
ln p ( y| A, H, kτ ) ∝ − tr HH AAH H + 2< tr Z (kτ ) AH HH .
(5)
where Ex|y [.] denotes the expectation w.r.t. x conditioned ˆ (n) converges to a stationary on y. It has been shown that b point of the LLF under fairly general conditions [7]. To avoid the convergence problems associated with discrete parameters [11], we propose the following solution [9]: let b = {bd , bc }, where bd and bc denote the discrete and
n n o n oo ˜ d, b ˆc b ˜d , b ˜ d, b ˆc b ˜d Q b
(7) ˆ ˆ while the final estimate of bc is given by bc bd . Note that no initial estimate of bd is required. The EM algorithm can easily be extended to acquire the Maximum a Posteriori (MAP) estimate of b, by taking the a priori distribution p(b) into account in (2).
(3)
a
˜ b ˆ (i) Q b,
continuous components of b, respectively. We assume that bd can only take on values in a finite set Bd . We keep bd ˜ d ∈ Bd and iteratively update bc : fixed to some value b
The E-step in the EM algorithm can be written as: n
o n o ˆ (i) k˜τ , H , kˆτ , H = (8) (i) (i) − tr HH AAH H + 2< tr Z k˜τ AH HH
Q
0
with
10
AH
(i) (i)
= =
i h ˆ (i) Ea AAH y, b i h ˆ (i) Ea AH y, b
i h ˆ (i) and Eq. (8) contains terms of the form Ea ak | y, b i h ˆ (i) . The marginal posterior probabilities of y, b E a ak aH k ˆ (i) the coded symbol vectors P ak y, b can be computed by a MAP detector. Although a true MAP detector is often prohibitively complex, we may resort to sub-optimal iterative detectors, such as described in [4]. Thanks to the presence of the interleaver at transmitter side, the marginal posterior probabilities of the coded symbol vectors can be easily expressed in terms of the received signal and the so-called extrinsic probabilities of the codes bits. We will denote by ID the number of iterations within the detector.
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10
FER
AAH
Perfect EM 0 Pilot DA 4 Pilot DA 8 Pilot
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0
1
E /N (dB) b
2
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0
Figure 3: FER frame synchronization when CSI is known compared to perfect synchronization and CSI.
B. Maximization step
The maximization problem (6) can be solved as follows (for a fixed value of k˜τ ∈ {0, . . . , Mτ − 1}) (i) −1 (i) H (i+1) ˜ H ˜ ˆ H (kτ ) = Z(kτ )A . AA
The updated channel matrix estimate is now forwarded to ˆ k˜τ ) is achieved, the the detector. Once convergence of H( parameter kτ is determined according to (7). n kˆτ = arg maxk˜τ −tr HH (k˜τ )AAH H(k˜τ ) o . +2< tr Z k˜τ AH HH (k˜τ ) V. Performance results We have carried out computer simulations for a rate 1/3 turbo coded system with BPSK signaling on a 2 transmit, 3 receive antennas set-up. The considered turbo code is a parallel concatenation of two recursive systematic binary convolutional codes with octal generators (37,33) and a total frame length of 840 bits. In order to reduce the computational complexity of the receiver, only one internal decoding iteration is performed for each EM-iteration (i.e., ID = 1). Iterations between estimation and detection are carried out until convergence of the EM algorithm. Although this approach requires more EM iterations to achieve convergence, the total number of decoding iterations and thus the global complexity will substantially decrease. Data and pilot symbols (the latter are needed to obtain a DA channel estimate to initiate the iterations) are randomly chosen in all simulations. The channel matrix is changed randomly for each frame, with each coefficient having a zero-mean complex gaussian distribution with variance 1. The delay kτ is chosen uniformly within {0, 1, 2} for each frame. The number of pilot symbols mentioned in
the sequel corresponds to the number of pilot symbols per transmit antenna. The considered performance measure will be the frame error rate (FER) and the mean square error (MSE) of the channel estimates. In all figures, ’perfect’ refers to perfect channel state information (CSI) and perfect frame synchronization. We consider the following situations: 1. frame synchronization with perfect CSI 2. channel estimation with perfect frame synchronization 3. Joint frame synchronization and channel estimation. In the first scenario the channel matrix H is considered to be known. As illustrated in Fig. 3, DA frame synchronization requires at least 8 pilot symbols to achieve low degradations. Application of the EM algorithm results in almost perfect FER performance, even when no pilot sequence is present. Due to the inherent randomness of the turbo-code, time-shifted versions of a codeword are very unlikely to be another codeword. Hence frame-synchronization is possible, based on information from the coded data symbols only. In the other two scenarios, channel estimation is involved. As explained in section III, the EM algorithm reˆ (0) ). This estimate is quires an initial channel estimate (H obtained from a short pilot sequence. Let us consider the second scenario (kτ is known). We observe in the left part of Fig. 4 that the EM algorithm yields a significant performance gain compared to DA estimation. For example, the EM algorithm using 8 pilot symbols to provide the initial channel estimate gives rise to a performance gain of almost 3 dB as compared to the DA channel estimator. However, it should be noted that the EM algorithm using 8 pilot symbols for the initial estimate
0
0
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10
Perfect EM 8 Pilot EM 16 Pilot DA 8 Pilot DA 16 Pilot
−1
EM 8 Pilot EM 16 Pilot DA 8 Pilot DA 16 Pilot MCRB
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FER
MSE
10
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E /N (dB) b
1
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E /N (dB) b
1
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0
Figure 4: FER (left) and MSE (right) for channel estimation with perfect frame synchronization. still loses 1 dB in FER performance, compared to perfect CSI. The right part of Fig. 4 indicates that the channel matrix MSE approaches the modified Cramer-Rao bound (MCRB) [12] above a certain SNR threshold, provided that the initial channel estimate is sufficiently accurate. Using 16 pilot symbols, for example, the MCRB is reached for Eb /N0 > 2 dB. In this case, only a small FER degradation is encountered compared to perfect CSI (Fig. 4, left part). We have verified (results not shown) that DA channel estimation requires more than 64 pilot symbols to attain the same FER performance as EM estimation initiated with 16 pilot symbols. Finally, we investigate joint frame synchronization and channel estimation. Again, pilot symbols are necessary to provide an initial channel matrix estimate for every trial value kτ . Fig. 5 shows the FER performance using 8 pilot symbols. We also include the FER performance when the EM algorithm is used solely to provide an estimate of H, while frame synchronization is performed exploiting only the pilot symbols (i.e., kτ is excluded from the EM algorithm). We observe that the latter scheme results in little or no degradation as compared to joint EM frame synchronization and channel estimation. This means that a significant reduction in computational complexity can be achieved (namely a factor Mτ ) without any FER degradation, by using DA frame synchronization and EM-based channel estimation. VI. Conclusions
• joint frame synchronization and channel estimation results in little or no performance degradation compared to channel estimation with perfect frame synchronization. • since a substantial number of pilot symbols are necessary to provide a good initial channel estimate, frame synchronization can be carried out based solely on pilot symbols. As a result, code-aided frame synchronization is especially attractive when conventional NDA and/or DA estimation algorithms fail to provide reliable estimates. In addition, we have shown how the computational complexity may be reduced by performing only one decoding operation per EM iteration. Acknowledgments This work has been supported by the Interuniversity Attraction Poles Programme - P5/11 - Belgian Science Policy. The second author also acknowledges the support from the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen). References [1] G.J. Foshini and M.J. Gans. ”On limits of wireless communication in a fading environment when using multiple amtennas”. Wireless Personal Communications, 6(3):311–335, March 1998. [2] V. Tarokh, N. Seshadri and A. R. Calderbank. ”Space-time codes for high data rate wireless communications: Performance criterion and code construction”. IEEE Trans. Comm., 44(2):744–765, March 1998.
In this contribution we have proposed an EM-based receiver that performs joint detection, frame synchronization and channel estimation. We made the following observations:
[3] C. Cozzo and B.L. Hughes. ”Joint Channel Estimation and Data Detection in Space-Time Communications”. IEEE Trans. Comm., 51(8):1266–1270, aug 2003.
• for turbo codes, frame synchronization can be performed without the use of any pilot symbols when the channel is perfectly known.
[4] J.J. Boutros, F. Boixadera and C. Lamy. ”Bit-interleaved coded modulations for multiple-input multiple-output channels”. In Proceedings of the IEEE 6th Int. Symp. on Spread-Spectrum Tech. & Appli., pages 123–126, New Jersey, USA, Sept 2000.
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Figure 5: FER for 8 pilot symbols for EM joint frame synchronization and channel estimation compared to DA frame synchronization with EM channel estimation [5] A. Grant. ”Joint Decoding and Channel Estimation for Linear MIMO Channels”. In In proceedings of Wireless Comm. and Networking Conf., pages 1009–1012, Chicago, USA, sep 2000. [6] R. Visoz and A.O. Berthet. ”Iterative Decoding and Channel Estimation for Space-Time BICM over MIMO Block Fading Multipath AWGN Channel”. IEEE Trans. Comm., 51(8):1358–1367, aug 2003. [7] A.P. Dempster, N.M. Laird and D.B. Rubin. ”Maximum likelihood from incomplete data via the EM algorithm”. Journal of the Royal Statistical Society, 39(1):pp. 1–38, 1977. Series B. [8] N. Noels, V. Lottici, A. Dejonghe, H. Steendam, M. Moeneclaey, M. Luise and L. Vandendorpe. ”A theoretical framework for softinformation-based synchronization in iterative (turbo) receivers”. submitted to IEEE Trans. Comm. unpublished. [9] H. Wymeersch and M. Moeneclaey. ”ML frame synchronization for turbo and LDPC codes”. In Proc. 7th Int. Symp. on DSP and Comm. Systems, Coolangatta, Australia, December 2003. [10] M. Oerder and H. Meyr. ”Digital filter and square timing recovery”. IEEE Trans. Comm., 36:605–611, May 1988. [11] P. Spasojevic and C.N. Georghiades. ”On the (non) convergence of the EM algorithm for discrete parameter estimation”. In Proc. Allerton Conference, Monticello, Illinois, Oct. 2000. [12] A. N. D’Andrea, U. Mengali and R. Reggiannini. ”The modified Cramer-Rao bound and its application to synchronization parameters,”. IEEE Trans. Comm., 42:1391–1399, Feb/Mar/Apr 1994.
