Estimation for MIMO Wireless Communications. Jongsoo Choi, Martin Bouchard and Tet Hin Yeap. School of Information Technology and Engineering.
Adaptive Filtering-Based Iterative Channel Estimation for MIMO Wireless Communications Jongsoo Choi, Martin Bouchard and Tet Hin Yeap School of Information Technology and Engineering University of Ottawa Ottawa, Ontario, Canada, K1N 6N5 Email:{jchoi,bouchard,tet}@site.uottawa.ca
Abstract— Accurate channel state information (CSI) is a prerequisite for diversity and multiplexing gains in multiple-input multiple-output (MIMO) wireless systems. More refined CSI can be attained by bootstrapping channel estimation techniques. In this paper, we propose adaptive filtering-based iterative channel estimators with the incorporation of an iterative receiver over a flat-fading MIMO wireless link. The performance of three different algorithms with the aid of hard decision feedback is evaluated and compared through simulations. We show that the least-mean-square algorithm is a reasonable choice among the other considered algorithms in terms of the performance and feasibility. Keywords – Adaptive filtering, channel estimation, iterative receiver, MIMO wireless link.
I. I NTRODUCTION The use of a multiple-input multiple-output (MIMO) wireless link has made the promise of higher data rates and increased spectral efficiency through increased computational complexity [1]. Future wireless communications will operate under a MIMO antenna link to support a wide range of services requiring high transmission data rates. Accurate channel state information (CSI) is a prerequisite for the full exploitation of the advantages of MIMO systems. However, in real wireless scenarios the perfect CSI is never known to the receiver. Obviously channel estimation becomes a key topic for future applications of the MIMO system. Recently, iterative (turbo) receivers for MIMO wireless links, which perform joint detection and decoding, have received great attention due to their performance gain achieved. A popular estimation technique incorporated in an iterative receiver is iterative channel estimation [2],[3]. In an iterative channel estimation method, both pilot symbols and soft (or hard) estimates of the data symbols are used to improve the channel quality in a semiblind manner which distinguishes itself from a blind procedure and a supervised procedure. Adaptive filtering techniques may be a natural choice for iterative channel estimation, because of their computational efficiency compared to one-shot approaches [3],[4], such as least-squares (LS) and linear minimum mean-square error (MMSE) methods which need matrix inversions. At a cost of complexity, both recursive least-squares (RLS) [5],[6] and Kalman filter (KF) [7],[8] algorithms are extensively used for channel estimation in wireless communications, while a least-mean-square (LMS) algorithm is rarely used despite its
computational efficiency and feasibility. In [5] and [6], RLS methods are used for iterative channel estimation of MIMO systems, but only the soft estimate of data symbols was evaluated. Although in [7] and [8], KF techniques are applied for tracking MIMO channels, they are not in an iterative estimation mode. The performance of adaptive filtering-based iterative channel estimators is compared in [9], but it is performed in a single transmit and single receive link. One should note that soft iterative channel estimation does not always guarantee the refinement of channel estimates, especially at a lower signal-to-noise ratio (SNR). The authors [10] have revealed that hard iterative channel estimation can attain a better performance gain than soft iterative channel estimation under a BPSK modulation. In this paper, three adaptive filtering-based iterative channel estimation techniques using hard decisions are evaluated with the incorporation of an iterative receiver, turbo-BLAST [11]. The performance of three algorithms is compared through Monte Carlo simulations. II. S IGNAL M ODEL A ND T RANSCEIVER S TRUCTURE A. Signal Model We consider a MIMO wireless link with M transmit and N receive antennas in a flat-fading environment. We assume that the channel matrix is modeled as Rayleigh random processes and the fading is to be uncorrelated across antennas where each individual realization of the channel path is independent for all time steps k. The received signal vector r ∈ C N ×1 on the receiver at symbol time k can be written as r[k] = H[k]s[k] + v[k]
(1)
where H ∈ C (N ×M ) denotes the channel matrix, s ∈ C M ×1 is the symbol vector simultaneously transmitted by the M transmit antennas, and v ∈ C N ×1 denotes the complex additive white Gaussian noise vector with zero mean and covariance matrix σv2 IN . Throughout the paper we assume that transmission and receiving are perfectly synchronized with symbol timing. B. Transmitter Structure A high-level description of a turbo-BLAST transmitter [11] employing the random layered space-time codes is depicted in Fig. 1(a). User information stream b is demultiplexed into
M substreams {bm }M m=1 of an equal data rate. Each data substream is independently block-encoded, where the block encoders use the same linear block codes with code generator G(D). The encoded substreams are represented as C = [b1 G b2 G . . . bM G] = [c1 c2 . . . cM ]T
(2)
where {bm }M m=1 is J-dimensional information sequences, {cm }M m=1 is K-dimensional code sequences, and G is (J × K) binary code generator. The encoded substreams are bitinterleaved using a pre-designed space-time random interleaver, Π, and the interleaved substreams are denoted by A = Π(C). The space-time interleaved substreams are independently mapped into symbol matrix S = f (A), where f (·) relies on a modulation scheme. The modulated substreams are transmitted using the M transmit antennas. b1 b2 b
Demux (1:M)
c1
G(D)
c2
G(D)
.. .
Space-Time Interleaver (Π)
.. .
Encoder
bM
a1
cM
G(D)
a2
Mod Mod
s1 s2
.. . aM
Mod
Π
L(aM;i)
L(c1; p)
+
+ _
L(cM;e)
+
+
. . .
. . .
^ H
L(cM; p)
r1
.. .
SISO Detector (MMSE)
rN
.. .
-1
Π
L(c1;i) L(cM;i)
L(aM;e)
.. .
SISO Dec 1
.. . ^ b b^1
SISO Dec M
Mux (M:1)
^ b
(6)
(7)
This extrinsic information yields the intrinsic information for the inner SISO detector after interleaving as follows
M
L(A; i) = Π(L(C; e))
Outer Decoders
Inner Detector
where L(ck ; i) and L(cm ; e) denote a priori (or intrinsic) information and extrinsic information of the coded bit cm , respectively. The extrinsic information of the inner SISO detector, L(A; e), is deinterleaved to compensate for the pseudorandom interleaving used in the transmitter before feeding to the outer SISO decoders, yielding
L(C; e) = L(C; p) − L(C; i)
_
L(a1;e)
(5)
In turn, the M parallel SISO decoders process the soft input L(cm ; e) and compute the refined estimates of soft information on both code bits cm and information bits bm based on the trellis structure of the channel code constraint. The extrinsic information of code bits from the M outer decoders is given by
. . . L(c1;e)
L(a1;i)
L(cm ; p) = L(cm ; i) + L(cm ; e)
L(C; i) = Π−1 (L(A; e))
sM
(a) Iterative Channel Estimator
where wm is a linear filter for the mth substream and E{sm } is the statistical expectation of interfering substreams. The expectations of the transmitted data are computed using the soft outputs of the symbols, which are log-likelihood ratios (LLRs), provided by the outer SISO decoders. 2) Outer SISO Decoders: The outer SISO decoders for convolutionally encoded symbols perform symbol-by-symbol log-MAP decoding realized by the BCJR algorithm [12]. The a posteriori LLR of a coded bit cm , m = 1, 2, . . . , M , conditioned on the received symbol vector r, is defined as
(8)
(b) Fig. 1.
Turbo-BLAST: (a) Transmitter and (b) Receiver.
C. Iterative Receiver Structure Fig. 1(b) illustrates the iterative detection and decoding receiver of turbo-BLAST with a channel estimator. 1) Inner SISO Detector: A maximum a posteriori (MAP) algorithm can be used as an optimal detector. However, due to the computational complexity of the MAP detector, a multistream detector based on minimum mean-square error (MMSE) and parallel soft-interference cancelation (PSIC) is an alternative choice. This MMSE soft-input soft-output (SISO) detector optimizes jointly the weights of the linear detector and the co-antenna interference (CAI) estimate. The CAI can be removed from the linear beamformer output ym , and the improved estimate xm of a transmitted symbol sm using PSIC based on the channel information is given by [11] xm
H H = wm r − wm Hm E{sm } H 2 −1 H ∼ (h h + σ hm (r − Hm E{sm }) = m m v)
(3) (4)
The expectations of the transmitted bits are computed from L(A; i) as follows L(am ; i) , m = 1, 2, . . . , M (9) E{am } = tanh 2 These soft estimates are used to produce E{sm } in (4). An estimate of the information bit matrix B is obtained by slicing the LLR, L(B; p), at the outer decoders b = D(L(B; p)) B
(10)
b = D(L(A; i)) A
(11)
where D(·) is a decision device. The hard decision of the soft information L(A; i) from the SISO decoders is obtained by,
b is reorganized to constitute the estimated The hard decision A b which is used in iterative channel estimasymbol matrix, S, tion.
III. A DAPTIVE F ILTERING -BASED C HANNEL E STIMATION
C. RLS-based Estimation
Iterative channel estimation is performed using the dedicated pilot symbols located in a preamble and the estimated code symbols fed back from the decoders. As shown in Fig. 1(b), the hard decisions of the extrinsic information from the decoders is devoted to iterative channel estimation. At the first iteration, an initial CSI is estimated using only known pilot symbols, where we employ the LS estimation due to no need of the knowledge of the noise power a priori [10]. As the iterations proceed between the SISO detector and the SISO decoders, both the pilot and the estimated code symbols are contributed to the iterative channel estimator.
The performance index to be minimized by the RLS algorithm is defined as an exponentially windowed sum of the squared error: k X J [k] = λk−i |en [i]|2 i=1
ˆ n [k] minimizing J [k] can be realized by The CSI vector h a recursive form. It can be noted that the standard RLS recursion is modified to an extended RLS form, when the process equation with the 1st-order AR model described in (12) is assumed. The extended RLS algorithm is represented as follows [13]:
A. State-Space Model
kn [k]
Under the assumption of spatially uncorrelated channels for sufficiently large antenna spacings, an N -by-M MIMO channel can be separated into N multiple-input single-output (MISO) sub-channels. Thus channel estimation can be performed independently for each receive antenna, leading to computational efficiency in implementing the following adaptive filters. In what follows, it is convenient to use the notation of the channel vector h[k] , vec{HH [k]} = [h1 [k] h2 [k] . . . hN [k]] of size M × N that is obtained by stacking all columns of the Hermitian transpose of the channel matrix H[k]. The wireless channel can be modeled as a stochastic 1st-order autoregressive (AR) process of the form:
en [k]
hn [k] = Fhn [k − 1] + wn [k] ∈ C (M ×1)
(12)
where n = 1, 2, · · · , N and wn [k] is the driving noise vector of the process, and the state transition matrix F modeling the spatio-temporal correlations of the channel is defined as
ˆ n [k + 1|k] h Pn [k] Pn [k + 1, k]
= α2 Pn [k] + Rn [k]
D. KF-based Estimation ˆ n [k], the KF algorithm based on one-step To compute h prediction is represented as follows: Kn [k] ˆ n [k + 1|k] h Pn [k] Pn [k + 1, k]
= F[k + 1, k]Pn [k, k − 1]sH [k]Cn [k] ˆ n [k|k − 1] + Kn [k]en [k] = F[k + 1, k]h = I − F[k, k − 1]Kn [k]sH [k] Pn [k, k − 1]
= F[k + 1, k]Pn [k]FH [k + 1, k] + Rn [k]
where Cn [k] en [k]
B. LMS-based Estimation
n
ˆ n [k|k − 1] + kn [k]e∗ [k] = αh n = Pn [k, k − 1] − Bn [k]
where Rn [k] means correlation matrix of the process noise vector, and −1 An [k] = sH [k]Pn [k, k − 1]s[k] + 1 Pn [k, k − 1]s[k]sH [k]Pn [k, k − 1] Bn [k] = 1 + sH [k]Pn [k, k − 1]s[k]
F = αI ∈ R(M ×M ) with α = Jo (2πfD Ts ) ≤ 1, where J0 (·) is the 0th-order Bessel function of the first kind, and fD and Ts denote the Doppler spread and the reciprocal bandwidth (e.g., symbol period), respectively.
= αPn [k, k − 1]s[k]An [k] ˆ H [k|k − 1]s[k] = rn [k] − h
−1 sH [k]Pn [k, k − 1]s[k] + Qn [k] ˆ n [k|k − 1] = rn [k] − sH [k]h =
The received signal in (1) at the n-th antenna can be rewritten as follows:
In the recursion above, Rn denotes the correlation matrix of the process noise, and Qn denotes the correlation matrix of the measurement noise which has to be estimated.
rn [k] = sH [k]hn [k] + vn [k] ∈ C (1×1)
IV. P ERFORMANCE E VALUATION
(13)
where n = 1, 2, · · · , N . The error signal at time k can be defined as ˆ H [k]ˆs[k] en [k] = rn [k] − h (14) n where ˆs[k] denotes the hard decision of the code symbol ˆ n are provided by the outer decoders. Channel state vectors h updated according to the LMS solution ˆ n [k + 1] = h ˆ n [k] − µs[k]e∗ [k] h n where µ is a step-size parameter.
(15)
To evaluate the performance of adaptive filtering-based iterative channel estimators, Monte Carlo simulations are performed in a MIMO link with M = 4 transmit and N = 4 receive antennas. At the transmit end, each substream of 100 information bits is independently encoded with a rate-1/2 convolutional code generator (7, 5)Oct , and then interleaved via a space-time interleaver. The interleaved substreams are modulated with BPSK. Pilot symbols of length 10 are dedicated to channel estimation. In simulations, we assume that in (12) α = 1 and zero process noise, which makes a quasi-static fading channel. The quality of the CSI estimates is evaluated
0
10
improvement is achieved by iterative channel estimation. As supported by the channel estimation error, the LMS estimator has attained an equivalent BER performance with the RLS and the KF estimators. As more iterations proceed the performance is improved, but it may not be significant because the estimators reach a steady state within three or four iterations and the short block length is used.
LMS RLS KF SNR: 0 [dB] −1
ANSE
10
SNR: 4 [dB]
V. C ONCLUSIONS −2
10
SNR: 8 [dB]
−3
10
1
1.5
2
Fig. 2.
2.5
3 Iterations
3.5
4
4.5
5
Channel Estimation Errors.
0
10
After 1st Iteration Using Initial LS Estimation (Pilot Only)
−1
10
R EFERENCES
After 1st Iteration Using Perfect CSI −2
BER
10
−3
10
−4
10
LMS RLS KF Known
After 3rd Iteration Using Adaptive Filters (Pilot & Data Estimates Used)
−5
10
0
1
Fig. 3.
2
3
4 SNR [dB]
5
6
We have compared the performance of adaptive filteringbased iterative channel estimation techniques with the incorporation of turbo-BLAST. The channel estimate can be improved using an iterative estimation process with the aid of hard decision feedback. Selecting an adaptive filtering algorithm depends on system environments such as the fading rate and the frame structure, and it depends on the parameters to be carefully chosen to implement themselves. In a quasi-static fading MIMO channel, we conclude that the LMS estimator is a reasonable choice in terms of computational efficiency, without the loss of performance gain compared to the RLS and the KF estimators.
7
8
Performance Comparison for Bit Error Rates.
by the average normalized square error (ANSE) [4] defined as ˆ k2 / k h k2 }. E{k h − h Fig. 2 shows the channel estimation error vs. the iteration for different SNR values. In the simulation, the ANSE is ensemble-averaged by 100 trials where µ = 0.005 and λ = 0.999 for the LMS and the RLS filter, respectively. The ANSE at the first iteration depicts an initial estimate by the LS algorithm using only the pilot symbols. We observe that the RLS and the KF estimators converge faster than the LMS estimator until the second iteration. At the third iteration and beyond, the LMS estimator reaches almost identical ANSEs in steady-state performance, compared with the RLS and the KF estimators. The BER performance of the three estimators is depicted in Fig. 3 where 5000 data blocks in total are transmitted. As seen, the iteration gain is large. We observe that after only three iterations, the BER performance gap between the receiver with the known CSI and the receivers with the iterative channel estimators is within 0.5 dB. In other words, more than 1 dB
[1] D. Gesbert, M. Shafi, D. Shiu, P. Smith, and A. Naguib, “From theory to practice: An overview of MIMO space-time coded wireless systems,” IEEE Journal on Selected Areas in Communications, vol. 21, pp. 281– 302, April 2003. [2] P. Alexander and A. Grant, “Iterative channel and information sequence estimation,” in Proceedings of IEEE Symposium on Spread Spectrum Technology and Application, pp. 593–597, Parsippany, NJ, USA, September 2000. [3] M. Lonˇcar, R. R. M¨uller, J. Wehinger, C. F. Mecklenbr¨auker, and T. Abe, “Iterative channel estimation and data detection in frequency-selective fading mimo channels,” European Transactions on Telecommunications, vol. 15, July/August 2004. [4] J. Wehinger, C. F. Mecklenbr¨auker, R. R. M¨uller, T. Zemen, and M. Lonˇcar, “On channel estimators for iterative CDMA multiuser receivers in flat Rayleigh fading,” in Proceedings of IEEE International Conference on Communications, Paris, France, May 2004. [5] T. Abe and T. Matsumoto, “Space-time turbo equalization in frequencyselective mimo channels,” IEEE Transactions on Vehicular Technology, vol. 52, pp. 469–475, May 2003. [6] Y. Sun, M. Yee, and M. Sandell, “Iterative channel estimation with MIMO MMSE-turbo equalization,” in Proceedings of IEEE Semiannual Vehicular Technology Conference, Orlando, FL, USA, October 2003. [7] C. Komninakis, C. Fragouli, A. H. Sayed, and R. D. Wesel, “Multi-input multi-output fading channel tracking and equalization using Kalman estimation,” IEEE Transactions on Signal Processing, vol. 50, pp. 1065– 1076, May 2002. [8] D. Schafhuber, G. Matz, and F. Hlawatsch, “Kalman tracking of timevarying channels in wireless MIMO OFDM systems,” in Proceedings of Asilomar Conf. Signals, Syst. Comput., Pacific Grove, CA, USA, November 2003. [9] R. Otnes and M. T¨uchler, “Soft iterative channel estimation for turbo equalization: Comparison of channel estimation algorithms,” in Proceedings of IEEE International Conference on Communications, Singapore, November 2002. [10] J. Choi, M. Bouchard, and T. H. Yeap, “Iterative channel estimators for turbo-BLAST in flat-fading MIMO channels,” Submitted to ICC 2005. [11] M. Sellathurai and S. Haykin, “Turbo-BLAST for wireless communications: Theory and experiments,” IEEE Transactions on Signal Processing, vol. 50, pp. 2538–2546, October 2002. [12] J. Hagenauer, E. Offer, C. M´easson, and M. M¨orz, “Decoding and equalization with analog non-linear networks,” European Transactions on Telecommunications, pp. 107–128, October 1999. [13] S. Haykin, Adaptive Filter Theory, 4th Ed. Upper Saddle River, NJ: Prentice Hall, 2002.
