algorithm [4] is used to find the maximum a posteriori (MAP) estimation of the ... maximum likelihood (ML) channel estimation and simple non-iterative data ...
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Iterative Channel Estimation and Decoding for Coded MIMO System in Unknown Spatially Correlated Noise Wei Mo (contact author), Zhengdao Wang, and Aleksandar Dogandˇzi´c
Abstract — We present an iterative channel estimation and decoding scheme for a coded system through multi-input multioutput (MIMO) Rayleigh block fading channels in spatially correlated noise. An expectation-maximization (EM) algorithm is utilized to find the maximum likelihood (ML) estimates of the channel and the spatial noise covariance matrix, based on which the soft information of coded symbols is computed and sent to the error-control decoder. Extrinsic information produced by the decoder is fed back to the channel estimators to refine the channel estimation. Several iterations are performed between the channel estimators and the decoder. We also develop a deterministic ML channel estimator that uses soft information from the decoder to update the channel and the noise covariance estimation. There is negligible difference between the performance with the EM-based channel estimation and that with the true channel at bit error rate of 10−6 . Keywords: MIMO, iterative channel estimation and decoding, Expectation-Maximization, turbo codes, spatially correlated noise
I. I NTRODUCTION Communication systems utilizing multiple transmit and receive antennas have attracted much attention in recent years, because of its significantly increased capacity, see e.g., [8], [16]. Estimation of the multi-input multi-output (MIMO) fading channel is a major challenge for multiple antenna systems because the detection of information symbols depends critically on the availability of full or partial channel state information. Recently there have been increasing interests in iterative channel estimation and data decoding [2], [5], [14], where data decision obtained from the decoding, either hard or soft, is used as additional information to refine the channel estimation. Expectation-Maximization (EM) algorithm [4] is used to find the maximum a posteriori (MAP) estimation of the channel in [2]. Leastsquare (LS) estimation together with hard and soft decision feedback is studied in [14]. All of these methods assume that the additive noise is both temporally and spatially white. The channel estimation for MIMO systems in spatially correlated noise has been studied in [7], [11]. In [11], the deterministic maximum likelihood (ML) channel estimation and simple non-iterative data decoding was proposed. Unlike the white noise scenario treated in [2], we propose an iterative channel estimation and decoding scheme via EM algorithm for spatially correlated noise with unknown covariance matrix. Instead of using MAP estimation as in [2], where the second-order statistic of the channel is assumed to be known at the The authors are with the Dept. of Electrical and Computer Engr., Iowa State University, Ames, Iowa 50011; Tel: (515)294-2307; Emails: {mowei, zhengdao, ald}@iastate.edu
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receiver, we estimate both the channel and the spatial noise covariance without prior statistical knowledge of the channel statistics. This work generalizes our results for single-input multi-output systems in [6] to the coded MIMO scenario. We will also compare the proposed method with the weighted least-square estimation using soft-decision feedback in [3]. II. S YSTEM MODELING We consider a coded MIMO system equipped with nT transmit antennas and nR receive antennas in a frequency-flat slow Rayleigh fading environment. We will use turbo code as an example of the error control code. Any other code can also be used. The transmitter, shown in Fig. 1 on page 9, consists of the following parts: (1) a turbo encoder, which encodes information bits u into coded bits c; (2) a modulator, which generates symbols from M -ary phase shift keying (PSK) constellation; (3) a channel interleaver; (4) a space-time encoder, which generates space-time codewords X with pilot codewords X p embedded. Suppose a packet of L space-time codewords of size nT × K each are transmitted. Denote y l (k) the
nR × 1 data vector received at time k of the lth space-time codeword in the packet, where k = 1, 2, . . . , K
and l = 1, 2, . . . , L. The received space-time data matrix for the lth codeword Y l = [y l (1) · · · y l (K)] can
be modeled as where •
Y l = H · X l + El,
(1)
H is an unknown nR × nT channel response matrix;
•
X l is the lth transmitted space-time codeword;
•
E l = [el (1) · · · el (K)] is the lth noise matrix;
•
l = 1, 2, . . . , L,
el (k), k = 1, 2, . . . , K, l = 1, 2, . . . , L is temporally white and circularly symmetric zero-mean complex Gaussian noise vector with unknown positive definite spatial covariance matrix Σ . It models co-channel interference (CCI) and receiver noise. This is a standard model for a communication channel, subject to (unstructured) interference and jamming, see e.g. [7], [11].
In this paper, we assume that space-time orthogonal design is used [1], [9], [15]. But with suitable minor modifications, the proposed method can also be applied to general space-time codes. Let each (l)
(l)
space-time codeword X l be a linear function of the corresponding set of symbols Sl = {s1 , . . . , sK },
which is defined as
K X (l) (l) Xl = Re sk Ak + j · Im sk B k k=1
where Re(·) and Im(·) denote the real and imaginary parts, j =
√
(2)
−1, and Ak and B k are fixed real-valued
“elementary” code matrices, satisfying the orthogonality conditions [9], [15] as follows: Ak ATk = I nT , Ak ATt = −At ATk ,
B k B Tk = I nT
B k B Tt = −B t B Tk ,
Ak B Tt = B t ATk
k 6= t
(3)
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where (·)T denotes transpose. To allow unique estimation of the channel H (i.e., to resolve the phase ambiguity associated with PSK modulation), we further assume that Lp known pilot space-time codewords X T,ℓ , ℓ = 1, . . . , Lp , are inserted at the beginning of the packet, and denote the corresponding data blocks received by the array as Y T,ℓ . We assume that the channel H and the noise covariance matrix Σ remain constant for each packet of (L + Lp ) codewords, i.e., K(L + Lp ) time intervals, and change from one packet to another. Since turbo codes need long data block length to achieve good error performance, and the length of each packet is limited by the coherent time of fading channel, the turbo encoder implements coding across R packets, see e.g., [11]. Then, the decoding needs the estimates of channels and noise covariance matrices for all R packets. Any imperfect estimate tends to produce burst errors, while turbo codes are more effective with uncorrelated errors [10]. Therefore, a channel interleaver is used here to spread the effect of imperfect channel estimates across the whole R packets [17]. Note that this interleaver is not the nonuniform interleaver required in the turbo encoder. III. C HANNEL ESTIMATOR VIA EM ALGORITHM The channel estimation problem can be formulated as follows: Given a packet of received data [Y T,1 , . . . , Y T,Lp , Y 1 , . . . , Y L ], and the pilot space-time codewords [X T,1 , . . . , X T,Lp ], find the ML estimates of the channel H and the noise covariance matrix Σ for this packet. The EM algorithm is a general iterative method for computing ML estimates in the scenarios where ML estimation cannot be easily performed by directly maximizing the likelihood function of the observed data [4]. Each EM iteration consists of maximizing the expected complete-data log-likelihood function, where the expectation is computed with respect to the conditional distribution of the unobserved data given the observed data. A good choice of unobserved data allows easy maximization of the expected complete-data log-likelihood. For our channel estimation problem, the unknown space-time codewords {X l }Ll=1 are modeled as the
unobserved (or missing) data. Based on the data model in Section II, we can derive the following EM iteration: E step: H (i+1)
# " L Lp X X 1 (i) (i) = Y T,ℓ X H Y l EX l |Y l (X H T,ℓ l ;H ,Σ ) + (L + Lp )K l=1 ℓ=1
(4)
M step: Σ (i+1) = Ryy − H (i+1) · (H (i+1) )H where (·)H denotes the conjugate transpose and # " L Lp X X 1 Ryy = Y T,ℓ Y H Y lY H T,ℓ . l + (L + Lp )K l=1 ℓ=1
(5)
(6)
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To ensure positive definiteness (with probability one) of the estimates of Σ , the following condition needs to be satisfied: (L + Lp )K ≥ (nT + nR ),
(7)
see e.g., [12, Theorems 10.1.1 and 3.1.4]. Since there is a one-to-one mapping between the set of transmitted symbols Sl and the codeword X l , conditioning on Sl is equivalent to conditioning on X l .
Following a similar derivation to [11, eq. 7], the likelihood function of the received data Y l can then be written as
f (Y l |H, Σ , X l ) = f (Y l |H, Σ , Sl ) K n Y H −1 (l) o −1 = const · exp 2Re Re Tr Y H Σ HA + j · Im Tr Y Σ HB sk k k l l = const ·
k=1 K Y
k=1
(l)
f1 (Y l |H, Σ , sk )
(8) (l)
where const is a normalizing constant that does not depend on sk . The second equality in above formula uses the orthogonality conditions in (3), which decouples the likelihood function of space-time codeword (l)
into likelihood functions of transmitted symbols, f1 (Y l |H, Σ , sk ), omitting the normalizing constants. (l)
Assume that the transmitted symbols sk , k = 1, . . . , K, l = 1, . . . , L, are independent and have a prior (l)
(l)
probability mass functions p(sk ) = p(sk = sm ), m = 1, . . . , M , then the E step of the EM iteration can be formulated as H (i+1)
" L K i h XX 1 (l) = Y l Re Es(l) |Y sk ; H (i) , Σ (i) AH k l k (L + Lp )K l=1 k=1 # Lp i h X (l) Y T,ℓ X H −j · Im Es(l) |Y sk ; H (i) , Σ (i) B H + k T,ℓ ,
(9)
l
k
ℓ=1
where
Es(l) |Y k
l
h
(l)
i
sk ; H (i) , Σ (i) =
M X
(l)
sm p(sk = sm )f1 (Y l |H (i) , Σ (i) , sm )
m=1 M X n=1
.
(10)
(l)
p(sk = sn )f1 (Y l |H (i) , Σ (i) , sn )
(l)
The prior probabilities p(sk ) will be coming from the error control decoder. In the (i + 1)st iteration, E step requires computing (Σ (i) )−1 , which can be done using the matrix inversion lemma: −1 (i) (i) −1 (Σ (i) )−1 = R−1 · [I nT − (H (i) )H R−1 · (H (i) )H R−1 yy + Ryy H yy H ] yy ,
(11)
where R−1 yy needs to be evaluated only once, before the iteration starts. The above equation is useful when nT < nR since it requires inversion of an nT × nT matrix, which is computationally more efficient than inverting the nR × nR matrix Σ (i) .
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IV. I TERATIVE SPACE - TIME RECEIVER We derive in this section an iterative receiver based on the channel estimation and turbo decoding. Because the EM channel estimator utilizes the a priori symbol probabilities, and can produce symbol a posteriori probabilities using the estimated channel and noise covariance matrix, it is well suited for iterative processing, which can refine the channel estimation based on the soft information from the decoding stage, and vice versa. The iterative receiver model is shown in Fig. 2 on page 9. It consists of two modules: 1) a bank of channel estimators developed in the previous section, and 2) a turbo decoder. The two modules are connected by channel deinterleaver and interleaver.
n
The received data matrices Y are first divided into R packets [Y
r r r r T,1 , . . . , Y T,Lp , Y 1 , . . . , Y L ]
oR
of
r=1
length (L + Lp )K, and then fed into R channel estimators. Based on the pilot codewords and a priori c r and noise probabilities of the transmitted symbols, each channel estimator estimates the channel H
b r , then computes the a posteriori log-likelihoods of the transmitted symbols covariance matrix Σ (l) (l) cr , Σ b r , s(l) ) Λ1r [sk ] = const + log p(sk ) + log f1 (Y rl |H k (l)
(l)
12 = const + λ21 r [sk ] + λr [sk ]
r = 1, . . . , R,
l = 1, . . . , L,
(12)
k = 1, . . . , K.
(l)
(l)
where const denotes terms independent of sk . The second term λ21 r [sk ] represents the a priori log(l)
likelihood of the transmitted symbol sk , which is computed by the turbo decoder in the previous iteration, interleaved, and then fed back to the channel estimator. For the first iteration, we assume (l)
equally likely symbols, i.e., no prior information available. The third term λ12 r [sk ] in (12) represents the extrinsic information produced by the channel estimator, based on the received data Y r , pilot codewords, and the prior information about all other symbols in the packet. All the extrinsic information metrics n oR,K,L (l) λ12 [s ] are reassembled together, deinterleaved, and sent into the turbo decoder, as the a r k r=1,k=1,l=1
priori information for the decoding.
With the extrinsic information of the transmitted symbols coming from channel estimators and the structure of the turbo codes, the turbo decoder computes the a posteriori log-likelihood of each symbol as: n oR,K,L (n) (l) (l) Λ2r [sk ] = const + log p(sk | λ12 [s ] r k
r=1,k=1,l=1
= const +
(l) λ12 r [sk ]
+
; code constraints) (13)
(l) λ21 r [sk ]
(n)
It is seen from (13) that the output of turbo decoder consists of the prior information λ12 r [sk ] provided (n)
by the channel estimators, and the extrinsic information λ21 r [sk ] interleaved and delivered to the channel (n)
estimators in the next iteration. This extrinsic information is the information about the symbol sk obtained n oR,K,L 12 (l) from the prior information about the other symbols λr [sk ] based on the code constraints. r=1,k=1,l=1,l6=n
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The turbo decoder also outputs the a posteriori probability APP(ui ) of every information bit, which is used to do the decision at the last iteration. The whole iterative channel estimation and turbo decoding scheme is performed as follows: Step 1. Divide the received data Y into R packets of length (L + Lp )K, and send them to R channel estimators; Step 2. For rth channel estimator, r = 1, . . . , R, do the following steps: a) Estimate channel and noise covariance matrix according to the EM algorithm derived in c r and Σ b r; Section III to obtain H (l)
b) Compute the extrinsic information about the symbols {sk }K,L k=1,l=1 in rth packet: h i h i (l) r H b −1 c r H b −1 c 12 (l) λr [sk ] = 2 · Re Re Tr (Y l ) Σ r H r Ak + j · Im Tr (Y l ) Σ r H r B k sk
(14)
Step 3. Reassemble R sets of extrinsic information from channel estimators into a whole sequence, then pass it through the channel deinterleaver; Step 4. The turbo decoder uses the deinterleaved extrinsic information from channel estimation module o n 21 (l) as a priori information to compute the extrinsic information λr [sk ] about the transmitted symbols;
Step 5. Divide the interleaved extrinsic information from turbo decoder into R packets, and feed back to the channel estimators as a priori information, then go to Step 2 and repeat; Step 6. Final decision: at the last iteration, estimate the information bits ui ’s as follows 0, if APP(ui = 0) > APP(ui = 1) uˆi = 1, otherwise.
(15)
V. D ISCUSSION
A few remarks for the proposed algorithms are presented in the following. A. Initial values of the channel estimation Although the EM algorithm increases (or at least does not decrease) the likelihood function at each iteration, it is often trapped at the local maximum when the initial values are too far from the true parameters. So we need a more robust method to give rough estimates of the channel and noise covariance, which are used as the initial values of our EM algorithm. For this purpose, we choose the iterative weighted least-square with projection method for space-time coding system proposed in [13]. For completeness, we summarize below our implementation of this method: c and compute Step 1: Fix H = H
cl = proj H H R−1 Y l , X yy
l = 1, . . . , L,
(16)
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c1 , . . . , X L = X cL and compute Step 2: Fix X 1 = X # " L Lp X X 1 c= Y T,ℓ X H Y lX H · H T,ℓ . l + (L + Lp )K ℓ=1 l=1
(17)
Go to step 1 and repeat.
where proj[·] denotes projecting onto the nearest space-time codeword. This method is initialized with the least-square estimate using the pilot codewords c LS = H
Lp 1 X Y T,ℓ X H T,ℓ . Lp K ℓ=1
(18)
After several iterations, we obtain a rough estimate of the channel , then the estimate of the noise covariance matrix is computed using (5). Both of them will be used to initialize our EM algorithm. B. Sliding window technique The complexity of the EM algorithm proposed in this paper is higher than that of the weighted leastsquare method mentioned above, and increases linearly with the packet length. A sliding window technique can be used to decrease the complexity of EM algorithm and the overhead of the pilot sequence. Suppose the channel does not change during N time intervals, which is called a “window”. The channel is estimated within this window. After sliding the window one step forward in time, the channel estimation will be updated only using the new data within the window via our EM algorithm. A number of pilot symbols can be inserted at the beginning of the transmission to get a good enough initial channel estimation, then the channel is estimated step by step via the EM algorithm, using the channel estimation obtained in previous step as the initial value. Using this technique, the complexity of the EM algorithm does not increase with the window width, but depends on the step size. VI. S IMULATION R ESULTS Using numerical simulations, we evaluated the performance of the proposed iterative channel estimation and decoding scheme for a turbo coded MIMO system in a frequency-flat correlated Rayleigh fading environment, with nT = 2 transmit and nR = 2 receive antennas. The Alamouti transmission scheme [1] was used to generate the space-time codewords X l , which meant K = 2 and A1 =
"
1 0 0 1
#
A2 =
# " 0 −1 1
0
B1 =
"
1
0
0 −1
#
B2 =
"
0 1 1 0
#
,
(19)
(l)
where the transmitted symbols {sk } were generated from a 4-PSK constellation (i.e., M = 4) with
normalized energy. The space-time codewords were transmitted in R packets as one frame, and each
packet consisted of Lp = 2 pilot codewords followed by L = 32 data codewords. The signal was corrupted by additive complex Gaussian noise with spatial noise covariance matrix Σ whose (p, q)th element is Σ p,q = σ 2 · 0.9|p−q| · exp[j(π/2)(p − q)],
(20)
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which is the noise covariance model used in [18] (see also references therein). The bit signal-to-noise ratio (SNR) per receiver antenna was defined as L + Lp L + Lp 1 1 SNR = = · 10 log10 2 · 10 log10 L σ · log2 (M ) L 2σ 2
(dB).
(21)
There had been 3 iterations carried out between the EM channel estimators and the turbo decoder in our proposed iterative receiver scheme. We compare our proposed EM-based scheme with another iterative receiver using deterministic ML channel estimation with soft decision feedback. The difference between them is that, the channel estimation algorithm in the second method is the deterministic ML that uses the expectations of the coded symbols computed from the extrinsic information produced by the decoder. Fig. 3 shows the mean-square error (MSE) of the channel estimates and bit-error-rate (BER) performance versus bit SNR per receive antenna with R = 16. It is seen from Fig. 3(a) that the soft information about transmitted symbols fed back from the turbo decoder can improve the channel estimation significantly for both methods. At MSE = 10−3 , our proposed EM algorithm outperforms the deterministic ML method by about 0.6 dB after three iterations between the channel estimation and turbo decoding. Similarly, it can be observed from Fig. 3(b) that the iterations between the channel estimation and turbo decoding can also improve the error performance, since the channel estimation becomes better. As a comparison, the result for coherent detector which assumed known H and Σ is also presented here. After three iterations, our method outperforms the deterministic ML by about 0.6 dB at BER = 10−4 , and comes within about 2 dB of the performance of the coherent detector. To study the effects of the frame length R, similar simulations were run again with R = 64. Fig. 4 shows the corresponding results. Compared with Fig. 3, it is seen that longer frame length improves both the channel estimation and error performance. After 3 iterations, the gaps between our method and the deterministic ML increase from 0.6 dB to 1 dB for both the channel estimation and the error performance, at MSE = 10−3 and BER = 10−5 respectively. Another important observation is that, at a BER of 10−6 , the difference of the performance between our method and the coherent detector becomes negligible. R EFERENCES [1] S. M. Alamouti,
“A simple transmit diversity technique for wireless communications,”
IEEE Journal on Selected Areas in
Communications, vol. 16, no. 8, pp. 1451–1458, Oct. 1998. [2] Z. Baranski, A.M. Haimovich, and J. Garcia-Frias, “EM-based iterative receiver for space-time coded modulation with noise variance estimation,” in Proc. of GLOBECOM, Taipei, Taiwan, Nov. 2002. [3] P. Bohlin and M. Lundberg, “A blind and robust space-time receiver for a turbo coded system,” in Proceedings of the Asilomar Conference on Signals, Systems & Computers, 2000, vol. 1, pp. 532–536. [4] A.P. Dempster, N.M. Laird, and D.B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc., vol. 39, pp. 1–38, July 1977. [5] X. Deng, A.M. Haimovich, and J. Garcia-Frias, “Decision directed iterative channel estimation for MIMO systems,” in Proc. of Intl. Conf. on Com., 2003, vol. 4, pp. 2326–2329.
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[6] A. Dogandˇzi´c, W. Mo, and Z. Wang, “Maximum likelihood semi-blind channel and noise estimation using the EM algorithm,” in Proc. of 37th Conf. on Info. Sciences and Systems, Johns Hopkins University, March 12-14 2003. [7] A. Dogandzic and A. Nehorai, “Space-time fading channel estimation and symbol detection in and unknown spatially correlated noise,” Proc. of Intl. Conf. on ASSP, vol. 50, no. 3, pp. 457–474, Mar. 2002. [8] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Communications, vol. 6, no. 3, pp. 311–335, Mar. 1998. [9] G. Ganesan and P. Stoica, “Space-time block codes: a maximum SNR approach,” IEEE Trans. Inform. Theory, vol. 47, no. 4, pp. 1650–1656, Apr. 2001. [10] E.K. Hall and S.G. Wilson, “Design and analysis of turbo codes on rayleigh fading channels,” IEEE Journal on Selected Areas in Communications, vol. 16, no. 2, pp. 160–174, Feb. 1998. [11] E.G. Larsson, P. Stoica, and Jian Li, “Orthogonal space-time block codes: maximum likelihood detection for unknown channels and unstructured interferences,” IEEE Trans. Signal Processing, vol. 51, no. 2, pp. 362–372, Feb. 2003. [12] R.J. Muirhead, Aspects of Multivariate Statistical Theory, Wiley, New York, 1982. [13] A. Ranheim, A.P. des Rosiers, and P.H. Siegel, “An iterative receiver algorithm for space-time encoded signals,” in Proceedings of the Asilomar Conference on Signals, Systems & Computers, 2000, vol. 1, pp. 516–520. [14] M. Sandell, C. Luschi, P. Strauch, and R. Yan, “Iterative channel estimation using soft decision feedback,” in Proc. of GLOBECOM, 1998, vol. 6, pp. 3728–3733. [15] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inform. Theory, vol. 45, no. 5, pp. 1456–1467, July 1999. [16] I. Emre Telatar, “Capacity of multi-antenna Gaussian channels,” European Trans. on Telecomm., vol. 10, no. 6, pp. 585–596, Nov.–Dec. 1999. [17] M.C. Valenti and B.D. Woerner, “Iterative channel estimation and decoding of pilot symbol assisted and turbo codes over flat-fading channels,” IEEE Journal on Selected Areas in Communications, vol. 19, no. 9, pp. 1697–1705, Sept. 2001. [18] M. Viberg, P. Stoica, and B. Ottersten, “Maximum likelihood array processing in spatially correlated noise and fields using parameterized signals,” Proc. of Intl. Conf. on ASSP, vol. 45, no. 4, pp. 996–1004, Apr. 1997.
u
Turbo encoder
c
Modulator
s’
s Channel interleaver
ST encoder
Xc
Pilot insertion
X
Xp Fig. 1.
Discrete-time transmitter model
packet 1 EM Channel Λ 11 [ s (kl ) ] estimator 1 +
Fig. 2.
λ121[ s (kl ) ]
+
P/S
Channel deinterleaver
λ12R[ s (kl ) ]
-
The receiver with iterative channel estimation and decoding
Turbo decoder
Λ r2 [ s (kl ) ] - λ r2 [ s (kl ) ] Channel + + interleaver
APP( u) ^ u
S/P
. . .
packet R EM Channel Λ 2R[ s (Rl ) ] estimator R +
+ . . .
S/P
. . .
Y
-
10
0 iter 1 iter 2 iter 3 iter
−1
10
0 iter 1 iter 2 iter 3 iter
−1
10
−2
10 −2
10
Bit Error Rate
Average MSE of ML channel estimates
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−3
10
−3
10
−4
10
dashed lines: deterministic ML solid lines: EM dotted line: coherent
−4
10
dashed lines: deterministic ML solid lines: EM −5
−4.5
−4
−3.5
−3
−2.5
−2
10 −4.5
−1.5
−4
Bit SNR per receiver antenna
−3.5
−3
−2.5
−2
−1.5
Bit SNR per receiver antenna
(a) MSE for the channel estimates using the EM algorithm and
(b) BER of the EM-based, deterministic ML with soft-decision
the deterministic ML with soft-decision feedback method.
feedback based, and coherent detectors.
MSE for the channel estimates and BER versus the bit SNR per receive antenna for L = 32, Lp = 2, K = 2, R = 16.
0 iter 1 iter 2 iter 3 iter
0 iter 1 iter 2 iter 3 iter
−1
10
−1
10
−2
10
Bit Error Rate
Average MSE of ML channel estimates
Fig. 3.
−2
10
−3
10
−4
10
−3
10
−5
10
dashed lines: deterministic ML solid lines: EM dotted line: coherent
dashed lines: deterministic ML solid lines: EM −6
−6
−5.5
−5
−4.5
−4
−3.5
Bit SNR per receiver antenna
10
−6
−5.5
−5
−4.5
−4
−3.5
Bit SNR per receiver antenna
(a) MSE for the channel estimates using the EM algorithm and
(b) BER of the EM-based, deterministic ML with soft-decision
the deterministic ML with soft-decision feedback method.
feedback based, and coherent detectors.
Fig. 4.
MSE for the channel estimates and BER versus the bit SNR per receive antenna for L = 32, Lp = 2, K = 2, R = 64.
