SAGE based joint timing-frequency offsets and channel estimation in distributed MIMO systems

Computer Communications 33 (2010) 2125–2131

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SAGE based joint timing-frequency offsets and channel estimation in distributed MIMO systems Yuan Tian *, Xia Lei, Yue Xiao, Shaoqian Li National Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China, No. 2006, Xiyuan Ave., West Hi-tech Zone, Chengdu, Sichuan Province 611731, PR China

a r t i c l e

i n f o

Article history: Available online 29 July 2010 Keywords: Channel estimation CRB Frequency offset estimation MIMO Timing offset estimation

a b s t r a c t This paper proposes a timing-frequency offset estimation algorithm joint with channel estimation in distributed multi-input multi-output (MIMO) systems, in which different timing and frequency offsets are considered for each pair of transmit and receive antennas. Different from most of the existing methods which estimate the timing or frequency offsets separately, the proposed scheme makes a joint estimation with a maximum-likelihood (ML) estimation model. Furthermore, an iterative space-alternating generalized expectation-maximization (SAGE) estimator is proposed to provide a solution to the multi-dimensional extreme-value problem. And the Cramér-Rao bound (CRB) for multiple parameter estimation is also derived. Simulations show that the estimation performance of the proposed scheme can reach the CRB effectively. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Multi-input multi-output (MIMO) technology has attracted much attention in wireless communications, since it offers significant increases in system capacity, data rate and link range without additional bandwidth or transmit power. It achieves these merits with higher spectral efficiency, link reliability and diversity [1]. Due to these properties, MIMO has been proposed to be one of the important candidate techniques for the 4G radio network. However, the MIMO capacity gain would be limited in applications where the channel rank and multipath richness are insufficient to support the number of antennas utilized [2]. To solve this problem, distributed MIMO, which is shown to enjoy cooperative diversity and improved throughput [3,4], is proposed to improve the capacity. In distributed MIMO system, synchronization errors would generate inter-symbol interference (ISI) and decrease system capacity, so timing and frequency offsets estimation plays an important role. Although there have been many estimation methods for MIMO systems, most of them are concerned with centralized systems [5–8]. Since there are multiple timing and frequency offsets to be estimated in distributed systems, these methods can not be used. For the estimation methods for distributed MIMO systems [9–13], current methods focus on estimating the timing or frequency offsets alternatively assuming the other factors having been perfectly estimated. [9] proposes a joint frequency offset and channel estimation method * Corresponding author. Tel.: +86 2861830300; fax: +86 2861830284. E-mail address: [email protected] (Y. Tian). 0140-3664/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2010.07.011

which has better performance than the methods in [10–12] and its mean-square-error (MSE) performance can reach the Cramér-Rao bound (CRB) if no timing error exist. But when only small timing offsets occur, its estimation performance will deteriorate. A joint small timing offset and channel estimation method is proposed in [9], but it cannot get the accurate timing points with frequency offsets exist. In general, few researches have considered estimating all the parameters (channel estimation, timing and frequency offsets estimation) in a whole. The fact is that estimation on one single parameter cannot achieve the best system performance with other unknown parameters. In this case, accurate joint parameter estimation remains a problem. In this paper, we propose a joint channel, timing and frequency offsets estimation algorithm in distributed MIMO flat-fading channels in which multiple channel coefficients and timing-frequency offsets are concerned. Firstly a maximum-likelihood (ML) unbiased estimation model is established. Then we expand the space-alternating generalized expectation–maximization (SAGE) based algorithm in [9] to perform the multi-dimensional estimation in a practical way (we expand the channel and frequency offset estimation algorithm to the channel and timing-frequency offset estimation). The proposed algorithm solves the problems of the methods in [9,13] simultaneously. Furthermore, the CRB is derived to evaluate the proposed algorithm. The simulation results show that the MSE performance of the proposed scheme can reach the CRB effectively. Notation: The small bold italic letters denote row/column vectors and the capital bold letters denote matrices. The operators Re() and Im() correspond to the real part and the imaginary part of a complex number. diag(x) denotes a diagonal matrix with the

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elements of x located on the main diagonal. ()*, ()T and ()H denote the conjugate, the transpose and the conjugate transpose operators.  denotes the element-wise product of two vectors/matrices. kxk represents the L2 norm of vector x.

Let us consider a distributed MIMO system with Nt transmit and Nr receive antennas. The propagation channel is assumed to be quasi-static flat-fading [14]. The received signal at the lth receive antenna can be written as [13] Nt X

hkl ejwkl t

k¼1

L0 X þLg 1

pðyjhÞ ¼ ðpr Þ

exp 

ky  Khk2

r2

) :

xk ðiÞgðt  iT  ekl TÞ þ nl ðtÞ;

ð1Þ

i¼Lg

where xk(i) is the sequence of training symbols transmitted from the kth transmit antenna. We assume the training sequences transmitted from each transmit antenna are uncorrelated. hkl and wkl are the unknown channel coefficient and frequency offset between the kth transmit antenna and the lth receive antenna. hkl’s are statistically independent between different transmit and receive antennas. hkl and wkl remain constant during the interval in which xk(i) is transmitted. ekl is the unknown timing offset normalized to the symbol duration T, and g(t) is the pulse shaping filter. L0 denotes the observation length and Lg denotes the number of symbols affected by the inter-symbol interference introduced by one side of g(t). nl(t) is a noise sequence of zero-mean, independent and identically distributed complex-valued Gaussian random variables with a variance of r2. Noise sequences at different receive antennas are statistically independent. The timing offset ekl can be broken up into ekl = rkl + skl, with skl 2 [0, 1) and rkl 2 {0, 1, 2 . . . }. Different methods have been proposed to estimate rkl, such as the methods in [15,16]. So we will focus on the estimation of skl, assuming that rkl is known. The problem to be solved is the estimation of skl, wkl and hkl. It can be considered as Nr independent estimation problems for Nr multi-input single-output (MISO) systems. Hence, in the following of the paper, without special declaration, we consider the parameter estimation from all the transmit antennas to the lth receive antenna. In this case, the suffix l is omitted. Assume the received signal is oversampled at a rate of Q, the sample interval is Ts = T/Q. By stacking L0Q received samples, we define T

y , ½yð0Þ; yðT s Þ; . . . ; yððL0 Q  1ÞT s Þ ;

ð2Þ

h , ½h1 ; h2 ; . . . hNt T ;

ð3Þ

n , ½nð0Þ; nðT s Þ; . . . nððL0 Q  1ÞT s ÞT ;

ð4Þ

K , ½k1 ; k2 ; . . . ; kNt ;

ð5Þ

where

ð6Þ

Wk , diag ð1; ejwk =Q ; . . . ; ejðL0 Q 1Þwk =Q Þ;

ð7Þ

dsk , Ask xk ;

ð8Þ

xk , ½xk ðLg Þ; . . . ; xk ð0Þ; . . . ; xk ðL0 þ Lg  1ÞT ;

ð9Þ

Ask , ½aLg ðsk Þ; . . . ; a0 ðsk Þ; . . . ; aL0 þLg 1 ðsk Þ;

ð10Þ

ai ðsk Þ , ½gðiT  sk TÞ; gðiT þ T s  sk TÞ; . . . ; gðiT þ ðL0 Q  1ÞT s  sk TÞT :

ð11Þ

Then the received signal vector is given by

y ¼ Kh þ n:

ð12Þ ½hT1 hT2

   hTNt T ,

Let the parameter of interest be h , where hk , [skwkhk]T. The ML estimation of h is given by maximizing

ð14Þ

And for the given value of w and s, the minimizing of (14) with respect to h is

 1 ^ ¼ KH Kw;s KHw;s y: h w;s

ð15Þ

Substitute (15) back into (14), the estimation of w and s is obtained as

 1 ^ s ^Þ ¼ arg max yH Kw;s KHw;s Kw;s ðw; KHw;s y: w;s

ð16Þ

(16) is a 2Nt-dimensional maximization problem with high computational complexity. Especially, large numbers of trial values of wk’s are needed to achieve (16). 3. Proposed iterative estimation algorithm The expectation–maximization (EM) algorithm [17] is an iterative method which enables approximating ML estimation when a direct calculation of this estimation is computationally prohibitive. But one disadvantage of EM algorithm is its slow convergence rate. In this case, the SAGE algorithm [18] has been proposed for accelerating the convergence of the EM algorithm. Based on EM and SAGE algorithms, a joint channel coefficients and frequency offsets estimation method for distributed MIMO system has been proposed in [9]. The performances of this algorithm achieve the CRBs for both channel and frequency offsets estimation. But a shortage of this algorithm is the desire of perfect timing. When residual timing offsets (skl 2 [0, 1)) exist, the performance of the method in [10] will deteriorate and can not achieve the CRB any more. Based on the SAGE algorithm in [9], we propose a joint channel, timing and frequency offsets estimation method in flat-fading distributed MIMO systems. As mentioned in Section 2, we consider a MISO system in (1) and (12), and define

sk ðt  sk TÞ ,

L0 QþL g Q1 X

xk ðiÞgðt  iT s  sk TÞ;

ð17Þ

i¼Lg Q

sk , ½sk ðsk TÞ; sk ðT s  sk TÞ; . . . ; sk ððL0 Q  1ÞT s  sk TÞT ; jwk =Q

ek , ½1; e

jðL0 Q 1Þwk =Q T

;...;e

 :

ð18Þ ð19Þ

Then the received signal vector y in (12) can be expressed as

y¼

kk , Wk dsk ;

ð13Þ

It is equivalent to minimizing

ky  Khk2 :

2. System model and ML estimation

yl ðtÞ ¼

( 2 L0 Q

Nt X ðsk  ek Þhk þ n:

ð20Þ

k¼1

In our SAGE algorithm, y is the incomplete data space. The parameter to be estimated is h (which has been defined above). The hidden data space is defined as z , ½z1 ; z2 ; . . . ; zNt T , where

zk , ðsk  ek Þhk þ n:

ð21Þ

For the sake of increasing the convergence rate, all the noise is associated to the hidden data space [9]. The SAGE algorithm is an itera½m ½m ½m tive method, let ^ h½m , ½^ h2 . . . ^ h1 ^ hNt T be the estimated value of h ½m ^ ½m ^½m T ^½m obtained after the (m-1)th iteration, where ^ hk ,½s k wk hk  . As for Nt transmit antennas, h is divided into Nt groups of hk, k = 1, 2, . . . , Nt. When updating one group of hk, the other groups remain fixed. The mth iteration contains Nt times the following expectation step (E-step) and maximization step (M-step).

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3.1. E-step

Step 3.In this step, the updated value of hk is determined while ½mþ1 sk and wk are fixed at s½mþ1 and wk . hk is updated by k

Compute the expectation

  n  n o J hk j^h½m ,E log f zk jhk ; ^h½m m



m–k

jy; ^h½m

o

ð22Þ

 ^½mþ1 ¼ arg min k^z½m  ðs  e Þh k2  h k k k k k



m–k

1 2 ÞL0 Q

ðpr

(

2

exp 

kzk  ðsk  ek Þhk k

:

r2

  ½mþ1 2 ^½m  ^ ^½mþ1 T ejiwk =Q  : zk ðiÞ  sk i  s k

ð32Þ

i¼0

  ½mþ1  ^z½m ^½mþ1 T ejiw^ k =Q k ðiÞsk i  sk : PL0 Q 1 jsk ðiÞj2 i¼0

ð33Þ

The foregoing statement is the iteration procedure of one receive antenna. By repeating it Nr times, all the desired parameters for the MIMO system can be estimated.

r

2 1   ½m  ¼ C 2  2 ^zk  ðsk  ek Þhk  ;

ð24Þ

r

where

n o   ^½m ^½m ¼ ^s½m  ^e½m h ^z½m k , E z k jy; h k k k ! N Nt t X X  ½m ^½m ^½m ^½m ¼ y  ^s½m ^sm  ^e½m þ y m  em hm m hm ; m¼1

^½mþ1 ¼ h k

ð23Þ

      1 J hk j^h½m ¼ C 1  E 2 kzk  ðsk  ek Þhk k2 y; ^h½m

ð25Þ

m¼1;m–k

h      iT ^½m ^½m ^½m ^s½m ; k ¼ sk sk T ;sk T s  sk T ;...;sk ðL0 Q  1ÞT s  sk T h i T ^ ½m ^ ½m jw v =Q ;...;ejðL0 Q1Þw v =Q ^e½m v ¼ 1;e

ð26Þ ð27Þ

4. Cramér-Rao bound for the multi-dimensional parameter estimation In this section, the CRBs of the joint channel, timing and frequency offsets estimation are analyzed. Considering a MISO system defined in Section 2, we rearrange the vector of the parameters of interest as H , [Re{h}TIm{h}T wTsT]T, where w , ½w1 w2 . . . wNt T and s , ½s1 s2 . . . sNt T . The conditional likelihood function of y is shown in (13). We define

lðtÞ ,

Nt X

hk ejwk t=Q

L0 Q X þLg Q 1

xk ðiÞgðt  iT s  sk TÞ;

ð34Þ

i¼Lg Q

k¼1

and C1 and C2 are constants independent of h.

l , ½lð0Þ; lð1Þ; . . . ; lðL0 Q  1ÞT ;

3.2. M-step

ð35Þ

then

Update the parameter vector hk to

½mþ1 hk

l ¼ Kh:

according to

 2    ½m  ½m ¼ arg max J hk j^hk ¼ arg min ^zk  ðsk  ek Þhk  : hk

hk

ð28Þ

The updating process contains three small steps. For each small step, one of the three parameters (sk wk hk) is updated, while the others remain unchanged. Then the M-step of the SAGE algorithm consists of the following three small steps. Step 1.In this step, the updated value of sk is determined while ½m ½m wk and hk are fixed at wk and hk :sk is updated by

s^½mþ1 ¼ arg k ¼ arg

min

sk 2½0;1=Q ;...;ðQ 1Þ=Q 

max

sk 2½0;1=Q ;...;ðQ 1Þ=Q 

 ½m 2 k^zk  ðsk  ek Þhk k 

½m

½m

^ ^ ;hk ¼h wk ¼w k k

L0X Q 1

Re

 n o ½m ^ ½m : ^zk½m ðiÞ sk ði  sk TÞejiw^ k =Q h k

i¼0

ð29Þ Step 2.In this step, the updated value of wk is determined while ½m sk and hk are fixed at s½mþ1 and hk . wk is updated by k ^ ½mþ1 w k

i¼0

PL0 Q1

)

Substituting (23) into (22), it can be obtained

^h½mþ1 k

hk

From (15), it can be obtained

¼ f ðzk jhk Þ ¼

LX 0 Q 1 

¼ arg min

Due to the statistical independence of n(t)’s, the conditional probability density function of zk is given by

 n o f zk jhk ; ^h½m m

^ ½mþ1 sk ¼s^½mþ1 ;wk ¼w k k

hk



 ½m ¼ argmin k^zk  ðsk  ek Þhk k2  w k

¼ argmax wk

L0X Q1

Define

P , ½p1 ; p2 ; . . . ; pNt ;

ð37Þ

g , ½g1 ; g2 ; . . . ; gNt ;

ð38Þ

where

pk , ½pk ð0Þ; pk ð1Þ; . . . ; pk ðL0 Q  1ÞT ; pk ðtÞ , ejwk t=Q

^ ½m sk ¼s^½mþ1 ;hk ¼h k k

ð30Þ

L0 QþL g Q 1 X

xk ðiÞgðt  iT s  sk TÞ;

ð40Þ

gk , ½gk ð0Þ; gk ð1Þ; . . . ; gk ðL0 Q  1ÞT ;

ð41Þ

@p ðtÞ gk ðtÞ , k : @ sk

ð42Þ

Then the Fisher information matrix (FIM) for the estimation of H can be shown in (43), where Dt , diagð1; 2; . . . ; L0 Q Þ; H, diag ðh1 ; h2 ; . . . ; hNt Þ. See the proof in Appendix A.

2

ReðPH PÞ ImðPH PÞ

ImðPH PÞ ReðPH PÞ

ImðPH DPHÞ ReðPH DPHÞ

ReðPH gHÞ ImðPH gHÞ

3

7 6 7 26 7 6 7 H H H H H H 2 H H r2 6 4ImðH P DPÞ ReðH P DPÞ ReðH P D PHÞ ImðH P DgHÞ5 H H H H H H H H ReðH g PÞ ImðH g PÞImðH g DPHÞ ReðH g gHÞ ð43Þ

i¼0

The CRBs of the estimation of H are given by the diagonal elements of F1

Using the method in [9], it can be derived

^ ½m ^ ½mþ1 ¼w w k k

  h  i ½m =Q ^½m ejiw^ ½m ^½mþ1 k Q i¼0 iIm ^zk ðiÞ sk i  s h k k   h  i  P ½m L0 Q1 2 =Q ^½m ejiw^ ½m ^½mþ1 k i Re ^zk ðiÞ sk i  s h i¼0 k k

ð39Þ

i¼Lg Q

F¼

n o    ½m ^½m : ^½mþ1 Re ^zk ðiÞ  sk i  s T ejiwk =Q h k k

ð36Þ

CRBðhk Þ ¼ ðF1 Þk;k þ ðF1 ÞNt þk;Nt þk ;

PL0 Q1

1

ð31Þ

ð44Þ

CRBðwk Þ ¼ ðF Þ2Nt þk;2Nt þk ;

ð45Þ

CRBðsk Þ ¼ ðF1 Þ3Nt þk;3Nt þk :

ð46Þ

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Y. Tian et al. / Computer Communications 33 (2010) 2125–2131

Note that, strictly speaking, the CRB derived above for the timing estimation is not applicable at low SNR, since it does not condider the prior information that the timing offsets are within [0, 1), [19]. To this point, the Weighted Bayesian CRB (WBCRB) of the timing offsets has been proposed, and it has been proved to be a valid lower bound [13,20]. 5. Simulation results In this section, the performances of the proposed algorithm and the algorithm in [9,13] are assessed by Monte Carlo simulations and then compared with the theoretical lower bounds derived in Section 4, the CRBs. As described in Section 2, the synchronization problem of a distributed MIMO system can be divided into that of several MISO systems. So in all the simulations, we consider systems with two transmit antennas and one receive antenna. A pair of Hadamard sequences are assigned to the transmit antennas respectively as the training sequences. The length of the training sequences is 32. The pulse shaping filter g(t) is root-raised cosine filter with roll-off factor 0.22 and R þ1 normalized energy 1 g 2 ðtÞdt ¼ 1, Lg = 4. The simulations were performed in the particular fixed channel (h = [h1 h2]T = [0.2929 + 0.5169i 0.1074  0.9303i]T) and the fading channel. In the fading channel, the channel coefficients hk’s of the repetitious simulation statistics obey the complex Gaussian distribution. Here the synchronization performance results for the pair of the second transmit antenna and the receive antenna are presented. Firstly, the effect of the frequency offsets to the method in [13] was studied in the fixed channel and the fading channel. The simulation was done under the condition that the SNR is set to be 20 dB, the frequency offset between the first transmit antenna and the receive antenna w1 is fixed at 2p  0.015, and the timing offsets are set to be s = [s1s2] T = [1/4 3/4]T. We can see in Fig. 1 that the correct timing probability of s2 decreases with the increase of the frequency offset w2. Though the method in [13] is based on a ML estimation model, it does not take the frequency offsets into account. When the frequency offsets exist, the estimation is not accurate. The simulation results show this method can not initiate the timing offsets for the method in [9]. For the simulations of the proposed method and the method in [9], the method in [11] is adopted to initialize the frequency offsets w and the channel coefficients h for the iterative calculation. The iteration stops when the difference between log-likelihood function of the two consecutive iterations is less than 0.001. The frequency offsets are set to be w = [w1 w2]T = 2p[0.010 0.015]T.

Fig. 1. Correct timing probability of s2 of the method in [13] with Q = 4, SNR = 20 dB, w1 = 2p  0.010, s = [1/4 3/4]T.

The influence of the timing offsets to the method in [9] was studied in the fixed channel. The CRB is derived when there are no timing offsets. Without timing estimation and compensation procedure, it is shown in Fig. 2, the performance of the method in [9] degrades with the increase of timing offset s2 (where s1 = 0). There is about 5 dB loss when s2 = 2/4, and 10 dB loss when s2 = 3/4. That means this method can only work under the rigorous condition that the timing offsets have been well estimated. The simulation results also show this method can not initiate the frequency offsets for the method in [13]. Then the performance of our proposed algorithm was presented. The simulations were performed both in the fixed channel and the fading channel. Fig. 3 shows the correct timing probability of s2 with the oversample rate Q = 4. The proposed method can achieve accurate timing offset estimation at reasonable SNR. Due to the deep fading, the performance declines at low SNR in the fading channel. Figs. 4 and 5 show the MSE performance of w2 and h2 with Q = 4. The performances of the method in [9] are also provided for comparison. The CRBs are derived using the method in Section 4. The CRB in the fading channel represents the average CRB for the many different channel coefficients h’s in statistic. It is shown that the performances of the proposed algorithm reach the CRBs when SNRs are large enough. This proves the validity of our method. At low SNR, the proposed method may converge to the wrong estimation values, so the performance curves deviate the CRBs. However, the method in [9] behaves badly with the same conditions. Fig. 6 shows the average number of iterations needed to reach convergence. The proposed algorithm can reach the desired estimation values after finite number of iterations. Our algorithm needs a higher number of iterations for it estimates more parameters, the timing offsets, than [9]. We also took the simulation in the case of Q = 8. Fig. 7 shows the correct timing probability of s2. Figs. 8 and 9 show the MSE performance of w2 and h2. The simulations still show the good performance of our method. Fig. 10 shows the average number of iterations. In this section, we provide the simulation results of the proposed method and those in [9,13]. When timing and frequency offsets exist simultaneously, the methods in [9,13] cannot obtain the desired performances. On the contrary, the proposed method takes a joint estimation on all the unknown synchronization parameters, and the simulations prove the validity of the proposed algorithm. At reasonable SNRs, its MSE performances can reach the theoretical lower bounds (CRBs).

Fig. 2. MSE performances of w2 of the method in [9] with different timing offset s2.

Y. Tian et al. / Computer Communications 33 (2010) 2125–2131

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Fig. 3. Correct timing probability of s2 of the proposed method with Q = 4, s = [1/ 4 3/4]T.

Fig. 6. Average number of iterations with Q = 4, s = [1/4 3/4]T.

Fig. 4. MSE performance of w2 with Q = 4, s = [1/4 3/4]T.

Fig. 7. Correct timing probability of s2 of the proposed method with Q = 8, s = [3/ 8 6/8]T.

Fig. 5. MSE performance of h2 with Q = 4, s = [1/4 3/4]T.

Fig. 8. MSE performance of w2 with Q = 8, s = [3/8 6/8]T.

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Acknowledgment This work was supported in part by National Natural Science Foundation of China under Grant No. 60602009, ‘‘863” Project under Grant No. 2009AA01Z236, Chinese Important National Science & Technology Specific Projects under Grant No. 2009ZX03005-003 and ‘‘973” Project under Grant No. 2007CB310604. Appendix A. Derivation of the FIM for the estimation of H Here gives the derivation of the FIM for the estimation of

H , [Re{h}TIm{h}T wTsT]T. The FIM for the estimation of H is given by

F¼

Fig. 9. MSE performance of h2 with Q = 8, s = [3/8 6/8]T.

2

r

2

Re

 @ lH @ l :  T @H @H

ð47Þ

Then we use the similar mathematical derivations as in [10] to get F. The (l, q)th element of F is derived by

Fðl; qÞ ¼

H 

  L0 Q 1 @l @l 2 X @ l ðtÞ @ lðtÞ ¼ : Re  Re  @ Hl @ Hq r2 @ Hl @ Hq r2 t¼0 2

ð48Þ

Now the elements of F are calculated one by one. From (34), it is derived

@ lðtÞ ¼ pl ðtÞ; @Reðhl Þ @ lðtÞ ¼ jpl ðtÞ; @Imðhl Þ @ lðtÞ ¼ jthl pl ðtÞ; @wl @ lðtÞ ¼ hl gl ðtÞ: @ sl

ð49Þ ð50Þ ð51Þ ð52Þ

For l,q = 1, 2, . . . , Nt, it can be obtained

Fðl; qÞ ¼

2

r2

 Re pHl pq ;

ð53Þ

 Fðl; q þ Nt Þ ¼  2 Im pHl pq ; 2

Fig. 10. Average number of iterations with Q = 8, s = [3/8 6/8]T.

6. Conclusion In this paper, we propose a joint channel, timing and frequency offsets estimation algorithm for distributed MIMO system in flatfading channels in which multiple timing and frequency offsets exist. Conventional methods estimate timing and frequency offsets apart. In fact, the unsolved parameters will influence the performances of these methods. In the simulations, we show the effect of the frequency offsets on the timing offsets estimation method [13] and the effect of the timing offsets on the frequency offsets estimation method [9], both are not negligible. So we consider the parameters in a whole. We firstly establish a joint unbiased ML estimator for the multiple parameters. Due to the high computational complexity of ML estimation, we then propose an applicable iterative method based on the SAGE algorithm. For the sake of evaluating the performance of the proposed method, the CRB of the joint estimation is also derived. The simulation was done both in the fixed channel and the fading channel. The results show, at the reasonable SNRs, the proposed method can estimate the timing offsets precisely, and the MSE performances of the frequency offsets and channel coefficients estimation achieve the CRBs. And our iterative method will converge within a finite number of iterations..

r

 2 Fðl; q þ 2Nt Þ ¼  2 Im hq pHl Dt pq ; r   2 Fðl; q þ 3Nt Þ ¼ 2 Re hq pHl gq ;

r

Fðl þ Nt ; q þ Nt Þ ¼

2

r2

 Re pHl pq ;

 Re hq pHl Dt pq ;   2 Fðl þ Nt ; q þ 3Nt Þ ¼ 2 Im hq pHl gq ; r   2  Fðl þ 2Nt ; q þ 2Nt Þ ¼ 2 Re hl hq pHl D2t pq ; r   2  Fðl þ 2Nt ; q þ 3Nt Þ ¼ 2 Im hl hq pHl Dt gq ; r   2  Fðl þ 3Nt ; q þ 3Nt Þ ¼ 2 Re hl hq gHl gq ;

Fðl þ Nt ; q þ 2Nt Þ ¼

2

r2

r

ð54Þ ð55Þ ð56Þ ð57Þ ð58Þ ð59Þ ð60Þ ð61Þ ð62Þ

then the FIM can be derived and shown in (43). References [1] D. Gesbert, M. Shafi, D.-S. Shiu, From theory to practice: an overview of MIMO space-time coded wireless systems, IEEE Journal on Selected Areas in Communications 21 (3) (2003) 281–302. [2] A. Goldsmith, S.A. Jafar, N. Jindal, Capacity limits of MIMO channels, IEEE Journal on Selected Areas in Communications 21 (5) (2003) 684–702.

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