Time, Frequency Synchronization, and Equalization for ... - IEEE Xplore

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 7, JULY 2007

2593

Time, Frequency Synchronization, and Equalization for Asynchronous Multiuser MIMO Systems Yonghong Zeng, Senior Member, IEEE, and A. Rahim Leyman

Abstract— An asynchronous multiuser MIMO system with multipath channels and carrier frequency offsets (CFOs) or Doppler shifts is remodeled into another system. A blind method based on second order statistics is then used for estimating the shaped channels with ambiguity. Based on the shaped channels, specially designed pilots are proposed to find the CFOs. The same pilots are also used to resolve the time delays and the ambiguity in the estimated channels. After the CFOs, time delays and ambiguity are found, an equalization method based on the minimum mean square error (MMSE) criteria is obtained to recover the transmitted symbols. Only upper bounds for the channel orders and time delays are needed for implementing the algorithms, which makes them applicable to practical multiuser MIMO systems. Index Terms— MIMO, CFO, frequency offset, time delay, equalization, asynchronous, uplink, blind, semi-blind, order overestimation, doubly selective.

I. I NTRODUCTION

I

T is widely recognized that multiple input multiple output (MIMO) will be a major technique for future wireless communication systems. To fully exploit the potential of MIMO in multiuser systems, several problems must be addressed. First, since there are multiple users (antennas), and signals propagate through different channels usually have different time delays, a multiuser MIMO system is usually an asynchronous system. It is virtually impossible to precisely synchronize a multiuser MIMO system. Therefore, the time delays must be precisely acquired. Secondly, in most cases, there is a carrier frequency offset (CFO) or Doppler shift between each pair of transmitter and receiver, which turns the channels to be time selective. Due to the multiuser nature, different pair of transmitter and receiver may have different CFOs. This causes the CFO estimation much more difficult than that for a single input single output (SISO) system. Finally, for high speed transmission, channels are usually frequency selective (multipath), which causes serious inter-symbol interference (ISI). To combat ISI and multiuser interference (MUI), channel estimation and equalization are indispensable for recovering the transmitted signals. Both time delay acquisition and CFO estimation for a multiuser MIMO with unknown multipath channels are challenging problems. Since there are multiple time delays and multiple Manuscript received December 4, 2005; revised May 22, 2006 and October 26, 2006; accepted December 4, 2006. The associate editor coordinating the review of this paper and approving it for publication was Z. Tian. This work is supported by A*STAR EHS Research Grant (Number: 0221060041), Singapore. The authors are with the Institute for Infocomm Research, A*STAR, 21 Heng Mui Keng Terrace, Singapore 119613 (e-mail: {yhzeng, larahim}@i2r.a-star.edu.sg). Digital Object Identifier 10.1109/TWC.2007.05940.

CFOs to be estimated, while MUI and ISI exist, estimation methods for SISO systems [1]–[13] cannot be used here. Although there have been extensive researches on MIMO systems, most of them are concerned on synchronous systems [14]–[33] (due to the limitation of space, many others cannot be listed here). For the few investigations on asynchronous systems [34]–[37], at least one of the following restrictions is placed: (1) there is no CFO or only one CFO (all channels have the same CFO) [34]–[37]; (2) block transmission mode (such as OFDM) is assumed (the redundancy in blocks leads to some tolerance in time delay estimations) [36], [37]. Few researches, if any, have considered all the above-mentioned problems (time delay acquisition, CFO estimation and channel estimation) in a whole. Most known research works solve them separately (considering only one or two of them by assuming that the others have been solved). In practical wideband wireless communication based on MIMO, we need a holistic solution for asynchronous systems with doubly selective channels. That is, we must have a comprehensive method to find the time delays, CFOs and channels, and subsequently recover the symbols. In this paper, an uplink asynchronous multiuser MIMO system with multipath channels and frequency offsets is considered. After shifting the channels and embedding the CFOs, we remodel the system into a form in which only few time delays and CFOs need to be estimated. A blind method based on second order statistics (SOS) is then used for estimating the shaped channels with ambiguity. Based on the estimated shaped channels, we turn the MIMO system with frequency selective channels into a MIMO system with flat fading channels only. Then specially designed pilots are proposed to find the CFOs. The same pilots are also used for resolving the time delays and the ambiguity in the estimated channels. After the CFOs, time delays and ambiguity are found, symbols are recovered based on a linear minimum mean square error (MMSE) method. Some major features of the method are: (1) it is applicable to asynchronous multiuser MIMO systems with multiple time delays, multiple CFOs and multipath channels; (2) only an upper bound for the channel orders is needed (knowledge of true channel orders is unnecessary); (3) time delays for the multiple channels can be different, and only an upper bound for all the time delays is required; (4) different channels are allowed to have different CFOs. Simulations show that the algorithms are effective and robust. The rest of the paper is organized as follows. In Section 2, the system model and its transformation are discussed. A channel identification algorithm is presented in Section 3. Section 4

c 2007 IEEE 1536-1276/07$25.00

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where hij (k) is the channel response from user j to antenna i, Nij is the order of channel hij (k) (hij (0) = 0, hij (Nij ) = 0), A2j is the transmission power √ of user j, ηi (n) is the channel noise, and ω = ej2π (j = −1). The variances of the symbols are normalized to 1. Aj can be incorporated into hij (k) and therefore we simply use hij (k) to denote hij (k)Aj . In order to recover the transmitted symbols, it is usually necessary to know the time delays dij , the frequency offsets f (j) and the channels hij (k). Both the time delay and frequency offset estimations in the MIMO uplink are challenging problems. In the following, we will transform the system into another form where the CFOs are embedded into channels and the inputs are shifted by the minimum delays. The new form opens a new gate to solve the problems. Defining hij (k) = 0 for k < 0 or k > Nij and dj = min(dij ), we have

η1 (n)

h11 (k − d11 ) ? CFO f (1) - x1 (n) sP 1 CFO f (1) h (k − d ) @PP 12 12 PP @ CFO f (1)PP h1P (k − d1P ) P @ P PPh (k − d )η2 (n) CFO f (2)@ 21 P PP 21 ? q P @ - x2 (n) s2 (n) s Q CFO f (2) @ (k − d ) h 3 22 22 Q @ Q CFO f (2)Q @ Q h2P (k − d2P ) @ Q Q @ .. hM 1 (k − dM 1 ).. Q . CFO f (P ) . QQ@ @ Q (P ) Q@ CFO f Q@ ηM (n) )@ hM 2 (k − dM 2Q CFO f (P ) R ? s Q - xM (n) sP (n) s s1 (n)

hM P (k − dM P )

i

Fig. 1. Asynchronous multiuser MIMO uplink with multipath channels and frequency offsets.

xi (n) =

P +∞

ωf

(j)

n

hij (k − dij )sj (n − k) + ηi (n)

j=1 k=−∞

=

proposes a method for CFO estimations. The resolving of the time delays and the ambiguity, and equalization are discussed in Section 5. Some simulation results and comparisons are given in Section 6. Finally, conclusions are drawn in Section 7. Some notations are used in the following: superscripts T, † and ∗ stand for transpose, Hermitian (transconjugate), and conjugate, respectively. Iq is the identity matrix of order q and ⊗ is the Kronecker product of matrices.

P +∞

ωf

(j)

n

hij (k − dij + dj )

j=1 k=−∞

·sj (n − k − dj ) + ηi (n) =

P +∞

ωf

(j)

k

hij (k − dij + dj )

j=1 k=−∞ (j)

·ω f (n−k) sj (n − k − dj ) + ηi (n) P +∞ ˆ ij (k)ˆ = sj (n − k) + ηi (n), (2) h j=1 k=−∞

where

II. A SYNCHRONOUS MULTIUSER MIMO AND RE - MODELING A asynchronous multiuser MIMO uplink with multipath channels and frequency offsets is depicted in Figure 1. Assume that there are P users with each user sending a symbol sequence: sj (n) (j = 1, 2, · · · , P ) and M receivers (antennas) in the base-station with received signal: xi (n) (i = 1, 2, · · · , M ). In an asynchronous MIMO system, different channels may have different time delays. Let the time delay of the channel from user (transmitting antenna) j to receiving antenna i be dij . In general, there is a carrier frequency mismatch or Doppler shift between the user j and antenna i, which causes a CFO f (i,j) . It is reasonable to assume that all the antennas in the base-station share the same oscillator. Hence, f (i,j) = f (j) for all i. For notation simplicity, we normalize the CFO by the symbol time duration Ts , that is, f (j) = f (j) Ts . Thus, if the CFOs and channels are quasi-static (invariant within a certain time period), the received signal in baseband can be described as xi (n) =

Nij P

=

ˆ

xi (n) =

Nj P

ˆ ij (k)ˆ sj (n − k) + ηi (n). h

(4)

j=1 k=0

We assume that M > P . Letting x(n) = [x1 (n), x2 (n), · · · , xM (n)]T , ˆ 1j (n), h ˆ 2j (n), · · · , h ˆ Mj (n)]T , ˆ j (n) = [h h η(n) = [η1 (n), η2 (n), · · · , ηM (n)]T ,

(5)

we can express (4) into vector form as x(n) =

Nj P

ˆj (k)ˆ h sj (n − k) + η(n), n = 0, 1, · · · . (6)

j=1 k=0

ωf

(j)

n

Considering L consecutive outputs and defining

hij (k)Aj sj (n − k − dij ) + ηi (n),

x ˆ(n) = [xT (n), xT (n − 1), · · · , xT (n − L + 1)]T ,

j=1 k=0 +dij P Nij

ˆ ij (k) = ω f (j) k hij (k − dij + dj ), sˆj (n) = ω f (j) n sj (n − dj ). h (3) ˆj = max(Nij + dij ) − dj . Then it is obvious that Let N i ˆ ij (k) = 0 if k < 0 or k > N ˆj . Hence (2) can be written as h

ωf

(j)

n

hij (k − dij )Aj sj (n − k) + ηi (n),

j=1 k=dij

n = 0, 1, · · · ,

(1)

ηˆ(n) = [η T (n), η T (n − 1), · · · , η T (n − L + 1)]T , ˆ s(n) = [ˆ s1 (n), sˆ1 (n − 1), · · · , sˆ1 (n − N1 − L + 1), · · · , sˆP (n), sˆP (n − 1), · · · , sˆP (n − NP − L + 1)]T ,

(7)

ZENG and LEYMAN: TIME, FREQUENCY SYNCHRONIZATION, AND EQUALIZATION FOR ASYNCHRONOUS MULTIUSER MIMO SYSTEMS

we get x ˆ(n) = Hˆ s(n) + ηˆ(n), ˆ + P L) (N ˆ = where H is an M L × (N

(8)

P ˆj ) matrix defined N j=1

as ⎡ ⎢ Hj = ⎣

ˆj (0) h 0

··· .. . ···

H = [H1 , H2 , · · · , HP ], ⎤ ˆj (N ˆj ) · · · h 0 ⎥ .. ⎦. . ˆj (N ˆj ) ˆj (0) · · · h h

(9)

III. B LIND CHANNEL IDENTIFICATION The remodeled system (4) does not explicitly include the time delays and CFOs. Hence some known blind methods, such as [14], [15], [17], [19], [21], [22], [24], [26], [28], [31], could be used to identify the shaped channels. However, some methods, such as the subspace method [14], are very sensitive to channel order over-estimation [31], [38]–[40]. In practice, it is very difficult to obtain the exact channel orders (especially for multiuser MIMO where different channels usually have different channel orders). Only an upper bound for the channel orders is available (based on the distance of the users, it is possible to know the maximum delay spread). Hence, we choose to use the method in [31], which is proved to be robust to channel order over-estimation. Some properties (assumptions) on the statistics of the signals and channel noises are assumed as follows. (A1) Transmitted symbols have zero expectations and are independently and identically distributed. (A2) Noises are white and uncorrelated. (A3) Noises and transmitted signals are uncorrelated. A critical requirement in [31] is that the matrix H must be of full column rank. If M > P and the smoothing factor ˆ /(M − P ), the matrix has more rows than columns. L>N ˆj (0) = 0 Therefore, it is most likely of full column rank if h ˆ ˆ and hj (Nj ) = 0 (j = 1, 2, · · · , P ). This condition is indeed guaranteed and proved in the following. ˆj (0) = 0 and h ˆj (N ˆj ) = 0 (j = 1, 2, · · · , P ). Lemma 1: h Proof. From the definition of dj , there exists an i, say, i0 , ˆ i j (0) = hi j (0) = 0, which leads to such that dj = di0 j . So, h 0 0 ˆ hj (0) = [h1j (dj −d1j ), · · · , hMj (dj −dMj )]T = 0. Similarly, ˆj = Nij + dij − dj , and therefore there exists an i such that N ˆj (N ˆj ) = 0. h Hence, matrix H can be assumed to be of full column rank, and the method in [31] can be directly used for identifying the ˆ ij (k) (with ambiguities), which are time shaped channels h delayed and phase shifted versions of hij (k). However, to finally obtain the symbols sj (n), the delays dj (not all dij ), the CFOs f (j) and the ambiguities must be found too. The resolving of the dj , f (j) and the ambiguities will be discussed in the following sections. For convenience of reading, the ˆ ij (k) (with ambiguities) is given algorithm for identifying the h in the following. Algorithm 1: Blind channel identification for asynchronous MIMO systems with multipath channels and frequency offsets

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It is assumed that an upper bound for all the channel ˆj Nupp (j = orders, that is, a number Nupp such that N 1, 2, · · · , P ), is known or estimated. Choose a smoothing factor L such that M L > P Nupp + P L. L+L −1 Step 1. Compute R = L1s n=L s x ˆ(n)ˆ x† (n), where Ls is the number of samples used. Compute the eigenvalue decomposition (EVD) of R. Use the MDL method [41] to estimate the rank of H (only some simple manipulation of the eigenvalues are involved). Let the estimated rank be r. Average the least M L − r eigenvalues of R to get an estimation (¯ ση2 ) 2 ¯ for the noise variance. Let R = R − σ ¯η IML . 1 L+Ls −1 x ˆ(n)ˆ x† (n − 1) and Step 2. Compute Q = Ls n=L 2 ¯ Q = Q−σ ¯η (JL ⊗ IM ), where JL is an L × L down shift matrix, that is, it is a lower triangular Toeplitz matrix with its first column being [0, 1, 0, · · · , 0]T . Then compute the singular ¯ Let K1 = M L−r + value decomposition (SVD) of matrix Q. P . Choose K1 vectors ui (i = 1, 2, · · · , K1 ) corresponding ¯ and denote a matrix to the K1 least left singular values of Q U = [u1 , u2 , · · · , uK1 ]. ¯ and the EVD of A. Let V1 Step 3. Compute A = U† RU be the matrix of size K1 × P whose columns are eigenvectors corresponding to nonzero eigenvalues of A, and U1 = UV1 . Step 4. For k = 0, 1, · · · , Nupp + L − 1, compute

Rx (k), k = 0 , Rz (k) = Rx (k) − σ ¯η2 IM , k = 0 k+L −1 where Rx (k) = L1s n=k s x(n)x† (n − k). Step 5. For k = 0, 1, · · · , Nupp , let G(k) = Rz (k) Rz (k + 1) · · · Rz (k + L − 1) U1 . (10) ˆ The MIMO channel matrix is then H(k) = G(k)B−1 , where B is a P × P matrix to be determined, and ⎡ ⎤ ˆ 11 (k) · · · h ˆ 1P (k) h ⎢ ⎥ .. .. ˆ H(k) =⎣ (11) ⎦. . ··· . ˆ MP (k) ˆ M1 (k) · · · h h IV. CFO ESTIMATION In the last section, we have obtained an estimation for the shaped channels (with ambiguity) using the second order statistics (SOS) of the received signals. The shaped channels (with ambiguity) can be used to simplify the time delay and CFO estimations as shown in the following. Let Wn = diag(ω f

(1)

n

, ωf

(2)

n

, · · · , ωf

(P )

n

)

(12)

¯ s(n) = [s1 (n − d1 ) s2 (n − d2 ) · · · sP (n − dP )] .

(13)

T

Based on the shaped channels, a linear minimum mean square error (MMSE) equalization can be constructed as ¯ †γ R−1 x Wn−γ¯ s(n − γ) = H ˆ(n),

(14)

¯ γ is a where γ is a delay, 0 γ Nupp + L − 1, and H matrix of size M L × P which is constructed from the shaped ˆ channels H(k) (see (11) for definition) as follows (see [18],

2596


where bj is the j-th column of matrix B† , and n = (j − 1)Jp + k or n = (P + j − 1)Jp + k (It k Jp − 1). Noticing the periodical property of the pilots, we have

page 341), ⎡ ⎢ ⎢ ⎢ ¯ Hγ = ⎢ ⎢ ⎢ ⎣

ˆ H(γ) .. . ˆ H(0) .. . ˆ H(−(L − 1 − γ))

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

(15)

(16)

B† Wn¯ s(n) = y(n),

(17)

¯ † R−1 x y(n) = G ˆ(n + γ). γ

(18)

that is,

where Please note that y(n) can be computed from the outputs and estimated channels (with ambiguity) G(k). For high SNR, the matrix R tends to be singular. To avoid inverting the matrix, we use FΣ−1 F† to replace R−1 , where the columns of F are r eigenvectors of R corresponding to the largest r eigenvalues and Σ is a diagonal matrix with diagonal elements being the r largest eigenvalues (this is somewhat similar to the subspace approach in [42]). Here r is the detected rank of H obtained in the Step 1 of Algorithm 1. F is found as a byproduct of Algorithm 1. (17) can be treated as a multiuser MIMO with flat fading channels. In (17), the ambiguity matrix B, the CFOs and the time delays dj are unknowns. In this section, a pilotbased method is proposed to resolve the CFOs. The delays and ambiguity resolvation will be discussed in the following section. We assume that an upper bound for the time delays is known or estimated (for example, a coarse synchronization has been done such that the time delays are within a certain window). Let It be the upper bound, that is, It ≥ dj (j = 1, 2, · · · , P ). We use specially designed pilots to resolve the CFOs. Assume that the 2P Jp symbols at the beginning of a data packet are pilots, where Jp > It . The pilots are designed as follows.

sj (n) = 0, (j − 1)Jp n jJp − 1 sj (n) = 0, otherwise and 0 n P Jp − 1 sj (n + P Jp ) = sj (n), 0 n P Jp − 1. (19) Based on the pilot structure, it is obvious that, at time n = (j − 1)Jp + k or n = (P + j − 1)Jp + k (0 k Jp − 1), only the user j is active while the other users are silent. This eliminates the multiuser interferences in (17). Without multiuser interferences, (17) is turned to y(n) = ω

n

sj (n − dj )bj ,

P Jp

y(n),

(21)

Similar to [5], defining

¯ † R−1 x s(n − γ) = G ˆ(n), B† Wn−γ¯ γ

f

(j)

(j − 1)Jp + It n jJp − 1.

ˆ Please note that, similarly as before, H(k) = 0 for k < 0 or k > Nupp . It is possible to find the best delay γ (minimizing the equalization error) (see [18]). However, the minimization process needs very high computational complexity. For sim¯ γ be defined similarly as plicity, we choose γ = Nupp . Let G ¯ γ from G(k) (see (10)). From Algorithm 1 (Step 5), it is H ¯γ = G ¯ γ B−1 . Therefore, obvious that H

(j)

y(n + P Jp ) = ω f

(20)

1 ψj = Jp − It

jJp −1

y† (n)y(n + P Jp ),

(22)

n=(j−1)Jp +It

from (21), we thus obtain an estimation for f (j) as fˆ(j) =

1 arg(ψj ), 2πP Jp

(23)

where arg means the argument of a complex number. It is obvious that the method is valid only if |f (j) | < 2P1Jp . The algorithm for estimating the CFOs is summarized as follows. Algorithm 2: Estimation of the CFOs Assume that the 2P Jp pilots are inserted in a data packet. Step 1. Use Algorithm 1 to get G(k). ˆ γ from G(k) and compute y(n) = Step 2. Construct G † −1 ¯ Gγ R x ˆ(n + γ) for (j − 1)Jp + It n jJp − 1 and (P + j − 1)Jp + It n (P + j)Jp − 1, where R has been obtained in Algorithm 1. Step 3. Compute ψj =

1 Jp − It

jJp −1

y† (n)y(n + P Jp ).

n=(j−1)Jp +It

Step 4. The estimation for f (j) is fˆ(j) =

1 2πP Jp

arg(ψj ).

V. T IMING ACQUISITION , AMBIGUITY RESOLVATION AND EQUALIZATION

Given the CFOs, we now consider estimating the time delays. For simplicity of using notations, we assume that the estimated CFOs are equal to the true CFOs. Let (j) (24) αj (k, n) = y(n)ω −f n s∗j (n − k), (j − 1)Jp + It n jJp − 1, 0 k It . From (20), we know that αj (k, n) = bj sj (n − dj )s∗j (n − k),

(25)

(j − 1)Jp + It n jJp − 1, 0 k It . Summarize the terms to define jJp −1

βj (k) =

αj (k, n)

n=(j−1)Jp +It jJp −1

= bj

n=(j−1)Jp +It

sj (n − dj )s∗j (n − k), 0 k It . (26)


If we choose the pilot sequence to have the best autocorrelation, that is, jJp −1

sj (n − dj )s∗j (n − k) = 0, k = dj , 0 k It

n=(j−1)Jp +It

(27) the delay dj is uniquely determined to be the k which maximizes the power of βj (k) for 0 k It . Since the time delay dj is unknown, condition (27) is untestable. Therefore, we should use the following sufficient condition (may not necessary) for the best pilots:

2597

Assume that the Nupp and It are known or estimated. Choose a smoothing factor L > P Nupp /(M − P ). Step 1. Blind identify the channels by Algorithm 1. Step 2. Use Algorithm 2 to find the CFOs. Step 3. Apply Algorithm 3 to obtain the time delays. Step 4. Determine the ambiguity matrix B as discussed above. Step 5. Equalize the symbols by equation (14). Please note that Steps 1 to 4 are used only once in a time slot. When the channels, the CFOs and delays are estimated, only Step 5 is needed for equalization.

jJp −1

sj (n − l)s∗j (n − k) = 0, k = l, 0 k, l It .

n=(j−1)Jp +It

(28) It is difficult to give a general solution for the best pilots. However we can use computers to search for the best ones. Numerical experiments show that in most cases there exists binary best pilots. Even if the pilot is not the best (the best pilot may not exist in some cases), dj is still determinable provided that ||βj (dj )||2 > ||βj (k)||2 , k = dj . It is guaranteed for any a sequence that ||βj (dj )||2 ≥ ||βj (k)||2 , k = dj based on the well-known Schwarz theorem, but the strict inequality needs some properties of the sequence. Obviously using the best pilot leads the estimation less vulnerable to noise and roundoff errors. The algorithm for estimating dj is outlined in the following. Algorithm 3: Estimation of the time delays Provided that there are 2P Jp pilots in a data packet defined in the last section. Step 1. Compute jJp −1

βj (k) =

ˆ(j) n ∗ sj (n − k), k

y(n)ω −f

= 0, 1, · · · , It

n=(j−1)Jp +It

Step 2. Choose dj to be the k such that ||βj (k)||2 is maximized. When the CFOs and time delays are resolved, we now consider the estimation of the ambiguity matrix B. The same pilots for resolving the CFOs and time delays can be used here. In fact, from (20), we know that

VI. S IMULATIONS AND COMPARISONS In the simulations, only two parameters are assumed known as a priori: (1) an upper bound, Nupp , for all the channels orders; (2) an upper bound, It , for all the time delays. We do not need the exact channel orders, which are usually unavailable in an MIMO system. Let Nmax be the exact upper ˆj . To bound of all the channel orders, that is, Nmax = max N j

verify the robustness of the algorithm to the channel order overestimation, we test four cases of the upper bound (Nupp ): Nmax (exact upper bound), Nmax +2, Nmax +4 and Nmax +6, respectively. Unless otherwise stated, the smoothing factor is chosen as L = P Nupp /(M − P ) + 1. Although larger L sometimes gives better performance, smaller L means lower computational complexity. So, we choose the smallest possible L. 1 In the following, a Monte Carlo realization means: random Rayleigh fading channels (tap coefficients are random 2 complex numbers with Gaussian distribution) are created; 3 for each user, a packet with random CFOs are generated; 1200 symbols (including 1056 random 4-QAM data samples and 144 pilots) is transmitted through the wireless channel with Gaussian white noises. The signal-noise-ratio (SNR) is defined as the ratio of the average received signal power to the average noise power SNR =

E(||x(n) − η(n)||2 ) , E(||η(n)||2 )

(32)

(31)

where E(·) is the mathematical expectation. The algorithms are tested by simulations on a large number of different channels, time delays and CFOs. Simulations show that the algorithms are truly robust to channel order and time delay overestimation, noise and round-off errors. We give the simulation results for a 2-user 4-antenna system (M = 4, P = 2) in the following. The real channel orders are 3 (for user 1) and 4 (for user 2) respectively (we do not use this information for simulation). The time delays are ⎡ ⎤ 0 1 ⎢ 0 1 ⎥ ⎥ (dij ) = ⎢ ⎣ 0 2 ⎦. 0 1

After the CFOs, the time delays and the ambiguity have been resolved, equalization can be done by (14). The algorithm is outlined as follows. Algorithm 4: Semi-blind equalization for doubly selective asynchronous MIMO systems

ˆ 2 = 5, Nmax = 5 and d1 = 0, d2 = 1. The ˆ1 = 3, N Hence N CFOs, f (1) and f (2) , are different random numbers evenly distributed in (−0.004, 0.004). That is, if the symbol time duration is Ts = 0.1μs, the CFOs are in (−104 , 104 )Hz (a mobile unit moving at the speed of 100 km/h can only

y(n)ω −f

(j)

n ∗ sj (n

− dj ) = |sj (n − dj )|2 bj ,

(29)

(j − 1)Jp + It n jJp − 1. A least square (LS) estimation for bj is then jJp −1 −f (j) n ∗ sj (n − dj ) n=(j−1)Jp +It y(n)ω . bj = jJp −1 2 n=(j−1)Jp +It |sj (n − dj )| Thus the ambiguity matrix is B† = b1 · · ·

bP

.

(30)

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−4

10

Nupp=5 Nupp=7 Nupp=9 Nupp=11

−1

10

Error rates of the delay estimations

RMSE of the estimated CFOs


−2

10

−3

10

−5

10

−4

5

10

15

20

25

30

35

10

40

5

10

15

20

SNR (dB)

Fig. 2.

RMSE of CFO versus SNR (Ls = 1000).

25

30

35

40

SNR (dB)

Fig. 3.

Error of delay estimation versus SNR (Ls = 1000). −5

3.8

N =5 upp Nupp=7 Nupp=9 Nupp=11

3.7

3.6


produce a Doppler shift up to about 500Hz when the carrier frequency is 5GHz). The upper bound for time delays is chosen as It = 4 (the time delays are overestimated by at least 3). The number of pilots is 144 (Jp = 36). We use the best pilot sequence satisfying (28) as (−1, 1, 1, −1, −1, 1, 1, 1, 1, −1, 1, 1, 1, 1, 1, −1, −1, −1, 1, −1, −1, 1, −1, 1, 1, 1, 1, −1, 1, −1, 1, −1, −1, 1, 1, −1). All the results are averaged over 2000 Monte Carlo realizations. Since the identification of the shaped channels is based on the method in [31], the results are similar to those in [31] and therefore not shown here. 1. CFO and delay estimations We use the root mean square error (RMSE) to measure the accuracy of the estimated CFOs. The RMSE between the estimated and true channel CFOs is defined as ⎞ ⎛ P RMSE = E ⎝ |f (j) − fˆ(j) |2 ⎠ . (33)

x 10

3.5

3.4

3.3

3.2

3.1

3

2.9 400

500

600

700

800

900

1000

1100

1200

1300

1400

Sample size

Fig. 4.

RMSE of CFO versus sample size (SNR=25dB).

j=1

The accuracy for delay estimation is measured by error rate, that is, the number of wrong estimations to the number of whole estimations. Note that we do not need the exact error of a delay estimation, because any wrong estimation, no matter what the error is, has similar (catastrophic) impact to the equalization. The RMSE (for CFO) versus SNR is shown in Figure 2. The error rates of delay estimations are shown in Figure 3. For these two figures, 1000 samples are used for estimating the statistical correlation matrices. In Figure 4 and 5, the SNR is fixed to 25dB, while the sample size used is changed from 300 to 1200. From the figures, we see that the algorithm works well when only upper bounds for all the channel orders and time delays are known. Other simulations show that the smoothing factor does not affect the RMSE significantly. To save space, we do not include the simulation results here. 2. Equalization Figure 6 shows the bit error rate (BER) (average of the two users) versus SNR, where 1000 samples are used for estimating the statistical correlation matrices, where the transmitted symbols are 4-QAM modulated. For comparison, the BER

results using the true channels, true CFOs and true delays (the line with no mark) and those using the true channels and true delays but without CFO estimation (the line with mark ) are also given. Since the test is based on finite number of samples, very small (< 10−5 ) BER is meaningless and therefore it is not shown. From the figure, it is clear the algorithms are robust to channel order and time delay overestimations. Furthermore, the assumed upper bounds can be much larger than the exact upper bounds. It also shows that the CFO, if not estimated, can cause catastrophic effect. The accuracy of time delay estimations is even more crucial. Due to the errors in time delay estimations (about 1 error in 2000 tests at 30dB SNR), the BERs are much higher than the ideal ones. 3. Comparisons Very few researches, if any, have considered the time delay acquisition, CFO estimation and channel estimation in a whole for a multiuser MIMO system with multipath channels. Hence, we choose to compare our method with the following pilot based approach. First, letting the P users transmit their pilots at different time (when one user is transmitting pilots, other users are quiet), we turn the MIMO system into SISO systems.


2599

−3

5

x 10

Nupp=5 Nupp=7 Nupp=9 N =11

N =5 upp Nupp=7 Nupp=9 N =11

4.5

upp

4



upp

3.5

3

2.5

2

−4

10

1.5 −5

10 1

400

500

600

700

800

900

1000

1100

1200

1300

5

1400

10

15

20

Fig. 5.

Fig. 7.

Error of delay estimations versus sample size. Real channel Nupp=5 Nupp=7 Nupp=9 N =11

−1

10

25

30

35

40

SNR (dB)

Sample size

RMSE of CFO versus SNR (pilot based).


−1

10

Bit Error Rate (BER)

−2

10

−3

10


upp

No cfo estimation

−2

10

−4

10

−3

−5

10

10 5

10

15

20

25

5

30

10

15

Fig. 6.

BER versus SNR (Ls = 1000).

Fig. 8.

At a time period, if only user j is transmitting, the received signal at antenna i is xi (n) =

Nij

ω

f (j) n

hij (k)sj (n − k − dij ) + ηi (n).

20

25

30

35

40

SNR (dB)

SNR (dB)

(34)

Equation (34) can be turned to xi = Bij hij + ηi ,

(35)

where

k=0

This is a SISO system with multipath channel. Secondly, methods for timing and frequency synchronization for SISO systems are used. There are quiet a few methods to choose at this moment [3], [4], [6]. In general, timing and frequency synchronization, especially timing synchronization, is still much more difficult than those for the SISO systems with flat-fading channels. For timing synchronization, the unknown multipath channel will cause many methods unreliable [3], [4]. To overcome this difficulty, the recently proposed generalized ML in [3] and the GAIC in [4] jointly estimate the timing and channel order. Both methods need two-dimensional exhaustive search which is very complex. We choose to compare our method with the generalized ML (though its complexity may be too high for practical application). For convenience of reading, the method is described in the following.

Error of delay estimations versus SNR.

Bij

=

sj (n − dij ) sj (n + 1 − dij ) .. . sj (n + Np − 1 − dij )

··· ··· ···

sj (n − Nij − dij ) sj (n + 1 − Nij − dij ) .. . sj (n + Np − 1 − Nij − dij )

,

(36)

xi hij ηi

= [xi (n), xi (n + 1), · · · , xi (n + Np − 1)]T , = [hij (0), hij (1), · · · , hij (Nij )]T , = [ηi (n), ηi (n + 1), · · · , ηi (n + Np − 1)]T . (37)

Here NP is the pilot length. Note that the real channel order is also unknown. The generalized ML jointly estimate the time delay dij and the channel order Nij . The estimation is as

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TABLE I P ILOT STRUCTURES FOR THE TWO USERS user 1 user 2

(Nupp + 1) zeros

s0

s0

72 zeros (Nupp + 1) zeros

72 zeros s0 s0

follows (see equation (60) in [3]). Define a cost function

= (−Np +

Ψ(dij , Nij ) † −1 † Nij + 2) log(||xi − Bij (Bij Bij ) Bij xi ||2 ) −(Nij + 1) log(2) − log(det(B†ij Bij )).

(38)

The estimation for (dij , Nij ) is to find the point that maximizes the cost function Ψ(dij , Nij ) (search dij ∈ {0, 1, · · · , It } and Nij ∈ {0, 1, · · · , Nupp }). To convert the MIMO system into SISO systems, the pilot structure is designed in Table I, where s0 is a random sequence (BPSK) of length (72 − Nupp − 1)/2. Before the timing synchronization, the method in [6] is used for frequency synchronization. The RMSE (for CFO) versus SNR is shown in Figure 7. The error rates of delay estimations are shown in Figure 8. Comparing Figure 2 and Figure 3 with Figure 7 and Figure 8, we see that the two methods achieve comparable results for frequency synchronization. But the proposed method obtains much better timing synchronization than the pilot based method. VII. C ONCLUSIONS With upper bounds for channel orders and time delays as a priori, methods have been proposed to estimate the CFOs and time delays for asynchronous multiuser MIMO systems with doubly selective channels. The methods use the statistical information as well as pilot symbols. After finding the CFOs and time delays, a linear MMSE method has been presented for equalization. The mild constraints on the methods make them applicable to practical multiuser MIMO systems.

ACKNOWLEDGMENT Thanks to the associate editor and the anonymous reviewers for their invaluable comments. R EFERENCES [1] Z. Tian and G. B. Giannakis, “A GLRT approach to data-aided timing acquisition in UWB radios–part i: algorithms,” IEEE Trans. Wireless Commun., vol. 4, no. 6, pp. 2956–2967, 2005. [2] Z. Tian and G. B. Giannakis, “A GLRT approach to data-aided timing acquisition in UWB radios–part ii: training sequence design,” IEEE Trans. Wireless Commun., vol. 4, no. 6, pp. 2994–3004, 2005. [3] Y. C. Wu, K. W. Yip, T. S. Ng, and E. Serpedin, “Maximumlikelihood symbol synchronization for IEEE 802.11a WLANs in unknown frequency-selective fading channels,” IEEE Trans. Wireless Commun., vol. 4, no. 6, pp. 2751–2763, 2005. [4] E. Larsson, G. Liu, J. Li, and G. B. Giannakis, “Joint symbol timing and channel estimation for OFDM based WLANs,” IEEE Commun. Lett., vol. 5, no. 8, pp. 325–327, 2001. [5] J. Van de Beek, M. Sandell, and P. O. B¨ orjesson, “ML estimation of time and frequency offset in OFDM systems,” IEEE Trans. Signal Processing, vol. 45, no. 7, pp. 1800–1805, 1997.

[6] T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun., vol. 45, no. 12, pp. 1613– 1621, 1997. [7] M. Morelli and U. Mengali, “An improved frequency offset estimator for OFDM applications,” IEEE Commun. Lett., vol. 3, no. 3, pp. 75–77, 1999. [8] M. Ghogho, A. Swami, and G. B. Giannakis, “Optimized null-subcarrier selection for CFO estimation in OFDM over frequency selective fading channels,” in Proc. IEEE GlobeCom, pp. 202–206, 2001. [9] X. Ma, C. Tepedelenlioglu, G. B. Giannakis, and S. Barbarossa, “Nondata-aided carrier offset estimators for OFDM with null subcarriers: identifiability, algorithms, and performance,” IEEE J. Sel. Areas Commun., vol. 19, no. 12, pp. 2504–2515, 2001. [10] J. Lei and T. S. Ng, “A consistent OFDM carrier frequency offset estimator based on distinctively spaced pilot tones,” IEEE Trans. Wireless Commun., vol. 3, no. 3, pp. 588–599, 2004. [11] U. Tureli, P. J. Honan, and H. Liu, “Low-complexity nonlinear least square carrier offset estimator for OFDM: identifiability, diversity and performance,” IEEE Trans. Signal Processing, vol. 52, no. 9, pp. 2441– 2452, 2004. [12] A. B. Salberg and A. Swami, “Doppler and frequency-offset synchronization in wideband OFDM,” IEEE Trans. Wireless Commun., vol. 4, no. 6, pp. 2870–2881, 2005. [13] Y. H. Zeng and A. R. Leyman, “Pilot-based fast estimation of frequency offset and channel for OFDM,” in Proc. IEEE VTC-Spring (Australia), May 2006. [14] K. Abed-Meraim, P. Loubaton, and E. Moulines, “A subspace algorithm for certain blind identification problem,” IEEE Trans. Inf. Theory, vol. 32, no. 2, pp. 499–511, 1997. [15] K. Abed-Meraim, P. Loubaton, and E. Moulines, “Prediction error method for second-order blind identification,” IEEE Trans. Signal Processing, vol. 45, no. 3, pp. 694–705, 1997. [16] O. Besson and P. Stoica, “On parameter estimation of MIMO flatfading channels with frequency offsets,” IEEE Trans. Signal Processing, vol. 51, no. 3, pp. 602–613, 2003. [17] I. Bradaric, A. P. Petropulu, and K. I. Diamantaras, “On blind identifiability of FIR-MIMO systems with cyclostationary inputs using second order statistics,” IEEE Trans. Signal Processing, vol. 51, no. 2, pp. 434– 441, 2003. [18] Z. Ding and Y. Li, Blind Equalization and Identification. New York: Marcel Dekker, Inc., 2001. [19] Z. Ding, “Matrix outer product decomposition method for blind multiple channel identification,” IEEE Trans. Signal Processing, vol. 45, no. 12, pp. 3053–3061, 1997. [20] G. B. Giannakis, Y. Hua, P. Stoica, and L. Tong, Signal Processing Advances in Wireless & Mobile Communications, vol. 1. Prentice Hall PTR, 2001. [21] A. Gorokhov and P. Loubaton, “Blind identification of MIMO-FIR systems: a generalized prediction approach,” Signal Processing, vol. 73, pp. 105–124, 1999. [22] T. P. Krauss and M. D. Zoltowski, “Bilinear approach to multiuser second-order statistics-based blind channel estimation,” IEEE Trans. Signal Processing, vol. 48, no. 9, pp. 2473–2486, 2000. [23] Y. Li and K. J. R. Liu, “Adaptive blind source separation and equalization for multiple-input/multiple-output systems,” IEEE Trans. Inf. Theory, vol. 44, no. 11, pp. 2864–2876, 1998. [24] R. Lopez-Valcarce and S. Dasgupta, “Blind channel equalization with colored sources based on second-order statistics: a linear prediction approach,” IEEE Trans. Signal processing, vol. 49, no. 9, pp. 2050– 2059, 2001. [25] J. Shen and Z. Ding, “Direct blind MMSE channel equalization based on second-order statistics,” IEEE Trans. Signal Processing, vol. 48, no. 4, pp. 1015–1022, 2000. [26] D. Slock, “Blind fractionally-spaced equalization, perfect reconstruction filterbanks, and multilinear prediction,” in Proc. IEEE ICASSP Conf. (Adalaide, Australia), 1994. [27] N. Thirion-Moreau and E. Moreau, “Generalized criteria for blind multivariate signal equalization,” IEEE Signal Processing Lett., vol. 9, no. 2, pp. 72–74, 2002. [28] J. K. Tugnait and B. Huang, “Multistep linear predictors-based blind identification and equalization of multiple-input multiple-output channels,” IEEE Trans. Signal Processing, vol. 48, no. 1, pp. 26–38, 2000. [29] C. S. Xiao and Y. R. Zheng, “Ergodic capacity of doubly selective rayleigh fading MIMO channels,” in Proc. IEEE Wireless Communications and Networking Conference (WCNC) (Taoyuan, Taiwan), vol. 1, pp. 345–350, March 2004. [30] L. Q. Yang, X. L. Ma, and G. B. Giannakis, “Optimal training for MIMO fading channels with time- and frequency-selectivity,” in Proc. IEEE


[31] [32] [33] [34] [35] [36] [37] [38] [39]

[40] [41] [42]

International Conference on Acoustics, Speech, and Signal Processing (ICASSP), vol. 3, pp. 821–824, May 2004. Y. H. Zeng and T. S. Ng, “Blind estimation of MIMO channels with an upper bound for channel orders,” Signal Processing, vol. 82, no. 2, pp. 212–222, 2006. Y. H. Zeng and T. S. Ng, “A semi-blind channel estimation method for multi-user multi-antenna OFDM systems,” IEEE Trans. Signal Processing, vol. 52, no. 5, pp. 1419–1429, 2004. J. Zhu, Z. Ding, and X. R. Cao, “Column-anchored zero-forcing blind equalization for multiuser wireless FIR channels,” IEEE J. Sel. Areas Commun., vol. 17, pp. 411–423, 1999. Y. Inouye and R. W. Liu, “A system-theoretic foundation for blind equalization of an FIR MIMO channel system,” IEEE Trans. Circuits Systems -I, vol. 49, no. 4, pp. 425–436, 2002. Y. L. C. de Jong and T. J. Willink, “Iterative trellis search detection for asynchronous MIMO systems,” in Proc. IEEE Vehicular Technology Conference (VTC 2003-Fall), vol. 1, pp. 503–507, 2003. A. N. Mody and G. L. Stuber, “Synchronization for MIMO OFDM systems,” in Proc. IEEE GlobeCom, pp. 509–513, 2001. G. L. Stuber, J. R. Barry, S. W. McLaughlin, Y. Li, M. A. Ingram, and T. G. Pratt, “Broadband MIMO-OFDM wireless communications,” Proc. of the IEEE, vol. 92, no. 2, pp. 271–294, 2004. W. Qiu and Y. Hua, “Performance analysis of the subspace method for blind channel identification,” Signal Processing, vol. 50, pp. 71–81, 1996. A. P. Liavas, P. A. Regalia, and J. P. Delmas, “On the robustness of the linear prediction method for blind channel identification with respect to effective channel undermodeling/overmodeling,” IEEE Trans. Signal Processing, vol. 48, no. 5, pp. 1477–1481, 2000. H. Gazzah, P. A. Regalia, J. P. Delmas, and K. Abed-Meraim, “A blind multichannel identification algorithm robust to order overestimation,” IEEE Trans. Signal Processing, vol. 50, no. 6, pp. 1449–1458, 2002. M. Wax and T. Kailath, “Detection of signals by information theoretic criteria,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 33, pp. 387–392, Apr. 1985. X. Wang and H. V. Poor, “Blind multiuser detection: a subspace approach,” IEEE Trans. Inf. Theory, vol. 44, pp. 677–690, March 1998.

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Yonghong Zeng received the B.S. degree from the Peking University, Beijing, China, and the M.S. degree and the Ph.D. degree from the National University of Defense Technology, Changsha, China. He worked as an associate professor in the National University of Defense Technology before July 1999. From Aug. 1999 to Oct. 2004, he was a research fellow in the Nanyang Technological University, Singapore and the University of Hong Kong, successively. Since Nov. 2004, he has been working in the Institute for Infocomm Research, A*STAR, Singapore, as a scientist. His current research interests include signal processing and wireless communication, especially on cognitive radio and software defined radio, channel estimation, equalization, detection, and synchronization. He has co-authored six books, including Transforms and Fast Algorithms for Signal Analysis and Representation (Springer-Birkh¨auser, Boston, 2003), and more than 60 refereed journal papers. He is an IEEE senior member. He received the ministry-level Scientific and Technological Development Awards in China four times.

A. Rahim Leyman obtained his PhD from University of Strathclyde in 1994. He was an Assistant Professor at Nanyang Technological University from 1995 - 2001. Since 2001 he is a Senior Scientist at the Institute for Infocomm Research (I2R), A*STAR. Currently, he is an Adjunct Associate Professor at National University of Singapore, Department of Electrical and Computer Engineering. His current research interests are in applications of statistical signal processing to communication and biomedical signal processing, blind source separation, sensor array processing, higher order statistics and non-Gaussian signal processing. He publishes widely in IEEE Transactions on Signal Processing (TSP) and he is currently an Associate Editor of TSP. He is an active reviewer for IEEE Signal Processing Letters, IEEE Transactions on Signal Processing, IEEE Communications Letters, and IEEE Transactions on Communications, and ICASSP conferences. He was the co-chairman for the IEEE Signal Processing Society Workshop on Statistical Signal Processing, 2001.