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Asymptotically Efficient Reduced Complexity Frequency Offset and Channel Estimators for Uplink MIMO-OFDMA Systems Serdar Sezginer, Student Member, IEEE, Pascal Bianchi*, Member, IEEE

Abstract In this paper, we address the joint data-aided estimation of frequency offsets and channel coefficients in uplink MIMO-OFDMA systems. As the Maximum Likelihood (ML) estimator is impractical in this context, we introduce a family of suboptimal estimators with the aim of exhibiting an attractive tradeoff between performance and complexity. The estimators do not rely on a particular subcarrier assignment schemes (SAS) and are thus valid for a large number of OFDMA systems. As far as complexity is concerned, the computational cost of the proposed estimators is shown to be significantly reduced compared to existing estimators based on ML. As far as performance is concerned, the proposed suboptimal estimators are shown to be asymptotically efficient, i.e., the covariance matrix of the estimation error achieves the Cram´er-Rao bound when the total number of subcarriers increases. Simulation results sustain our claims. Index Terms Asymptotic efficiency, frequency synchronization, channel estimation, MIMO, OFDMA.

EDICS category: SSP-PERF; SPC-SYNC; SPC-MULT I. I NTRODUCTION Orthogonal Frequency Division Multiple Access (OFDMA) has recently become very popular in wireless communications and already been included in IEEE 802.16 specifications for fixed and mobile broadband wireless access. In an OFDMA system, each user modulates a certain group of subcarriers, following S. Sezginer and P. Bianchi are with the Telecommunications Department, Sup´elec, Plateau de Moulon, F-91192 Gif-sur-Yvette, France (e-mail: {serdar.sezginer, pascal.bianchi} @supelec.fr, tel: +33 (0)1 69 85 14 55, fax: +33 (0)1 69 85 14 69). * Corresponding Author. November 25, 2006

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a given subcarrier assignment scheme (SAS). The signal transmitted by a given user is impaired by a frequency selective channel and by a certain frequency offset. Along with its attractive features, the drawbacks associated with Orthogonal Frequency Division Multiplexing (OFDM) are directly inherited by OFDMA: in OFDM, it is well-known that the presence of frequency offsets introduces intercarrier interference (ICI) which is damaging in terms of symbol detection. In addition to ICI, inaccurate frequency offset estimation furthermore introduces multiple-access interference (MAI) in uplink OFDMA systems which degrades as well the overall system’s performance. Accurate estimation of the unknown frequency offsets and channel coefficients is therefore a crucial issue in OFDMA. Estimation of frequency offset and/or channel coefficients for (single-user) OFDM systems has been thoroughly investigated in the literature (see, e.g., [1], [2], and references therein). Moreover, a number of works has been devoted to the analysis of the performance of frequency offset estimators in a single user context [3]–[5]. In the single user case, it is well-known that accurate frequency offset and channel estimation can be achieved with very reasonable complexity. In the uplink OFDMA case, frequency offset and channel estimation is unfortunately a more difficult task. In particular, parameter estimation based on the direct maximization of the log-likelihood function is impractical. Therefore, suboptimal algorithms have been proposed. Most of these estimators are non data-aided and are designed for a specific SAS. In case of clustered SAS, i.e., when each user modulates a group of contiguous subcarriers, [6] proposes an estimator based on the presence of non-modulated subcarriers between groups of modulated subcarriers. Other examples include estimators based on a cyclic prefix method proposed in [7] and a Kurtosis maximization presented in [2]. In case of the equispaced (also called the interleaved) SAS, i.e., when adjacent subcarriers are modulated by different users, [8] proposes a subspace method based on the periodic structure of the signals transmitted by each user. Although non data-aided techniques for synchronization and channel estimation have recently taken considerable attention, the recently standardized multicarrier systems employ training blocks and thus encourage the design of data-aided estimators. In parallel, current trends for transmission in OFDMA systems bring to the fore the importance of more flexible estimators which do not rely on particular SAS. Recent works [9] and [10] investigate the data-aided estimation of frequency offsets in OFDMA uplink for general SAS. In [9], the estimation of carrier frequency and timing offsets of a new user entering an OFDMA system is addressed, assuming that the other users have already been synchronized. On the other hand, [10] proposes an ML-based alternating-projection algorithm. At each iteration of the algorithm, the procedure mainly consists in estimating the parameter of only one user while keeping the estimates of the other users at their most updated values. This method allows to replace the multiDRAFT

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dimensional exhaustive search initially required by the direct likelihood maximization with a succession of one-dimensional exhaustive searches. The complexity is therefore reduced compared to the rigorous ML estimator whereas the performance remains close to the performance of the ML estimator. However, the algorithm of [10] is still computationally demanding. For instance, the algorithm is difficult to implement in situations where the number of subcarriers is large. It is therefore of practical interest to propose suboptimal estimators which are likely to be implemented in such contexts, i.e., which have reasonable complexity and, on the otherhand, which have a performance close to the ML performance. In this paper, we consider an uplink OFDMA system with K users in a Multiple-Input MultipleOutput (MIMO) context. We address the issue of the estimation of the set of K frequency offsets ω = [ω1 , . . . , ωK ]T corresponding to each of these K users, and the set of K vectors of channel coefficients. The

signal model is described in Section II. We briefly recall the principle of the ML estimator in Section III. It is shown that the computational burden of the ML estimator is mainly due to the fact that the evaluation ML ˜ ˜ → JN of the log-likelihood function ω (ω) requires the inversion of a certain matrix for each trial value

˜ of the set of frequency offsets ω . The proposed family of estimators is introduced in Section IV. It is ω

based on the observation that the costly matrix inverse involved in the ML estimation can be approximated by an other matrix which is simpler to evaluate. This approximation is motivated by previous works [11] and is particularly accurate when the number of subcarriers is large. Although the proposed estimation method is suboptimal, we prove in Section V that the proposed estimator is asymptotically efficient. In other words, its performance coincides with the performance of the ML algorithm provided that the number of subcarriers is large enough. Simulation results are presented in Section VII. II. S IGNAL M ODEL We consider an uplink MIMO-OFDMA transmission. We assume that K users share N subcarriers. Each user has NT transmit antennas. One symbol sequence is sent by each transmit antenna t (t = 1, . . . , NT ) of each user k (k = 1, . . . , K ) using an OFDM modulator. The OFDM symbol transmitted by user k at a (t)

(t)

given antenna t in the frequency domain is represented by sequence sN,k (0), . . . , sN,k (N − 1). We omit the block index for the sake of simplicity. In the sequel, we assume that for each k and for each t, sequence (t)

(sN,k (j))j is known by the receiver (training sequence). It is worth noting that in usual OFDMA systems,

only a subset of the N available subcarriers is effectively modulated by a given user k , following a given (t)

SAS. For each j = 0, . . . , N − 1, we simply consider that sN,k (j) = 0 in the case where subcarrier j is not (t)

modulated by user k . However, we do not specify any SAS. In our model, training sequences (sN,k (j))j (t0 )

and (sN,k (j))j sent at different antennas t and t0 are possibly different. For a given user k and a given November 25, 2006

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(t)

(t)

antenna t, we denote by (aN,k (n))n the inverse discrete Fourier transform of sequence (sN,k (j))j : (t) aN,k (n)

N −1 nj 1 X (t) =√ sN,k (j)e2ıπ N N j=0

(1)

for each integer n. Cyclic prefix is added to the above time-domain version of the OFDM block and the resulting sequence is transmitted over a multipath channel. We denote by NR the number of receive antennas at the base station. For each r = 1, . . . , NR , the complex envelope of the signal received by antenna r is sampled at symbol rate. After cyclic prefix removal, the corresponding received samples can be written for each n = 0, . . . , N − 1 as (r)

yN (n) =

K X

eıωk n

NT L−1 X X

(t,r)

hk

(t)

(l)aN,k (n − l) + v (r) (n).

(2)

t=1 l=0

k=1

For each user k , parameter ωk is defined as ωk = 2πδfk T , where δfk denotes the frequency offset (t,r)

corresponding to user k and where T denotes the sampling period. Parameter hk

(l) represents lth tap of

the channel impulse response between tth transmit antenna of user k and rth receive antenna of the base station. Each channel is assumed to have no more than L nonzero taps, where integer L does not depend on k and does not exceed the length of the cyclic prefix. Sequence (v (r) (n))n denotes a white Gaussian noise of variance σ 2 . Note that equation (2) implicitly assumes that all users are quasi-synchronous in time: all delays of signals transmitted by all users are within the length of cyclic prefix. In equation (2), we also assume that the (angular) frequency offset ωk is constant with respect to (w.r.t.) antenna pairs (t, r). We mention that in certain MIMO systems, different frequency offsets may be associated with

each transmit-receive antenna pair. This case is usually considered in macro-diversity systems [12]. In the present paper, we consider the classical case (see, e.g., [2] and references therein) where ωk is constant w.r.t. antenna pairs (t, r). In the sequel, it is convenient to make use of a compact matrix representation (t,r)

of (2). To that end, we introduce the following notations. Define hk (r)

(1,r) T

and hk = [hk

(NT ,r) T T ] ,

, . . . , hk

(t,r)

= [hk

(t,r)

(0), . . . , hk

(L − 1)]T

where (·)T represents the transpose operator. Stacking all N samples

(r)

(r)

(r)

(r)

yk (n) received by antenna r into one column vector yN = [yN (0), . . . , yN (N − 1)]T , one obtains: (r) yN

=

K X

(r)

(r)

(3)

ΓN (ωk )AN,k hk + vN ,

k=1 (r)

where ΓN (ωk ) = diag(1, eıωk , . . . , eıωk (N −1) ) and vN = [v (r) (0), . . . , v (r) (N − 1)]T Here, the matrices (1)

(N )

(t)

AN,k are defined as AN,k = [AN,k , . . . , AN,kT ] where for each antenna t, AN,k is an N × L matrix

containing the time-domain training sequence sent at the tth transmit antenna of user k . More precisely, ³ ´ (t) (t) (t) AN,k = aN,k (i − j) 0≤i≤N −1 . Note that AN,k is a circulant matrix because of the cyclic prefix insertion 0≤j≤L−1

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at the transmitter side. We finally stack the samples received by all antennas into a single N NR × 1 vector (1) T

yN = [yN

(NR ) T T ]

, . . . , yN

given by yN =

K X

[INR ⊗ (ΓN (ωk )AN,k )] hk + vN ,

(4)

k=1

where INR denotes the NR × NR identity matrix, ⊗ stands for the Kronecker product, and vN = (1) T

[vN

(NR ) T T ]

, . . . , vN

is an additive noise vector with independent complex circular Gaussian random entries (1) T

of variance σ 2 . Here, vector hk = [hk

(NR ) T T ]

, . . . , hk

contains all channel coefficients of a given user

k . Denoting by ω = [ω1 , . . . , ωK ]T the vector containing all frequency offsets and by h = [h1 , . . . , hK ]T

the vector of channel coefficients, (4) can be written under the following compact form yN = QN (ω) h + vN ,

(5)

where QN (ω) represents the following NR N × KNR NT L matrix QN (ω) = [INR ⊗ (ΓN (ω1 )AN,1 ) , . . . , INR ⊗ (ΓN (ωK )AN,K )] .

(6)

In the sequel, we investigate the data-aided ML estimation of the frequency offsets and the channel parameters associated with all K users. III. E XACT ML

ESTIMATION OF FREQUENCY OFFSETS AND CHANNEL PARAMETERS

We describe the exact ML estimator of the unknown deterministic parameter vector [ω, h]T . The loglikelihood function for the unknown parameters ω, h has the form ° °2 ° ˜ =− 1 ° ˜ ˜ h) ˜ ΛN (ω, y − Q ( ω) h ° N ° + CN , N σ2

(7)

˜ = [h ˜ 1, . . . , h ˜ K ]T are trial values of ω and h, respectively. Here, CN ˜ = [˜ where ω ω1 , . . . , ω ˜ K ]T and h

designates a constant which does not depend on the parameter vector. Notation kxk2 stands for xH x where ˆ ML ˆ ML x is any column vector. The ML estimates ω N , hN of parameters ω, h are defined as the argument of

the maximum of the log-likelihood function. The latter maximization can be simplified as follows. For any ˜ is maximum when h ˜ coincides with ˜ , the log-likelihood ΛN (ω, ˜ h) fixed value of ω ¡ ¢−1 H ˆ N (ω) ˜ N. ˜ = QH ˜ N (ω) ˜ QN (ω)y h N (ω)Q

(8)

ˆ ω)) ˜ → ΛN (ω, ˜ h( ˜ . Substituting Therefore, the maximum of (7) coincides with the maximum of function ω ˜ in (7), we conclude that the log-likelihood function is maximum for ω ˆ ML the above expression of h N = ML ˜ arg maxω˜ JN (ω), where

¡ ¢−1 H ML H ˜ N. ˜ = yN ˜ QH ˜ N (ω) ˜ QN (ω)y JN (ω) QN (ω) N (ω)Q November 25, 2006

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In practice, the exact ML estimation of the whole set of parameters can thus be divided in two steps. Firstly, ML estimates of the frequency offsets are obtained by maximization of (9). Secondly, ML estimates of the ˆ ML = h ˆ N (ω ˆ ˜ is defined as in (8). ˆ ML channel coefficients are obtained by h N ), where function h(ω) N ˆ ML However, the determination of ω N requires not only a K dimensional search but also an KNT NR L × ˜ . This makes the rigorous ML estimation impractical. KNT NR L matrix inversion for each trial value ω

Hence, simpler estimators are needed having similar performance to the ML estimator with lower complexity. IV. P ROPOSED E STIMATOR In this section, we first provide some preliminary remarks which allow to motivate the introduction of ¡ ¢−1 ˜ ˜ the proposed estimate. Basically, our approach consists in replacing the inner factor QH ( ω)Q ( ω) N N of the righthand side of (9) with an appropriate approximation of the latter matrix. This approximation should be chosen such that the corresponding “approximated ML criterion” becomes easier to compute ML ˜ than the exact ML criterion JN (ω). On the other hand, the approximation should be fine enough so that

the estimation remains (almost) optimal in the ML sense. Thus, our aim is to define a simplified estimation algorithm whose performance becomes identical to the performance of the exact ML algorithm when the number N of subcarriers increases. Remark 1: In our model, as N tends to infinity we assume that i) the number K of users remains constant and ii) the number of antennas remains constant. We also assume that when N tends to infinity, the overall bandwidth is constant. In other words, the sampling rate the subcarrier spacing

1 NT

1 T

remains constant and as a result,

decreases to zero.

A. The Main Idea Here, we recall some results of [11] and explain how these results can be used to exhibit a simple ¡ ¢−1 ˜ N (ω) ˜ ˜ . Since our aim is to find an approximation of QH . Consider a given (fixed) value of ω N (ω)Q ¡ H ¢−1 ˜ N (ω) ˜ equivalent form for matrix QN (ω)Q as N tends to infinity, we now study the asymptotic behavior ˜ N (ω) ˜ . Using (6), one obtain of QH N (ω)Q



I ⊗ UN,1,1 . . .  NR 1 H ..  ˜ N (ω) ˜ = QN (ω)Q . N  INR ⊗ UN,K,1 . . .

 INR ⊗ UN,1,K  ..   , .  INR ⊗ UN,K,K

(10)

where for each k, l = 1, . . . , K , UN,k,l = DRAFT

1 H A ΓN (˜ ωl − ω ˜ k )AN,l . N N,k

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It has been shown in [11] that, under some mild assumptions which will be detailed in the sequel, •

if k 6= l, UN,k,l tends to the null matrix,

•

if k = l, UN,k,k tends to a certain deterministic matrix Rk which is related to the statistics of the (t)

training sequence (sN,k (j))j of user k . Before defining more accurately the above limit Rk , we briefly explain why the above lemma has an important consequence. Using this result, one immediately deduces that matrix

1 N

˜ N (ω) ˜ converges QH N (ω)Q

to a block diagonal matrix diag (INR ⊗ R1 , . . . , INR ⊗ RK ). Consequently, one could easily think about ¡ ¢−1 ˜ N (ω) ˜ replacing the complicated matrix QH in criterion (9) with the asymptotically equivalent N (ω)Q ¡ ¢ −1 matrix N1 diag INR ⊗ R−1 1 , . . . , INR ⊗ RK . Although the estimator which is proposed in this paper turns out to be slightly more involved, it is based on the same kind of idea. Before introducing this novel estimator in Section IV-C, it is necessary to recall in more details the result of [11] which provides the limit of matrices UN,k,l as N tends to infinity. B. A Former Result From now on, we make the following assumptions on the training sequences transmitted by all users. (t)

Assumption 1: For a given antenna t of a given user k , (sN,k (j))j is a sequence of independent random variables with zero mean. However, we do not assume that training symbols are identically distributed. In particular, the variance (t)

E[|sN,k (j)|2 ] of the j th training symbol depends on j . As we consider the OFDMA context, a certain

number of subcarriers may not be modulated by user k . If j is one of these subcarriers, we simply (t)

(t)

(t0 )

consider that E[|sN,k (j)|2 ] = 0. Furthermore, training sequences (sN,k (j))j and (sN,k (j))j transmitted by two different antennas t and t0 of a given user k are possibly correlated (due to the possible use of a (t)

(t0 )

beamformer). Therefore, the cross-correlation E[sN,k (j)sN,k (j)? ] may be nonzero, where x? denotes the conjugate of x. Assumption 2: Training sequences sent by two different users k 6= l are independent. (t)

Assumption 3: 16th-order moments1 of random variables (sN,k (j))j are uniformly bounded, i.e., ·¯ ¯16 ¸ ¯ (t) ¯ sup max E ¯sN,k (j)¯ <M N

j

for each t, where M is a constant independent of N . 1

the assumption is somewhat stronger than in [11] where it was only assumed that the 8th-order moments are bounded. In this

paper, stronger Assumption 3 is needed for the purpose of the asymptotic analysis of the proposed estimate. November 25, 2006

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We now introduce the following tool in order to be able to characterize the limit of matrices UN,k,l as N tends to infinity. For each k = 1, . . . , K , define the following matrix-valued measure [13] µN,k defined for any Borel set A of [0, 1] by N −1 i? 1 X h j µN,k (A) = E sN,k (j)sN,k (j)H IA ( ), N N

(12)

j=0

where IA stands for the indicator function of set A (i.e., IA (f ) = 1 if f ∈ A, IA (f ) = 0 otherwise) and (1)

(N )

where vector sN,k (j) = [sN,k (j), . . . , sN,kT (j)]T contains training symbols sent by all antennas of user k at a given subcarrier j . We assume as in [11] that Assumption 4: For each k , there is a matrix-valued measure µk such that µN,k converges weakly to µk as N tends to infinity. We are now able to study the asymptotic behavior of matrices UN,k,l as N tends to infinity. Lemma 1: Define vector e(f ) = [1, e2ıπf , . . . , e2ıπf (L−1) ]T for each f ∈ [0, 1]. For each k, l = 1, . . . , K lim UN,k,l = δ(k − l)Rk a.s.

N →∞

(13)

where notation a.s. stands for almost surely and where coefficient δ(k − l) is equal to 1 if k = l and to zero otherwise. Here, matrix Rk denotes the following LNT × LNT matrix Z 1 £ ¤ Rk = µk (df ) ⊗ e(f )e(f )H .

(14)

0

We refer to [11] for a proof of the above lemma. Using (10), the following result is the immediate consequence of Lemma 1. ˜, Lemma 2: For each ω

1 N

˜ N (ω) ˜ converges almost surely to a deterministic matrix R which is QH N (ω)Q

˜ and defined by independent of ω R = diag (INR ⊗ R1 , . . . , INR ⊗ RK ) .

(15)

Thanks to the above lemma, we are now able to provide a relevant approximation of the exact ML criterion.

C. Proposed estimate of frequency offsets Our aim is now to make use of Lemma 2 in order to propose a relevant and simple approximation ML ˜ of criterion JN (ω) defined by (9). We recall that the main obstacle in the computation of the exact ¡ ¢−1 ˜ N (ω) ˜ ML estimate is the calculation of the inverse matrix QH in (9). Using Lemma 2, it is N (ω)Q ¡1 H ¢−1 ˜ N (ω) ˜ converges a.s. to R−1 as N tends to infinity. Therefore, straightforward to show that N QN (ω)Q

a simple idea would be to simply replace the inner factor of the righthand side of (9) by its limit R−1 . DRAFT

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Following such an idea, the corresponding suboptimal estimate of ω would coincide with the maximum of the following cost function H ˜ −1 QH ˜ N. yN QN (ω)R N (ω)y

(16)

Note that matrix R−1 can be calculated beforehand. Thus, such a criterion does not involve any matrix inversion during the estimation step. Estimation of ω becomes practical. Unfortunately, as discussed in the sequel, the performance of such an estimate is far from achieving the performance of the exact ML estimate, even in the asymptotic regime (i.e., when N is large). In particular, a detailed asymptotic analysis would reveal that it is not asymptotically efficient. This unfortunate behavior is due to the fact that the substitution of R with matrix

1 H ˜ ˜ N (ω) N QN (ω)Q

is actually a too sharp approximation. In the sequel, we

propose a finer estimate. ˜ N (ω) ˜ verifies: Due to Lemma 2, matrix QH N (ω)Q 1 H ˜ N (ω) ˜ = R + EN (ω), ˜ Q (ω)Q N N

(17)

˜ converge almost surely to zero as N tends to infinity. Intuitively, since EN (ω ˜ ) is close to where EN (ω) ˜ −1 by its first zero for sufficiently large values of N , it is reasonable to approximate matrix (R + EN (ω)) ˜ −1 and to construct a “simplified” ML criterion by substituting order expansion [14] R−1 − R−1 EN (ω)R

this first order expansion with the inner factor of (9). The proposed cost function is thus defined as follows: ¡ ¢ H ˜ = yN ˜ R−1 − R−1 EN (ω)R ˜ −1 QH ˜ N JN (ω) QN (ω) N (ω)y

(18)

˜ , where EN (ω) ˜ is defined by (17). The proposed estimate of ω can be obtained by for each trial vector ω ˜ . For large values of N , it is reasonable to believe that this novel criterion is nearly maximization of JN (ω) ˜ , we now simplify the expression of JN (ω) ˜ . equal to the ML criterion. Using the definition of EN (ω) µ ¶ 1 H −1 ˜ = yN ˜ 2R−1 − R−1 QH ˜ ˜ ˜ N JN (ω) QN (ω) ( ω)Q ( ω)R QH N N N (ω)y N H ˜ + N yN = −N JN (ω) yN ,

where

(19)

°2 °µ ¶ ° ° 1 −1 H ° ˜ ˜ − IN NR yN ° ˜ =° QN (ω)R QN (ω) JN (ω) ° . N

(20)

Note that the maximization of (18) is of course equivalent to the minimization of (20). Finally, the proposed estimate of the set of frequency offsets is obtained as follows. Proposed estimate : ˆ N = arg min JN (ω). ˜ ω ˜ ω

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˜ is given by (20). where JN (ω)

Remark 2: A whole family of criteria can be derived from the same kind of idea. Generally speaking, ˜ −1 , one may approximate instead of using the first order expansion of the inverse matrix (R + EN (ω)) ¡ ¢m −1 P m R−1 E (ω) ˜ −1 = M this inversion with its M th order expansion (R + EN (ω)) R . Then, N ˜ m=0 (−1)

after some algebra, µ (M ) ˜ JN (ω)

M

= (−1)

H yN

1 ˜ −1 QH ˜ − IN N R QN (ω)R N (ω) N (M )

We refer to [15] for the detailed derivation of the above expression of JN (M )

¶M +1

(22)

yN .

˜ . Intuitively, if M > M 0 , the (ω)

(M 0 )

criterion JN is expected to lead to a better performance than JN , because the M th-order expansion ¡ ¢−1 ˜ N (ω) ˜ of N1 QH is a more accurate approximation than the M 0 th-order expansion. In the sequel, N (ω)Q (1)

we will mainly focus on the most simple one of these criteria, namely, the criterion JN = JN obtained with M = 1 and given by (20). If this simple criterion is shown to perform well, it is a fortiori reasonable to conjecture that the other (more involved) criteria also do. This claim will be confirmed by simulations.

D. Proposed estimate of channel coefficients Once vector ω has been obtained via the above minimization of JN , it is straightforward to compute an estimate of channel coefficients using a relation similar to (8). The most immediate way to estimate ˜ = ω ˆ N , where ω ˆ N is the proposed estimate (21) of is of course to use directly expression (8) with ω

frequency offsets. Such a procedure is of course likely to be implemented and provides very satisfying ¡ ¢−1 ˆ N )QN (ω ˆ N ) . In certain situations, results. However, it still requires the computation of matrix QH N (ω this can still be too computationally demanding, especially when the number K of users and/or the number L of channel coefficients is significant even for SISO case.

Using the same idea as in the previous Section, we propose a simpler channel estimate based on ¡ ¢−1 ˆ N )QN (ω ˆ N ) . As in the previous section, we replace the approximation of the above matrix QH N (ω ¡ H ¢−1 ˆ N )QN (ω ˆN) QN (ω by its first order expansion µ ¶−1 µ ¶ 1 H 1 H −1 −1 ˆ N )QN (ω ˆN) ˆ N )QN (ω ˆ N ) − R R−1 . Q (ω 'R −R Q (ω N N N N Based on (8), this leads to the following estimate of channel coefficients: µ ¶ 1 1 −1 H −1 −1 ˆ ˆ N )QN (ω ˆ N )R ˆ N )yN . hN = 2R − R QN (ω QH N (ω N N

(23)

Of course, matrix R−1 can be calculated beforehand, so that in practice, no matrix inversion is required (r)

(r)

ˆ for the computation of the above channel estimate. Denoting by h N,k the estimate of hk for each antenna DRAFT

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r = 1, . . . , NR of each user k = 1, . . . , K , equation (23) is equivalent to à ! K X 1 (r) (r) −1 H H ˆ h ωN,k ) 2 IN − TN (ˆ ωN,l ) yN , N,k = N Rk AN,k ΓN (ˆ

11

(24)

l=1

where TN (ˆ ωN,l ) =

H 1 H ωN,l )AN,l R−1 ωN,l ). l AN,l ΓN (ˆ N ΓN (ˆ

It is worth mentioning that, using an approach ¡ H ¢−1 ˆ N )QN (ω ˆN) similar to the previous paragraph, the inverse matrix QN (ω can as well be approximated with it M th order expansion with the aim of constructing an even more accurate estimate of the channel coefficient. V. A SYMPTOTIC S TUDY OF THE P ROPOSED E STIMATOR We now study the asymptotic behavior of the estimation error associated with the frequency offsets ˆ N,k − hk as the number N of subcarriers tends to infinity. ω ˆ N,k − ωk and with the channel coefficients h

The estimates of the frequency offsets and the channel coefficients are respectively given by (21) and (24). As the estimate of the channel is a simple function (24) of the estimate of the frequency offsets, it is natural ˆ N at first. Once the asymptotic behavior of ω ˆ N has been characterized, the asymptotic study to focus on ω

of channel estimates becomes possible. A. The main result ˆ N,k the estimate of the real The main result is given in the following theorem. Here, we denote by θ h T iT T ˆN = θ ˆ ,...,θ ˆ the estimate of the whole parameter vector θ k = [ωk , hTk,R , hTk,I ]T and by θ N,1 N,K ¤ £ T T including all users. In the above definition, K(1 + 2LNR NT ) × 1 parameter vector θ = θ 1 , . . . , θ TK hk,R and hk,I respectively represent the real and the imaginary parts of vector hk . We denote the whole set h iT of training symbols of all users by sN = sTN,1 (0) . . . sTN,1 (N − 1), . . . , sTN,K (0) . . . sTN,K (N − 1) . In the ³ ´ ˆ N − θ , where sequel, we study the asymptotic behavior of the normalized estimation error vector WN θ T , . . . , wT ) WN is the K(1+2LNR NT )×K(1+2LNR NT ) diagonal normalization matrix WN = diag(wN N £ 3/2 1/2 ¤ T T 1/2 where wN denotes the row vector wN = N , N , . . . , N of length (1 + 2LNR NT ).

Theorem 1: For almost any realization of the training symbols sN , the normalized estimation error ³ ´ ˆ N − θ converges in distribution to a Gaussian random vector of zero mean and covariance matrix WN θ Σ defined by Σ = diag (Σ1 , . . . , ΣK ) where for each k , Σk is the (1 + 2LNR NT ) × (1 + 2LNR NT )

matrix equal to

Σk =



12 γk  2 σ  6 hk,I  2  γk −6 hk,R γk

November 25, 2006

Re

£¡

INR £¡ Im INR

6 hT k,I γk ¢¤ hk,I hT ⊗ R−1 + 3 γk k,I k ¢¤ hk,R hT ⊗ R−1 − 3 γk k,I k



6 hT k,R γk  £¡ ¢¤  hk,I hT k,R . −Im INR ⊗ R−1 − 3 k γk  T £¡ ¢¤ h h k,R k,R Re INR ⊗ R−1 + 3 k γk

−

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³ ´ ˆ More precisely, the conditional distribution of WN θ N − θ w.r.t. sN converges to the Gaussian distri-

bution of zero mean and covariance matrix Σ with probability one. The proof of Theorem 1 is provided in Section V-C. We now make the following comments. Comments •

The proposed estimate is almost surely asymptotically normal.

•

Theorem 1 implies in particular that for each user k , £ ¤ 6σ 2 lim N 3 EN (ˆ ωN,k − ωk )2 = a.s. N →∞ γk ·° °2 ¸ ¡ ¢ 3σ 2 hH °ˆ ° k hk lim N EN °hN,k − hk ° = NR σ 2 tr R−1 + a.s. k N →∞ 2 γk

(26) (27)

where EN [·] designates the conditional expectation w.r.t. the training sequence sN . In particular, the MSE associated with the frequency offset of a given user k tends to zero at convergence speed

•

1 N3

while the MSE associated with the channel coefficients tends to zero at convergence speed N1 . ³ ´ ˆN − θ For large values of N , the covariance matrix of the normalized estimation error WN θ converges to a block diagonal matrix. This means that the estimation errors on the parameters of two different users k 6= l become independent as N tends to infinity. Furthermore, it is worth noting that the mean square errors corresponding to the parameters of a given user k does not depend on the number K of users. Therefore, as long as N is large enough, the performance of the proposed estimate is not affected by the presence of other users: it is identical to the performance that one would have observed if only one user was transmitting.

B. Asymptotic efficiency of the proposed estimate The proposed estimate of ω is suboptimal compared to the rigorous ML estimate, because it is based on an approximation of the log-likelihood function. However, it is explained in this subsection that the performance of the estimate is asymptotically optimal. More precisely, when the number N of subcarriers is large enough, there is no degradation of the performance when using the proposed suboptimal estimate instead of the ML estimate. This result is the immediate consequence of Theorem 1 along with the results of [11]. Denote by CRBN (sN ) the Cram´er-Rao Bound (CRB) associated with θ . We recall that the CRB can ˆ N of θ : be interpreted as a lower bound on the covariance matrix of any unbiased estimate θ h i ˆ N − θ)(θ ˆ N − θ)T < CRBN (sN ), EN (θ DRAFT

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where notation A < B means that the matrix A − B is a non-negative square matrix. In [11], the following result has been shown: lim WN CRBN (sN )WN = Σ a.s.

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N →∞

where Σ is precisely the matrix defined in Theorem 1. In other words, the limit Σ of the covariance matrix of the estimation error corresponding to the proposed estimate coincides with the asymptotic CRB. As a result, the proposed estimate is almost surely asymptotically efficient. This remark is of practical importance. Indeed, in spite of the fact that the proposed estimator is suboptimal compared to the ML estimator, it turns out that its performance becomes equal to the performance of the ML algorithm when N increases. There is no loss of optimality when using the proposed simple estimator instead of the rigorous ML algorithm, provided that N is large enough.

C. Proof of Theorem 1 In the present section, we provide the proof of Theorem 1. 1) Step 1. Consistency of the estimate of frequency offsets: The first step of the asymptotic analysis ˆ N converges to ω provided that the number of training symbols consists in showing that the estimate of ω

increases. This property is established by Theorem 2 below. Note that it is necessary to prove the consistency of the proposed estimate in order to be able to further study the asymptotic behavior of the estimation error. Due to the lack of space, the following statement is not proved in the present paper but the proof is available in [15]. Note that the proof is based on results previously introduced by [16]. Theorem 2: The following two properties hold: ˆ N = ω a.s. lim ω

N →∞

ˆ N − ω) = 0 a.s. lim N (ω

N →∞

Here, recall that notation a.s. stands for almost surely. 2) Step 2. Asymptotic study of the estimate of frequency offsets: We now study the behavior of the ˆ N − ω as the number N of subcarriers tends to infinity. According to Assumption 4, for estimation error ω

each Borel set A in [0, 1], µN,k (A) → µk (A) as N tends to infinity. In order to say that our asymptotic results are valid, we need furthermore to ensure the convergence of µN,k (A) to µk (A) “quickly enough” (that is, at least at speed

√1 ). N (t,t0 )

(t,t0 )

(t,t0 )

(t,t0 )

Assumption 5: For each k and (t, t0 ), denote by |µN,k − µk | the total variation of µN,k − µk √ (t,t0 ) (t,t0 ) We assume that N |µN,k − µk |([0, 1]) ≤ C for certain constant C . November 25, 2006

[17].

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Due to [17], this implies that for a given function F continuous on [0, 1], µZ ¶ Z √ (t,t0 ) (t,t0 ) N F (f )µN,k (df ) − F (f )µk (df ) ≤ C 0 , where C 0 is a certain constant (which depends on F ). Note that such a technical assumption encompasses usual subcarrier and power allocation schemes. We now study the asymptotic behavior of the estimation error. We define the gradient vector of cost i h ˜ T ˜ (ω) ˜ as ∇JN (ω) ˜ = ∂J∂Nω˜(1ω) ˜ as function JN (ω) , . . . , ∂J∂Nω˜ K and the Hessian matrix associated with JN (ω) h 2 i ˜ ˜ = ∂∂ ω˜JkN∂(ω˜ω) . We then make use of the Taylor-Lagrange expansion the K × K matrix HN (ω) l k,l=1,...,K

˜ at the true value ω of the parameter vector. There exists a real number ω ˜ N such that the of ∇JN (ω) √ ˆ N − ω) can be written as (normalized) estimation error N N (ω µ ¶−1 µ ¶ √ 1 1 √ ∇JN (ω) , ˆ N − ω) = − ˜N) N N (ω HN (ω N3 N N ˜ N can be written as ω ˜ N = λω ˆ N + (1 − λ)ω for a certain λ ∈ [0, 1]. Therefore, the asymptotic where ω

analysis of the estimation error reduces to the separate study of the gradient vector and the Hessian matrix associated with JN . We now provide the following lemmas. ˜ = ω verifies Lemma 3: The gradient vector of JN at point ω 1 √ ∇JN (ω) = −ζ N + oas (1), N N

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where components of random vector ζ N = [ζN,1 , . . . , ζN,K ]T are defined for each k = 1, . . . , K by à " ! # ¡ ¢ H 2 AH N,k IN − N DN ΓN (ωk ) H √ ζN,k = Im hk INR ⊗ vN , (31) N where DN = diag (0, 1, . . . , N − 1). ˜ N converges almost surely to a diagonal Lemma 4: The normalized Hessian matrix of JN at point ω

matrix as N tends to infinity: 1 1 ˜ N ) = diag (γ1 , . . . , γK ) a.s. HN (ω 3 N →∞ N 6 lim

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where for each k = 1, . . . , K , γk represents the following deterministic constant: γk = hH k (INR ⊗ Rk ) hk .

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The proofs of the above lemmas are provided in the Appendices II and III. We now give some insights on the consequences of these lemmas. Lemmas 3 and 4 indicate that the normalized estimation error h iT √ ˆ N − ω) has the same asymptotic behavior as vector γ61 ζN,1 , . . . , γ6K ζN,K . Indeed, it is straightN N (ω forward to show from Lemmas 3 and 4 that √ 6 N N (ˆ ωN,k − ωk ) = ζN,k + oas (1). γk DRAFT

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Note that for each k , ζN,k depends only on the training sequence of user k (via matrix AN,k ). Recall that the training sequences of two distinct users k 6= l are independent. Therefore, we already have the √ √ insight that the estimation errors N N (ˆ ωN,k − ωk ) and N N (ˆ ωN,l − ωl ) corresponding to users k and l respectively become independent as N tends to infinity.

3) Step 3. Asymptotic study of the estimate of channel coefficients: We now study the asymptotic behavior of the estimation error associated with the proposed estimate of channel coefficients given by (24). We (r)

first focus on the estimate of the channel coefficients hk corresponding to a given user k and associated (r)

with a given receive antenna r. For the purpose of the asymptotic analysis, it is convenient to write hk as follows: ˆ (r) h N,k

=

(r) hk

R−1 + √k N

"

# (r) K H X AH ωN,k )vN N,k ΓN (ˆ (r) (r) √ + ∆N,k,l hl + ²N,k . N l=1

(r)

Here, the quantities ∆N,k,l and ²N,k are given by ∆N,k,l = PN,k,l (ωl − ω ˆ N,k ) − PN,k,l (ˆ ωN,l − ω ˆ N,k ) and (r) ²N,k

K R−1 X PN,k,l (ˆ ωN,l − ω ˆ N,k )R−1 = − √k l N l=1

where, for each x, PN,k,l (x) =

√ N

³

Ã

K X

(r) !

PN,l,l0 (ωl0 −

l =1 0

1 H N AN,k ΓN (x)AN,l

(r) ω ˆ N,l )hl0

H AH ωN,l )vN N,l ΓN (ˆ √ + N

´ − δ(k − l)Rk . Using an approach similar to

(r)

Appendix II, it can be shown that ²N,k converges almost surely to zero as N tends to infinity. We now further simplify the terms ∆N,k,l . To that end, we provide the following lemma. Lemma 5: For each k, l = 1, . . . , K , 3iζN,k Rk + oas (1). γk The proof of the above lemma is omitted due to lack of place but a sketch of the proof is available in [15]. ∆N,k,l = −δ(k − l)

It can also be shown using a similar proof that vector totic behavior as

(r) √1 R−1 AH ΓH (ωk )v . N,k N N N k

(r) √1 R−1 AH ΓH (ˆ N,k N ωN,k )vN N k

has the same asymp-

We finally obtain the following expression of the normalized

estimation error. ´ √ ³ (r) (r) ˆ N h − h = N,k k

H H R−1 3iζN,k (r) k AN,k ΓN (ωk ) (r) √ hk + oas (1). vN − γk N

Due to the presence of random variable ζN,k in both expressions (34) and (35), we have the insight that the estimation error corresponding to channel coefficients hk is correlated to the estimation error corresponding to frequency offset ωk . The last step of the asymptotic analysis consists in putting all pieces (34) and (35) together, and to provide a compact asymptotic expression of the performance associated with the whole set of parameters [ω, h]. November 25, 2006

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4) Step 4. Asymptotic performance of the global estimate: Our aim is to study the asymptotic behavior of ˆ N − θ . Equations (34) and (35) suggest to rather study the normalized the vector-valued estimation error θ ³ ´ ˆ N − θ . Using (34) and (35), it is straightforward to obtain an asymptotically estimation error WN θ ³ ´ ˆ equivalent expression of the normalized estimation error WN θ N − θ . Replacing ζN,k with its defini-

tion (31), we obtain directly that ³ ´ ˆ N − θ = Im [ZN vN ] + oas (1), WN θ

(35)

£ ¤T where ZN is the following block matrix ZN = ZN,1 T , . . . , ZN,K T and where for each k ,   6 H γk hk ΦN,k     ZN,k =  ıΨN,k + γ3 hk,I hH Φ N,k  , k k   ΨN,k − γ3k hk,R hH Φ N,k k

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¡ ¢ H 2 AH N,k IN − N DN ΓN (ωk ) √ ΦN,k = INR ⊗ , N H H R−1 k AN,k ΓN (ωk ) √ . ΨN,k = INR ⊗ N ³ ´ ˆ N − θ has the same asymptotic behavior We deduce from (35) that the normalized estimation error WN θ

as random vector Im [ZN vN ]. It can be easily shown that, for a fixed value of training symbols (i.e., a fixed value of matrices AN,k ), the latter Im [ZN vN ] has a Gaussian distribution. In other words, the conditional distribution of Im [ZN vN ] w.r.t. the set of training symbols sN is a Gaussian distribution (whose covariance matrix depends on sN ). This claim can be easily shown by noticing that for a fixed value of sN , Im [ZN vN ] (r)

(r)

is a linear function of random vectors ΓH N (ωk )vN for all r = 1, . . . , NR . Since vector vN has i.i.d. zero mean complex circular Gaussian entries, then the above vectors have as well i.i.d. zero mean complex circular Gaussian entries. Thus, Im [ZN vN ] is Gaussian distributed. Consequently, for a fixed value of training symbols sN , Im [ZN vN ] converges in distribution to Gaussian distributed random vector whose covariance matrix Σ coincides with the limit of the covariance matrix of Im [ZN vN ]. In other words, ° i h ° Σ = lim E Im [ZN vN ] Im [ZN vN ]T ° sN . N →∞

The last technical point is to calculate the above limit to prove Theorem 1. This is done in Appendix IV. VI. P RACTICAL I MPLEMENTATION AND C OMPUTATIONAL C OMPLEXITY (M )

given

ˆ iN ω

at the

In order to implement the estimator of ω based on criterion JN (or on its generalized version JN in (22)), we propose to make use of a Newton search algorithm. The frequency offsets estimate DRAFT

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ˆ iN )∇JN (ω ˆ iN ), where H−1 ˆ iN ) and ∇JN (ω ˆ iN ) ˆ i+1 ˆ iN + H−1 ith iteration is updated as follows: ω N (ω N (ω N =ω

respectively represent the inverse Hessian matrix and the gradient vector associated with JN . Of course, a more simple gradient search algorithm is also likely to be used in order to avoid the computation of the inverse Hessian matrix. The above Newton (or gradient) search algorithm requires a relevant initial ˆ 0N in order to converge toward the desired value. For example, such an initial estimate can estimate ω ˜ w.r.t. ω ˜ on a grid. Simpler criteria such as the suboptimal be obtained by minimizing criterion JN (ω)

criterion (16) or the “single user” estimator of [18] can also be proposed for this task. We refer to [10] for a discussion on such issues. We now briefly investigate the complexity of the above Newton search algorithm and compare the latter with existing estimates. Gradient vector ∇JN can be computed using the formula given in Appendix II. We first remark that the matrix R−1 which is involved in this expression can be computed beforehand, so that it generates no additional computational burden. It can be shown that at each iteration of the Newton algorithm, the evaluation of the Gradient and the Hessian matrix both require less than O(N K 2 (LNT )2 ) operations. The overall complexity of the proposed algorithm is thus bounded by O(ni (N K 2 (LNT )2 + K 3 )), where ni denotes the number of iterations of the Newton algorithm and where the presence of the term K 3 is due to the inversion of the Hessian matrix. Note that in practice, only a few iterations are needed (typically ni = 2, 3, 4). In the case where M > 1, the complexity analysis (M )

of the estimator base on the generalized criterion JN

is more difficult. However, simulations indicate

that the computational burden remains at the same order at least for M = 3 and 5. It is interesting to compare the complexity of the proposed algorithm with existing estimators. As the rigorous ML estimator is known to be impractical anyway, it is more interesting to draw a comparison with respect to more practical estimators such as the ML-based algorithm proposed by [10]. In this paper, authors use the so-called alternating projection frequency estimation (AFPE) procedure in order to estimate the argument of the maximum of a simplified version of the log-likelihood criterion. The performance of such an algorithm is shown to be very attractive. Unfortunately, the computational burden of the APFE method is still very significant so that this estimator may be impractical in certain situations, especially for large values of N . For each iteration and for each user k = 1, . . . , K , the APFE algorithm requires to evaluate a certain cost function (which depends on k ) for each ω ˜ k on a grid of size ng . It can be shown that the complexity of the APFE method is about O(ni (ng K + K 4 )(N (LNT )2 + (LNT )3 )). In practice, the complexity of the above algorithm depends on the number ng of points in the grid which can be considerable, especially when N is large. Note finally that the proposed channel estimate can as well be implemented with low computational cost. Unlike the estimators [9][10] based on expression (8) and which require the inversion of a KNR NT L × November 25, 2006

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KNR NT L matrix, the proposed channel estimate (23) avoids such an inversion.

VII. S IMULATION R ESULTS We consider an uplink OFDMA system with QPSK signaling. We investigate either the SISO case or the MIMO case with NR = NT = 2. For each user and for each transmit-receive antenna pair, we consider a multipath fading channel with 8 independent paths. Complex gains associated with each path are assumed to be circular complex Gaussian random variables with zero mean and unit variance. For each user k , the £ ¤ 0.4 value of δfk is randomly chosen in the interval −0.4 N T , N T . In the sequel, without loss of generality, we focus on the results corresponding to the first user k = 1 and suppose that average transmitted powers Pk are equal for all users. All results are averaged over 1000 realizations of the training sequences, the

frequency offsets and the channel parameters. The SAS which is considered in the simulations is the following. We assume that the total bandwidth is divided into groups of four consecutive subcarriers. Each group is modulated by only one user. Groups of subcarriers are assigned to the different users following an equispaced assignment scheme. For instance, user 1 modulates the first group, the (K + 1)th group, the (2K + 1)th group, etc. All subcarriers are modulated with equal power and training sequences sent at different antennas are uncorrelated. The proposed estimates of the frequency offsets are obtained by minimization of criterion J (M ) defined by (22) for M = 1 and M = 3. The estimates are computed using the Newton algorithm depicted in Section VI and are compared with the AFPE proposed by [10]. Both algorithms are used with ni = 2 iterations as in [10]. Estimates of channel parameters are obtained using (23) when M = 1 and using the generalized version of (23) when M = 3 as discussed in Section IV-D. Finally, the performance is compared with the exact and asymptotic CRB derived in [11]. We first study the behavior of the mean square error (MSE) as the number N of subcarriers increases. Figure 1(a) represents the (empirical evaluation of the) MSE E[(ˆ ωN,1 − ω1 )2 ] as a function of N , when P1 σ2

= 10dB and when NR = NT = 1. Here, since the main focus is on the accuracy of the fine search of

the parameters, Newton algorithms are initialized with the true value of the parameters. We observe that the proposed estimator fits to the CRB, even for moderate values of N . As expected, the estimation of the parameters based on criterion J (M ) with M = 3 provides more accuracy compared to M = 1. This is due to the fact that criterion J (3) is finer approximation of the log-likelihood criterion. Similar observations can ˆ N,1 − h1 k2 ]: again, the be drawn from Figure 1(b) which represents the MSE of the channel estimate E[kh

estimates fit to the CRBs. This sustains the claim that the proposed estimators are asymptotically efficient. In the 2 × 2 MIMO case, Figures 2(a) and 2(b) indicate that the proposed estimates are as well close to the CRB. However, when the number of antennas or the number of users increases, we observe that the DRAFT

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19

asymptotic regime is reached for a larger number of subcarriers. Again, the increase in the value of the expansion order M results in a faster convergence to the CRB. Figures 3, 4 represent the performance as a function of the signal to noise ratio (SNR)

P1 σ2

when the

number N of subcarriers is constant. Figure 3 is obtained for N = 1024 for both SISO and MIMO case and for either 2 or 4 users. It again sustains the theoretical claim that the MSE of the estimate are close to the CRB, especially when M ≥ 3. However, when M = 1, we remark that the performance of the P1 σ2 .

This can be intuitively explained as √ follows. The asymptotic study shows that the (normalized) estimation error N N (ˆ ωN,1 − ω1 ) coincides estimate does not fit to the CRB for high values of the SNR

with

6 γ1 ζN,1

+ oas (1). For moderate values of the SNR, the term oas (1) is negligible compared to ζN,1 .

However, ζN,1 is a linear function of the noise vector. Thus, at high SNR, components of the noise vector are close to zero, so that the modulus of ζN,1 may be very slight: for large but finite N , the modulus of ζN,1 may be about as small as the second term oas (1), which is thus no longer negligible. Therefore, at

high SNR, this intuitively indicates that the asymptotic regime is reached for larger values of N compared to the low SNR case. Finally, in Figure 4, we compare the proposed estimates with the APFE method. Here, both Newton algorithms and the APFE method are initialized using the result of a preliminary coarse search. The simple suboptimal “single user” estimator of [18] is used for this coarse search as suggested in Section VI. The results of the fine search are presented after the elimination of the possible outliers produced by the coarse search. For instance, when N = 256 and

P1 σ2

= 15dB, the percentage of outliers which have been

eliminated in order to plot the results is equal to %0.6 for APFE and %0.1 for the proposed method. Note that the number of outliers increases at low SNR. We refer to [3] for a detailed discussion on this issue. Note that the APFE method is difficult to implement for large values of N as discussed in Section VI. Therefore, we consider the case N = 256. We also focus on the SISO case in order to implement the APFE method described in [10]. Figure 4(a) represents the MSE corresponding to frequency offset estimates while Figure 4(b) represents the MSE corresponding to channel estimates. As far as the APFE ˜ in the algorithm is concerned, the channel estimation is performed using expression (8) (only replacing ω

latter with the APFE estimate of ω ). Although the value of N is moderate, Figures 4(a) and 4(b) show that when M = 3, the proposed estimator performs similarly to the APFE method: both estimators fit to the CRB curve. As discussed previously, the estimator based on the first order expansion M = 1 does not fit to the CRB. The asymptotic regime is not reached for N = 256 in this range of SNRs. For the sake of completeness, Figures 4(a) and 4(b) also include the MSE of the estimates obtained from the single user estimator [18]. As expected, such estimates are far from achieving the CRB since they are not specially November 25, 2006

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designed for the multi-user context. VIII. C ONCLUSIONS A family of estimators of frequency offsets and MIMO channel coefficients has been proposed in the case where K users are transmitting toward a single receiver using an orthogonal frequency division multiple access scheme. The estimators do not rely on a particular SAS and can be used in a general OFDMA context. The implementation of such estimators can be achieved with very reasonable complexity compared to existing estimators based on the maximization of the likelihood function. In addition to their reduced complexity, the estimators are shown to be asymptotically efficient. Indeed, when the number N of subcarriers increases, the covariance matrix of the estimation error becomes identical to the Cram´er-Rao bound. Simulations sustain our theoretical claims and illustrate the attractive performance of the estimators even for moderate values of the number of subcarriers. A PPENDIX I S KETCH OF THE PROOF OF THE CONSISTENCY OF THE ESTIMATE The proof is based on the following preliminary lemma. Lemma 6: Consider a given antenna pair t, t0 . For each p, q = 0, . . . , L − 1, define (t)

(t0 )

(t)

(t0 )

? ? ep,q N,k,l (n) = aN,k (n − p) aN,l (n − q) − E[aN,k (n − p) aN,l (n − q)].

Define for each real number δ and for each u = 0, 1, 2, p,q,u SN,k,l (δ) =

N −1 u + 1 X u p,q n eN,k,l (n)eınδ . N u+1

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n=0

¯ ¯ ¯ p,q,u ¯ Then, supδ ¯SN,k,l (δ)¯ converges almost surely to zero as N tends to infinity.

The proof of the above lemma is not provided here due to the lack of space but more details are available in [15]. In the sequel, we focus on the case NT = NR = 1 but the proof can be extended without any difficulty to the MIMO case. For each N , define the estimation error associated with the k th user as δN,k = ωk − ω ˆ N,k modulo 2π . Our aim is to prove that for each k = 1, . . . , K , N δN,k converges to zero

with probability one. To that end, we first prove the following lemma. Lemma 7: The following result holds: K

X ¡ ¢ 1 ˆ N ) − JN (ω)) + (JN (ω hk H Rk hk 1 − |qN (δN,k )|2 = 0 a.s. N →∞ N lim

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k=1

where qN (δ) = DRAFT

1 N

PN −1 n=0

eın δ . November 25, 2006

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The proof of the above lemma is based on preliminary Lemma 6. Again, the proof is omitted due to the lack of space, but is available in [15]. Since |qN (δ)| ≤ 1 for every δ , it is clear that both terms of the righthand side of (38) are non-negative numbers. Therefore, both terms converge to zero. In other words, ¡ ¢ for each k = 1, . . . , K , hk H Rk hk 1 − |qN (δN,k )|2 tends almost surely to zero. As a consequence, for each k , we obtain that limN →∞ qN (δN,k ) = 1 a.s. Following [16], we now make use of the following lemma from [19]: Lemma 8: Consider a real-valued sequence cN ∈ (−π, π] such that limN →∞ cN = c. •

if c 6= 0, then qN (cN ) tends to 0,

•

if c = 0 and N |cN − c| → ∞, then qN (cN ) tends to 0,

•

if c = 0 and N |cN − c| → β ∈ R, then qN (cN ) tends to eı 2

β

sin(β/2) β/2 .

Now consider a sequence δN,k such that qN (δN,k ) tends to 1 as N tends to infinity. As δN,k belongs to the bounded interval (−π, π], there exist a convergent subsequence δϕ(N ),k extracted from δN,k . The above lemma along with condition qϕ(N ) (δϕ(N ),k ) → 1 implies that δϕ(N ),k converges to zero. Thus, every convergent subsequence extracted from δN,k converges to zero. Therefore, δN,k converges to zero. We now show that N δN,k is a bounded sequence. If N δN,k is not bounded, one can extract a subsequence ϕ(N )δϕ(N ),k such that ϕ(N )δϕ(N ),k tends to infinity as N → ∞. Using again Lemma 8, we conclude that qϕ(N ) (δϕ(N ),k ) tends to 0. This is in contradiction with qN (δN,k ) → 1. Thus, sequence N δN,k is bounded.

Again, consider a sequence ϕ(N )δϕ(N ),k extracted from N δN,k such that ϕ(N )δϕ(N ),k tends to a certain β

β ∈ R, then qϕ(N ) (δϕ(N ),k ) tends to eı 2

sin(β/2) β/2 .

Using again condition qϕ(N ) (δϕ(N ),k ) → 1, we conclude

that β = 0. Since every convergent subsequence extracted from sequence N δN,k converges to zero, it follows that N δN,k converges to zero. We conclude that limN →∞ N δN,k = 0 with probability one.

A PPENDIX II P ROOF OF L EMMA 3 Consider a given user k ∈ {1, . . . , K}. We derive the k th component of the gradient vector

∂JN ∂ω ˜k .

Due

˜ = zN (ω) ˜ H zN (ω) ˜ where to (20), cost function JN can be written as JN (ω) µ ¶ 1 −1 H ˜ = ˜ ˜ − IN NR yN . zN (ω) QN (ω)R QN (ω) (39) N h i ˜ ˜ ˜ H ∂z∂Nω˜(kω) ˜ can be written as follows: Therefore, ∂J∂Nω˜(kω) = 2Re zN (ω) . Now, the derivative of vector zN (ω) ˜ ∂zN (ω) = −ı INR ⊗ (TN,k (˜ ωk )DN − DN TN,k (˜ ωk )) yN , ∂ω ˜k November 25, 2006

(40) DRAFT

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H 1 ωk )AN,k Rk −1 AH ωk ). N,k ΓN (˜ N ΓN (˜ ˜ ∂JN (ω) 1 √ = AN,k + BN,k + CN,k where ˜k N N ∂ω

where DN = diag(0, 1, . . . , N −1) and where TN,k (˜ ωk ) = the derivative of JN can be written as the sum AN,k =

NR X K X

2 N 3/2

BN,k = − CN,k = −

· Im

(r) H (r) yN TN,l (˜ ωl )TN,k (˜ ωk )DN yN

Using (40),

¸

(41)

r=1 l=1 NR X K X

2 N 3/2

N 3/2

(42)

r=1 l=1 NR X

4

· ¸ (r) H (r) Im yN TN,l (˜ ωl )DN TN,k (˜ ωk )yN

· ¸ (r) H (r) Im yN TN,k (˜ ωk )DN yN .

(43)

r=1

˜ = ω . The first step of the We now study the asymptotic behaviors of the above three variables at point ω

proof consists in re-expressing the inner terms of the above equations as a function of more convenient quantities, whose asymptotic behavior can be easily characterized. Due to the definition of matrices TN,k , it turns out that AN,k , BN,k and CN,k mainly depend on the two quantities (r)

R−1 k

H AH N,k ΓN (ωk )yN

N

(r)

2R−1 k

and

H AH N,k ΓN (ωk )DN yN

N2

(44)

.

It is thus convenient to express the above quantities in a more compact way. For each u = 0, 1, Ã ! (r) AH ΓH (ωk )DuN yN AH ΓH (ωk )DuN X (r) (r) −1 N,k N −1 N,k N (u + 1)Rk ΓN (ωl )AN,l hl + vN , = (u + 1)Rk N u+1 N u+1 l

(r),u xN,k

(r) = hk + √ , N

(45)

where (r),u

xN,k

= R−1 k

PuN,k,l

√ = N

K X

(r)

PuN,k,l hl

(r),u

l=1

(r),u

wN,k

=

µ

(46)

+ wN,k

u+1 H A ΓN (ωl − ωk )DuN AN,l − δ(k − l)Rk N u+1 N,k u + 1 −1 H H (r) √ Rk AN,k ΓN (ωk )DuN vN . u N N

¶

(47) (48)

Decomposition (45) is particularly useful because it stresses the fact that the key quantities (44) simply (r)

coincide with hk

up to a term which (as we shall prove in the sequel) tends to zero as N tends to (r)

infinity. Using these notations, we may rewrite AN,k , BN,k and CN,k as simple functions of hk (r),u xN,k ,

DRAFT

and

u = 0, 1. As an example, it is straightforward to show that  H   (r),1 (r),0 N R √ X xN,k  xN,k  (r) (r) CN,k = −2 N Im hk + √  Rk hk + √  . N N r=1 November 25, 2006

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Using (47) and summing the three terms AN,k + BN,k + CN,k , we obtain after some algebra  H   (r),0 (r),0 N 1 R √ XX xN,l x PN,l,k ˜ 1 ∂JN (ω)  (r) h(r) + √N,k  √ Im hl + √  √ =− N  k ˜k N N ∂ω N N N r=1 l6=k +

√ N

 H Ã  ! (r),1 (r),0 0 x xN,l PN,l,k  (r) h(r) + √N,k  Im hl + √  δ(k − l)Rk + √ . k N N N l=1

NR X K X r=1

(49)

The rest of the proof simply consists in expanding the above expression and in only keeping the dominant √ terms, i.e., the terms which do not contain any factor 1/ N or 1/N . In order to motivate this, we make use of the following lemma. Lemma 9: For each u, v = 0, 1, for each k, l, l0 = 1, . . . , K , the three following random matrices (r),v

PuN,l,k PuN,l,k R−1 PvN,k,l0 PuN,l,k wN,k √ , √k √ , N N N

converge to zero as N tends to infinity with probability one. Before providing the proof of the above lemma, we first show how Lemma 9 allows to complete the proof (r),1 √ of the final result. Using Lemma 9, it becomes clear that terms such as (typically) P0N,l,k xN,k / N (r),0 H (r),1 √ or xN,k xN,k / N tend to zero as N tends to infinity. Consequently, if one expand all factors of the righthand side of ¸equation (49), only a few of them do not converge to zero. Remarking also that · √ (r) H (r) N Im hk Rk hk = 0, we obtain: ˜ 1 ∂JN (ω) √ ˜k N N ∂ω

=

NR X

 (r) H

Im hk

 ³ ´ X ¢ ¡ H (r) (r) (r),0 (r),1 hk P0N,k,l − P1N,k,l hl  + oas (1) Rk xN,k − xN,k −

r=1

=

NR X

l6=k

· Im

(r) H hk Rk

³ ´¸ (r),0 (r),1 wN,k − wN,k + oas (1),

r=1 (r) H

where we used the fact that for k = l, the imaginary part of hk (r),0

³ ´ (r) P0N,k,k − P1N,k,k hk is equal to zero.

(r),1

Replacing wN,k and wN,k with the corresponding expressions, we obtain the final result. We now prove Lemma 9.

√ Proof: The almost sure convergence to zero of the first random matrix PuN,l,k / N has been proved

in [11]. We refer to [11] for a proof of this result. Here, we only address the most difficult part, which √ v is the almost sure convergence to zero of the second random matrix PuN,l,k R−1 k PN,k,l0 / N . We focus, without restriction, on the case u = v = 0 in order to keep the notations simple. The general case can be treated similarly, without more difficulties. Moreover, we focus on the single transmit antenna case NT = 1 in order to avoid the heavy indices t, t0 representing the antenna numbers. The generalization to November 25, 2006

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NT > 1 is straightforward. Firstly, we write matrix P0N,l,k as the sum of a random and a deterministic

matrix P0N,l,k = PaN,l,k + PbN,l,k , where

³ ´ H Γ (ω − ω )A AH Γ (ω − ω )A − E A N N l k N,l l k N,l N,k N,k √ PaN,l,k = N ´  ³  E AH A √ N,k N,k PbN,l,k = δ(k − l) N  − Rk  N ³ ´ Here, we used the fact that E AH Γ (ω − ω )A l k N,l = 0 when k 6= l due to the independence of N,k N

matrices AN,k and AN,l . In the above definitions, superscript “0” is omitted for the sake of simplicity. Matrix PbN,l,k is equal to zero for k 6= l. For k = l, the component (p, q) of matrix PbN,k,k can be written as PbN,k,k (p, q)

Ã

! N −1 1 X ? E[aN,k (n − p) aN,k (n − q)] − Rk (p, q) N n=0   N −1 X √ j 1 N E[|sN,k (j)|2 ]e2ıπ(p−q) N − Rk (p, q) = N j=0 ¶ µZ 1 Z 1 √ = e2ıπ(p−q)f µN,k (df ) − e2ıπ(p−q)f µk (df ) . N √ N =

0

0

According to Assumption 5, the above term is bounded with a constant which does not depend on N . Using √ P0N,k,l0 √ the almost sure convergence to zero of the first random matrix PuN,l,k / N , PbN,l,k R−1 becomes the k N product between a deterministic matrix which bounded componentwise and a matrix which tends almost surely to zero. The product thus converges almost surely to zero. With such observations, it turns out that √ v the proof of the almost sure convergence to zero of PuN,l,k R−1 N simply reduces to the proof P / 0 N,k,l k √ −1 that PaN,l,k Rk PaN,k,l0 / N → 0 a.s. To that end, it is sufficient to prove that for each (p, q), (p0 , q 0 ), ξN =

√1 Pa (p, q)PaN,k,l0 (p0 , q 0 ) N N,l,k

converges almost surely to zero. After some algebra, we obtain

N −1 N −1 1 X X 0 PaN,l,k (p, q) = √ σN,l,k (i, j)ψN (j − i + N δflk ) N i=0 j=0 PN −1 ınx/N −ωk 0 (x) = 1 where δflk = ωl2π , ψN and where σN,l,k (i, j) = sN,k (i)? sN,l (j)−E [sN,k (i)? sN,l (j)] . n=0 e N

We now derive the fourth-order moment of ξN . By simply expanding the product |ξN |4 , taking the expectation and finally using the triangular inequality, we obtain ¯ " #¯ 4 8 ¯ Y Y X X ¯¯ 1 ¯ σN,l,k (in , jn )?n σN,k,l0 (in , jn )?n ¯ E[|ξN |4 ] ≤ 6 ¯E ¯ ¯ N n=1 n=5 i1 ,...,i8 j1 ,...,j8 ¯ ¯ 4 8 ¯ ¯Y Y ¯ ¯ 0 0 ψN (jn − in + N δflk ) ψN (jn − in + N δfkl0 )¯ , ׯ ¯ ¯ n=1

DRAFT

(50)

n=5

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25

where x?n = x if n is even and x?n = x? if n is odd. Due to the fact that, sN,k (i) is a sequence of independent random variables, it can be seen that the expectation inside the sum in the above equation is zero for most 16-uplets i1 , i2 , . . . , j8 . Only a limited number of terms are non zero. For instance, one of these terms is the one obtained for i1 = i2 , j1 = j2 , i3 = i4 , j3 = j4 , i5 = i6 , j5 = j6 , i7 = i8 , j7 = j8 . We now study the latter term in more details. This term can be written as 1 N6

X

αN (i1 , . . . , j7 )

1 3 Y ¯ 0 ¯ 0 ¯ Y ¯ ¯ψN (j2n+1 − i2n+1 + N δflk )¯2 ¯ψN (j2n+1 − i2n+1 + N δfkl0 )¯2 n=0

i1 ,i3 ,i5 ,i7 ,j1 ,j3 ,j5 ,j7

n=2

(51) ¤ where αN (i1 , i3 , . . . , j7 ) = E |σN,l,k (i1 , j1 )|2 |σN,l,k (i3 , j3 )|2 |σN,k,l0 (i5 , j5 )|2 |σN,k,l0 (i7 , j7 )|2 is bounded ¡ £ ¤ £ ¤ £ ¤ £ ¤¢ 1 by E |σN,l,k (i1 , j1 )|8 E |σN,l,k (i3 , j3 )|8 | E |σN,k,l0 (i5 , j5 )|8 | E |σN,k,l0 (i7 , j7 )|8 4 by the Cauchy£

Schwarz inequality. By expanding the fourth order moment of σN,l,k and by using Cauchy-Schwarz and Jensen inequalities, it is straightforward to show that αN (i1 , . . . , j7 ) is bounded with a constant, say C 0 , which does not depend on N . This implies that the term (51) is less than C0 N6

X

1 3 Y ¯ 0 ¯ Y ¯ 0 ¯ ¯ψN (j2n+1 − i2n+1 + N δflk )¯2 ¯ψN (j2n+1 − i2n+1 + N δfkl0 )¯2

i1 ,i3 ,...,j7 n=0

n=2

C0 = 2 N

Ã

N −1 X

¯ 0 ¯ ¯ψN (m + N δflk )¯2

!2 Ã

m=−N +1

N −1 X

¯ 0 ¯ ¯ψN (m + N δfkl0 )¯2

!2

m=−N +1

¯ 0 ¯2 P −1 ¯ ¯ Finally, we use the fact that for each k, l, the sum N m=−N +1 ψN (m + N δflk ) is bounded with a constant

which does not depend on N [11]. Finally, (51) is less than

C 00 N2 ,

where C 00 is a certain constant. Using the

same kind of study for all other non zero terms of the sum (50), we conclude that E[|ξN |4 ] ≤

C N2 ,

where

C is a constant which does not depend on N . Using Chebyshev inequality followed by the Borel-Cantelli

lemma, we finally obtain that ξN converges almost surely to zero. This completes the proof. A PPENDIX III P ROOF OF L EMMA 4 We first consider the case k = l and we derive shown that

1 ∂ 2 JN ˜ N 3 ∂ω ˜ k2 (ω N )

∂ 2 JN ∂ω ˜ k2

˜ =ω ˜ N . After some algebra, it can be at point ω

can be written as the sum of a certain number of terms. For instance, the first

of these terms coincides with ¸ · NR X K 2 X (r) H 2 (r) ˜ AN,k = − 3 Re yN TN,l (˜ ωN,l )TN,k (˜ ωN,k )DN yN . N r=1 l=1

Other terms can be written under a similar form. Their expressions are not provided here due to the lack of space, but a more detailed proof is available in [15]. Clearly, the asymptotic study of November 25, 2006

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reduces to the study of terms such as A˜N,k . Here, we only focus on the study of A˜N,k for the sake of conciseness. Replacing the above matrices TN,k (˜ ωN,k ) with definition of matrices TN,k , we obtain A˜N,k

  NR X K (r) H H Γ (˜ H Γ (˜ 2 y(r) A 3A ω − ω ˜ )A ω )D Γ (˜ ω )A y N N 2X N,k N,l N,k N,k N N,l N,l −1 N,l N N  N,k Re  N =− Rl R−1 k 3 N N N3 r=1 l=1

(52) The limit of A˜N,k can be directly obtained by making use of the following lemma. Lemma 10: For each u = 0, 1, for each k, l, lim

(u + 1)AH ωN,l − ω ˜ N,k )DuN AN,l N,k ΓN (˜ N u+1

N →∞

= δ(k − l)Rk a.s.

(53)

(r)

lim

(u + 1)AH ωN,k )H DuN yN N,k ΓN (˜ N u+1

N →∞

(r)

= Rk hk

a.s.

(54)

P R (r) (r) By straightforward application of the above lemma, A˜N,k converges to − 32 N Rk hk = − 32 γk r=1 hk H

as N tends to infinity. Using similar arguments for all components such as A˜N,k , we obtain that converges to

γk 6

∂ 2 JN ∂ω ˜ k2

as N tends to infinity. The rest of the proof consists in deriving the non diagonal terms

˜ N and in proving that for each k 6= l, of the Hessian matrix at point ω

1 ∂ 2 JN (ω) ˜N) N 3 ∂ω ˜k ∂ω ˜ l (ω

converges a.s.

to zero as N tends to infinity. The proof follows exactly the same approach. This completes the proof of Lemma 4. We now provide a sketch of the proof of Lemma 10. Proof: Again, in order to simplify the proof and to avoid the overabundance of indices, we provide the proof in the single antenna case NT = 1 without loss of generality. We focus on the proof of (53). A more detailed proof is available in [15]. The element (p, q) of the matrix in the righthand side of (53) coincides with N −1 u+1 X aN,k (n − p)? aN,l (n − q)nu eın(˜ωN,l −˜ωN,k ) N u+1 n=0

N −1 u+1 X p,q,u E [aN,k (n − p)? aN,l (n − q)] eın(˜ωN,l −˜ωN,k ) + SN,k,l = u+1 (˜ ωN,l − ω ˜ N,k ) N n=0

−1 u (0) N X £ ¤ j δ(k − l)qN p,q,u E |sN,k (j)|2 e2ıπ N (q−p) + SN,k,l = (˜ ωN,l − ω ˜ N,k ), N j=0

p,q,u u (x) = where SN,k,l is defined by (37) and where qN

u+1 N u+1

PN −1 n=0

p,q,u nu eınx . Due to Lemma 6, SN,k,l (˜ ωN,l −

ω ˜ N,k ) converges a.s. to zero as N tends to infinity. On the otherhand, the first term of the above equation

converges to δ(k − l)Rk (p, q). This completes the proof. DRAFT

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A PPENDIX IV D ERIVATION OF THE LIMIT COVARIANCE MATRIX OF Im [ZN vN ] We first remark that h i EN Im [ZN vN ] Im [ZN vN ]T = H Matrix ZN ZH N is composed of K × K blocks, i.e., ZN ZN

£ ¤ σ2 Re ZN ZH N . 2 h i = ZN,k ZH . We study one of these N,l k,l=1,...,K

blocks. Due to the definition (36) of ZN,k , the product ZN,k ZH N,l can be written as a simple function of H H H the following four matrices: ΦN,k ΦH N,l , ΨN,k ΨN,l , ΦN,k ΨN,l and ΨN,k ΦN,l . It is thus sufficient to study

the limit of each of these matrices as N tends to infinity. Clearly, ΨN,k ΨH N,l

= INR ⊗

R−1 k

H AH N,k ΓN (ωl − ωk )AN,l

N

R−1 k

−1 −1 converges almost surely to INR ⊗ R−1 k (δ(k − l)Rk ) Rk = δ(k − l)INR ⊗ Rk . Similarly, ¡ ¢ H 2 AH N,k IN − N DN ΓN (ωl − ωk )AN,l −1 H ΦN,k ΨN,l = INR ⊗ Rk N ! Ã H H D ΓH (ω − ω )A 2A AH Γ (ω − ω )A N l k N,l l k N,l N N,k N,k N − R−1 = INR ⊗ k N N2

and thus converges almost surely to zero as both terms enclosed in the parenthesis of the above equation converge almost surely to δ(k − l)Rk . Finally, ¡ ¢2 H 2 AH ΓN (ωl − ωk )AN,l N,k IN − N DN H ΦN,k ΦN,l = INR ⊗ N Ã H H AN,k ΓN (ωl − ωk )AN,l = INR ⊗ N −

H 4AH N,k DN ΓN (ωl − ωk )AN,l

N2

+

2 H 4AH N,k DN ΓN (ωl − ωk )AN,l

!

N3

converges a.s. to INR ⊗ (1 − 2 + 43 )δ(k − l)Rk = 31 δ(k − l)INR ⊗ Rk . These results imply that ZN,k ZH N,l converges to zero if k 6= l. For k = l, the use of the above results along with (36), it is straightforward h i £ ¤ 2 σ2 H to show that σ2 Re ZN,k ZH N,k converges almost surely to Σk . Finally, 2 Re ZN ZN converges almost surely to diag (Σ1 , . . . , ΣK ). R EFERENCES [1] L. Hanzo, M. M¨unster, B. J. Choi, and T. Keller, OFDM and MC-CDMA for broadband multi-user communications, WLAN’s and broadcasting, IEEE Press, Wiley, 2003. [2] Y. Yao and G. B. Giannakis, “Blind carrier frequency offset estimation in SISO, MIMO, and multiuser OFDM systems,” IEEE Transactions on Communications, vol. 53, no. 1, pp. 173-183, January 2005. November 25, 2006

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[3] Ph. Ciblat and L. Vandendorpe, “On the maximum-likelihood based data-aided joint frequency offset and channel estimation,” in Proceedings EUSIPCO’02, Toulouse, France, September 3-6, 2002, pp. 627–630. [4] Ph. Ciblat, Ph. Loubaton, E. Serpedin, and G. B. Giannakis, “Performance of blind carrier-offset estimation for non-circular transmissions through frequency-selective channels,” IEEE Transactions on Signal Processing, vol. 50, no. 1, pp. 130–140, January 2002. [5] Y. Wang, Ph. Ciblat, E. Serpedin, and Ph. Loubaton, “Performance analysis of a class of non-data aided carrier frequency offset and symbol timing delay estimators for flat-fading channels,” IEEE Transactions on Signal Processing, vol. 50, no. 9, pp. 2295–2305, September 2002. [6] S. Barbarossa, M. Pompili, and G.B. Giannakis, “Channel-independent synchronization of orthogonal frequency division multiple access systems,” IEEE Journal on Selected Areas in Communications, vol. 20, no. 2, pp. 474–486, February 2002. [7] J. J. Van de Beek, P. O. Borjesson, M. L. Boucheret, D. Landstram, J. M.Arenas, P. Odling, C. Ostberg, M.Wahlqvist, and S. K.Wilson, “A time and frequency synchronization scheme for multiuser OFDM,” IEEE Journal on Selected Areas in Communications, vol. 17, pp. 1900–1914, November 1999. [8] Z. Cao, U. Tureli, and Y. D. Yao, “Deterministic multiuser carrier-frequency offset estimation for interleaved OFDMA uplink,” IEEE Transactions on Communications, vol. 52, no. 9, pp. 1585–1594, September 2004. [9] M. Morelli, “Timing and frequency synchronization for the Uplink of an OFDMA system,” IEEE Transactions on Communications, vol. 52, no. 2, pp. 296–306, February 2004. [10] M. Pun, M. Morelli and C. J. Kuo, “Maximum likelihood synchronization and channel estimation for OFDMA uplink transmissions”, IEEE Transactions on Communications, vol. 54, no. 4, pp. 726–736, April 2006. [11] S. Sezginer, P. Bianchi, and W. Hachem, “Asymptotic Cram´er-Rao Bounds and Training Design for Uplink MIMO-OFDMA Systems with Frequency Offsets,” to appear in IEEE Transactions on Signal Processing. [12] C. Ibars and Y. Bar-Ness, “Inter-carrier interference cancellation for OFDM systems with macrodiversity and multiple frequency offsets,” Wireless Personal Communications, no. 26(4), pp. 285–304, September 2003. [13] P. Billingsley, Probability and Measure, 3rd edition, Wiley, New York, 1995. [14] R. A. Horn and C. R. Johnson, Matrix Analysis, 7rd edition, Cambridge University Press, New York, 1999. [15] S. Sezginer, A study of OFDMA, Ph.D. Thesis, Sup´elec, 2006, available at http://www.supelec.fr/ecole/radio/sezginer.html. [16] P. Ciblat, Ph. Loubaton, E. Serpedin, and G. B. Giannakis, “Asymptotic analysis of blind cyclic correlation-based symbol-rate estimators,” IEEE Transactions on Information Theory, vol. 48, no. 7, pp. 1922–1934, July 2002. [17] W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill International Editions, Singapore, 1986. [18] M. Morelli and U. Mengali, “Carrier frequency estimation for transmissions over selective channels,” IEEE Transactions on Communications, vol. 48, no. 9, pp. 1580–1589, September 2000. [19] T. Hasan, “Non linear time series regression for a class of amplitude modulated cosinusoids,” Journal of Time Series Analysis, vol. 3, no. 2, pp. 109–122, January 1982.

DRAFT

November 25, 2006

REDUCED COMPLEXITY FREQUENCY OFFSET AND CHANNEL ESTIMATION FOR UPLINK MIMO-OFDMA SYSTEMS

29

−1

−3

10

10

Proposed estimator, M = 1, K = 2 Proposed estimator, M = 1, K = 4 Proposed estimator, M = 3, K = 2 Proposed estimator, M = 3, K = 4 Exact CRB, K = 2 Exact CRB, K = 4 Asymptotic CRB

Proposed estimator, M = 1, K = 2 Proposed estimator, M = 1, K = 4 Proposed estimator, M = 3, K = 2 Proposed estimator, M = 3, K = 4 Exact CRB, K = 2 Exact CRB, K = 4 Asymptotic CRB

−4

10

−5

10

−6

MSE

MSE

10

−7

−2

10

10

−8

10

−9

10

−3

10

−10

10

64 128

256

512 N

1024

64 128

256

(a) frequency offset estimation Fig. 1.

1024

(b) channel estimation

Convergence behavior of the proposed estimator : NR = NT = 1,

P1 σ2

= 10dB.

0

−3

10

10

Proposed estimator, M = 1, K = 2 Proposed estimator, M = 1, K = 4 Proposed estimator, M = 3, K = 2 Proposed estimator, M = 3, K = 4 Exact CRB, K = 2 Exact CRB, K = 4 Asymptotic CRB

Proposed estimator, M = 1, K = 2 Proposed estimator, M = 1, K = 4 Proposed estimator, M = 3, K = 2 Proposed estimator, M = 3, K = 4 Exact CRB, K = 2 Exact CRB, K = 4 Asymptotic CRB

−4

10

−5

10

−6

10

−1

MSE

MSE

512 N

−7

10

10

−8

10

−9

10

−2

10 −10

10

64 128

256

512 N

1024

64 128

256

(a) frequency offset estimation Fig. 2.

Convergence behavior of the proposed estimator : NR = NT = 2,

November 25, 2006

512 N

1024

(b) channel estimation P1 σ2

= 10dB.

DRAFT

30

SUBMITTED FOR PUBLICATION IN IEEE TRANSACTIONS ON SIGNAL PROCESSING

−7

−7

10

10 Proposed estimator, M = 1, K = 2 Proposed estimator, M = 1, K = 4 Proposed estimator, M = 3, K = 2 Proposed estimator, M = 3, K = 4 Exact CRB, K = 2 Exact CRB, K = 4

−8

−8

10

MSE

MSE

10

−9

10

−10

−9

10

−10

10

10

−11

10

Proposed estimator, M = 1, K = 2 Proposed estimator, M = 1, K = 4 Proposed estimator, M = 3, K = 2 Proposed estimator, M = 3, K = 4 Exact CRB, K = 2 Exact CRB, K = 4

−11

0

5

10 P1 /σ 2 (dB)

15

20

10

0

5

(a) NR = NT = 1 Fig. 3.

10 P1 /σ 2 (dB)

15

20

15

20

(b) NR = NT = 2

Performance of the proposed estimator for frequency offset estimation with N = 1024.

−5

−1

10

10 Single user estimator Proposed estimator, M = 1 Proposed estimator, M = 3 APFE Exact CRB Asymptotic CRB

−6

10

−2

MSE

MSE

10

−7

10

Single user estimator Proposed estimator, M = 1 Proposed estimator, M = 3 APFE Exact CRB Asymptotic CRB

−3

10 −8

10

0

5

10 P1 /σ 2 (dB)

15

(a) frequency offset estimation Fig. 4.

DRAFT

20

0

5

10 P1 /σ 2 (dB)

(b) channel estimation

Comparison of the proposed estimator with APFE : NR = NT = 1, K = 4, N = 256.

November 25, 2006