MULTIPLE-ANTENNA wireless systems have received

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1 2074 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 6, JUNE 2005 Timing Estimation in Multiple-Antenna Systems Over Rayleigh Flat-Fading Channels Yong Liu, Tan F. Wong, Senior Member, IEEE, and Ashish Pandharipande Absact The use of multiple ansmit and receive antennas can offer substantial performance improvement to a wireless communication system by providing spatial diversity and supporting high data rate services. Many of the current space-time coding schemes proposed for multiple-antenna systems assume perfect timing information to achieve the expected performance gain. The lack of timing synchronization between the ansmit and receive signals could degrade the system performance. In this paper, we investigate the problem of timing estimation in multiple-antenna systems with the aid of aining signals. A slow, independent and identically disibuted Rayleigh flat-fading channel model is considered. We derive two maximum likehood timing estimators based on two different approaches, namely, eating the channel deterministic and random and present the corresponding Cramér Rao bounds (s). Then, the optimal designs of aining signals based on some figures of merit associated with the s are discussed. Index Terms Cramér Rao bound, MIMO system, optimal aining sequence, Rayleigh fading, timing estimation. I. INTRODUCTION MULTIPLE-ANTENNA wireless systems have received considerable attention over the past several years. These systems can provide significantly higher capacity as compared with single-antenna systems without requiring an increase in system bandwidth [1], [2]. An issue that has not been sufficiently explored is timing synchronization in multiple-antenna systems. Inaccuracies in timing synchronization can degrade the performance of such communication systems. For instance, many of the current space-time coding schemes proposed for multiple-antenna systems assume perfect knowledge of timing and channel gains at the receiver in order to be able to achieve the promised diversity gain and capacity improvement. The performance of these systems may be limited by the accuracy of timing estimation. The subject of this paper is to study the problem of timing estimation for a wireless communication system employing multiple ansmit and receive antennas. Manuscript received July 18, 2003; revised June 25, This work was supported in part by the Office of Naval search under Grant N and the National Science Foundation under Grant ANI Part of this material was presented at the 37th Asilomar Conference on Signals, Systems and Computers, Nov The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Vikram Krishnamurthy. Y. Liu and T. F. Wong are with the Wireless Information Networking Group, Department of Elecical and Computer Engineering, University of Florida, Gainesville, FL USA ( yongliu@dsp.ufl.edu; twong@ece.ufl.edu). A. Pandharipande is with the Samsung Advanced Institute of Technology, Giheung, Korea ( pashish@ieee.org). Digital Object Identifier /TSP Previous related work was primarily resicted to acquisition in spread specum systems with multiple receive antennas [3] [5]. In [3] and [5], the maximum likelihood (ML) estimator of the received code lag was obtained, and the error probability for the acquisition system was derived. A deterministic but unknown channel was considered in [3], whereas a flat Rayleigh fading channel with known statistics was assumed in [5]. An optimal estimator for code acquisition was derived in [4] for spatially correlated channels. In [6], the performance of code acquisition in a direct-sequence code-division multiple-access system employing multiple ansmit antennas was analyzed. Through simulations, it was shown that the presence of multiple ansmit antennas improved the code acquisition performance, relative to that of a single-antenna system. Issues related to parameter estimation of signals received by an array of antennas have also been eated in the radar array signal processing literature [7], [8]. Time delay and spatial signature estimation of known signals received by an array of antennas was investigated in [9]. ML algorithms and the Cramér Rao bounds (s) for time delay and array calibration estimation were developed, and some computationally efficient approximations of the ML algorithms were proposed. In [10], ML methods were developed for space-time fading channel estimation with an antenna array in spatially correlated noise. The s for the unknown directions of arrival, time delays, and Doppler shifts were derived, under a suctured and unsuctured array response model. In the present paper, we consider a wireless communication system with multiple ansmit and receive antennas in a slow, independent and identically disibuted (i.i.d.) Rayleigh flat-fading environment. The goal of this paper is to investigate the problem of timing estimation in such a system with the aid of aining signals. One of the main objectives is to find the optimal aining signal design. We investigate the timing estimation problem under two approaches. In the first approach, the channel is assumed to be unknown and deterministic where joint estimation of the channel and delay is carried out. We derive an ML estimator for joint channel and timing estimation and compute the associated. Then, we discuss the optimal aining signals with respect to two performance measures based on the : the outage probability that the is larger than a threshold and the average. We show that the optimal aining scheme is one, wherein orthogonal aining signals from multiple ansmit antennas are used. In the second approach, the channel is assumed to be unknown but random with known statistics. We use the likelihood function averaged over all random channel realizations to obtain the ML estimator for the delay. We derive the associated and study the X/$ IEEE

2 LIU et al.: TIMING ESTIMATION IN MULTIPLE-ANTENNA SYSTEMS OVER RAYLEIGH FLAT-FADING CHANNELS 2075 TABLE I NOTATIONS optimal aining scheme in terms of minimizing this. We show that perfectly correlated aining signals employed at different ansmit antennas constitute the optimal ansmit scheme in conast to orthogonal aining signals in the first approach. The rest of the paper is organized in the following manner. The system model is inoduced in Section II. In Section III, we consider the timing estimation problem when the channel is assumed to be unknown but deterministic. In Section IV, we study the problem of timing estimation with the assumption that the channel is random but with known statistics. In both sections, we derive the ML timing estimators and compute the associated s. Optimal aining signal designs are discussed based on the corresponding s. In Section V, some discussions comparing these two timing estimation approaches are provided. The notation used in this paper is summarized in Table I for clarity, and most derivations and proofs of the results are relegated to the Appendices. II. SYSTEM MODEL the th ansmit antenna, and, with denoting the channel gain from the th ansmit antenna to the th receive antenna. Define the channel vector as. In addition, is a complex, circularsymmeic, white Gaussian noise process with zero mean and covariance maix. The symbol denotes the unknown, deterministic delay to be estimated. This model assumes that the delays between all pairs of ansmit and receive antennas are the same. This corresponds to the case in which the distance between the ansmit and receive antenna arrays is much larger than the sizes of the arrays. We consider the Rayleigh flat-fading channel model in which the channel coefficients are i.i.d. complex, circular-symmeic, zero-mean Gaussian random variables with the disibution, i.e., for and We consider a single-user multiple-input multiple-output (MIMO) system with ansmit antennas and receive antennas. We assume a quasistatic (block fading) channel where the channel varies slowly enough to be considered invariant over a block. However, the channel changes to an independent value from block to block. By using the unsuctured array model [8], the received baseband signals at the receive antennas are given in vector form by The conditional likelihood function of and, can be written as, given the unknown (2) where is the received signal vector from the receive antenna array, is the ansmitted aining signal from (1) where we have assumed that the aining signals, for,havefinite durations, and the observation interval is larger than the sum of the maximum aining signal duration and the maximum possible value of. Thus, the whole ansmitted aining signals are observed at the receiver.

3 2076 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 6, JUNE 2005 We can simplify the exponent of the likelihood function to find the sufficient statistics for the estimation of the delay : From (4) and (6), the conditional likelihood function of, given the unknowns and, can be written as const (7) const Let Im, Im, Im and. By using the isomorphism between real and complex maices [11], we have Im and. In terms of these real quantities, the conditional likelihood function of is then const (8) where the term const represents the part that does not depend on the delay and the channel. In addition, the last equality holds due to the assumption that is larger than the sum of the maximum aining signal duration and the maximum possible delay. Denote the matched filter output corresponding to the th ansmit signal by III. TIMING ESTIMATION WITH UNKNOWN DETERMINISTIC CHANNEL In this section, we will eat as unknown but deterministic in the estimation process and consider the joint estimation of the delay and the channel vector. provides suf-. With this notation, we then Note that ficient statistics for estimating have (3) A. ML Estimate In this subsection, we develop the ML estimator for the joint estimation of the timing and the channel vector. The joint ML estimate of and maximizes the conditional likelihood function (8) as a function of and : (9) (4) Denote the crosscorrelation between the aining signals from the th and th ansmit antennas as (5) Alternatively, we can maximize the log-likelihood function given by const (10) As suggested in (9), we first maximize the log likelihood function over. Taking the first derivative of with respect to (w.r.t.) gives which forms the th element of the correlation maix. Let. Then, we have (6) By letting as, we get the ML estimate of the channel (11)

4 LIU et al.: TIMING ESTIMATION IN MULTIPLE-ANTENNA SYSTEMS OVER RAYLEIGH FLAT-FADING CHANNELS 2077 where we have assumed that, i.e.,, is nonsingular to obtain a unique estimation of the channel. Then, substituting (11) into (10) gives the ML estimate of the delay in the form (12) To implement the ML estimator in general, we need to conduct a line search over all possible values of to maximize the above meic. B. The gives a lower bound on the variance of any unbiased estimator [11], [12]. It has been widely used to lower bound the mean square error (MSE) of symbol timing estimators [13], [14]. It is well known [11], [12] that ML estimators, under mild regularity conditions and with i.i.d. observations, are asymptotically unbiased and efficient. It can be easily verified that the elements of given in (3), corresponding to different receive antennas are i.i.d. observations. Thus, for a particular realization of the channel, the ML estimator is asymptotically efficient, i.e., it approaches the, as the number of receive antennas becomes large. Hence, the is a suitable performance measure for the ML estimator of the delay. We will also verify the suitability of employing the as a performance meic by computer simulation examples. The main result of this subsection on the is contained in the following theorem. Theorem 3.1 (): Suppose that the first and second derivatives of the aining signals, for, exist and that they are uniformly continuous on. Together with the standard regularity conditions in [11], [12], the for the estimation of the delay for a given realization of the channel is given by where and with with (13) (14) (15) Proof: See Appendix A. We note that the varies with different choices of aining signals. By carefully choosing the aining signals to minimize a suitable measure associated with the, we can potentially improve the estimation performance. C. Optimal Training Scheme Communication systems often employ the same symbol waveforms for both the aining and data phases. The choice of the symbol waveform is mainly decided by the performance required by data ansmissions. In this section, we will make the following simplifying but practically reasonable assumptions on the aining signals. Assumption 1: Let be the aining sequence assigned to the th ansmit antenna, and on this antenna, the aining signal waveform is of the form (16) where is the number of aining symbols, and is the symbol waveform. We call the maix the aining sequence maix. Assumption 2: The symbol waveform is time-limited to a single symbol period so that adjacent symbols do not interfere with each other. In addition, is sufficiently smooth to guarantee the existence of uniformly continuous first and second derivatives. This condition is satisfied for most symbol waveforms of practical interest. Two typical examples are the time-domain raised-cosine pulse and the half-sine pulse. Assumption 3: is nonsingular, and hence, and are also nonsingular. We note that this implies that. Under the assumptions stated above, the for the timing estimation can be simplified to the expression summarized in the following corollary. Corollary 3.1 (): Given Assumptions 1 3, the for the estimation of for a particular realization of the channel reduces to (17) where,,, and. Proof: See Appendix B. As a result, the dependence of the on the aining signals, for, simplifies into that on the aining sequence maix and the symbol waveform. In the following two subsections, we optimize the aining sequence maix in terms of two performance measures, namely, the outage probability that the is larger than a threshold and the average over all channel realizations. 1) Outage Probability: In this subsection, the outage probability that the is larger than the threshold, i.e.,, is used as a performance measure with respect to which the aining signals from different ansmit antennas are optimized. Write the specal decomposition of as, where is a unitary maix, and diag the positive eigenvalues of is the diagonal maix containing. The design of the optimal

5 2078 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 6, JUNE 2005 aining scheme can now be formulated as the following optimization problem: subject to (18) where specifies a consaint on the total ansmit power. First, we consider a simple but important case: two ansmit antennas and one receive antenna. In this case, the optimization problem (18) can be simplified as follows. Starting from Corollary 3.1, we have (19) With the specal decomposition of,, where. Since is a random vector with i.i.d. complex, circular-symmeic, zero-mean Gaussian elements and is a unitary maix, is also a complex Gaussian random vector with the same disibution as. We note that has the exponential disibution with. Let and, for, 2; then (20) where and are independent random variables with exponential disibution, and. The total power consaint is equivalent to. Hence, the optimization problem can be rewritten in the following simple form: subject to and (21) In order to solve the above optimization problem, we employ the following result on the Schur-convexity 1 of the disibution function of the linear combination of two exponential random variables [15]. Lemma 3.1: Let and be independent random variables with exponential disibution, and. Then, the function where and 1 A detailed description on Schur-convexity and majorization can be found in [16]. is Schur convex on (, )if, and it is Schur concave on (, )if. Using the above lemma and considering the region in which the threshold, the optimization cost function in (21) is Schur convex on. Thus, minimization of the cost function occurs if and only if, i.e., [16]. This implies that the optimal is such that. The optimal aining scheme is summarized in the following theorem. Theorem 3.2: Suppose that the threshold. The aining sequence maix such that minimizes the outage probability of the for a system with two ansmit antennas and one receive antenna. That is, the optimal aining sequences from different ansmit antennas are orthogonal to each other and have equal powers. We will see from the discussion in the next subsection on the average (Corollary 3.2) that the value is exactly one half of the average over all channel realizations. Thus, it is reasonable to consider the stated region of the threshold. It seems natural that a result analogous to the one in Lemma 3.1 be ue for the more general case. While the proof of such a result remains open, there is song evidence regarding the Schur convexity of the function, where, for, are independent random variables with unit-mean exponential disibution. The following conjecture has been advanced in [15] and supported by some song numerical results. Conjecture 3.1: The family of unimodal disibution functions is increasing with respect to the variance (i.e., Schur-convex) for small values and decreasing (i.e., Schur-concave) for large values of. Based on the above conjecture, we conjecture that the result in Theorem 3.2 extends to the case of arbiary numbers of ansmit and receive antennas. Conjecture 3.2: When, the outage probability of the is minimized if the threshold is not too small. Thus, the optimal aining sequences from different ansmit antennas, in terms of minimizing the outage probability, are orthogonal to each other and have equal powers. In [17], the authors assumed perfect timing estimation and studied the problem of choosing the optimal aining sequences for channel estimation to maximize a lower bound on the capacity of the channel that was learned by aining. The optimal aining sequences for channel estimation turned out to have the same sucture as those we get here for timing estimation. To illusate our conjecture on the optimality of orthogonal sequences, we have carried out a large number of numerical calculations. In the broad region of in which we are interested, we have not observed the existence of any other schemes that can achieve a lower outage probability than the orthogonal aining signals. In Fig. 1, we plot, for instance, the outage probabilities for a system with four ansmit antennas and a single receive antenna employing different aining signal sets. Note that since is the total ansmit power consaint,

6 LIU et al.: TIMING ESTIMATION IN MULTIPLE-ANTENNA SYSTEMS OVER RAYLEIGH FLAT-FADING CHANNELS 2079 More precisely, the outage probability for orthogonal aining signals is given by (22) where the second equality is obtained from the fact that is -disibuted [18]. From (22), it is not hard to see that when the SNR is large, i.e.,, the outage probability is approximately given by Fig. 1. Outage probabilities achieved using different aining signal sets for a system with four ansmit and one receive antennas. The unit of the threshold is T. Fig. 2. Outage probabilities achieved using orthogonal aining signals for different numbers of ansmit antennas. One receive antenna is employed. The unit of the threshold is T. the signal-to-noise ratio (SNR) here should be understood as the total SNR for the whole aining period instead of the SNR for one symbol period. The time-domain raised-cosine pulse is used as the symbol waveform. The results in the figure suggest that the orthogonal aining signals are optimal and can provide a significant performance gain over the other aining signals. In Fig. 2, we compare the outage performance of orthogonal aining sequences for different numbers of ansmit antennas. The results in the figure show that the use of multiple ansmit antennas can offer substantial estimation performance improvement over a single-antenna system. For example, if we consider the outage probability, the two-ansmit antenna system can achieve a 4-dB performance gain, and the four-ansmit antenna system can achieve a 6-dB performance gain. The performance gap grows with decreasing outage probability. (23) Equation (23) indicates that the outage probability decreases with the th power of the reciprocal of the SNR. The power is usually referred to as the diversity order of the system [18]. Thus, we conclude that the use of multiple ansmit and receive antennas (with orthogonal aining signals) provides spatial diversity for timing estimation in the same way as spacetime coding does for demodulation [1], [2]. An important remaining issue is whether the ML estimator can achieve the outage probability of the. For each realization of the channel, the ML estimator is asymptotically efficient with increasing number of receive antennas. We note that, where is the indicator function. Because the indicator function is a bounded function, the dominated convergence theorem [19] implies that the ML estimator can achieve the outage probability of the asymptotically. To verify the suitability of using the outage probability as a performance meic when the number of receive antennas is small, we evaluate the performance of the ML estimator via Monte Carlo simulations. In Fig. 3, we plot the outage probabilities of the ML estimator obtained from simulation and calculated using the, respectively, for a system with two ansmit antennas and employing orthogonal aining sequences. It can be seen that the ML estimator gives an outage probability performance very close to that predicted by the even for small values of, 2, and 4. Hence, the simulation results verify that the outage probability of the also provides an effective performance meic when the number of receive antennas is small. 2) Average : In this subsection, we use the averaged over the Rayleigh flat-fading channel as an alternate performance measure based on which the aining signals from the ansmit antennas are optimized. After averaging over the Rayleigh flat-fading channel, the average is given as (24)

7 2080 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 6, JUNE 2005 Fig. 3. Comparison of outage probabilities of the ML estimator obtained from simulation and calculated from the. The number of ansmit antennas K is 2, and =10 T. The design of the optimal aining scheme can now be formulated as the following optimization problem: subject to (25) The following theorem specifies the optimal aining sequence that minimizes the average. Theorem 3.3: When, the average over the Rayleigh flat-fading channel is minimized. That is, the optimal aining sequences from different ansmit antennas, in terms of minimizing the average, are orthogonal to each other and have equal powers. Proof: See Appendix C. With the optimal aining sequences, we can provide an explicit expression for the average which is described in the next corollary. Corollary 3.2 (Average ): Using the optimal aining scheme, the average over the Rayleigh flat-fading channel is given by (26) when. Proof: See Appendix D. With the optimal (orthogonal) aining sequences, the average is a simple function of the constant, which only depends on the symbol waveform, the signal-to-noise ratio, the number of ansmit antennas, and the number of receive antennas. Note that the average in the limit of large or large can be approximated as (27) Fig. 4. Comparison of the MSE of the ML estimator obtained from simulation and the average. The number of ansmit antennas K is 2. The unit in the vertical axis is T. which is inversely proportional to the number of receive antennas. When is symmeic about, such as the time-domain raised-cosine pulse and the half-sine pulse, becomes zero. Then, the average for the estimation of the delay with orthogonal aining signals can be written as where (28) is known as the root-mean-square bandwidth [12] of the symbol waveform. Here, is the Fourier ansform of. We note that the average can be decreased by increasing the bandwidth of the symbol waveform. As before, we would like to know whether the ML estimator can achieve the average. Because the function is not a bounded function, thus unlike the outage probability of the, the ML estimator may not achieve the average asymptotically (see further discussion in Section V-A). However, the average provides a lower bound for the variance of any unbiased timing estimator averaged over the channel realizations. Again, we employ Monte Carlo simulations to evaluate the performance of the ML estimator with a small number of receive antennas. In Fig. 4, we compare the MSE achieved by the ML estimator and the average given by (26) for a system with two ansmit antennas and employing orthogonal aining sequences. For a single receive antenna system, the performance of the ML estimator deviates significantly from the average. This is due to the events in which all the channel coefficients are very small simultaneously, causing the estimation performance to be very poor. The large estimation errors caused by these events dominate the MSE of the ML estimator. We can see from the figure that the effect of these events diminishes as the number of receive antennas or the SNR increases. In the former case, the error-dominating events become rarer

8 LIU et al.: TIMING ESTIMATION IN MULTIPLE-ANTENNA SYSTEMS OVER RAYLEIGH FLAT-FADING CHANNELS 2081 as the number of receive antennas increases. In the latter case, the estimation errors, and hence the effect of the error dominating events, get smaller as SNR increases. For a reasonably small value of, e.g., 4, and a reasonably high SNR, e.g., 20 db, we see that the average is still a rather appropriate performance meic. IV. TIMING ESTIMATION WITH RANDOM CHANNEL cently, differential space-time coding schemes [20] [22] have been developed where channel estimates are not required at the receiver. For this situation, we only need to consider the estimation of the delay. A reasonable model to represent this scenario is that the channel is random with known statistics. A. ML Estimator call that the conditional likelihood function of in terms of real vectors and maices is given by (8). With the assumption of i.i.d. Rayleigh flat-fading channels between the ansmit and receive antennas, we have and. The joint probability density function of the channel vector is given as We assume that is known to the receiver for the implementation of the ML estimator. We note that the maix is always invertible. Therefore, unlike the resiction in Section III, can be singular, which implies the aining signals from different ansmit antennas can be linearly dependent. B. The for the timing estimation based on the random channel model is summarized in the following theorem. Theorem 4.1 (): Suppose that the first and second derivatives of the aining signals exist and that they are uniformly continuous on. Together with the standard regularity conditions in [11] and [12], the for the estimation of the delay over the i.i.d Rayleigh flat-fading channel model is given by where, and the th element of is (34) (29) We can average over all realizations of to obtain the unconditional likelihood function as const (30) where we have used the integral result from [23, a]. The natural logarithm of is the log-likelihood function: const (31) By using the relationship between real and complex maices [11], the log-likelihood function can be written in terms of complex quantities as const Hence, the ML estimator for the delay is given by (32) for,. Proof: See Appendix E. C. Optimal Training Scheme (35) In this subsection, we impose Assumptions 1 and 2 made in Section III-C on the form of the aining signals. With these two assumptions, can be simplified to Hence, we have. Thus, the for the timing estimation can be simplified as given in the following corollary. Corollary 4.1: Given Assumptions 1 and 2, the for the estimation of the delay over the i.i.d Rayleigh flat-fading channel model reduces to (33) (36)

9 2082 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 6, JUNE 2005 Moreover, in terms of the eigenvalues of, we have (37) Thus, the minimization of the is equivalent to the following optimization problem: subject to (38) It can be easily verified that the cost function is a convex function on. Then, the following theorem specifies the optimal aining sequences [24]. Theorem 4.2: The is minimized by choosing the aining sequence maix such that, and. That is, the optimal aining sequences from different ansmit antennas are perfectly correlated. We note that the rank of the optimal aining sequence maix is 1. This implies that we can choose an arbiary subset of ansmit antennas to ansmit the aining signals as long as the aining sequences from the chosen ansmit antennas are perfectly correlated with each other. A common choice is to use the same aining sequence and evenly assign the power to each ansmit antenna. With the optimal choice of aining sequences, the corresponding minimum is given by Fig. 5. Comparison of s obtained using orthogonal aining sequences and perfectly correlated aining sequences for different numbers of ansmit antennas. Note that the of the system with the perfectly correlated aining sequences is the same for any number of ansmit antennas. (39) On the other hand, when orthogonal aining signals are employed, i.e.,, the is maximized to the value (40) Conary to the previous case of joint estimation of the channel and delay where orthogonal aining sequences are optimal, they are the worst in terms of the value for estimating the delay under the random channel model. Fig. 5 compares the s of the system with the perfectly correlated aining sequences and that with the orthogonal aining sequences. Note that the of the system with the perfectly correlated aining sequences is the same for any number of ansmit antennas. We see that the performance gain achieved by the optimal scheme is obvious when the SNR is low. For any fixed, the performance gap vanishes as the SNR becomes sufficiently large. In Fig. 6, we compare the MSE achieved by the ML estimator and the given in (39) for a system with two ansmit antennas and employing perfectly correlated aining sequences. The perfect correlation is obtained by using the same aining sequence and evenly assign the power to each ansmit antenna. As will be discussed in Section V-B, no knowledge of signal-tonoise ratio is needed to implement the ML delay estimator for Fig. 6. Comparison of the MSE of the ML estimator obtained from simulation and the. The number of ansmit antennas K is 2. The unit in the vertical axis is T. this choice of perfectly correlated aining sequences. We observe from the figure that for a reasonably small value of, e.g. 4, and a reasonably high SNR, e.g. 20 db, the is a tight lower bound on the MSE performance of the ML estimator. This, together with the asymptotic achievability of the, suggest that the is an appropriate performance meic. V. DISCUSSIONS AND CONCLUSIONS In the previous two sections, we have studied the problem of timing estimation in multiple-antenna systems from two different approaches. In Section III, the channel is assumed to be unknown but deterministic, and joint ML estimation of and the delay is performed. In conast, in Section IV, we assume that the channel is random but with known statistics and use the likelihood function averaged over all channel realizations

10 LIU et al.: TIMING ESTIMATION IN MULTIPLE-ANTENNA SYSTEMS OVER RAYLEIGH FLAT-FADING CHANNELS 2083 to consuct the ML estimator for the delay. These two approaches lead to two different optimal aining signal designs. For the deterministic channel approach, we see that orthogonal aining sequences minimize the outage probability as well as the average. For the random channel approach, perfectly correlated aining sequences minimize the. Here, we compare these two approaches in terms of the resulting ML estimators, s, and suitability of the outage and average performance measures. A. Orthogonal Training Signals When orthogonal aining signals are employed, both the ML estimators of the delay under the deterministic and random channel approaches, respectively, reduce to (41) Thus, the equal gain combination of the received signals from the receive antennas is the ML estimator for both approaches. Under the deterministic channel approach, the average has the value (42) Under the random channel approach, the has the value (43) As discussed before, the in (43) is asymptotically achievable by the ML estimator when the number of receive antennas goes to infinity. In addition, the limiting ratio between (42) and (43), when approaches infinity, is, which is smaller than 1. This implies that the average in (42) is not achievable by the ML estimator asymptotically when approaches infinity. On the other hand, for small values of, the value in (42) can be larger than the value of (43) when the SNR is large enough. More precisely, this happens when. Thus, in this case, the average in (42) actually gives a tighter bound on the performance of the ML estimator. The simulation results in Fig. 4 are in agreement with this observation. In this sense, the average may not be as good a performance measure as the outage probability in the deterministic channel approach since the latter is asymptotically achievable, starting at very small values of, by the ML estimator. However, for small values of and at high SNR, the average may still be a reasonable performance meic. to be exactly the same as the one for orthogonal aining sequences given in (41). We note that the knowledge of the SNR is not needed to implement the above ML estimator. Comparing the results in Figs. 4 and 6, the MSE obtained by the ML estimator with the perfectly correlated aining sequences is smaller than that obtained by the ML estimator with orthogonal aining sequences for all cases considered in the simulation studies. This observation is in agreement with the aining sequence optimization result based on the that the perfectly correlated sequences are superior to the orthogonal sequences under the random channel approach. In general, the SNR information is needed to implement the ML estimator. We also note that perfectly correlated aining signals are not applicable in the deterministic channel approach since they cannot be used to estimate the channel vector. C. Deterministic versus Random Channel Approaches The results and discussions in the previous sections provide some guidelines of whether to use the deterministic or random channel approaches in estimating the timing parameter. If the design consideration is the outage probability, i.e., neglecting a small percentage of the worst-case channel realizations, one would employ the deterministic channel approach with orthogonal aining signals. On the other hand, if the average estimation (over all channel realizations) error is the main design criterion, one would employ the random channel approach with perfectly correlated aining signals. We note that the perfectly correlated aining signals cannot be used for channel estimation. Thus, they may be more suitable for space-time coding schemes that do not require the channel information. In addition, the advantage of the perfectly correlated aining signals over orthogonal signals vanishes at high SNR in the random channel approach. Thus, when the number of ansmit antennas is not very large and at high SNR, one could employ orthogonal aining signals for either of the two approaches. APPENDIX A. Proof of Theorem 3.1 The for the estimation of is given as (44) where is the Fisher information maix for the joint estimation of the channel and the delay, which is defined as (45) B. Perfectly Correlated Training Signals Under the random channel approach employing perfectly correlated aining signals, we have, where is an arbiary vector with.for instance, when we use the same aining sequence and evenly assign the power to each ansmit antenna. By using the maix inversion formula, the ML delay estimator for this choice of perfectly correlated sequences is reduced Since and,wehave Moreover (46) (47)

11 2084 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 6, JUNE 2005 Letting, then Im. The th block of can be computed from B. Proof of Corollary 3.1 With the three assumptions on the aining signals, we have (53) Then, (14) can be written in terms of the aining sequences as (54) Thus The fact that the noise is zero-mean gives (55) (48) Hence. Moreover, (15) can also be simplified in terms of the aining sequences as Finally, the fact that. Similarly, can be computed from (49) Thus, have (56). Similarly, we (57) Applying the standard result on the inverse of a partitioned maix to (44) and (45) gives and. Hence, (51) can be written as (50) By using the relationship between real and complex maices [11], we get (58) (51) Then, the for the estimation of the delay is Then, the of the estimation of the delay is (52) (59)

12 LIU et al.: TIMING ESTIMATION IN MULTIPLE-ANTENNA SYSTEMS OVER RAYLEIGH FLAT-FADING CHANNELS 2085 By using some standard properties of the Fourier ansform similar to the Parseval s theorem, we have,, and, where is the Fourier ansform of. Then, according to the Cauchy Schwarz inequality, we have (60) Since, we have, which implies that the expression of the given in (59) is nonivial. C. Proof of Theorem 3.3 Let, where diag contains the positive eigenvalues of the Hermitian maix, and is a unitary maix. Consider the following optimization problem: Since are i.i.d., they are exchangeable random variables. Since are exchangeable random variables and is a symmeic Borel-measurable convex function, the function is Schur-convex by the lemma. Moreover, since is majorized by whenever,, we know [16] that is minimized with. We note that this choice of, also satisfies the consaints in the minimization problem in (25). Thus, it is also a solution to the original minimization problem. Thus, the optimal aining sequence maix should satisfy, which implies that the aining sequences from different ansmit antennas are orthogonal to each other and have equal powers. D. Proof of Corollary 3.2 From Theorem 3.3 and its derivation in Appendix C, the average under the optimal aining scheme is given as (63) where are i.i.d. complex circular-symmeic Gaussian random variables with the disibution. Let. Then, is -disibuted with the probability density function (p.d.f.) for subject to (61) Let. The p.d.f. of is given as (64) Note that for The expectation of can be computed as (62) where. As before, is a complex Gaussian random vector with the same disibution as. Let, where are assumed to be fixed constants. It is not hard to show that is a convex function on for.in order to solve the above optimization problem, we employ the following result from the theory of majorization [16]. Lemma 5.1: If are exchangeable random variables and the multivariable, single-valued function is a symmeic Borel-measurable convex function, then the function is Schur convex. When [25], we have (65) (66) Then, from (63) and (66), the average can be written in a simplified way as (67)

13 2086 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 6, JUNE 2005 E. Proof of Theorem 4.1 Then To derive the, we start from the log-likelihood function in (32): const The second derivative of the log likelihood function w.r.t. is (70) call that the channel gain vector is assumed to have i.i.d complex circular symmeic Gaussian elements, i.e.,,, and,. Thus, we have The expectation of the above is (68) (71) Write the th block can be computed as, where As a result, we have, where the th element of is given by (69)

14 LIU et al.: TIMING ESTIMATION IN MULTIPLE-ANTENNA SYSTEMS OVER RAYLEIGH FLAT-FADING CHANNELS 2087 Similarly, we also have of is given by, where the th element for,. As a result, we note that does not depend on the noise. The of the timing estimation is given as (74) for,. Letting, then (72) where the second equality is obtained by using, and the third equality is obtained from the property [26]. Let. Since differentiating the left side twice w.r.t. gives const Then, using the above equality, the th element of becomes (73) REFERENCES [1] I. E. Telatar, Capacity of multi-antenna Gaussian channels, Eur. Trans. Telecom., vol. 10, pp , Nov [2] G. J. Foschini, Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas, Bell Labs. Tech. J., vol. 1, no. 2, pp , [3] D. M. Dlugos and R. A. Scholtz, Acquisition of spread specum signals by an adaptive array, IEEE Trans. Acoust., Speech, Signal Process., vol. 37, no. 8, pp , Aug [4] P. Shamain and L. B. Milstein, Acquisition of direct sequence spread specum signals with correlated fading, IEEE J. Sel. Areas Commun., vol. 19, no. 12, pp , Dec [5] M. Z. Win and R. K. Mallik, Acquisition of spread specum signals in Rayleigh fading, in Proc. IEEE GLOBECOM, vol. 1, 2002, pp [6] Y. Zhang and S. L. Miller, Acquisition performance in ansmission diversity CDMA systems, in Proc. IEEE GLOBECOM, vol. 2, 2001, pp [7] H. Krim and M. Viberg, Two decades of array signal processing research, IEEE Signal Process. Mag., vol. 13, no. 4, pp , Jul [8] A. L. Swindlehurst and P. Stoica, Maximum likelihood methods in radar array signal processing, Proc. IEEE, vol. 86, no. 2, pp , Feb [9] A. L. Swindlehurst, Time delay and spatial signature estimation using known asynchronous signals, IEEE Trans. Signal Process., vol. 46, no. 2, pp , Feb [10] A. Dogandzic and A. Nehorai, Space-time fading channel estimation and symbol detection in unknown spatially correlated noise, IEEE Trans. Signal Process., vol. 50, no. 3, pp , Mar [11] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Englewood Cliffs, NJ: Prentice-Hall, [12] H. V. Poor, An Inoduction to Signal Detection and Estimation. New York: Springer-Verlag, [13] F. M. Gardner, Demodulator ference covery Techniques Suited for Digital Implementation, Eur. Space Agency Final p., ESTEC Conact 6847/86/NL/DG, May [14] M. Moeneclaey, A fundamental lower bound on the performance of practical joint carrier and bit synchronizers, IEEE Trans. Commun., vol. 32, no. 9, pp , Sep [15] M. Merkle and L. Peovic, On Schur-convexity of some disibution functions, Publ. Inst. Math., vol. 56, no. 7, pp , [16] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications. New York: Academic, [17] B. Hassibi and B. M. Hochwald, How much aining is needed in multiple-antenna wireless links, IEEE Trans. Inf. Theory, vol. 49, no. 4, pp , Apr [18] J. G. Proakis, Digital Communications, Fourth ed. Englewood Cliffs, NJ: Prentice-Hall, [19] W. Rudin, Principles of Mathematical Analysis. New York: McGraw- Hill, [20] B. L. Hughes, Differential space-time modulation, IEEE Trans. Inf. Theory, vol. 46, no. 7, pp , Nov [21] V. Tarokh and H. Jafarkhani, A differential detection scheme for ansmit diversity, IEEE J. Sel. Areas Commun., vol. 18, no. 7, pp , Jul [22] B. M. Hochwald and W. Sweldens, Differential unitary space-time modulation, IEEE Trans. Commun., vol. 48, no. 12, pp , Dec [23] H. Cramér, Mathematical Methods of Statistics. Princeton, NJ: Princeton Univ. Press, [24] E. K. P. Chong and S. H. Zak, An Inoduction to Optimization. New York: Wiley, [25] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. New York: Academic, [26] A. Graham, Kronecker Products and Maix Calculus With Application. New York: Wiley, 1981.

15 2088 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 6, JUNE 2005 Yong Liu received the B.Eng. degree in information engineering from Zhejiang University, Hangzhou, China, in 1999 and the M.Sc. degree in elecical and computer engineering from the University of Florida, Gainesville, in He is currently working toward the Ph.D. degree with the Department of Elecical and Computer Engineering, the University of Florida. His research interests include wireless communications, networking, and signal processing. Tan F. Wong (SM 03) received the B.Sc. degree (with first-class honors) in eleconic engineering from the Chinese University of Hong Kong in 1991 and the M.S.E.E. and Ph.D. degrees in elecical engineering from Purdue University, West Lafayette IN, in 1992 and 1997, respectively. He was a research engineer working on the highspeed wireless networks project with the Department of Eleconics at Macquarie University, Sydney, Ausalia. He also served as a post-doctoral research associate with the School of Elecical and Computer Engineering, Purdue University. Since August 1998, he has been with the University of Florida, where he is currently an associate professor of elecical and computer engineering. Dr. Wong serves as Editor for Wideband and Multiple Access Wireless Systems for the IEEE TRANSACTIONS ON COMMUNICATIONS and as the Editor for the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. Ashish Pandharipande was born in India in He received the B.Eng. degree in eleconics and communications engineering from the College of Engineering, Osmania University, Osmania, India, in He pursued his graduate education at the University of Iowa, Iowa City, where he received the M.Sc. degrees in elecical and computer engineering and mathematics in 2000 and 2001, respectively, and the Ph.D. degree in elecical and computer engineering in He was with the Elecical and Computer Engineering Department, University of Florida, Gainesville, as a post-doctoral researcher in He is currently a senior researcher on the technical research staff at Samsung Advanced Institute of Technology, Giheung, Korea. His research interests are in the areas of Multicarrier (OFDM) and MIMO communications, multirate signal processing, and signal processing techniques in communications.