Doubly Selective Channel Estimation Using ... - Semantic Scholar

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 3, MARCH 2010

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Doubly Selective Channel Estimation Using Exponential Basis Models and Subblock Tracking Jitendra K. Tugnait, Fellow, IEEE, Shuangchi He, and Hyosung Kim

Abstract—Three versions of a novel adaptive channel estimation approach, exploiting the over-sampled complex exponential basis expansion model (CE-BEM), is presented for doubly selective channels, where we track the BEM coefficients rather than the channel tap gains. Since the time-varying nature of the channel is well captured in the CE-BEM by the known exponential basis functions, the time variations of the (unknown) BEM coefficients are likely much slower than those of the channel, and thus more convenient to track. We propose a “subblockwise” tracking scheme for the BEM coefficients using time-multiplexed (TM) periodically transmitted training symbols. Three adaptive algorithms, including a Kalman filtering scheme based on an assumed autoregressive (AR) model of the BEM coefficients, and two recursive least-squares (RLS) schemes not requiring any model for the BEM coefficients, are investigated for BEM coefficient tracking. Simulation examples illustrate the superior performance of our approach over several existing doubly selective channel estimators. Index Terms—Adaptive channel estimation, basis expansion models, doubly selective channels, Kalman filtering, recursive least-squares.

I. INTRODUCTION UE to multipath propagation and Doppler spread, wireless channels are characterized by frequency- and timeselectivity [1]. Knowledge of channel state information (CSI) is often a prerequisite for many physical layer approaches. Accurate modeling of the temporal evolution of the channel plays a crucial role for estimation and tracking purpose. Among various models, the autoregressive (AR) process, particularly the first-order AR model, is often regarded as a tractable model to describe a time-varying channel, where the channel is assumed to be Markovian, that is, for the current channel sample, the effect of channel samples other than the immediately preceding one is negligible [2]. This Markovian assumption has been justified for Rayleigh fading channels in [2], by considering the mutual information between successive channel samples. The AR models, however, have their own drawback. When time-multiplexed (TM) training

D

Manuscript received January 21, 2009; accepted October 01, 2009. First published November 06, 2009; current version published February 10, 2010. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ut-Va Koc. This work was supported by the NSF by Grants ECS-0424145 and ECCS-0823987. The material in this paper was presented in part at the 50th IEEE Global Telecommunications Conference (GLOBECOM), Washington, DC, November 2007 and the 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, Las Vegas, NV, March–April 2008. J. K. Tugnait and H. Kim are with the Department of Electrical and Computer Engineering, Auburn University, Auburn, AL 36849 USA. S. He is with the School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TSP.2009.2036047

is used, channel tracking may not perform well during information-symbol transmissions (information sessions) since the information data are unknown. During information sessions, channel estimates can only be obtained based on the results from previous training sessions [3]. This channel prediction strategy is not effective in a fast-fading channel, where the AR model may lead to high estimation variance resulting in erroneous symbol detections [4]. Potential solutions lie in exploiting the detected symbols for channel tracking: In [5], a Kalman filter is used in a decision-feedback mode during information sessions; per survivor processing (PSP) is used in [6], which embeds data aided channel estimation into the Viterbi algorithm; in [7], joint channel estimation and data detection is implemented via extended Kalman filtering. Although channel tracking can be improved by such means during information sessions, error propagation due to incorrect detections can be pronounced for fast-fading channels. More accurate channel modeling becomes necessary for tracking fast-fading channels. Recently, basis expansion models (BEMs) have been widely investigated to represent doubly selective channels in wireless applications [10]–[14], in which the time-varying channel taps are expressed as superpositions of time-varying basis functions in modeling Doppler effects, weighted by time-invariant coefficients. Candidate basis functions include complex exponential (Fourier) functions [10], [13], polynomials [11], wavelets [12], and discrete prolate spheroidal sequences [14]. In contrast to AR models that describe temporal variation on a symbol-by-symbol update basis, a BEM depicts the evolution of the channel over a period (block) of time. Intuitively the coefficients of the BEM approximations should evolve much more slowly in time than the channel tap gains, and hence are more convenient to track in a fast-fading environment. In this paper, we present a tracking approach for doubly selective channels using TM training, which exploits the complex exponential BEM (CE-BEM) for the overall channel variations in each block of received signal. The slow-varying BEM coefficients, rather than the fast-varying channel taps, are tracked and updated at each training session; during information sessions, channel estimates are generated by the CE-BEM using the estimated BEM coefficients. Since the time variations of the channel are typically well described by the CE-BEM, better performance can be achieved for fast-fading channels than using symbol-wise AR models. In addition, as the detected information symbols are not involved in the tracking procedure, the typical issue of error propagation is avoided for fast-fading channels. Three adaptive algorithms are considered to track the BEM coefficients “subblockwise”. We first assume that the BEM coefficients (rather than the time-varying channel tap gains) follow

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a first-order AR model, and then Kalman filtering is used to track the coefficients. This first-order AR assumption, however, is not necessarily true and possibly incurs significant modeling errors in estimation. We then seek adaptive channel estimation schemes with no a priori models for the BEM coefficients. Two adaptive filtering algorithms with finite memory are considered: the exponentially weighted recursive least-squares (RLS) algorithm and the sliding-window RLS algorithm. Decision-directed channel tracking using a polynomial BEM has been investigated in [15], where the BEM coefficients are updated every block via the RLS algorithm within a sliding window. Channel estimation using Kalman filtering and polynomial or CE-BEMs for OFDM systems has been explored in [16]–[18], among which decision-directed tracking is considered in [16] and [18]. All these contributions consider block-by-block updating unlike our contribution where we exploit subblockwise updating. The distinction is as follows: One block comprises several subblocks. For parameter identifiability, one needs the number of subblocks to be at least as large as the number of basis functions used for channel modeling. Data in just one subblock do not satisfy the parameter identifiability requirements. Also, unlike the blockwise schemes where the receiver has to collect the whole block of data in order to generate channel estimates and perform equalization, in the proposed approaches the receiver is able to accomplish the two tasks after every subblock. We focus on LS and related channel estimation approaches, as in [10]–[19]. In [13] and [26], minimum mean-square error (MMSE) channel estimation approaches using CE-BEMs have also been considered where one needs knowledge of channel tap correlation function, including the power delay profile. No such knowledge is assumed in this paper (indeed, typically it would be unavailable); therefore, we do not consider MMSE-related channel estimation and confine our attention to LS channel estimation. The remainder of the paper is organized as follows. Section II introduces the channel model, including the AR and the CE-BEM representations. The Kalman filtering-based tracking of the BEM coefficients based on an AR assumption on the BEM coefficients is the subject of Section III. We then discuss the finite-memory RLS tracking in Section IV, which assumes no a priori models for the BEM coefficients. Computational complexity aspects of the proposed approaches are discussed in Section V. Simulation examples are presented in Section VI, and Section VII concludes the paper. Notations: Superscripts and denote the complex conjugation, transpose, complex conjugate transpose, and Moore-Penrose pseudoinverse, respectively. is the identity matrix, and is the null vector. We reserve for integer ceiling and for integer floor. The symbol denotes expectation, denotes the Kronecker product, and is the trace of a square matrix . is the Kronecker delta, that is, for , and otherwise. II. SYSTEM MODEL AND BACKGROUND A. Received Signal Consider a scalar time- and frequency-selective (doubly selective), finite impulse response (FIR) linear channel.

Let denote a scalar sequence that is input to the time-varying channel with discrete-time response (channel response at time to a unit input at time ). Then the symbol-rate noisy channel output is given by (1) is white complex Gaussian noise with zero mean where and variance . We assume that represents a widesense stationary uncorrelated scattering (WSSUS) channel [1]. In TM training schemes, can be either a training or an information symbol. We assume that if it is an information symbol, is independent and identically distributed (i.i.d.) with zero mean and variance . B. Channel Models 1) Autoregressive (AR) Model for Channel Variations: It is possible to accurately represent a WSSUS channel by a large order AR model; see [4], [7] and references therein. Let (2) where model,

is , for

column vector. Then a is given by

th order AR

(3)

AR coefficient matrices, where s are the is also and the i.i.d. driving noise sequence is zero-mean with identity covariance matrix. Suppose that we know the correlation function for lags . The following Yule-Walker equation holds for (3) [8]: (4) , and the fact that Using (4) for , one can estimate s. Using the estimated s and (4) for one can find , from which one can find (nonunique) by computing its “square root” [9, p. 358]. As noted in [4, Sec. II.B], this procedure amounts to matching the first lags of the autocorrelation function of . In [4], [7] only AR(1) or AR(2) models have been used. In Section VI we will use AR models for some simulation comparisons where various channel tap gains are assumed to be mutually statistically independent. In this case we have an independent AR process for each channel tap gain. Furthermore, following [4], [7], we only consider first-order AR model for each tap gain, given by (5) is the AR coefficient for the th channel tap gain and where the driving noise is zero-mean white with variance .

TUGNAIT et al.: DOUBLY SELECTIVE CHANNEL ESTIMATION

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If we assume that is also zero-mean with variance , then one picks (to match the correlation functions at lags 0 and 1) [3]

that by using the over-sampled CE-BEM, the basis functions are no longer orthogonal (but can be orthogonalized).

(6)

Here we summarize the time-multiplexed training approach of [13]. In Section VI we provide simulation comparisons with results of [13]. We also employ the TM training scheme of [13] in our subblock-wise tracking approaches. In [13] each transmitted block consisting of symbols for is segmented into subblocks of equal length symbols. Every subblock consists of an information session of symbols and a succeeding training session of symbols . If denotes a column-vector composed of , then is arranged as

(7) , and in (3). It must be noted that in practice, one would not know . 2) Complex Exponential Basis Expansion Model (CE-BEM): In contrast with the symbol-wise AR model, a BEM assigns temporal variations to basis functions, facilitating parsimonious representation of rapidly fading channels with coherence time as small as a few tens of symbols [10]. In the CE-BEM [10], [13], [19], over the th block consisting of an observation window of symbols, the channel impulse response is represented by (for and )

Using (5)–(7), we have

(8) where one chooses (9) (10) (11) (12) and are the delay spread and the Doppler spread, respectively, and is the symbol duration. The BEM coefficients ’s remain invariant during each block, but are allowed to change at the next block; the Fourier basis functions are common for every block. If the delay spread and the Doppler spread of the channel (or at least their upper bounds) are known, one can infer the basis functions of the CE-BEM [13]. Treating the basis functions as known parameters, estimation of a timevarying process is reduced to estimating the invariant coefficients over a block of symbols. Note that the BEM period is whereas the block size is symbols. If (e.g., or ), then the Doppler spectrum is said to be over-sampled [19] compared to the case where the Doppler spectrum is said to be critically sampled. In [10], [13] only (henceforth called critically sampled CE-BEM) is considered whereas [19] considers (henceforth called over-sampled CE-BEM). CE-BEM has a FIR structure in both time and frequency domains [20]. This unique time-frequency duality makes it a widely used model depicting the temporal variations of wireless channels. For , the rectangular window of this truncated discrete Fourier transform (DFT)-based model introduces spectral leakage [21]. The energy at each individual frequency leaks to the full frequency range, resulting in significant amplitude and phase distortion at the beginning and the end of the observation window [14]. To mitigate this leakage, the over-sampled CE-BEM with or 3 has been considered in [19]. Note

C. Block-Adaptive Channel Estimation Using CE-BEM [13]

(13) is a column of information where symbols and is a column of training symbols. Given the system model (1) and the CE-BEM (8), [13] has shown that for (the critically sampled CE-BEM) and , the optimal training session contains an impulse guarded by zeros (silent periods), which has the structure (14) . Thus, given a transmission block of size symbols have to be devoted to training and the remaining symbols are available for information symbols. Let denote the location of the training impulse (the nonzero ) of the th training session. By (1), at time with , the received signal (assuming timing synchronization) is given by with

(15) ’s, one can uniquely Using the CE-BEM (8) in these ’s via a least-squares approach. The channel estisolve for mates are given by the CE-BEM (8) using the estimated BEM coefficients. Remark 1: Identifiability: As shown in [13] and [26], one needs number of subblocks to be to uniquely identify the unknown BEM parameters. By (10), as increases, increases (in discrete steps) for a given value of . Since the subblock size , therefore, , as increases, also increases. Typically, one would first pick (which would fix with for oversampled CE-BEM), and then pick to satisfy ; for a given value of or does not depend upon . If turns out to be “too small,” training overhead is large since symbols out of are needed for training. On the other hand, suppose one first picks to achieve a certain training overhead . Then in order to have parameter identifiability, one needs , i.e., . In Section VI, in Example 1, we used s, Hz and . In this case there exist no integer with for which we can satisfy whereas satisfies this inequality when ( and ). This

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=

Fig. 1. Block/subblock structure: T KT for oversampled CE-BEM. CE-BEM, K

2

;K

= 1 for critically sampled

implies that for blockwise updating, we cannot use oversampled CE-BEM, only the less accurate critically sampled CE-BEM. D. Objectives Suppose that we collect the received signal over a time interval of symbols. We wish to estimate the time-varying channel using a channel model and TM training (such as that discussed in Section II-C and [13]), and subsequently estimate the information symbols using the estimated channel. For the CE-BEM, if we choose as the block size, then in general by (10) the number of basis functions will be very large requiring estimation of a large number of parameters, thereby degrading the channel estimation performance. If we divide into blocks of (much smaller) size each (see Fig. 1), and then fit the CE-BEM block by block, we need smaller ; however, ’s is now based on a shorter observation size estimation of of symbols which might also degrade channel estimation performance. Thus, one has to strike a balance between the estimation variance and the block size. Such considerations do not apply to the symbol-wise AR channel model fitting. In the sequel, in Section III, we propose a novel Kalman filtering-based subblockwise tracking approach to CE-BEM channel estimation where we update estimates of ’s every subblock based on all past training sessions. In Section IV we investigate RLS approaches that do not require any models for the BEM coefficients, but we still update estimates of ’s every subblock. As we shall see later, use of subblock updating avoids some of the limitations of blockwise estimation. In subblock tracking there is no longer a strict definition of the block size unlike blockwise estimation, allowing for a more flexible choice of parameters such as subblock size and training overhead. Intuitively, updating more often as in subblock tracking should be superior to (or at least as good as) less frequent updating as in blockwise estimation. As noted earlier in Remark 1 of Section II-C, the parameter identifiability requirement of for blockwise approach imposes certain restrictions on the selection of the block and subblock sizes, which, in turn, may preclude the use oversampled CE-BEM, only the less accurate critically sampled CE-BEM may be used. Our simulation results will illustrate this fact together with the fact that subblockwise updating encounters no such problem; see the results shown in Figs. 6 and Fig. 7. We also note that the same considerations as above prevent the use of in the blockwise scheme because this would

. Alternatively, to make the blockwise scheme lead to update the BEM parameters as frequently as the subblockwise schemes, one may take but to achieve parameter identifiability, one has to insert at least “clusters” of zero-padded training impulses (as in Section II-C). To make this more concrete, let us consider Example 2 of Section VI where we take and . With , one obtains ; note that for doubly selective channels, when using CE-BEM, minimum . Thus, we need 3 clusters of zero-padded training impulses with each cluster of length . Therefore, one assigns 51 symbols out 80 for training leaving just 29 information symbols—a poor choice resulting in poor spectral efficiency. Such a design has 63.75% training overhead whereas our proposed subblock schemes in Example 2 of Section VI have a training overhead of 21.25% (17 out of 80 symbols for training). III. CHANNEL ESTIMATION VIA SUBBLOCK-WISE KALMAN TRACKING By exploiting the invariance of the coefficients of the CE-BEM over each block (and hence each of the subblocks per block of symbols), we consider two overlapping blocks (each of symbols) that differ by just one subblock: the “past” block beginning at time and the “present” block beginning at time . Since the two blocks overlap so significantly, one would expect the BEM coefficients to vary only “a little” from the past block to the present overlapping one. Therefore, rather than estimating ’s anew with every nonoverlapping block as in Section II-C and [13], we propose to track the BEM coefficients (rather than the channel tap gains) subblock by subblock using a first-order AR model for their variations. In subblock tracking, we will use (8) for all times , not just the particular block of size symbols, by allowing ’s to change with time (subblockwise). the coefficients Stack the channel coefficients in (8) into vectors (16) (17) , respectively. The coefficients of size and and the coefficient vectors in (16) and (17) of the th overlapwill be denoted by , ping block and , respectively. Again, we emphasize that the th block and the st block differ by just one subblock. We further assume that the channel coefficients of each block follow an AR model. One could fit a general model with a high value of (as in Section II-B–1 for channel variations), but we seek a “simple” AR(1) model given by (18) is the AR coefficient matrix and the driving noise where vector is zero-mean white with identity covariance. Using the th channel tap gains over one block, define

(19)


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Consider two overlapping blocks that differ by just one subblock: and , with and , respectively, as the corresponding BEM coefficients. Define (20) Using (20), we further define

(21) (22) where

is

and

is

. It then follows that (23) (24)

Collecting all channel tap gains over one block, further define

(25) It then follows that (26) (27) where

(28) If (18) holds, then using the Yule-Walker equation we have (29) where using (26) and (27) we have

(30) (31) and and can be calculated using (19) and (25) if we know the channel correlation function (as defined in Section II–B–1). As

in Section II–B–1, this procedure results in matching the corat lags 0 and 1. [One could also fit relation function of a higher-order AR model for following the method discussed in Section II-B–1.] Typically will not be available. Therefore, to simplify we will assume that (implying that all tap gains have the same Doppler spectrum), and , leading to (32) If the channel is stationary (WSSUS) and coefficients ’s are independent (as assumed in [13]), then by (32) and Yule-Walker equations, we have (33), shown at the bottom of the page, and for a uniform power delay profile, where . Since the coefficients evolve slowly, we will have (but for tracking). It is seen from (33) that as increases, increases, and vice versa. Note that the results of (33) are similar to (6) except that in the former changes occur every subblock whereas in the latter the changes occur . every symbol. Note that (33) also requires knowledge of In order to avoid this, one can somewhat arbitrarily pick a value of such that but ; this has been done in, e.g., [31] (in a different but similar context). To gain more insight, let us consider a specific channel tap following the Jakes’ spectrum (also used in Section VI in simulation examples). When , and , one gets and for and 40, respectively; i.e., dependence between subblocks decreases with increasing . With held at 0.01, when and are halved to 200 and 100, respectively, (leading to ), one gets and 0.8791 for and 40, respectively; i.e., as increases, increases, and vice versa. When , and (leading to ), one gets and 0.9605 for and 40, respectively; i.e., as decreases, increases and vice versa. These numerical results are consistent with one’s intuition: faster channel variations call for smaller values (less dependence), and vice versa. Under this formulation, we do not need a “strict” definition of the block size —although the channel is still represented by ’s are updated (8) for arbitrary time , the BEM coefficients every subblock based on the training symbols. A key parameter now is the CE-BEM period , not the block size . Later we use (8) for all times , not just the particular block of size symbols, by allowing the coefficients ’s to change with time (subblockwise).

(33)

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A. Subblock Tracking Using Kalman Filtering

• Measurement update:

If at time the th subblock is being received, by (1), (8)–(12), and (16)–(17), the received signal can be written as (34) where . Treating (32) and (34) as the state and the measurement equations respectively, Kalman filtering can be applied to track the coefficient vector for each subblock. We will employ the TM training scheme proposed in [13] (see symbols) Section II-C) where each subblock (of equal length consists of a data block (of length symbols) and a succeeding training block (of length symbols). By (1), (8), and (15)–(17), the received signal at time is given by (recall that denotes the coefficient vector in (16) for the th subblock)

where the vector is the estimate of given the observations , and is the error covariance matrix of , defined as (41) Now we generate the channel for the entire th subblock by the estimate via the CE-BEM (8) as (42) . The definition of

for (35) We intend to use only training blocks for subblockwise channel tracking. Define

..

.

(36)

(37)

is similar to (16). B. Minimum Mean-Square Error Decision Feedback Equalizer (MMSE-DFE) [23], [25] Using the estimated channel, the information symbols are detected by an FIR MMSE-DFE [23], [25]. Given the lengths of the feedforward and the feedback filters as and , respectively, the estimate of the information symbol at time with equalization delay can be written as

and (38) By (35), it follows that (39) which forms a basis to estimate the BEM coefficients using training symbols (sessions). With this training scheme (optimal for critically sampled CE-BEMs), the measurement (34) is now simplified as (39). We have obtained a linear discrete-time system represented by (32) and (39). Treating (32) and (39) as the state and the measurement equations, respectively, the Kalman filtering is applied to track the coefficient vector for each subblock, via the following steps [22] (note that ): 1) Initialization:

(43) ’s and ’s are the taps of the feedforward and where the feedback time-varying filters at time , respectively, and is the hard decision of . The DFE output is also fed into the quantizer to obtain the symbol decision . Design of the filter coefficients for the chosen filter lengths and is considered in [25] for doubly selective channels, following the general approach of [23]. We follow the approach of [25] where we use the channel estimates obtained by (42) and valid over a subblock, in lieu of the unavailable true channel response , to design the MMSE-DFE; details are omitted and may be found in [25, Section III.B]. Using the estimated channel one may rewrite (1) as

(40) 2) Kalman recursion for • Time update:

• Kalman gain:

(44)

where the “effective” noise is now instead of as in (1). In order to account for the additional signal-dependent component in , for simulations presented in Section VI, we took the variance of in (44) to be instead of , the variance of .


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In [24], a different approach to the design of MMSE-DFE for doubly selective channels has been presented using a CE-BEM representation for the equalizer also (in addition to the channel). As shown in [25], compared to [24], the approach of [25] yields an improved bit error rate at a lower computational cost. Therefore, in this paper, we confine our attention to the approach of [25].

TABLE I THEORETICAL VALUES OF FOR EW-RLS CHANNEL TRACKING

IV. CHANNEL ESTIMATION VIA SUBBLOCK-WISE RLS TRACKING The Kalman tracking approach in Section III assumes that each BEM coefficient follows a first-order AR process [see (32)], which is not necessarily true, and possibly incurs modeling errors in channel estimation. In this section, we seek adaptive channel estimation schemes with no a priori models for the BEM coefficients. As before, we propose to track the BEM coefficients (rather than the channel tap gains) subblock by subblock; to this end, we investigate two RLS schemes. Since the CE-BEM (8) is periodic with period , the memory of the algorithm should be less than to avoid periodicity of the model influencing the estimation results as the actual channel is not periodic. Therefore, adaptive algorithms with finite memory are preferred. Two “finite-memory” algorithms based solely on the formulation (39) are considered in subblockwise channel tracking: the exponentially weighted RLS (EW-RLS) algorithm and the sliding-window RLS (SW-RLS) algorithm. A. Exponentially Weighted RLS Tracking Based on (39), we can apply the EW-RLS algorithm [27, Ch. 12] to tracking the channel BEM coefficients. In this approach, as applied to (39), one chooses to minimize the cost function (45) is a regularization parameter and is where the forgetting factor. [Note that the forgetting factor is used to give more weight to recent data and less weight to past data. For the subblockwise updating, we take to be (much) smaller than one (e.g., 0.65) although it is very close to one (e.g., 0.99) for the “standard” symbol-wise updating. More on this later in this section (and the Appendix) where we provide some guidelines for the selection of ]. Mimicking [27, Algorithm 12.3.1], the EW-RLS algorithm has the following steps: 1) Initialization (recall that ): (46) 2) RLS recursion: For

where

is the estimate of .

Now we generate the channel for the entire th subblock by the estimate via the CE-BEM (8) as (47) . The definition of is similar to (16). 1) Choice of for EW-RLS Tracking: Here we consider the theoretical choices of the forgetting factor for EW-RLS tracking. Under some reasonable conditions detailed in the Appendix (derivation of results are also provided therein), it turns out that for

(48) where is a prespecified parameter defined in the Appendix. Given the CE-BEM period , some theoretical choices of are shown in Table I for subblock sizes of = 20 and 40 (see Section II-C). In Section VI we compare the theoretical choices with simulation-based empirical choices. B. Sliding-Window RLS Tracking Compared with the EW-RLS algorithm that exponentially reduces the weight of past data, the SW-RLS algorithm only utilizes the data in a sliding window of length subblocks ([27], Chapter 12). Since the BEM coefficients are invariant within a block of symbols, we set so that only the present subblock and the past subblocks within one block are used for adaptation. To this end, each iteration consists of a downdating stage that removes the oldest received subblock training segment from the window, and an updating stage that inserts the next received subblock training segment into the window. The cost function in this case is (49) where if , we set , and is a regularization parameter. The SW-RLS algorithm has the following steps [27, Problem 12.7]: 1) Initialization: For , conduct channel tracking as in the exponentially weighted RLS algorithm, but with . Then set

given the observations 2) RLS recursion: For

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TABLE II EW-RLS: FLOP COUNT FOR CHANNEL ESTIMATION OVER ONE SUBBLOCK OF

m

TABLE III FLOP COUNT FOR CHANNEL ESTIMATION OVER ONE BLOCK OF

Downdating:

Updating:

M = Q(L + 1))

SYMBOLS (

T

SYMBOLS

matrix has flops. When the explicit inverse of matrix with matrix , i.e., , is solved by LU factorization using Gaussian elimination with partial pivoting [28, p. 121], one needs flops. The flop count for various steps in the EW-RLS algorithm is shown in Table II. The total flop count for various approaches is summarized in Table III where the counts are derived following the procedure used to deduce Table II. Discussion of specific numbers and relative complexity is deferred to Section VI where we revisit the flop count in the context of specific examples and numerical values. VI. SIMULATION EXAMPLES A. Example 1

Now

is the estimate of based on data . Channel estimates are generated for the th subblock using (47) by setting . V. COMPUTATIONAL COMPLEXITY In this section we compare computational complexity of the three proposed schemes and the block-adaptive scheme of Section II-C using the floating point operation (flop) counts for channel estimation over one block (equivalently, subblocks). [As in [28] a flop is one floating point operation.] Following [28], general multiplication of matrix and

A random time- and frequency-selective Rayleigh fading channel is considered. We assume is zero-mean, complex Gaussian with , the same for all taps implying a uniform power delay profile. We take (3 taps) in (1), and . For different ’s, ’s are mutually independent and satisfy Jakes’ model. To this end, we simulate each single tap following [29], [14, a correction in the Appendix]. We consider a communication system with carrier frequency of 2 GHz, data rate of 40 kBd (kilo-Bauds), therefore, s, and a varying Doppler spread in the range of 0 to 400 Hz, or the normalized Doppler spread from 0 to 0.01 (corresponding to a maximum mobile velocity from 0 to 216 km/h).


The additive noise was zero-mean complex white Gaussian. The (receiver) SNR refers to the average energy per symbol over one-sided noise spectral density. The TM training scheme of [13] described in Section II-C is adopted, where during information sessions the symbols are transmitted using the quadrature phase-shift keying (QPSK) modulation with unit power. The training session is described by (14) (of length symbols) with so that the average symbol power of training sessions is equal to that of information sessions. We first select the period of the CE-BEM symbols, and, hence, by (10) corresponding to maximum Hz. Note that even as we vary , the value of used is held fixed at corresponding to the assumed maximum Hz. We set the subblock size or 40 symbols. Six estimation and tracking schemes are compared: 1) The blockwise channel estimation scheme in [13] (see Section II-C), where the transmitted symbols are segmented into consecutive blocks each of symbols. Every block consists of subblocks as in Section II-C. For each nonoverlapping block, we estimate the BEM coefficients anew via a least-squares approach, and obtain the channel estimates over this block by the CE-BEM. We use an over-sampled CE-BEM with when subblock size with since the over-sampled CE-BEM approximates WSSUS channels much better than critically sampled CE-BEM [19], whereas for subblock size , an over-sampled CE-BEM is not possible (cannot obtain ), so that we choose . [For parameter identifiability, one needs , i.e., . With , Hz and , there exists no integer with for which we can satisfy . This implies that for blockwise updating with , we cannot use oversampled CE-BEM, only the less accurate critically sampled CE-BEM.] We used a regularized least-squares approach (with a regularization parameter , as in (45) and (49)) to estimate of BEM coefficients, and then obtain the channel estimates over this block via the CE-BEM. This is to be fair since the recursive subblock-tracking approaches use the same regularization parameter of . Furthermore, the results were worse in the over-sampled CE-BEM case when no regularization was used, caused by possible ill-conditioning of certain matrices when the basis functions are not orthogonal. In the figures, this scheme is denoted by blockwise LS. 2) The channel tracking scheme in [3] using the symbol-wise first-order AR model and Kalman filtering, where the time-varying channel tap gains are assumed to follow (5). Channel tracking is performed at training sessions only. During the information sessions, the receiver updates the channel via

We assume that only the upper bound of the Doppler spread is known. Then by (6) and Jakes’ model, for , where

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denotes the zeroth Bessel function of the first kind. This scheme is denoted by symbol-wise KF in the figures. 3) We also implement the approach of joint channel estimation and data detection via extended Kalman filtering in [7], where the channel tap gains are also described by the first-order AR model (5). For fairness, the turbo equalization procedure in [7] is omitted. The AR coefficient of the channel also follows (6), as suggested by [7]. This scheme is denoted by symbol-wise JKF. 4) Our proposed subblockwise tracking scheme using CE-BEM and Kalman filtering, which is denoted by subblockwise KF. For and 40, we take and 0.97, respectively (these values were selected empirically via simulations; the corresponding theoretical values via (33) are and 0.943, respectively). 5) Our proposed exponentially weighted RLS algorithm with . For and 40, we take the forgetting factor and 0.5, respectively (these values were determined empirically by simulations—see also the comments in Section IV-A-1 —later we compare theoretical choices discussed in Section IV-A-1 with our empirical choices). In the figures, this scheme is denoted by subblockwise EW-RLS. 6) Our proposed sliding-window RLS tracking with . We take by exploiting the over-sampled CE-BEM, so that for and 40 the window size is given by and 5 subblocks, respectively. It is denoted by subblockwise SW-RLS in the figures. We evaluate the performance of the six schemes by considering the normalized channel mean-square error (NCMSE) and the bit error rate (BER). The NCMSE is defined as

(50) is the “true” channel and is the estiwhere mated channel at the th run, among total runs. The BERs for the schemes other than the symbol-wise JKF [7] are evaluated by employing an MMSE-DFE described in Section III-B with feedforward length , feedback length , and equalization delay , using the channel estimates obtained by each estimation scheme. In each run, a symbol sequence of length 4000 is generated and fed into the random doubly selective channel. The first 200 symbols are treated as training overhead and thus discarded in evaluations, leading to in (50). All simulation results are based on 500 runs. Subblockwise KF is initialized as in (40) and the EW-RLS and SW-RLS algorithms are initialized as in (46), i.e., they all are initialized over one subblock; however, in performance evaluations the first 200 symbols (one block when ) were discarded. In Figs. 2 and 3, the performances of the six schemes over different Doppler spread ’s are compared. We set dB and symbols, so that 25% of the transmitted symbols are dedicated to training. As the Doppler spread increases from 0 to 400 Hz, in the case of the two tracking schemes based on the symbol-wise AR model (symbol-wise

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Fig. 2. Normalized channel mean-square error NCMSE (50) versus Doppler spread f for SNR = 20 dB, subblock size m = 20, CE-BEM period T = 400.

20, CE-BEM period T = 400.

KF and JKF) [3], [7], we can clearly see that their performances deteriorate sharply due to the modeling inadequacies of the symbol-wise AR representation. In contrast, the NCMSEs and BERs of the four schemes based on CE-BEM vary more gradually with increasing Doppler spreads, because the modeling error of the CE-BEM is much smaller. Noting that the CE-BEM used in the schemes is based solely on the maximum normalized Doppler shift , it follows that the performances of the CE-BEM-based schemes are not sensitive to the “actual” Doppler spread if it is no larger than the assumed maximum value. That is, we do not have to know the exact Doppler spread of the channel—an upper bound of it suffices. Since more parameters are involved in the CE-BEM-based schemes and hence result in higher estimation variance, these schemes are slightly inferior to the symbol-wise AR-based schemes for small values of ; but as increases, the CE-BEM-based schemes outperform the other two. Our subblockwise LS schemes also outperform the blockwise LS estimation of [13] whereas the subblockwise Kalman tracking scheme is only slightly better than the blockwise LS scheme when BER is the performance criterion and is worse when average channel MSE is the performance criterion. It is worth pointing out that superior average channel MSE does not necessarily translate into superior average BER since a monotone relationship between average MSE (averaged over all taps and runs) and average BER does not necessarily exist. The two finite-memory RLS schemes have similar performances, and they are both better than the subblockwise Kalman tracking scheme, since the latter assumes the BEM coefficients follow a first-order AR model and this may introduce additional modeling errors. In Figs. 4 and 5, the performances of the six schemes, with the Doppler spread and the subblock size symbols, are compared for different SNRs. Note that while subblockwise approaches produce channel estimates and detected bits every symbols, blockwise approach does so every symbols. In such a fast-fading environment,

our LS-based subblockwise tracking schemes achieve much better performance than the other schemes at higher SNRs. As in Figs. 2 and 3, although subblockwise Kalman tracking yields channel MSE that is higher than that for blockwise LS scheme, its BER is distinctly better. [To further investigate this “counter-intuitive” behavior, we plotted histograms (not shown in the paper) of the channel MSE and the BER for the two schemes (LS and KF) for a specific case corresponding to dB in Figs. 4 and 5. The histograms show that the channel MSE of the LS scheme has a spread around the mean value that is more than twice that for the KF scheme. This, in turn, translates into a larger BER for the LS scheme in quite a few runs, compared to the KF scheme.] In Figs. 6 and 7, longer information sessions are used to achieve a more spectrally efficient transmission, where we take the subblock size symbols (12.5% of the symbols are now devoted to training). For the blockwise LS scheme, an error floor can be seen due to spectral leakage (since an over-sampled CE-BEM cannot be used in this case as discussed in Remark 1 of Section II-C). Our subblockwise schemes maintain satisfactory performances, while the two symbol-wise AR-based schemes have severely deteriorated performances. [One could use a regularized blockwise LS solution with oversampled CE-BEM even though parameter identifiability does not hold. We did try this and the obtained results were very close to that shown for the blockwise LS scheme with critically sampled CE-BEM—they are not shown in Figs. 6 and 7.] Table IV shows the specific flop counts for various approaches corresponding to the results shown in Figs. 2–5. Considering Figs. 2–5 and Table IV, subblockwise EW-RLS scheme appears to provide a good complexity-versus-performance trade-off at higher SNRs (e.g., better than an order of magnitude improvement in BER for SNR dB at the cost of an increase of flop counts by a factor of 5.5) whereas at lower SNR values such an increase in flop count may not warrant the use of subblockwise tracking schemes.

Fig. 3. BER versus Doppler spread f for SNR = 20 dB, subblock size m =


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Fig. 6. NCMSE versus SNR for f Fig. 4. NCMSE versus SNR for f

= 400 Hz, m = 20; T = 400.

Fig. 7. BER versus SNR for f Fig. 5. BER versus SNR for f

= 400Hz; m = 20; T = 400.

In Figs. 8 and 9, we compare our subblockwise schemes by using different CE-BEM periods . Now we take the period of the CE-BEM , as well as as in the previous cases. By (10), is required for while for we need . Fewer BEM coefficients can reduce the computational complexity of our tracking scheme, with a (small) loss in performance, as can be seen in Figs. 8 and 9. Now we turn to a comparison of the theoretical choices for the forgetting factor for the EW-RLS algorithm (discussed in Section IV-A-1) with simulation-based empirical choices. In Fig. 10, the channel estimation performance of the EW-RLS algorithm is shown for the subblock sizes of and based on 100 Monte Carlo runs for the simulation example discussed earlier, under different values of . In our simulations discussed earlier we picked the forgetting factor and 0.5 for and 40 respectively, which satisfy the theoretical condition in (48). Indeed, yields

= 400 Hz, m = 40; T = 400.

= 400 Hz, m = 40; T = 400.

that are close to the values suggested values of by the minima in Fig. 10. B. Example 2 The random fading channel in this example is as in Example 1, except that compared to Example 1, we change the channel length parameter to 8 (from 2) leading to 9 taps, and the power delay profile now is exponential satisfying where the constant is . We consider a communicapick to satisfy tion system with carrier frequency of 2 GHz, data rate of 0.1 MBd (mega-Bauds), therefore s, and a Doppler spread Hz, or the normalized Doppler spread (corresponding to a maximum mobile velocity of 135 km/h). The training session is described by (14) (of length symbols) with so that the average symbol power of training sessions is equal to that of information sessions. We select the period of the CE-BEM symbols, and hence by (10) and a block size

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TABLE IV COMPARATIVE FLOP COUNT FOR CHANNEL ESTIMATION OVER ONE BLOCK OF T

Fig. 8. NCMSE versus SNR for f T or 400.

= 200

Fig. 9. BER versus SNR for T or 400.

= 200

f

SYMBOLS

10. NCMSE versus forgetting factor for EW-RLS with m = 20 or 40; = 400 Hz, m = 20, and CE-BEM period Fig. SNR = 20 dB, 3-tap channel with normalized Doppler spread f T = 0:01.

= 400 Hz, m = 20, and CE-BEM period

blockwise SW-RLS. The BERs are evaluated by employing an MMSE-DFE described in Section III-B with feedforward length , feedback length , and equalization delay , using the channel estimates obtained by each estimation scheme. For the subblockwise KF scheme we picked and for the subblockwise EW-RLS scheme we picked . In Figs. 11 and 12, the performances of the four schemes are compared for different SNRs. Superiority of the subblockwise schemes over the blockwise scheme is evident at higher SNRs. Table IV shows the specific flop counts for various approaches corresponding to the results shown in Figs. 11 and 12, and as in Example 1, subblockwise EW-RLS scheme appears to provide a good complexity-versus-performance trade-off at higher SNRs (e.g., better than an order of magnitude improvement in BER for SNR dB at the cost of an increase of flop counts by a factor of 3.68) whereas at lower SNR values such an increase in flop count may not warrant the use of subblockwise tracking schemes. VII. CONCLUSION

for oversampled CE-BEM. We set the subblock size symbols. Four estimation and tracking schemes are compared: blockwise LS, subblockwise KF, subblockwise EW-RLS and sub-

Three versions of an adaptive channel estimation approach, exploiting TM training and subblockwise tracking, were proposed for frequency-selective and time-varying channels. We employ the CE-BEM to describe the channel. Rather than


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assumption for the BEM coefficients used in the latter scheme may introduce additional modeling errors. Finally, unlike the blockwise schemes where the receiver has to collect the whole block of data in order to generate channel estimates and perform equalization, in the proposed approaches the receiver is able to accomplish the two tasks after every subblock. APPENDIX Here we provide details regarding the theoretical choices of the forgetting factor for EW-RLS tracking, first discussed in Section IV-A-1. The cost function for the EW-RLS algorithm as can be rewritten for “large” (51) Fig. 11. MSE versus SNR for f .

800

= 250 Hz, T = 10 s, m = 80; T = denote the number of subblocks on which we would like Let to base the estimate of the BEM coefficients . It is clear that is periodic with period as the CE-BEM is periodic with period , that is, for any integer . In practice, we would like to have the memory length (in symbols) to be less than the model period (recall that the channel is by no means periodic) so that there are no deleterious effects due to the use of (8) for all time, i.e., in order to avoid this periodicity. Let us pick . What other restriction should we impose on ? A least-square solution for to minimize is given by [30, p. 796] (52) where (53)

Fig. 12. BER versus SNR for f .

800

(54)

= 250 Hz, T = 10 s, m = 80; T =

track each channel tap gain, we estimate the BEM coefficients subblock by subblock, and then (re)generate the channel via the CE-BEM with the estimated BEM coefficients. In this way, the modeling mismatch introduced by the conventionally used symbol-wise AR channel model can be greatly reduced, and hence better performance can be achieved in fast-fading environments. Three adaptive algorithms were proposed for this tracking task: the Kalman filtering with a first-order AR modeling of the BEM coefficients, the exponentially weighted RLS algorithm, and the sliding-window RLS algorithm, where the latter two assume no a priori models for the channel BEM coefficients. Simulation results showed superior performance of our subblockwise tracking approach compared to the conventional symbol-wise or blockwise estimation schemes. Also, the two subblockwise RLS scheme with finite memory outperform the subblockwise Kalman tracking scheme, since the AR

In order to analyze the behavior of we need a model for for all subblocks . To this end, for the analysis presented in this subsection, we assume the following “simplified” model [recall (39)]: (55) where

is the “true” BEM coefficient vector satisfying

(56) That is, there exist some true BEM parameters that are “fixed” over one BEM period of symbols (therefore, subblocks), and are allowed to change over nonoverlapping periods. In this setup, we are trying to estimate the most recent

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true BEM coefficient vector . influenced by Using the periodicity for and as (for )

via

without it being unduly

, we can rewrite (53)

(57)

Similarly, using (55), (56) and (57), after some manipulations, we obtain (58) (59) From (52), (57), and (58), we have

(60) For memory length considerations, we will consider only the noisefree case, and hence, we set in (60). With this restriction together with , we obtain (61) For WSSUS channels, hence, norm in estimating

and via

have the same statistics, . The normalized error is, therefore, given by

(62) where is prespecified for the We would like to keep desired performance. This leads to (63) REFERENCES [1] T. S. Rappaport, Wireless Communications: Principles and Practice, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 2002. [2] H. S. Wang and P.-C. Chang, “On verifying the first-order Markovian assumption for a Rayleigh fading channel model,” IEEE Trans. Veh. Technol., vol. 45, pp. 353–357, May 1996. [3] Z. Liu, X. Ma, and G. B. Giannakis, “Space-time coding and Kalman filtering for time-selective fading channels,” IEEE Trans. Commun., vol. 50, pp. 183–186, Feb. 2002. [4] C. Komninakis, C. Fragouli, A. H. Sayed, and R. D. Wesel, “Multi-input multi-output fading channel tracking and equalization using Kalman estimation,” IEEE Trans. Signal Process., vol. 50, no. 5, pp. 1065–1076, May 2002. [5] W. Chen and R. Zhang, “Estimation of time and frequency selective channels in OFDM systems: A Kalman filter structure,” in Proc. IEEE GLOBECOM’04, Dallas, TX, Nov. 29–Dec. 3 2004, vol. 2, pp. 800–803.

[6] R. Bosisio, M. Nicoli, and U. Spagnolini, “Kalman filter of channel modes in time-varying wireless systems,” in Proc. IEEE ICASSP’05, Philadelphia, PA, Mar. 28–23, 2005, vol. 3, pp. 785–788. [7] X. Li and T. F. Wong, “Turbo equalization with nonlinear Kalman filtering for time-varying frequency-selective fading channels,” IEEE Trans. Wireless Commun., vol. 6, pp. 691–700, Feb. 2007. [8] S. M. Kay, Modern Spectral Analysis: Theory and Application. Englewood Cliffs, NJ: Prentice-Hall, 1988. [9] P. Stoica and R. Moses, Spectral Analysis of Signals. Upper Saddle River, NJ: Pearson Prentice-Hall, 2005. [10] G. B. Giannakis and C. Tepedelenlio˘glu, “Basis expansion models and diversity techniques for blind identification and equalization of time-varying channels,” Proc. IEEE, vol. 86, pp. 1969–1986, Nov. 1998. [11] D. Borah and B. Hart, “Frequency-selective fading channel estimation with a polynomial time-varying channel model,” IEEE Trans. Commun., vol. 47, pp. 862–873, Jun. 1999. [12] M. Martone, “Wavelet-based separating kernels for array processing of cellular DS/CDMA signals in fast fading,” IEEE Trans. Commun., vol. 48, pp. 979–995, Jun. 2000. [13] X. Ma, G. B. Giannakis, and S. Ohno, “Optimal training for block transmissions over doubly selective channels,” IEEE Trans. Signal Process., vol. 51, no. 5, pp. 1351–1366, May 2003. [14] T. Zemen and F. Mecklenbräuker, “Time-variant channel estimation using discrete prolate spheroidal sequences,” IEEE Trans. Signal Process., vol. 53, no. 9, pp. 3597–3607, Sep. 2005. [15] D. Borah and B. Hart, “Frequency-selective fading channel estimation with a polynomial time-varying channel model,” IEEE J. Sel. Areas Commun., vol. 17, no. 11, pp. 1863–1875, Nov. 1999. [16] K. A. D. Teo, S. Ohno, and T. Hinamoto, “Kalman channel estimation based on oversampled polynomial model for OFDM over doubly selective channel,” in Proc. IEEE Workshop Signal Process. Adv. Wireless Commun., New York, Jun. 2005, pp. 116–120. [17] R. C. Cannizzaro, P. Banelli, and G. Leus, “Adaptive channel estimation for OFDM systems with Doppler spread,” in Proc. IEEE Workshop Signal Process. Adv. Wireless Commun., Cannes, France, Jul. 2–5, 2006. [18] P. Banelli, R. C. Cannizzaro, and L. Rugini, “Data-aided Kalman tracking for channel estimation in Doppler-affected OFDM systems,” in Proc. IEEE ICASSP 2007, Honolulu, HI, Apr. 2007, vol. 3, pp. 133–136. [19] G. Leus, “On the estimation of rapidly time-varying channels,” in Proc. European Signal Process. Conf., Vienna, Austria, Sep. 6–10, 2004, pp. 2227–2230. [20] A. M. Sayeed and B. Aazhang, “Joint multipath-Doppler diversity in mobile wireless communications,” IEEE Trans. Commun., vol. 47, pp. 123–132, Jan. 1999. [21] J. G. Proakis and D. G. Manolaks, Digital Signal Processing, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996. [22] M. D. Srinath, P. K. Rajasekaran, and R. Viswanathan, Introduction to Statistical Signal Processing With Applications. Upper Saddle River, NJ: Prentice-Hall, 1996. [23] N. Al-Dhahir and J. M. Cioffi, “MMSE decision-feedbck equalizers: Finite-length results,” IEEE Trans. Inf. Theory, vol. 41, no. 4, pp. 961–975, Jul. 1995. [24] I. Barhumi, G. Leus, and M. Moonen, “Time-varying FIR decision feedback equalization of doubly selective channels,” in Proc. 2003 IEEE GLOBECOM Conf., Dec. 2003, vol. 4, pp. 2263–2268. [25] L. Song and J. K. Tugnait, “On time-varying FIR decision feedback equalization of doubly selective channels,” in Proc. 2008 IEEE Int. Conf. Commun., Beijing, China, May 19–23, 2008, pp. 558–562. [26] I. Barhumi, G. Leus, and M. Moonen, “Estimation and direct equalization of doubly selective channels,” EURASIP J. Appl. Signal Process., vol. 2006, pp. 1–15. [27] A. H. Sayed, Fundamentals of Adaptive Filtering. New York: Wiley, 2003. [28] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins Univ. Press, 1996. [29] Y. R. Zheng and C. Xiao, “Simulation models with correct statistical properties for Rayleigh fading channels,” IEEE Trans. Commun., vol. 51, no. 6, pp. 920–928, Jun. 2003. [30] S. Haykin, Adaptive Filter Theory, 4th ed. Upper Saddle River, NJ: Prentice-Hall, 2001. [31] K. Huber and S. Haykin, “Improved Bayesian MIMO channel tracking for wireless communications: Incorporating a dynamical model,” IEEE Trans. Wireless Commun., vol. 5, pp. 2468–2476, Sep. 2006.


Jitendra K. Tugnait (M’79–SM’93–F’94) was born in Jabalpur, India, on December 3, 1950. He received the B.Sc.(Hons.) degree in electronics and electrical communication engineering from the Punjab Engineering College, Chandigarh, India, in 1971, the M.S. and the E.E. degrees from Syracuse University, Syracuse, NY, and the Ph.D. degree from the University of Illinois, Urbana-Champaign, in 1973, 1974, and 1978, respectively, all in electrical engineering. From 1978 to 1982, he was an Assistant Professor of Electrical and Computer Engineering, University of Iowa, Iowa City. He was with the Long Range Research Division of the Exxon Production Research Company, Houston, TX, from June 1982 to September 1989. He joined the Department of Electrical and Computer Engineering, Auburn University, Auburn, AL, in September 1989 as a Professor. He currently holds the title of James B. Davis Professor. His current research interests are in statistical signal processing, wireless and wireline digital communications, multiple sensor multiple target tracking, and stochastic systems analysis. Dr. Tugnait is a past Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, the IEEE TRANSACTIONS ON SIGNAL PROCESSING, and IEEE SIGNAL PROCESSING LETTERS. He is currently an Editor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS.

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Shuangchi He received the B.E. and M.S. degrees in electronic engineering from Tsinghua University, Beijing, China, in 2000 and 2003, respectively, and the Ph.D. degree in electrical engineering from Auburn University in August 2007. From August 2003 to August 2007, he was a Graduate Research Assistant and then a Vodafone Fellow with the Department of Electrical and Computer Engineering, Auburn University, Auburn, AL. His research interests include channel estimation and equalization, multiuser detection, and statistical and adaptive signal processing and analysis.

Hyosung Kim received the B.A. degree in electrical engineering from Soongsil University, Seoul, Korea, in 1996 and the M.S. degree in computer science from Korea National Defense University, Seoul, in 2004. He is currently working toward the Ph.D. degree in electrical engineering at Auburn University, Auburn, AL. Since 2007, he has been a Graduate Research Assistant with the Department of Electrical and Computer Engineering, Auburn University. His research interests include channel estimation and equalization, multiuser detection, wireless security, and statistical and adaptive signal processing and analysis.