Maximum likelihood blind channel estimation in the ... - IEEE Xplore

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 6, JUNE 1999

1559

Maximum Likelihood Blind Channel Estimation in the Presence of Doppler Shifts Hakan A. Cirpan and Michail K. Tsatsanis, Member, IEEE

Abstract— Transmitter/receiver motion in mobile radio channels may cause frequency shifts in each received path due to Doppler effects. Most blind equalization methods, however, assume time-invariant channels and may not be applicable to fading channels with severe Doppler spread. In this paper, we address the problem of simultaneously estimating the Doppler shift and channel parameters in a blind setup. Both deterministic and stochastic maximum likelihood methods are developed and iterative solutions proposed. The stochastic maximum likelihood solution is based on the modified version of the Baum–Welch algorithm, which originated in the study of hidden Markov models. The proposed methods are well suited for short data records appearing in TDMA systems. Identifiability and performance analysis issues are discussed, and Cram´er–Rao bounds are derived. In addition, some illustrative simulations are presented.

I. INTRODUCTION

A

DAPTIVE equalizers do not usually model the channel variations explicitly. In this way, they opt for generality at the expense of potential performance gains when a channel variation model is exploited. For example, coherent transmission in underwater channels became possible due to pioneering work of [4], where the channel variation due to platform motion was modeled as a Doppler shift, and a phaselocked loop (PLL) was introduced to complement the adaptive equalizer and compensate for the frequency shift. Compensation of the channel variations due to Doppler shifts is a crucial problem in achieving high-speed coherent communications. Traditionally, fast adaptive algorithms of the RLS family are employed in fading environments to track the channel variations and equalize the received symbols [1], [2]. Aided by periodic retraining and (when possible) diversity combining, the adaptive equalizer will track the channel variations if they are sufficiently slow (compared with the algorithm’s convergence time) [3]. This issue is even more important when blind channel estimation is considered. The very long convergence time of Manuscript received December 13, 1996; revised September 29, 1998. This work was supported by the National Science Foundation under Grant NSFNCR 9706658 and CAREER CCR-9733048 and the Army Research Office under Grant DAAG55-98-1-0224. Part of the results of this paper were presented at the 13th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, Nov. 3–6, 1996. The associate editor coordinating the review of this paper and approving it for publication was Prof. Pierre Comon. H. A. Cirpan was with the Electrical and Computer Engineering Department, Stevens Institute of Technology, Hoboken, NJ 07030 USA. He is now with the Department of Electrical Engineering, Instanbul University, Istanbul, Turkey. M. K. Tsatsanis is with the Electrical and Computer Engineering Department, Stevens Institute of Technology, Hoboken, NJ 07030 USA (e-mail: [email protected]). Publisher Item Identifier S 1053-587X(99)03653-3.

blind adaptive algorithms (e.g., [5]) renders them impractical for all but the most slowly fading channels. For this reason, the Godard algorithm [5] is mostly studied for time-invariant (TI) channels. As current trends of increased data rates and higher mobility persist, however, the need for explicitly taking into account the time-varying (TV) nature of the channel becomes more apparent. In this paper, we consider a channel model where a few resolvable paths exists, each with a different Doppler shift. This model can be thought of as a generalization of the model of [4], where all paths experienced the same frequency shift. It follows naturally in situations where a small number of remote dominant scatterers introduce time delays that change (approximately) linearly with time. Similar models have been widely used in radar processing [6] and are gaining attention in wireless communications [7]–[9]. There exist alternative viewpoints under different conditions. One approach regards channel’s taps as random processes [10], [11]. Another alternative approach would be an adaptive maximum likelihood (ML) sequence estimation method that models the channel coefficients as time varying [12]. However, these methods may not be preferable when more structured variations exist. In this paper, we are interested in blind solutions for the estimation of the multipath parameters and associated Doppler frequencies. We target TDMA1 systems with short data records, and hence, we avoid solutions based on the higher order statistics that require large data records [9]. We would rather focus on maximum likelihood (ML) solutions, develop the estimation algorithms, and derive Cram´er–Rao bounds. We treat both deterministic and stochastic ML approaches. The first approach (deterministic ML) results in joint estimation of the unknown parameter vector and the input sequence. The second approach, which is known as stochastic ML, treats the input as a random quantity and averages the cost function over the input’s p.d.f. Stochastic ML methods offer good performance but result in complicated cost functions with computationally demanding solutions [13]. Fortunately, at least in the TI case, the computationally efficient Baum–Welch algorithm can be used to solve that problem [15]. The Baum–Welch algorithm originated in the study of hidden Markov models (HMM) in the context of speech processing. It has been applied to communication [16], [17] and source separation [18] problems for estimating linear and nonlinear parameters. 1 Time-division

1053–587X/99$10.00  1999 IEEE

multiple access

1560

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 6, JUNE 1999

In the current framework, the Baum–Welch algorithm is not directly applicable. A HMM description of the channel is possible, but the p.d.f. of the observations at each state changes with time. For this reason, we have modified the algorithm to optimize the multipath parameters and Doppler frequencies at each step of the re-estimation procedure, as we will explain later. In this way, we avoid overparametrization, which is usually associated with HMM modeling of communication channels [19], [20]. The rest of the paper is organized as follows. In the next section, the model for the mobile radio channel with Doppler shifts is derived, and the problem statement is presented. In Section III, both deterministic and stochastic ML blind estimation methods are developed. The performance of the stochastic ML method is analyzed in Section IV, whereas identifiability issues for both DML and SML are discussed in Section V. Finally, some illustrative simulations are presented in Section VI.

constants for a symbol period, (4) yields the discrete-time model

II. PROBLEM STATEMENT In the sequel, we develop a model for the mobile radio channel with multipath, assuming (as a first approximation) that the path delays change linearly with time, i.e., the vehicle velocity with respect to the transmitter and the reflectors is constant. If we consider linear modulation, the transmitted signal is of the form

(5) is the deterministic correlation of the spectral where is assumed Gaussian. It is shaping pulse. The noise term is zero for white, provided that the correlation function . Otherwise, a prewhitening filter needs to be employed [21]. to process Assuming that the path delays change linearly with time, , we can rewrite (5) is the form i.e.,

(6) which can be further simplified by observing that is approximately constant with compared with the rate of change of the exponential. Therefore, the discrete-time equivalent model can be rewritten compactly as (7) where the TV discrete-time impulse response of the channel is (8)

(1) where carrier frequency; spectral shaping filter2; i.i.d., discrete-time, complex symbol sequence that takes values from a finite alphabet set . Due to the multipath environment, multiple copies of the arrive at the receiver with some transmitted signal attenuation and propagation delay (2)

are complex constants, where is the Doppler frequency associated with the and th path. is not a sufficient statistic for It should be noted that are zero [21]. Therefore, detection unless all the path delays a symbol spaced system will experience some SNR loss. Extensions to fractionally spaced systems are straightforward but will not be pursued in this paper. has infinite order in Although the impulse response general, we follow the common practice in communications and approximate it by a finite impulse response of some order , yielding the finitely parameterized discrete-time model of Fig. 1

and are, respectively, the attenuation and where is additive time delay corresponding to the th path, and white Gaussian noise. If we substitute (1) into (2), we obtain

(9) For

a

collection

received data points , the matrix formulation

of

(3) where

of (9) is obtained as

is the complex envelope

(10) (4) After demodulation, matched filtering, and sampling at the , as receiver and approximating 2 We

use subscript c to denote continuous-time signals.

where

.. .

(11)

CIRPAN AND TSATSANIS: MAXIMUM LIKELIHOOD BLIND CHANNEL ESTIMATION IN THE PRESENCE OF DOPPLER SHIFTS

1561

algorithm [21]. In the current problem, the parameters enter in a linear fashion (given ); however, the frequencies do not, requiring the solution of a nonlinear LS problem for each given . The Viterbi algorithm is still applicable. The following steps summarize the algorithm. and . Initialization: Start with initial estimates of Step 1: Employ linear least squares to estimate of

Fig. 1. Equivalent discrete-time model.

denotes Kronecker product, stands for transpose, , , , , , , , , , . and The estimation problem we deal with in this paper can be formulated as follows. Given the observations , estimate the complex constants and the Doppler shifts , or equivalently, estimate the parameter vector (12)

(16) in (14) and solve Step 2: Substitute using gradient descent to obtain the estimate of (17) is the estimate of after iterations, and is the where step size. provided from the th itStep 3: Given the estimate of eration, employ the computationally efficient Viterbi algorithm to obtain ML estimate for the input vector

III. BLIND ESTIMATION

(18)

The problem we address in this section is the ML estimation of TV channel parameters using only the received data (blind estimation). Two different assumptions on the input sequence lead to corresponding ML solutions. A. Deterministic Maximum Likelihood Blind estimation algorithms have been proposed for the TI case that do not make any statistical assumption on the input sequence and model it as a deterministic signal [22]–[25]. In this section, a maximum likelihood method is developed as an along these lines, which treats the input sequence unknown but deterministic quantity. Therefore, the p.d.f. of is also indexed by in addition to the unknown parameter vector . Similar methods have been widely used in array processing (see [26] and references therein) and are referred as deterministic ML (DML). They have also been proposed for estimating TI communication channels [23], [25]. If the noise is Gaussian, then the p.d.f. of is

Step 4: Repeat Steps 1–3 until the algorithm converges. Note that in the deterministic maximum likelihood method, the size of the unknown parameter vector increases as the observation interval increase. This implies that the parameter vector cannot be estimated consistently [14]. Finally, the step size influences the stability and rate of convergence of the algorithm. B. Stochastic Maximum Likelihood An alternative to the DML approach is to treat the input sequences as random quantities and not include them in the parameter set. In this case, the probability density function is indexed only by and is obtained after averaging over . Under the model assumptions made for , the probability density function of the received data sequence (indexed only by the unknown parameter vector ) can be written as

(13) and its negative log-likelihood function (after neglecting unnecessary terms) is given by (14)

(19)

ML estimation of the parameter vector and input sequence involves finding those values of and that minimize

Since the transmitted data sequence is not available at the can be estimated receiver, the unknown parameter vector , which can be from the probability density function over obtained, after averaging

(15)

(20)

Typically, the minimization problem of (15) is solved by alternatively minimizing with respect to (w.r.t.) and while keeping the other parameters fixed [27]. This is convenient in the TI case because the minimization w.r.t. is a linear LS problem that can be solved in closed form. Then, the minimization w.r.t. , given , is implemented using the Viterbi

is the set of all possible transmitted sequences. where Then, the ML estimator of is the one that minimizes the average negative log-likelihood (21)

1562

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 6, JUNE 1999

The direct minimization of (20) requires the evaluation of (19) over all possible transmitted sequences, and its complexity increases exponentially with the data length . In the sequel, we integrate ideas from the theory of HMM [15] in order to provide a computationally efficient solution to this minimization problem. Let us first develop an HMM description for the problem at hand. 1) HMM Framework: An FIR communications channel is described by a finite length state vector

by the means [16], [19], in which case, the HMM estimation procedures are directly applicable (Baum–Welch algorithm). Unfortunately, in this setup, the are time-varying, and hence, we cannot avoid means . Thereparametrizing the p.d.f. by the vector3 fore, the Baum–Welch algorithm has to be modified for the current scenario. 2) Baum–Welch Algorithm: The Baum–Welch or, equivalently, the expectation/maximization (EM) algorithm has successfully been applied in a diversity of problems, especially where there are multiple parameters and nonlinear likelihood functions. It is based on the maximization of an auxiliary function related to the Kullback–Leibler information measure. The auxiliary function is defined as a function of two set of , parameters

(22) takes on values from a finite alphabet of size Since each , it is clear that the channel can be described by a finite state states, i.e., the state machine of (23)

(30) , and is the th entry where . of The Markov model’s state sequence is not directly observed but can only be inferred from the (noisy) output measurements; hence, an HMM should be considered. In order to complete the HMM description, the following parameters of the HMM need to be defined. i) the transition probability of the underlying Markov chain (24) or, equivalently, the transition matrix , where , and for each ; ii) initial probability vector

,

The theorem that forms the basis for the Baum–Welch algorithm explains the reason why the Kullback–Liebler information measure can be used instead of the average likelihood. implies Theorem: . For the proof of the theorem, see [28]. The iterative algorithm for the evaluation of the estimates . At the th iteration, it starts with an initial guess takes estimate

from the

th iteration as a initial value

and uses it to find the re-estimate over

by maximizing

(31)

(25) iii) the p.d.f. of the observations given the state, i.e., (26)

This procedure is repeated until the parameters converge to a stable point. Let us now derive the auxiliary function for the estimation th of the TV channel. The auxiliary function at the iteration has the form

Then, the HMM is completely specified by the triplet . In the current setup, due to the i.i.d. and equiprobable nature , all permissible state transitions have the of the input same probability, i.e.,

To obtain the explicit form for the auxiliary function, we start with

(27)

(33)

if state leads to state and 0 otherwise. Moreover, for the same reasons, the initial state probability is the same for all . Finally, due to the Gaussianity of the noise, states the p.d.f. of the observations can be written as

is Since the input sequences are equiprobable, constant. For the second term, we use the fact that the noise samples are independent and obtain

(32)

(28) (29) (34) and are predetermined for the current model, the Since only model parameters left to be estimated are the param. In the TI case, it is common to parametrize eters of

3 We now include  2 in the parameter vector v Baum–Welch re-estimation procedure.

because it is needed for the

CIRPAN AND TSATSANIS: MAXIMUM LIKELIHOOD BLIND CHANNEL ESTIMATION IN THE PRESENCE OF DOPPLER SHIFTS

where when Substitution of (34) in (32) yields

and 0 otherwise.

1563

Using the definitions of backward and forward variables, (38) can be easily calculated as (42) C. Proposed Method The objective here is to obtain estimates of from the observed sequence . It is shown that for the model (8) and (9), the auxiliary function has the form

(35) In [17], it was shown that the second part of (35) is equal to .

(43) The iterative estimation formulas can be obtained by setting the gradient of the auxiliary function to zero as in

(36) In order to recursively compute (36), we can rewrite it for any as given

(44) (37) Now, applying the Markovian property, we obtain

(38) The computation of the joint likelihood (38) can be recursively and backward obtained by defining the forward variables that are the probability of the partial observation sequences, given the state at time

(39) and

can be obtained recursively, as in [15], as

(45)

(46) Based on these results, the proposed algorithm consists of the following steps. Initialization: Set the parameters to some initial value . Step 1: Compute the forward and backward variables using . (40) to obtain using gradient-descent Step 2: Solve

for (47)

for

(40)

with initial conditions for

for

is the estimate of after iterations, and is where the step size. from the re-estimation formula (45), Step 3: Obtain which yields (48)

(41)

where we have (49) and (50), shown at the bottom of the next page.

1564

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 6, JUNE 1999

Step 4: Solve (46) for

to obtain

The evaluation of the exact form of the CRB requires the Hessian matrix for . Since the problem at hand to be stochastic, the conditional likelihood funcconsiders needs to be averaged over all possible input tion sequences

as

(51) converges. Step 5: Repeat Steps 1–4 until Step 6: Employ the Viterbi algorithm to recover the transmitted symbols (52) . or use the statistics The proposed method implements a ML solution and is therefore expected to exhibit superior performance, especially when only short data records are available. On the other hand, due to the nonlinear nature of the problem, global convergence is not guaranteed, and a suboptimal method is needed to provide reasonable initial estimates [34], [35]. In practice, we did not observe local minima problems when we initialized the parameters according to the suggestions of [19] (large value and small values for and ). for

(55) making the problem intractable. The exact CRB is then (56) However, a valid bound can still be obtained from the con. Since the logarithmic ditional likelihood function function is concave, we can employ Jensen’s inequality [31] and obtain (57) The right-hand side provides a CRB, which may not be tight but is much easier to compute. Since the noise is Gaussian, the conditional likelihood function is

IV. PERFORMANCE ANALYSIS In this section, the performance of the stochastic ML method is evaluated based on the derivation of the CRB for the unbiased estimates of the nonrandom parameters. , the vector of The parameter vector consists of channel coefficients, the vector of Doppler shifts, and the noise . The CRB variance and has the dimension of depends on the information on vector parameter quantified by the Fisher information matrix (FIM) and provides a lower bound on the variance of the unbiased estimate. Let the FIM , whose th entry is for the vector parameter be given by

(58) To evaluate the FIM, partial derivatives of the conditional likelihood with respect to parameter vector are needed. Then, the FIM can be written as (59) It turns out from (58) that

(53) The CRB for an unbiased estimator inverse of the FIM

is bounded by the

(60)

(54)

th element of by and th If we denote the , then the expressions for the entries of element of by

(49)

(50) .. .

.. .

.. .

CIRPAN AND TSATSANIS: MAXIMUM LIKELIHOOD BLIND CHANNEL ESTIMATION IN THE PRESENCE OF DOPPLER SHIFTS

the individual submatrices are defined as for else (61)

(62) (63)

(64) In the SML method, the cost function is averaged over the input’s p.d.f. [cf. (20)]. However, for high SNR, the summation in (20) is dominated by only one term. Therefore, it is expected ) the SML cost function approaches the that (as SNR DML one and yields similar performance [13]. It is more interesting to note that the looser CRB provided in this section becomes tight as SNR increases for of the same reason. Simulation results presented in Section VI also confirm this fact.

Definition: A system is locally identifiable if such that in an -neighborhood of and for each , no for some . system exists with The following proposition establishes the necessary conditions for local identifiability. be Proposition 1: Let the data generated by the system (9) and (10) under the following conditions. are distinct. (C1) All frequencies , , cannot be identically zero (C2) For a given . (C3) The data length . Then, is locally identifiable. Before proceeding to prove Proposition 1, let us briefly discuss the imposed conditions. C1) excludes the trivial case of two paths having identical frequencies, C2) excludes the trivial case where one frequency is not present at the received data at all (all ’s are zero), and C3) is a mild length condition. Proof: We show the proof for BPSK signals, whereas generalizations to other constellations are straightforward. The in (65) can be written from (9) and (10) as mean of .. .

V. IDENTIFIABILITY In this section, we study the conditions under which the system is uniquely identifiable from the likelihood function, i.e., the conditions under which different systems do not result in identical p.d.f.’s for the received data. The investigation of this issue is worthwhile since system identifiability is a prerequisite for the consistency and asymptotic efficiency of the proposed ML methods. We study separately the deterministic and stochastic ML approach.

1565

.. .

.. . (66)

In the deterministic ML approach, the p.d.f. is indexed by and [cf. (14), (28), (29)]

with obvious notation. Based on C2) and the fact that can only change in discrete steps (finite alphabet property), it is clear from (66) that no can be compensated by a local change in . change in We can therefore, without loss of generality, consider and study whether it is possible that for and for each . Taking all possibilities for the state vector , we obtain from (66)

(65)

(67)

Since the p.d.f. of the data depends on the particular realization of the input , we consider a system identifiable if there , does not exist another system such that for each for some . The p.d.f. of (65) and . It is clear from (14) depends on the parameters and , and (29) that if we change the sign of the remains unchanged, and , disproving identifiability. This should be no surprise as it is well known that all blind methods suffer from a sign or more general phase shift ambiguity. In the current method, the phase ambiguity can only take discrete values (depending on the constellation used). For this reason, we are only concerned in this section with local identifiability, i.e., whether small changes in result in different p.d.f.’s. A more precise definition of local identifiability follows.

is a vector of all possible system responses where is a matrix having as at time , whereas rows all possible state vectors. Requiring , we obtain from (67)

A. DML Identifiability

(68) Since

has full rank, (68) yields for

(69)

or equivalently (70)

1566

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 6, JUNE 1999

Fig. 2. Estimation error for the multipath parameters versus SNR for Case 1.

Fig. 3. Estimation error for the Doppler frequencies versus SNR for Case 1.

where

is defined as (71)

similarly. and Let , share a set of common frequencies let us partition them (and the corresponding ) as

, and

.. . .. .

(72)

Based on (72), (70) can be expressed as

.. .

.. .

(73) Fig. 4.

From (C3), (C1), and the Vandermonde property of the second matrix in (73), we conclude that it has full row rank. Therefore, the first matrix must be identically equal to zero. In particular, we have the following. , i.e., the coefficients of the common i) frequencies are identical. , i.e., the coefficients of the noncomii) mon frequencies are zero. Point ii) contradicts C2); hence, the two models cannot have noncommon frequencies. Finally, if all frequencies are common, point i) is what we needed to show. B. SML Identifiability In the stochastic ML approach, the likelihood is indexed only by the and can be written as (74)

Performance Evaluation: BER versus SNR Case 1.

It represents a mixture of Gaussian densities and is parameand the means . However, the same sign trized by or constellation ambiguity exists since (74) is symmetric in . Therefore, we are still only concerned with local identifiability. It turns out that (under the same condition) the system is locally identifiable by (74). The following proposition summarizes that result. be Proposition 2: Let the data generated by the system (9) and (10) under the conditions C1)–C3). Then, the system is locally identifiable, i.e., there ) does not exist in an -neighborhood of (for some . such that Proof: Since is a continuous function of , we can only have equal densities for a local change in , , if all means are equal for each in the mixture that is (74). However, according to Proposition 1, . this is only true if

CIRPAN AND TSATSANIS: MAXIMUM LIKELIHOOD BLIND CHANNEL ESTIMATION IN THE PRESENCE OF DOPPLER SHIFTS

Fig. 5. Estimation error for the channel impulse response versus SNR for Case 2.

1567

Fig. 7. Performance evaluation: BER versus SNR for Case 2.

ment. The data were generated based on this model (9), where the multipath parameters were chosen (75)

Fig. 6. Estimation error for the Doppler frequency versus SNR for Case 2.

VI. SIMULATIONS In this section, we illustrate some simulation results to evaluate the effectiveness of the proposed ML methods. Two different cases were tested, and results were compared with the Godard algorithm. In all simulations, an i.i.d. BPSK sequence was used as input. The performance of the of size proposed methods was evaluated as a function of SNR based on Monte Carlo simulations. Both DML and SML, as well as the Godard algorithm, were tested for 100 Monte Carlo trials per SNR point across a range of SNR’s. In each trial, the magnitude of the estimation error from DML and SML for were recorded. the parameters of the TV model Moreover, the average number of symbol errors from the same Monte Carlo simulations for DML, SML, and Godard were computed. Case 1: We consider a periodically TV channel model (9) that arises because of Doppler shifts in a multipath environ-

and its associated Doppler frequencies (direct path and one ; therefore, the system is periodic reflector) with period equal to 50 samples. In Figs. 2 and 3, we have plotted the estimation error for the multipath parameters and the associated Doppler frequencies , respectively, as well as the corresponding CRB’s. Fig. 4 illustrates average bit error rate (BER) for the DML, SML, and Godard algorithm. Some remarks and observations are now in order. i) Estimation error versus SNR and average BER versus SNR figures show that SML yields slightly better performance for low SNR; however, SML and DML perform almost identically for high SNR, as expected. ii) It is known that both the SML and the DML algorithms perform well for short data records. This fact is also observed from simulations. Therefore, the proposed methods are well suited for TDMA systems. iii) We tried the Godard algorithm to compare our results since it is an adaptive technique that could perhaps track the channel’s variations. Unfortunately, it could not equalize the channels due to its fast variations and available short data records. Case 2: In the second case, we consider a simpler TV channel model, where all paths experienced the same Doppler shift; therefore, the received data has the form (76)

Note that model (76) is the same model used in carrier tracking corresponds to a time-varying problems. In this problem,

1568

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 6, JUNE 1999

carrier phase shift due to frequency offset and phase jitter, corresponds to the channel impulse response. whereas Therefore, the proposed methods in this paper are also directly applicable to joint carrier tracking and channel estimation problems. To evaluate the performance of the estimator , the CRB’s derived in Section IV can be easily modified for the model (76). Due to lack of space, we only give the terms of interest (diagonal entries) of the FIM matrix

[10] A. W. Fuxjaeger and R. A. Iltis, “Acquisition of timing and Doppler shift in a direct sequence spread spectrum system,” IEEE Trans. Commun., vol. 42, pp. 2870–2879, Oct. 1994. [11] M. K. Tsatsanis, G. B. Giannakis, and G. Zhou, “Estimation and equalization of fading channels with random coefficients,” Signal Process., vol. 53, no. 2/3, pp. 211–229, Sept. 1996. [12] R. Raheli, A. Polydoros, and T. Ching-Kae, “Per-survivor processing: A general approach to MLSE in uncertain environments,” IEEE Trans. Commun., vol. 43, pp. 354–364, Feb.–Apr. 1995. [13] B. Halder, B. C. Ng, A. Paulraj, and T. Kailath, “Unconditional maximum likelihood approach for blind estimation of digital signals,” in Int. Conf. Acoust, Speech, Signal Process. (ICASSP), Atlanta, GA, May 1996, vol. II, pp. 1081–1084. [14] P. Stoica and A. Nehorai, “MUSIC, maximum likelihood and Cramer–Rao bound,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, pp. 720–741, May 1989. [15] L. R. Rabiner, “A tutorial on hidden Markov models and selected applications in speech recognition,” Proc. IEEE, vol. 77, pp. 257–274, Feb. 1989. [16] J. A. R. Fonollosa and J. Vidal, “Application of hidden Markov models to blind channel characterization and data detection,” in Int. Conf. Acoust., Speech, Signal Process. (ICASSP-94), Adelaide, Australia, Apr. 1994, vol. IV, pp. 185–188. [17] K. Kaleh and R. Vallet, “Joint parameter estimation and symbol detection for linear or nonlinear unknown channels,” IEEE Trans. Commun., vol. 42, pp. 2406–2413, July 1994. [18] A. Belouchrani and J.-F. Cardoso, “Maximum likelihood source separation for discrete sources,” in Proc. EUSIPCO, Edinburgh, U.K., Sept. 1994, pp. 768–771. [19] M. Erkurt and J. G. Proakis, “Joint data detection and channel estimation for rapidly fading channels,” in Proc. IEEE Global Telecommun. Conf. (Globecom), Chicago, IL, 1992, pp. 1277–1279. [20] C. A. Anton-Haro, J. A. R. Fonollosa, and J. R. Fonollosa, “On the inclusion of channel’s time dependence in a hidden Markov model for blind chnanel estimation,” in Proc. 8th IEEE Signal Process. Workshop Stat. Array Process., Corfu, Greece, June 1996, pp. 164–167. [21] G. D. Forney, Jr., “Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference,” IEEE Trans. Inform. Theory, vol. IT-18, pp. 363–377, May 1972. [22] G. Xu, H. Liu, L. Tong, and T. Kailath, “A least-squares approach to blind channel identification,” IEEE Trans. Signal Processing, vol. 43, pp. 2982–2993, Dec. 1995. [23] Y. Hua, “Fast maximum likelihood for blind identification of multiple FIR channels,” IEEE Trans. Signal Processing, vol. 44, pp. 661–672, Mar. 1996. [24] E. Moulines, P. Duhamel, J. Cardoso, and S. Mayrargue, “Subspace methods for the blind identification of multichannel FIR filters,” IEEE Trans. Signal Processing, vol. 43, pp. 516–525, Feb. 1995. [25] D. T. M. Slock, “Blind fractionally-spaced equalization, perfect reconstruction filter banks and multichannel linear prediction,” in Proc. Intl. Conf. Acoust., Speech, Signal Process., Adelaide, Australia, Apr. 1994, vol. VI, pp. 585–588. [26] H. Krim and M. Viberg, “Two decades of array signal processing research,” IEEE Signal Processing Mag., vol. 13, pp. 67–94, July 1996. [27] S. Talwar, M. Viberg, and A. Paulraj, “Blind estimation of multiple cochannel digital signals using an antenna array,” IEEE Signal Processing Lett., vol. 1, pp. 29–31, Feb. 1994. [28] L. E. Baum, T. Petrie, G. Soules, and N. Weiss, “A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains,” Ann. Math. Stat., vol. 41, no. 1, pp. 164–171, 1970. [29] M. Feder and J. A. Catipovic, “Algorithms for joint channel estimation and data recovery-application to equalization in underwater communications,” IEEE J. Oceanic Eng., vol. 16, Jan. 1991. [30] J. Preising and D. Brady, “Adaptive equalization for underwater wireless communications,” in Proc. Int. Conf. Acoust., Speech, Signal Process., Atlanta, GA, May 1996, vol. II, pp. 1077–1080. [31] B. Porat, Digital Processing of Random Signals. Englewood Cliffs, NJ: Prentice-Hall, 1994. [32] O. Shalvi and E. Weinstein, “Maximum likelihood and lower bounds in system identification with non-Gaussian inputs,” IEEE Trans. Inform. Theory, vol. 40, pp. 328–339, Mar. 1994. [33] C. W. Therrien, Discrete Random Signals and Statistical Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1992. [34] M. K. Tsatsanis and G. B. Giannakis, “Equalization of rapidly fading channels: Self-recovering methods,” IEEE Trans. Commun., vol. 44, May 1996. , “Subspace methods for blind estimation of time-varying FIR [35] channels,” IEEE Trans. Signal Processing, vol. 45, pp. 3084–3093, Dec. 1997.

(77)

For this case, the data were generated from (76) with model parameters (78) and the corresponding simulation results are illustrated. In Figs. 5 and 6, the estimation error for the impulse response of the channel and the Doppler shift (or carrier phase) as well as the CRB’s (77) are plotted, respectively. Finally, average bit error rate (BER) for the DML, SML and Godard algorithms is plotted in Fig. 7. VII. CONCLUSIONS We presented the DML and SML iterative approaches for the joint estimation of frequency shifts and channel parameters. Since the proposed methods provide ML solutions, they exhibit superior performance over higher order statisticsbased methods and available blind adaptive methods for TI channels, especially when only a short data record is available. Our theoretical and experimental results indicate that the two methods are equivalent for high SNR, whereas the SML method is slightly superior at low SNR. REFERENCES [1] R. Sharma, W. D. Grover, and W. A. Krzymien, “Forward-error control (FEC)-assisted adaptive equalization for digital cellular mobile radio,” IEEE Trans. Veh. Technol., vol. 42, pp. 94–102, Feb. 1993. [2] G. D’Aria, F. Muratore, and V. Palestini, “Simulation and performance of the pan-European land mobile radio system,” IEEE Trans. Veh. Technol., vol. 41, pp. 177–189, May 1992. [3] J. G. Proakis, Digital Communications. New York: McGraw Hill, 1995. [4] M. Stojanovic, J. Catipovic, and J. G. Proakis, “Adaptive multichannel combining and equalization for underwater acoustic communications,” J. Acoust. Soc. Amer., pp. 1621–1631, Sept. 1993. [5] D. Godard, “Self recovering equalization and carrier tracking in twodimensional data communications systems,” IEEE Trans. Commun., vol. COMM-28, pp. 1867–1875, Nov. 1980. [6] A. W. Rihaczek, Principles of High-Resolution Radar. San Francisco, CA: Peninsula, 1985. [7] M. C. Jeruchim, P. Balaban, and K. S. Shanmugan, Simulation of Communication Systems. New York: Plenum, 1992. [8] H. Liu, G. B. Giannakis, and M. K. Tsatsanis, “Time-varying system identification: A deterministic blind approach using antenna arrays,” in Proc. 30th Conf. Inform. Sci. Syst. (CISS’96), Princeton Univ., Princeton, NJ, Mar. 20–22, 1996. [9] M. K. Tsatsanis and G. B. Giannakis, “Modeling and equalization of rapidly fading channels,” Int. J. Adaptive Contr. Signal Process., vol. 10, pp. 159–176, 1996.

CIRPAN AND TSATSANIS: MAXIMUM LIKELIHOOD BLIND CHANNEL ESTIMATION IN THE PRESENCE OF DOPPLER SHIFTS

Hakan A. Cirpan was born in Instanbul, Turkey, in 1968. He received the M.Sc. degree in electrical engineering from the Instanbul University in 1992 and the Ph.D. degree in electrical engineering from the Stevens Institute of Technology, Hoboken, NJ, in 1997. In 1997, he joined the Department of Electrical Engineering, Instanbul University, as an Assistant Professor. His research interests lie in the areas of statistical signal processing, system identification, and wireless communications.

1569

Michail K. Tsatsanis (M’93) was born in Patras, Greece, in 1964. He received his diploma degree in electrical engineering from the National Technical University of Athens, Athens, Greece, in 1987 and the M.Sc. and Ph.D degrees in electrical engineering from the University of Virginia, Charlottesville, in 1990 and 1993, respectively. From 1986 until 1988, he was with Binary Logic Applications, Athens, where he worked in the design and development of digital systems for industrial control. From 1994 to 1995, he worked as a Research Associate at the Department of Electrical Engineering, University of Virginia. In 1995, he joined the Department of Electrical Engineering and Computer Science, Stevens Institute of Technology, Hoboken, NJ, as an Assistant Professor. His general research interests lie in the areas of statistical signal and array processing, system identification, pattern recognition, higher order statistics, and wavelet theory. His current interests focus on signal processing techniques for wireless communications including blind equalization, multiuser detection, fading channel estimation and tracking, and signal processing methods for networking problems. Dr. Tsatsanis is a Member of the IEEE Technical Committee on SPCOM. He has served as a Member of the Organizing Committee for the 1996 IEEE Signal Processing Workshop on SSAP and is the Technical Co-Chair of the Organizing Committee for the 1999 IEEE Workshop on Signal Processing Advances in Wireless Communications. He received the NSF CAREER award in 1998 and the IEEE Signal Processing Society Best Paper Young Author Award in 1999. He is an Associate Editor for the IEEE COMMUNICATIONS LETTERS and the IEEE TRANSACTIONS ON SIGNAL PROCESSING.