2008Noise Robust Multichannel Frequency-Domain LMS Algorithms for Blind Channel Identification

IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008305Noise Robust Multichannel Frequency-Domain LMS Algorithms for Blind Channel IdentificationMohammad Ariful Haque and Md Kamrul Hasan, Senior Member, IEEEAbstract—A number of multichannel least mean square (LMS)-type algorithms have been proposed in the literature to identify single-input multi-output finite impulse response channels. All of these algorithms share the common characteristic of good initial convergence followed by a rapid misconvergence in the presence of noise. This misconvergence characteristic is due to the nonuniform spectral attenuation of the estimated channel coefficients as reported in some research results. In this letter, we formulate a novel cost function that inherently oppose such spectral attenuation resulting from the noisy update vector. We show analytically that the gradient of the proposed penalty term enforces uniform distribution of the estimated channel spectral energy over the entire frequency band and thus contribute to ameliorating the misconvergence of these multichannel algorithms in the presence of noise. The robustness of the proposed algorithm is verified using numerical examples for different channels in a wide range of signal-to-noise ratios. Index Terms—Blind channel identification, multichannel least mean square (LMS) algorithm, noise robustness.I. INTRODUCTION LIND channel identification (BCI) is a signal processing technology aimed to identify the unknown channel impulse responses solely from the observations often corrupted by noise. Both single- and multichannel identification schemes are reported in the literature by many researchers. Multichannel identification schemes, however, are increasingly becoming popular due to their suitability in removing the unknown channel effects more effectively than their single-channel counterparts. Among the various multichannel BCI schemes reported so far, e.g., least-squares approach [1], subspace method [2], maximum-likelihood method [3], the normalized multichannel frequency-domain LMS (NMCFLMS) [4], and the variable step-size multichannel frequency-domain LMS (VSS-MCFLMS) [5], algorithms are attractive as they are computationally efficient and effective for BCI. It is shown in [6] that the VSS-MCFLMS algorithm is more robust as compared to the NMCFLMS algorithm in the presence of noise. However, the NMCFLMS algorithm has higher speed of convergence in acoustic multichannel systems followed by a rapid misconvergence. Therefore, if the robustness of the NMCFLMS algorithm could be ensured, it would be an attractive algorithm for acoustic channel estimation. A constrained NMCFLMS algorithm has been described in [7] for improving its robustness to additive noise. However, theBimplementation of the algorithm assumes that the positions and amplitudes of these dominant components, e.g., direct-path coefficients of the acoustic impulse responses, are known, which restricts its use in practical situations. Moreover, the algorithm is not suitable for random coefficients systems where there are no such dominant components. Recently, it has been reported in [8] and [9] that the misconverged estimates of the channel impulse responses have narrowband characteristics in the frequency-domain. An energy constrained NMCFLMS algorithm for blind identification of acoustic channels is proposed in [9] which is based on achieving the spectral energy balance between the low- and high-frequency subbands of the estimated channel. However, the technique is not generalized to include the total misconvergence scenario. For example, if the narrowband phenomenon appears both in the low- and high-frequency bands, the constraint does not work. Moreover, we have observed that the technique is not useful to stop the misconvergence when the channel coefficients are white random sequence. In this letter, we propose a robust technique that can stop the misconvergence of the NMCFLMS and VSS-MCFLMS algorithms in blind identification of both acoustic and random channels. We formulate a mathematically tractable penalty function to enforce that the spectral energy is approximately uniformly distributed over all the frequencies in the estimated channel response. Thus, the gradient of this penalty function works against the nonuniform spectral attenuation effect of the original noisy gradient and hence ensures robustness of the adaptive blind multichannel algorithms. This penalty term can be directly calculated from the estimated impulse responses, and therefore, it is implementable without a priori information of the channel and observation noise power. II. PROBLEM FORMULATION The input–output relationship of a single-input multipleoutput (SIMO) finite impulse response (FIR) channel is given by (1) (2) where is the number of sensors, is the length of the im, , , , and denote, respecpulse response, tively, the common source signal, th channel output, th channel output corrupted by background noise, observation noise, and impulse response of the source to the th sensor. Using vector notation, (1) can be written as (3) where, “ ” denotes transpose is and the impulseManuscript received October 24, 2007; revised December 15, 2007. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Yue (Joseph) Wang. The authors are with the Department of Electrical and Electronic Engineering, Bangladesh University of Engineering and Technology, Dhaka, Bangladesh (e-mail: arifulhoque@eee.buet.ac.bd; khasan@eee.buet.ac.bd). Color versions of one or more of the figures in this paper are available online at . Digital Object Identifier 10.1109/LSP.2008.9178031070-9908/$25.00 © 2008 IEEE306IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008response vector of the estimates observations the data length.,th channel and . A BCI algorithm solely from the noisy , where denotesnorm in the time-domain. Imposing this constraint in the frequency-domain, we have the update equation as (9) It is observed that both the NMCFLMS and VSS-MCFLMS algorithms give good initial estimate of the channels followed by rapid divergence from this better estimate in the presence of additive noise. This misconvergence is associated with the nonuniform spectral attenuation of the estimated channel impulse response [8], [9]. They proposed a modified cost function , where and are the original and penalty cost functions, respectively, which are cou. was formupled through the Lagrange multiplier, , where denotes the frelated as quency band over where spectral energy has been concentrated indicates due to misconvergence. Though minimization of that energy in a particular band is minimized, but this does not mean that energy will be uniformly distributed over the entire is minimized, but the frequency band. It is not unlikely that shape of the spectrum is still narrow band. Therefore, we propose a novel penalty function that ensures spectral flatness of the estimated channel coefficients in the presence of additive noise. The penalty function that can ameliorate the misconvergence of the MCFLMS-type algorithms is defined in this work as maximize subject to (11) where (11) is ensured by the unit norm constraint imposed on the update equation. Now substituting (11) into (10), we obtain (10)III. ROBUST MCFLMS-TYPE ALGORITHMS We first briefly review the multichannel frequency-domain LMS-type algorithms for blind channel identification. The cost function in the frequency-domain is defined as (4) is the frequency-domain block error signal bewhere tween the th and th channels and the superscript “ ” denotes Hermitian transpose. The update equation of the NMCFLMS algorithm [4] can be expressed as(5) where algorithm and is the step-size for the NMCFLMS , and . Here, “ ” indicates estimate, is the frame index, and denotes the DFT matrix. is given by The frequency-domain error function (6) is the DFT of the th frame data The diagonal matrix block for the th channel, i.e., see the equation at the bottom of the page. Again, the update equation of the VSS-MCFLMS algorithm [5] can be expressed as (7) is the gradient of the cost function and is where the step-size. Here, the step-size is adapted so that the distance and is minimum at each iteration, and it is between expressed as (8)(12) Differentiating (12) with respect to , we get(13) The penalty function is maximized when (13), we see that it is only possible when . From(14) is the norm and is a small positive real where number used to prevent singularity. To avoid a trivial estimate with all zero elements, at each step, the estimated filter coefficient vector is constrained to have unit If any other term in (13) is equated to zero, we obtain , which denotes the minimum value of the cost function. As we intend to maximize the penalty function, this is a trivial solution. simultaneous In a similar approach, we can obtainHAQUE AND HASAN: NOISE ROBUST MULTICHANNEL FREQUENCY-DOMAIN LMS ALGORITHMS307linear equations of the same form as (14) for each value of . Adding all such equations, we get, is estimated such that the total The coupling factor, in the steady-state condigradient becomes zero , and premultiplying tion. This gives both sides by , we can obtain as (19) Similarly, the update equation for the robust NMCFLMS (RNMCFLMS) algorithm can be written as(15) Subtracting (15) from (14), we obtain the condition for penalty . Thus, the function maximization as penalty function will be maximum when the estimated channel coefficients have uniform magnitude spectra in the frequencydomain. Therefore, to combat nonuniform spectral attenuation problem in the misconvergence phase, spectral flatness can be , attached as a constraint with the original cost function, , using the Lagrange multiplier. Unvia the penalty term, like the modified cost function proposed in [9], the total regularized cost function to be minimized can be defined as . The negative sign before enis minimized, is maximized. The sures that while adaptive update rule for this constrained minimization can be readily obtained as(20) where is estimated similar to (19) but using the NMCFLMS algorithm update parameters. The extra computational cost required to implement the proposed penalty term is not significant. For example, the total number of multiplications and divisions required by the NMCFLMS algorithm is per iteration, whereas the increase in the com. putational cost due to the added penalty term is only IV. SIMULATION RESULTS In this section, we investigate the performance of the proposed RNMCFLMS and RVSS-MCFLMS algorithms in (20) and (16) for both random and acoustic multichannel systems. We also compare the performance of the proposed method with that of the original NMCFLMS, VSS-MCFLMS, and energy constrained NMCFLMS (CNMCFLMS) [9] algorithms in different noisy environments to show the robustness of our method. In all cases, the step-size parameter, , for the NMCFLMS and RNMCFLMS algorithms was fixed to 0.5, unless otherwise stated. The performance index used for measurement of improvement and comparison is the normalized projection misalignment (NPM) as mentioned in [9]. A. Random Multichannel System We now present blind identification results for a channel random coefficient impulse response system. The impulse responses were generated using the “randn” function of MATLAB. The length of each channel impulse response is . The source signal was Gaussian white noise. Fig. 1 shows the results of channel estimation for all the algorithms at dB. In terms of final misalignment error, the RNMSNR CFLMS algorithm shows the best performance while the original NMCFLMS algorithm without the penalty term demonstrates the poorest performance. The CNMCFLMS algorithm though performs slightly better than the NMCFLMS and is not robust for random coefficient channel as mentioned before in Section I. The results in Fig. 1 also demonstrate that the VSS-MCFLMS algorithm is more robust to noise as compared to the NMCFLMS algorithm in the blind identification of random channels. As it is not showing misconvergence at this SNR value for random coefficient channel estimation, the RVSS-MCFLMS does not bring further improvement in this case.(16) The beauty of the proposed penalty function is that its gradient remains almost inactive as compared to the original signal gradient in the initial phase of iterations. This phenomena stems from the fact that the true channel vector, whether it is acoustic or random, is spectrally wide band. Thus, the original cost function is expected to be better minimized with a wide band estimate of the channel. This leads to almost negligible gradient of the penalty term, thereby making no noticeable effect on the update equation. However, when the misconvergence starts because of nonuniform spectral attenuation in the estimate, the initially dormant gradient of the penalty term becomes active, enforces spectral flatness, and eventually eradicates misconvergence. In order to simplify the expression of the penalty gradient, we take natural a logarithm on both sides of (10). This does not relax the functionality of the penalty term. Therefore, we can rewrite the penalty cost function as (17) Now the penalty function gradient is obtained aswherecan be expressed aswhere Therefore, we can writeand as.(18) where , is a diagonal matrix with diagonal elements .308IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008Fig. 1. NPM profile for M = 5 channels L = 32 random coefficients SIMO channel identification at SNR = 10 dB.Fig. 4. NPM profile for a M = 5 channel L = 128 coefficients acoustic channel identification with speech input at SNR = 15 dB.file of the estimated channel using speech input at SNR 15 dB in Fig. 4. In case of the NMCFLMS and VSS-MCFLMS algorithms, we see good initial convergence. With increased iterations, the NMCFLMS completely misconverges. As stated earlier, the VSS-MCFLMS is more robust than the NMCFLMS and shows a slow misconverging trend. To the contrary, the proposed RNMCFLMS and RVSS-MCFLMS algorithms show no sign of misconvergence with noise.Fig. 2. NPM profile for a M = 5 channel L = 256 coefficients acoustic channel identification with white Gaussian input at SNR = 10 dB.V. CONCLUSION A noise robust technique for adaptive blind channel estimation using multichannel frequency-domain LMS-type algorithms has been presented in the letter. We have derived a penalty function whose gradient effectively works against the nonuniform spectral attenuation phenomenon in the estimated channel impulse responses caused by the noisy gradient of the conventional cost function. Computer simulation tests have demonstrated that the proposed technique with complementary cost functions gives reasonably good final misalignment error and improved robustness to additive noise without sacrificing the speed of convergence. REFERENCES[1] G. Xu, H. Liu, L. Tong, and T. Kailath, “A least-squares approach to blind channel identification,” IEEE Trans. Signal Process., vol. 43, no. 12, pp. 2982–2993, Dec. 1995. [2] S. Gannot and M. Moonen, “Subspace methods for multimircophone speech dereverberation,” EURASIP J. Appl. Signal Process., vol. 2003, no. 11, pp. 1074–1090, 2003. [3] Y. Hua, “Fast maximum likelihood for blind identification of multiple FIR channels,” IEEE Trans. Signal Process., vol. 44, no. 3, pp. 661–672, Mar. 1996. [4] Y. Huang and J. Benesty, “A class of frequency-domain adaptive approaches to blind multichannel identification,” IEEE Trans. Signal Process., vol. 51, no. 1, pp. 11–24, Jan. 2003. [5] M. A. Haque and M. K. Hasan, “Variable step size frequency domain multichannel LMS algorithm for blind channel identification with noise,” in Proc. Communication Systems, Networks and Digital Signal Processing, 2006. [6] M. A. Haque and M. K. Hasan, “Performance comparison of the blind multi channel frequency domain normalized LMS and variable step-size LMS with noise,” in Proc. European Signal Processing Conf., 2007. [7] M. K. Hasan, J. Benesty, P. A. Naylor, and D. B. Ward, “Improving robustness of blind adaptive multichannel identification algorithms using constraints,” in Proc. Eur. Signal Processing Conf., 2005. [8] M. K. Hasan and P. A. Naylor, “Effect of noise on blind adaptive multichannel identification algorithms: Robustness issue,” in Proc. Eur. Signal Processing Conf., 2006. [9] M. A. Haque, M. Bashar, P. Naylor, K. Hirose, and M. K. Hasan, “Energy constrained frequency-domain normalized LMS algorithm for blind channel identification,” in J. Signal, Image and Video Process.. London, U.K.: Springer, Apr. 2007, pp. 203–213.Fig. 3. NPM versus SNR profile for M = 5 channel L = 128 coefficients acoustic channel identification with white Gaussian input.B. Acoustic Multichannel System The dimension of the room was taken to be (5 4 3) m. microphones with uniform A linear array consisting of along the -axis was used in the exseparation of periment. The first microphone and source were positioned at (1.0,1.5,1.6) m and (2.0,1.2,1.6) m, respectively. The impulse responses were generated using the well-known image model . The sampling frequency for reverberation time was 8 kHz. dB The results of acoustic channel estimation at SNR is shown in Fig. 2. It can be observed that the penalty term ensures robustness of both the algorithms without sacrificing the speed of convergence for the reason explained after (16). It can also be seen that the step-size parameter, , acts as a trade-off between the convergence speed and final misalignment error for the proposed RNMCFLMS algorithm. Now, to observe the result of channel estimation at different SNRs, we present the final NPM versus SNR profile in Fig. 3. As can be seen, the proposed algorithms give significantly lower final misalignment error as compared to the original ones. We now present the NPM pro-。