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IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 12, DECEMBER 2007

A Time/Frequency-Domain Unified Delayless Partitioned Block Frequency-Domain Adaptive Filter Yin Zhou, Student Member, IEEE, Jialu Chen, and Xiaodong Li

Abstract—In this letter, a delayless counterpart of the partitioned frequency-domain adaptive filter algorithm is straightforwardly derived from the block least-mean-square (BLMS) algorithm. To improve its tracking capability, a time/frequency-domain unified (TFU) adaptation scheme is proposed. It is further shown that other delayless realizations can be seen as simplified implementations of the TFU technique. Simulations are presented to support the analyses and demonstrate the good convergence and tracking performance of the proposed algorithm.

to improve the tracking ability, and it implements FD block adaptation to ensure fast convergence rate comparable to that of the PBFDAF algorithm. The TD and FD weights are not disjunctive but tightly connected to introduce feedback into each other so as to achieve overall optimized adaptation. We also compare the DL-PBFDAF and TFU-PBFDAF algorithms with other delayless realizations and reveal their connections.

Index Terms—Acoustic echo cancellation, delayless, frequencydomain adaptive filter, tracking capability.

II. DERIVATION OF THE DL-PBFDAF ALGORITHM

I. INTRODUCTION

with Let us consider a fullband filter coefficient vector taps defined as . It -tap partitions. The output signal can be separated into can then be expressed as

REQUENCY-DOMAIN (FD) techniques are widely used for long-tap adaptive filters to reduce the computational complexity and improve the convergence performance. The FD adaptive filter (FDAF) algorithm is an efficient realization of the block least-mean-square (BLMS) algorithm by using fast Fourier transform (FFT) filtering techniques [1]–[5]. The partitioned block FDAF (PBFDAF) algorithm [6], [7], also known as the multidelay adaptive filter (MDF) algorithm [8], [9], is developed to achieve a tradeoff between signal path delay, tracking capability, convergence rate, and computational complexity. Generally speaking, for colored input signals with appropriate weight constraint and step normalization, a larger block size would result in lower computational complexity and improved convergence but at the expense of increased signal path delay and degraded tracking capability. To mitigate this conflict, Buchner et al. proposed an extended multidelay filter by taking the inter-partition correlations into account [10]. This algorithm allows using very small block size to achieve low delay and is shown to still have good convergence and tracking performance. In this letter, we propose to alleviate this conflict in another way. We first straightforwardly derive from the BLMS algorithm a delayless PBFDAF (DL-PBFDAF) algorithm. This algorithm eliminates the undesired signal path delay and retains the favored convergence performance, but its tracking capability still needs to be improved. We further propose a time/frequency-domain unified (TFU) DL-PBFDAF (TFU-PBFDAF) algorithm, which implements sample-by-sample time-domain (TD) adaptation

F

Manuscript received March 25, 2007; revised July 22, 2007. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Ljubisa Stankovic. The authors are with the Communication Acoustics Laboratory, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100080, China (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2007.906627

(1) where is the output of the th partition, and and are, respectively, the th partitions of the fullband coefficient and input signal vectors, defined as and . The output signal can also be written as , where is the output of the first partition, and (2) is the summation of the outputs of the remaining partitions. Define the vectors

(3) where and are DFT and IDFT matrixes, respectively. Based on the relationship between linear convolution can be calculated as and circular convolution, (4) where

denotes element-wise multiplication. Defining , using (2)–(4), we have

1070-9908/$25.00 © 2007 IEEE

(5)

ZHOU et al.: A TIME/FREQUENCY-DOMAIN UNIFIED DELAYLESS PARTITIONED BLOCK FREQUENCY-DOMAIN ADAPTIVE FILTER

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Fig. 1. Block diagram of the DL-PBFDAF algorithm.

Noting from (3) that

, we have (6)

Here, block contains samples . Defining the th element of as , , we have . Therefore, the output in block can be obtained as (7) in block can be It is known from (6) and (7) that efficiently precalculated without waiting for the accumulation can be calculated in real time of new incoming data, and through sample-by-sample linear convolution in the first partition, so that there should be no signal path delay if (7) is used to calculate the output. The remaining problem is how to calculate the TD weights in (7) and the FD weights , , in (6). It is known from (3) that , , and can be directly related as (8) , as the Therefore, we can adapt once every samPBFDAF algorithm does, update ples by (8), and delaylessly calculate the output signal by (7). Using this method, the DL-PBFDAF algorithm is straightrepresents forwardly derived, as depicted in Fig. 1, where block size unit delay, and thick lines represent FD vectors; further explanation and details follow below. III. TIME/FREQUENCY-DOMAIN UNIFIED ADAPTATION The DL-PBFDAF is a delayless counterpart to the PBFDAF algorithm. It eliminates the signal path delay at the expense of a moderate increase in the computational complexity and has the same convergence. However, its tracking capability, the same as that of the PBFDAF algorithm, still needs to be improved, especially if the block size is large and the system impulse response changes frequently. To solve this problem, we propose a TFU adaptation scheme, as depicted in Fig. 2. Comparing Figs. 1 and 2, we can find that the TFU-PBFDAF differs from the DL-PBFDAF in the procedure to calculate the

Fig. 2. Block diagram of the TFU-PBFDAF algorithm.

adaptive weights in the TD partition. In the DL-PBFDAF, the weight adaptations are all implemented in the FD, whereas in the TFU-PBFDAF, the FD and TD adaptations are unified. The DL-PBFDAF and TFU-PBFDAF algorithms are summarized as follows. At the beginning of block , that is , , , and the output the FD weights have already been calculated, (7) of the FD partitions is used to calculate the output signal , and the error signal is obtained by , where is the desired response. The TD weights in the first partition at time are transformed from the FD weights by (9) In the DL-PBFDAF algorithm, keeps unchanged in the entire block, whereas in the TFU-PBFDAF algorithm, is to updated sample-by-sample from time , described as (10) , is a normalized where step size, is the input signal is a variable regularization parameter. At the vector, and , that is , The FD weights end of block in the first partition are transformed from the TD weights by (11) , that is At the beginning of the new block is calculated as follows: the step size vector

, (12) (13) (14)

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IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 12, DECEMBER 2007

where , is the th element of , is the forgetting factor of the power estimate, is a normalized step size, and , , are the power estimate, a variable regularization and parameter, and the step size for the th frequency-bin in block , respectively. The weight adaptation and constraints for are described as the FD partitions in block

(15) where partition in block

. The weight adaptation for the first is described as (16)

By iterating (9) and (11)–(16) for each new block, the DL-PBFDAF is obtained, whereas by iterating (9)–(16) for each new block, the TFU adaptation scheme is obtained. In the TFU adaptation scheme, the first partition is not only adapted sample-by-sample by (10) in the TD but is also block adapted by (16) in the FD, and the TD and FD weights are tightly connected with (9) and (11). The merits of the TFU adaptation scheme are obvious. In practical scenarios, the most significant fullband coefficients and impulse response variations usually exist in the first partition [11]. Implementing sample-by-sample, TD adaptations in this partition will improve the tracking capability. On the other hand, implementing block-by-block FD adaptations will ensure overall convergence speed as fast as that of the PBFDAF. More importantly, the TD and FD adaptations are not disjunctive but rather are tightly connected through (9) and (11) to introduce feedback into each other so as to achieve overall optimized adaptation. It is suggested that a relatively large optimized step is used for the FD adaptation, and a relatively small step is used for the TD adaptation. In this manner, the de-correlation effects introduced by the Fourier transform and the independent adaptations for each frequency bins are maintained. Thus, the overall convergence direction is determined by the FD adaptation, and TD adaptation serves as an efficient complementary adjustment. Several other methods are also proposed for the PBFDAF algorithm to eliminate the signal path delay. Merched and Sayed derive the DFT-MDF algorithm through an embedding approach [12]. Its best convergence and tracking performance is essentially the same as the DL-PBFDAF since they are both delayless counterparts to the PBFDAF algorithm. However, the DFT-MDF employs a structure very similar to the closed-loop delayless subband adaptive filter [13], and it uses the same approach to implement the fullband convolution. In this manner, decreasing the rate of subband/fullband transformation will considerably reduce the computational complexity, but it introduces weight update delay, resulting in degraded convergence and tracking performance [12], [13]. Bendel et al. propose the delayless frequency domain (DLFD) algorithm [11], which implements sample-by-sample TD adaptation for the first partition while using a block FD update for the remaining partitions. However, the first partition and the remaining partitions are adapted separately without tight connections. In this manner,

the de-correlation effects introduced by the Fourier transform are weakened, and the independent adaptations for each frequency bins are impaired, leading to degraded convergence. It turns out that the DL-PBFDAF, DFT-MDF, and DLFD algorithms can all be recognized as simplified implementations of the TFU-PBFDAF algorithm. As observed in Fig. 2, if the TD adaptation for the first partition is frozen, the TFU-PBFDAF becomes the same as the DL-PBFDAF and similar to the DFTMDF. On the other hand, if the FD adaptation in the first partition is frozen, the TFU-PBFDAF reduces to the DLFD. If the TD and FD adaptations are unified, we get the TFU-PBFDAF, which is expected to have not only fast convergence but also good tracking capability. If large impulse response variations occur in partitions other than the first, no improvement in the tracking performance can be obtained by implementing TFU adaptations only in the first partition. This problem can be solved by simply generalizing the TFU adaptation scheme in the first partition to any other partitions. Strategies to decide which partitions should be instantaneously adapted may be included to reduce the computational burden. These techniques will not be further discussed since they are beyond the scope of this letter and could be considered as further research topics. If the input signal is real, the approximate number of real multiplications per sample for the PBFDAF is , and the approximate numbers for the for the DL-PBFDAF, various delayless realizations are for the DLFD, and for the TFU-PBFDAF. Therefore, these delayless realizations have similar computational complexity. IV. SIMULATIONS The performance of the TFU-PBFDAF algorithm and other delayless realizations are evaluated by computer simulations for an acoustic echo cancellation application. The signal is samis trunpled at 8 kHz, and a 1030-tap impulse response cated from an acoustical impulse response measured in an of, fice room. We choose the fullband filter length as , and the number of partitions as the block length as . Strategies to appropriately choose the forgetting factor in (12) and regulation parameters in (10) and (13) can be found in [5], [14], and [15]. The filter starts to adapt to at , so that the initial convergence can be obto 9 s, the first points of , which served. From form the first partition, nonlinearly increase by multiplying with , while the other points are held constant, and from to 15 s, all points of are held constant, so that the , is raditracking capability can be observed. At cally changed by being shifted right by 20 points, and then it , so that the convergence and keeps unchanged until tracking performance can be examined. The echo return loss en, where hancement LPF denotes a lowpass filter with a single pole at 0.999, and the is used to system error evaluate the performance [5]. The input signal is either colored noise generated by passing a white Gaussian noise through an AR system or a speech signal from the TIMIT database.

ZHOU et al.: A TIME/FREQUENCY-DOMAIN UNIFIED DELAYLESS PARTITIONED BLOCK FREQUENCY-DOMAIN ADAPTIVE FILTER

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Fig. 3. Performance comparison for the DL-PBFDAF, DLFD, and TFU-PBFDAF algorithms with (a) colored noise as input and (b) speech signal as input. The ERLE and SysE curves are shown in the upper and lower parts of both figures, respectively.

The simulation results are presented in Fig. 3(a) and (b). It is observed from both figures that the DL-PBFDAF has fast convergence rate but suffers from low ERLE when impulse response variations occur in the first partition. This poor tracking capability will limit its application in practical scenarios. It is also observed that the DLFD has the lowest convergence rate but can quickly track impulse response variations in the first partition. This good tracking capability makes it favored over the DL-PBFDAF for practical applications. It is further observed that the TFU-PBFDAF has the fastest convergence rate and best tracking capability. Therefore, although the DL-PBFDAF has fast overall convergence rate not only for a colored stationary input but also for a highly nonstationary input with a large dynamic spectral range, its tracking capability needs to be improved. The DLFD improves the tracking capability but has degraded convergence. In contrast, the TFU-PBFDAF has good tracking and also rapidly converges close to the minimum residual SysE. So the TFU-PBFDAF is preferred to the DL-PFBDAF and DLFD algorithms because it obtains better convergence and tracking performance at the expense of a slight increase in the computational complexity. V. CONCLUSIONS In this letter, the DL-PBFDAF algorithm is derived to eliminate the signal path delay of the PBFDAF algorithm. The TFUPBFDAF algorithm is further proposed to improve the tracking capability. It is shown that the TFU-PBFDAF algorithm has the best convergence and tracking performance, and other delayless realizations can all be recognized as its special versions. ACKNOWLEDGMENT The authors would like to thank Dr. D. R. Morgan for proofreading the draft of this manuscript and making a number of valuable comments and suggestions. The authors would also

like to thank the anonymous reviewers for their constructive comments. REFERENCES [1] J. J. Shynk, “Frequency-domain and multirate adaptive filtering,” IEEE Signal Process. Mag., vol. 9, no. 1, pp. 14–37, Jan. 1992. [2] G. A. Clark, S. R. Parker, and S. K. Mitra, “A unified approach to time- and frequency-domain realization of FIR adaptive digital filters,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-31, no. 5, pp. 1073–1083, Oct. 1983. [3] B. Farhang-Boroujeny, Adaptive Filters: Theory and Applications. Chichester, U.K.: Wiley, 1998. [4] S. Haykin, Adaptive Filter Theory. Englewood Cliffs, NJ: PrenticeHall, 2002. [5] J. Benesty, T. Gänsler, D. R. Morgan, M. M. Sondhi, and S. L. Gay, Advances in Network and Acoustic Echo Cancellation. Berlin, Germany: Springer-Verlag, 2001. [6] J. M. P. Borrallo and M. G. Otero, “On the implementation of a partitioned block frequency domain adaptive filter (PBFDAF) for long acoustic echo cancellation,” Signal Process., vol. 27, pp. 301–315, Jun. 1992. [7] K. S. Chan and B. Farhang-Boroujeny, “Analysis of the partitioned frequency-domain block LMS (PFBLMS) algorithm,” IEEE Trans. Signal Process., vol. 49, no. 9, pp. 1860–1874, Sep. 2001. [8] J. S. Soo and K. K. Pang, “Multidelay block frequency-domain adaptive filter,” IEEE Trans. Acoust., Speech, Signal Process., vol. 38, no. 2, pp. 373–376, Feb. 1990. [9] E. Moulines, O. A. Amrane, and Y. Grenier, “The generalized multidelay adaptive filter: Structure and convergence analysis,” IEEE Trans. Signal Process., vol. 43, no. 1, pp. 14–28, Jan. 1995. [10] H. Buchner, W. Kellermann, and J. Benesty, “An extended multidelay filter: Fast low-delay algorithms for very high-order adaptive systems,” in Proc. IEEE ICASSP, Apr. 2003, vol. 5, pp. 385–388. [11] Y. Bendel, D. Burshtein, O. Shalvi, and E. Weinstein, “Delayless frequency domain acoustic echo cancellation,” IEEE Trans. Speech Audio Process., vol. 9, no. 5, pp. 589–597, Jul. 2001. [12] R. Merched and A. Sayed, “An embedding approach to frequency-domain and subband adaptive filtering,” IEEE Trans. Signal Process., vol. 48, no. 9, pp. 2607–2619, Sep. 2000. [13] D. R. Morgan and J. C. Thi, “A delayless subband adaptive filter architecture,” IEEE Trans. Signal Process., vol. 43, no. 8, pp. 1819–1830, Aug. 1995. [14] E. Hänsler and G. Schmidt, Acoustic Echo and Noise Control: A Practical Approach. New York: Wiley, 2004. [15] J. Benesty, H. Rey, L. R. Vega, and S. Tressens, “A nonparametric VSS NLMS algorithm,” IEEE Signal Process. Lett., vol. 13, no. 10, pp. 581–584, Oct. 2006.

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