声学回波抵消自适应算法的研究
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摘要
随着VoIP技术的不断发展和WLAN的日益成熟,VoIP技术和WLAN技术的融合是通信技术发展的必然趋势,可以预计WLAN上VoIP技术将是下一代通信领域中的关键技术之一。与传统电话相比,VoIP的致命的弱点就是语音质量较差。影响VoIP语音质量的因素是多方面的,关键因素之一是回声的影响。因此,要提高VoIP的语音质量,就必须在VoIP通信系统中进行消除回声的处理。
目前通信系统中大多采用自适应回声抵消技术来实现回声消除。通信双方同时讲话(双端对讲)情况下的回声抵消是自适应回声抵消技术的主要难题之一。传统的自适应回声抵消技术都是通过在双端对讲情况下冻结滤波器系数调整的方法来解决这一问题。具体地说,就是通过信号检测技术确定通信双方的讲话状态,一旦检测到双端对讲,就禁止或放慢自适应滤波器系数的调整更新。这种基于双端对讲检测的自适应回声抵消技术会导致两个问题:一是由于自适应滤波器系数在双端对讲期间不调整,一旦回声路径在双端对讲期间发生变化,自适应滤波器就不能跟踪回声路径的变化,导致回声抵消效果下降;二是高性能的双端对讲检测算法大都比较复杂,双端对讲检测运算的延时性,有可能使自适应滤波器的系数被错误地冻结在已偏离最优解的滤波器系数上。
回波抵消中的双端对讲检测器是用于监测远端和近端同时有话音存在的情况。它的作用是在近端有语音信号存在的时候停止滤波器的系数调整,以防止滤波器系数发散。许多文献资料中介绍了多种关于双端对讲检测的算法,其中包括能量检测法、互相关检测法和相干检测法,在这篇论文中我们将阐述这几种算法并辅以实验数据。
With the fast development of VoIP (Voice over IP) and WLAN (Wireless Local Area Network) technology, the integration of WLAN with VoIP has become the mainstream for the future of communication. It is anticipated that VoIP over WLAN will be one of the most critical technologies employed in next generation communication system. Compared with traditional telephone, the fatal disadvantage of VoIP is its worse speech quality. The influence factors are various and one of the most important is the echo. So, in order to enhance the speech quality of VoIP, echo cancellation plays a critical role in the VoIP systems.
At present, echo cancellation is mostly implemented by adaptive echo cancellation techniques. However, double talk, in which the near-end and the far-end speakers talk simultaneously, causes a serious problem in the adaptive estimation of the adaptive echo cancellation filter coefficients. To tackle this problem, almost all current adaptive echo cancellation techniques first detect the presence of double talk and then take appropriate actions such as disabling the adaptive algorithm during the double-talk period. But these techniques result in two problems: 1) Since the coefficients of the filter are not updated during the double-talk, once echo path changes during this period, it is impossible for the adaptive filter to trace the change of echo path and reduces the echo cancellation effect; 2) Most of these detectors are too complex in order to achieve satisfactory performance. Therefore, it is possible to make the coefficients of the filter diverge from the optimum due to the time required by the detector to detect double-talk.
A doubletalk detector (DTD) is used with an echo canceler to sense when far-end speech is corrupted by near-end speech. Its role is to freeze the adaptation of the model filter when near-end speech is present in order to avoid divergence of the adaptive algorithm. Several authors have proposed to describe many different ways of doing so ,including the Giegel Algorithm, the cross-correlation algorithm and the coherence algorithm. We show in this paper that all the algorithms and the calculated data of the algorithm.
目前通信系统中大多采用自适应回声抵消技术来实现回声消除。通信双方同时讲话(双端对讲)情况下的回声抵消是自适应回声抵消技术的主要难题之一。传统的自适应回声抵消技术都是通过在双端对讲情况下冻结滤波器系数调整的方法来解决这一问题。具体地说,就是通过信号检测技术确定通信双方的讲话状态,一旦检测到双端对讲,就禁止或放慢自适应滤波器系数的调整更新。这种基于双端对讲检测的自适应回声抵消技术会导致两个问题:一是由于自适应滤波器系数在双端对讲期间不调整,一旦回声路径在双端对讲期间发生变化,自适应滤波器就不能跟踪回声路径的变化,导致回声抵消效果下降;二是高性能的双端对讲检测算法大都比较复杂,双端对讲检测运算的延时性,有可能使自适应滤波器的系数被错误地冻结在已偏离最优解的滤波器系数上。
回波抵消中的双端对讲检测器是用于监测远端和近端同时有话音存在的情况。它的作用是在近端有语音信号存在的时候停止滤波器的系数调整,以防止滤波器系数发散。许多文献资料中介绍了多种关于双端对讲检测的算法,其中包括能量检测法、互相关检测法和相干检测法,在这篇论文中我们将阐述这几种算法并辅以实验数据。
With the fast development of VoIP (Voice over IP) and WLAN (Wireless Local Area Network) technology, the integration of WLAN with VoIP has become the mainstream for the future of communication. It is anticipated that VoIP over WLAN will be one of the most critical technologies employed in next generation communication system. Compared with traditional telephone, the fatal disadvantage of VoIP is its worse speech quality. The influence factors are various and one of the most important is the echo. So, in order to enhance the speech quality of VoIP, echo cancellation plays a critical role in the VoIP systems.
At present, echo cancellation is mostly implemented by adaptive echo cancellation techniques. However, double talk, in which the near-end and the far-end speakers talk simultaneously, causes a serious problem in the adaptive estimation of the adaptive echo cancellation filter coefficients. To tackle this problem, almost all current adaptive echo cancellation techniques first detect the presence of double talk and then take appropriate actions such as disabling the adaptive algorithm during the double-talk period. But these techniques result in two problems: 1) Since the coefficients of the filter are not updated during the double-talk, once echo path changes during this period, it is impossible for the adaptive filter to trace the change of echo path and reduces the echo cancellation effect; 2) Most of these detectors are too complex in order to achieve satisfactory performance. Therefore, it is possible to make the coefficients of the filter diverge from the optimum due to the time required by the detector to detect double-talk.
A doubletalk detector (DTD) is used with an echo canceler to sense when far-end speech is corrupted by near-end speech. Its role is to freeze the adaptation of the model filter when near-end speech is present in order to avoid divergence of the adaptive algorithm. Several authors have proposed to describe many different ways of doing so ,including the Giegel Algorithm, the cross-correlation algorithm and the coherence algorithm. We show in this paper that all the algorithms and the calculated data of the algorithm.
引文
[1] Christina B., Pia D., Eberhard H., Andreas M., Bernhard N., Henning P., Thomas S., Gerhard S. and Jan T., Acoustic Echo Control an Application of Very-High-Order Adaptive Filters, IEEE SIGNAL PROCESSING MAGAZINE, 1053-5888/99, JULY 1999.
[2] George-Othon Glentis, Kostas Berberidis and Sergios Theodoridis, Efficient Least Squares Adaptive Algorithms For FIR Transversal Filtering, IEEE SIGNAL PROCESSING MAGAZINE, 1053-5888/99, JULY 1999.
[3] Ki-Ho Kim and Edward J. Powers, Analysis of Initialization and Numerical Instability of Fast RLS Algorithms, CH2977-7/91/0000-1861,1991 IEEE.
[4] André Gilloire; Thierry Pétillon and Sergios Theodordis, Acoustic Echo Cancellation Using Fast RLS Adaptive Filters with Reduced Complexity, 0-7803-0593-0/92, 1992 IEEE.
[5] Alberto Carini and Enzo Mumolo, A Numerically Stable Fast RLS Algorithm for Adaptive Filtering and Prediction Based on the UD Factorization, 1053–587X/99, 1999 IEEE
[6] I.D.Skidmore and I.K.Proudler, KAGE: A New FAST RLS Algorithm, 0-7803-7041-4/10, 2001 IEEE
[7]Erlendur Karlsson, and Monson H. HAYES, Least Squares ARMA Modeling of Linear Time-Varying Systems: Lattice Filter Structures and .Fast RLS Algorithms, 0096-3518/87/0700-0994, 1987 IEEE
[8] 何振亚,自适应信号处理[M],北京:科学出版社,2002
[9] ITU-T G.165, Echo cancellers [S], 1993.
[10] M. M. Sondhi and D. A. Berkley, Silencing echo on the telephone network [J],Proc. IEEE, Vol.68, No.8, pp. 948-963, Aug. 1980.
[11] Steven L. Gay, Jacob Benesty, Acoustic Signal Processing forTelecommunication [M], Kluwer Academic Publishers, Boston Hardbound,2000.
[12] D. R. Morgan and S. G. Kratzer, On a Class of Computationally Efficient,Rapidly Converging, Generalized NLMS Algorithm [J], IEEE Signal Processing Letters, vol.3, No.8, pp.245-247, Aug. 1996.
[13] Yasukawa H., Shimada S., Furukrawa I., etc. Acoustic echo canceller with high speech quality [J], Proc. ICASSP’87, pp.2125-2128, 1987.
[14] Gitlin R. D., Weinstein S. D., The effects of large interference on the tracking capability of digitally implemented echo cancellers [J], IEEE Trans. on COM, No.6, pp.833-839, Jun. 1978.
[15] Raymond H. Kwong, Edward W. Johnston, A Variable Step Size LMS Algorithm[J], IEEE Trans. on Signal Processing, vol. 40, No.9, pp.1633-1642, Jul. 1992.
[16] Tyseer Aboulnast and K. Mayyas, A Robust Variable Step-Size LMS-Type Algorithm: Analysis and Simulations [J], IEEE Trans. on Signal Processing, vol.45, No.3, pp.631-639, Mar. 1997.
[17] H. Ye and B. X. Wu, A new double-talk detectors algorithm based on theorthogonality theorem [J], IEEE Trans. Commun., Vol.39, pp.1542-1545, Nov. 1991.
[18] Jacob Benesty, Dennis R. Morgan, and Jun H. Cho, A family of Doubletalk Detectors Based on Cross-Correlation [J], Proc. IWAENC, pp.108-111, Sep.1999.
[19] J. H. Cho, Dennis R. Morgan, and Jacob Benesty, An objective technique for evaluating doubletalk detectors in acoustic cancellers [J], IEEE Trans. on Speech and Audio Processing, Vol.7, No.11, pp.718-724, Nov. 1999.
[20] J. C. Jenq and S. F. Hsieh, Acoustic echo cancellation using iterative-maximal-length correlation and double-talk detection [J], IEEE Trans. Speech Audio Processing, vol.9, No.11, pp.932-942, Nov. 2001.
[21] J. Borish and J. B. Angell, An efficient algorithm for measuring the impulse response using pseudorandom noise [J], J. Audio Eng. Soc., vol. 31, No.7, pp.478-488, Jul. 1983.
[22] Ajay Divakaran, William A. Pearlman, A Closed Form Expression for an Efficient Class of Quadrature Mirror Filters and Its FIR Approximation [J], IEEE Trans. on Circuits and Systems II: Analog and Digital Signal Processing, vol.43, No.3, pp.207-219, March 1996.
[23] C. K. Goh, and Y. C. Lim, An efficient algorithm to design weighted minimax perfect reconstruction quadrature mirror filter banks [J], IEEE Trans. on Signal Processing, vol.47, No.12, pp.3303-3314, Dec. 1999.
[24] A. Guiloire, M. Viterli, Adaptive filtering in subband with critical sampling:Analysis, experiments and application to acoustic echo cancellation [J], IEEE Trans. Signal processing, Vol.40, No.8, pp.1862-1875, Aug. 1992.
[25] Petraglia M. R., Alves R. G., Diniz P. S. R., New structures for adaptive filtering in subbands with critical sampling [J], Signal Processing, IEEE Trans. On Acoustics, Speech, and Signal Processing, vol.48, No.12, pp.3316-3327, Dec.2000.
[26] Harteneck, M., Weiss, S., Stewart, R.W., Design of near perfect reconstruction oversampled filter banks for subband adaptive filters [J], IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol.46, No.8, pp.1081–1085, Aug. 1999.
[27] Dimitris G. Manolakis, Vinay k. Ingle and Stephen M. Kogon, Statistical and Adaptive Signal Processing, 电子工业出版社,2003年5月
[28] B. C. J. Moore, An Introduction to the Psychology of Hearing [M], Academic Press, London, 1989.
[29] A. Gilloire and V. Turbin, Using auditory properties to improve the behavior of stereophonic acoustic echo cancellers [J], Proc. ICASSP, pp.3681-3684, 1998.
[30] M. G. Bellanger, Adaptive digital filters and signal analysis [M], Marcel Dekker, Inc., 1987.
[31] T. Hoya, Y. Loke, J. A. Chambers and P. A. Naylor, Application of the leaky extended LMS (XLMS) algorithm in stereophonic acoustic echo cancellation [J], Signal Processing, vol.64, pp.87-91, 1998.
[32] J. Benesty, D. R. Morgan and M. M. Sondhi, Stereophonic acoustic echo cancellation using nonlinear transformations and comb filtering [J], Proc. ICASSP’98, pp.3673-3676, 1998.
[33] M. Ali., Stereophonic acoustic echo cancellation system using time-varyingAll-pass filtering for signal decorrelation [J], Proc. ICASSP’98, pp.3689-3692, 1998.
[34] Y. Joncour and A. Sugiyama, A Stereo echo canceller with pre-processing for correct echo-path identification [J], Proc. ICASSP’98, p.3677-3680, 1998.
[35] T. Gaensler and P. Eneroth, Influence of audio coding on stereophonic acoustic cancellation [J], IEEE, ICASSP, pp.3649-3652, 1998.
[36] T. N. Yensen, R. A. Goubran and I. Lambadaris, Synthetic stereo acoustic echo cancellation structure for multiple participant VoIP conferences [J], IEEE Transactions on Speech and Audio Processing, vol. 9, No. 2, 2001, pp. 168-174.
[37] T. Yensen, J. Ryan, R. Goubran and I. Lambadaris, Synthetic stereo acoustic echo cancellation structure with microphone array beamforming for VoIP conferences [J], Proc. IEEE ICASSP2000, pp.817-820, 2000.
[38] T. Okuno and M. O. Tokhi, Stereophonic acoustic echo cancellation using blind source separation as post-processing [J], Processings of Seventh International Workshop on Acoustic Echo and Noise Control, Darmstadt, Germany, pp.75-78, September 2001.
[39] T. Okuno and M. O. Tokhi, Stereophonic acoustic echo cancellation using blind source separation [J], Processing of Ninth International Congress on Sound and Vibration, Florida, July 2002.
[40] S. Hemami and R. M. Gray, Subband filters optimized for lost coefficient reconstruction [J], IEEE Transactions on Signal Processing, vol.45, No.3, pp.763-767, March 1997.
[41] Ali N. Akansu , Mark J. T. Smith, Subband and Wavelet Transforms: Design and Applications [M], Kluwer Academic Publishers, 1996.
[42] Kellermann W, Analysis and Design of Multirate Systems for Cancellation of Acoustical Echoes [J], Proc. IEEE ICASSP, New York, vol.5, pp.2570—2573, 1988.
[43] Pierre Moulin, Mihai Anitescu, K.O. Kortanek, Florian Potra, The Role of Linear Semi-Infinite Programming in Signal-Adapted QMF Bank Design [J], vol.45, No.9, pp.2160-2174, Sep. 1997.
[44] H. Xu, W. S Lu, and A. Antoniou, Efficient iterative design method for cosine-modulated QMF banks [J], IEEE Trans. Signal Processing, vol.44, No.7, pp.1657-1667, July 1996.
[45] Ajay Divakaran, William A. Pearlman, A Closed Form Expression for an Efficient Class of Quadrature Mirror Filters and Its FIR Approximation [J], IEEE Trans. on Circuits and Systems II: Analog and Digital Signal Processing, vol.43, No.3, pp.207-219, March 1996.
[46] C. K. Goh, and Y. C. Lim, An efficient algorithm to design weighted minimax perfect reconstruction quadrature mirror filter banks [J], IEEE Trans. on Signal Processing, vol.47, No.12, pp.3303-3314, Dec. 1999.
[47] A. Guiloire, M. Viterli, Adaptive filtering in subband with critical sampling: Analysis, experiments and application to acoustic echo cancellation [J], IEEE Trans. Signal processing, Vol.40, No.8, pp.1862-1875, Aug. 1992.
[48] Petraglia M. R., Alves R. G., Diniz P. S. R., New structures for adaptive filtering in subbands with critical sampling [J], Signal Processing, IEEE Trans. On Acoustics, Speech, and Signal Processing, vol.48, No.12, pp.3316-3327, Dec. 2000.
[49] Harteneck, M., Weiss, S., Stewart, R.W., Design of near perfect reconstruction oversampled filter banks for subband adaptive filters [J], IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol.46, No.8, pp.1081–1085, Aug. 1999.
[2] George-Othon Glentis, Kostas Berberidis and Sergios Theodoridis, Efficient Least Squares Adaptive Algorithms For FIR Transversal Filtering, IEEE SIGNAL PROCESSING MAGAZINE, 1053-5888/99, JULY 1999.
[3] Ki-Ho Kim and Edward J. Powers, Analysis of Initialization and Numerical Instability of Fast RLS Algorithms, CH2977-7/91/0000-1861,1991 IEEE.
[4] André Gilloire; Thierry Pétillon and Sergios Theodordis, Acoustic Echo Cancellation Using Fast RLS Adaptive Filters with Reduced Complexity, 0-7803-0593-0/92, 1992 IEEE.
[5] Alberto Carini and Enzo Mumolo, A Numerically Stable Fast RLS Algorithm for Adaptive Filtering and Prediction Based on the UD Factorization, 1053–587X/99, 1999 IEEE
[6] I.D.Skidmore and I.K.Proudler, KAGE: A New FAST RLS Algorithm, 0-7803-7041-4/10, 2001 IEEE
[7]Erlendur Karlsson, and Monson H. HAYES, Least Squares ARMA Modeling of Linear Time-Varying Systems: Lattice Filter Structures and .Fast RLS Algorithms, 0096-3518/87/0700-0994, 1987 IEEE
[8] 何振亚,自适应信号处理[M],北京:科学出版社,2002
[9] ITU-T G.165, Echo cancellers [S], 1993.
[10] M. M. Sondhi and D. A. Berkley, Silencing echo on the telephone network [J],Proc. IEEE, Vol.68, No.8, pp. 948-963, Aug. 1980.
[11] Steven L. Gay, Jacob Benesty, Acoustic Signal Processing forTelecommunication [M], Kluwer Academic Publishers, Boston Hardbound,2000.
[12] D. R. Morgan and S. G. Kratzer, On a Class of Computationally Efficient,Rapidly Converging, Generalized NLMS Algorithm [J], IEEE Signal Processing Letters, vol.3, No.8, pp.245-247, Aug. 1996.
[13] Yasukawa H., Shimada S., Furukrawa I., etc. Acoustic echo canceller with high speech quality [J], Proc. ICASSP’87, pp.2125-2128, 1987.
[14] Gitlin R. D., Weinstein S. D., The effects of large interference on the tracking capability of digitally implemented echo cancellers [J], IEEE Trans. on COM, No.6, pp.833-839, Jun. 1978.
[15] Raymond H. Kwong, Edward W. Johnston, A Variable Step Size LMS Algorithm[J], IEEE Trans. on Signal Processing, vol. 40, No.9, pp.1633-1642, Jul. 1992.
[16] Tyseer Aboulnast and K. Mayyas, A Robust Variable Step-Size LMS-Type Algorithm: Analysis and Simulations [J], IEEE Trans. on Signal Processing, vol.45, No.3, pp.631-639, Mar. 1997.
[17] H. Ye and B. X. Wu, A new double-talk detectors algorithm based on theorthogonality theorem [J], IEEE Trans. Commun., Vol.39, pp.1542-1545, Nov. 1991.
[18] Jacob Benesty, Dennis R. Morgan, and Jun H. Cho, A family of Doubletalk Detectors Based on Cross-Correlation [J], Proc. IWAENC, pp.108-111, Sep.1999.
[19] J. H. Cho, Dennis R. Morgan, and Jacob Benesty, An objective technique for evaluating doubletalk detectors in acoustic cancellers [J], IEEE Trans. on Speech and Audio Processing, Vol.7, No.11, pp.718-724, Nov. 1999.
[20] J. C. Jenq and S. F. Hsieh, Acoustic echo cancellation using iterative-maximal-length correlation and double-talk detection [J], IEEE Trans. Speech Audio Processing, vol.9, No.11, pp.932-942, Nov. 2001.
[21] J. Borish and J. B. Angell, An efficient algorithm for measuring the impulse response using pseudorandom noise [J], J. Audio Eng. Soc., vol. 31, No.7, pp.478-488, Jul. 1983.
[22] Ajay Divakaran, William A. Pearlman, A Closed Form Expression for an Efficient Class of Quadrature Mirror Filters and Its FIR Approximation [J], IEEE Trans. on Circuits and Systems II: Analog and Digital Signal Processing, vol.43, No.3, pp.207-219, March 1996.
[23] C. K. Goh, and Y. C. Lim, An efficient algorithm to design weighted minimax perfect reconstruction quadrature mirror filter banks [J], IEEE Trans. on Signal Processing, vol.47, No.12, pp.3303-3314, Dec. 1999.
[24] A. Guiloire, M. Viterli, Adaptive filtering in subband with critical sampling:Analysis, experiments and application to acoustic echo cancellation [J], IEEE Trans. Signal processing, Vol.40, No.8, pp.1862-1875, Aug. 1992.
[25] Petraglia M. R., Alves R. G., Diniz P. S. R., New structures for adaptive filtering in subbands with critical sampling [J], Signal Processing, IEEE Trans. On Acoustics, Speech, and Signal Processing, vol.48, No.12, pp.3316-3327, Dec.2000.
[26] Harteneck, M., Weiss, S., Stewart, R.W., Design of near perfect reconstruction oversampled filter banks for subband adaptive filters [J], IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol.46, No.8, pp.1081–1085, Aug. 1999.
[27] Dimitris G. Manolakis, Vinay k. Ingle and Stephen M. Kogon, Statistical and Adaptive Signal Processing, 电子工业出版社,2003年5月
[28] B. C. J. Moore, An Introduction to the Psychology of Hearing [M], Academic Press, London, 1989.
[29] A. Gilloire and V. Turbin, Using auditory properties to improve the behavior of stereophonic acoustic echo cancellers [J], Proc. ICASSP, pp.3681-3684, 1998.
[30] M. G. Bellanger, Adaptive digital filters and signal analysis [M], Marcel Dekker, Inc., 1987.
[31] T. Hoya, Y. Loke, J. A. Chambers and P. A. Naylor, Application of the leaky extended LMS (XLMS) algorithm in stereophonic acoustic echo cancellation [J], Signal Processing, vol.64, pp.87-91, 1998.
[32] J. Benesty, D. R. Morgan and M. M. Sondhi, Stereophonic acoustic echo cancellation using nonlinear transformations and comb filtering [J], Proc. ICASSP’98, pp.3673-3676, 1998.
[33] M. Ali., Stereophonic acoustic echo cancellation system using time-varyingAll-pass filtering for signal decorrelation [J], Proc. ICASSP’98, pp.3689-3692, 1998.
[34] Y. Joncour and A. Sugiyama, A Stereo echo canceller with pre-processing for correct echo-path identification [J], Proc. ICASSP’98, p.3677-3680, 1998.
[35] T. Gaensler and P. Eneroth, Influence of audio coding on stereophonic acoustic cancellation [J], IEEE, ICASSP, pp.3649-3652, 1998.
[36] T. N. Yensen, R. A. Goubran and I. Lambadaris, Synthetic stereo acoustic echo cancellation structure for multiple participant VoIP conferences [J], IEEE Transactions on Speech and Audio Processing, vol. 9, No. 2, 2001, pp. 168-174.
[37] T. Yensen, J. Ryan, R. Goubran and I. Lambadaris, Synthetic stereo acoustic echo cancellation structure with microphone array beamforming for VoIP conferences [J], Proc. IEEE ICASSP2000, pp.817-820, 2000.
[38] T. Okuno and M. O. Tokhi, Stereophonic acoustic echo cancellation using blind source separation as post-processing [J], Processings of Seventh International Workshop on Acoustic Echo and Noise Control, Darmstadt, Germany, pp.75-78, September 2001.
[39] T. Okuno and M. O. Tokhi, Stereophonic acoustic echo cancellation using blind source separation [J], Processing of Ninth International Congress on Sound and Vibration, Florida, July 2002.
[40] S. Hemami and R. M. Gray, Subband filters optimized for lost coefficient reconstruction [J], IEEE Transactions on Signal Processing, vol.45, No.3, pp.763-767, March 1997.
[41] Ali N. Akansu , Mark J. T. Smith, Subband and Wavelet Transforms: Design and Applications [M], Kluwer Academic Publishers, 1996.
[42] Kellermann W, Analysis and Design of Multirate Systems for Cancellation of Acoustical Echoes [J], Proc. IEEE ICASSP, New York, vol.5, pp.2570—2573, 1988.
[43] Pierre Moulin, Mihai Anitescu, K.O. Kortanek, Florian Potra, The Role of Linear Semi-Infinite Programming in Signal-Adapted QMF Bank Design [J], vol.45, No.9, pp.2160-2174, Sep. 1997.
[44] H. Xu, W. S Lu, and A. Antoniou, Efficient iterative design method for cosine-modulated QMF banks [J], IEEE Trans. Signal Processing, vol.44, No.7, pp.1657-1667, July 1996.
[45] Ajay Divakaran, William A. Pearlman, A Closed Form Expression for an Efficient Class of Quadrature Mirror Filters and Its FIR Approximation [J], IEEE Trans. on Circuits and Systems II: Analog and Digital Signal Processing, vol.43, No.3, pp.207-219, March 1996.
[46] C. K. Goh, and Y. C. Lim, An efficient algorithm to design weighted minimax perfect reconstruction quadrature mirror filter banks [J], IEEE Trans. on Signal Processing, vol.47, No.12, pp.3303-3314, Dec. 1999.
[47] A. Guiloire, M. Viterli, Adaptive filtering in subband with critical sampling: Analysis, experiments and application to acoustic echo cancellation [J], IEEE Trans. Signal processing, Vol.40, No.8, pp.1862-1875, Aug. 1992.
[48] Petraglia M. R., Alves R. G., Diniz P. S. R., New structures for adaptive filtering in subbands with critical sampling [J], Signal Processing, IEEE Trans. On Acoustics, Speech, and Signal Processing, vol.48, No.12, pp.3316-3327, Dec. 2000.
[49] Harteneck, M., Weiss, S., Stewart, R.W., Design of near perfect reconstruction oversampled filter banks for subband adaptive filters [J], IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol.46, No.8, pp.1081–1085, Aug. 1999.