本論文提出一個時域欠定的多通道濾波器(time-domain underdetermined multichannel inverse filtering, TUMIF)應用於前饋式架構下的主動式管路消噪。在傳統方法例如濾波x最小均方誤差演算法(filtered-x least-mean-square , FXLMS)，基於單一通道下的前饋式主動消噪控制可被表示成一個超定(overdetermined)的反濾波器問題，這導致了殘餘的噪音，為了處理此問題，一個多通道的控制方法依據著名的多進多出反濾波器原理(multiple-input/output inverse theorem, MINT)被修正從向量子空間的觀點與模型匹配架構著手，靠著多個次要聲源，此問題被轉化成一個欠定(underdetermined)的問題，這類問題擁有無限多組精確解而沒有殘餘噪音，但是由最小平方誤差法(least-square method, LS)得到的有限的脈衝響應(finite impulse response, FIR)濾波器係數過長而無法即時實現，為了克服這問題，壓縮感知(compressed sensing , CS)技術像是least absolute shrinkage and selection operator (LASSO)演算法與orthogonal matching pursuit (OMP)被用來降低控制器的階數，模擬和實驗證明時域欠定的多通道濾波器可以達到更好的消噪效果且比起傳統方法有著較低的複雜度。

This thesis proposes a time-domain underdetermined multichannel inverse filtering (TUMIF) technique for active feedforward control of noise in ducts. In traditional approaches such as the filtered-x least-mean-square (FXLMS), the feedforward control problem is formulated as an overdetermined inverse filtering problem which generally leads to non-zero residual noise. To address the problem, a multichannel control approach based on the celebrated multiple-input/output inverse theorem (MINT) is revised from the perspectives of vector subspaces and model-matching framework. By introducing multiple secondary sources, the problem can be reformulated into an underdetermined system, which admits infinite number of exact solutions with zero residual noise. However, the finite impulse response (FIR) filter coefficients obtained using the least-square (LS) method tend to be too many to admit real-time implementation. To overcome this difficulty, a compressed sensing (CS) techniques, the least absolute shrinkage and selection operator (LASSO) algorithm and the orthogonal matching pursuit (OMP), are exploited to reduce the controller orders. Simulation and experiment results demonstrated that the TUMIF approach has achieved significantly higher noise reduction with much lower complexity than the conventional approaches.

[1] S. M. Kuo and D. R. Morgan, Active Noise Control Systems: Algorithms and DSP Implementations. New York, USA: Wiley, 1996, pp. 58-76.
[2] S. M. Kuo and D. R. Morgan, “Review of DSP algorithms for active noise control,” Proceedings of the 2000. IEEE International Conference on Control Applications, pp. 243-248, Sep. 2000.
[3] C. M. Harris, Handbook of Acoustical Measurements and Noise Control. New York, USA: McGraw-Hill, 1991.
[4] Process of silencing sound oscillations, by P. Lueg. (1936, June 9). U.S. Patent 2043416.
[5] P. A. Nelson and S. J. Elliott, Active Control of Sound. San Diego, CA: Academic, 1992, pp. 161-203.
[6] N. Miyazaki and Y. Kajikawa, “Adaptive feedback ANC system using virtual microphones,” 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, Vancouver, BC, pp. 383-387, May 2013.
[7] B. Rafaely and M. Jones, “Combined feedback-feedforward active noise-reducing headset - the effect of the acoustics on broadband performance,” Journal of the Acoustical Society of America, vol. 112, no. 3, pp.981-989, 2002.
[8] T. K. Roy and M. Morshed, “Active noise control using filtered-x LMS and feedback ANC filter algorithms,” 2013 2nd International Conference on Advances in Electrical Engineering (ICAEE), Dhaka, pp. 7-12, 2013.
[9] W. K. Tseng, B. Rafaely and S. J. Elliott, “Combined feedback-feedforward active control of sound in a room,” Journal of the Acoustical Society of America, vol. 104, no. 6, pp. 3417-3425, 1988.
[10] B. Widrow and S. D. Stearns, Adaptive Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1985, pp. 99-114.
[11] M. Miyoshi and Y. Kaneda, “Inverse filtering of room acoustics,” IEEE Trans. Acoustics, Speech, and Signal Processing, vol. 36, no. 2, pp. 145-152, Feb. 1988.
[12] A. N. Tikhonov, “Solution of nonlinear integral equations of the first kind,” Soviet Math. Dokl, vol. 5, pp. 835-838, 1964.
[13] M. Bertero, T. A. Poggio and V. Torre, “Ill-posed problems in early vision,” in Proceedings of the IEEE, vol. 76, no. 8, pp. 869-889, Aug. 1988.
[14] J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Processing Mag, vol. 25, no. 2, pp. 21-30, 2008.
[15] G. F. Edelmann and C. F. Gaumond, “Beamforming using compressive sensing,” The Journal of the Acoustical Society of America, vol. 130, no. 4, pp. EL232-EL237, 2011.
[16] R. Tibshirani, “Regression Shrinkage and Selection via the Lasso,” Journal of the Royal Statistical Society, vol. 58, no. 4, pp. 267-288, 1996.
[17] T. T. Cai and L. Wang, “Orthogonal Matching Pursuit for Sparse Signal Recovery With Noise,” in IEEE Transactions on Information Theory, vol. 57, no. 7, pp. 4680-4688, July 2011.
[18] Y. C. Pati, R. Rezaiifar and P. S. Krishnaprasad, “Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition,” Proceedings of 27th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, USA, pp. 40-44, 1993.
[19] G. Strang, Linear algebra and its applications. Boston, USA: Cengage Learning, 2006, pp. 141-151.
[20] S. Boyd and L. Vandenberghe, Convex Optimization. New York, USA: Cambridge University Press, 2004, pp. 69-102.
[21] M. R. Bai and C. C. Chen, “Application of Convex Optimization to Acoustical Array Signal Processing,” J. Sound Vibration, vol. 332, no. 5, pp. 6596-6616, 2013.
[22] M. Grant, and S. Boyd. (2013 Sept). CVX, version 2.0 beta, MATLAB software for disciplined convex programming. [Online]. Available: http://cvxr.com/cvx.
[23] T. Habib, M. T. Akhtar and M. Arif, “Acoustic Feedback Path Modeling and Neutralization in Active Noise Control Systems,” 2006 IEEE International Multitopic Conference, Islamabad, pp. 89-93, 2006.
[24] T. Habib and M. Tufail, “Online acoustic feedback path modeling in multi-channel active noise control systems using Variable Step-Size algorithm,” 2009 IEEE/SP 15th Workshop on Statistical Signal Processing, Cardiff, pp. 177-180, 2009.
[25] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, Upper Saddle River, USA: Pearson, 2007, pp. 347-350.