Proportionate Adaptive Filters From a Basis Pursuit Perspective

In this letter, we show that the normalized least-mean-square (NLMS) algorithm and the affine projection algorithm (APA) can be decomposed as the sum of two orthogonal vectors. One of these vectors is derived from an ℓ2-norm optimization problem while the other one is simply a good initialization vector. By replacing this optimization with the basis pursuit, which is based on the ℓ1-norm optimization, we derive the proportionate NLMS (PNLMS) algorithm and the proportionate APA (PAPA). Many other adaptive filters can be derived following this approach, including new ones.

[1]  Kazuhiko Ozeki,et al.  An adaptive filtering algorithm using an orthogonal projection to an affine subspace and its properties , 1984 .

[2]  Shoji Makino,et al.  Exponentially weighted stepsize NLMS adaptive filter based on the statistics of a room impulse response , 1993, IEEE Trans. Speech Audio Process..

[3]  Bhaskar D. Rao,et al.  Adaptive filtering algorithms for promoting sparsity , 2003, 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2003. Proceedings. (ICASSP '03)..

[4]  Donald L. Duttweiler,et al.  Proportionate normalized least-mean-squares adaptation in echo cancelers , 2000, IEEE Trans. Speech Audio Process..

[5]  Michael A. Saunders,et al.  Atomic Decomposition by Basis Pursuit , 1998, SIAM J. Sci. Comput..

[6]  S. Haykin,et al.  Adaptive Filter Theory , 1986 .

[7]  Akihiko Sugiyama,et al.  A fast convergence algorithm for adaptive FIR filters under computational constraint for adaptive tap-position control , 1996 .

[8]  Jacob Benesty,et al.  Sparse Adaptive Filters for Echo Cancellation , 2010, Synthesis Lectures on Speech and Audio Processing.

[9]  Akihiko Sugiyama,et al.  A generalized proportionate variable step-size algorithm for fast changing acoustic environments , 2004, 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[10]  Dennis R. Morgan,et al.  On a class of computationally efficient, rapidly converging, generalized NLMS algorithms , 1996, IEEE Signal Processing Letters.

[11]  Emre Ertin,et al.  On the Relation Between Sparse Reconstruction and Parameter Estimation With Model Order Selection , 2010, IEEE Journal of Selected Topics in Signal Processing.

[12]  Iven M. Y. Mareels,et al.  LMS estimation via structural detection , 1998, IEEE Trans. Signal Process..