Optimal Data Compression and Filtering: The Case of Infinite Signal Sets

Authors: Anatoli Torokhti, Phil Howlett

Abstract:

We present a theory for optimal filtering of infinite sets of random signals. There are several new distinctive features of the proposed approach. First, we provide a single optimal filter for processing any signal from a given infinite signal set. Second, the filter is presented in the special form of a sum with p terms where each term is represented as a combination of three operations. Each operation is a special stage of the filtering aimed at facilitating the associated numerical work. Third, an iterative scheme is implemented into the filter structure to provide an improvement in the filter performance at each step of the scheme. The final step of the concerns signal compression and decompression. This step is based on the solution of a new rank-constrained matrix approximation problem. The solution to the matrix problem is described in this paper. A rigorous error analysis is given for the new filter.

Keywords: stochastic signals, optimization problems in signal processing.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1076784

References:

[1] T. L. Boullion and P. L. Odell, Generized Inverse Matrices, John Willey & Sons, Inc., New York, 1972.
[2] N. Dunford and J. T. Schwartz, Linear Operators, Part 1, General Theory, Wiley Classics Library, Wiley, New York, 1988.
[3] C. Eckart and G. Young, The Approximation of One Matrix by Another of Lower Rank, Psychometrika, 1, 211-218, 1936.
[4] J.H. Ferziger, Numerical Methods for Engineering Applications, Wiley, New-York, 1999.
[5] V. N. Fomin and M. V. Ruzhansky, Abstract optimal linear filtering, SIAM J. Control Optim., 38, pp. 1334-1352, 2000.
[6] S. Friedland and A. P. Torokhti, Generalized rank-constrained matrix approximations, SIAM J. Matrix Anal. Appl., 29, issue 2, pp. 656-659, 2007.
[7] G.H. Golub and C.F. Van Loan, Matrix Computation, Johns Hopkins Univ. Press, 3rd Ed., 1996.
[8] S. Haykin, Adaptive Filter Theory, Prentice-Hall, Englewood Cliffs, N. J., 1991.
[9] V. Hutson and J.S. Pym, Applications of Functional Analysis and Operator Theory, Academic Press, London, 1980.
[10] I.T. Jolliffe, "Principal Component Analysis," Springer Verlag, New York, 1986.
[11] A.N. Kolmogorov and S.V. Fomin, Elements of the Theory of Functions and Functional Analysis, Dover, New York, 1999.
[12] J. Manton and Y. Hua, Convolutive reduced rank Wiener filtering, Proc. of ICASSP-01, 6, pp. 4001-4004, 2001.
[13] V. J. Mathews and G. L. Sicuranza, Polynomial Signal Processing, J. Wiley & Sons, 2001.
[14] L.L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis,New York: Addison-Wesley Publishing Co., 1990.
[15] A. Torokhti and P. Howlett, Optimal fixed rank transform of the second degree, IEEE Trans. on Circuits and Systems. Part II, Analog & Digital Signal Processing, 48, 309-315, 2001.
[16] A. Torokhti and P. Howlett, Method of recurrent best estimators of second degree for optimal filtering of random signals, Signal Processing, 83, 5, 1013 - 1024, 2003.
[17] A. Torokhti and P. Howlett, Best approximation of identity mapping: the case of variable memory, J. Approx. Theory, 143, 1, 111-123, 2006.
[18] A. Torokhti and P. Howlett, Computational Methods for Modelling of Nonlinear Systems, Elsevier, 2007.
[19] N. Wiener, The Extrapolation, Interpolation and Smoothing of Stationary Time Series with Engineering Applications, Academic Press, New York, 1949.