Generic Filtering of Infinite Sets of Stochastic Signals
A theory for optimal filtering of infinite sets of random
signals is presented. There are several new distinctive features of the
proposed approach. First, a single optimal filter for processing any
signal from a given infinite signal set is provided. 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 scheme 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.
[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] S. Friedland and A. P. Torokhti, Generalized rank-constrained matrix
approximations, SIAM J. Matrix Anal. Appl., 29, issue 2, pp. 656-659,
2007.
[5] G.H. Golub and C.F. Van Loan, Matrix Computation, Johns Hopkins
Univ. Press, 3rd Ed., 1996.
[6] S. Haykin, Adaptive Filter Theory, Prentice-Hall, Englewood Cliffs, N.
J., 1991.
[7] V. Hutson and J.S. Pym, Applications of Functional Analysis and Operator
Theory, Academic Press, London, 1980.
[8] I.T. Jolliffe, "Principal Component Analysis," Springer Verlag, New York,
1986.
[9] J. Manton and Y. Hua, Convolutive reduced rank Wiener filtering, Proc.
of ICASSP-01, 6, pp. 4001-4004, 2001.
[10] V. J. Mathews and G. L. Sicuranza, Polynomial Signal Processing, J.
Wiley & Sons, 2001.
[11] L.L. Scharf, Statistical Signal Processing: Detection, Estimation, and
Time Series Analysis,New York: Addison-Wesley Publishing Co., 1990.
[12] 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.
[13] 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.
[14] A. Torokhti and P. Howlett, Computational Methods for Modelling of
Nonlinear Systems, Elsevier, 2007.
[15] N. Wiener, The Extrapolation, Interpolation and Smoothing of Stationary
Time Series with Engineering Applications, Academic Press, New
York, 1949.
[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] S. Friedland and A. P. Torokhti, Generalized rank-constrained matrix
approximations, SIAM J. Matrix Anal. Appl., 29, issue 2, pp. 656-659,
2007.
[5] G.H. Golub and C.F. Van Loan, Matrix Computation, Johns Hopkins
Univ. Press, 3rd Ed., 1996.
[6] S. Haykin, Adaptive Filter Theory, Prentice-Hall, Englewood Cliffs, N.
J., 1991.
[7] V. Hutson and J.S. Pym, Applications of Functional Analysis and Operator
Theory, Academic Press, London, 1980.
[8] I.T. Jolliffe, "Principal Component Analysis," Springer Verlag, New York,
1986.
[9] J. Manton and Y. Hua, Convolutive reduced rank Wiener filtering, Proc.
of ICASSP-01, 6, pp. 4001-4004, 2001.
[10] V. J. Mathews and G. L. Sicuranza, Polynomial Signal Processing, J.
Wiley & Sons, 2001.
[11] L.L. Scharf, Statistical Signal Processing: Detection, Estimation, and
Time Series Analysis,New York: Addison-Wesley Publishing Co., 1990.
[12] 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.
[13] 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.
[14] A. Torokhti and P. Howlett, Computational Methods for Modelling of
Nonlinear Systems, Elsevier, 2007.
[15] N. Wiener, The Extrapolation, Interpolation and Smoothing of Stationary
Time Series with Engineering Applications, Academic Press, New
York, 1949.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:51338", author = "Anatoli Torokhti and Phil Howlett", title = "Generic Filtering of Infinite Sets of Stochastic Signals", abstract = "A theory for optimal filtering of infinite sets of random
signals is presented. There are several new distinctive features of the
proposed approach. First, a single optimal filter for processing any
signal from a given infinite signal set is provided. 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 scheme 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 = "Optimal filtering, data compression, stochastic signals.", volume = "3", number = "6", pages = "417-13", }