## Optimal Deconvolution based on Polynomial Methods

IEEE Transactions on Acoustics, Speech and Signal Processing,
vol ASSP-37, pp 217-226, February 1989. © 1989 IEEE.

Paper available in PDF (718k)

Outline:
Deconvolution, or input estimation, is intricate for at least two reasons: measurements are usually noise corrupted and the system is frequently nonminimum phase. These properties restrict the use of the simplest deconvolution filter, namely the inverse system. The paper discussed a systematic Wiener design of deconvolution estimators which utilizes polynomial equations.

Abstract:
The problem of estimating the input to a known linear system is treated in a shift operator polynomial formulation. The mean square estimation error is to be minimized. The input and a colored measurement noise are described by independent ARMA processes. The filter is calculated by performing a spectral factorization and solving a polynomial equation.

The approach covers input prediction, filtering and smoothing problems, and the use of prefilters in the quadratic criterion. It is applicable to nonminimum phase as well as unstable systems. This is illustrated by two examples. The possible applications range from seismic signal processing and linear MMSE equalization to numerical differentiation of noisy signals.

Matlab m-files
for polynomial equation design of scalar IIR Wiener deconvolution estimators:
fnfilt.m Function for estimator design.
abstar.m Used by fnfilt.m.
addcent.m Used by fnfilt.m.
spefac2.m Spectral factorization, by polynomial roots, used by fnfilt.m
polysolve.m Solution of Diophantine equation. used by fnfilt.m

Related publications:
Conference paper in SPIE'91 on adaptive deconvolution based on spectral decomposition.
Paper in IEEE Trans. SP 1991 on differentiation of noisy data.
Paper in IJC 1993 on the duality between deconvolution and feedforward control.
Paper in Automatica 1993, which includes robust design based on averaged MSE criteria.
Paper in IEEE Trans. AC 1995, which discussed robust design of multivariable estimators.

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