Identifiability of the Deconvolution Problem

Anders Ahlén

Automatica, Special Issue on Identification, vol 26, pp 177-181, January 1990.

Outline:

A deconvolution filter estimates the input to a linear system. In some situations, the system dynamics may be known, while the properties of the input signal and the noise have to be estimated from output data, before a filter can be designed. The paper derives necessary and sufficient conditions for this to be possible.

Abstract:

A deconvolution problem is stated and its identifiability properties are analysed. The information assumed available consists of the spectral density of the output measurements and a dynamical description of the system. The input to the system and the measurement noise are modelled as independent ARMA processes.

In order to obtain an optimal estimate (in a mean square sense) of the input signal, the parameters of the input model and measurement noise are required either to be known a priori or correctly estimated. Since the normal situation requires estimation of the parameters, a successful estimate of the input signal depends crucially on the identifiability properties.

Hence, necessary and sufficient conditions for parameter identifiability are derived, in terms of model orders, with the aid of spectral factorization. An example shows that when these conditions are violated, only generic results can be obtained. A sufficient condition for generic parameter identifiability is given.

Related publications:

Conference paper , IFAC World Congress 1988, on the above theme.
Paper in IEEE Trans. ASSP 1989 on the design of linear scalar deconvolution estimators.
Conference paper in IFAC ACASP 1989 on adaptive deconvolution.
Conference paper in SPIE 1991 on adaptive deconvolution.