Hashemi-Zahan, Saeid
(1998)
Inversion of non-minimum phase systems in signal processing.
*University of Southampton, Doctoral Thesis*.

## Abstract

The work presented in this thesis is concerned with inversion of non-minimum phase systems, in both time and frequency domains, but with the emphasis on the time domain. The inversion of a causal linear system in the time domain may be described in matrix terms, the system matrix so formed has a lower triangular Toeplitz form. The problem is to find the inverse matrix or an approximation to it. Various difficulties may arise in any application. The most fundamental issue relates to the fact that the operator may be ill-conditioned. Ill-conditioning relates to the fact that results are highly sensitive to perturbations in data and system.

The inverse of the system operator may be created using singular value decomposition, and the condition number quantifies the "sensitivity" of the inversion. The magnitude of singular values of system matrix dramatically reflects the minimum or non-minimum phase structure of the system allowing clear interpretation of the magnitude of the condition number.

The problem of ill-conditioning of non-minimum phase systems and the relation of the phase structure of the system to the singular values of its system matrix is discussed. A related problem is that the system operator may only be known approximately. Further, noise may be present in the measurement of the output.

An extension to the Least Squares approximations of solving a set of linear equations in presence of noise is the Total Least Squares approach. Among several other alternatives to the least squares technique is the use of absolute value error criteria, or L_{1}-norm minimisation. L_{2}-norm minimisation technique is less effective than the L_{1}-norm at recovering a spike train.

Finally, the application of these techniques to data from an actual mechanical system consisting of a cantilever beam, driven by an electromagnetic shaker is demonstrated.

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