Finite Frequency $H_{\infty }$ Deconvolution With Application to Approximated Bandlimited Signal Recovery

Abstract

This paper investigates a finite frequency ${H_{\infty }}$ deconvolution problem for single-input–single-output discrete-time systems. Motivated by the fact that many practical signals are located in a finite frequency range, this paper is focused on developing a design method for achieving better deconvolution performance over a finite frequency range. First, a finite frequency ${H_{\infty }}$ deconvolution problem is formulated such that the deconvolution specification is characterized by a modified $H_{\mathbf {\infty }}$ index associated with the corresponding finite frequency range. By an adapted generalized Kalman–Yakubovich–Popov lemma and an auxiliary vector approach, a matrix inequality condition is obtained for parameterizing a required deconvolution filter, based on which a simple iterative algorithm is then constructed for optimizing the filter performance. Multiobjective deconvolution is further discussed in the same framework. Compared with the traditional frequency weighting strategy, the proposed method can directly analyze and synthesize the deconvolution performance within finite frequency ranges, the advantage of which is demonstrated in a bandlimited signal recovery example.