Reduced-Order Generalized $H_{\infty}$ Filtering for Linear Discrete-Time Systems With Application to Channel Equalization

This paper investigates the problem of reduced-order generalized $H_{\infty}$ filtering for linear discrete-time systems. Generalized $H_{\infty}$ filtering covers standard $H_{\infty}$ filtering as a special case and can make use of the frequency-domain characteristics of practical signals. Moreover, reduced-order filters are more attractive than full-order ones regarding implementability. By virtue of the generalized bounded real lemma, a new necessary and sufficient condition is proposed for analyzing the generalized $H_{\infty}$ performance of the filtering error system. To compute reduced-order generalized $H_{\infty}$ filters, a necessary and sufficient condition is then derived in terms of matrix inequalities, to solve which iterative linear matrix inequality algorithms are further developed. In addition, the proposed filter design method is applied to cope with the channel equalization problem. Examples are finally presented for illustrating the proposed method.