Min-Max Design of FIR Digital Filters by Semidefinite Programming

Robustness is a fundamental issue in signal processing; unmodeled dynamics and unexpected noise in systems and signals are inevitable in designing systems and signals. Against such uncertainties, min-max optimization, or worst case optimization is a powerful tool. In this light, we propose an efficient design method of FIR (finite impulse response) digital filters for approximating and inverting given digital filters. The design is formulated by min-max optimization in the frequency domain. More precisely, we design an FIR filter which minimizes the maximum gain of the frequency response of an error system. This design has a direct relation with H∞ optimization (Francis, 1987). Since the space H∞ is not a Hilbert space, the familiar projection method in conventional signal processing cannot be applied. However, many studies have been made on the H∞ optimization, and nowadays the optimal solution to the H∞ problem is deeply analysed and can be easily obtained by numerical computation. Moreover, as an extension of H∞ optimization, a min-max optimization on a finite frequency interval has been proposed recently (Iwasaki & Hara, 2005). In both optimization, the Kalman-Yakubovich-Popov (KYP) lemma (Anderson, 1967; Rantzer, 1996; Tuqan & Vaidyanathan, 1998) and the generalized KYP lemma (Iwasaki & Hara, 2005) give an easy and fast way of numerical computation; semidefinite programming (Boyd & Vandenberghe, 2004). Semidefinite programming can be efficiently solved by numerical optimization softwares. In this chapter, we consider two fundamental problems of signal processing: FIR approximation of IIR (infinite impulse response) filters and inverse FIR filtering of FIR/IIR filters. Each problems are formulated in two types of optimization: H∞ optimization and finite-frequency min-max one. These problems are reduced to semidefinite programming in a similar way. For this, we introduce state-space representation. Semidefinite programming is obtained by the generalized KYP lemma. We will give MATLAB codes for the proposed design, and will show design examples.