**Min-Max Design of FIR Digital Filters by Semidefinite Programming**

Masaaki Nagahara *Kyoto University Japan*

### **1. Introduction**

*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*<sup>∞</sup> *optimization* (Francis, 1987). Since the space *H*<sup>∞</sup> 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*<sup>∞</sup> optimization, and nowadays the optimal solution to the *H*<sup>∞</sup> problem is deeply analysed and can be easily obtained by numerical computation. Moreover, as an extension of *H*<sup>∞</sup> 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*<sup>∞</sup> 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.
