Image Processing Reference
In-Depth Information
10
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 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.
2. Preliminaries
In this chapter, we frequently use notations in control systems. For readers who are not
familiar to these, we here recall basic notations and facts of control systems used throughout
the chapter. We also show MATLAB codes for better understanding.
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