Design of a Class of Multirate Systems Using a Maximum Relativè 2 -error Criterion an Example of a Rate-changing Multirate System. 1. Introduction

A criterion for designing the class of multirate systems for rate-changing is presented. This criterion arises from a model-matching perspective with maximum relativè 2-error over a general class of inputs and is a natural extension of the standard Chebyshev method for lter design. Using multirate and convex analysis techniques, the criterion is shown to lead to a convex matrix-valued-function approximation problem. An algorithm using convex optimization is proposed to solve the problem. An example illustrates the use of the algorithm and eeectiveness over standard lter-design techniques. The problem of designing the multirate system shown in Figure 1 arises when dealing with discrete-time signals at diierent sampling rates. In this paper, we focus on the design of such structures along with a related generalization|the class of systems, M, which are linear, continuous, and satisfy S L M = MS M where S is the shift operator and L 6 = M. We call M a rate-changing system. We explore a criterion for the design of rate-changing systems. Criteria using a speciic function class 1], stochastic inputs 1, 2, 3], and approximating function methods 4, 5] have been considered for general multirate systems. If little is known about the class of inputs, a general approach is preferrable. In order to address this concern, we explore thè 2 operator norm as a criterion for the design of these systems. This error criterion was recently examined for general multirate systems design 6, 7] and has been used for the design of decimators and interpolators 7]; we analyze its application to rate-changing systems. This error measure also appears in control theory 8] where it is used for control related design; H 1 optimization methods have recently been applied to the design of multirate lter banks 9]. Our design method is diierent and is based upon a modern convex optimization algorithm. Section 2 explores the structure of the problem. We motivate rate-changing systems by presenting two possible applications. Rate-changing systems are then presented in a matrix form. The relation between this matrix system and multirate systems is given. We remark that the general process of representing multirate systems as matrix systems has been studied by many authors 10, 11, 12]. Our analysis establishes the relations between several structures|the matrix form, the commutator form, and the standard form. 3 Section 3 presents a new analysis of a criterion for the design of rate-changing systems. We begin by …