## Electronic Theses and Dissertations

2010

Dissertation

Ph.D.

#### Department

Electrical and Computer Engineering

Kwan, Hon (Electrical and Computer Engineering)

#### Keywords

Engineering, Electronics and Electrical.

CC BY-NC-ND 4.0

#### Abstract

Digital filters play an important role in digital signal processing and communication. From the 1960s, a considerable number of design algorithms have been proposed for finite-duration impulse response (FIR) digital filters and infinite-duration impulse response (IIR) digital filters. Compared with FIR digital filters, IIR digital filters have better approximation capabilities under the same specifications. Nevertheless, due to the presence of the denominator in its rational transfer function, an IIR filter design problem cannot be easily formulated as an equivalent convex optimization problem. Furthermore, for stability, all the poles of an IIR digital filter must be constrained within a stability domain, which, however, is generally nonconvex. Therefore, in practical designs, optimal solutions cannot be definitely attained. In this dissertation, we focus on IIR filter design problems under the weighted least-squares (WLS) and minimax criteria. Convex optimization will be utilized as the major mathematical tool to formulate and analyze such IIR filter design problems. Since the original IIR filter design problem is essentially nonconvex, some approximation and convex relaxation techniques have to be deployed to achieve convex formulations of such design problems. We first consider the stability issue. A sufficient and necessary stability condition is derived from the argument principle. Although the original stability condition is in a nonconvex form, it can be appropriately approximated by a quadratic constraint and readily combined with sequential WLS design procedures. Based on the sufficient and necessary stability condition, this approximate stability constraint can achieve an improved description of the nonconvex stability domain. We also address the nonconvexity issue of minimax design of IIR digital filters. Convex relaxation techniques are applied to obtain relaxed design problems, which are formulated, respectively, as second-order cone programming (SOCP) and semidefinite programming (SDP) problems. By solving these relaxed design problems, we can estimate lower bounds of minimum approximation errors, which are useful in subsequent design procedures to achieve real minimax solutions. Since the relaxed design problems are independent of local information, compared with many prevalent design methods which employ local search, the proposed design methods using the convex relaxation techniques have an increased chance to obtain an optimal design.

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