# 0.11 Introduction to infinite impulse response filters

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Chapter [link] introduced the problem of designing ${l}_{p}$ FIR filters, along with several design scenarios and their corresponding design algorithms. This chapter considers the design of ${l}_{p}$ IIR filters and examines the similarities and differences compared to ${l}_{p}$ FIR filter design. It was mentioned in [link] that ${l}_{p}$ FIR design involves a polynomial approximation. The problem becomes more complicated in the case of IIR filters as the approximation problem is a ratio of two polynomials. In fact, the case of FIR polynomial approximation is a special form of IIR rational approximation where the denominator is equal to 1.

Infinite Impulse Response (or recursive ) digital filters constitute an important analysis tool in many areas of science (such as signal processing, statistics and biology). The problem of designing IIR filters has been the object of extensive study. Several approaches are typically used in designing IIR filters, but a general procedure follows: given a desired filter specification (which may consist of an impulse response or a frequency specification), a predetermined approximation error criterion is optimized. Although one of the most widely used error criteria in Finite Impulse Response (FIR) filters is the least-squares criterion (which in most scenarios merely requires the solution of a linear system), least-squares ( ${l}_{2}$ ) approximation for IIR filters requires an optimization over an infinite number of filter coefficients (in the time domain approximation case). Furthermore, optimizing for an IIR frequency response leads to a rational (nonlinear) approximation problem rather than the polynomial problem of FIR design.

As discussed in the previous chapter, a successful IRLS-based ${l}_{p}$ algorithm depends to a large extent in the solution of a weighted ${l}_{2}$ problem. One could argue that one of the most important aspects contrasting FIR and IIR ${l}_{p}$ filter design lies in the ${l}_{2}$ optimization step. This chapter presents the theoretical and computational issues involved in the design of both ${l}_{2}$ and ${l}_{p}$ IIR filters and explores several approaches taken to handle the resulting nonlinear ${l}_{2}$ optimization problem. [link] introduces the IIR filter formulation and the nonlinear least-squares design problem. [link] presents the ${l}_{2}$ problem more formally, covering relevant methods as a manner of background and to lay down a framework for the approach proposed in this work. Some of the methods covered here date back to the 1960's, yet others are the result of current active work by a number of research groups; the approach employed in this work is described in [link] . Finally, [link] considers different design problems concerning IIR filters in an ${l}_{p}$ sense, including IIR versions of the complex, frequency-varying and magnitude filter design problems as well as the proposed algorithms and their corresponding results.

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