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Designing IIR Filters with a Given 3 dB Point
[chapter]

Ricardo A. Losada, Vincent Pellissier

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Streamlining Digital Signal Processing
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O ften in infinite impulse response (IIR) filter design, our critical design parameter is the cutoff frequency at which the filter's power decays to half (−3 dB) the nominal passband value. This article presents techniques that aid in the design of discrete-time Chebyshev and elliptic filters given a 3-dB attenuation frequency point. These techniques place Chebyshev and elliptic filters on the same footing as Butterworth filters, which traditionally have been designed for a given 3-dB point.
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... result is that it is easy to replace a Butterworth design with either a Chebyshev or an elliptic filter of the same order and obtain a steeper rolloff at the expense of some ripple in the passband and/or stopband of the filter. PROBLEM STATEMENT We begin by presenting a technique that solves the problem of designing discrete-time Chebyshev type I and II IIR filters given a 3-dB attenuation frequency point. Traditionally, to design a lowpass Chebyshev (type I) IIR filter, we start with the following set of desired specifications: {N, ω p , A p }, where N is the filter order, ω p is the passband-edge frequency, and A p is the desired attenu-ation at ω p (see Figure 1 ). Yet, the problem is that it's impractical to set A p = 3 dB and design for the specification set {N, ω p , 3}; due to the filter's equiripple behavior, all the ripples in the passband would reach the −3 dB point, yielding intolerable passband ripple. To solve this problem, our designs are based on analytic relations that can be found in the analog domain between the passband-edge frequency and the 3-dB cutoff frequency in the case of type I Chebyshev filters and between the stopband-edge frequency and the 3-dB cutoff frequency in the case of type II Chebyshev filters. We use the inverse bilinear transformation to map the specifications given in the digital domain into analog specifications. We then use the analytic relation between the frequencies mentioned earlier to translate a set of specifications into another set that we know how to handle. Finally, we use the bilinear transformation to map the new set of specifications back to the digital domain. In the case of high-pass, bandpass, and bandstop filters, we here show a "trick" in which we use an arbitrary prototype low-pass filter, along with the Constantinides spectral transformations, to build upon what was done in the low-pass case and enable the design of these other types of filters with given 3-dB points. We then turn our attention to elliptic filters. For this case, we don't use analytic relations between frequency points. Instead, we work strictly in the digital domain and rely on systematically determining the 3-dB point of a prototype filter and using the Constantinides spectral transformations to obtain the desired filter. We will use to indicate analog frequency (rad/s) and a set such as {N, c , A p } to indicate analog design specifications. Normalized (or "digital") frequency is denoted by ω (rad/sample or simply radians), and we will use a set such as {N, ω c , A p } to indicate discretetime or digital design specifications. The various analog and digital frequencydomain variables used in this article are illustrated in Figure 1 . Example design specifications are described below. The low-pass case is described first and then the high-pass, bandpass, and bandstop cases are developed.

doi:10.1002/9780470170090.ch4
fatcat:2365z3dkgfe7varyyrj35b3ejy