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to IIR filter design is relatively simple. The second approach to design IIR digital filters is to use an algorithmic design procedure, which generally requires the use of a computer to solve a set of linear or nonlinear equations. These methods may be used to design digital filters with arbitrary frequency response characteristics for which no analog filter prototype exists or to design filters when other types of constraints are imposed on the design. In this section, we consider the approach of mapping analog filters into digital filters. Initially, the focus will be on the design of digital low-pass filters from analog low-pass filters. Techniques for transforming these designs into more general frequency selective filters will then be discussed.

Fig. 18-11

To design an IIR digital low-pass filter from an analog low-pass filter, we must first know how to design an analog low-pass filter. Historically, most analog filter approximation methods were developed for the design of passive systems having a gain less than or equal to 1. Therefore, a typical set of specifications for these filters is as shown in Fig. 9-5(a),with the passband specifications having the form

(a) 270 clockwise or 90 counter-clockwise about the origin, P(x, y) A Pr( y, x) (b) 180 about the origin (either clockwise or counter-clockwise), P(x, y) A Ps( x, y) (c) 45 clockwise, P(x, y) A P-(x cos 45 P-a 22 x 2 22 22 y, y 2 2 22 xb 2 y sin 45, y cos 45 x sin 45 )

((I)

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18.11 Naming rotations Name the transformation that rotates clockwise about the origin: (a) 20 (b) 30 (c) 60 (d) 75

and 6,.

Fig. 7-7 Sometimes we deal with intervals that are unbounded on one side. [a, ) is made up of all x such that a x.

Fig. 9-5. Tivo different conventions for specifying the passband and stopband deviations for an analog

(a) P(x, y) A Pr(x cos 20 (b) P(x, y) A Pr(x cos 30 Pr(0.866x y sin 20 , y cos 20 y sin 30 , y cos 30 0.5x) x sin 20 ) x sin 30 ) Pr(0.9397x Pr a 23 x 2 0.3420y, 0.9397y 1 23 y, y 2 2 1 xb 2 0.3420x)

low-pass filter. Another convention that is commonly used is to describe the passband and stopband constraints in terms of the parameters E and A as illustrated in Fig. 9-5(h). Two auxiliary parameters of interest are the dist.riminatior7 factor,

x sin 60 )

k ="

EXAMPLE The range of the function graphed in Fig. 7-6 is (0, ). The domain of the function graphed in Fig. 7-5(b) is ( , l]. 7.3 EVEN AND ODD FUNCTIONS A function f is called even if, for any x in the domain of f , x is also in the domain of f and f ( x) = f (x). EXAMPLES

x sin 75 )

The three most commonly used analog low-pass filters are the Butterworth, Chebyshev, and elliptic filters. These filters are described below.

0.9659x)

The parameter N is the order of the filter (number of poles in the system function), and Q,. is the 3-dB cutoff frequency. The magnitude of the frequency response may also be written as

where The frequency response of the Butterworth filter decreases monotonically with increasing 0, and as the filter order increases, the transition band becomes narrower. These properties are illustrated in Fig. 9-6, which shows IH,(jQ)l for Butterworth filters of orders N = 2 , 4 , 8 , and 12. Because

A figure has rotational symmetry if it can be rotated around its center by fewer than 360 and look the same as it did originally. In Fig. 18-12, there is (a) a figure that looks the same under a 72 rotation, (b) a figure that looks the same under a 120 rotation, (c) a figure that looks the same under a 180 , and (d) a figure without rotational symmetry.

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