## Frequency spacing This response function turns out to be advantageous

from three standpoints: more gain can be achieved for the same bandwidth; the approximation to constant gain is usually better for steady-state applications than is the maximally flat response; and the selectivity ratio is better.

The analytic expression of the amplitude response of Fig. 9-10 is given below. It contains Chebyshev polynomials, and because of this the approximation function is sometimes called the "Chebyshev response."

ripple (db)

C3(x) = 4xs - 3x Ct>(x) = 32zH - 48x4 + 18x2 - 1

There is also a convenient expression for the general polynomial in terms of trigonometric functions,

The property the Chebyshev polynomials have which is of interest to us is shown in Fig. 9-11; for x in the range of ±1, the maximum and minimum values of Cn(x) are also ±1, and Cn2(x) oscillates between 0 and 1 in the "approximation band"--1 < x < +1. Therefore inspection of Eq.

(9-19) shows that in this band the maximum value of A(x) is 1 and the minimum value of A(x) is l/yT+l, as shown graphically in Fig. 9-10 (for n = 5). Far outside the approximation band Cn2(x) increases like x2n. Therefore the gain function decreases outside the passband much as it did for the maximally flat case [cf. Eq. (9-1)].

Since Eq. (9-19) seems to provide a very useful gain function, what we now need to know is where to locate the poles of the gain function to realize this gain function along the ju axis. From these pole locations we may later determine the necessary element values in the interstages.

The simplest gain function leading to the equal-ripple amplitude response is an all-pole one, with the poles situated around a semiellipse, as opposed to a semicircle for the maximally flat response. This is depicted in Fig. 9-12 and was previously shown in Fig. 8-10. The actual value of the pole locations may be found by the process used to factor Eq. (9-1) and resulting in Eq. (9-14).

The band between x = 1 and —1, the foci of the ellipse, corresponds to the bandwidth B\ in Fig. 9-10 and is the band within which the gain stays

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