Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

Approximation 2-31TABLE 2.4 Zeros of Elliptic RationalFunctionsi0 0.00001 0.59232 0.85883 0.93524 0.9500v iEvaluating the zeros and points of maximum deviation of the Chebyshev rational function numericallyusing Equation 2.129, one obtains the values shown in Table 2.4. Thus, the required elliptic rationalfunction isð0:3508 v2Þð0:8746 v 2 ÞR 4 (v) ¼ð1 0:3508v 2 Þð1 0:8746v 2 Þ(2:135)Finally, the maximum error is given by Equation 2.130:e ¼ ðv 1 v 2 Þ 2 ¼ (0:5923 0:9352) 2 ¼ 0:3078 (2:136)Figure 2.26 shows the resulting gain plot. Observe the transmission zero at v ¼ 1=0.9352 ¼ 1.069,corresponding to the pole of the Chebyshev rational function located at the inverse of the largestpassband zero. Also, as anticipated, the minimum gain in the passband isand the maximum stopband gain is11A p ¼ pffiffiffiffiffiffiffiffiffiffiffiffi¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi¼ 0:9558 (2:137)1 þ e 2 1 þ (0:3078) 211A s ¼ qffiffiffiffiffiffiffiffiffiffiffi¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi¼ 0:2942 (2:138)1 þ 1 1 þ 1e 2(0:3078) 2Perhaps a summary of the various filter types is in order at this point. The elliptic filter has moreflexibility than the Butterworth or the Chebyshev because one can adjust its transition band rolloffindependently of the passband ripple. However, as the sharpness of this rolloff increases, as a moredetailed analysis shows, the stopband gain increases and the passband ripple increases. Thus, if these11A(ω) =√1 + R 2 4(ω)0.500 1 2 3 4 ω 5FIGURE 2.26Gain plot for the example elliptic filter.