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

2-14 Passive, Active, and Digital Filters1A(ω)1A(ω)0.50.995(a)0 0.5ω100 0.5 1 1.5 ω 2(b)FIGURE 2.14Frequency response of a fifth-order Chebyshev lpp: (a) passband and (b) overall.1A min ¼ p ffiffiffiffiffiffiffiffiffiffiffiffi(2:50)1 þ e 2The passband ripple is usually specified as the peak to peak variation in dB. Since the maximum value isone, that is 0 dB, this quantity is related to the ripple parameter e through the equationpffiffiffiffiffiffiffiffiffiffiffiffipassband ripple in dB ¼ 20 log 1 þ e 2(2:51)The Chebyshev approximation is the best possible among the class of all-pole filters—over the passband.But what about its stopband behavior? As was pointed out previously, it is desirable that—in addition toapproximating zero in the passband—the characteristic function should go to infinity as rapidly aspossible in the stopband. Now it is a happy coincidence that the Chebyshev polynomial goes to infinityfor v > 1 faster than any other polynomial of the same order. Thus, the Chebyshev approximation is thebest possible among the class of polynomial, or all-pole, filters.The basic definition of the Chebyshev polynomial works fine for values of v in the passband, wherev 1. For larger values of v, however, cos 1 (v) is a complex number. Fortunately, there is an alternateform that avoids complex arithmetic. To derive this form, simply recognize the complex nature ofcos 1 (v) explicitly and writex ¼ cos 1 (v) (2:52)One then has*v ¼ cos(x) ¼ cos[ j(jx)] ¼ cosh(jx) (2:53)soThus, one can also writejx ¼ cosh 1 (v) (2:54)T n (v) ¼ cos[jn(jx)] ¼ cosh[n(jx)] ¼ cosh[n cosh 1 (v)] (2:55)* Since cos(x) ¼ cosh(jx).