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

2-26 Passive, Active, and Digital Filters(0; jvj v pK(v) ¼1; jvj 1 v p(2:114)with the real and even or odd (for realizability) rational function R mn (v) having the aforementionedreciprocal property. The Weierstrass theorem equiripple property for such rational functions demandsthat the total number of error extrema* on the compact set be m þ n þ 2. (This assumes the degree ofboth polynomials to be relative to the variable v 2 .)Based on the preceding discussion, one has the following form for R mn (v):R 2n,2n (v) R 2n (v) ¼ v2 1v 2 v 2 3v 2 v 2 2n 1v 2ð1 v 2 1 v2 Þð1 v 2 2 v2 Þð1 v 2 2n 1 v2 ÞR 2nþ1,2n (v) R 2nþ1 (v) ¼ v v2 2 v 2 v 2 4 v 2 v 2 2n v 2ð1 v 2 2 v2 Þð1 v 2 4 v2 Þð1 v 2 2n v2 Þ(2:115)(2:116)The first is clearly an even rational function and latter odd. The problem now is to find the location of thepole and zero factors such that equiripple behavior is achieved in the passband. The even case isillustrated in Figure 2.21 for n ¼ 2. Noticep ffiffithat the upper limit of the passband frequency interval hasbeenptaken for convenience to be equal to k ; hence, the lower limit of the stopband frequency interval is1= ffiffik . Thus,k ¼ v pv s(2:117)R 2n (ω)1=1√kn = 2ε–εω 0ω 1ω p1ω31ω11ε1εω 2ω 3ω p = √k1–ωFIGURE 2.21Typical plot of an even-order Chebyshev rational function.* There can be degeneracy in the general approximation problem, but the constraints of the problem being discussed herepreclude this from occurring.