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

19-14 Passive, Active, and Digital Filterswhere8Q Np i>: (1 þ e 2 )1=2QNp ii¼1N is oddN is evenp i ¼ l p sinh (f) sin 2i 1 2Np þ jl p cosh (f) cos" #f ¼ 1 N ln 1 þ (1 þ e2 ) 1=2e 2i 12Np(19:33)(19:34)(19:35)The value of C normalizes the magnitude so that the maximum magnitude is 1. Note that C is always apositive constant. The poles are on the left-half s plane, lying on an ellipse centered at the origin with aminor radius of l p sinh (f) and major radius of l p cosh (f). Except for one pole when N is odd, all thepoles have a complex–conjugate pair. Specifically, p i ¼ p* N iþ1 , i ¼ 1, 2, , N=2 or(N 1)=2. Combiningeach complex–conjugate pair in Equation 19.32 yields a second-order factor with real coefficients.The magnitude response can be computed from Equations 19.33 through 19.35 with s ¼ jl. Its squarecan also be written asjH a (l)j 2 ¼11 þ e 2 T 2 N (l=l p)(19:36)where T N (x) is the Nth degree Chebyshev polynomial of the first kind, which is shown in Table 19.2 andalso given recursively byT 0 (x) ¼ 1T 1 (x) ¼ xT nþ1 (x) ¼ 2xT n (x) T n 1 (x) n 1(19:37)Notice that TN 2 (1) ¼ 1. Therefore, we have from Equation 19.36 that the magnitude square passesthrough 1=(1 þ e 2 )atl ¼ l p , i.e., jH a (l p )j 2 ¼ 1=(1 þ e 2 ). Note also that T N (0) ¼ ( 1) N=2 for even Nand it is 0 for odd N. Therefore, jH a (0)j 2 equals 1=(1 þ e 2 ) for even N and it equals 1 for odd N. Figure19.10 shows some examples of magnitude response square.TABLE 19.2 Coefficients of Chebyshev PolynomialsT n (x) of the First Kind, of Order n, in AscendingPowers of Variable xn1 0, 12 1, 0, 23 0, 3, 0, 44 1, 0, 8, 0, 85 0,5,0, 20, 0, 16Coefficients of T n (x)6 1, 0, 18, 0, 48, 0, 327 0, 7, 0, 56, 0, 112, 0, 648 1, 0, 32, 0, 160, 0, 256, 0, 1289 0,9,0, 120, 0, 432, 0, 576, 0, 25610 1, 0, 50, 0, 400, 0, 1120, 0, 1280, 0, 512