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

IIR Filters 19-1910.90.80.7Magnitude response0.60.50.40.30.20.100 0.5 1 1.5 2 2.5 3 3.5 4λ/λpFIGURE 19.12Magnitude response of elliptic filter. z ¼ (1 þ l r s 2 ) 1þ s2l rnh ¼ i N oddi 0:5 N even(19:56)Pc i ¼ 2b1=4 1k¼0 ( 1)k b k(kþ1) sin [(2k þ 1)ph=N]1 þ 2 P 1k¼1 ( 1)k b k2 cos [2kph=N](19:57) m i ¼ (1 l r c 2 i ) 1 c 2 1=2il r(19:58)p 0 ¼ l c sB i ¼ l2 cc 2 iA i1 ¼ 2l csm i1 þ s 2 c 2 iA i2 ¼ l 2 s 2 m 2 i þ zc 2 ic 1 þ s 2 c 2 2(19:59)iThe infinite summations above converge very quickly, so that only a few terms are needed in actualcalculation. A simple program can be written to compute the values in Equations 19.52 through 19.59.The filter order required to satisfy Equation 19.27 is calculated fromN 1 log (b) log e 2 16(s 2 1)(19:60)where b is given by Equation 19.53. An example of the magnitude response is plotted in Figure 19.12.We see that there are ripples in both the passband and the stopband.19.2.2.5 ComparisonIn comparing the filters given above, the Butterworth filter requires the highest order and the elliptic filterrequires the smallest order to satisfy the same passband and stopband specifications. The Butterworthfilter and the inverse Chebyshev filter have nicer (closer to linear) phase characteristics in the passband