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

22-12 Passive, Active, and Digital FiltersR 1 = 1.5π/4.5R 2 = 3π/4.5N 1 = N 2 = 9Amplitude responseLog magnitude responseIn-band ripple = 0.08Peak attenuation = 32.5 dBFIGURE 22.5 Frequency response of the circularly symmetric filter obtained by using the frequency samplingmethod. (From Hu, J.V. and Rabiner, L.R., IEEE Trans. Audio Electroacoust., 20, 249, 1972. With permission.ß 1972 IEEE.)One design approach is to minimize the L p norm of the error01e p ¼ @4p 2ð p pð p p11pjE(v 1 , v 2 ) j p dv 1 dv 2A : (22:28)Filter coefficients are selected by a suitable algorithm. For p ¼ 2 Parseval’s relation implies thate 2 2 ¼X1X 1n 1¼ 1 n 2¼ 1[h(n 1 , n 2 ) h id (n 1 , n 2 )] 2 : (22:29)By minimizing Equation 22.29 with respect to the filter coefficients, h(n 1 , n 2 ), which are nonzero only in afinite-extent region, I, one getsh(n 1 , n 2 ) ¼ h id(n 1 , n 2 ), (n 1 , n 2 ) 2 I,0, otherwise,(22:30)which is the filter designed by using a straightforward rectangular window. Due to the Gibbs phenomenonit may have large variations at the edges of passband and stopband regions. A suitable weightingfunction can be used to reduce the ripple [2], and an approximately equiripple solution can be obtained.For the general case of p 6¼ 2 [32], the minimization of Equation 22.28 is a nonlinear optimizationproblem. The integral in Equation 22.28 is discretized and minimized by using an iterative nonlinearoptimization technique. The solution for p ¼ 2 is easy to obtain using linear equations. This serves as anexcellent initial estimate for the coefficients in the case of larger values of p. Asp increases, the solutionbecomes approximately equiripple. The error term, E(v 1 , v 2 ), in Equation 22.28 is nonuniformlyweighted in passbands and stopbands, with larger weight given close to band-edges where deviationsare typically larger.