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

Two-Dimensional IIR Filters 23-45Step 1. Construct a parameterized analog transfer function of the 2-D IIR filter, by using theHurwitz polynomial D(s 1 , s 2 , y kl )H(s 1 , s 2 , y kl , a ij ) ¼ p(s 1, s 2 )D(s 1 , s 2 , y kl )(23:127)wherep(s 1 , s 2 ) ¼ XN1i¼1X N2j¼1a ij s i 1 sj 2is an arbitrary 2-V polynomial in s 1 and s 2 with degree in each variable not greater than thecorresponding degree of the denominator.Step 2. Perform the double bilinear transformation to the parameterized analog transfer functionobtained in step 1.H(z 1 , z 2 , y kl , a ij ) ¼ p(s 1, s 2 )D(s 1 , s 2 , y kl ) j s i ¼2(z i 1)=T i (z i þ1), i¼1, 2 (23:128)Step 3. Construct an objective function according to the given design specifications and theparameterized discrete transfer function obtained in step 2.J(x) ¼ X n 1Xn 2[M(n 1 , n 2 ) M I (n 1 , n 2 )] p (23:129)where p is an even positive, M(n 1 , n 2 ) and M I (n 1 , n 2 ) are the actual and desired amplituderesponses, respectively, of the required filter at frequencies (v 1n1 , v 1n2 ), and x is the vectorconsisting of parameters {y kl :1< k < l N} and {a ij ,0 i N 1 ,0 j N 2 }.Step 4. Apply an optimization algorithm to find the optimal vector x that minimizes the objectivefunction and substitute the resulting x into Equation 23.128 to obtain the required transfer functionH(z 1 , z 2 ).Example 23.8 [33]By using the preceding approach, design a 2-D circularly symmetric low-pass filter of order (5, 5) withv p ¼ 0.2p, assuming that v s1 ¼ v s2 ¼ 1.2p.Solution1. Construct the desired amplitude response of the desired filterwhere( M I (v 1n1 , v 2n2 ) ¼ 1 for v2 1n 1þ v 2 2n 2 0:2p0 otherwisev 1n1 ¼ 0:01pn 2 for 0 n 1 200:01pn 1 for 21 n 1 24