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

1-D Multirate Filter Banks 24-25duration is infinite. For the causal stable case the function extends to positive infinity and for thenoncausal case the function extends to infinity in both directions.A versatile building block for IIR filters is the all-pass filter a(z) which has the following two properties:1a(z) ¼ a(z 1 ) and ja(e jv )j¼1 (24:68)The all-pass filter a(z) can be considered a generalization of the unit delay z 1 since je jv j¼ja(e jv )j¼1,i.e., both have the same magnitude response. The unit delay has a linear phase response but the all-passfilter can have a variety of phase responses. A wide family of IIR filters can be obtained by variousinterconnections of all-pass filters [7–9]. The simplest interconnection (described below) can give a largeclass of useful IIR filter banks.24.7.1 All-Pass Filter Design MethodsConsider analysis filters obtained by the weighted sum and difference of two all-pass filters, a 0 and a 1 :H 0 (z) ¼ p 1 ffiffi [a 0 (z) þ a 1 (z)]2H 1 (z) ¼ p 1 ffiffi [a 0 (z)2a 1 (z)](24:69)This can be considered an extension of the FIR Haar filter pair in Equation 24.2 to IIR filters. The terms 1and z 1 (simple all-pass) have been replaced with a 0 and a 1 , general all-pass filters. The Haar filters arequadrature mirror versions of each other. If the two all-pass filters are chosen to be of the forma 0 (z) ¼ A 0 (z 2 ), a 1 (z) ¼ z 1 A 1 (z 2 ), then the analysis filters, given byH 0 (z) ¼ 1 pffiffiA 0 (z 2 ) þ z 1 A 1 (z 2 )2H 1 (z) ¼ 1 pffiffiA 0 (z 2 ) z 1 A 1 (z 2 )2(24:70)will have the quadrature mirror characteristics:jH 1 (e jv )j¼jH 0 (e j(p v) )j, i.e., be mirrored about v ¼ p 2(24:71)This means that if H 0 (e jv ) has a low-pass frequency response, H 1 (e jv ) will automatically have a high-passresponse.How should the all-pass filters be designed so that H 0 (e jv ) is a good low-pass filter with half of the fullbandwidth?By denoting the phase response of the all-pass filters as f 0 (v) ¼ffA 0 (e jv ) and f 1 (v) ¼ffA 1 (e jv ), themagnitude response of the low-pass filter can be written asjH 0 (e jv )j¼ 1 pffiffie jf0(2v) þ e j(f 1(2v) v) 2¼ 1 pffiffie jf 0(2v) 1 þ ej(f 1 (2v) f 0 (2v) v) 2¼ 1 pffiffi j(u(2v) v)1 þ e where u(v) f 1 (v) f 0 (v) (24:72)2