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

1-D Multirate Filter Banks 24-13Now we define the low-pass product filter:and substitute relations in Equation 24.42 into Equation 24.40 to getP(z) ¼ H 0 (z)G 0 (z) (24:44)G 0 (z)H 0 (z) þ G 1 (z)H 1 (z) ¼ G 0 (z)H 0 (z) þ H 0 ( z)G 0 ( z)¼ P(z) þ P( z) 2 (24:45)Hence satisfying Equation 24.40 requires that all P(z) terms in even powers of z be zero, except the z 0term which must be 1. However the P(z) terms in odd powers of z may take any desired values sincethey cancel out in Equation 24.45. If P is low-pass, this type of filter is often known as a halfband filtersince P(e jv ) þ P(e j(vþp) ) ¼ a constant, and so the bandwidth of P must be half of the input bandwidth.A further commonly applied constraint on P(z) is that it should be zero phase,* in order to minimizethe magnitude of any distortions due to samples from the high-pass filters being suppressed (perhaps as aresult of quantization or denoising). Hence P(z) should be of the form:P(z) ¼þp 5 z 5 þ p 3 z 3 þ p 1 z þ 1 þ p 1 z 1 þ p 3 z 3 þ p 5 z 5 þ (24:46)The design of a set of PR filters H 0 , H 1 and G 0 , G 1 can now be summarized as1. Choose a set of coefficients p 1 , p 3 , p 5 to give a zero-phase low-pass product filter P(z) withdesirable characteristics. (This is nontrivial and is discussed below.)2. Factorize P(z) into H 0 (z) and G 0 (z), preferably so that the two filters have similar low-passfrequency responses.3. Calculate H 1 (z) and G 1 (z) from Equation 24.42.It can help to simplify the tasks of choosing P(z) and factorizing it if, based on the zero-phaserequirement, we transform P(z) into P t (Z) such thatP(z) ¼ P t (Z) ¼ 1 þ p t,1 Z þ p t,3 Z 3 þ p t,5 Z 5 þ where Z ¼ 1 2 (z þ z 1 ) (24:47)To calculate the frequency response of P(z) ¼ P t (Z), let z ¼ e jvTs .Therefore; Z ¼ 1 2 (e jvTs þ e jvTs ) ¼ cos (vT s ) (24:48)This is a purely real function of v, varying from 1 at v ¼ 0to 1atvT s ¼ p (half the samplingfrequency). Hence we may substitute cos (vT s ) for Z in P t (Z) to obtain its frequency response directly.24.6.3 Some Simple Filters=Wavelets (Haar and LeGall)As discussed in Section 24.5, in order to achieve smooth wavelets after many levels of the binary tree,the low-pass filters H 0 (z) and G 0 (z) must both have a number of zeros at half the sampling frequency(at z ¼ 1). These will also be zeros of P(z), and so P t (Z) will have zeros at Z ¼ 1 (each of which willcorrespond to a pair of zeros at z ¼ 1).* See the end of Section 24.1 for a discussion of causality assumptions related to this.