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

1-D Multirate Filter Banks 24-9Haar 2-tap waveletsFrequency responses10h143.5H0000−1h013H00012.5−2−3h001h000121.51H001H01H1−4−5h0000−5 0 5Sample number0.50−0.50 1 2 3 4Frequency* (8/f s )FIGURE 24.5Impulse responses and frequency responses of the 4-level tree of Haar filters.p ffiffisum to the value 2 , i.e., the DC frequency response is H0 (e j0 ) ¼ H 0 (1) ¼ P n h p0(n) ¼ffiffi2 . (This scalinghas the effect of making the DWT tend to be an energy preserving transformation.)Central to the principle of the DWT are the two-scale equations that link the discrete time filters of thefilter bank to continuous time functions known as the scaling function and the mother wavelet. Theseequations exist both for the analysis side of the filter bank and for the synthesis side:1. Analysis side equations:p~f(t) ¼ffiffi X2 h 0 (k) f(2t ~ k) (24:21)kp~c(t) ¼ffiffi X2 h 1 (k) f(2t ~ k) (24:22)kwhere ~ f(t) and ~c(t) are the analysis scaling function and mother wavelet, respectively. Thecoefficients of filters H 0 (z) and H 1 (z) are h 0 (k) and h 1 (k):i.e.,H 0 (z) ¼ X kh 0 (k)z k and H 1 (z) ¼ X kh 1 (k)z k (24:23)2. Synthesis side equations:pf(t) ¼ffiffi X2 g 0 (k)f(2t k) (24:24)k