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

FIR Filters 18-39F(z L )z −LN F /2Inz −Lz −Lz −Lz −L++−G 2 (z)z −Lz −Lz −Lz −L+++++Outf(0) f(1) f(2)Nf ( F−1)N2 f ( F2)+++ +G 1 (z)FIGURE 18.21Efficient implementation for a filter synthesized using the frequency-response masking approach.andG 2 (z) ¼ z M 2X N 2n¼0g 2 (n)z n , g 2 (N 2 n) ¼ g 2 (n) (18:61d)Here, N F is even, whereas both N 1 and N 2 are either even or odd. For N 1 N 2 , M 1 ¼ 0 andM 2 ¼ (N 1 N 2 )=2, whereas for N 1 < N 2 , M 1 ¼ (N 2 N 1 )=2 and M 2 ¼ 0. These selections guarantee thatthe delays of both of the terms of H(z) are equal. An efficient implementation for the overall filter isdepicted in Figure 18.21, where the delay term z LN F=2 is shared with F(z L ). Also, G 1 (z) and G 2 (z) can sharetheir delays if a transposed direct-form implementation (exploiting the coefficient symmetry) is used.The frequency response of the overall filter can be written asH(e jv ) ¼ H(v)ej(LNFþmax [N1,N2])v = 2(18:62)where H(v) denotes the zero-phase frequency response of H(z) and can be expressed asH(v) ¼ H 1 (v) þ H 2 (v)(18:63a)whereH 1 (v) ¼ F(Lv)G 1 (v)(18:63b)andH 2 (v) ¼ [1 F(Lv)]G 2 (v) (18:63c)withF(v) ¼ f (N F =2) þ 2 XNF=2f (N F =2 n) cos nv (18:63d)n¼1