/966 3-parallel FIR H。z3H2z3H 「X Y 三 H H X H.H HO y(3k) x(3k) H1 y(3k+1) H2 y3k+2) HO x(3k+1) ■Requires3N H1 multiplications H2 D D:23 and 3(N-1) HO additions. x(3k+2) H1 D 2021年2月 H2 D
2021年2月 3-parallel FIR Requires 3N multiplications and 3(N-1) additions
/966 L-parallel FIR X(z)= ∑z*x(Lk+),i=0,1,…,L-1 k=0 N/L-1 H(2)= ∑zkh(Lk+)j=0,1,…,L-1 k=0 Thus X=21∑H,X+2H,X,0sk≤L-2 i=k+1 Y-E8X i=0 2021年2月 12
2021年2月 12 L-parallel FIR Thus ,0 2 0 1 1 Y z H X H X k L k i i k i L i k i L k i L k 1 0 1 1 L i YL Hi X L i
/986 This can also be expressed in Matrix form as Y=H.X 6 Ho 2Hia … 2H H H … 2H X Y. HLA H-2 … Ho X- Requires L2 sub-filtering operations,each of which is of length N/L and requires N/L multiply-add operations. Hence,the L-parallel FIR filter requires L2*N/L=LN multiply- add operations. 2021年2月 13
2021年2月 13 This can also be expressed in Matrix form as Requires L2 sub-filtering operations, each of which is of length N/L and requires N/L multiply-add operations. Hence, the L-parallel FIR filter requires L2*N/L=LN multiplyadd operations
9.2.2 Fast FIR algorithms (FFA) /986 From the work of Winograd,2 polynomials of degree L-1 can be multiplied using 2L-1 product terms; The reduction in the number of multiplications comes at the expense of increasing the number of additions required for implementation; This implies that the parallel FIR filter can be realized using approximately(2L-1)FIR filter of length N/L. This structure require (2L-1)*N/L=2N-N/L multiplications. 2021年2月 14
2021年2月 14 9.2.2 Fast FIR algorithms (FFA) From the work of Winograd, 2 polynomials of degree L-1 can be multiplied using 2L-1 product terms; The reduction in the number of multiplications comes at the expense of increasing the number of additions required for implementation; This implies that the parallel FIR filter can be realized using approximately (2L-1) FIR filter of length N/L. This structure require (2L-1)*N/L=2N-N/L multiplications
9.2.2.1 Two-parallel FFAs 96 Y(z2)=X(z2)H(z2)+z2X1(22)H(z2) Y(z2)=X(22)H(z2)+X(z2)H(z2) Yo=XoHo+zX H Y=(Ho+H(Xo+X)-XoHo-X H V(2k) HO x(2k) y(2k+1) H0+H1 x(2k+1) H1 2021年2月 15
9.2.2.1 Two-parallel FFAs 2021年2月 15 1 0 1 0 1 0 0 1 1 1 1 2 0 0 0 Y (H H )(X X ) X H X H Y X H z X H ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 0 2 1 2 1 2 0 2 1 2 1 2 1 2 2 0 2 0 2 0 Y z X z H z X z H z Y z X z H z z X z H z