10.3.1 Pipelining for 1st-Order IIR Filters /96 Look-ahead pipelining adds canceling poles and zeroes to the transfer function,such that the coefficients of {z(M-1)in the denominator of the transfer function are zero. Then,the output sample /n)can be computed using the inputs and the output sample n-M) such that there are Mdelay elements in the critical loop,which in turn can be used to pipeline the critical loop by Mstages and the sample rate can be increased by a factor M. 2021年2月 11
2021年2月 11 10.3.1 Pipelining for 1st-Order IIR Filters Look-ahead pipelining adds canceling poles and zeroes to the transfer function, such that the coefficients of {z-1 , .., z-(M-1)} in the denominator of the transfer function are zero. Then, the output sample y(n) can be computed using the inputs and the output sample y(n-M) such that there are M delay elements in the critical loop, which in turn can be used to pipeline the critical loop by M stages and the sample rate can be increased by a factor M
966 Example: Consider the 1st-order filter,( 1-az ⑧ which has a pole at z=a (a<1). A 3-stage pipelined equivalent stable filter can be derived by adding poles and zeroes at,==ae*02/3).and is given by (1-aeJ2a8z1)1-ae2a/3z1) 1+az1+a2z2 H(=)=(1-g)(1-de1-ae) 1-a3z-3 u(n) D u(n-1) D u(n-2) a 只) retime 3D y(n) yn-3) 2021年2月 12
2021年2月 12 Example: Consider the 1st-order filter, , which has a pole at z=a (a<1). A 3-stage pipelined equivalent stable filter can be derived by adding poles and zeroes at , , and is given by 1 1 1 ( ) az H z ( j2 / 3) z ae 3 3 1 2 2 1 2 /3 1 2 /3 1 2 /3 1 2 /3 1 1 1 (1 )(1 )(1 ) (1 )(1 ) ( ) a z az a z az ae z ae z ae z ae z H z j j j j retime
10.3.2 Look-Ahead Pipelining with Power-of-2 986 Decomposition With power-of-2 decomposition,an M-stage (for power- of-2 M)pipelined implementation for 1st-order IIR filter can be obtained by log,Msets of transformations. ■ Example:Consider a 1st-order recursive filter transfer function described by H(2)= bz-1 1-az-i The equivalent pipelined transfer function can be described using the decomposition technique as follows b-1+ad2z”) non-recursive section H(z)= 1-aMz-M recursive section 2021年2月 13
2021年2月 13 10.3.2 Look-Ahead Pipelining with Power-of-2 Decomposition With power-of-2 decomposition, an M-stage (for powerof-2 M) pipelined implementation for 1st-order IIR filter can be obtained by log2M sets of transformations. Example: Consider a 1st-order recursive filter transfer function described by The equivalent pipelined transfer function can be described using the decomposition technique as follows 1 1 1 ( ) az bz H z M M M i a z bz a z H z i i 1 (1 ) ( ) log 1 0 1 2 2 2 non-recursive section recursive section
/966 This pipelined implementation is derived by adding(M-1) poles and zeros at identical locations. The original transfer function has a single pole at z=a. s The pipelined transfer function has poles at the following locations a,aeMaeM,aeae-) The i-th stage of the decomposed non-recursive portion implements 2i zeros located at: z=ae2m+r/2,n=0,l.(2-1) i-th 1+a2z2 0-th 1+az1 z=aei 1st 1+a2z2 z=aei2 aei3/2 2nd 1+a4z4 z-aei4,aej3/4,z-aej5x/4 aei7/4 2021年2月 14
2021年2月 14 This pipelined implementation is derived by adding (M-1) poles and zeros at identical locations. The original transfer function has a single pole at z=a. The pipelined transfer function has poles at the following locations The i-th stage of the decomposed non-recursive portion implements 2i zeros located at: { , , , ,.., } j2 / M j2(2 ) / M j3(2 ) / M j(M 1) (2 ) / M a ae ae ae ae , 0,1,..(2 1) (2 1) / 2 j n i z ae n i i-th 1+a2 i z -2 i z 0-th 1+az-1 z=aejπ 1st 1+a2z -2 z=aejπ/2,aej3π/2 2nd 1+a4z -4 z=aejπ/4,aej3π/4 , z=aej5π/4,aej7π/4
Im[z] Im(z] U ja ⊕ /966 a 1.0 Refz) Re[z] 1.0 ⊕ -ja ⊕ 2 j2π 1 2π *1-ae 1-ae M- 1 *1-ae MZ- 1-az 2 1-az -1 1-ae2a21-ae272 r1 a) b) 1-ae M z- Im(z] 1m(z] Re(z] 1.0 1.0 Retz] a 1+az-1 1+a2z-2 Im(z] Im(z] Re(z] Re[z] 1.0 1.0 1+a4z 1 -ja 1-a8z8 u(m)1+az→1+a2z2 -4 1+a'z 1-a3z8 ◆y(n) 0-th stage 1-th stage 2-th stage 2021年2月 15
2021 年 2 月 15 1 1 1 az 1 1 2 1 1 2 1 2 2 1 2 2 1 1 2 1 1 2 1 11 ...* 11* 11* 1 1 ae z ae z ae z ae z ae z ae z az MM j MM j M j M j M j M j 1 1 az 2 2 1 a z 4 4 1 a z 8 8 1 1 a z bz - 1 0 -th stage 1 -th stage 2 -th stage