5.2.2 Inverse Systems systems with rational H system functions: ∏-c=) ∏(-d=) H()= 0k=l1 H(=) 0|k=1 dn k=1 +ROC of H(z)and ROC of H,()must overlap for convolution theorem to hold h小*h[]=5H(2)H1(z)=1
27 5.2.2 Inverse Systems ( ) ( ) ( ) = − = − − − = N k k M k k d z c z a b H z 1 1 1 1 0 0 1 1 ( ) ( ) ( ) = − = − − − = N k k M k k i c z d z b a H z 1 1 1 1 0 0 1 1 ◆ROC of and ROC of must overlap, for convolution theorem to hold: H(z) H (z) i ( ) H(z) H z i 1 = systems with rational system functions: h n h n n = i ( ) ( ) 1, H z H z i =
Ex 5. 4 analyse Inverse System for First-Order System 1-0.5z ROC:|z|>0.9 1-0.9z Solution 1-0.9z H1(z)= ROC:z|>0.5 1-0.5z H: (z: stable and causal h四]}=(0.5y-0.905n-1]
28 Ex. 5.4 analyse Inverse System for First-Order System ( ) 1 1 1 0.5 : 1 0. 0 9 .9 z H z ROC z z − − − = − ( ) 1 1 1 0.9 : 1 0. 0.5 5 i z H z ROC z z − − − = − (0.5) 0.9(0.5) 1 1 = − − − h n u n u n n n i ( ): i H z stable and causal Solution:
EX 5.5 find Inverse for System with a zero in the roc 0.5 H ROC:|z|>0.9 1-0.9z Solution 0.9 2 1-0.9z 2+1.8z H 0.51-2 1)ROC:|<2b[小]=2(2)u-n-1]-182)-[n H,(z):stable, not causal 2)ROC:>2h2]=-2(2)u]+1.8(2)[m- H (z): causal, not stable
29 Ex. 5.5 find Inverse for System with a Zero in the ROC ( ) 1 1 0.5 : 1 0. 0.9 9 z H z ROC z z − − − = − ( ) 1 1 1 1 1 2 2 1.8 0.5 1 0.9 − − − − − − + = − − = z z z z H z i ( ) ( ) 1 1 : 2 2 1 1.8 2 2 n n ROC h n u i z n u n − = − − − − ( ) ( ) 1 2 : 2 2 1.8 2 2 1 n n ROC h n u n u n i z − = − + − Solution: 0.9 2 ( ): , i H z stable not causal 1) 2) ( ): , i H z causal not stable
Minimum-phase Systems CA LTI system is stable and causal and also has a stable and causal inverse if and only if both the poles and the zeros of H(z)are inside the unit circle 1-0.5z H ROC:|z>0.9 1-0.9z 1-0.9z ROC:|z|>0.5 1-0.5z A Such systems are referred as minimum phase systems
30 Minimum-phase Systems ◆A LTI system is stable and causal and also has a stable and causal inverse if and only if both the poles and the zeros of are inside the unit circle H(z) ◆Such systems are referred as minimumphase systems 1 ( ) : 0.9 1 0.9 1 0.5 1 1 − − = − − ROC z z z H z ( ) : 0.5 1 0.5 1 0.9 1 1 − − = − − ROC z z z H z i
5.2.3 Impulse Response for RatIonal System Functions ◆ For a Lti system ∑ H k=0 M≥N A ∑a-k H(2)=∑Bz+∑ 0 k N f causi小∑B-小+∑4ll =0 k=1 Ccan be infinite impulse response(iIr) Cor finite impulse response FIR) 31
31 5.2.3 Impulse Response for Rational System Functions ◆For a LTI system ( ) = − − = − − = + N k k k M N r r r d z A H z B z 1 1 0 1 = − = = − + N k n k k M N r h n Br n r A d u n 0 1 ◆can be infinite impulse response (IIR) ◆or finite impulse response (FIR) If causal, ( ) 0 0 , M N M k k k N k k k b z H z a z − = − = =