Stability For a stable system the impulse response must be absolutely summable, i.e. ∑列 <0 <oo for z=1 ROC of H(z) includes the unit circle
22 Stability n=− h n =1 =− − h n z for z n n ◆ ROC of H(z) includes the unit circle ◆For a stable system The impulse response must be absolutely summable, i.e
EX 5.3 Determine the Roc stability and causality for LTI system n-11+yn-2|=x Solution H z+z 2z poles: 1/2, 2; Zeros(two): 0
23 Ex. 5.3 Determine the ROC, Stability and causality for LTI system: yn− yn −1+ yn − 2= xn 2 5 ( ) ( ) 1 2 1 1 1 2 2 1 1 1 2 5 1 1 − − − − − − = − + = z z z z H z ◆poles: 1/2, 2; zeros(two)：0 Solution:
Example 5.3 Determining the ROC z> 2: causal not stable J∮ Z-plane Unit circle <z< 2 2 Re stable, not causal )zk<: not causal, not stable 3
24 Example 5.3 Determining the ROC z causal not stable 2: , 1 2 : 2 , z stable not causal z : not causal, not stable 2 1 1) 2) 3)
Causal and stable system Causal: Roc must be outside the outermost pole Stable: roc includes the unit circle Causal and stable: all the poles of the system function are inside the unit circle ROC is outside the outermost pole, and includes the unit circle
25 Causal and Stable system ◆Causal: ROC must be outside the outermost pole ◆Stable: ROC includes the unit circle ◆Causal and stable: all the poles of the system function are inside the unit circle. ROC is outside the outermost pole, and includes the unit circle
5.2.2 Inverse Systems For a lti system h(z the inverse system H, (z)which cascaded with H(z)satisfies 对一H(2)H(z)ym G(x)=H()(=)=1,H()= ◆ Time domain:gl]=h]*h四= ◆ Frequency response H(el') w Not all systems have an inverse. Ideal LPF hasn 't 26
26 5.2.2 Inverse Systems ( ) ( ) ( ) 1, G z H z H z = = i ( ) ( ) 1 H z i H z = ◆Time domain: gn hn h n n = i = ◆Not all systems have an inverse. Ideal LPF hasn’t ◆For a LTI system , the inverse system which cascaded with satisfies: H z( ) H z i ( ) H(z) ( ) ( ) j w j w i H e H e 1 ◆Frequency response = x[n] h n y [n] i H z( ) h n( ) H z i