5.2 LTI System Characterized by linear Constant-coefficient Difference equation N M k]=∑bxm- k=0 k=0 If a system is not LTI then the following Z-transform cannot be derived.(see P37, example 2.16, for xIn]=koin yin]=an+ic t Kanulnl, for all n xin] have z-transform K yIn] have no z-transform.) ∑4z(z)=∑bzX(z) k=0 k=0 17
17 5.2 LTI System Characterized by Linear Constant-coefficient Difference equation = = − = − M k k N k k a y n k b x n k 0 0 ( ) ( ) = − = − = M k k k N k k k a z Y z b z X z 0 0 If a system is not LTI, then the following Z-transform cannot be derived. (see P37, example 2.16, for x[n]=kδ[n], y[n]=an+1c + Kanu[n], for all n, x[n] have z-transform K, y[n] have no z-transform.)
5.2 System Functions For Systems Characterized by linear Constant coefficient Difference equation ◆ For an Lti system B()=y()么=x CZ k k=1 ∑q24(/1(-d=) k=0 k=1 ◆ its poles and zeros: 0→z=C,:zerO 1-dkz=0→z=dk:pole 18
18 1 1 0 : k k c z z c zero − − = = ( ) ( ) ( ) ( ) ( ) 1 0 0 1 1 0 0 1 1 1 M M k k k k k N N k k k k k b z c z Y z b H z X z a a z d z − − = = − − = = − = = = − 1 1 0 : k k d z z d pole − − = = ◆For an LTI system: ◆its poles and zeros: 5.2 System Functions For Systems Characterized by Linear Constantcoefficient Difference equation
Ex 5.2 find difference equation for second-order System function Solution 1+z 1+2z-1+z Y H 1+-z 1+ +-Z ()=(1+22+z2)x(2) 48 y7+ yn yn-2]=x{z]+2xn-1]+rn-2]
19 Ex. 5.2 find difference equation for second-order System function ( ) ( ) 2 1 1 1 1 1 3 1 1 2 4 z H z z z − − − + = − + z z Y(z) ( z z )X (z) 1 2 1 2 1 2 8 3 4 1 1 − − − − = + + + − 2 2 1 2 8 3 1 4 1 y n + y n − − y n − = x n + x n − + x n − ( ) ( ) 1 2 1 2 1 2 1 3 1 4 8 z z Y z X z z z − − − − + + = = + − Solution:
5.2.1 Stability and Causality The difference equation does not uniquely specify the impulse response of a linear time-invariant system Each possible choice for the roc of the system function will lead to a different impulse response but they will all correspond to the same difference equation
20 5.2.1 Stability and Causality ◆The difference equation does not uniquely specify the impulse response of a linear time-invariant system. ◆Each possible choice for the ROC of the system function will lead to a different impulse response, but they will all correspond to the same difference equation
Causality 4 For a causal system the impulse response h/n must be right-sided sequence h[n]=0.,orn<0 .The region of convergence(ROC)of H(z) must be outside the outermost pole 21
21 Causality ◆For a causal system the impulse response must be right-sided sequence. hn ◆The region of convergence (ROC) of must be outside the outermost pole. H(z) h n for n = 0, 0