Causality Condition of a Discrete- Time LTI System et xi[n] and xln be two input sequences 21」forn O The corresponding output samples atn=no of an lti system with an impulse response hnd are then given by Copyright C 2001, S.K. Mitra
Copyright © 2001, S. K. Mitra 6 Causality Condition of a Discrete-Time LTI System • Let and be two input sequences with • The corresponding output samples at of an LTI system with an impulse response {h[n]} are then given by x [n] 1 x [n] 2 x [n] x [n] 1 = 2 n no for n = no
Causality Condition of a Discrete-Time LTI System n[=∑hx1[no-k]=∑ hk ] iln-k k=-00 k=0 +∑k]x[-k k y2m]=∑kx2[no-k]=∑hkx2{no-k k=-00 k=0 +∑kx2[no-k] k=-∞0 Copyright C 2001, S.K. Mitra
Copyright © 2001, S. K. Mitra 7 Causality Condition of a Discrete-Time LTI System = =− = − = − 0 2 2 2 k o k o o y [n ] h[k]x [n k] h[k]x [n k] − =− + − 1 2 k o h[k]x [n k] = =− = − = − 0 1 1 1 k o k o o y [n ] h[k]x [n k] h[k]x [n k] − =− + − 1 1 k o h[k]x [n k]
Causality Condition of a Discrete-Time LTI System If the lti system is also causal, then O Asx[n]=x2[m]forn≤mo ∑k]x1[n0-k]=∑kx2[n- k=0 k=0 This implies ∑1{m-]=∑hk]x2{m-k 〓一00 k Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 8 Causality Condition of a Discrete-Time LTI System • If the LTI system is also causal, then • As • This implies x [n] x [n] 1 = 2 n no for [ ] [ ] o no y n y 1 = 2 = = − = − 0 2 0 1 k o k o h[k]x [n k] h[k]x [n k] − =− − =− − = − 1 2 1 1 k o k o h[k]x [n k] h[k]x [n k]
Causality Condition of a Discrete-Time LTI System As xi[n]+x2[n] for n>no the only way the condition ∑ h(k]xno k]=∑k]x2{-h k=-0o k=-0o will hold if both sums are equal to zero, Which is satisfied if h[k]=o for k<0 Copyright C 2001, S.K. Mitra
Copyright © 2001, S. K. Mitra 9 Causality Condition of a Discrete-Time LTI System • As for the only way the condition will hold if both sums are equal to zero, which is satisfied if x [n] x [n] 1 2 n no − =− − =− − = − 1 2 1 1 k o k o h[k]x [n k] h[k]x [n k] h[k] = 0 for k < 0
Causality Condition of a Discrete-Time LTI System A discrete-time LTI system is causal if and only if its impulse response hin is a causal sequence Example- The discrete-time system defined y]=a1x1n]+a2x[n-1]+3xn-2]+a4xn-3] is a causal system as it has a causal impulse response hn]}={(10234 Copyright C 2001, S.K. Mitra
Copyright © 2001, S. K. Mitra 10 Causality Condition of a Discrete-Time LTI System • A discrete-time LTI system is causal if and only if its impulse response {h[n]} is a causal sequence • Example - The discrete-time system defined by is a causal system as it has a causal impulse response [ ] [ ] [ 1] [ 2] [ 3] y n = 1 x n +2 x n − +3 x n − +4 x n − { [ ]} { } h n = 1 2 3 4