Causality Condition of a Discrete-Time LTI System Let xi[n] and x2In]be two input sequences i[]=x2n]forn≤no The corresponding output samples at n=n of an lti system with an impulse response thin 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[m]=∑k]x[-k]=∑k]x1[-k k=-∞ k=0 +∑hk]x1{m-k] k y2[no=∑hk]x2[m-k]=∑hkx2[m-k k: k=0 +∑hk]x2{o-k] 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 yIno]=y2lnol ASx1m]=x2 n for n≤no ∑h[kxo-k]=∑hk]x2{r-k] k=0 k=0 This implies ∑k-k]=∑kx2[r-k k: k 8 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 ∑k]x[-k]=∑hk]x2{To-k] k 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 hn is a causal sequence Example- The discrete-time system defined yn]=1x{]+x2x{n-1]+a3x[n-2]+4x[n-3] is a causal system as it has a causal impulse response {h4n]}={a1a234} 10 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