Causality condition of a Discrete-Time LTI System Example- The discrete-time accumulator d etnea bV yn=∑(l is a causal system as it has a causal impulse response given b Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 11 Causality Condition of a Discrete-Time LTI System • Example - The discrete-time accumulator defined by is a causal system as it has a causal impulse response given by [ ] [ ] n y n x =− = h[n] [ ] [n] n = = =−
Causality condition of a Discrete-Time LTI System Example- The factor-of-2 interpolator defined by yn]=x1]+1(xn[m-1+x1[n+1) is noncausal as it has a noncausal impulse response given by {hn]}={0.510.5} 12 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 12 Causality Condition of a Discrete-Time LTI System • Example - The factor-of-2 interpolator defined by is noncausal as it has a noncausal impulse response given by [ ] [ ] ( [ 1] [ 1]) 2 1 y n = xu n + xu n − + xu n + {h[n]} ={0.5 1 0.5}
Causality Condition of a Discrete-Time LTI System Note: A noncausal LTI discrete-time system with a finite-length impulse response can often be realized as a causal system by inserting an appropriate amount of delay Example -a causal version of the factor-of- 2 interpolator is obtained by delaying the input by one sample period yn]=x1n-1]1+1(x1n-2]+x[m]) 13 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 13 Causality Condition of a Discrete-Time LTI System • Note: A noncausal LTI discrete-time system with a finite-length impulse response can often be realized as a causal system by inserting an appropriate amount of delay • Example - A causal version of the factor-of- 2 interpolator is obtained by delaying the input by one sample period: [ ] [ 1] ( [ 2] [ ]) 2 1 y n = xu n − + xu n − + xu n
Finite-Dimensional Discrete Time LTI Systems An important subclass of discrete-time lti systems is characterized by a linear constant coefficient difference equation of the form ∑dky{n-k]=∑pkx[n-k] k=0 k=0 x[n] and yln are, respectively, the input and the output of the system Rdk and pki are constants characterizing the system 14 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 14 Finite-Dimensional DiscreteTime LTI Systems • An important subclass of discrete-time LTI systems is characterized by a linear constantcoefficient difference equation of the form • x[n] and y[n] are, respectively, the input and the output of the system • and are constants characterizing the system { } dk { } pk = = − = − M k k N k k d y n k p x n k 0 0 [ ] [ ]
Finite-Dimensional Discrete Time LTI Systems The order of the system is given by max(n, M, which is the order of the difference equation It is possible to implement an LTI system characterized by a constant-coefficient difference equation since the computation involves two finite sums of products Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 15 Finite-Dimensional DiscreteTime LTI Systems • The order of the system is given by max(N,M), which is the order of the difference equation • It is possible to implement an LTI system characterized by a constant-coefficient difference equation since the computation involves two finite sums of products