Properties of LTI System

In the preceding chapters,we have already derived expressions for discrete as well as continuous time convolution operations.

Discrete :	Continuous :

We shall now discuss the important properties of convolution for  LTI  systems.

1) Commutative property :

By the commutative property,the following equations hold true :

a) Discrete time:

 

Proof : We know that

Hence we make the following substitution (n - k = l )

 The above expression can be written as

So it is clear from the derived expression that

 

 Note :

1. 'n' remains constant during the convolution operation so 'n' remains constant in the substitution “n-k = l” even as 'k' and 'l' change.

2. “l” goes from  to +, this would not have been so had 'k' been bounded.( e.g :- 0 < k < 11 would make n < l < n – 11)