O'Reilly logo

Signals and Systems by Smarajit Ghosh

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

7.2 PROPERTIES OF LINEAR CONVOLUTION

The linear convolution is the convolution process for the continuous time systems. The convolution of continuous time (CT) signals possesses the following properties:

Commutative Property: x1(t)*x2(t) = x2(t)*x1(t)

Associative Property : x1(t)*[x2(t) *x3(t)] = [x1(t)*x2(t)] *x3(t)

Distributive Property : x1(t)*[x2(t) + x3(t)] = x1(t)*x2(t) + x1(t)*x3(t)

If any function is convolved with impulse function d (t), it results the function itself. Again, convolution of any function with δ (tt0), it results the shifted function. If the original function is impulse, this is also true. Mathematically, we can write

 

x(t)* δ (t) = x(t)
x(t)* δ (t t0) = x(t t0)
δ (t)* δ (t) = δ (t)
δ (t)* δ (t – t0) = δ (

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required