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 (t – t0), 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 (

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