Convolution – Properties:

Property 1

The convolution operation is commutative, i.e., x(n) * h(n) = h(n) * x(n).

A linear shift-invariant system with the input x(n) and unit sample response h(n) will have same output as the linear shift-invariant system with input h(n) and unit sample response x(n).

Property 2 The convolution of h(n) with δ(n) is, equal to h(n).

That is, convolution of any function with δ function gives back the original function. Stated another way: The identity sequence for convolution operator is the unit sample, or

x(n) * δ(n) = δ(n) * x(n) = x(n)

Property 3 The convolution of the delayed unit sample sequence with x(n)

x(n) * δ(n-k) = x(n-k)

Property 4 Associativity

x(n) * (y(n) * z(n)) = (x(n) * y(n)) * z(n) = x(n) * y(n) * z(n)

Property 5 Distributivity over the sequence addition.

x(n) * (y(n) + z(n)) = x(n) * y(n) + x(n) * z(n)

Block diagram manipulation The properties of associativity, commutativity, and distributivity enable us to determine impulse response of series or the parallel combinations of systems in the terms of their individual impulse responses, as shown below:

(1) The following 3 LSI systems have similar unit sample response:

(2) The following 2 LSI systems are equivalent:

(3) The combination of above 2 is the series parallel combination:

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