Application of linearity - Convolution

An arbitrary sequence, x(n), can be written as the weighted sum of delayed unit sample functions:

x(n) = ...+ x(-2) δ(n+2) + x(-1) δ(n+1) + x(0) δ(n) + x(1) δ(n-1) +...

So the response of a linear system to input x(n) can be written down using the linearity principle, that is, linear superposition. For a linear shift-invariant system the impulse response of which is T[δ(n)] = h(n) the reasoning goes like this

• For the input δ(n) output is h(n). For the input x(0) δ(n) the output is x(0) h(n) by the virtue of scaling.

• For the input δ(n-1) the output is h(n-1) by virtue of shift invariance. For an input x(1) δ(n-1) the output is x(1) h(n-1) by virtue of scaling.

• Thus for the input of x(0) δ(n) + x(1) δ(n-1) output is x(0) h(n) + x(1) h(n- 1) by virtue of the additivity property.

This reasoning is extended to cover all the terms which make up x(n). Generally the response to x(k) δ(n-k) can be given by x(k) h(n-k).

Given that

h(n) = T[δ(n)],  and x(n) = 

we have

As T[.] is linear we can apply linearity of the countable infinite number of times to write

In the equation stated above since the system is shift-invariant we write T[δ(n-k)] = h(n-k). Otherwise write hk(n) or h(n, k) in place of h(n-k). Therefore for a linear shift-invariant system

Note that if the system is not specified to be shift-invariant we would leave the above result in the form

Then if the shift-invariance is invoked we replace h(n, k) with h(n-k).

As in the case of continuous-time systems, impulse response, h(n), can be determined considering that the system has no initial energy; else the linearity property does not hold, so that y(n), as determined using the above equation, corresponds to the forced response of the system only.

The sum x(k )h(n, k ) is called as convolution sum, and is denoted x(n) * h(n).

A discrete-time linear shift-invariant system is characterized by its unit sample response h(n).

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