Convolution Sum

5.6 CONVOLUTION SUM

The convolution (linear convolution) relates the input, output and unit sample response of the discrete time systems.

5.6.1 Discrete Time Signal as Weighted Impulses

Consider the arbitrary discrete time signal x(n) of five samples:

x(n) = {2, 1, 3, -2, 1}

↑

Here

x(-2) = 2, x(-1) = 1, x(0) = 3, x(1) = -2 and x(2) = 1. This signal is graphically shown in figure

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5.6.2 Convolution Sum Formula

Consider the equation.

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Step 1:

If x(n) is applied as an input to the discrete time system, then response y(n) of the system is given as,

y(n) = T[x(n)]

Putting for x(n) in above equation

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Step 2:

The above equation can be expanded as,

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Step 3:

Since the system is linear, the above equation can be written as,

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Step 4:

In the above equation the sample values ... + x(-3), x(-2), x(-1), x(0), x(1), x(2), x(3) ... etc, are constants.

Hence with the help of scaling property of linear systems we can write above equation as.

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Step 5:

The response of the system to unit sample sequence δ(n) is given as.

T[δ(n)] = h(n)

.'. T[δ(n - k)] = (n-k) For shift invariant system

putting for T[δ(n − k)] = h(n−k) in equation (3) we get,

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The equation gives the response of linear shift invariant (LTI) system or LTI system to an input x(n). The above equation is basically linear convolution of x(n) an h(n). This linear convolution gives y(n). Thus,

Convolution sum: y(n) = x(n) * h(n).

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The convolution is commutative.

The convolution sum consists of following four operations:

(i) Folding: Sequence h(k) is folded at k = 0 get h(-k)

(ii) Shifting: h(-k) is shifted depending upon the value of 'n' in y(n).

(iii) Multiplication: x(k) and h(n-k) are then multiplied on sample to sample basis.

(iv) Summation: The product sequence obtained by multiplication of x(k) and h(n-k) is added over all values of 'k' to get value of h(n).

These operations will be more clear through the following example.