Convolution of two sequences X₁[n] and X₂[n] is represented as

Concept:

Convolution of signals:

Convolution is a mathematical way of combining two signals to form a third signal.

Convolution in the time domain is equal to multiplication on the frequency domain and vice versa.

If X(ω) and Y(ω) are the Fourier transforms of x(t) and y(t), then

x(t) * y(t) = X(ω) × Y(ω)

In terms of Z transform we can write:

x(n) * y(n) = X(z) × Y(z)

Analysis:

x₁(n) * x₂(n) = X₁(z) X₂(z)

Important Points

For the two discrete-time sequences defined in the interval as shown:

n1 ≤ x1(n) ≤ n2

n3 ≤ x2(n) ≤ n4

The convolution of the two will be in the interval:

y(n) = x1(n) ⊕  x2(n) with n1 + n3 ≤ x1(n) ≤ n2 + n4