Convolution Theorem

CONVOLUTION THEOREM

The convolution theorem plays an important role in the solution of difference equations and in probability problems involving sums of two independent random variables.

Definition: Convolution of sequences :

1. The convolution of two sequences

2. The convolution of two functions f(t) and g(t) is defined as

State and prove convolution theorem on Z-transform.

Statement :

XI. Z-transform of f (n) * g (n) type.

1. Find the Z-transform of the convolution of

2. Find the Z-transform of f(n) * g(n), where f(n) = u(n) and g(n) = δ(n) + (1/2) ⁿ u(n).

Solution: By convolution theorem,

3. Find the Z-transform of f(n) * g(n) where f(n) g(n) = (1/2)ⁿ and g(n) cos n л

Solution: By convolution theorem,

XII. Use convolution theorem to find the inverse Z-transform of

1(a) Find Z-¹ [z²/ (z-a) (z - b)

1(d) Using convolution theorem evaluate inverse Z-transform of