Transforms and its Applications: UNIT 2: Z Transform
Z Transform: Convolution Theorem
Z Transform: Definition ofConvolution of sequences. State and prove convolution theorem on Z‒transform. Example Important Solved Problems with formula, steps, derivation and answer based on Z Transform Convolution Theorem. Use convolution theorem to find the inverse Z‒transform.
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
{x(n)} and {y(n)} is defined as
(i)
{x(n)*y(n)} =
f(K) g(n − K)
if the sequences are non‒causal
(ii)
{x(n) *y(n)} = 
f(K) g(n − K)
if the sequences are causal.
2.
The convolution of two functions f(t)
and g(t) is defined as
f(t) *g(t) = f(KT) g[(n‒K) T], where T is the
sampling period.
State and prove convolution theorem on Z‒transform.
Statement :
If
Z‒1[X(z)] = xn and Z‒1[Y(z)] = yn then
Z‒1[X(z)
Y(z)] = 
xmyn-m = xn * yn where
* denotes the convolution operation.
Proof :
We know that,
=
X (z). Y (z)
1. Problems on - Z‒transform of f(n) * g (n) type
2. Use convolution theorem to find the inverse Z‒transform of
Transforms and its Applications: UNIT 2: Z Transform : Tag: Engineering mathematics, Maths : - Z Transform: Convolution Theorem
Transforms and its Applications: UNIT 2: Z Transform
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