Transforms and its Applications: UNIT 4: Fourier Transform
Fourier Transform: Convolution Theorem Parseval's Identity
Fourier Transform: Example Important Solved Problems with formula, steps, derivation and answer based on Convolution Theorem Parseval's Identity.
CONVOLUTION
THEOREM PARSEVAL'S IDENTITY
Definition: Convolution
The
convolution of two functions f(x) and
g(x) is defined as
Convolution Theorem:
The
Fourier transform of the convolution of f(x)
and g (x) is the product of their Fourier transforms.
(i.e.,) F[f(x)
*g(x)] = F(s) G(s) = F [f(x)] F[g(x)]
Proof:
We know that, F [f (x)]
by
changing the order of integration, we get
=
G (s)F(s) = F(s)G (s)
Note:
F‒1 [F(s) G (s)] = f(x) *g (x)
=
F‒1[F(s)] * F‒1[G
(s)]
PARSEVAL'S IDENTITY:
If
F(s) is the Fourier transform of f(x), then
Proof:
By convolution theorem,
F[f(x)
* g(x)] = F(s). G (s)
f(x)
* g (x) = F−1 [F(s). G
(s)]
Note:
In the same way, we can prove Parseval's identity for Fourier sine and cosine
transforms.
If
Fs [f(s)] = FS[s] and Fc[g
(x)] = Fc(s) then
Transforms and its Applications: UNIT 4: Fourier Transform : Tag: Engineering mathematics, Maths : - Fourier Transform: Convolution Theorem Parseval's Identity
Transforms and its Applications: UNIT 4: Fourier Transform
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