Transforms and its Applications: UNIT 4: Fourier Transform
Properties of Fourier Transforms
Fourier Transform: Example Important Solved Problems with formula, steps, derivation and answer based on Properties of Fourier Transforms.
TRANSFORMS OF SIMPLE FUNCTIONS
PROPERTIES
OF FOURIER TRANSFORMS
1. Linear property
F [af(x)+bg(x)]
= aF[f(x)] + bF[ g(x) ]
where
a and b are real numbers.
Proof:
=
aF[f(x)] + bF(g(x)]
2. Change of scale property
For any non‒zero real
a, F [f(ax)] = 1/|a| F [s/a]
Proof:
3. Shifting property
(i) F [f(x
− a)] = eias F(s)
(ii) F [eiax f(x)] = F [s+a]
Proof :
4. Modulation Property:
Modulation theorem:
If F(s) is the Fourier transform of f(x), then
F [f(x)
cos ax] = 1/2 (F (s + a) + F(s − a) ]
Proof:
We know that,
5. F [xnf(x)] = (‒i)n ( dn
F(s) / dsn)
Proof:
We know that,
Differentiating
both sides n times w.r.to s, we get
6. (i) F
[f ‘(x)] = −is F(s) if f(x) → 0 as x→ ± ∞
(ii) F [f(n)(x)] = (‒i)nsn
F(s) if f (x), f ‘(x), ...
f(n − 1)(x) → 0 as x→ ± ∞.
Proof:
We know that,
= ‒is
F(s)
Similarly,
F [f(n) (x)] = (‒is)n
F(s) if f, f ', f '', ... f(n‒1) → 0 as x→ ±∞.
7. F[aʃx
f(x) dx] = F(s) / (‒is)
Proof:
8. 
Proof:
We know that,
F(s)
= F [f(x)]
9. F [f(x)] = F(‒s)
Proof:
We know that,
Transforms and its Applications: UNIT 4: Fourier Transform : Tag: Engineering mathematics, Maths : - Properties of Fourier Transforms
Transforms and its Applications: UNIT 4: Fourier Transform
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Transforms and its Applications
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