Fourier Transform - Chennai Tuition Centre,Home tuition in chennai Chennai Tuition Centre

Fourier Integral Theorem:

If f(x) is a given function defined in (-l, l) and satisfies dirichlet’s conditions then

F(x) = 1/π   0-∞f(t) cosλ (t-x) dt dλ

This is known as Fourier integral Theorem or Fourier integral formula.

 

Point of discontinuity

{F(x+0) – F(x-0)}/2 = 0-∞f(t) cosλ (t-x) dt dλ

Fourier Sine integral:

 

F(x) = 2/π   0 sin λx 0f(t) sin λt dt dλ

 

 Fourier cosine integral:

 

F(x) = 2/π   0cos λx 0f(t) cos λt dt dλ

 

Complex form of Fourier integral:

 

F(x) = 1/2π   -∞e-iλx -∞f(t) eiλt dt dλ

 

Fourier Transforms

Complex Fourier Transforms and its inversion formula

 

F[ f(x) ] = 1/√2π -∞f(t) eist dt is called the complex Fourier transform of f(x).

 

f(x) =  1/√2π -∞F[ f(t) ] e-isx ds is called the inversion formula for the complex Fourier transform of F[ f(t) ]

 

Fourier Sine Transform:


Fs[ f(x) ] = √(2/π) 0f(t) sin st dt   is called the Fourier sine transform of the function f(x).

 

f(x)=  √(2/π) 0 Fs[ f(x) ]  sin sx ds    is called the inversion formula for the Fourier Sine Transform.

 

Fourier Cosine Transform:

 

Fc[ f(x) ] = √(2/π) 0f(t) cos st dt   is called the Fourier cosine transform of the function f(x).

 

f(x)=  √(2/π) 0 Fc[ f(x) ]  cos sx ds    is called the inversion formula for the Fourier Cosine Transform.

 

 

Convolution of two functions:

 

If f(x) and g(x) are any two functions defined in (-∞, ∞) then the convolution of these two functions is defined by

 

1/√2π -∞f(t) g(x-t) dt and is denoted by

 

 f * g = 1/√2π -∞f(t) g(x-t) dt

 

Convolution Theorem for Fourier Transforms:

 

F[(f * g) x ] = F(s) . G(s)

1/√2π -∞(f * g) x eisx  dx = {1/√2π -∞f(x) eisx  dx} {1/√2π -∞g(x) eisx  dx }

 

Parseval’s Identity:

 

If f(x) is a given function defined in (-∞, ∞) then it satisfy the identity

 

-∞[ f(x) ]2 dx =  -∞[ f(s) ]2  ds  where F(s) is the fourier transform of f(x).

 

 









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