# FOURIER TRANSFORM

### DEFINITION OF FOURIER TRANSFORM

• Let  ƒ( x ) be defined in ( a , b ). The integral transform of ƒ( x ) is denoted by I ( ƒ( x ) ) and it is defined as ……………..(1)
• If  the integral exists. k ( s,x) is known as a function of s and x called the kernel of the transform . s is called the parameter of the transform and ƒ( x ) is the inverse transform of F( s ).

LAPLACE TRANSFORM

• • INFINITE FOURIER TRANSFORM

• • INFINITE FOURIER COSINE TRANSFORM

• cos sx dx
• cos sx ds

INFINITE FOURIER SINE TRANSFORM

• sin sx dx
• sin sx ds

## PROPERTIES OF FOURIER TRANSFORM

1. Linear property
• F [ a ƒ( x ) + b g ( x ) ] =  a F [ ƒ( x ) ] + b F [ g ( x ) ]  where a, b are constants.
1. Shifting property
•  F(s)

3. 4 . Change of scale property

• If then    5. If then    6. If then is F(s) as 7. If then  8. If then where bar (-) denote the complex conjugate.

9. If then where the bar represent the complex conjugate.

1. Modulation theorem

If then   ###### CONVOLUTION OF TWO FUNCTIONS
•   If ƒ(x) and g(x) are any two functions defined in (−∞ , ∞ ) then the convolution of these two functions is denoted by ƒ(x) g(x) and it is defined by  • For convolution is commutative of ###### CONVOLUTION THEOREM ( OR )  FALTUNG THEOREM
• If  F [ ƒ (x) ] = F (s) and  F[g (x)] = G(s) then
• F [ ƒ (x) ⋇ g(x) ] = F(s) . G(s) = F [ ƒ (x) . F[ g(x)] is the Fourier transform converts a convolution into product of their Fourier Transforms
###### PARSEVAL’S  IDENTITY
• If F [ ƒ (x) ] = F (s) then = SELF RECIPROCAL

• If the transform of ƒ(x) is ƒ(s), then the function ƒ(x) is said to be reciprocal. consider now then # PROPERTIES OF FOURIER SINE & COSINE TRANSFORM

1. PROPERTY

• • 2. PROPERTY

•  •  3. PROPERTY

•  •  4. PROPERTY

•  •  •    •    5. Fourier Sine and Cosine transform of derivatives:

• as •   as 6. If ƒ(x) and g(x) are given Function are and their Fourier Cosine transform and and are their  Fourier Sine transform

7. PROPERTY

•  •  8.   Parseval’s identity for Fourier Sine and Cosine Transforms

•   ###### FINITE FOURIER SINE TRANSFORM
• If ƒ(x) is piecewise continuous function defined in the interval ( 0,l ) , then the finite Fourier Sine transform of ƒ(x) in 0<x<l is defined as
• where n is an integer.
• The inverse finite Fourier Sine transform of ƒ(x) in the interval 0<x<l is given by
• ###### FINITE FOURIER COSINE TRANSFORM
• If ƒ(x) is piece wise continuous function defined in the interval ( 0, l ) then the finite Fourier Cosine transform is ƒ(x) in 0<x<l is defined as
• where n is an integer
• The inverse finite Fourier cosine transform of ƒ(x) is given by
• 