# 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**

**Linear property**

- F [ a ƒ( x ) + b g ( x ) ] = a F [ ƒ( x ) ] + b F [ g ( x ) ] where a, b are constants.

**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.

- 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