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 I[f(x)] = F (s) = \int_{a}^{b} f(x) k (s.x)dx……………..(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

  • F(s) = \int_{0}^ { \infty}  e^{-sx}  f(x)dx
  • F(x) = \frac{1}{\2 \p i}\int_{c- i\infty}^{c+ i\infty} F(s)  e^{sx}ds

INFINITE FOURIER TRANSFORM 

  • F(s) = \frac{1}{\sqrt {\2 \pi}}\int_{-\infty}^{+\infty} f(x)  e^{-isx}ds
  • f(x)= \frac{1}{\sqrt {\2 \pi}}\int_{-\infty}^{+\infty} F(s)  e^{-isx}ds

INFINITE FOURIER COSINE TRANSFORM

  • F_{c}(s) = \sqrt{\frac{2}{{\Pi }}}\int_{0}^{\infty}  f(x) cos sx dx
  • f(x) = \sqrt{\frac{2}{{\Pi }}}\int_{0}^{\infty} F_{{c}}(s) cos sx ds

INFINITE FOURIER SINE TRANSFORM

  • F_{s}(s) = \sqrt{\frac{2}{{\Pi }}}\int_{0}^{\infty}f(x) sin sx dx
  • F(x) = \sqrt{\frac{2}{{\Pi }}}\int_{0}^{\infty} F_{{s}}(s) 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[f(x)] = F(s) then F [f(x-a)]= e^{ias} F(s)

     3.  F[ e^{iax} f(x)] = F(s+a)

     4 . Change of scale property

  • If F[f(x)] = F(s) then F[f(ax)]=\frac{1}{\left \|a  \right \|}[\frac{s}{{a}}], a\gt 0

     5. If F[f(x)] = F(s) then F[x^{n} f(x)] =  (-i)^{n}\frac{  d^{n}}{d  s^{n}} [F (s)]

     6. If F[f(x)] = F(s) then F [f'(x)] =  is F(s) f(x) \rightarrow0 as x\pm{\infty}

     7. If F[f(x)] = F(s) then F [\int_{a}^{x} f(x) dx ] \frac{F(s)}{-is}

     8. If F[f(x)] = F(s) then F=[\overline{f(x)}]= \overline {F(-s)where bar (-) denote the complex conjugate.

    9. If F[f(x)] = F(s) then F=[\overline{f(-x)}]= \overline {F(s) where the bar represent the complex conjugate.                                 

  1. Modulation theorem

   If F[f(x)] = F(s) then F[f(x) cos ax] \frac{1}{2}[F(s-a) +F(s+a)]

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)\astg(x) and it is defined by f(x)\ast g(x) = \frac{1}{\sqrt {2\pi}} \int_{- \infty}^{\infty} f(t)g(x-t)dt
  • For convolution is commutative of f(x)\ast g(x) = g(x) \ast f(x)
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

        \int_{-\infty}^{ \infty}    \left \| f(x) \right \|^2 dx = \int_{-\infty}^{ \infty}    \left \| F(s) \right \|^2 ds

 SELF RECIPROCAL

  • If the transform of ƒ(x) is ƒ(s), then the function ƒ(x) is said to be reciprocal. consider  f(x) = e ^{\frac{-x^2}{2}} now F  [e^{\frac{-x^2}{2}}] = e^{\frac{-x^2}{2}} = f(s) then  f(x) = e ^{\frac{-x^2}{2}}

 PROPERTIES OF FOURIER SINE & COSINE TRANSFORM

     1. PROPERTY

  • F_{{s}} [a f(x) + b g(x)] = a F_{{s}} [f(x)] + b F_{{s}} [g(x)]
  • F_{{c}} [a f(x) + b  g(x)] = a F_{{c}}[f(x)] +b F_{{c}}[g(x)]

      2. PROPERTY

  • F_{{s}} [f(x) cos ax = \frac{1}{2} [F_{s} (s+a) + F_{s} (s-a)]
  • F_{{c}} [f(x) cos ax = \frac{1}{2} [F_{c} (s+a) + F_{c} (s-a)]

      3. PROPERTY

  • F_{{s}}[f(x) sin  ax = \frac{1}{2} [F_{s} (s+a) + F_{s} (s-a)]
  • F_{{c}}[f(x) sin ax = \frac{1}{2} [F_{c} (s+a) + F_{c} (s-a)]

      4. PROPERTY

  • F_{{s}} [f(ax)] = \frac{1}{a} F_{{s}} \left [ \frac{s}{a} \right ] , a\gt 0
  • F_{{c}} [f(ax)] = \frac{1}{a} F_{{c}} \left (\frac{s}{a}  \right ),     a\gt 0
  • F_{{s}} \left [ f \left ( \frac{x}{a} \right ) \right ] = a F_{{s}} [as] ,a\gt 0
  • F_{{c}}  \left [ f \left ( \frac{x}{a} \right ) \right ] = a F_{{c}} [as] ,a\gt 0

      5. Fourier Sine and Cosine transform of derivatives:

  • F_{{s}} [f'(x)] = -s F_{{c}} [f(x)] if f(x)\rightarrow0  as x\rightarrow \infty
  • F_{{c}}[f'(x)] =\sqrt{\frac{2}{\pi}} f(0)+s F_{s}f(x)






if f(x)\rightarrow0 as x\rightarrow \infty

      6. If ƒ(x) and g(x) are given Function are F_{{c}} [f(x)] and G_{{c}}[f(x)] their Fourier Cosine transform and F_{{s}}[f(x)]  and G_{{s}}[f(x)]are their  Fourier Sine transform

    7. PROPERTY

  • F_{{s}}[ x f(x)] = -\frac{d}{ds}F_{{c}} [f(x)]
  • F_{{c}}[ x f(x)] = -\frac{d}{ds}F_{{s}} [f(x)]

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

  • \int_{0}^{\infty}  \left \| F_{c} (s) \right \|^2 ds =\int_{0}^{\infty}  \left \| F_{s} (s) \right \|^2 ds=\int_{0}^{\infty}  \left \| f(x) \right \|^2 dx
       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                                                                                   
  •     F_{{s}}[f(x)] =\int_{0}^{l}   f(x) sin \frac{n \pi x}{l} dx  where n is an integer.                       
  • The inverse finite Fourier Sine transform of ƒ(x) in the interval 0<x<l is given by
  • f(x)= \frac{2}{l}  \sum_{n=1}^{\infty}  F_{{s}}[f(x)] sin \frac{n \pi x}{l}
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 
  • F_{{c}}[f(x)] = \int_{0}^{l}  f(x) cos \frac{n \pi x}{l} dx where n is an integer
  • The inverse finite Fourier cosine transform of ƒ(x) is given by
  • f(x)=\frac{1}{l}F_{{c}} (0) + \frac{2}{l}\sum_{n=1}^{\infty} F_{{c}}[f(x)] cos \frac{n\pi x}{l}

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