UNIT – IV FOURIER TRANSFORMS

Some Important Formulae

Fourier Cosine Transform

Fourier Transform Pair Fourier Sine Transform Pair
Pair
  
1 2 2
Transform F[f(x)]=

2  
f ( x )e isx dx FC[f(x)]=  f ( x) cos sxdx
 0
FS[f(x)]=
  f ( x) sin sxdx
0
 F(s)  FC (s )  FS (s)
  
1 2 2
Inverse
Transform
f(x) = 
2  
F(s)e i sx ds f(x) =
  FC (s) cos sxds
0
f(x) =
  F (s) sin sxds
S
0

Fourier Fourier Cosine Fourier Sine

Integral Formula Integral Formula Integral Formula
   
f(x) = 1   f (t ) cos( x  t ) s dtds 2 2
 0 
f ( x) 
 
0 0
f (t ) cos sx cos st dt ds f ( x) 
  f (t )sin sx sin st dt ds
0 0

Convolution of two functions:


1
f ( x) * g ( x) 
2 

f (t ) g ( x  t ) dt

Convolution Theorem:
If F[f(x)]= F(s) and F[g(x)]= G(s) then

F  f ( x)* g ( x)  F  s   G  s 
Note: F 1  F  s  G  s   f ( x)* g ( x)

Parseval’s Identity for Fourier transform:

If F[ f ( x)]  F (s) then
 

 f ( x) dx  
2 2
F ( s) ds
 
Parseval’s Identity for Fourier cosine transform:
If Fc [ f ( x)]  Fc (s) and Fc [g( x)]  Gc (s) then
 

 f ( x) dx   Fc ( s) ds and
2 2

0 0
 


0
f ( x) g ( x) dx   Fc ( s) Gc ( s) ds
0

Parseval’s Identity for Fourier sine transform:

If Fs [ f ( x)]  Fs (s) and Fs [g( x)]  Gs (s) then
 

 f ( x) dx   Fs ( s) ds and
2 2

0 0
 


0
f ( x) g ( x) dx   Fs ( s) Gs ( s) ds
0
 PROPERTIES

S.No Property Complex F.T Fourier Cosine Fourier Sine

F[af(x) + bg(x)]= FC[af(x)+bg(x)]= FS[af(x) + bg(x)]=
1. Linearity
a F[f(x)] +b F[g(x)] aFC[f(x)]+bFC[g(x)] aFS[f(x)]+bFS[g(x)]
Shifting F[f(x-a)]=eiasF[f(x)]
2. - -
Theorem = eiasF(s)
Shifting in
3. F[eiaxf(x)] = F(s-a) - -
respect to s
Change of 1
4. F[f(ax)] = F(s/a) FC[f(ax)] = 1/a FC (s/a) FS[f(ax)] = 1/a FS (s/a)
Scale a
d
F[xf(x)] = (-i) [F(s)] d
Derivative of ds FC[xf(x)] = [FS (s)] d
5. ds FS[xf(x)] =  [Fc (s)]
Transform ds
F[xnf(x)]= (-i)ndnF/dsn (Relation between FC & FS)
F[f ’(x)] = (-is)F(s)
Transform 2
6. FC[f ’(x)] =  f(0) + sFS(s) FS [f ‘(x)] = - sFc(s)
of derivative n n
F[d f/dx ] = (-is) F(s) n


i) F  f ( x)   F ( s)
Conjugate
7. symmetry ii) F  f ( x)   F (s) - -
property
iii)F[f(-x)] = F(-s)
i) F[f(x)cosax]= i) Fc[f(x)cosax]=
i) Fs[f(x)cosax]=
1 1
[F(s+a)+F(s–a)] [Fc(s+a)+Fc(s-a)] 1
2 2 [Fs(s+a)+Fs(s– a)]
Modulation 2
8. ii) F[f(x)sinax]= ii) Fc[f(x)sinax]=
Theorem ii) Fs[f(x)sinax]=
1 1
[F(s+a)-F(s-a)] [Fs(s+a)-Fs(s-a)] 1
2i 2 [Fc(s-a)-Fc(s+ a)]
2
Fourier x  F(s)
9. Transform F  f ( x )dx  = - -
of Integral a  (-is)