Fourier Transform Pairs
Derived Functions (using basic functions and properties)
Some Basic Functions
x(t)
xt 
X(ω)
1 
X()e jt d

2 
(Synthesis)
X()  


x(t)e  jt dt
(Analysis)
 t 
1
(impulse)
(constant)
0,
t  
1,
t 1 2
t 1 2
 
sinc  
 2 
x(t)
 t  
0,
   
 Tp  1,
  
X(ω)
t  Tp 2
t  Tp 2
(time scaled rectangular pulse, width=Tp)
  

 p 


p
 t
sinc  p 
2
 2 
(rectangular pulse in ω)
1
2  
t 1
 0,
t  
1  t , t  1
(constant)
(impulse)
(triangular pulse, width=2)
(unit rectangular pulse, width=1)
(sinc)


e j0 t
2    0 
 t Tp
(complex exponential)
(shifted impulse)
(scaled triangular pulse, width=2Tp)
et  (t)
1
j  
cos(0 t)
(causal exponential)
(same as Laplace w/ s=jω)
t2
 2
1
e 2
 2
(Gaussian)
e

 2 2
2
(Gaussian)
 T 
Tp sinc  p 
 2 
 
sinc2  
 2 
 T 
Tp sinc 2  p 
 2 
      0       0  
sin(0 t)
j      0       0  
γ(t)
(unit step function)
1
   
j
Using these functions and some Fourier Transform Properties (next page),
we can derive the Fourier Transform of many other functions.
Information at http://lpsa.swarthmore.edu/Fourier/Xforms/FXUseTables.html
© 2015, Erik Cheever
Fourier Transform Properties
Name
Linearity
Time Scaling
Time Delay (or
advance)
Time Domain
  x1 (t)    x 2 (t)
xt a
Frequency Domain
  X1 ()    X 2 ()
aX  a 
f (t  a)
X()e  ja
Complex Shift
Time Reversal
Convolution
x(t)e j0 t
x(t)
x(t) * h(t)
X(  0 )
X()
X()H()
Multiplication
x1 (t)x 2 (t)
1
X1 () * X 2 ()
2
d n x(t)
dt n
( j) n X()
 x()d
1
X()  X(0)()
j
d
jn n X()
d
Differentiation
Integration
Time
multiplication
Parseval’s
Theorem
Duality
t
n
t x(t)
Energy  


2
x(t) dt Energy 
1 
2
X() d

2 
X(t)
2x()
1
X( t)
2
x()
Symmetry Properties
x(t)
X(ω)
X()  X* ()
Re  X()   Re  X() 
x(t) is real
Im  X()    Im  X() 
X()  X()
X()  X()
Real part of X(ω) is even,
imaginary part is odd
X()  X()
x(t) real,
even
Im  X()   0
X(ω) is real and even
x(t) real,
odd
Re  X()   0
Im  X()    Im  X() 
X(ω) is imaginary and odd
Relationship between Transform and Series
If xT(t) is the periodic extension of x(t) then:
cn 
1
X  n0 
T
Where cn are the Fourier Series coefficients of xT(t)
and X(ω) is the Fourier Transform of x(t)
Furthermore

XT    2  cn     n0 
n 