1. Prepared By:- Nisarg Amin
Topic:- Properties Of Fourier Transform

2. Properties Of Fourier Transform
•There are 11 properties of Fourier Transform:
i. Linearity Superposition
ii. Time Scaling
iii. Time Shifting
iv. Duality Or Symmetry
v. Area Under x(t)
vi. Area Under X(f)
vii. Frequency Shifting
viii. Differentiation In Time Domain
ix. Integration In Time Domain
x. Multiplication In Time Domain
xi. Convolution In Time Domain

3. Linearity Superposition
• If
and
Then
• This follows directly from the definition of the
Fourier Transform (as the integral operator is
linear). It is easily extended to a linear
combination of an arbitrary number of signals
)()( 11 fXtx
F

)()( 12 fXtx
F

)()()()( 212211 fXafXatxatxa
F


4. Time Scaling
• Let x(t) and X(f) be Fourier Transform pairs and let
‘α’ be a constant. Then time scaling property states
that
• x(αt) represents a time scaled signal and
X(f/ α) represents frequency scaled signal.
• For α<1, x(αt) represents compressed signal but X(f/
α) represents expanded version of X(f).
• For α>1, x(αt) will be an expanded signal in the time
domain. But its Fourier Transform X(f/ α)
represents version of X(f).









f
Xtx
||
1
)( F

5. Time Shifting
• The time shifting property states that if x(t)
and X(f) form a Fourier transform pair then,
• Here the signal is time shifted signal.
• It is the same signal x(t) only shifted in time.
 fXettx dftj
d
2
)( 
 F
)( dttx 

6. Duality Or Symmetry
• This property states that,
if
then
• The duality theorem tells us that the shape of
the signal in the time domain and the shape of
the spectrum an be interchanged.
)()( fXtx F

)()( fxtX F


7. Area Under x(t)
• This property states that the area under the
curve x(t) equals the value of its Fourier
Transform at f=0
i.e. if
Then x(t)=X(0)
)()( fXtx
F


8. Area Under X(f)
• This property states that the area under the
curve X(f) equals the value of signal x(t) at t=0.
i.e. if
Then x(0)=X(f)
)()( fXtx F


9. Frequency Shifting
• The frequency shifting characteristics states
that if x(t) and X(f) form a Fourier Transform
pair then,
• Fc is a real constant.
)()(2
c
Ftfj
ffXtxe c


10. Differentiation In Time Domain
• This property is applicable if and only if the
derivative of x(t) is Fourier Transformable.
)(2)( ffXjtx
dt
d F


11. Integration In Time Domain
• Integration in time domain is equivalent to
dividing the Fourier Transform by (j2πf).
• If and provided that X(0)=0)()( fXtx F

)(
2
1
)( fX
fj
dx
t
F




12. Multiplication In Time Domain
• The multiplication theorem states that:
If and are the two
Fourier Transform pairs then,
• This means that multiplication of two signals in
time domain gets transformed into convolution
of their Fourier Transform .
)()( 11 fXtx
F
 )()( 12 fXtx
F

 dfXXtxtx F
)()()()( 2121  


)()()()( 2121 fXfXtxtx F


13. Convolution In Time Domain
• This property states that the convolution of
signals in the time domain will be transformed
into the multiplication of their Fourier
Transform in the frequency domain.
i.e.
)()()()( 2121 fXfXtxtx F
