Fourier Transformation

2. Department Of
Mathematics
Program : MS Mathematics

3. Group Members
1. S.Wasim Shah
2. Salman Yousaf
3. Iftikhar khan

4. Fourier Transform
Let f(w) be defined on (-∞,∞) and is piecewise
continuous, differentiable in each finite interval
and is absolutely integral on (-∞,∞), then
Fourier transform of f(w) is defined as:
1
( ) ( )
2
iwx
f w f x e dx






5. Mean of Fourier transformation
The Fourier transform decomposes
a function of time into the
frequencies that make it up, in a
way similar to how a musical chord
can be expressed as the
frequnecies of its notes

6. Need of Fourier Transform
• The Fourier transform decomposes a
function of time into frequencies
that make it up in the way similar to
how a musical chord can be
expressed as the frequencies of its
nodes

7. Physical Significance of Fourier
Transform
The Fourier transform convert a set of
time domain data vector into a sets of
frequency domain vector. Imagine you
wanted to know changes in the
temperature of soil.

8. • Now suppose we to measure the
temperature of this soil in your
garden at dawn, midday and dusk
every day of the year.

9. • we would have a list of real
numbers representing this soil
temperature. The Fourier
transform provide a mean of
manipulating or transform this
raw data into alternative set of
data.

10. Mathematical Modeling of Fourier
Transform
The Fourier transform is a generalization of the
Fourier Series.
It is applied to continues and a periodic
functions, but the use of the impluse function
allow the use of discrete signals .

11. The Fourier transform is defined as
Where w is transform parameter
1
( ) ( )
2
iwx
f w f x e dx






12. Example
• Find the Fourier transform of f(x)=
if x>0 and f(x)=0 if x>0 with a>0.
• Solution
( )ax
F e ax
e
1
( ) ( )
2
iwx
f w f x e dx






13. 0
0
( )
0
1 1
0( ) ( )
2 2
1
2
ax iwx ax
iw a x
e dx e e dx
e dx
 


  


 
  


14. ( )
0
1
[ ]
( )2
1 1
[ ]
( )2
a iw x
e
a iw
a iw


 

 


15. Which is required result
1 1
( )2 a iw 

16. Applications
• Designing and using anntennas
• Image processing and filtering
• Transformation, representation and encoding.
• Smoothing and sharpening

17. • Restoration blur removal.
• Data processing and analysis
• High pass, low pass and band pass filters.
• Signal and noise estimation.

18. •Thanks a lots dear
teacher and fellows.
•Any Question ?