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Communication Systems: Fourier Analysis

This document discusses Fourier analysis and Fourier series. It defines the Fourier series representation of a function f(t) as the sum of a constant term and sinusoidal terms with coefficients an and bn. It describes how the coefficients are calculated from the function and discusses symmetry properties that result in certain coefficients being zero. The document also covers the discrete phase spectrum obtained from the Fourier series and definitions for average power and energy of periodic and aperiodic signals.

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Mubashir Ghaffar
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© © All Rights Reserved
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100 views11 pages

Communication Systems: Fourier Analysis

This document discusses Fourier analysis and Fourier series. It defines the Fourier series representation of a function f(t) as the sum of a constant term and sinusoidal terms with coefficients an and bn. It describes how the coefficients are calculated from the function and discusses symmetry properties that result in certain coefficients being zero. The document also covers the discrete phase spectrum obtained from the Fourier series and definitions for average power and energy of periodic and aperiodic signals.

Uploaded by

Mubashir Ghaffar
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Communication

Systems

Fourier Analysis

By
Engr. Jawwad Ahmad (Ph.D)

1
Today’s Goal
 Fourier Summary

 Symmetry

 Energy & Power Signals

Engr. Dr. Jawwad Ahmad 2


Summary
N
f (t )  ao    an cos n1t  bn sin n1t
n 1
2
1   2 f1
T
T
1
ao 
T  f (t ) dt
0
T T
2 2
an   f (t ) cos n1t dt bn   f (t ) sin n1t dt
T 0 T 0

Engr. Dr. Jawwad Ahmad 3


Summary
Magnitude = 2 2
a n  bn

Magnitude = =
a n |a
2
n|0
(Even Symmetry)

2
Magnitude = 0 = b|b
n n
|
(Odd Symmetry)

Engr. Dr. Jawwad Ahmad 4


Summary
In addition to the amplitude spectrum, one also constructs a
discrete Phase Spectrum. At the frequency now, the combine
cosine and sine terms are used to determine the phase angle  n.

an cos no t  bn sin not  a n2  bn2 cos not  tan1   bn 


 
  a n 

  tan   bn a 
1

 n

Engr. Dr. Jawwad Ahmad 5


Summary

Fourier
Series

Engr. Dr. Jawwad Ahmad 6


Symmetry
 The function f (t) has Even Symmetry if:

f(t) = f (–t)

Engr. Dr. Jawwad Ahmad 7


Symmetry
 The function f (t) has Odd Symmetry if:

f (t)= – f (–t)

Engr. Dr. Jawwad Ahmad 8


Average Power
 The average value of periodic function/signal is,
T
1
 V (t )   V (t ) dt
T 0
T /2
1
 V (t )   | V (t ) | dt
T T / 2
 For A-periodic function/signal ,
T
1
 V (t )  lim  V (t ) dt
T 0

Engr. Dr. Jawwad Ahmad 9


Average Energy
 The average energy for periodic function/signal is,
T /2
E 
2
| V (t ) | dt
T / 2

 For A-periodic function/signal ,



E   | V (t ) |2 dt


Engr. Dr. Jawwad Ahmad 10


Thank you

Engr. Dr. Jawwad Ahmad 11

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