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Communication Systems: Fourier Analysis

The document discusses Fourier analysis and the coefficients of a Fourier series. It defines a Fourier series as a function expressed as a sum of sines and cosines with constant coefficients. It then describes how to calculate the coefficients ao, an, and bn by integrating the function over a full period using properties of trigonometric integrals. Specifically, ao is the average value of the function, an is calculated from the integral of f(t) multiplied by the cosine, and bn is calculated from the integral of f(t) multiplied by the sine. The document provides the equations for calculating each of the Fourier series coefficients.

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Mubashir Ghaffar
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
44 views14 pages

Communication Systems: Fourier Analysis

The document discusses Fourier analysis and the coefficients of a Fourier series. It defines a Fourier series as a function expressed as a sum of sines and cosines with constant coefficients. It then describes how to calculate the coefficients ao, an, and bn by integrating the function over a full period using properties of trigonometric integrals. Specifically, ao is the average value of the function, an is calculated from the integral of f(t) multiplied by the cosine, and bn is calculated from the integral of f(t) multiplied by the sine. The document provides the equations for calculating each of the Fourier series coefficients.

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 Analysis

 Integration

 Coefficients of Fourier Series

Engr. Dr. Jawwad Ahmad 2


Fourier Series
 In terms of Angular Frequency,

N
f (t )  a0    an cos nt  bn sin nt
n 1
 or,
N
f (t )  a0    an cos n1t  bn sin n1t
n 1
1
where ao, an, bn are the constant co-efficient
for the corresponding frequency which
depends upon f(t).
Engr. Dr. Jawwad Ahmad 3
Fourier Analysis
 Equation 1 is the trigonometric form of Fourier series of f(t) and
the process of determining the values of the constant co-efficient
ao, an, bn is called Fourier analysis.

 Or before calculating these co-efficient or constant co-efficient, we


will use some trigonometric integral having two sets of integers 'k'
and 'n' (i.e. 1, 2, 3, … ) in the following integrations, where '0'
and 'T' are used as integration limits.
T T

 sin n0 tdt  0  cos n


0
0 tdt  0
0
2 3
Engr. Dr. Jawwad Ahmad 4
Fourier Analysis
 If; k ≠ n andk ≠ –n
T T

 sin k t cos n tdt  0


o o  sin k t sin n tdt  0
o o
0
4 0
5
T

 cos k t cos n tdt  0


0
o o

6
 For square trigonometric function;
T
T T
T
0 cos notdt  2
2
0 sin notdt  2
2

7 8

Engr. Dr. Jawwad Ahmad 5


Fourier Analysis

The evaluation of the known co-efficient or


constant co-efficient may be calculated by using
above 7-integrals.
Engr. Dr. Jawwad Ahmad 6
Coefficients of Fourier Series
 For ao integrate each side of Fourier series (equation-1) over a full
period.
T T T N

 f (t ) dt   a o dt     an cos n1 t  bn sin n1 t dt


0 0 0 n 1

 Using equation 2 and equation 3,


T T

 f (t )dt   a dt
0 0
o

T T

 f (t )dt  a  dt
0
o
0

Engr. Dr. Jawwad Ahmad 7


Coefficients of Fourier Series
T
T

0
f (t )dt  a o t
o
T

 f (t )dt  a
0
o (T )
T
1
ao 
T  f (t )dt
0

 Notice that this constant ao is simply the arrange value of f(t) over
a period.
 Thus ao is describe as DC component of f(t).

Engr. Dr. Jawwad Ahmad 8


Coefficients of Fourier Series
 To evaluate the constant coefficient an multiplying equation-1 by
cos kω1t and then integrate it for whole period.
T T T N

 f (t ) cos k t dt   a
1 o cos k1t dt    an cos n1t cos k1t dt
0 0 0 n 1
T N
   bn sin n1t cos k1t dt
0 n 1
 For, k = n, Using equation 3 and equation 6 above equation
becomes;
T T


0
f (t ) cos n1t dt  an  cos 2 n1t dt
0

Engr. Dr. Jawwad Ahmad 9


Coefficients of Fourier Series
 Using equation 8,
T
T

0
f (t ) cos n1tdt  an
2

T
2
an   f (t ) cos n1tdt
T 0

Engr. Dr. Jawwad Ahmad 10


Coefficients of Fourier Series
 To evaluate the constant coefficient bn multiplying equation-1 by
sin kω1t and then integrate it for whole period.
T T T N

 f (t ) sin k t dt   a
1 o sin k1t dt    an cos n1t sin k1t dt
0 0 0 n 1
T N
   bn sin n1t sin k1t dt
0 n 1
 For, k = n, Using equation 2 and equation 4 above equation
becomes;
T T


0
f (t ) sin n1t dt  bn  sin 2 n1t dt
0

Engr. Dr. Jawwad Ahmad 11


Coefficients of Fourier Series
 Using equation 7,

T
T

0
f (t ) sin n o tdt  bn
2
T
2
bn   f (t ) sin n o tdt
T 0

Engr. Dr. Jawwad Ahmad 12


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 13


Thank you

Engr. Dr. Jawwad Ahmad 14

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