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Communication Systems: Complex Fourier Transform

The document is a lecture on the complex form of the Fourier series given by Engr. Dr. Jawwad Ahmad. It begins by expressing a function f(t) as an infinite sum of cosine and sine terms. It then develops this into an exponential form using Euler's formula. This leads to expressing f(t) as an infinite sum of complex exponential terms, where each term is multiplied by a complex constant cn. It derives an expression for evaluating the values of these complex constants cn. The document concludes by expressing the complex form of the Fourier series as an infinite sum of complex exponential terms multiplied by cn, ranging from negative to positive infinity.

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Mubashir Ghaffar
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56 views10 pages

Communication Systems: Complex Fourier Transform

The document is a lecture on the complex form of the Fourier series given by Engr. Dr. Jawwad Ahmad. It begins by expressing a function f(t) as an infinite sum of cosine and sine terms. It then develops this into an exponential form using Euler's formula. This leads to expressing f(t) as an infinite sum of complex exponential terms, where each term is multiplied by a complex constant cn. It derives an expression for evaluating the values of these complex constants cn. The document concludes by expressing the complex form of the Fourier series as an infinite sum of complex exponential terms multiplied by cn, ranging from negative to positive infinity.

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

Complex Fourier
Transform
By
Engr. Jawwad Ahmad (Ph.D)

1
Today’s Goal
 Complex Form of the Fourier Serious

 Drill Problem

Engr. Dr. Jawwad Ahmad 2


Complex Form of the Fourier Serious
 We know that

f (t )  ao    an cos not  bn sin not
n 1

 Since θ = n ωo t

f (t )  ao    an cos  bn sin  
n 1

 Now develop an exponential function


2 a n cos   bn sin  
a n cos   bn sin  
2

Engr. Dr. Jawwad Ahmad 3


Complex Form of the Fourier Serious
a n cos   bn sin  a n cos   bn sin 
a n cos   bn sin   
2 2

an cos   jbn cos   ja n sin   bn sin 


an cos   bn sin  
2
an cos   jbn cos   ja n sin   bn sin 

2

an cos   bn sin  
 an  jbn  cos    ja n  bn  sin 
2
 a  jbn  cos     ja n  bn  sin 
 n
2

Engr. Dr. Jawwad Ahmad 4


Complex Form of the Fourier Serious
an cos   bn sin  
 an  jbn  cos    an  jbn  j sin 
2

 an  jbn  cos    an  jbn  j sin 
2

an cos   bn sin  
 an  jbn 
 cos   j sin   
 an  jbn 
 cos   j sin  
2 2

  e j
 Since cos   j sinand cos   j sin   e  j

an cos   bn sin  
 an  jbn 
e j

 an  jbn 
e  j
2 2

Engr. Dr. Jawwad Ahmad 5


Complex Form of the Fourier Serious
 Since θ = n ωo t

an cos no t  bn sin n t 


 an  jbn 
e jnot

 an  jbn 
e  jnot
o
2 2
 Thus

  an  jbn  jn o t  an  jbn   jn o t 
f (t )  ao    e  e 
n 1  2 2 
 Now define a complex constant cn.
1
cn   an  jbn  where n = 1, 2, 3, ….
2

Engr. Dr. Jawwad Ahmad 6


Complex Form of the Fourier Serious
 Suppose n is replaced by –n, then
a n  an
 But
bn  bn
 So,
1
cn   an  jbn 
2
 Or
1
c n   an  jbn 
 Also 2
cn  c*n
 And
co  a o

Engr. Dr. Jawwad Ahmad 7


Complex Form of the Fourier Serious
 Now f(t) becomes

 

f (t )  co   cn e jn o t  c n e  jn o t
n 1
 
f (t )   cn e jn o t
  c n e  jn o t

n0 n 1
 If the limits are changed then,
 
f (t )   cn e jn o t  n
c e jn o t

n0 n  1

Engr. Dr. Jawwad Ahmad 8


Complex Form of the Fourier Serious
 Or 
f (t )  n
c e jn o t

n  

 Where, 1
cn   an  jbn 
2
 Now the value of cn can be evaluated,
T T
2
1 1 2
cn   f (t ) cos notdt  j  f (t ) sin notdt
T T T T
2 2
T
 Therefore, 1 2

 jn o t
cn  f (t ) e dt
T T
2
Engr. Dr. Jawwad Ahmad 9
Thank you

Engr. Dr. Jawwad Ahmad 10

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