FOURIER SERIES

CONTD.,
Exponential Fourier Series
A compact way of expressing the Fourier
series is to put it in exponential form.
This is done by representing the sine and
cosine functions in exponential form using
Euler’s law.
1 jn0t
cos n0t   e  e  jn0t 
2
1 jn0 t  jn0t
sin n0t  
 e e 
2j
2
Exponential Fourier Series
This can be rewritten as:

f t   ce
n 
n
jn0t

This is the complex or exponential Fourier

series representation.
The values of cn are:
T
1
cn   f  t  e  jn0t dt
T 0
3
 COMPLEX FORM OF FOURIER SERIES


f ( x)   n
c
n  
e inx

in which
1
cn  (an  ibn ),
2
1
c n  (an  ibn ), n  0,
2
1
and c0  a0 .
2
From the earlier definitions of an , bn we can show that

1

T
cn  f ( x)e  in x
dx,   2 / T ,
T 0
Exponential Fourier Series
The exponential Fourier series of a periodic
function describes the spectrum in terms of
the amplitude and phase angle of ac
components at positive and negative
harmonic frequencies.
The co effcients of the three forms of
Fourier series (sine-cosine, amplitude-
phase, and exponential form) are related by:
An n  an  jbn  2cn
5
Example
 Fourier Series and Frequency Spectra

We can plot the frequency spectrum or line

spectrum of a signal
 In Fourier Series k represent harmonics
 Frequency spectrum is a graph that shows the
amplitudes and/or phases of the Fourier Series
coefficients Ck.
 Amplitude spectrum |Ck|
 Phase spectrum k

 The lines |Ck| are called line spectra because we

indicate the values by lines
Example 1. Find the Fourier series of
the following periodic function.
f(x)

x
 0  3 5 7 9

f x  x 2
when    x  
f   2   f  
  
f  x  dx 
1 1
a0   
2
x dx
2  2 
x 
1 x  3
 2
   
2  3  x   3
 
f  x  cos nx dx
1
an  
  

1  2 

 
  
x cos nxdx

 4
an  2 cos n
n
4
an   2 when n is odd
n
4
an  2 when n is even
n
  
f t   a0   an cos nt   bn sin nt
n 1 n 1

This is an even function.

Therefore, bn  0
The corresponding Fourier series is

 2
 cos 2 x cos 3 x cos 4 x 
 4 cos x     
 
2 2 2
3 2 3 4
 Example 2. Find the Fourier series of
the following periodic function.

f(t)

3T/4

0 t
T/4

-T/2 T/2 T 2T

f t   t when   t 
T T
4 4
T T 3T
 t  when t
2 4 4
 f t  T   f t 
This is an odd function. Therefore, an  0
 2n 
T
bn   f t  sin
2
t dt
T 0  T 
T

 2n 
2
f t  sin
4

T 
0  T 
t dt
 T

 2n 
4
4
bn 
T   T 
t sin t  dt
0
T

T   2n 
2
4 

T T   t  2  sin T t dt
4
Use integration by parts.
   T 
4
2
 n 
bn   2.  sin 
  2n 
T  2 
2T  n 
 2 2 sin 
n  2 

bn  0 when n is even.
Therefore, the Fourier series is

2T   2  1  6  1  10  
 sin  T t   2 sin  T t   2 sin  T t   
2    3   5   
 Example 3

Determine the Fourier series of the waveform shown

below. Obtain the amplitude and phase spectra

18
 Solution:
1, 0  t  1
f (t )   and f (t )  f (t  2)
0, 1  t  2

2 T  2 / n , n  odd
an   f (t ) cos(n0t )dt  0 and An  
T 0  0, n  even
2 T  2 / n , n  odd   90, n  odd
bn   f (t ) sin( n0t )dt   n   a) Amplitude and
T 0 0, n  even  0, n  even b) Phase spectrum

1 2  1
f (t )    sin( nt ), n  2k  1
2  k 1 n

Truncating the series at N=11

19
Example.4

 x, 0 x 
f ( x)  
 x  2 ,   x  2 .
 1 
an 
  t cos ntdt  0,


1  2 
bn 
  t sin ntdt   
 0
t sin ntdt

2  
 t cos nt 0   cos ntdt 


n  0 
2
 ( 1) n1 ,
n
So, the expansion of f(x) reads

 sin 2 x sin 3x sin nx 

f ( x)  x  2sin x      (1) n1  . (7.15)
.
 2 3 n 
Figure 7.1 Fourier representation of sawtooth wave
Ex.5

f ( x)  sin t , 0  t   ,
f ( x)   sin t ,    t  0.
 1 1 
  sin td (t )  
0
a0  sin td (t )
 
  0

2  4

 sin td (t )  ,
0 
2 
an 
 
0
sin t cos ntd (t )

2 2
 , n  even
 n 1
2

0 n  odd .

2 cos nt
4 
f (t )   
  n2, 4,6, n  1
2
.
Example 4
Find the Fourier series expansion of f(t) given
below.

2 
1 n   n 
Ans: f (t )   
 n 1 n 
1  cos  sin 
2   2 
t

25
 Example 5 determine Fourier Series and N = ?

f m  4 and T  s
2

 2
T  0   4 rad/s
2 T
To obtain the most advantages form of symmetry,
we choose t1 = 0 s  Odd & Quarter-wave
All an = 0 and bn = 0 for even values of n and a0 = 0
8 T /4
bn   f (t )sin n0t dt ; for odd n
T 0
 4
fm 4 fm
f (t )  t t ; 0  t T /4
T /4 T

32 2
 f (t )  t ; 0  t T /4

8  32  T / 4
bn     t sin n0t dt
T  0
T /4
512  sin n0t t cos n0t 
 2  2 2  
  n 0 n0  0
32 n
 2 2 sin ; for odd n
 n 2
 The Fourier Series is
N
1 n
f (t )  3.24 2 sin sin n0t ; for odd n
n 1 n 2
32
2
The first 4 terms (upto and including N = 7)
1 1 1
f (t )  3.24(sin 4t  sin12t  sin 20t  sin 28t )
9 25 49
Next harmonic is for N = 9 which has magnitude
3.24/81 = 0.04 < 2 % of b1 ( = 3.24)
Therefore the first 4 terms (including N = 7) is enough for
the desired approximation
• FIND THE TRIGNOMETRIC FOURIER SERIES
Common Functions

30
Common Functions

31
Exponential Fourier Series
This can be rewritten as:

f t   ce
n 
n
jn0t

This is the complex or exponential Fourier

series representation.
The values of cn are:
T
1
cn   f  t  e  jn0t dt
T 0
32
COMPLEX EXPONENTIAL FOURIER
SERIES
• FIND THE COMPLEX EXPONENTIAL FOURIER SERIES