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Fourier analysis
Fourier series: A review
Let f (t ) be a real valued periodic function with time period T0 that satisfies
following conditions [Dirchelet’s conditions]. Then, it can be expressed in either of the
following equivalent forms.
a) f (t )  a 0 

a
n 1

n
Where  0 
cos n 0 t   bn sin n 0 t
n 1
2
is called fundamental frequency and n0 , n th harmonic where
T0
n - is an integer.
b) f (t ) 
n  
c e
n  
jn 0t
n
In (a), the coefficients ( a i and bi ) are real valued, while in (b), coefficient cn may be
complex. Both forms (a) and (b) are equivalent. (b) can be obtained from (a) by following
substitutions.
e jn0t  e  jn0t
cos n 0 t 
2
jn0t
e
_ e  jn0t
sin n0t 
2j
Thus, a n , bn and cn are related as follows.
c0  a 0
an  jbn 

2

an  jbn 

2 
cn 
c n
n0
In other words, c n  c * n , the usage of exponential form also introduces the concept of
negative frequency. It can be interpreted in the following way.

e jn0t
(n  0) represents a phasor of unit magnitude rotating in anticlockwise direction at speed of n0 rad/s.

e  jn0t
represents a phasor of unit magnitude rotating in clockwise
direction at speed of n0 rad/s.
The coefficients a n , bn and cn can be computed using the following expressions.
2
T0
an 

0

2
2
T0
T0

1
2
1
=
0
bn 
=
T0

0

0
f (t ) cos n 0 t dt
f ( 0 t ) cos n 0 t d ( 0 t )
f (t ) sin n0 t dt
f ( 0 t ) sin n 0 t d ( 0 t )
T
1
1
a0   f (t ) dt 
T0
2
2
 f ( t ) d t
0
0
0
T
1
cn   f (t ) e  jn0t dt
T0

1
2
2
 f (t ) e
 jn0t
d0 t
0
Since, many of the voltage and current phasor estimation methods are based upon
Least square estimation, it is a worthwhile intellectual exercise to show that Fourier series
truncated at some ‘n’ can be interpreted through Least square estimation theory.
We define the concept of mean square error (MSE). MSE is defined as
2
r
1 2

MSE 
f
(
t
)



(
t
)

i
i
 dt
t 2  t1 t1 
i 1

t
The interval t 2  t1 for periodic function should be time period T0 , while i (t ) should
be sine/cosine function (or) exponential function of the Fourier series. Our job is to find
the coefficient  i ' s so as to minimize the mean square error. Thus, the optimization
problem is given by
min MSE
Consider f (t ) approximated by n  functions  i as follows.
f (t )  11 (t )   22 (t )       rr (t )
(1)
In truncated Fourier series, i (t ) belong to the set 1, cos m0 t , sin m0 t ; m integer}.
To evaluate the quality of approximation,
Now,
2
t
1 2
 f (t )  11 (t )   22 (t )     rr (t ) dt
MSE 
t 2  t1 t1
2
2 2
2 2
2 2
1 2 [ f (t )  1 1 (t )   2 2 (t )       r r (t )

t 2  t1 t1  21 f (t )1 (t )  2 2 f (t )2 (t )    2 r f (t )r (t )] dt
t
t
1 r r 2

  j ii (t ) j (t ) dt
t 2  t1 j 1 i 1 t1
i j
(2)
Now, if we choose i (t ) to be either cos n 0 t or sin n0 t from the trigonometric form
of Fourier series, then the expression for the MSE simplifies drastically. It can be verified
that with i (t )  cos n0 t and  j (t )  cos m 0 t (m  n)
T0
  (t ) 
i
j
(t ) dt
0
T0

= cos n0 t cos m0 t dt
0
T
1 0
=  [cos n  m 0 t  cos n  m 0 t ] dt
20
= 0;
This, infact is analogous to the statement that we cannot have non-zero average
power exchange over a time-interval (T0 ) if the voltages and currents belong to different
frequencies of the harmonic spectrum.
Similarly, with i  cos n0 t and  j  sin m 0 t , it can be verified that
T0
  (t ) 
i
0
T0
j
(t ) dt =  cos n0 t sin m0 t dt
0
T0
=
1
[sin n  m 0t  sin n  m 0t ] dt = 0;
2 0
(3)
Equation (3) holds true even when n  m ; when n  m ; the remarks made earlier on
interaction of harmonic frequencies apply. When n  m , we have the analogy of
interaction between voltage and current phasors separated by 90 , which leads to zero
average power.
The concept of zero average power interaction at (1) sines and cosines harmonic at
different frequencies and (2) sines and cosines at same frequency can be generalized by
the concept of orthogonal functions.
Orthogonal functions:
A set of complex valued functions i (t ),  j (t ), t  [a, b] , are said to be
orthogonal if
b
  (t )
i
*
j
(t )  0
i j
a
The definition obviously applies to real valued functions where in  *j  i . Clearly, the
functions used in Fourier series (both in trigonometric and complex exponential form) are
orthogonal. Many more set of orthogonal functions like Walsh, Harr etc and
corresponding approximate series can be found in the literature of signal processing.
They also have applications in power quality. However, for our use, Fourier series
suffices.
From the discussion so far, we conclude that in case of Fourier series over a time
period T0 , the second term in (2) vanishes. Hence
2
2 2
2 2
2 2
1 0 [ f (t )  1 1 (t )   2 2 (t )       n n (t )
MSE 
T0 0  21 f (t )1 (t )  2 2 f (t )2 (t )    2 n f (t )n (t )] dt
T
Now, for dc signal (n  0, cos n0 t )
T
1 0 2
 (t ) dt  1
T0 0
Also, for  (t ) to be sin m0 t , or cos n 0 t ;
T
1 0 1  cos 2n0 t
1 0 2
dt = 1
sin n0 t dt =
2
T0 0
2
T0 0
T
(4)
Hence,
T
  12   22       n2
1  2
MSE    f (t ) dt  
T0  0
2

2
 1
T0

1
T0
T

0
T0

0
2
f (t )1 (t ) dt    n
T0
T0
 f (t )
n
(t ) dt
0
 2 2 T0

 2 2
f 2 (t ) dt   1  1  f (t ) 1 (t ) dt        n  n
T0 0
T0
 2

 2
T0
 f (t ) 
0
n

(t ) dt 

At the minimum, the partial derivative of MSE with variable  i should be zero.
Now,
MSE
2
 0  i 
 i
T0
 i 
2
T0
T0
 f (t )  (t ) dt  0
i
0
T0
 f (t )  (t ) dt
i
0
Thus, when we substitute i (t )  cos n0 t , we get coefficient a n , similarly by
substituting i (t )  sin n0 t , we get coefficient bn
Finally, for dc signal, which is defined by i (t )  1
2
0 
T0
T0
 f (t ) dt
0
These coefficients infact are nothing but the coefficients of Fourier series. This leads us
to a very important conceptual result viz. coefficients in Fourier series minimize the
MSE. This establishes the linkage between Fourier series and Least square methods.
Similar series can be developed for other set of orthogonal functions. [See Q:2]
U
Review Questions:
1) For the square periodic wave form as shown in Fig-1
4
 n  odd
Show that C n  n 
n  even
0 
Interpret you answers in terms of a n and bn .
2) A class of function known as Walsh function are defined as follows
0 t   1, 0  t  1
1

1 0t  2
1 t  
1
 1
 t 1
2

1 3

1
0

t

,
 t 1

1
4
4
 2 t  
1
 1
 t 1
4

1
1
3

1 0t  ,
t 

4
2
4
 22  t  
1
1
3
 1
t  ,
 t 1
4
2
4

1

k



2
t
0

t

m

2
m( 2k11) t  
1
(1) k 1  m( k ) 2t  1
 t 1
2

m  1, 2, 3,...
k  1, 2,....,2 m 1
1

mk 2t 
0t 

2
m( 2k1) t  
1
(1) k m( k ) 2t  1
 t 1
2

a) Plot the functions
b) Show that they are orthogonal
c) What advantages can they have in DSP
3) Function
n d 

Sin ( 0 ) 

A
2
Ans : Fn  d 

t  n 0 d 
2


4) State Dirchelet’s conditions