Copyright 20 13 Dorling Kindersley (India) Pvt. Ltd

v t
v t
o
( )
[ ( )
(
)]
=
− −
1
2
is called the 
odd part 
of 
v
(
t
).
How will even and odd symmetry affect the exponential Fourier series coefficients? If v(t) is even on t,

v
T
v t e
dt
T
v t e
e
dt
T
v
n
jn t
T
T
jn t
jn
t
o
o
o
=
=
+
=
−
−
−
−
−
∫
1
1
2
2
2
( )
( )[
]
(
)
w
w
w
(( ) cos
.
t
n t dt
o
T
T
w
0
2
0
2
∫
∫
Thus, its Fourier series coefficients 

v
n
will be real numbers. This indicates that its trigonometric 
Fourier series will contain only cosine terms.

---

Waveform Symmetry and Fourier Series Coefficients 
9.13
If 
is 
on then is real and 
for all 
v t
even
t
v
b
n
n
n
( )
,
,
=
0
∴
=
+
=
=
=
∞
∑
v t
a
a
n t
a
T
v t dt
a
T
v t
n t dt
o
n
o
n
o
n
o
T
( )
cos
( )
( ) cos
w
w
1
0
2
2
4
and
∫∫
∫
0
2
T
The coefficients of exponential Fourier series of an 
even
periodic waveform on 
t
will be 
real
and its trigonometric Fourier series will contain only 
cosine
terms.
This is quite reasonable since sum of odd functions will produce only an odd function, and hence 
an even v(t) cannot contain even a single sine term.
It can be shown similarly that, 
If 
is 
on , then
is pure imaginary and 
v t
odd
t
v
j
T
v
n
( )
(
= −
2
tt
n t dt a
n
o
n
T
)sin
,
,
w
=
∫
0
0
2
for all 
and v(t) will contain only sine terms in its trigonometric Fourier series. The DC content in the waveform 
has to be zero since DC value is an even function of time.
If 
is 
on then is imaginary and 
for a
v t
odd
t
v
a
n
n
( )
,
,
=
0
lll n
∴
=
=
=
∞
∑
∫
v t
b
n t
b
T
v t
n t dt
n
o
n
n
o
T
( )
sin
( )sin
w
w
1
0
2
4
and 
The coefficients of exponential Fourier series of an 
odd
periodic waveform on 
t
will be 
imaginary
and its trigonometric Fourier series will contain only 
sine
terms.
Another kind of symmetry exhibited by waveforms is called half-wave symmetry. A periodic 
waveform v(t) is half-wave symmetric if one half-cycle of the waveform has the same appearance of 
the other half cycle inverted about time-axis. This is expressed as 
Half-wave symmetry 
⇒
v t
v t
T
t
( )
,
.
= −
±




1
2
for all 
v
(
t
)
(a)
t
T
2
T
2
T
2
T
2
v
(
t
)
(b)
t
Fig. 9.5-3 
Waveforms showing half-wave symmetry

---

9.14
Dynamic Circuits with Periodic Inputs – Analysis by Fourier Series
Figure 9.5-3 shows three waveforms exhibiting half-wave symmetry. The waveform in Fig. 9.5-3 (a) 
shows half-wave symmetry, but it does not possess even or odd symmetry. The waveform represented by 
solid curve in Fig. 9.5-3 (b) shows half-wave symmetry along with even symmetry and the one represented 
by dotted curve shows half-wave symmetry along with odd symmetry. When a periodic waveform 
possesses both half-wave symmetry and even or odd symmetry it is said to have quarter-wave symmetry. 
The effect of half-wave symmetry on Fourier series coefficients is derived below in the following:

v
T
v t e
dt
T
v t e
dt
T
v t e
dt
n
jn t
jn t
jn t
T
o
o
o
=
=
+
=
−
−
−
−
∫
1
1
1
1
2
0
( )
( )
( )
w
w
w
II
I
t
t
T
T
T
T
1
2
0
1
2
1
2
1
2
2
+
′ = +
∫
∫
−
Substitute 
in the second integraal
I
T
v t e
dt v t
T
e
e
dt
jn t
jn t
jn
T
T
T
o
o
o
2
2
0
0
1
2
1
1
2
1
2
=
′ −
′
=
−
−
′ −
−
∫
∫
( )
(
)
w
w
w
T
T
v t e
e
dt
v t
jn t
jn
T
o
o
− ′
′
−
′
( )
(
( )
)
w
w
2
0
1
Q
is half-wave symmetric
22
1
2
1
2
1
1
0
T
jn t
jn
o
T
n
T
v t e
e
dt
T
T
v t e
o
∫
∫
=
− ′
′
=
= −
− ′
−
′
−
( )
(
)
( )
( )
w
p
w
p
Q
jjn t
jn
T
o
dt
e
n
n
w
p
′
′
−
=
∫
(
)
(
Q
is 1, for even and 
for odd 
1
0
1
2
−−
′
′ = −
+
+
−
′
1
1
1
1
1
)
( )
n
jn t
T
v t e
dt
I
n
I
n
o
w
for even and 
for odd 
00
1
2
T
∫
∴ =





=
=
−
∫

v
n
T
v t e
dt
n
a
a
n
jn t
T
o
n
o
0
2
0
0
1
2
for even 
for odd 
( )
w
00
4
0
0
1
2
for even 
for odd 
and 
f
n
T
v t
n t dt
n
b
o
T
n
( ) cos
w
∫





=
oor even 
for odd 
n
T
v t
n t dt
n
o
T
4
0
1
2
( )sin
w
∫





the coefficients of exponential Fourier series of an 
even
periodic waveform on 
t
will be 
real
and its trigonometric Fourier series will contain only 
cosine
terms.
the coefficients of exponential Fourier series of an 
odd
periodic waveform on 
t
will be 
imaginary
and its trigonometric Fourier series will contain only 
sine
terms.
A periodic waveform with 
half-wave symmetry 
does not contain any average value (or DC 
content) and does not contain any even harmonics.
If a periodic waveform is 
even
on 
t
and is 
half-wave symmetric
, its Fourier series expansion 
will contain only 
cosine functions
at 
odd harmonic frequencies
.
If a periodic waveform is 
odd
on 
t
and is 
half-wave symmetric
, its Fourier series expansion 
will contain only 
sine functions
at 
odd harmonic frequencies
.

---

Properties of Fourier Series and Some examples 
9.15
9.6 
ProPertIes of fourIer serIes and some examPles
We take up some examples of Fourier series and develop the important properties of Fourier series 
through them. The waveforms used in the examples that follow are very important signal waveforms 
that appear frequently in electrical and electronic engineering applications and are not just some 
functions used to illustrate Fourier series. The waveforms appearing in the examples are important in 
their own right.
The first property of Fourier series is almost self-evident. It is the property of linearity. It states 
that if v
1
(t) and v
2
(t) are two periodic waveforms with same period T and v
3
(t) 
= 
a
1
v
1
(t) 
+ 
a
2
 v
2
(t), 
then, 



v
a v
a v
n
n
n
n
3
1 1
2 2
=
+
for all , where 
 

v v
v
n
n
n
1
2
3
,
and 
are the coefficients of exponential Fourier 
series of v
1
(t), v
2
(t) and v
3
(t), respectively. This may be proved easily.
example: 9.6-1
Find the exponential Fourier series of v(t) 
= 
d
(
)
t k
k
−
=−∞
∞
∑
and derive the trigonometric Fourier series 
from it.
Solution
This waveform is a periodic sequence of unit impulses with a period of 1 s. It is shown in Fig. 9.6-1.

v
T
v t e
dt
v t e
dt
t e
dt
t dt
n
jn t
jn
t
jn
t
o
=
=
=
=
=
−
−
−
1
1
1
1
2
2
( )
( )
( )
( )
w
p
p
d
d
00
0
0
0
1
2
1
2
1
2
1
2
−
+
−
+
∫
∫
∫
∫
−
−
T
T
1
–1
1
2
3
4
Time(s)
(
t
)
v
–2
–3
4
– 
Fig. 9.6-1 
Waveform 
v 
(
t
) in example: 9.6-1 
The waveform has zero value at all points in the first period [
-
0.5,0.5] except between 0
-
and 0
+
.
In that interval it is an impulse of unit magnitude. And the value of exponential in that interval is 1. 
Therefore all coefficients in the exponential Fourier series are equal to 1.
∴
−
=
=−∞
∞
=−∞
∞
∑
∑
d
p
(
)
t k
e
j
nt
n
k
2
is the exponential Fourier series of v(t).
We obtain the trigonometric Fourier series from exponential Fourier series by taking two components 
with harmonic order –n and 
+
n together. 
∴
−
=
=
+
+
= +
⋅ ⋅
=−∞
∞
−
=
∞
∑
∑
d
p
p
p
p
p
(
)
cos
t k
e
e
e
e
j
nt
j
t
n
j
nt
j
nt
n
2
2 0
2
2
1
1
2
2
nnt
n
k
=
∞
=−∞
∞
∑
∑
1
(9.6-1)

---

9.16
Dynamic Circuits with Periodic Inputs – Analysis by Fourier Series
The waveform contains a DC component of 1 unit. This should be so since the total area content 
of the waveform under one period is the area content of unit impulse, i.e. unity. This area divided by T 
must be the DC content in the waveform. T in the example is 1 s.
The waveform contains only cosine terms. This too is expected since v(t) is an even function of t.
But most important aspect to be noted is that periodic impulse train contains all harmonics with 
equal strength, the amplitude of harmonic components does not show any let up as the frequency 
of harmonic component goes up. Sinusoids at all harmonic frequencies with uniform amplitude are 
required to synthesise the periodic impulse train.
But didn’t we stretch the concept of Fourier series a bit too far? Impulse is a highly discontinuous 
waveform. In fact, the v(t) in this example violates the Dirichlet’s condition that, discontinuity, if 
present, must be of finite value. Hence, though we have a found a Fourier series for v(t), will it really 
converge to v(t) at all t ?
The answer, strictly speaking, is that, it will not. As usual, faced with such mathematical difficulties, 
we just change our viewpoint and make the Fourier series we derived in this example a useful one! We 
simply view the impulse as rectangular pulse of large height and small width and unit area. Then we 
argue that over the width of rectangular pulse in the first period (which is centered around t
= 
0) the 
exponential factor 
e
jk
t
o
-
w
is close to 1 and may be approximated as such. But won’t the approximation 
fail if k becomes very large though t is small? It may, but we will not let it fail; we will state that 
we will compress the pulse a little more while keeping its area at unity! Hence the Fourier series in
Eqn. 9.6-1 represents the Fourier series of a periodic rectangular pulse train with each pulse containing 
unit area as the width of the pulse is made infinitesimal and height of the pulse is made infinitely large. 
We keep Dirichlet happy that way!