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9.5 
Waveform symmetry and fourIer serIes coeffIcIents
Waveforms can exhibit symmetry about the vertical axis. Figure 9.5-1 shows two waveforms that 
exhibit the so-called even symmetry.
Fig. 9.5-1 
Waveforms exhibiting even symmetry
If the right-side portion of the waveform, – i.e. the portion of waveform for t > 0, is mirror 
reflected about vertical axis, the reflection coincides with the portion of waveform located in the 
left side of time-axis. This can be expressed as v(
-
t) 
= 
v(t) for any t. A function v(t) that exhibits 
this kind of symmetry is said to possess the property of even symmetry and is called an even 
function.
Examine the two waveforms shown in Fig. 9.5-2. If the right-side portion of the waveform is 
mirror reflected about vertical axis, the reflection is equal and opposite to the waveform located 
in the left side of time-axis. This can be expressed as v(
-
t) 
= -
v(t) for any t. A function v(t) that 
exhibits this kind of symmetry is said to possess the property of odd symmetry and is called an odd 
function.

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9.12
Dynamic Circuits with Periodic Inputs – Analysis by Fourier Series
t
t
Fig. 9.5-2 
Waveforms exhibiting odd symmetry
Even and odd symmetry in waveforms is dependent on choice of t 
= 
0 point in the horizontal axis. 
If the vertical axis is moved to the right or left of its current position all the waveforms in Figs. 9.5-1 
and 9.5-2 will lose their symmetry properties.
Consider a v(t) which does not exhibit even or odd symmetry. Let us express v(t) in the following 
equivalent form.
v t
v t
v t
v t
v t
( )
( )
( )
( )
( )
=
+ −
+
− −
2
2
. v(t) is expressed here as sum of two functions. The first function 
is an even function. So we call it 
v t
e
( ).
v t
v t
v t
v
t
v t
v t
v t
v t
e
e
e
e
( )
( )
( )
.
( )
( )
( )
( )
( )
=
+ −
− =
− +
=
∴
2
2
Then, 
is aan 
function on for any 
even
t
v t
( )
The second function is an odd function. So we call it 
v t
o
( ).
v t
v t
v t
v
t
v t
v t
v t
v t
o
o
o
o
( )
( )
( )
.
( )
( )
( )
( )
( )
=
− −
− =
− −
= −
∴
2
2
Then,
is an 
function on for any 
odd
t
v t
( )
Any time-function 
v
(
t
) can be expressed as the sum of an even function and an odd 
function. 
v t
v t
v t
e
( )
[ ( )
(
)]
=
+ −
1
2
is termed as the 
even part
of 
v
(
t
) and 
v t