Copyright 20 13 Dorling Kindersley (India) Pvt. Ltd

9.10 
normalIsed PoWer In a PerIodIc Waveform and Parseval’s theorem
The concept of normalised power in a periodic waveform is often employed in communication 
engineering and allied areas as a measure of signal strength. It is defined as the average power that will 
be delivered to 1 
W
resistance if the periodic waveform is thought of as a voltage waveform applied to 
that resistor. The averaging is done over any interval equal to the period of the waveform.
P
T
v t
dt
n
T
T
=
2
−
∫
1
0 5
0 5
[ ( )]
.
.
(9.10-1)
If we consider v
3
(t) 
= 
[v(t)]
2
as a new time-function, the term on the right-hand side of Eqn. 9.10-1 
can be identified as the DC component of v
3
(t). Hence P
n
must be equal to the exponential Fourier 
series coefficient of v
3
(t), 

v
k
3
, for k 
= 
0.
We developed the multiplication-in-time property of Fourier series in Example 9.6-9. This property 
states that if v
1
(t) and v
2
(t) are two periodic waveforms with same period and v
3
(t) 
= 
v
1
(t)
×
v
2
(t), then 
the exponential Fourier series coefficients of v
3
(t) is given by 

 
v
v v
k
k
n
k n
n
3
1
2
=
− ∞ < < ∞
−
=−∞
∞
∑
(
)
,
, for 
where 


v
v
n
n
1
2
and 
are the exponential Fourier series coefficients of v
1
(t), and v
2
(t) respectively. We use 
this property with v
1
(t) 
= 
v
2
(t) 
= 
v(t) and evaluate the exponential Fourier series coefficient of [v(t)]
2
, 
for k
= 
0 as

 
v
v v
n
n
n
30
=
−
=−∞
∞
∑
.
Therefore, 
P
T
v t
dt
v v
v v
v
n
T
T
n
n
n
n n
n
n
=
=
=
=
−
−
=−∞
∞
=−∞
∞
∫
∑
∑
1
2
0 5
0 5
[ ( )]
| |
.
.
*
 
 

22
2
2
1
2
=
+
=
∞
=−∞
∞
∑
∑
| |
| |


v
v
o
n
n
n
This is Parseval’s Theorem on normalised power of periodic waveforms.
The trigonometric Fourier series for v(t) is 
v t
a
a
n t
b
n t
a
v
T
v t dt
o
n
o
n
o
n
n
o
o
( )
cos
sin
( )
=
+
+
=
=
=
∞
=
∞
∑
∑
w
w
1
1
1
where 
,,
Re( )
( ) cos
*
−
−
−
∫
=
+
=
+
=
=
T
T
n
n
n
n
n
n
o
T
T
a
v
v
v
v
v
T
v t
n t dt
2
2
2
2
2
2
w
∫∫
=
= − +
= − +
=
=
−
, for 
n
a
v
v
v
v
v
T
v t
n
n
n
n
n
n
n
1 2 3
2
2
, ,
Im( )
( )sin
*
…
w
w
o
T
T
t dt
n
−
∫
=
2
2
1 2 3
, for 
, ,
…
Therefore, Parseval’s Theorem can be expressed in terms of trigonometric Fourier series coefficients 
as 
P
a
a
b
n
o
n
n
n
=
+
+




=
∞
∑
2
2
2
1
2
.

---

Normalised Power in a Periodic Waveform and Parseval’s theorem 
9.39
The second form of trigonometric Fourier series is shown in the following:
∴
=
+
−
=
=
+
=
=
∞
∑
v t
c
c
n t
c
v c
a
b
v v
o
n
o
n
n
o
o
n
n
n
n
( )
cos(
)
,
w
f
1
2
2
2
where
nn
n
n
n
n
n
v
b
a
v
n
*
| |
tan
, ,
=
=
= −∠
=
−
2
1 2 3
1
…
and
of , for 
f
.
∴
=
+
=
∞
∑
P
c
c
n
o
n
n
2
2
1
2
the normalised power of a periodic waveform 
v
(
t
), 
P
n
, is given by
P
v v
v v
v
v
v
a
a
b
n
n
n
n n
n
o
n
o
n
n
n
=
=
=
=
+
=
+
+
−
=
 
 



*
| |
| |
| |
(
)
2
2
2
2
2
2
1
2
2
∞
∞
=
∞
=
∞
=−∞
∞
=−∞
∞
=−∞
∞
∑
∑
∑
∑
∑
∑
=
+
c
c
o
n
n
n
n
n
n
2
2
1
1
2
Though the multiplication-in-time property easily led us to Parseval’s theorem, it does not help 
us to see the significance of this theorem. Neither does it tell us how this total normalised power is 
distributed among various frequency components. Hence, we use the trigonometric Fourier series 
v t
c
c
n t
o
n
o
n
n
( )
cos(
)
=
+
−
=
∞
∑
w
f
1
for further appreciation of P
n
.
Consider a simpler situation in which v(t) contains just three components.
v t
c
c
m t
c
k
t
k
m
o
m
o
m
k
o
k
( )
cos(
)
cos(
),
=
+
−
+
−
w
f
w
f
and are integers
∴
=
+
−
+
−
+
[ ( )]
cos (
)
cos (
)
cos(
v t
c
c
m t
c
k
t
c c
m
o
m
o
m
k
o
k
o m
2
2
2
2
2
2
2
w
f
w
f
w
oo
m
o k
o
k
m k
o
m
o
k
t
c c
k
t
c c
m t
k
t
−
+
−
+
−
−
f
w
f
w
f
w
f
)
cos(
)
cos(
) cos(
)
2
2
∴
=
+
+
+
−
+
[ ( )]
cos (
)
cos (
v t
c
c
c
c
m t
c
k
t
o
m
k
m
o
m
k
o
2
2
2
2
2
2
1
2
1
2
1
2
2
1
2
2
w
f
w
−−
+
−
+
−
+
+
f
w
f
w
f
w
k
o m
o
m
o k
o
k
m k
o
c c
m t
c c
k
t
c c
m k
t
)
cos(
)
cos(
)
cos[(
)
2
2
−−
+
+
−
−
−
(
)]
cos[(
)
(
)]
f
f
w
f
f
m
k
m k
o
m
k
c c
m k
t
k and m are integers. Thus, if k 
≠
 m, the frequencies m
w
o
, k
w
o
, 2m
w
o,
2k
w
o
, (m
-
k)
w
o 
and (m
+ 
k)
w
o
are 
integer multiples of 
w
o
. Hence, all the cosine waves in [v(t)]
2
will have integer number of cycles in T s 
where T is the period of v(t). Therefore, their average over one T will be zero.
∴
=
=
+
+




×
=
+
−
∫
P
T
v t
dt
T
c
c
c
T
c
n
o
m
k
T
T
o
1
1
1
2
1
2
1
2
2
2
2
2
0 5
0 5
2
[ ( )]
.
.
cc
c
c
c
c
m
k
o
m
k
2
2
2
2
2
1
2
2
2
+
=
+ 



+ 




---

9.40
Dynamic Circuits with Periodic Inputs – Analysis by Fourier Series
Generalising the result for infinite term Fourier series,
∴
=
=
+




=
=
∞
−
∑
∫
P
T
v t
dt
c
c
i e P
n
o
n
n
T
T
n
1
2
2
2
2
1
0 5
0 5
[ ( )]
. .,
(
.
.
DC coomponent
rms value of 
harmonic component
)
(
)
2
2
1
+
=
∞
∑
n
th
n
(9.10-2)
Since the rms value of a DC component is same as its value, we can express this as
P
n
n
th
n
=
=
∞
∑
(
)
rms value 
harmonic component
2
0
(9.10-3)
Square root of this quantity will give the rms value of v(t) itself.
rms value of 
rms value of 
harmonic component
v t
n
th
n
( )
(
)
=
=
2
00
∞
∑
(9.10-4)
The normalised power of a particular harmonic component with amplitude c
n
when acting alone 
will be 0.5 c
n
2
. Equations 9.10-2 shows that it contributes the same amount to the total power even 
when it is acting along with other harmonics.
Consider two arbitrary waveforms v
1
(t) and v
2
(t). Let average of [v
1
(t)]
2
and [v
2
(t)]
2
over some 
interval be a
1
and a
2
, respectively. Will the average of [v
1
(t)
+ 
v
2
(t)]
2
over the same interval be a
1
+ 
a
2
? 
The answer depends on whether the average of 2v
1
(t)v
2
(t) in that interval is zero or not. In general, it 
is not zero, and average of [v
1
(t)
+ 
v
2
(t)]
2
is not the same as the sum of averages of [v
1
(t)]
2
and [v
2
(t)]
2
. 
However, if v
1
(t) and v
2
(t) are two sinusoids with different frequencies and if their frequencies are 
integer multiples of some basic frequency, then the average of 2v
1
(t)v
2
(t) in an interval that is equal to 
the period corresponding to the basic frequency is zero.Therefore, if v(t) is a mixture of harmonically 
related sinusoids and DC, the normalised power contributions from each component is unaffected 
by the presence of other components. Hence, the normalised power of the waveform is the sum of 
normalised power of individual components.
Now we understand that each harmonic component in the trigonometric Fourier series of a 
waveform contributes to normalised power. We can ascribe the power contributed by a particular 
component to its frequency and plot the information against as a line spectrum. This spectral plot is 
called discrete power spectrum. However, it will be a single-sided spectrum since we derived it from 
trigonometric Fourier series. Spectral lines will be located at 0, 
w
o
, 2
w
o, 
3
w
o
etc. and the length of the 
spectral line will be proportional to 0.5 c
n
2
. By Parseval’s theorem,
P
T
v t
dt
v v
v v
v
n
n
n
n n
n
n
n
n
=
=
=
=
−
=−∞
∞
=−∞
∞
=−∞
∞
−
∑
∑
∑
1
2
2
0
[ ( )]
| | .
*
 
 

..
.
5
0 5
T
T
∫
Therefore, we can draw the two-sided discrete power spectrum by plotting two lines of height 
proportional to 
| |

v
n
2
at n
w
o
and 
-
n
w
o
. We had noted earlier that two spectral components located at 
±
n
w
o
in the two-sided magnitude and phase spectra based on exponential Fourier series have to be 
thought of as an integral unit rather as individual components. Those two components always go 
together and form a real sinusoid. Similarly, it is understood that the power spectral components 
located at 
±
n
w
o
in the two-sided power spectrum always go together to make a total contribution 
of 2
| |

v
n
2
to P
n
.

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Normalised Power in a Periodic Waveform and Parseval’s theorem 
9.41