1.4
Signal Power
We have found how to calculate the amplitudes (voltages) of the waves that make up a signal.
It is also useful, for example in radio broadcasting, to know the power that is in our signal
at each frequency. Given a voltage V, we know that
P
∝ 
V

2
=
V V
*
. As such, you would
expect to be able to obtain a measure of the signal power simply by squaring the frequency
domain representation of the signal. This is essentially what is done.
Think about a sine wave like the one on the left in Fig. 28. This has en equation
x
(
t
) =
A
sin(2
πf
0
t
)
.
If we want to represent it as an amplitude spectrum in the frequency
domain, we can do so as shown in the right hand plot in the ±gure. The wave has a frequency
of
f
0
and an amplitude of
A
. We see that this is represented by a spike of height
A/
2 at
frequency
f
0
and another such spike at frequency

f
0
. The reason for this splitting is rooted
in the equation
sin(2
πf
0
t
) =
e
j
2
πf
0
t

e

j
2
πf
0
t
2
where we can see that a sine wave can be split into two complex exponentials, one with
positive frequency and one with negative frequency. It is not really important now though,
and for practical purposes, negative and positive frequencies both contribute the same wave
to a signal but the amplitude gets split between them. This plot shows how the voltage is
Time (Seconds)
Amplitude (Volts)
A

A
0
0
1
f
T
=
t
Frequency (Hz)
0
f
0
f

2
A
Frequency (Hz)
Power (Watts)
0
f
0
f

4
2
A
Figure 28: Left: a sine wave,
A
sin(2
πf
0
t
); right: frequency domain representation of the
wave; bottom: power spectrum of the wave.
broken down between frequencies, ie. half the voltage is at frequency

f
0
and half at +
f
0
.
If we went to know how the power is distributed, all we need to do is square the values in
the plot. This is shown in the lower plot in Fig. 28. We can see the height of each spike is
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A
2
/
4 = (
A/
2)
2
, and the units are now in Watts to indicate power. This plot is called
the power spectrum, or power spectral density of our function.
You may recall how to calculate the RMS power of a sine wave from an introductory
course. To Fnd the total power, we square the wave and average over one period. ±or the
sine wave in the plot, that would be
P
=
1
T
0
i
T
0
/
2

T
0
/
2
(
A
sin(2
πf
0
t
))
2
dt
=
1
T
0
i
T
0
/
2
0
A
2
1

cos(4
πf
0
t
)
2
dt
=
1
T
0
b
A
2
t

A
2
4
πf
0
sin(4
πf
0
t
)
B
t
=
T
0
/
2
t
=0
=
A
2
2
.
If we add up all the power in the spectrum at the bottom plot of ±ig. 28, we get
P
=
A
2
/
4 +
A
2
/
4 =
A
2
/
2 which is of course the same power we obtained by integrating in the
time domain.
1.4.1
Square Wave
Lets look at a more complex example, starting with trying to represent the function shown
in ±ig. 29 in the frequency domain. The plot shows a square wave which repeats for ever.
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 Summer '09
 Osama
 Fourier Series, Square wave, duty cycle

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