Energy and Power
Summary: Energy and power for analog and discrete time signals
From physics we've learned that energy is work and power is work per time
unit. Energy was measured in Joule (J) and work in Watts(W).
In signal processing energy and power are defined more loosely without
any necessary physical units, because the signals may represent very
different physical entities. We can say that energy and power are a
measure of the signal's "size".
Signal Energy
Analog signals
Since we often think of a signal as a function of varying
amplitude through time, it seems to reason that a good
measurement of the strength of a signal would be the area
under the curve. However, this area may have a negative part.
This negative part does not have less strength than a positive
signal of the same size. This suggests either squaring the signal or taking its absolute
value, then finding the area under that curve. It turns out
that what we call the energy of a signal is the
area under the squared signal, see Figure 1
Energy - Analog signal:
E a = ∫ - ∞ ∞ | x t | 2 d t
E a
t
x
t
2
Note that we have used squared magnitude(absolute value) if the signal
should be complex valued. If the signal is real, we can leave out the magnitude
operation.
 Subfigure 1.1: Signal x(t)  Subfigure 1.2: The energy of x(t) is the shaded region Figure 1: Sketch of energy calculation |
Discrete signals
For time discrete signals the "area under the squared signal"
makes no sense, so we will have to use another energy definiton.
We define energy as the sum of the squared magnitude of the samples.
Mathematically
Energy - Discrete time signal:
E d = ∑ n = - ∞ ∞ | x n | 2
E d
n
x
n
2
Example 1
Given the sequence
y l = b l u l
y
l
b
l
u
l
We recognize y(l) as a geometric series. Thus we can use the formula for
the sum of a geometric series and we obtain the energy,
E d = ∑ l = 0 ∞ y l 2 = 1 1 - b 2
E d
l 0
y
l
2
1
1
b
2
| b | < 1
b
1
Signal Power
Our definition of energy seems reasonable, and it is.
However, what if the signal does not decay fast enough? In this case we
have infinite energy for any such signal. Does this mean that
a fifty hertz sine wave feeding into your headphones is as
strong as the fifty hertz sine wave coming out of your outlet?
Obviously not. This is what leads us to the idea of
signal power, which in such cases is a more
adequate description.
 Figure 2: Signal with inifinite energy |
Analog signals
For analog signals we define power as
energy per time interval.
Power - analog signal:
P a = 1 T 0 ∫ - T 0 2 T 0 2 | x t | 2 d t
P a
1
T 0
t
T 0 2
T 0 2
x
t
2
Discrete signals
For time discrete signals we define power as energy per sample.
Power - Discrete time:
P d = 1 N ∑ n = N 1 N 1 + N - 1 | x n | 2
P d
1
N
n N 1
N 1 N
1
x
n
2
Example 2
Given the signals
x 1 t = sin 2 π t
x 1 t
2
t
x 2 n = sin π 1 10 n
x 2 n
1
10
n
For the analog sine we have
P a = 1 1 ∫ 0 1 sin 2 2 π t d t = 1 2
P a
1
1
t 0 1
2
t
2
1
2
For the discrete sine we get
P d = 1 20 ∑ n = 1 20 sin 2 1 10 π n = 0.500
P d
1
20
n 1 20
1
10
n
2
0.500
 Subfigure 3.1: Analog sine  Subfigure 3.2: Discrete time sine Figure 3: Analog and discrete time sine. |
Matlab files
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This work is licensed by Anders Gjendemsjø, Adam Blair, Richard Baraniuk, and Melissa Selik. See the Creative Commons License about permission to reuse this material.
Last edited by Anders Gjendemsjø on Feb 20, 2004 12:17 pm US/Central.