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12. Signal Energy and Power
12.1. Energy and power for continuous-time signals
The terms signal energy and signal power are used to characterize a signal.
They are not actually measures of energy and power. The definition of signal
energy and power refers to any signal x ( t ), including signals that take on
complex values.
Definition 1
The signal energy in the signal x ( t )is
∞
−∞
E = xt d t
2
. (12.1)
The signal power in the signal x ( t )is
∫^ ( )
−
→∞
T
T
T
xt t T
P d 2
lim
2
. (12.2)
If 0 < E <∞, then the signal x ( t )is called an energy signal. However, there are
signals where this condition is not satisfied. For such signals we consider the
power. If , then the signal is called a power signal. Note that the power
for an energy signal is zero
0 < P < ∞
( P = 0 )and that the energy for a power signal is
infinite ( E =∞). Some signals are neither energy nor power signals.
Let us consider a periodic signal x ( t )with period. The signal energy in
one period is
T 0
∫ −
2
2
2 1
0
0
d
T
T
E xt t
and energy in n periods is
∫^ ( ) −
2
2
2 1
0
0
d
T
T
En nE n xt t.
The power of this signal over all periods is given by
∫ −
→∞
2
2
2
0
1 0 0
1
0
0
d
lim
T
n T
xt t T
E
nT T
nE P. (12.3)
If the signal energy over one period is larger than zero but finite, then the total
energy is infinite and the signal power is finite. Therefore, the signal is a power
signal.
If the signal energy in one period is infinite, then both the power and the total
energy are infinite. Consequently, the signal is neither an energy signal nor a
power signal.
Consider a current signal i ( t )flowing through a transmission line represented
by resistance R. The energy loss in the line is
E R = R ( it ) t = REi
∫
∞
−∞
( ) d
2
where E i is the signal energy in the signal i ( t ). If the current i ( t )is periodic
with period T 0 , the average power loss in the line is given by
∫ −
2
2
2
0
0
0
() d
T
T
av Rit t RPi T
P
where Pi is the power of the periodic current signal i ( t ).
( ) ( ) ( ) ( ) ( )
x ( ) t X ( ) t
xt X j t xt X t
E xt t xtx t t xt X t
t
t t
j e d d 2
j e d d 2
e d d 2
d d ( j ) d
j
j j
(^2) - 1
∫ ∫
∫ ∫ ∫ ∫
∫ ∫ ∫
∞
−∞
∞
−∞
∗ −
∞
−∞
−∞
∞
∗ −
∞
−∞
∞
−∞
∗
∞
−∞
∗
∞
−∞
∞
−∞
∗
ω
ω ω
F
where the conjugate property (9.26) has been used.
Now we reverse the order of integration
X^ (^ )^.
E X xt t X X
t
∫
∫ ∫ ∫
∞
−∞
∞
−∞
∗
∞
−∞
∞
−∞
∗ −
ω
j d 2
j j d 2
j e d d 2
2
j
Thus, the equation
∫ ∫
∞
−∞
∞
−∞
x t dt X j d
2 2
2
holds. Expression (12.4) is known as Parseval’s relation.
Parseval’s relation states that the total energy may be determined either by
integrating (^ )^
2
x t over all time or by integrating (^ )^
2 j 2
ω π
X over all
frequencies. Therefore (^ )^
2
X j ω is interpreted as an energy spectral density of
signal x ( ) t.
12.2. Energy and power for discrete-time signals
The definition of signal energy and power for discrete signals parallel similar
definitions for continuous signals.
Definition 2
The signal energy in the discrete-time signal x ( n )is
∑^ ( )
∞
=−∞
n
E xn
2
. (12.5)
The signal power in the signal x ( n )is
= (^) ∑ →∞ =−
N
N n N
xn N
P
2
2 1
lim.^ (12.6)
A discrete-time energy signal is defined as one for which 0 < E <∞ and a
discrete-time power signal is defined as one for which 0 < P <∞. It is possible
for a discrete-time signal to be neither an energy signal nor a power signal.
Example 12.
Compute the signal energy and signal power for the discrete-time signal
x ( ) n u ( ) n
n
⎟ ⎠
We apply relationship (12.5)
∑ ( )^ ∑ ∑
∞
=
∞
=
∞
=−∞
0 0
2 2
16
n
n
n
n
n
E xn.
The expression on the right hand side is a geometric series; hence, we have
E =.
Since 0 < E <∞, signal x ( ) n is an energy signal, consequently, P = 0.
Example 12.
Compute the signal power and signal energy for the discrete-time signal
x ( n ) u ( n )
10 j n = e.
