In signal processing, the energy
E
s
{\displaystyle E_{s}}
of a continuous-time signal x(t) is defined as the area under the squared magnitude of the considered signal i.e., mathematically
E
s
=
⟨
x
(
t
)
,
x
(
t
)
⟩
=
∫
−
∞
∞
|
x
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t
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2
d
t
{\displaystyle E_{s}\ \ =\ \ \langle x(t),x(t)\rangle \ \ =\int _{-\infty }^{\infty }{|x(t)|^{2}}dt}
The units of
E
s
{\displaystyle E_{s}\,}
will be
(
[
units of
x
(
t
)
]
2
⋅
s
)
{\displaystyle \left(\left[{\text{units of}}\ x(t)\right]^{2}\cdot {\text{s}}\right)}
.
And the energy
E
s
{\displaystyle E_{s}}
of a discrete-time signal x(n) is defined mathematically as
E
s
=
⟨
x
(
n
)
,
x
(
n
)
⟩
=
∑
n
=
−
∞
∞
|
x
(
n
)
|
2
{\displaystyle E_{s}\ \ =\ \ \langle x(n),x(n)\rangle \ \ =\sum _{n=-\infty }^{\infty }{|x(n)|^{2}}}
Contents
Relationship to energy in physics
Energy in this context is not, strictly speaking, the same as the conventional notion of energy in physics and the other sciences. The two concepts are, however, closely related, and it is possible to convert from one to the other:
E
=
E
s
Z
=
1
Z
∫
−
∞
∞
|
x
(
t
)
|
2
d
t
{\displaystyle E={E_{s} \over Z}={1 \over Z}\int _{-\infty }^{\infty }{|x(t)|^{2}}dt}
where Z represents the magnitude, in appropriate units of measure, of the load driven by the signal.
For example, if x(t) represents the potential (in volts) of an electrical signal propagating across a transmission line, then Z would represent the characteristic impedance (in ohms) of the transmission line. The units of measure for the signal energy
E
Spectral energy density
Similarly, the spectral energy density of signal x(t) is
E
s
(
f
)
=
|
X
(
f
)
|
2
{\displaystyle \ E_{s}(f)=|X(f)|^{2}}
where X(f) is the Fourier transform of x(t).
For example, if x(t) represents the magnitude of the electric field component (in volts per meter) of an optical signal propagating through free space, then the dimensions of X(f) would become volt·seconds per meter and
E
s
(
f
)
{\displaystyle E_{s}(f)}
would represent the signal's spectral energy density (in volts2·second2 per meter2) as a function of frequency f (in hertz). Again, these units of measure are not dimensionally correct in the true sense of energy density as defined in physics. Dividing
E
s
Parseval's theorem
As a consequence of Parseval's theorem, one can prove that the signal energy is always equal to the summation across all frequency components of the signal's spectral energy density.