In
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, the energy
of a continuous-time signal ''x''(''t'') is defined as the area under the squared magnitude of the considered signal i.e., mathematically
:
:Unit of
will be (unit of signal)
2.
:
:
And the energy
of a discrete-time signal ''x''(''n'') is defined mathematically as
:
Relationship to energy in physics
Energy in this context is not, strictly speaking, the same as the conventional notion of
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and the other sciences. The two concepts are, however, closely related, and it is possible to convert from one to the other:
:
: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
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
(in
volt
The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827).
Defi ...
s) of an electrical signal propagating across a transmission line, then ''Z'' would represent the
characteristic impedance (in
ohm
Ohm (symbol Ω) is a unit of electrical resistance named after Georg Ohm.
Ohm or OHM may also refer to:
People
* Georg Ohm (1789–1854), German physicist and namesake of the term ''ohm''
* Germán Ohm (born 1936), Mexican boxer
* Jörg Ohm (b ...
s) of the transmission line. The units of measure for the signal energy
would appear as volt
2·seconds, which is ''not'' dimensionally correct for energy in the sense of the physical sciences. After dividing
by ''Z'', however, the dimensions of ''E'' would become volt
2·seconds per ohm,
which is equivalent to
joule
The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force applie ...
s, the
SI unit for energy as defined in the physical sciences.
Spectral energy density
Similarly, the spectral energy density of signal x(t) is
:
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
The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827).
Defini ...
per meter) of an optical signal propagating through
free space
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
, then the dimensions of ''X''(''f'') would become volt·seconds per meter and
would represent the signal's spectral energy density (in volts
2·second
2 per meter
2) as a function of frequency ''f'' (in
hertz
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that on ...
). Again, these units of measure are not dimensionally correct in the true sense of energy density as defined in physics. Dividing
by ''Z''
o, the characteristic impedance of free space (in ohms), the dimensions become joule-seconds per meter
2 or, equivalently, joules per meter
2 per hertz, which is dimensionally correct in
SI units for spectral energy density.
Parseval's theorem
As a consequence of
Parseval's theorem
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates ...
, one can prove that the signal energy is always equal to the summation across all frequency components of the signal's spectral energy density.
See also
*
Signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
*
Parseval's theorem
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates ...
*
Spectral density
The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
*
Inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
References
{{Reflist
Signal processing