Energy (signal processing)
Encyclopedia
In signal processing
, the energy
of a continuous-time signal x(t) is defined as
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:
For example, if x(t) represents the potential
(in volt
s) of an electrical signal propagating across a transmission line, then Z would represent the characteristic impedance
(in ohm
s) of the transmission line. The units of measure for the signal energy
would appear as volt2-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 volt2-seconds per ohm, which is equivalent to joule
s, the SI
unit for energy as defined in the physical sciences.
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 per meter) of an optical signal propagating through free space, then the dimensions of X(f) would become volt-seconds per meter and
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
by Zo, the characteristic impedance of free space (in ohms), the dimensions become joule-seconds per meter2 or, equivalently, joules per meter2 per hertz, which is dimensionally correct in SI
units for spectral energy density.
, one can prove that the signal energy is always equal to the summation across all frequency components of the signal's spectral energy density.
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
, the energy
of a continuous-time signal x(t) is defined asEnergy in this context is not, strictly speaking, the same as the conventional notion of energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
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
Electric potential
In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...
(in volt
Volt
The volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...
s) of an electrical signal propagating across a transmission line, then Z would represent the characteristic impedance
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
(in ohm
Ohm
The ohm is the SI unit of electrical resistance, named after German physicist Georg Simon Ohm.- Definition :The ohm is defined as a resistance between two points of a conductor when a constant potential difference of 1 volt, applied to these points, produces in the conductor a current of 1 ampere,...
s) of the transmission line. The units of measure for the signal energy
would appear as volt2-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 volt2-seconds per ohm, which is equivalent to jouleJoule
The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...
s, the SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
unit for energy as defined in the physical sciences.
Spectral Energy Density
Similarly, the spectral energy densitySpectral density
In statistical signal processing and physics, the spectral density, power spectral density , or energy spectral density , is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per hertz...
of signal x(t) is
where X(f) is the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
of x(t).
For example, if x(t) represents the magnitude of the electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
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
would represent the signal's spectral energy density (in volts2-second2 per meter2) as a function of frequency f (in hertzHertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....
). Again, these units of measure are not dimensionally correct in the true sense of energy density as defined in physics. Dividing
by Zo, the characteristic impedance of free space (in ohms), the dimensions become joule-seconds per meter2 or, equivalently, joules per meter2 per hertz, which is dimensionally correct in SISi
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
units for spectral energy density.
Parseval's Theorem
As a consequence of Parseval's theoremParseval's theorem
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum of the square of a function is equal to the sum of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later...
, one can prove that the signal energy is always equal to the summation across all frequency components of the signal's spectral energy density.