The power spectrum $S\_(f)$ of a

^{2} Hz^{−1} for the PSD and V^{2} s Hz^{−1} for the ESD (''energy spectral density'') even though no actual "power" or "energy" is specified.
Sometimes one encounters an ''amplitude spectral density'' (ASD), which is the square root of the PSD; the ASD of a voltage signal has units of V Hz^{−1/2}. This is useful when the ''shape'' of the spectrum is rather constant, since variations in the ASD will then be proportional to variations in the signal's voltage level itself. But it is mathematically preferred to use the PSD, since only in that case is the area under the curve meaningful in terms of actual power over all frequency or over a specified bandwidth.
In the general case, the units of PSD will be the ratio of units of variance per unit of frequency; so, for example, a series of displacement values (in meters) over time (in seconds) will have PSD in units of m^{2}/Hz.
For random vibration analysis, units of ''g''^{2} Hz^{−1} are frequently used for the PSD of

^{2} Ω^{−1}, the energy $E(f)$ has units of V^{2} s Ω^{−1} = J, and hence the estimate $E(f)/\backslash Delta\; f$ of the energy spectral density has units of J Hz^{−1}, as required. In many situations, it is common to forget the step of dividing by $Z$ so that the energy spectral density instead has units of V^{2} Hz^{−1}.
This definition generalizes in a straightforward manner to a discrete signal with a countably infinite number of values $x\_n$ such as a signal sampled at discrete times $x\_n=x\_0\; +\; (n\backslash ,\backslash Delta\; t)$:
:$\backslash bar\_(f)\; =\; \backslash lim\_(\backslash Delta\; t)^2\; \backslash underbrace\_,$
where $\backslash hat\; x\_d(f)$ is the

_{''xx''}(''t'') can be reconstructed from its power spectrum S_{xx}(f) by using the

_{n}'' and ''y_{n}'', the relationship between the cross-spectral density and the cross-covariance is
:$S\_(f)=\backslash sum\_^\backslash infty\; R\_(\backslash tau\_n)e^\backslash ,\backslash Delta\backslash tau$

Fourier transform#REDIRECT Fourier transform
In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...

methods (such as the

Power Spectral Density Matlab scripts

{{DEFAULTSORT:Spectral Density Frequency-domain analysis Signal processing Waves Spectroscopy Scattering Fourier analysis Radio spectrum

time series
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

$x(t)$ describes the distribution of power
Power most often refers to:
* Power (physics)
In physics, power is the amount of energy
In , energy is the that must be to a or to perform on the body, or to it. Energy is a ; the law of states that energy can be in form, bu ...

into frequency components composing that signal. According to Fourier analysis
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal (including noise
Noise is unwanted sound
In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid.
In human physiology and psychology, sound is the ''reception'' of such waves and t ...

) as analyzed in terms of its frequency content, is called its spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum
Continuum may refer to:
* Continuum (measurement)
Continuum theories or models expla ...

.
When the energy of the signal is concentrated around a finite time interval, especially if its total energy is finite, one may compute the energy spectral density. More commonly used is the power spectral density (or simply power spectrum), which applies to signals existing over ''all'' time, or over a time period large enough (especially in relation to the duration of a measurement) that it could as well have been over an infinite time interval. The power spectral density (PSD) then refers to the spectral energy distribution that would be found per unit time, since the total energy of such a signal over all time would generally be infinite. Summation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

or integration of the spectral components yields the total power (for a physical process) or variance (in a statistical process), identical to what would be obtained by integrating $x^2(t)$ over the time domain, as dictated by Parseval's theorem.
The spectrum of a physical process $x(t)$ often contains essential information about the nature of $x$. For instance, the pitch
Pitch may refer to:
Acoustic frequency
* Pitch (music), the perceived frequency of sound including "definite pitch" and "indefinite pitch"
** Absolute pitch or "perfect pitch"
** Pitch class, a set of all pitches that are a whole number of octaves ...

and timbre
In music, timbre ( ), also known as tone color or tone quality (from psychoacoustics
Psychoacoustics is the branch of psychophysics
Psychophysics quantitatively investigates the relationship between physical stimulus (physiology), stimuli a ...

of a musical instrument are immediately determined from a spectral analysis. The color
Color (American English
American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of varieties of the English language native to the United States. Currently, American Engli ...

of a light source is determined by the spectrum of the electromagnetic wave's electric field $E(t)$ as it fluctuates at an extremely high frequency. Obtaining a spectrum from time series such as these involves the Fourier transform#REDIRECT Fourier transform
In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...

, and generalizations based on Fourier analysis. In many cases the time domain is not specifically employed in practice, such as when a dispersive prism
Lamps as seen through a prism
In optics
Optics is the branch of physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that st ...

is used to obtain a spectrum of light in a spectrograph
An optical spectrometer (spectrophotometer, spectrograph or spectroscope) is an instrument used to measure properties of light
Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that c ...

, or when a sound is perceived through its effect on the auditory receptors of the inner ear, each of which is sensitive to a particular frequency.
However this article concentrates on situations in which the time series is known (at least in a statistical sense) or directly measured (such as by a microphone sampled by a computer). The power spectrum is important in statistical signal processing
Signal processing is an electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics
The field of elec ...

and in the statistical study of stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a Indexed family, family of random variables. Stochastic processes are widely used as mathematical models of systems and phen ...

es, as well as in many other branches of physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

and engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

. Typically the process is a function of time, but one can similarly discuss data in the spatial domain being decomposed in terms of spatial frequency
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

.
Explanation

Any signal that can be represented as a variable that varies in time has a corresponding frequency spectrum. This includes familiar entities such asvisible light
Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that is visual perception, perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nano ...

(perceived as color
Color (American English
American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of varieties of the English language native to the United States. Currently, American Engli ...

), musical notes (perceived as pitch
Pitch may refer to:
Acoustic frequency
* Pitch (music), the perceived frequency of sound including "definite pitch" and "indefinite pitch"
** Absolute pitch or "perfect pitch"
** Pitch class, a set of all pitches that are a whole number of octaves ...

), radio/TV (specified by their frequency, or sometimes wavelength
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...

) and even the regular rotation of the earth. When these signals are viewed in the form of a frequency spectrum, certain aspects of the received signals or the underlying processes producing them are revealed. In some cases the frequency spectrum may include a distinct peak corresponding to a sine wave
A sine wave or sinusoid is any of certain mathematical curves that describe a smooth periodic oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value (often a point of Mechanical equilib ...

component. And additionally there may be peaks corresponding to harmonics
A harmonic is any member of the harmonic series
Harmonic series may refer to either of two related concepts:
*Harmonic series (mathematics)
*Harmonic series (music)
{{Disambig .... The term is employed in various disciplines, including music ...

of a fundamental peak, indicating a periodic signal which is ''not'' simply sinusoidal. Or a continuous spectrum may show narrow frequency intervals which are strongly enhanced corresponding to resonances, or frequency intervals containing almost zero power as would be produced by a notch filter
In signal processing
Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Signa ...

.
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

, the signal might be a wave, such as an electromagnetic wave
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

, an acoustic wave
Acoustic waves are a type of energy propagation through a medium by means of adiabatic compression and decompression. Important quantities for describing acoustic waves are acoustic pressure, particle velocity
Particle velocity is the velocity of ...

, or the vibration of a mechanism. The ''power spectral density'' (PSD) of the signal describes the power
Power most often refers to:
* Power (physics)
In physics, power is the amount of energy
In , energy is the that must be to a or to perform on the body, or to it. Energy is a ; the law of states that energy can be in form, bu ...

present in the signal as a function of frequency, per unit frequency. Power spectral density is commonly expressed in watt
The watt (symbol: W) is a unit of power
Power typically refers to:
* Power (physics)
In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equa ...

s per hertz
The hertz (symbol: Hz) is the unit
Unit may refer to:
Arts and entertainment
* UNIT
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action ...

(W/Hz).
When a signal is defined in terms only of a voltage
Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is the ...

, for instance, there is no unique power associated with the stated amplitude. In this case "power" is simply reckoned in terms of the square of the signal, as this would always be ''proportional'' to the actual power delivered by that signal into a given impedance. So one might use units of Vacceleration
In mechanics
Mechanics (Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...

. Here ''g'' denotes the g-force
The gravitational force equivalent, or, more commonly, g-force, is a measurement of the type of force per unit mass – typically acceleration – that causes a perception of weight
In science
Science () is a systematic ente ...

.
Mathematically, it is not necessary to assign physical dimensions to the signal or to the independent variable. In the following discussion the meaning of ''x(t)'' will remain unspecified, but the independent variable will be assumed to be that of time.
Definition

Energy spectral density

Energy spectral density describes how theenergy
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

of a signal or a time series
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

is distributed with frequency. Here, the term energy
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

is used in the generalized sense of signal processing; that is, the energy $E$ of a signal $x(t)$ is:
:$E\; \backslash triangleq\; \backslash int\_^\backslash infty\; ,\; x(t),\; ^2\backslash \; dt.$
The energy spectral density is most suitable for transients—that is, pulse-like signals—having a finite total energy. Finite or not, Parseval's theorem (or Plancherel's theorem) gives us an alternate expression for the energy of the signal:
:$\backslash int\_^\backslash infty\; ,\; x(t),\; ^2\backslash ,\; dt\; =\; \backslash int\_^\backslash infty\; ,\; \backslash hat(f),\; ^2\backslash \; df,$
where:
:$\backslash hat(f)\; \backslash triangleq\backslash int\_^\backslash infty\; e^x(t)\; \backslash \; dt$
is the value of the Fourier transform#REDIRECT Fourier transform
In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...

of $x(t)$ at frequency
Frequency is the number of occurrences of a repeating event per unit of time
A unit of time is any particular time
Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparen ...

$f$ (in Hz). The theorem also holds true in the discrete-time cases. Since the integral on the right-hand side is the energy of the signal, the integrand $\backslash left\; ,\; \backslash hat(f)\; \backslash right\; ,\; ^2$ can be interpreted as a density function
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

describing the energy contained in the signal at the frequency $f$. Therefore, the energy spectral density of $x(t)$ is defined as:
The function $\backslash bar\_(f)$ and the autocorrelation
Autocorrelation, also known as serial correlation, is the correlation
In , correlation or dependence is any statistical relationship, whether or not, between two s or . In the broadest sense correlation is any statistical association, th ...

of $x(t)$ form a Fourier transform pair, a result is known as Wiener–Khinchin theorem
In applied mathematics, the Wiener–Khinchin theorem, also known as the Wiener–Khintchine theorem and sometimes as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-s ...

(see also Periodogram In signal processing, a periodogram is an estimate of the spectral density of a signal. The term was coined by Arthur Schuster in 1898. Today, the periodogram is a component of more sophisticated methods (see spectral estimation). It is the most co ...

).
As a physical example of how one might measure the energy spectral density of a signal, suppose $V(t)$ represents the potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (ph ...

(in volt
The volt is the derived unit for electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work (physics), work energy needed to move a ...

s) of an electrical pulse propagating along a transmission line
In electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics
The field of electronics is a branch o ...

of impedance $Z$, and suppose the line is terminated with a matched resistor (so that all of the pulse energy is delivered to the resistor and none is reflected back). By Ohm's law
Ohm's law states that the current
Currents or The Current may refer to:
Science and technology
* Current (fluid)
A current in a fluid
In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an ap ...

, the power delivered to the resistor at time $t$ is equal to $V(t)^2/Z$, so the total energy is found by integrating $V(t)^2/Z$ with respect to time over the duration of the pulse. To find the value of the energy spectral density $\backslash bar\_(f)$ at frequency $f$, one could insert between the transmission line and the resistor a bandpass filter
File:Bandpass_Filter.svg, 300px, A medium-complexity example of a band-pass filter.
A band-pass filter or bandpass filter (BPF) is a device that passes frequency, frequencies within a certain range and rejects (attenuates) frequencies outside t ...

which passes only a narrow range of frequencies ($\backslash Delta\; f$, say) near the frequency of interest and then measure the total energy $E(f)$ dissipated across the resistor. The value of the energy spectral density at $f$ is then estimated to be $E(f)/\backslash Delta\; f$. In this example, since the power $V(t)^2/Z$ has units of Vdiscrete-time Fourier transform
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

of $x\_n.$ The sampling interval $\backslash Delta\; t$ is needed to keep the correct physical units and to ensure that we recover the continuous case in the limit $\backslash Delta\; t\backslash to\; 0.$ But in the mathematical sciences the interval is often set to 1, which simplifies the results at the expense of generality. (also see normalized frequency)
Power spectral density

The above definition of energy spectral density is suitable for transients (pulse-like signals) whose energy is concentrated around one time window; then the Fourier transforms of the signals generally exist. For continuous signals over all time, one must rather define the ''power spectral density'' (PSD) which exists forstationary process In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

es; this describes how power
Power most often refers to:
* Power (physics)
In physics, power is the amount of energy
In , energy is the that must be to a or to perform on the body, or to it. Energy is a ; the law of states that energy can be in form, bu ...

of a signal or time series is distributed over frequency, as in the simple example given previously. Here, power can be the actual physical power, or more often, for convenience with abstract signals, is simply identified with the squared value of the signal. For example, statisticians study the variance
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

of a function over time $x(t)$ (or over another independent variable), and using an analogy with electrical signals (among other physical processes), it is customary to refer to it as the ''power spectrum'' even when there is no physical power involved. If one were to create a physical voltage
Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is the ...

source which followed $x(t)$ and applied it to the terminals of a 1 ohm
The ohm (symbol: Ω) is the SI derived unit
SI derived units are units of measurement derived from the
seven SI base unit, base units specified by the International System of Units (SI). They are either dimensionless quantity, dimensionless o ...

resistor
A resistor is a passive
Passive may refer to:
* Passive voice, a grammatical voice common in many languages, see also Pseudopassive (disambiguation), Pseudopassive
* Passive language, a language from which an interpreter works
* Passivity (b ...

, then indeed the instantaneous power dissipated in that resistor would be given by $x(t)^2$ watt
The watt (symbol: W) is a unit of power
Power typically refers to:
* Power (physics)
In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equa ...

s.
The average power $P$ of a signal $x(t)$ over all time is therefore given by the following time average, where the period $T$ is centered about some arbitrary time $t=t\_$:
:$P\; =\; \backslash lim\_\; \backslash frac\; 1\; \backslash int\_^\; ,\; x(t),\; ^2\backslash ,dt$
However, for the sake of dealing with the math that follows, it is more convenient to deal with time limits in the signal itself rather than time limits in the bounds of the integral. As such, we have an alternative representation of the average power, where $x\_(t)=x(t)w\_(t)$ and $w\_(t)$ is unity within the arbitrary period and zero elsewhere.
:$P\; =\; \backslash lim\_\; \backslash frac\; 1\; \backslash int\_^\; ,\; x\_(t),\; ^2\backslash ,dt$
Clearly in cases where the above expression for P is non-zero (even as T grows without bound) the integral itself must also grow without bound. That is the reason that we cannot use the energy spectral density itself, which ''is'' that diverging integral, in such cases.
In analyzing the frequency content of the signal $x(t)$, one might like to compute the ordinary Fourier transform $\backslash hat(f)$; however, for many signals of interest the Fourier transform does not formally exist. Regardless, Parseval's theorem tells us that we can re-write the average power as follows.
:$P\; =\; \backslash lim\_\; \backslash frac\; 1\; \backslash int\_^\; ,\; \backslash hat\_(f),\; ^2\backslash ,df$
Then the power spectral density is simply defined as the integrand above.
From here, we can also view $,\; \backslash hat\_(f),\; ^2$ as the Fourier transform#REDIRECT Fourier transform
In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...

of the time convolution
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of $x\_^*(-t)$ and $x\_(t)$
:$,\; \backslash hat\_(f),\; ^2=\backslash mathcal\backslash left\; \backslash =\backslash int\_^\backslash infty\; \backslash left;\; href="/html/ALL/s/int\_^\backslash infty\_x\_^*(t-\backslash tau)x\_(t)\_dt\_\backslash right.html"\; ;"title="int\_^\backslash infty\; x\_^*(t-\backslash tau)x\_(t)\; dt\; \backslash right">int\_^\backslash infty\; x\_^*(t-\backslash tau)x\_(t)\; dt\; \backslash right$
Now, if we divide the time convolution above by the period $T$ and take the limit as $T\; \backslash rightarrow\; \backslash infty$, it becomes the autocorrelation
Autocorrelation, also known as serial correlation, is the correlation
In , correlation or dependence is any statistical relationship, whether or not, between two s or . In the broadest sense correlation is any statistical association, th ...

function of the non-windowed signal $x(t)$, which is denoted as $R\_(\backslash tau)$, provided that $x(t)$ is ergodic
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies that ...

, which is true in most, but not all, practical cases..
:$\backslash lim\_\; \backslash frac\; 1\; ,\; \backslash hat\_(f),\; ^2\; =\backslash int\_^\backslash infty\; \backslash left;\; href="/html/ALL/s/lim\_\_\backslash frac\_1\_\backslash int\_^\backslash infty\_x\_^*(t-\backslash tau)x\_(t)\_dt\_\backslash right.html"\; ;"title="lim\_\; \backslash frac\; 1\; \backslash int\_^\backslash infty\; x\_^*(t-\backslash tau)x\_(t)\; dt\; \backslash right">lim\_\; \backslash frac\; 1\; \backslash int\_^\backslash infty\; x\_^*(t-\backslash tau)x\_(t)\; dt\; \backslash right$
From here we see, again assuming the ergodicity of $x(t)$, that the power spectral density can be found as the Fourier transform of the autocorrelation function (Wiener–Khinchin theorem
In applied mathematics, the Wiener–Khinchin theorem, also known as the Wiener–Khintchine theorem and sometimes as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-s ...

).
Many authors use this equality to actually ''define'' the power spectral density.
The power of the signal in a given frequency band $;\; href="/html/ALL/s/\_1,\_f\_2.html"\; ;"title="\_1,\; f\_2">\_1,\; f\_2$statistical ensemble
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

s of realizations of $x(t)$ evaluated over the specified time window.
Just as with the energy spectral density, the definition of the power spectral density can be generalized to discrete timeIn Dynamical system, mathematical dynamics, discrete time and continuous time are two alternative frameworks within which to model Variable (mathematics), variables that evolve over time.
Discrete time
Image:Sampled.signal.svg, Discrete sampled sign ...

variables $x\_n$. As before, we can consider a window of $-N\backslash le\; n\backslash le\; N$ with the signal sampled at discrete times $x\_n=x\_0+(n\backslash ,\backslash Delta\; t)$ for a total measurement period $T=(2N+1)\; \backslash ,\backslash Delta\; t$.
:$S\_(f)\; =\; \backslash lim\_\backslash frac\backslash left,\; \backslash sum\_^N\; x\_n\; e^\backslash ^2$
Note that a single estimate of the PSD can be obtained through a finite number of samplings. As before, the actual PSD is achieved when $N$ (and thus $T$) approaches infinity and the expected value is formally applied. In a real-world application, one would typically average a finite-measurement PSD over many trials to obtain a more accurate estimate of the theoretical PSD of the physical process underlying the individual measurements. This computed PSD is sometimes called a periodogram In signal processing, a periodogram is an estimate of the spectral density of a signal. The term was coined by Arthur Schuster in 1898. Today, the periodogram is a component of more sophisticated methods (see spectral estimation). It is the most co ...

. This periodogram converges to the true PSD as the number of estimates as well as the averaging time interval $T$ approach infinity (Brown & Hwang).
If two signals both possess power spectral densities, then the cross-spectral density
The power spectrum S_(f) of a time series x(t) describes the distribution of Power (physics), power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discre ...

can similarly be calculated; as the PSD is related to the autocorrelation, so is the cross-spectral density related to the cross-correlation
In signal processing
Signal processing is an electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electron ...

.
Properties of the power spectral density

Some properties of the PSD include: * The power spectrum is always real and non-negative, and the spectrum of a real valued process is also aneven function
The cosine function and all of its Taylor polynomials are even functions. This image shows \cos(x) and its Taylor approximation of degree 4.
In mathematics, even functions and odd functions are function (mathematics), functions which satisfy par ...

of frequency: $S\_(-f)\; =\; S\_(f)$.
* For a continuous stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a Indexed family, family of random variables. Stochastic processes are widely used as mathematical models of systems and phen ...

x(t), the autocorrelation function ''R''inverse Fourier transformIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

* Using Parseval's theorem, one can compute the variance
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

(average power) of a process by integrating the power spectrum over all frequency:
:: $P=\backslash operatorname(x)\; =\; \backslash int\_^\backslash !\; S\_(f)\; \backslash ,\; df$
*For a real process ''x''(''t'') with power spectral density $S\_(f)$, one can compute the ''integrated spectrum'' or ''power spectral distribution'' $F(f)$, which specifies the average ''bandlimited'' power contained in frequencies from DC to f using:
:: $F(f)=\; 2\; \backslash int\; \_0^f\; S\_(f\text{'})\backslash ,\; df\text{'}.$
:Note that the previous expression for total power (signal variance) is a special case where ƒ → ∞.
Cross power spectral density

Given two signals $x(t)$ and $y(t)$, each of which possess power spectral densities $S\_(f)$ and $S\_(f)$, it is possible to define a cross power spectral density (CPSD) or cross spectral density (CSD). To begin, let us consider the average power of such a combined signal. :$\backslash begin\; P\; \&\; =\; \backslash lim\_\; \backslash frac\; 1\; \backslash int\_^\; \backslash left;\; href="/html/ALL/s/\_(t)+y\_(t)\backslash right.html"\; ;"title="\_(t)+y\_(t)\backslash right">\_(t)+y\_(t)\backslash right$ Using the same notation and methods as used for the power spectral density derivation, we exploit Parseval's theorem and obtain :$S\_(f)\; =\; \backslash lim\_\; \backslash frac\; 1\; \backslash left;\; href="/html/ALL/s/hat^*\_(f)\backslash hat\_(f)\backslash right.html"\; ;"title="hat^*\_(f)\backslash hat\_(f)\backslash right">hat^*\_(f)\backslash hat\_(f)\backslash right$cross-correlation
In signal processing
Signal processing is an electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electron ...

function.
:$S\_(f)\; =\; \backslash int\_^\; \backslash left;\; href="/html/ALL/s/lim\_\_\backslash frac\_1\_\_\backslash int\_^\_x^*\_(t-\backslash tau)\_y\_(t)\_dt\_\backslash right.html"\; ;"title="lim\_\; \backslash frac\; 1\; \backslash int\_^\; x^*\_(t-\backslash tau)\; y\_(t)\; dt\; \backslash right">lim\_\; \backslash frac\; 1\; \backslash int\_^\; x^*\_(t-\backslash tau)\; y\_(t)\; dt\; \backslash right$
:$S\_(f)\; =\; \backslash int\_^\; \backslash left;\; href="/html/ALL/s/lim\_\_\backslash frac\_1\_\_\backslash int\_^\_y^*\_(t-\backslash tau)\_x\_(t)\_dt\_\backslash right.html"\; ;"title="lim\_\; \backslash frac\; 1\; \backslash int\_^\; y^*\_(t-\backslash tau)\; x\_(t)\; dt\; \backslash right">lim\_\; \backslash frac\; 1\; \backslash int\_^\; y^*\_(t-\backslash tau)\; x\_(t)\; dt\; \backslash right$
where $R\_(\backslash tau)$ is the cross-correlation
In signal processing
Signal processing is an electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electron ...

of $x(t)$ with $y(t)$ and $R\_(\backslash tau)$ is the cross-correlation of $y(t)$ with $x(t)$. In light of this, the PSD is seen to be a special case of the CSD for $x(t)\; =\; y(t)$. For the case that $x(t)$ and $y(t)$ are voltage or current signals, their associated amplitude spectral densities $\backslash hat(f)$ and $\backslash hat(f)$ are strictly positive by convention. Therefore, in typical signal processing, the full CPSD is just one of the CPSDs scaled by a factor of two.
:$\backslash operatorname\_=2S\_(f)=2\; S\_(f)$
For discrete signals ''xEstimation

The goal of spectral density estimation is toestimate
Estimation (or estimating) is the process of finding an estimate, or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertainty, uncertain, or Instability, unstable. The value is nonetheles ...

the spectral density of a random signal
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a Indexed family, family of random variables. Many stochastic processes can be represented by time series. However, a stochast ...

from a sequence of time samples. Depending on what is known about the signal, estimation techniques can involve parametric or non-parametricNonparametric statistics is the branch of statistics that is not based solely on Statistical parameter, parametrized families of probability distributions (common examples of parameters are the mean and variance). Nonparametric statistics is based on ...

approaches, and may be based on time-domain or frequency-domain analysis. For example, a common parametric technique involves fitting the observations to an autoregressive model
In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data
Data (; ) are individual facts, statistics, or items of information, often numeric. In a mor ...

. A common non-parametric technique is the periodogram In signal processing, a periodogram is an estimate of the spectral density of a signal. The term was coined by Arthur Schuster in 1898. Today, the periodogram is a component of more sophisticated methods (see spectral estimation). It is the most co ...

.
The spectral density is usually estimated using Welch method
Welch's method, named after Peter D. Welch, is an approach for spectral density estimation.
It is used in physics, engineering, and applied mathematics for estimating the Electric power, power of a Signal (electrical engineering), signal at differ ...

), but other techniques such as the maximum entropy method can also be used.
Related concepts

* The spectral centroid of a signal is the midpoint of its spectral density function, i.e. the frequency that divides the distribution into two equal parts. * The spectral edge frequency of a signal is an extension of the previous concept to any proportion instead of two equal parts. * The spectral density is a function of frequency, not a function of time. However, the spectral density of a small window of a longer signal may be calculated, and plotted versus time associated with the window. Such a graph is called a ''spectrogram
A spectrogram is a visual representation of the spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a Continuum (theory), continuum. Th ...

''. This is the basis of a number of spectral analysis techniques such as the short-time Fourier transform
The Short-time Fourier transform (STFT), is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. In practice, the procedure for computing STFTs is to divid ...

and waveletsA wavelet is a wave
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion an ...

.
* A "spectrum" generally means the power spectral density, as discussed above, which depicts the distribution of signal content over frequency. This is not to be confused with the frequency response
In electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons
The electron is a subatomic particle
In physical sciences, subatomic particles ...

of a transfer function
In engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...

which also includes a phase
Phase or phases may refer to:
Science
* State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter)
In the physical sciences, a phase is a region of space (a thermodynamic system
A thermodynamic system is a ...

(or equivalently, a real and imaginary part as a function of frequency). For transfer functions, (e.g., Bode plot
In and , a Bode plot is a of the of a system. It is usually a combination of a Bode magnitude plot, expressing the magnitude (usually in s) of the frequency response, and a Bode phase plot, expressing the .
As originally conceived by in t ...

, chirp
A chirp is a signal
In signal processing
Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scie ...

) the complete frequency response may be graphed in two parts, amplitude versus frequency and phase versus frequency—the phase spectral density, phase spectrum, or spectral phase (or less commonly, as real and imaginary parts of the transfer function). The impulse response
In signal processing
Signal processing is an electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electroni ...

(in the time domain) $h(t)$, cannot generally be uniquely recovered from the amplitude spectral density part alone without the phase function. Although these are also Fourier transform pairs, there is no symmetry (as there is for the autocorrelation) forcing the Fourier transform to be real-valued. See Ultrashort pulse#Spectral phase, phase noise
In signal processing
Signal processing is an electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electroni ...

, group delay
In signal processing
Signal processing is an electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electroni ...

.
Applications

Electrical engineering

The concept and use of the power spectrum of a signal is fundamental inelectrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics
The field of electronics is a branch of physics and electrical enginee ...

, especially in electronic communication systems, including radio communication
Radio is the technology of signaling and communicating
Communication (from Latin ''communicare'', meaning "to share") is the act of developing Semantics, meaning among Subject (philosophy), entities or Organization, groups through the use ...

s, radar
Radar (radio detection and ranging) is a detection system that uses radio waves to determine the distance (''ranging''), angle, or velocity of objects. It can be used to detect aircraft, Marine radar, ships, spacecraft, guided missiles, motor ...

s, and related systems, plus passive remote sensing
Remote sensing is the acquisition of information about an object or phenomenon without making physical contact with the object, in contrast to in situ
''In situ'' (; often not italicized in English) is a Latin
Latin (, or , ) is a class ...

technology. Electronic instruments called spectrum analyzer
A spectrum analyzer measures the magnitude of an input signal versus frequency within the full frequency range of the instrument. The primary use is to measure the power of the spectrum of known and unknown signals. The input signal that most com ...

s are used to observe and measure the ''power spectra'' of signals.
The spectrum analyzer measures the magnitude of the short-time Fourier transform
The Short-time Fourier transform (STFT), is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. In practice, the procedure for computing STFTs is to divid ...

(STFT) of an input signal. If the signal being analyzed can be considered a stationary process, the STFT is a good smoothed estimate of its power spectral density.
Cosmology

Primordial fluctuations
Primordial fluctuations are density variations in the early universe which are considered the seeds of all large-scale structure of the cosmos, structure in the universe. Currently, the most widely accepted explanation for their origin is in the con ...

, density variations in the early universe, are quantified by a power spectrum which gives the power of the variations as a function of spatial scale.
See also

*Noise spectral density In communications, noise spectral density, noise power density, noise power spectral density, or simply noise density (''N''0) is the power spectral density
The power spectrum S_(f) of a time series x(t) describes the distribution of power i ...

* Spectral density estimation
In statistical signal processing, the goal of spectral density estimation (SDE) is to estimate the spectral density (also known as the power spectral density) of a random signal from a sequence of time samples of the signal. Intuitively speaki ...

* Spectral efficiency
Spectral efficiency, spectrum efficiency or bandwidth efficiency refers to the information rate
In telecommunications
Telecommunication is the transmission of information
Information can be thought of as the resolution of uncertaint ...

* Spectral power distribution
In radiometry
Radiometry is a set of techniques for measuring
'
Measurement is the numerical quantification of the attributes of an object or event, which can be used to compare with other objects or events. The scope and application of meas ...

* Brightness temperature
Brightness temperature or radiance temperature is the temperature at which a black body in thermal equilibrium with its surroundings would have to be in order to duplicate the observed Intensity (heat transfer), intensity of a grey body object at a ...

* Colors of noise
In audio engineering
An audio engineer (also known as a sound engineer or recording engineer) helps to produce a recording
A record, recording or records may refer to:
An item or collection of data Computing
* Record (computer science), ...

* Spectral leakage
The Fourier transform
In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequency, ...

* Window function
In and , a window function (also known as an apodization function or tapering function) is a that is zero-valued outside of some chosen , normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually t ...

* Bispectrum
* Whittle likelihood
Notes

References

External links

Power Spectral Density Matlab scripts

{{DEFAULTSORT:Spectral Density Frequency-domain analysis Signal processing Waves Spectroscopy Scattering Fourier analysis Radio spectrum