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The power spectrum S_(f) of 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 ...

time series
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 ...

spectrum
. 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 ...

Summation
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 ...

timbre
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 ...

color
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 ...

dispersive prism
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 ...

stochastic process
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 ...

physics
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 ...

engineering
. 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 as
visible 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 ...

color
), 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 ...

wavelength
) 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 ...

sine wave
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 ...

harmonics
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 ...

physics
, 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 ...

electromagnetic wave
, 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 ...

acoustic wave
, 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 ...

watt
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 ...

hertz
(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 ...

voltage
, 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 V2 Hz−1 for the PSD and V2 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 m2/Hz. For random vibration analysis, units of ''g''2 Hz−1 are frequently used for the PSD of
acceleration 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 ...

acceleration
. 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 ...

g-force
. 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 the
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 ...
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 ...

time series
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 \triangleq \int_^\infty , x(t), ^2\ 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: :\int_^\infty , x(t), ^2\, dt = \int_^\infty , \hat(f), ^2\ df, where: :\hat(f) \triangleq\int_^\infty e^x(t) \ 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 ...

frequency
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 \left , \hat(f) \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 \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 ...

Periodogram
). 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 ...

potential
(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 ...

volt
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 ...

transmission line
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 ...

Ohm's law
, 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 \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 (\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)/\Delta f. In this example, since the power V(t)^2/Z has units of V2 Ω−1, the energy E(f) has units of V2 s Ω−1 = J, and hence the estimate E(f)/\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 V2 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\,\Delta t): :\bar_(f) = \lim_(\Delta t)^2 \underbrace_, where \hat x_d(f) is the
discrete-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 \Delta t is needed to keep the correct physical units and to ensure that we recover the continuous case in the limit \Delta t\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 for
stationary 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 ...

variance
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 ...

voltage
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 ...

ohm
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 ...

resistor
, 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 ...

watt
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 = \lim_ \frac 1 \int_^ , x(t), ^2\,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 = \lim_ \frac 1 \int_^ , x_(t), ^2\,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 \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 = \lim_ \frac 1 \int_^ , \hat_(f), ^2\,df Then the power spectral density is simply defined as the integrand above. From here, we can also view , \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) : , \hat_(f), ^2=\mathcal\left \=\int_^\infty \left int_^\infty x_^*(t-\tau)x_(t) dt \right^ \ d\tau Now, if we divide the time convolution above by the period T and take the limit as T \rightarrow \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_(\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.. : \lim_ \frac 1 , \hat_(f), ^2 =\int_^\infty \left lim_ \frac 1 \int_^\infty x_^*(t-\tau)x_(t) dt \right^ \ d\tau=\int_^\infty R_(\tau)e^ d\tau 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 _1, f_2/math>, where 0, can be calculated by integrating over frequency. Since S_(-f) = S_(f), an equal amount of power can be attributed to positive and negative frequency bands, which accounts for the factor of 2 in the following form (such trivial factors depend on the conventions used): : P_\mathsf = 2 \int_^ S_(f) \, df More generally, similar techniques may be used to estimate a time-varying spectral density. In this case the time interval T is finite rather than approaching infinity. This results in decreased spectral coverage and resolution since frequencies of less than 1/T are not sampled, and results at frequencies which are not an integer multiple of 1/T are not independent. Just using a single such time series, the estimated power spectrum will be very "noisy"; however this can be alleviated if it is possible to evaluate the expected value (in the above equation) using a large (or infinite) number of short-term spectra corresponding to
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\le n\le N with the signal sampled at discrete times x_n=x_0+(n\,\Delta t) for a total measurement period T=(2N+1) \,\Delta t. :S_(f) = \lim_\frac\left, \sum_^N x_n e^\^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 ...

periodogram
. 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 ...

cross-correlation
.


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 an
even 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 ...

even function
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 ...

stochastic process
x(t), the autocorrelation function ''R''''xx''(''t'') can be reconstructed from its power spectrum Sxx(f) by using the
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 ...

variance
(average power) of a process by integrating the power spectrum over all frequency: :: P=\operatorname(x) = \int_^\! S_(f) \, 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 \int _0^f S_(f')\, df'. :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. : \begin P & = \lim_ \frac 1 \int_^ \left _(t)+y_(t)\right*\left _(t)+y_(t)\rightt \\ & = \lim_ \frac 1 \int_^, x_(t), ^2 + x^*_(t)y_(t) + y^*_(t)x_(t)+, y_(t), ^2dt \\ \end Using the same notation and methods as used for the power spectral density derivation, we exploit Parseval's theorem and obtain :S_(f) = \lim_ \frac 1 \left hat^*_(f)\hat_(f)\right\ \ \ \ \ \ \ S_(f) = \lim_ \frac 1 \left hat^*_(f)\hat_(f)\right/math> where, again, the contributions of S_(f) and S_(f) are already understood. Note that S^*_(f)=S_(f), so the full contribution to the cross power is, generally, from twice the real part of either individual CPSD. Just as before, from here we recast these products as the Fourier transform of a time convolution, which when divided by the period and taken to the limit T\to \infty becomes the Fourier transform of a
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 ...

cross-correlation
function. :S_(f) = \int_^ \left lim_ \frac 1 \int_^ x^*_(t-\tau) y_(t) dt \righte^ d\tau= \int_^ R_(\tau) e^ d\tau : S_(f) = \int_^ \left lim_ \frac 1 \int_^ y^*_(t-\tau) x_(t) dt \righte^ d\tau= \int_^ R_(\tau) e^ d\tau where R_(\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 ...

cross-correlation
of x(t) with y(t) and R_(\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 \hat(f) and \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. :\operatorname_=2S_(f)=2 S_(f) For discrete signals ''xn'' and ''yn'', the relationship between the cross-spectral density and the cross-covariance is :S_(f)=\sum_^\infty R_(\tau_n)e^\,\Delta\tau


Estimation

The goal of spectral density estimation is to
estimate 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 ...

periodogram
. The spectral density is usually estimated using
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
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 ...

spectrogram
''. 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 ...

transfer function
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 ...

Bode plot
,
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 ...

impulse response
(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 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 of physics and electrical enginee ...

electrical engineering
, 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 ...

radar
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 ...

remote sensing
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