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The power spectrum S_(f) of a
time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Ex ...
x(t) describes the distribution of power into frequency components composing that signal. According to
Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph ...
, 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 considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference aris ...
) 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. The word was first used scientifically in optics to describe the rainbow of colors ...
. 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, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, ma ...
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 and timbre of a musical instrument are immediately determined from a spectral analysis. The
color Color (American English) or colour (British English) is the visual perceptual property deriving from the spectrum of light interacting with the photoreceptor cells of the eyes. Color categories and physical specifications of color are associ ...
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 A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
, and generalizations based on Fourier analysis. In many cases the time domain is not specifically employed in practice, such as when a 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 over a specific portion of the electromagnetic spectrum, typically used in spectroscopic analysis to identify mate ...
, 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 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 family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
es, as well as in many other branches of
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
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.


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 that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 t ...
(perceived as
color Color (American English) or colour (British English) is the visual perceptual property deriving from the spectrum of light interacting with the photoreceptor cells of the eyes. Color categories and physical specifications of color are associ ...
), musical notes (perceived as pitch), radio/TV (specified by their frequency, or sometimes
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
) 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, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ...
component. And additionally there may be peaks corresponding to 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
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, the signal might be a wave, such as an
electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) ...
, an
acoustic wave Acoustic waves are a type of energy propagation through a medium by means of adiabatic loading and unloading. Important quantities for describing acoustic waves are acoustic pressure, particle velocity, particle displacement and acoustic intensi ...
, or the vibration of a mechanism. The ''power spectral density'' (PSD) of the signal describes the power 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 the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James ...
s per
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that o ...
(W/Hz). When a signal is defined in terms only of a
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
, 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. ''Energy spectral density'' (ESD) would have units would be V2 s Hz−1, since
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
has units of power multiplied by time (e.g., watt-hour). 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 meters squared per hertz, m2/Hz. In the analysis of random
vibration Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic, such as the motion of a pendulum—or random, su ...
s, units of ''g''2 Hz−1 are frequently used for the PSD of
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by ...
, where ''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, with a g-force of 1 g (not gram in mass measur ...
. 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, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
of a signal or a
time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Ex ...
is distributed with frequency. Here, the term
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
is used in the generalized sense of signal processing; that is, the energy E of a signal x(t) is: : E \triangleq \int_^\infty \left, 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 \left, \hat(f)\^2\ df, where: :\hat(f) \triangleq\int_^\infty e^x(t) \ dt is the value of the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
of x(t) at
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
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 value of\left, \hat(f) \^2 df can be interpreted as a
density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
multiplied by an infinitesimally small frequency interval, describing the energy contained in the signal at frequency f in the frequency interval f + df. Therefore, the energy spectral density of x(t) is defined as: The function \bar_(f) and the
autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...
of x(t) form a Fourier transform pair, a result also known as the Wiener–Khinchin theorem (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 c ...
). 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 to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
(in
volt The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827). Defin ...
s) of an electrical pulse propagating along a
transmission line In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmi ...
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 through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equa ...
, 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 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−2. 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 t_n=t_0 + (n\,\Delta t): :\bar_(f) = \lim_(\Delta t)^2 \underbrace_, where \hat x_d(f) is the
discrete-time Fourier transform In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the ...
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 processes; this describes how the power 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 and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
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, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
source which followed x(t) and applied it to the terminals of a one
ohm Ohm (symbol Ω) is a unit of electrical resistance named after Georg Ohm. Ohm or OHM may also refer to: People * Georg Ohm (1789–1854), German physicist and namesake of the term ''ohm'' * Germán Ohm (born 1936), Mexican boxer * Jörg Ohm (bor ...
resistor A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active e ...
, then indeed the instantaneous power dissipated in that resistor would be given by x(t)^2
watt The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James ...
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 A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
of the time
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of x_^*(-t) and x_(t) :\left, \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, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...
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 tha ...
, which is true in most, but not all, practical cases.. : \lim_ \frac \left, \hat_(f)\^2 = \int_^\infty \left lim_ \frac\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). 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 Onekama ( ) is a village in Manistee County in the U.S. state of Michigan. The population was 411 at the 2010 census. The village is located on the shores of Portage Lake and is surrounded by Onekama Township. The town's name is derived from "On ...
/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_\textsf = 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 ensembles 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 time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...
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 c ...
. 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 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, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
.


Properties of the power spectral density

Some properties of the PSD include:


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 \int_^ \left _T(t) + y_T(t)\right*\left _T(t) + y_T(t)\rightt \\ &= \lim_ \frac \int_^ , x_T(t), ^2 + x^*_T(t) y_T(t) + y^*_T(t) x_(t) + , y_T(t), ^2 dt \\ \end Using the same notation and methods as used for the power spectral density derivation, we exploit Parseval's theorem and obtain :\begin S_(f) &= \lim_ \frac \left hat^*_T(f) \hat_T(f)\right& S_(f) &= \lim_ \frac \left hat^*_T(f) \hat_T(f)\right\end 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, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
function. :\begin 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 \end where R_(\tau) is the
cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
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 Fourier transforms \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_\text = 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, uncertain, or unstable. The value is nonetheless usable because it is de ...
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 family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
from a sequence of time samples. Depending on what is known about the signal, estimation techniques can involve parametric or non-parametric 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, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...
. 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 c ...
. The spectral density is usually estimated using
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
methods (such as the Welch method), 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 (SEF), usually expressed as "SEF ''x''", represents the
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
below which ''x'' percent of the total power of a given signal are located; typically, ''x'' is in the range 75 to 95. It is more particularly a popular measure used in EEG monitoring, in which case SEF has variously been used to estimate the depth of
anesthesia Anesthesia is a state of controlled, temporary loss of sensation or awareness that is induced for medical or veterinary purposes. It may include some or all of analgesia (relief from or prevention of pain), paralysis (muscle relaxation), ...
and stages of
sleep Sleep is a sedentary state of mind and body. It is characterized by altered consciousness, relatively inhibited Perception, sensory activity, reduced muscle activity and reduced interactions with surroundings. It is distinguished from wakefuln ...
. * A spectral envelope is the envelope curve of the spectrum density. It describes one point in time (one window, to be precise). For example, in
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 or on-site observation. The term is applied especially to acquiring information about Ear ...
using a
spectrometer A spectrometer () is a scientific instrument used to separate and measure spectral components of a physical phenomenon. Spectrometer is a broad term often used to describe instruments that measure a continuous variable of a phenomenon where the ...
, the spectral envelope of a feature is the boundary of its
spectral ''Spectral'' is a 2016 3D military science fiction, supernatural horror fantasy and action-adventure thriller war film directed by Nic Mathieu. Written by himself, Ian Fried, and George Nolfi from a story by Fried and Mathieu. The film stars J ...
properties, as defined by the range of brightness levels in each of the
spectral bands Spectral bands are parts of the electromagnetic spectrum of specific wavelengths, which can be filtered by a standard filter. In nuclear physics, spectral bands are referred to the emission of polyatomic systems, including condensed materials, larg ...
of interest. * 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''. This is the basis of a number of spectral analysis techniques such as the short-time Fourier transform and
wavelets A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the nu ...
. * A "spectrum" generally means the power spectral density, as discussed above, which depicts the distribution of signal content over frequency. For transfer functions (e.g., Bode plot,
chirp A chirp is a signal in which the frequency increases (''up-chirp'') or decreases (''down-chirp'') with time. In some sources, the term ''chirp'' is used interchangeably with sweep signal. It is commonly applied to sonar, radar, and laser syste ...
) the complete frequency response may be graphed in two parts: power versus frequency and phase versus frequency—the phase spectral density, phase spectrum, or spectral phase. Less commonly, the two parts may be the real and imaginary parts of the transfer function. This is not to be confused with the '' frequency response'' of a transfer function, which also includes a phase (or equivalently, a real and imaginary part) as a function of frequency. The time-domain
impulse response In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (). More generally, an impulse response is the reac ...
h(t) cannot generally be uniquely recovered from the power spectral density alone without the phase part. Although these are also Fourier transform pairs, there is no symmetry (as there is for the
autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...
) forcing the Fourier transform to be real-valued. See Ultrashort pulse#Spectral phase,
phase noise In signal processing, phase noise is the frequency-domain representation of random fluctuations in the phase of a waveform, corresponding to time-domain deviations from perfect periodicity ( jitter). Generally speaking, radio-frequency eng ...
,
group delay In signal processing, group delay and phase delay are delay times experienced by a signal's various frequency components when the signal passes through a system that is linear time-invariant (LTI), such as a microphone, coaxial cable, amplifier, ...
. * 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.


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, and electromagnetism. It emerged as an identifiable occupation in the l ...
, especially in electronic communication systems, including
radio communication Radio is the technology of signaling and communicating using radio waves. Radio waves are electromagnetic waves of frequency between 30 hertz (Hz) and 300  gigahertz (GHz). They are generated by an electronic device called a trans ...
s,
radar Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, Marine radar, ships, spacecraft, guided missiles, motor v ...
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 or on-site observation. The term is applied especially to acquiring information about Ear ...
technology. Electronic instruments called spectrum analyzers are used to observe and measure the ''power spectra'' of signals. The spectrum analyzer measures the magnitude of the short-time Fourier transform (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, 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.


Climate Science

Power spectral-analysis have been used to examine the spatial structures for climate research. These results suggests atmospheric turbulence link climate change to more local regional volatility in weather conditions.


See also

*
Bispectrum In mathematics, in the area of statistical analysis, the bispectrum is a statistic used to search for nonlinear interactions. Definitions The Fourier transform of the second-order cumulant, i.e., the autocorrelation function, is the traditional ...
* Brightness temperature *
Colors of noise In audio engineering, electronics, physics, and many other fields, the color of noise or noise spectrum refers to the power spectrum of a noise signal (a signal produced by a stochastic process). Different colors of noise have significantly ...
* Least-squares spectral analysis * Noise spectral density *
Spectral density estimation In statistical signal processing, the goal of spectral density estimation (SDE) or simply spectral estimation is to estimate the spectral density (also known as the power spectral density) of a signal from a sequence of time samples of the sig ...
*
Spectral efficiency Spectral efficiency, spectrum efficiency or bandwidth efficiency refers to the information rate that can be transmitted over a given bandwidth in a specific communication system. It is a measure of how efficiently a limited frequency spectrum is ut ...
* Spectral leakage *
Spectral power distribution In radiometry, photometry, and color science, a spectral power distribution (SPD) measurement describes the power per unit area per unit wavelength of an illumination ( radiant exitance). More generally, the term ''spectral power distribution'' ...
* Whittle likelihood *
Window function In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the int ...


Notes


References


External links


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