This articlemay be too technical for most readers to understand. Pleasehelp improve it tomake it understandable to non-experts, without removing the technical details.(June 2024) (Learn how and when to remove this message)

Insignal processing, the power spectrum${\displaystyle S_{xx}(f)}$ of acontinuous timesignal${\displaystyle x(t)}$ describes the distribution ofpower into frequency components${\displaystyle f}$ composing that signal.^[1]Fourier analysis shows that any physical signal can be decomposed into a distribution of frequencies over a continuous range, where some of the power may be concentrated at discrete frequencies. The statistical average of the energy or power of any type of signal (includingnoise) as analyzed in terms of its frequency content, is called itsspectral density.

When the energy of the signal is concentrated around a finite time interval, especially if its total energy is finite, one may compute theenergy spectral density. More commonly used is thepower spectral density (PSD, or simplypower spectrum), which applies to signals existing overall 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 PSD then refers to the spectral power distribution that would be found, since the total energy of such a signal over all time would generally be infinite.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${\displaystyle x^2(t)}$ over the time domain, as dictated byParseval's theorem.^[1]

The spectrum of a physical process${\displaystyle x(t)}$ often contains essential information about the nature of${\displaystyle x}$. For instance, thepitch andtimbre of a musical instrument can be determined from a spectral analysis. Thecolor of a light source is determined by the spectrum of the electromagnetic wave's electric field${\displaystyle E(t)}$ as it oscillates at an extremely high frequency. Obtaining a spectrum from time series data such as these involves theFourier transform, and generalizations based on Fourier analysis. In many cases the time domain is not directly captured in practice, such as when adispersive prism is used to obtain a spectrum of light in aspectrograph, 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 instatistical signal processing and in the statistical study ofstochastic processes, as well as in many other branches ofphysics andengineering. Typically the process is a function of time, but one can similarly discuss data in the spatial domain being decomposed in terms ofspatial frequency.^[1]

Inphysics, the signal might be a wave, such as anelectromagnetic wave, anacoustic wave, or the vibration of a mechanism. Thepower spectral density (PSD) of the signal describes thepower density of the signal as a function of frequency. Power spectral density is commonly expressed in theSI unitwatt perhertz (W/Hz).^[2]

When a signal is defined in terms of only avoltage varying in time, for instance, there is no specific power associated with a given voltage. In this case "power" is simply reckoned in terms of the square of the signal, as this would always beproportional to the actual power delivered by that signal into a givenimpedance. So one might use the unit V²⋅Hz⁻¹ for the PSD.Energy spectral density (ESD) would have the unit V²⋅s⋅Hz⁻¹, sinceenergy is power multiplied by time (e.g.,watt-hour).^[3]

In the general case, the unit of PSD will be the ratio of unit of variance per unit of frequency; so, for example, a series of displacement values (in meters) over time (in seconds) will have PSD with the unit m²/Hz.In the analysis of randomvibrations, the unitg₀²⋅Hz⁻¹ may be used for the PSD ofacceleration, whereg₀ denotesstandard gravity.^[4]

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

A PSD can be either aone-sided function of only positive frequencies or atwo-sided function of both positive andnegative frequencies but with only half the amplitude. Noise PSDs are generally one-sided in engineering and two-sided in physics.^[5]

Insignal processing, theenergy of a signal${\displaystyle x(t)}$ is given by$${\displaystyle E\triangleq {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left|x(t)\right|}^{2}dt.}$$Assuming the total energy is finite (i.e.${\displaystyle x(t)}$ is asquare-integrable function) allows applyingParseval's theorem (orPlancherel's theorem).^[6] That is,$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|x(t){|}^{2}dt={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left|\hat{x}(f)\right|}^{2}df,}$$where$${\displaystyle \hat{x}(f)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-i2\pi ft}x(t)dt,}$$is theFourier transform of${\displaystyle x(t)}$ atfrequency${\displaystyle f}$ (inHz).^[7] The theorem also holds true in the discrete-time cases. Since the integral on the left-hand side is the energy of the signal, the value of${\displaystyle {\left|\hat{x}(f)\right|}^{2}df}$ can be interpreted as adensity function multiplied by an infinitesimally small frequency interval, describing the energy contained in the signal at frequency${\displaystyle f}$ in the frequency interval${\displaystyle f+df}$.

Therefore, theenergy spectral density of${\displaystyle x(t)}$ is defined as^[8]

${\displaystyle {\overline{S}}_{xx}(f)\triangleq {\left|\hat{x}(f)\right|}^2}$

Eq.1

The function${\displaystyle {\overline{S}}_{xx}(f)}$ and theautocorrelation of${\displaystyle x(t)}$ form a Fourier transform pair, a result also known as theWiener–Khinchin theorem (see alsoPeriodogram).

As a physical example of how one might measure the energy spectral density of a signal, suppose${\displaystyle V(t)}$ represents thepotential (involts) of an electrical pulse propagating along atransmission line ofimpedance${\displaystyle Z}$, and suppose the line is terminated with amatched resistor (so that all of the pulse energy is delivered to the resistor and none is reflected back). ByOhm's law, the power delivered to the resistor at time${\displaystyle t}$ is equal to${\displaystyle V(t{)}^2/Z}$, so the total energy is found by integrating${\displaystyle V(t{)}^2/Z}$ with respect to time over the duration of the pulse. To find the value of the energy spectral density${\displaystyle {\overline{S}}_{xx}(f)}$ at frequency${\displaystyle f}$, one could insert between the transmission line and the resistor abandpass filter which passes only a narrow range of frequencies (${\displaystyle \mathrm{\Delta }f}$, say) near the frequency of interest and then measure the total energy${\displaystyle E(f)}$ dissipated across the resistor. The value of the energy spectral density at${\displaystyle f}$ is then estimated to be${\displaystyle E(f)/\mathrm{\Delta }f}$. In this example, since the power${\displaystyle V(t{)}^2/Z}$ has the unit V²⋅Ω⁻¹, the energy${\displaystyle E(f)}$ has the unit V²⋅s⋅Ω⁻¹ =J, and hence the estimate${\displaystyle E(f)/\mathrm{\Delta }f}$ of the energy spectral density has the unit J⋅Hz⁻¹. In many situations, it is common to omit the step of dividing by${\displaystyle Z}$ so that the energy spectral density instead has the unit V²⋅s·Hz⁻¹.

This definition generalizes in a straightforward manner to a discrete signal with acountably infinite number of values${\displaystyle x_n}$ such as a signal sampled at discrete times${\displaystyle t_n=t_0+(n\mathrm{\Delta }t)}$:$${\displaystyle {\overline{S}}_{xx}(f)=\underset{N\to \mathrm{\infty }}{lim}(\mathrm{\Delta }t{)}^2\underset{{\left|{\hat{x}}_d(f)\right|}^2}{\underbrace{{\left|\sum _{n=-N}^N{x}_n{e}^{-i2\pi fn\mathrm{\Delta }t}\right|}^2}},}$$where${\displaystyle {\hat{x}}_d(f)}$ is thediscrete-time Fourier transform of${\displaystyle x_n.}$  The sampling interval${\displaystyle \mathrm{\Delta }t}$ is needed to keep the correct physical unit and to ensure that we recover the continuous case in the limit${\displaystyle \mathrm{\Delta }t\to 0}$. But in the mathematical sciences the interval is often set to 1, which simplifies the results at the expense of generality. (Also seeNormalized frequency (unit))

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 thepower spectral density (PSD) which exists forstationary processes; this describes how thepower 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 thevariance of a function over time${\displaystyle x(t)}$ (or over another independent variable), and using an analogy with electrical signals (among other physical processes), it is customary to refer to it as thepower spectrum even when there is no physical power involved. If one were to create a physicalvoltage source which followed${\displaystyle x(t)}$ and applied it to the terminals of a oneohmresistor, then indeed the instantaneous power dissipated in that resistor would be given by${\displaystyle x^2(t)}$watts.

The average power${\displaystyle P}$ of a signal${\displaystyle x(t)}$ over all time is therefore given by the following time average, where the period${\displaystyle T}$ is centered about some arbitrary time${\displaystyle t=t_0}$:$${\displaystyle P=\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_{t_0-T/2}^{t_0+T/2}{\left|x(t)\right|}^{2}dt}$$

Whenever it is more convenient to deal with time limits in the signal itself rather than time limits in the bounds of the integral, the average power can also be written as$${\displaystyle P=\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left|x_T(t)\right|}^{2}dt,}$$where${\displaystyle x_T(t)=x(t)w_T(t)}$ and${\displaystyle w_T(t)}$ is unity within the arbitrary period and zero elsewhere.

When${\displaystyle P}$ is non-zero, the integral must grow to infinity at least as fast as${\displaystyle T}$ does. That is the reason why we cannot use the energy of the signal, which is that diverging integral.

In analyzing the frequency content of the signal${\displaystyle x(t)}$, one might like to compute the ordinary Fourier transform${\displaystyle \hat{x}(f)}$; however, for many signals of interest the ordinary Fourier transform does not formally exist.^{[nb 1]} However, under suitable conditions, certain generalizations of the Fourier transform (e.g. theFourier–Stieltjes transform) still adhere toParseval's theorem. As such,$${\displaystyle P=\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|{\hat{x}}_T(f){|}^{2}df,}$$where the integrand defines thepower spectral density:^[9][10]

${\displaystyle S_{xx}(f)=\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}|{\hat{x}}_T(f){|}^2}$

Eq.2

Theconvolution theorem then allows regarding${\displaystyle |{\hat{x}}_T(f){|}^2}$ as theFourier transform of the timeconvolution of${\displaystyle x_T^{\ast }(-t)}$ and${\displaystyle x_T(t)}$, where * represents the complex conjugate.

In order to deduce Eq.2, we will find an expression for${\displaystyle [{\hat{x}}_T(f){]}^{\ast }}$ that will be useful for the purpose. In fact, we will demonstrate that${\displaystyle [{\hat{x}}_T(f){]}^{\ast }=\mathcal{F}\left\{x_T^{\ast }(-t)\right\}}$. Start by noting thatand let${\displaystyle z=-t}$, so that${\displaystyle z\to -\mathrm{\infty }}$ when${\displaystyle t\to \mathrm{\infty }}$ and vice versa. Sowhere, in the last line, use has been made of${\displaystyle z}$ and${\displaystyle t}$ being dummy variables.So, we haveq.e.d.

Now, let's demonstrate eq.2 by using the demonstrated identity. In addition, we will make the substitution${\displaystyle u(t)=x_T^{\ast }(-t)}$. In this way, we have:where the convolution theorem has been used when passing from the 3rd to the 4th line.

Now, if we divide the time convolution above by the period${\displaystyle T}$ and take the limit as${\displaystyle T\to \mathrm{\infty }}$, it becomes theautocorrelation function of the non-windowed signal${\displaystyle x(t)}$, which is denoted as${\displaystyle R_{xx}(\tau )}$, provided that${\displaystyle x(t)}$ isergodic, which is true in most, but not all, practical cases.^{[nb 2]}$${\displaystyle \underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\left|{\hat{x}}_T(f)\right|}^2={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left[\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}_T^{\ast }(t-\tau )x_T(t)dt\right]e^{-i2\pi f\tau }d\tau ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{R}_{xx}(\tau )e^{-i2\pi f\tau }d\tau }$$

Assuming the ergodicity of${\displaystyle x(t)}$, the power spectral density can be found once more as the Fourier transform of theautocorrelation function${\displaystyle R_{xx}}$, a property known as theWiener–Khinchin theorem.^[11]

${\displaystyle S_{xx}(f)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{R}_{xx}(\tau )e^{-i2\pi f\tau }d\tau ={\hat{R}}_{xx}(f)}$

Eq.3

Many authors use this relationship to define the power spectral density in terms of the autocorrelation function instead of the Fourier transform of the signal as we have done.^[12]

The power of the signal in a given frequency band${\displaystyle [f_1,f_2]}$, where, can be calculated by integrating over frequency. Since${\displaystyle S_{xx}(-f)=S_{xx}(f)}$, 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):$${\displaystyle P_{\mathsf{\text{band-limited}}}=2{\int }_{f_1}^{f_2}{S}_{xx}(f)df}$$More generally, similar techniques may be used to estimate a time-varying spectral density. In this case the time interval${\displaystyle T}$ is finite rather than approaching infinity. This results in decreased spectral coverage and resolution since frequencies of less than${\displaystyle 1/T}$ are not sampled, and results at frequencies which are not an integer multiple of${\displaystyle 1/T}$ 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 tostatistical ensembles of realizations of${\displaystyle x(t)}$ evaluated over the specified time window.

Just as with the energy spectral density, the definition of the power spectral density can be generalized todiscrete time variables${\displaystyle x_n}$. As before, we can consider a window of${\displaystyle -N\le n\le N}$ with the signal sampled at discrete times${\displaystyle t_n=t_0+(n\mathrm{\Delta }t)}$ for a total measurement period${\displaystyle T=(2N+1)\mathrm{\Delta }t}$.$${\displaystyle S_{xx}(f)=\underset{N\to \mathrm{\infty }}{lim}\frac{(\mathrm{\Delta }t{)}^2}{T}{\left|\sum _{n=-N}^N{x}_n{e}^{-i2\pi fn\mathrm{\Delta }t}\right|}^2}$$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${\displaystyle N}$ (and thus${\displaystyle T}$) 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 aperiodogram. This periodogram converges to the true PSD as the number of estimates as well as the averaging time interval${\displaystyle T}$ approach infinity.^[13]

If two signals both possess power spectral densities, then thecross-spectral density can similarly be calculated; as the PSD is related to the autocorrelation, so is the cross-spectral density related to thecross-correlation.

Some properties of the PSD include:^[14]

Given two signals${\displaystyle x(t)}$ and${\displaystyle y(t)}$, each of which possess power spectral densities${\displaystyle S_{xx}(f)}$ and${\displaystyle S_{yy}(f)}$, it is possible to define across power spectral density (CPSD) orcross spectral density (CSD). To begin, let us consider the average power of such a combined signal.

Using the same notation and methods as used for the power spectral density derivation, we exploit Parseval's theorem and obtainwhere, again, the contributions of${\displaystyle S_{xx}(f)}$ and${\displaystyle S_{yy}(f)}$ are already understood. Note that${\displaystyle S_{xy}^{\ast }(f)=S_{yx}(f)}$, so the full contribution to the cross power is, generally, from twice the real part of either individualCPSD. 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${\displaystyle T\to \mathrm{\infty }}$ becomes the Fourier transform of across-correlation function.^[16]where${\displaystyle R_{xy}(\tau )}$ is thecross-correlation of${\displaystyle x(t)}$ with${\displaystyle y(t)}$ and${\displaystyle R_{yx}(\tau )}$ is the cross-correlation of${\displaystyle y(t)}$ with${\displaystyle x(t)}$. In light of this, the PSD is seen to be a special case of the CSD for${\displaystyle x(t)=y(t)}$. If${\displaystyle x(t)}$ and${\displaystyle y(t)}$ are real signals (e.g. voltage or current), their Fourier transforms${\displaystyle \hat{x}(f)}$ and${\displaystyle \hat{y}(f)}$ are usually restricted to positive frequencies by convention. Therefore, in typical signal processing, the fullCPSD is just one of theCPSDs scaled by a factor of two.$${\displaystyle {\mathrm{CPSD}}_{\text{Full}}=2S_{xy}(f)=2S_{yx}(f)}$$

For discrete signalsx_n andy_n, the relationship between the cross-spectral density and the cross-covariance is$${\displaystyle S_{xy}(f)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{R}_{xy}({\tau }_n)e^{-i2\pi f{\tau }_n}\mathrm{\Delta }\tau }$$

The goal of spectral density estimation is toestimate the spectral density of arandom signal from a sequence of time samples. Depending on what is known about the signal, estimation techniques can involveparametric ornon-parametric approaches, and may be based on time-domain or frequency-domain analysis. For example, a common parametric technique involves fitting the observations to anautoregressive model. A common non-parametric technique is theperiodogram.

The spectral density is usually estimated usingFourier transform methods (such as theWelch method), but other techniques such as themaximum entropy method can also be used.

• Thespectral centroid of a signal is the midpoint of its spectral density function, i.e. the frequency that divides the distribution into two equal parts.
• Thespectral edge frequency (SEF), usually expressed as "SEFx", represents thefrequency below whichx 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 inEEG monitoring, in which case SEF has variously been used to estimate the depth ofanesthesia and stages ofsleep.^[17][18]
• Aspectral envelope is theenvelope curve of the spectrum density. It describes one point in time (one window, to be precise). For example, inremote sensing using aspectrometer, the spectral envelope of a feature is the boundary of itsspectral properties, as defined by the range of brightness levels in each of thespectral bands 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 aspectrogram. This is the basis of a number of spectral analysis techniques such as theshort-time Fourier transform andwavelets.
• A "spectrum" generally means the power spectral density, as discussed above, which depicts the distribution of signal content over frequency. Fortransfer functions (e.g.,Bode plot,chirp) the complete frequency response may be graphed in two parts: power versus frequency andphase versus frequency—thephase spectral density,phase spectrum, orspectral phase. Less commonly, the two parts may be thereal and imaginary parts of the transfer function. This is not to be confused with thefrequency response of a transfer function, which also includes a phase (or equivalently, a real and imaginary part) as a function of frequency. The time-domainimpulse response${\displaystyle h(t)}$ 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 theautocorrelation) forcing the Fourier transform to be real-valued. SeeUltrashort pulse#Spectral phase,phase noise,group delay.
• Sometimes one encounters anamplitude spectral density (ASD), which is the square root of the PSD; the ASD of a voltage signal has the unit V⋅Hz^−1/2.^[19] This is useful when theshape 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.

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 (perceived ascolor), musical notes (perceived aspitch),radio/TV (specified by their frequency, or sometimeswavelength) 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 asine wave component. And additionally there may be peaks corresponding toharmonics of a fundamental peak, indicating a periodic signal which isnot 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 anotch filter.

The concept and use of the power spectrum of a signal is fundamental inelectrical engineering, especially inelectronic communication systems, includingradio communications,radars, and related systems, plus passiveremote sensing technology. Electronic instruments calledspectrum analyzers are used to observe and measure thepower spectra of signals.

The spectrum analyzer measures the magnitude of theshort-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.

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.

• Bispectrum
• Brightness temperature
• Colors of noise
• Least-squares spectral analysis
• Noise spectral density
• Spectral density estimation
• Spectral efficiency
• Spectral leakage
• Spectral power distribution
• Whittle likelihood
• Window function

• Birolini, Alessandro (2007).Reliability Engineering. Berlin ; New York: Springer Science & Business Media.ISBN 978-3-540-49388-4.
• Brown, Robert Grover; Hwang, Patrick Y. C. (1997).Introduction to Random Signals and Applied Kalman Filtering with Matlab Exercises and Solutions. New York: Wiley-Liss.ISBN 978-0-471-12839-7.
• Davenport, Wilbur B. (Jr); Root, William L. (1987).An Introduction to the Theory of Random Signals and Noise. New York: Wiley-IEEE Press.ISBN 978-0-87942-235-6.
• Imtiaz, Syed Anas; Rodriguez-Villegas, Esther (2014)."A Low Computational Cost Algorithm for REM Sleep Detection Using Single Channel EEG".Annals of Biomedical Engineering.42 (11):2344–59.doi:10.1007/s10439-014-1085-6.PMC 4204008.PMID 25113231.
• Iranmanesh, Saam;Rodriguez-Villegas, Esther (2017). "An Ultralow-Power Sleep Spindle Detection System on Chip".IEEE Transactions on Biomedical Circuits and Systems.11 (4):858–866.Bibcode:2017ITBC...11..858I.doi:10.1109/TBCAS.2017.2690908.hdl:10044/1/46059.PMID 28541914.S2CID 206608057.
• Maral, Gerard (2004).VSAT Networks. West Sussex, England ; Hoboken, NJ: Wiley.ISBN 978-0-470-86684-9.
• Miller, Scott; Childers, Donald (2012).Probability and Random Processes. Boston, MA: Academic Press.ISBN 978-0-12-386981-4.OCLC 696092052.
• Norton, M. P.; Karczub, D. G. (2003).Fundamentals of Noise and Vibration Analysis for Engineers. Cambridge: Cambridge University Press.ISBN 978-0-521-49913-2.
• Oppenheim, Alan V.; Verghese, George C. (2016).Signals, Systems & Inference. Boston: Pearson.ISBN 978-0-13-394328-3.
• Risken, Hannes; Frank, Till (1996).The Fokker-Planck Equation. New York: Springer Science & Business Media.ISBN 978-3-540-61530-9.
• Stein, Jonathan Y. (2000).Digital Signal Processing. New York Weinheim: Wiley-Interscience.ISBN 978-0-471-29546-4.

• Power Spectral Density Matlab scripts

• Frequency-domain analysis
• Signal processing
• Waves
• Spectroscopy
• Scattering
• Fourier analysis
• Radio spectrum
• Spectrum (physical sciences)

• CS1: unfit URL
• Articles with short description
• Short description matches Wikidata
• Use American English from March 2019
• All Wikipedia articles written in American English
• Wikipedia articles that are too technical from June 2024
• All articles that are too technical