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Spectral density

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about signal processing and relation of spectra to time-series. For further applications in the physical sciences, see <a href="Spectrum_(physical_sciences)" title="Spectrum (physical sciences)">Spectrum (physical sciences)</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">"Spectral power density" redirects here; not to be confused with <a href="Spectral_power" class="mw-redirect" title="Spectral power">Spectral power</a>.</div>

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In <a href="Signal_processing" title="Signal processing">signal processing</a>, the power spectrum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{xx}(f)}">

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<annotation encoding="application/x-tex">{\displaystyle S_{xx}(f)}</annotation>

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</math><img src="./cab836aa57df700d834cc0f736e9f81152fa470e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.626ex; height:2.843ex;" alt="{\displaystyle S_{xx}(f)}" loading="lazy"> of a <a href="Continuous_time" class="mw-redirect" title="Continuous time">continuous time</a> <a href="Signal" title="Signal">signal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

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</math><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"> describes the distribution of <a href="Power_(physics)" title="Power (physics)">power</a> into frequency components <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> composing that signal.<a href="#cite_note-P_Stoica-1">[1]</a> <a href="Fourier_analysis" title="Fourier analysis">Fourier analysis</a> 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 (including <a href="Noise_(electronics)" title="Noise (electronics)">noise</a>) as analyzed in terms of its frequency content, is called its spectral density.

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 (PSD, 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 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. <a href="Summation" title="Summation">Summation</a> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}(t)}">

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</math><img src="./6f58eb8c006ec38666f7f0bb9106d809a3903345.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.033ex; height:3.176ex;" alt="{\displaystyle x^{2}(t)}" loading="lazy"> over the time domain, as dictated by <a href="Parseval's_theorem" title="Parseval's theorem">Parseval's theorem</a>.<a href="#cite_note-P_Stoica-1">[1]</a>

The spectrum of a physical process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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</math><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"> often contains essential information about the nature of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">. For instance, the <a href="Pitch_(music)" title="Pitch (music)">pitch</a> and <a href="Timbre" title="Timbre">timbre</a> of a musical instrument can be determined from a spectral analysis. The <a href="Color" title="Color">color</a> of a light source is determined by the spectrum of the electromagnetic wave's electric field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(t)}">

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</math><img src="./bf5be62d9f63f5df52e8bc156f950d41e131d99f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.425ex; height:2.843ex;" alt="{\displaystyle E(t)}" loading="lazy"> as it oscillates at an extremely high frequency. Obtaining a spectrum from time series data such as these involves the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a>, and generalizations based on Fourier analysis. In many cases the time domain is not directly captured in practice, such as when a <a href="Dispersive_prism" title="Dispersive prism">dispersive prism</a> is used to obtain a spectrum of light in a <a href="Spectrograph" class="mw-redirect" title="Spectrograph">spectrograph</a>, 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 <a href="Statistical_signal_processing" class="mw-redirect" title="Statistical signal processing">statistical signal processing</a> and in the statistical study of <a href="Stochastic_process" title="Stochastic process">stochastic processes</a>, as well as in many other branches of <a href="Physics" title="Physics">physics</a> and <a href="Engineering" title="Engineering">engineering</a>. Typically the process is a function of time, but one can similarly discuss data in the spatial domain being decomposed in terms of <a href="Spatial_frequency" title="Spatial frequency">spatial frequency</a>.<a href="#cite_note-P_Stoica-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Units">Units</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Fourier_transform#Units" title="Fourier transform">Fourier transform § Units</a></div>

In <a href="Physics" title="Physics">physics</a>, the signal might be a wave, such as an <a href="Electromagnetic_wave" class="mw-redirect" title="Electromagnetic wave">electromagnetic wave</a>, an <a href="Sound_wave" class="mw-redirect" title="Sound wave">acoustic wave</a>, or the vibration of a mechanism. The power spectral density (PSD) of the signal describes the <a href="Power_(physics)" title="Power (physics)">power</a> density of the signal as a function of frequency. Power spectral density is commonly expressed in the <a href="SI_unit" class="mw-redirect" title="SI unit">SI unit</a> <a href="Watt" title="Watt">watt</a> per <a href="Hertz" title="Hertz">hertz</a> (W/Hz).<a href="#cite_note-FOOTNOTEMaral2004-2">[2]</a>

When a signal is defined in terms of only a <a href="Voltage" title="Voltage">voltage</a> 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 be proportional to the actual power delivered by that signal into a given <a href="Electrical_impedance" title="Electrical impedance">impedance</a>. So one might use the unit V2⋅Hz−1 for the PSD. Energy spectral density (ESD) would have the unit V2⋅s⋅Hz−1, since <a href="Energy_(physics)" class="mw-redirect" title="Energy (physics)">energy</a> is power multiplied by time (e.g., <a href="Watt-hour" class="mw-redirect" title="Watt-hour">watt-hour</a>).<a href="#cite_note-FOOTNOTENortonKarczub2003-3">[3]</a>

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 random <a href="Vibration" title="Vibration">vibrations</a>, the unit g02⋅Hz−1 may be used for the PSD of <a href="Acceleration" title="Acceleration">acceleration</a>, where g0 denotes <a href="Standard_gravity" title="Standard gravity">standard gravity</a>.<a href="#cite_note-FOOTNOTEBirolini200783-4">[4]</a>

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.

<div class="mw-heading mw-heading3"><h3 id="One-sided_vs._two-sided">One-sided vs. two-sided</h3></div>

A PSD can be either a one-sided function of only positive frequencies or a two-sided function of both positive and <a href="Negative_frequency" title="Negative frequency">negative frequencies</a> but with only half the amplitude. Noise PSDs are generally one-sided in engineering and two-sided in physics.<a href="#cite_note-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Energy_spectral_density">Energy spectral density </h3></div>

<div role="note" class="hatnote navigation-not-searchable">"Energy spectral density" redirects here; not to be confused with <a href="Energy_spectrum" class="mw-redirect" title="Energy spectrum">Energy spectrum</a>.</div>

In <a href="Signal_processing" title="Signal processing">signal processing</a>, the <a href="Energy_(signal_processing)" title="Energy (signal processing)">energy</a> of a signal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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</math><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"> is given by

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<annotation encoding="application/x-tex">{\displaystyle E\triangleq \int _{-\infty }^{\infty }\left|x(t)\right|^{2}\ dt.}</annotation>

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Assuming the total energy is finite (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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</math><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"> is a <a href="Square-integrable_function" title="Square-integrable function">square-integrable function</a>) allows applying <a href="Parseval's_theorem" title="Parseval's theorem">Parseval's theorem</a> (or <a href="Plancherel's_theorem" class="mw-redirect" title="Plancherel's theorem">Plancherel's theorem</a>).<a href="#cite_note-FOOTNOTEOppenheimVerghese201660-6">[6]</a> That is,

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<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }|x(t)|^{2}\,dt=\int _{-\infty }^{\infty }\left|{\hat {x}}(f)\right|^{2}\,df,}</annotation>

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where

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<annotation encoding="application/x-tex">{\displaystyle {\hat {x}}(f)=\int _{-\infty }^{\infty }e^{-i2\pi ft}x(t)\ dt,}</annotation>

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is the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

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</math><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"> at <a href="Frequency" title="Frequency">frequency</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> (in <a href="Hz" class="mw-redirect" title="Hz">Hz</a>).<a href="#cite_note-FOOTNOTEStein2000108,_115-7">[7]</a> 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<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|{\hat {x}}(f)\right|^{2}df}">

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<annotation encoding="application/x-tex">{\displaystyle \left|{\hat {x}}(f)\right|^{2}df}</annotation>

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</math><img src="./e9cdfe3d7660f5c2ddae825eeaa9f52e90fd6a6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.26ex; height:3.343ex;" alt="{\displaystyle \left|{\hat {x}}(f)\right|^{2}df}" loading="lazy"> can be interpreted as a <a href="Density_function" class="mw-redirect" title="Density function">density function</a> multiplied by an infinitesimally small frequency interval, describing the energy contained in the signal at frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> in the frequency interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f+df}">

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<annotation encoding="application/x-tex">{\displaystyle f+df}</annotation>

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</math><img src="./4c1b25aa5ed1a0704987e497cd22a5a27ba8c8a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.613ex; height:2.509ex;" alt="{\displaystyle f+df}" loading="lazy">.

Therefore, the energy spectral density of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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</math><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"> is defined as<a href="#cite_note-FOOTNOTEOppenheimVerghese201614-8">[8]</a>

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</style><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {S}}_{xx}(f)\triangleq \left|{\hat {x}}(f)\right|^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\bar {S}}_{xx}(f)\triangleq \left|{\hat {x}}(f)\right|^{2}}</annotation>

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The function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {S}}_{xx}(f)}">

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</math><img src="./f43c62387fe5b01596533b3e850be9843e4e4618.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.773ex; height:3.176ex;" alt="{\displaystyle {\bar {S}}_{xx}(f)}" loading="lazy"> and the <a href="Autocorrelation" title="Autocorrelation">autocorrelation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

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</math><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"> form a Fourier transform pair, a result also known as the <a href="Wiener%E2%80%93Khinchin_theorem" title="Wiener–Khinchin theorem">Wiener–Khinchin theorem</a> (see also <a href="Periodogram#Definition" title="Periodogram">Periodogram</a>).

As a physical example of how one might measure the energy spectral density of a signal, suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V(t)}</annotation>

</semantics>

</math><img src="./383b47023708ac9df0e198448491048ec7acb2cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.436ex; height:2.843ex;" alt="{\displaystyle V(t)}" loading="lazy"> represents the <a href="Electric_potential" title="Electric potential">potential</a> (in <a href="Volt" title="Volt">volts</a>) of an electrical pulse propagating along a <a href="Transmission_line" title="Transmission line">transmission line</a> of <a href="Electrical_impedance" title="Electrical impedance">impedance</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

</semantics>

</math><img src="./1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy">, and suppose the line is terminated with a <a href="Impedance_matching" title="Impedance matching">matched</a> resistor (so that all of the pulse energy is delivered to the resistor and none is reflected back). By <a href="Ohm's_law" title="Ohm's law">Ohm's law</a>, the power delivered to the resistor at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(t)^{2}/Z}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V(t)^{2}/Z}</annotation>

</semantics>

</math><img src="./275d111396d456efa3ad9b9dc2e55de79e605bef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.333ex; height:3.176ex;" alt="{\displaystyle V(t)^{2}/Z}" loading="lazy">, so the total energy is found by integrating <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(t)^{2}/Z}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V(t)^{2}/Z}</annotation>

</semantics>

</math><img src="./275d111396d456efa3ad9b9dc2e55de79e605bef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.333ex; height:3.176ex;" alt="{\displaystyle V(t)^{2}/Z}" loading="lazy"> with respect to time over the duration of the pulse. To find the value of the energy spectral density <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {S}}_{xx}(f)}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {S}}_{xx}(f)}</annotation>

</semantics>

</math><img src="./f43c62387fe5b01596533b3e850be9843e4e4618.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.773ex; height:3.176ex;" alt="{\displaystyle {\bar {S}}_{xx}(f)}" loading="lazy"> at frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">, one could insert between the transmission line and the resistor a <a href="Bandpass_filter" class="mw-redirect" title="Bandpass filter">bandpass filter</a> which passes only a narrow range of frequencies (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f}</annotation>

</semantics>

</math><img src="./ff38db24bc80b80a0a1ecfce75f3dca3fc79d54e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.214ex; height:2.509ex;" alt="{\displaystyle \Delta f}" loading="lazy">, say) near the frequency of interest and then measure the total energy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(f)}</annotation>

</semantics>

</math><img src="./d733654dab86d76c9ed9ba7f4473250dfea33d3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.864ex; height:2.843ex;" alt="{\displaystyle E(f)}" loading="lazy"> dissipated across the resistor. The value of the energy spectral density at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is then estimated to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)/\Delta f}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(f)/\Delta f}</annotation>

</semantics>

</math><img src="./2e6da71905d2242140e6720b4c263f1fe8736ecd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.24ex; height:2.843ex;" alt="{\displaystyle E(f)/\Delta f}" loading="lazy">. In this example, since the power <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(t)^{2}/Z}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V(t)^{2}/Z}</annotation>

</semantics>

</math><img src="./275d111396d456efa3ad9b9dc2e55de79e605bef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.333ex; height:3.176ex;" alt="{\displaystyle V(t)^{2}/Z}" loading="lazy"> has the unit V2⋅Ω−1, the energy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(f)}</annotation>

</semantics>

</math><img src="./d733654dab86d76c9ed9ba7f4473250dfea33d3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.864ex; height:2.843ex;" alt="{\displaystyle E(f)}" loading="lazy"> has the unit V2⋅s⋅Ω−1 = <a href="Joule" title="Joule">J</a>, and hence the estimate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)/\Delta f}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(f)/\Delta f}</annotation>

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</math><img src="./2e6da71905d2242140e6720b4c263f1fe8736ecd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.24ex; height:2.843ex;" alt="{\displaystyle E(f)/\Delta f}" loading="lazy"> of the energy spectral density has the unit J⋅Hz−1. In many situations, it is common to omit the step of dividing by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

</semantics>

</math><img src="./1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> so that the energy spectral density instead has the unit V2⋅Hz−1.

This definition generalizes in a straightforward manner to a discrete signal with a <a href="Countably_infinite" class="mw-redirect" title="Countably infinite">countably infinite</a> number of values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation>

</semantics>

</math><img src="./7c5ea190699149306d242b70439e663559e3ffbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}" loading="lazy"> such as a signal sampled at discrete times <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n}=t_{0}+(n\,\Delta t)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{n}=t_{0}+(n\,\Delta t)}</annotation>

</semantics>

</math><img src="./06f68e578977964c65c052bda78bcdf7e0e5af40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.257ex; height:2.843ex;" alt="{\displaystyle t_{n}=t_{0}+(n\,\Delta t)}" loading="lazy">:

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</munder>

<msup>

</mrow>

</msup>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

</munder>

</mrow>

<msup>

<mrow>

<mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {S}}_{xx}(f)=\lim _{N\to \infty }(\Delta t)^{2}\underbrace {\left|\sum _{n=-N}^{N}x_{n}e^{-i2\pi fn\,\Delta t}\right|^{2}} _{\left|{\hat {x}}_{d}(f)\right|^{2}},}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {x}}_{d}(f)}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {x}}_{d}(f)}</annotation>

</semantics>

</math><img src="./4d7bb68246cf565870b81b6236217966b9ca1c81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.51ex; height:2.843ex;" alt="{\displaystyle {\hat {x}}_{d}(f)}" loading="lazy"> is the <a href="Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">discrete-time Fourier transform</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n}.}</annotation>

</semantics>

</math><img src="./3ee2d0d918f9085dbdaaf812726427a039846070.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.195ex; height:2.009ex;" alt="{\displaystyle x_{n}.}" loading="lazy">  The sampling interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation>

</semantics>

</math><img src="./8c28867ecd34e2caed12cf38feadf6a81a7ee542.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}" loading="lazy"> is needed to keep the correct physical unit and to ensure that we recover the continuous case in the limit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t\to 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta t\to 0}</annotation>

</semantics>

</math><img src="./e5c59b4f951f4bcd6954ba444dcd840185fe058a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.552ex; height:2.176ex;" alt="{\displaystyle \Delta t\to 0}" loading="lazy">. But in the mathematical sciences the interval is often set to 1, which simplifies the results at the expense of generality. (Also see <a href="Normalized_frequency_(unit)" class="mw-redirect" title="Normalized frequency (unit)">Normalized frequency (unit)</a>)

<div class="mw-heading mw-heading3"><h3 id="Power_spectral_density">Power spectral density</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Spectral_power_distribution" title="Spectral power distribution">spectral power distribution</a>.</div>

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 <a href="Stationary_process" title="Stationary process">stationary processes</a>; this describes how the <a href="Power_(physics)" title="Power (physics)">power</a> 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 <a href="Variance" title="Variance">variance</a> of a function over time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

</semantics>

</math><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"> (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 <a href="Voltage" title="Voltage">voltage</a> source which followed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

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</math><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"> and applied it to the terminals of a one <a href="Ohm" title="Ohm">ohm</a> <a href="Resistor" title="Resistor">resistor</a>, then indeed the instantaneous power dissipated in that resistor would be given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}(t)}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle x^{2}(t)}</annotation>

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</math><img src="./6f58eb8c006ec38666f7f0bb9106d809a3903345.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.033ex; height:3.176ex;" alt="{\displaystyle x^{2}(t)}" loading="lazy"> <a href="Watt" title="Watt">watts</a>.

The average power <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> of a signal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

</semantics>

</math><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"> over all time is therefore given by the following time average, where the period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> is centered about some arbitrary time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=t_{0}}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle t=t_{0}}</annotation>

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</math><img src="./be6d7492e2d48bf34fdd5dffa189b188c140820c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.832ex; height:2.343ex;" alt="{\displaystyle t=t_{0}}" loading="lazy">:

</mrow>

</munder>

<mfrac>

</mfrac>

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<msub>

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</msub>

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<msub>

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<msup>

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<mrow>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{t_{0}-T/2}^{t_{0}+T/2}\left|x(t)\right|^{2}\,dt}</annotation>

</semantics>

</math>

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

</mrow>

</munder>

<mfrac>

</mfrac>

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</msubsup>

<msup>

<mrow>

<mrow>

<msub>

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</msub>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }\left|x_{T}(t)\right|^{2}\,dt,}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{T}(t)=x(t)w_{T}(t)}">

<msub>

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</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{T}(t)=x(t)w_{T}(t)}</annotation>

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</math><img src="./ecf040c677a347ef0ecbf38bdbe3956e1ca34014.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.147ex; height:2.843ex;" alt="{\displaystyle x_{T}(t)=x(t)w_{T}(t)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{T}(t)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle w_{T}(t)}</annotation>

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</math><img src="./3f71e75940d49cc0bdd1bb79455c8af96870c166.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.702ex; height:2.843ex;" alt="{\displaystyle w_{T}(t)}" loading="lazy"> is unity within the arbitrary period and zero elsewhere.

When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> is non-zero, the integral must grow to infinity at least as fast as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

</semantics>

</math><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy">, one might like to compute the ordinary Fourier transform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {x}}(f)}">

<mover>

</mover>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {x}}(f)}</annotation>

</semantics>

</math><img src="./594eba73af7f4adf231cffd67129dbed8957e7e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle {\hat {x}}(f)}" loading="lazy">; however, for many signals of interest the ordinary Fourier transform does not formally exist.<a href="#cite_note-9">[nb 1]</a> However, under suitable conditions, certain generalizations of the Fourier transform (e.g. the <a href="Fourier_transform#Fourier–Stieltjes_transform_on_measurable_spaces" title="Fourier transform">Fourier–Stieltjes transform</a>) still adhere to <a href="Parseval's_theorem" title="Parseval's theorem">Parseval's theorem</a>. As such,

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }|{\hat {x}}_{T}(f)|^{2}\,df,}</annotation>

</semantics>

</math>

where the integrand defines the power spectral density:<a href="#cite_note-FOOTNOTEOppenheimVerghese2016422–423-10">[9]</a><a href="#cite_note-FOOTNOTEMillerChilders2012429–431-11">[10]</a>

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<annotation encoding="application/x-tex">{\displaystyle S_{xx}(f)=\lim _{T\to \infty }{\frac {1}{T}}|{\hat {x}}_{T}(f)|^{2}\,}</annotation>

</semantics>

</math><img src="./98910d03e3d1cbbab5ac14c87e1ff49634c9aed9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.569ex; height:5.343ex;" alt="{\displaystyle S_{xx}(f)=\lim _{T\to \infty }{\frac {1}{T}}|{\hat {x}}_{T}(f)|^{2}\,}" loading="lazy"></td> <td></td> <td class="nowrap">Eq.2</td></tr></tbody></table>

</div>

The <a href="Convolution_theorem" title="Convolution theorem">convolution theorem</a> then allows regarding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\hat {x}}_{T}(f)|^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle |{\hat {x}}_{T}(f)|^{2}}</annotation>

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</math><img src="./a59cd3d6ccab529c49019269f66388b69b3fbc0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.155ex; height:3.343ex;" alt="{\displaystyle |{\hat {x}}_{T}(f)|^{2}}" loading="lazy"> as the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of the time <a href="Convolution" title="Convolution">convolution</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{T}^{*}(-t)}">

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<annotation encoding="application/x-tex">{\displaystyle x_{T}^{*}(-t)}</annotation>

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</math><img src="./7fb1d39ac427e1dc9610a2e92d8b22e555687b20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.176ex; height:3.009ex;" alt="{\displaystyle x_{T}^{*}(-t)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{T}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle x_{T}(t)}</annotation>

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</math><img src="./509c25f2150c4c401a7d9874bf9b59df21ad5ab7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.368ex; height:2.843ex;" alt="{\displaystyle x_{T}(t)}" loading="lazy">, where * represents the complex conjugate.

In order to deduce Eq.2, we will find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\hat {x}}_{T}(f)]^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle [{\hat {x}}_{T}(f)]^{*}}</annotation>

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</math><img src="./41fa28f6e7948273506683203a9253ae63e5ef39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.155ex; height:2.843ex;" alt="{\displaystyle [{\hat {x}}_{T}(f)]^{*}}" loading="lazy"> that will be useful for the purpose. In fact, we will demonstrate that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\hat {x}}_{T}(f)]^{*}={\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}}">

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<annotation encoding="application/x-tex">{\displaystyle [{\hat {x}}_{T}(f)]^{*}={\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}}</annotation>

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</math><img src="./79a982b983e0f902913897adf500331009768225.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.453ex; height:3.176ex;" alt="{\displaystyle [{\hat {x}}_{T}(f)]^{*}={\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}}" loading="lazy">. Start by noting that

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}&=\int _{-\infty }^{\infty }x_{T}^{*}(-t)e^{-i2\pi ft}dt\end{aligned}}}</annotation>

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</math>

and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=-t}">

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<annotation encoding="application/x-tex">{\displaystyle z=-t}</annotation>

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</math><img src="./58b99b744d1707a023f6bba8fd709b440a32d2cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.834ex; height:2.176ex;" alt="{\displaystyle z=-t}" loading="lazy">, so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\rightarrow -\infty }">

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<annotation encoding="application/x-tex">{\displaystyle z\rightarrow -\infty }</annotation>

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</math><img src="./71c52bb6e2294389a4f8d341711fc48b20df4e4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.834ex; height:2.176ex;" alt="{\displaystyle z\rightarrow -\infty }" loading="lazy"> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\rightarrow \infty }">

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<annotation encoding="application/x-tex">{\displaystyle t\rightarrow \infty }</annotation>

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</math><img src="./b543f76f961ec3f52d78fa3d72c3d87a521dd3a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.777ex; height:2.009ex;" alt="{\displaystyle t\rightarrow \infty }" loading="lazy"> and vice versa. So

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }x_{T}^{*}(-t)e^{-i2\pi ft}dt&=\int _{\infty }^{-\infty }x_{T}^{*}(z)e^{i2\pi fz}\left(-dz\right)\\&=\int _{-\infty }^{\infty }x_{T}^{*}(z)e^{i2\pi fz}dz\\&=\int _{-\infty }^{\infty }x_{T}^{*}(t)e^{i2\pi ft}dt\end{aligned}}}</annotation>

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</math>

where, in the last line, use has been made of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

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</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> being dummy variables.

So, we have

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}&=\int _{-\infty }^{\infty }x_{T}^{*}(-t)e^{-i2\pi ft}dt\\&=\int _{-\infty }^{\infty }x_{T}^{*}(t)e^{i2\pi ft}dt\\&=\int _{-\infty }^{\infty }x_{T}^{*}(t)[e^{-i2\pi ft}]^{*}dt\\&=\left[\int _{-\infty }^{\infty }x_{T}(t)e^{-i2\pi ft}dt\right]^{*}\\&=\left[{\mathcal {F}}\left\{x_{T}(t)\right\}\right]^{*}\\&=\left[{\hat {x}}_{T}(f)\right]^{*}\end{aligned}}}</annotation>

</semantics>

</math>

q.e.d.

Now, let's demonstrate eq.2 by using the demonstrated identity. In addition, we will make the substitution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(t)=x_{T}^{*}(-t)}">

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<annotation encoding="application/x-tex">{\displaystyle u(t)=x_{T}^{*}(-t)}</annotation>

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</math><img src="./ce18a6c1feadfb3a0bfba270a7abfcb8fbd053ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.253ex; height:3.009ex;" alt="{\displaystyle u(t)=x_{T}^{*}(-t)}" loading="lazy">. In this way, we have:

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left|{\hat {x}}_{T}(f)\right|^{2}&=[{\hat {x}}_{T}(f)]^{*}\cdot {\hat {x}}_{T}(f)\\&={\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}\cdot {\mathcal {F}}\left\{x_{T}(t)\right\}\\&={\mathcal {F}}\left\{u(t)\right\}\cdot {\mathcal {F}}\left\{x_{T}(t)\right\}\\&={\mathcal {F}}\left\{u(t)\mathbin {\mathbf {*} } x_{T}(t)\right\}\\&=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }u(\tau -t)x_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau \\&=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }\ d\tau ,\end{aligned}}}</annotation>

</semantics>

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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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> and take the limit as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\rightarrow \infty }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle T\rightarrow \infty }</annotation>

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</math><img src="./e25dc989701049391fbd92c5a4daa9bb15db65a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.574ex; height:2.176ex;" alt="{\displaystyle T\rightarrow \infty }" loading="lazy">, it becomes the <a href="Autocorrelation" title="Autocorrelation">autocorrelation</a> function of the non-windowed signal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

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</math><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy">, which is denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{xx}(\tau )}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle R_{xx}(\tau )}</annotation>

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</math><img src="./595461d5cce972e6ef30d5e8903fcef2313223ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.888ex; height:2.843ex;" alt="{\displaystyle R_{xx}(\tau )}" loading="lazy">, provided that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

</semantics>

</math><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"> is <a href="Ergodic" class="mw-redirect" title="Ergodic">ergodic</a>, which is true in most, but not all, practical cases.<a href="#cite_note-12">[nb 2]</a>

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</munder>

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<annotation encoding="application/x-tex">{\displaystyle \lim _{T\to \infty }{\frac {1}{T}}\left|{\hat {x}}_{T}(f)\right|^{2}=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }\ d\tau =\int _{-\infty }^{\infty }R_{xx}(\tau )e^{-i2\pi f\tau }d\tau }</annotation>

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Assuming the ergodicity of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

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</math><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy">, the power spectral density can be found once more as the Fourier transform of the <a href="Autocorrelation_function" class="mw-redirect" title="Autocorrelation function">autocorrelation function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{xx}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle R_{xx}}</annotation>

</semantics>

</math><img src="./44c57d170e7cb2b17a74d1fec2427a2a26836b39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.877ex; height:2.509ex;" alt="{\displaystyle R_{xx}}" loading="lazy">, a property known as the <a href="Wiener%E2%80%93Khinchin_theorem" title="Wiener–Khinchin theorem">Wiener–Khinchin theorem</a>.<a href="#cite_note-FOOTNOTEMillerChilders2012433-13">[11]</a>

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<annotation encoding="application/x-tex">{\displaystyle S_{xx}(f)=\int _{-\infty }^{\infty }R_{xx}(\tau )e^{-i2\pi f\tau }\,d\tau ={\hat {R}}_{xx}(f)}</annotation>

</semantics>

</math><img src="./da9c6a2f23b198e55d8971514264d1ec3e24f42a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.993ex; height:6.009ex;" alt="{\displaystyle S_{xx}(f)=\int _{-\infty }^{\infty }R_{xx}(\tau )e^{-i2\pi f\tau }\,d\tau ={\hat {R}}_{xx}(f)}" loading="lazy"></td> <td></td> <td class="nowrap">Eq.3</td></tr></tbody></table>

</div>

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.<a href="#cite_note-14">[12]</a>

The power of the signal in a given frequency band <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [f_{1},f_{2}]}">

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<annotation encoding="application/x-tex">{\displaystyle [f_{1},f_{2}]}</annotation>

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</math><img src="./1ce110719ca89eeeac9a2786c1fce4e86800dd41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.715ex; height:2.843ex;" alt="{\displaystyle [f_{1},f_{2}]}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<f_{1}<f_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle 0<f_{1}<f_{2}}</annotation>

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</math><img src="./d35da78371f3b4fc8ad5906e8ead3a04a87e5cc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.746ex; height:2.509ex;" alt="{\displaystyle 0<f_{1}<f_{2}}" loading="lazy">, can be calculated by integrating over frequency. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{xx}(-f)=S_{xx}(f)}">

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<annotation encoding="application/x-tex">{\displaystyle S_{xx}(-f)=S_{xx}(f)}</annotation>

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</math><img src="./028da7805e56429af9ae8ace88b80fab9692f2f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.158ex; height:2.843ex;" alt="{\displaystyle S_{xx}(-f)=S_{xx}(f)}" loading="lazy">, 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):

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<mtext mathvariant="sans-serif">band-limited</mtext>

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<annotation encoding="application/x-tex">{\displaystyle P_{\textsf {band-limited}}=2\int _{f_{1}}^{f_{2}}S_{xx}(f)\,df}</annotation>

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More generally, similar techniques may be used to estimate a time-varying spectral density. In this case the time interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> is finite rather than approaching infinity. This results in decreased spectral coverage and resolution since frequencies of less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/T}">

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<annotation encoding="application/x-tex">{\displaystyle 1/T}</annotation>

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</math><img src="./ee06bfe8f48b840ea1c11f78977a90f661f2375e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.961ex; height:2.843ex;" alt="{\displaystyle 1/T}" loading="lazy"> are not sampled, and results at frequencies which are not an integer multiple of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/T}">

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<annotation encoding="application/x-tex">{\displaystyle 1/T}</annotation>

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</math><img src="./ee06bfe8f48b840ea1c11f78977a90f661f2375e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.961ex; height:2.843ex;" alt="{\displaystyle 1/T}" loading="lazy"> 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 <a href="Statistical_ensemble" class="mw-redirect" title="Statistical ensemble">statistical ensembles</a> of realizations of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

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</math><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"> evaluated over the specified time window.

Just as with the energy spectral density, the definition of the power spectral density can be generalized to <a href="Discrete_time" class="mw-redirect" title="Discrete time">discrete time</a> variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}">

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</math><img src="./7c5ea190699149306d242b70439e663559e3ffbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}" loading="lazy">. As before, we can consider a window of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -N\leq n\leq N}">

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<annotation encoding="application/x-tex">{\displaystyle -N\leq n\leq N}</annotation>

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</math><img src="./fa7a34ea9aa050c8697291eec5cdc576d23206f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.527ex; height:2.343ex;" alt="{\displaystyle -N\leq n\leq N}" loading="lazy"> with the signal sampled at discrete times <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n}=t_{0}+(n\,\Delta t)}">

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<annotation encoding="application/x-tex">{\displaystyle t_{n}=t_{0}+(n\,\Delta t)}</annotation>

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</math><img src="./06f68e578977964c65c052bda78bcdf7e0e5af40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.257ex; height:2.843ex;" alt="{\displaystyle t_{n}=t_{0}+(n\,\Delta t)}" loading="lazy"> for a total measurement period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=(2N+1)\,\Delta t}">

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<annotation encoding="application/x-tex">{\displaystyle T=(2N+1)\,\Delta t}</annotation>

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</math><img src="./980dd75b7530aaf351daa6cd351838d5cade30fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.936ex; height:2.843ex;" alt="{\displaystyle T=(2N+1)\,\Delta t}" loading="lazy">.

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<annotation encoding="application/x-tex">{\displaystyle S_{xx}(f)=\lim _{N\to \infty }{\frac {(\Delta t)^{2}}{T}}\left|\sum _{n=-N}^{N}x_{n}e^{-i2\pi fn\,\Delta t}\right|^{2}}</annotation>

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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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> (and thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy">) 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 <a href="Periodogram" title="Periodogram">periodogram</a>. This periodogram converges to the true PSD as the number of estimates as well as the averaging time interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> approach infinity.<a href="#cite_note-FOOTNOTEBrownHwang1997-15">[13]</a>

If two signals both possess power spectral densities, then the <a href="#Cross-spectral_density">cross-spectral density</a> can similarly be calculated; as the PSD is related to the autocorrelation, so is the cross-spectral density related to the <a href="Cross-correlation" title="Cross-correlation">cross-correlation</a>.

<div class="mw-heading mw-heading4"><h4 id="Properties_of_the_power_spectral_density">Properties of the power spectral density</h4></div>

Some properties of the PSD include:<a href="#cite_note-FOOTNOTEMillerChilders2012431-16">[14]</a>

<div><ul><li>The power spectrum is always real and non-negative, and the spectrum of a real valued process is also an <a href="Even_function" class="mw-redirect" title="Even function">even function</a> of frequency: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{xx}(-f)=S_{xx}(f)}">

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<annotation encoding="application/x-tex">{\displaystyle S_{xx}(-f)=S_{xx}(f)}</annotation>

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</math><img src="./028da7805e56429af9ae8ace88b80fab9692f2f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.158ex; height:2.843ex;" alt="{\displaystyle S_{xx}(-f)=S_{xx}(f)}" loading="lazy">.</li><li>For a continuous <a href="Stochastic_process" title="Stochastic process">stochastic process</a> x(t), the autocorrelation function Rxx(t) can be reconstructed from its power spectrum Sxx(f) by using the <a href="Inverse_Fourier_transform" class="mw-redirect" title="Inverse Fourier transform">inverse Fourier transform</a></li><li>Using <a href="Parseval's_theorem" title="Parseval's theorem">Parseval's theorem</a>, one can compute the <a href="Variance" title="Variance">variance</a> (average power) of a process by integrating the power spectrum over all frequency:

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<annotation encoding="application/x-tex">{\displaystyle P=\operatorname {Var} (x)=\int _{-\infty }^{\infty }\!S_{xx}(f)\,df}</annotation>

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</math></li><li>For a real process x(t) with power spectral density <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{xx}(f)}">

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<annotation encoding="application/x-tex">{\displaystyle S_{xx}(f)}</annotation>

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</math><img src="./cab836aa57df700d834cc0f736e9f81152fa470e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.626ex; height:2.843ex;" alt="{\displaystyle S_{xx}(f)}" loading="lazy">, one can compute the integrated spectrum or power spectral distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(f)}">

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</math><img src="./28b338d0157928ab4d2fbcaedb56bff674b9c8b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.829ex; height:2.843ex;" alt="{\displaystyle F(f)}" loading="lazy">, which specifies the average bandlimited power contained in frequencies from DC to f using:<a href="#cite_note-FOOTNOTEDavenportRoot1987-17">[15]</a>

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<annotation encoding="application/x-tex">{\displaystyle F(f)=2\int _{0}^{f}S_{xx}(f')\,df'.}</annotation>

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Note that the previous expression for total power (signal variance) is a special case where f → ∞.</li></ul></div>

<div class="mw-heading mw-heading3"><h3 id="Cross_power_spectral_density">Cross power spectral density </h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Coherence_(signal_processing)" title="Coherence (signal processing)">Coherence (signal processing)</a></div>

Given two signals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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</math><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(t)}">

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</math><img src="./397de1edef5bf2ee15c020f325d7d781a3aa7f50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle y(t)}" loading="lazy">, each of which possess power spectral densities <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{xx}(f)}">

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</math><img src="./cab836aa57df700d834cc0f736e9f81152fa470e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.626ex; height:2.843ex;" alt="{\displaystyle S_{xx}(f)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{yy}(f)}">

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</math><img src="./4c29b5f848b786dea29cf5f205fbbd8bc3498754.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.379ex; height:3.009ex;" alt="{\displaystyle S_{yy}(f)}" loading="lazy">, 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.

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P&=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }\left[x_{T}(t)+y_{T}(t)\right]^{*}\left[x_{T}(t)+y_{T}(t)\right]dt\\&=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }|x_{T}(t)|^{2}+x_{T}^{*}(t)y_{T}(t)+y_{T}^{*}(t)x_{T}(t)+|y_{T}(t)|^{2}dt\\\end{aligned}}}</annotation>

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Using the same notation and methods as used for the power spectral density derivation, we exploit Parseval's theorem and obtain

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S_{xy}(f)&=\lim _{T\to \infty }{\frac {1}{T}}\left[{\hat {x}}_{T}^{*}(f){\hat {y}}_{T}(f)\right]&S_{yx}(f)&=\lim _{T\to \infty }{\frac {1}{T}}\left[{\hat {y}}_{T}^{*}(f){\hat {x}}_{T}(f)\right]\end{aligned}}}</annotation>

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where, again, the contributions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{xx}(f)}">

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<annotation encoding="application/x-tex">{\displaystyle S_{xx}(f)}</annotation>

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</math><img src="./cab836aa57df700d834cc0f736e9f81152fa470e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.626ex; height:2.843ex;" alt="{\displaystyle S_{xx}(f)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{yy}(f)}">

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<annotation encoding="application/x-tex">{\displaystyle S_{yy}(f)}</annotation>

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</math><img src="./4c29b5f848b786dea29cf5f205fbbd8bc3498754.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.379ex; height:3.009ex;" alt="{\displaystyle S_{yy}(f)}" loading="lazy"> are already understood. Note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{xy}^{*}(f)=S_{yx}(f)}">

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<annotation encoding="application/x-tex">{\displaystyle S_{xy}^{*}(f)=S_{yx}(f)}</annotation>

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</math><img src="./4cc82db12ccd633a5695fa8533e068f6d1f5867b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.103ex; height:3.009ex;" alt="{\displaystyle S_{xy}^{*}(f)=S_{yx}(f)}" loading="lazy">, 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\to \infty }">

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<annotation encoding="application/x-tex">{\displaystyle T\to \infty }</annotation>

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</math><img src="./6d124f9781b2ebb80044d6fb64b3a5f7e1cafdd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.574ex; height:2.176ex;" alt="{\displaystyle T\to \infty }" loading="lazy"> becomes the Fourier transform of a <a href="Cross-correlation" title="Cross-correlation">cross-correlation</a> function.<a href="#cite_note-18">[16]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S_{xy}(f)&=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )y_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau =\int _{-\infty }^{\infty }R_{xy}(\tau )e^{-i2\pi f\tau }d\tau \\S_{yx}(f)&=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }y_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau =\int _{-\infty }^{\infty }R_{yx}(\tau )e^{-i2\pi f\tau }d\tau ,\end{aligned}}}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{xy}(\tau )}">

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<annotation encoding="application/x-tex">{\displaystyle R_{xy}(\tau )}</annotation>

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</math><img src="./b9f7ddb3b7b684e1d9d52b97ff377c88b88b0623.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.765ex; height:3.009ex;" alt="{\displaystyle R_{xy}(\tau )}" loading="lazy"> is the <a href="Cross-correlation" title="Cross-correlation">cross-correlation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

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</math><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(t)}">

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<annotation encoding="application/x-tex">{\displaystyle y(t)}</annotation>

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</math><img src="./397de1edef5bf2ee15c020f325d7d781a3aa7f50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle y(t)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{yx}(\tau )}">

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<annotation encoding="application/x-tex">{\displaystyle R_{yx}(\tau )}</annotation>

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</math><img src="./8db09cdbf6012ec80ae4693a60e9b6d846b2f616.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.765ex; height:3.009ex;" alt="{\displaystyle R_{yx}(\tau )}" loading="lazy"> is the cross-correlation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(t)}">

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<annotation encoding="application/x-tex">{\displaystyle y(t)}</annotation>

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</math><img src="./397de1edef5bf2ee15c020f325d7d781a3aa7f50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle y(t)}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

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</math><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy">. In light of this, the PSD is seen to be a special case of the CSD for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)=y(t)}">

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<annotation encoding="application/x-tex">{\displaystyle x(t)=y(t)}</annotation>

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</math><img src="./8e64fbe15b5c71503c97394db87bfb1922219c9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.881ex; height:2.843ex;" alt="{\displaystyle x(t)=y(t)}" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

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</math><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(t)}">

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<annotation encoding="application/x-tex">{\displaystyle y(t)}</annotation>

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</math><img src="./397de1edef5bf2ee15c020f325d7d781a3aa7f50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle y(t)}" loading="lazy"> are real signals (e.g. voltage or current), their Fourier transforms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {x}}(f)}">

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</math><img src="./594eba73af7f4adf231cffd67129dbed8957e7e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle {\hat {x}}(f)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {y}}(f)}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {y}}(f)}</annotation>

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</math><img src="./322b92da277b2088d4ebb87ceea703a7a3a0a1aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.39ex; height:2.843ex;" alt="{\displaystyle {\hat {y}}(f)}" loading="lazy"> are usually restricted to positive frequencies by convention. Therefore, in typical signal processing, the full CPSD is just one of the CPSDs scaled by a factor of two.

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For discrete signals xn and yn, the relationship between the cross-spectral density and the cross-covariance is

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<annotation encoding="application/x-tex">{\displaystyle S_{xy}(f)=\sum _{n=-\infty }^{\infty }R_{xy}(\tau _{n})e^{-i2\pi f\tau _{n}}\,\Delta \tau }</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Estimation">Estimation</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></div>

The goal of spectral density estimation is to <a href="Estimation_theory" title="Estimation theory">estimate</a> the spectral density of a <a href="Random_signal" class="mw-redirect" title="Random signal">random signal</a> from a sequence of time samples. Depending on what is known about the signal, estimation techniques can involve <a href="Parametric_statistics" title="Parametric statistics">parametric</a> or <a href="Non-parametric_statistics" class="mw-redirect" title="Non-parametric statistics">non-parametric</a> approaches, and may be based on time-domain or frequency-domain analysis. For example, a common parametric technique involves fitting the observations to an <a href="Autoregressive_model" title="Autoregressive model">autoregressive model</a>. A common non-parametric technique is the <a href="Periodogram" title="Periodogram">periodogram</a>.

The spectral density is usually estimated using <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> methods (such as the <a href="Welch_method" class="mw-redirect" title="Welch method">Welch method</a>), but other techniques such as the <a href="Maximum_entropy_spectral_estimation" title="Maximum entropy spectral estimation">maximum entropy</a> method can also be used.

<div class="mw-heading mw-heading2"><h2 id="Related_concepts">Related concepts</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Spectral_density_(physical_science)" class="mw-redirect" title="Spectral density (physical science)">spectral density (physical science)</a>.</div>

<ul><li>The <a href="Spectral_centroid" title="Spectral centroid">spectral centroid</a> of a signal is the midpoint of its spectral density function, i.e. the frequency that divides the distribution into two equal parts.</li>

<li> The spectral edge frequency (SEF), usually expressed as "SEF x", represents the <a href="Frequency" title="Frequency">frequency</a> 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 <a href="EEG" class="mw-redirect" title="EEG">EEG</a> monitoring, in which case SEF has variously been used to estimate the depth of <a href="Anesthesia" title="Anesthesia">anesthesia</a> and stages of <a href="Sleep" title="Sleep">sleep</a>.<a href="#cite_note-FOOTNOTEIranmaneshRodriguez-Villegas2017-19">[17]</a><a href="#cite_note-FOOTNOTEImtiazRodriguez-Villegas2014-20">[18]</a></li>

<li> A spectral envelope is the <a href="Envelope_curve" class="mw-redirect" title="Envelope curve">envelope curve</a> of the spectrum density. It describes one point in time (one window, to be precise). For example, in <a href="Remote_sensing" title="Remote sensing">remote sensing</a> using a <a href="Spectrometer" title="Spectrometer">spectrometer</a>, the spectral envelope of a feature is the boundary of its <a href="Electromagnetic_spectrum" title="Electromagnetic spectrum">spectral</a> properties, as defined by the range of brightness levels in each of the <a href="Spectral_bands" class="mw-redirect" title="Spectral bands">spectral bands</a> of interest.</li>

<li>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 <a href="Spectrogram" title="Spectrogram">spectrogram</a>. This is the basis of a number of spectral analysis techniques such as the <a href="Short-time_Fourier_transform" title="Short-time Fourier transform">short-time Fourier transform</a> and <a href="Wavelets" class="mw-redirect" title="Wavelets">wavelets</a>.</li>

<li> A "spectrum" generally means the power spectral density, as discussed above, which depicts the distribution of signal content over frequency. For <a href="Transfer_function" title="Transfer function">transfer functions</a> (e.g., <a href="Bode_plot" title="Bode plot">Bode plot</a>, <a href="Chirp#Relation_to_an_impulse_signal" title="Chirp">chirp</a>) the complete frequency response may be graphed in two parts: power versus frequency and <a href="Phase_(waves)" title="Phase (waves)">phase</a> versus frequency—the phase spectral density, phase spectrum, or spectral phase. Less commonly, the two parts may be the <a href="Real_and_imaginary_parts" class="mw-redirect" title="Real and imaginary parts">real and imaginary parts</a> of the transfer function. This is not to be confused with the <a href="Frequency_response" title="Frequency response">frequency response</a> of a transfer function, which also includes a phase (or equivalently, a real and imaginary part) as a function of frequency. The time-domain <a href="Impulse_response" title="Impulse response">impulse response</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(t)}">

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</math><img src="./66abbb8ae1d9f30bb529739b109e1e5bbe83c626.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.988ex; height:2.843ex;" alt="{\displaystyle h(t)}" loading="lazy"> 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 <a href="Autocorrelation" title="Autocorrelation">autocorrelation</a>) forcing the Fourier transform to be real-valued. See <a href="Ultrashort_pulse#Spectral_phase" title="Ultrashort pulse">Ultrashort pulse#Spectral phase</a>, <a href="Phase_noise" title="Phase noise">phase noise</a>, <a href="Group_delay" class="mw-redirect" title="Group delay">group delay</a>.</li>

<li> Sometimes one encounters an amplitude spectral density (ASD), which is the square root of the PSD; the ASD of a voltage signal has the unit V⋅Hz−1/2.<a href="#cite_note-21">[19]</a> 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.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Spectrum" title="Spectrum">Spectrum</a></div>

Any signal that can be represented as a variable that varies in time has a corresponding frequency spectrum. This includes familiar entities such as <a href="Visible_light" class="mw-redirect" title="Visible light">visible light</a> (perceived as <a href="Color" title="Color">color</a>), musical notes (perceived as <a href="Pitch_(music)" title="Pitch (music)">pitch</a>), <a href="Radio_frequency" title="Radio frequency">radio/TV</a> (specified by their frequency, or sometimes <a href="Wavelength" title="Wavelength">wavelength</a>) 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 <a href="Sine_wave" title="Sine wave">sine wave</a> component. And additionally there may be peaks corresponding to <a href="Harmonics" class="mw-redirect" title="Harmonics">harmonics</a> 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 <a href="Notch_filter" class="mw-redirect" title="Notch filter">notch filter</a>.

<div class="mw-heading mw-heading3"><h3 id="Electrical_engineering">Electrical engineering</h3></div>

The concept and use of the power spectrum of a signal is fundamental in <a href="Electrical_engineering" title="Electrical engineering">electrical engineering</a>, especially in <a href="Communication_systems" class="mw-redirect" title="Communication systems">electronic communication systems</a>, including <a href="Radio_communication" class="mw-redirect" title="Radio communication">radio communications</a>, <a href="Radar" title="Radar">radars</a>, and related systems, plus passive <a href="Remote_sensing" title="Remote sensing">remote sensing</a> technology. Electronic instruments called <a href="Spectrum_analyzer" title="Spectrum analyzer">spectrum analyzers</a> are used to observe and measure the power spectra of signals.

The spectrum analyzer measures the magnitude of the <a href="Short-time_Fourier_transform" title="Short-time Fourier transform">short-time Fourier transform</a> (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.

<div class="mw-heading mw-heading3"><h3 id="Cosmology">Cosmology</h3></div>

<a href="Primordial_fluctuations" title="Primordial fluctuations">Primordial fluctuations</a>, 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.

<ul><li><a href="Bispectrum" title="Bispectrum">Bispectrum</a></li>

<li><a href="Brightness_temperature" title="Brightness temperature">Brightness temperature</a></li>

<li><a href="Colors_of_noise" title="Colors of noise">Colors of noise</a></li>

<li><a href="Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li>

<li><a href="Noise_spectral_density" title="Noise spectral density">Noise spectral density</a></li>

<li><a href="Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li>

<li><a href="Spectral_efficiency" title="Spectral efficiency">Spectral efficiency</a></li>

<li><a href="Spectral_leakage" title="Spectral leakage">Spectral leakage</a></li>

<li><a href="Spectral_power_distribution" title="Spectral power distribution">Spectral power distribution</a></li>

<li><a href="Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li>

<li><a href="Window_function" title="Window function">Window function</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-9"><a href="#cite_ref-9">^</a> Some authors, e.g., (<a href="#CITEREFRiskenFrank1996">Risken & Frank 1996</a>, p. 30) still use the non-normalized Fourier transform in a formal way to formulate a definition of the power spectral density

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<annotation encoding="application/x-tex">{\displaystyle \langle {\hat {x}}(\omega ){\hat {x}}^{\ast }(\omega ')\rangle =2\pi f(\omega )\delta (\omega -\omega '),}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (\omega -\omega ')}">

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<annotation encoding="application/x-tex">{\displaystyle \delta (\omega -\omega ')}</annotation>

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</math><img src="./d9412fe50622d943fadaf2ab8bd1a7bd9d9c3b51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.275ex; height:3.009ex;" alt="{\displaystyle \delta (\omega -\omega ')}" loading="lazy"> is the <a href="Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>. Such formal statements may sometimes be useful to guide the intuition, but should always be used with utmost care.

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> The <a href="Wiener%E2%80%93Khinchin_theorem" title="Wiener–Khinchin theorem">Wiener–Khinchin theorem</a> makes sense of this formula for any <a href="Wide-sense_stationary_process" class="mw-redirect" title="Wide-sense stationary process">wide-sense stationary process</a> under weaker hypotheses: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{xx}}">

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<annotation encoding="application/x-tex">{\displaystyle R_{xx}}</annotation>

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</math><img src="./44c57d170e7cb2b17a74d1fec2427a2a26836b39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.877ex; height:2.509ex;" alt="{\displaystyle R_{xx}}" loading="lazy"> does not need to be absolutely integrable, it only needs to exist. But the integral can no longer be interpreted as usual. The formula also makes sense if interpreted as involving <a href="Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a> (in the sense of <a href="Laurent_Schwartz" title="Laurent Schwartz">Laurent Schwartz</a>, not in the sense of a statistical <a href="Cumulative_distribution_function" title="Cumulative distribution function">Cumulative distribution function</a>) instead of functions. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{xx}}">

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<annotation encoding="application/x-tex">{\displaystyle R_{xx}}</annotation>

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</math><img src="./44c57d170e7cb2b17a74d1fec2427a2a26836b39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.877ex; height:2.509ex;" alt="{\displaystyle R_{xx}}" loading="lazy"> is continuous, <a href="Bochner's_theorem" title="Bochner's theorem">Bochner's theorem</a> can be used to prove that its Fourier transform exists as a positive <a href="Measure_(mathematics)" title="Measure (mathematics)">measure</a>, whose distribution function is F (but not necessarily as a function and not necessarily possessing a probability density).

</li>

</ol></div></div>

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</style><cite id="CITEREFP_StoicaR_Moses2005" class="citation web cs1"><a href="Peter_Stoica" title="Peter Stoica">P Stoica</a> & R Moses (2005). <a rel="nofollow" class="external text" href="http://user.it.uu.se/~ps/SAS-new.pdf">"Spectral Analysis of Signals"</a> (PDF).</cite>

</li>

<li id="cite_note-FOOTNOTEMaral2004-2"><a href="#cite_ref-FOOTNOTEMaral2004_2-0">^</a> <a href="#CITEREFMaral2004">Maral 2004</a>.

</li>

<li id="cite_note-FOOTNOTENortonKarczub2003-3"><a href="#cite_ref-FOOTNOTENortonKarczub2003_3-0">^</a> <a href="#CITEREFNortonKarczub2003">Norton & Karczub 2003</a>.

</li>

<li id="cite_note-FOOTNOTEBirolini200783-4"><a href="#cite_ref-FOOTNOTEBirolini200783_4-0">^</a> <a href="#CITEREFBirolini2007">Birolini 2007</a>, p. 83.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFPaschotta2005" class="citation web cs1">Paschotta, Rüdiger (5 April 2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20240415070408/https://www.rp-photonics.com/power_spectral_density.html">"Power Spectral Density"</a>. rp-photonics.com. Archived from <a rel="nofollow" class="external text" href="https://www.rp-photonics.com/power_spectral_density.html">the original</a> on 2024-04-15. Retrieved 2024-06-26.</cite>

</li>

<li id="cite_note-FOOTNOTEOppenheimVerghese201660-6"><a href="#cite_ref-FOOTNOTEOppenheimVerghese201660_6-0">^</a> <a href="#CITEREFOppenheimVerghese2016">Oppenheim & Verghese 2016</a>, p. 60.

</li>

<li id="cite_note-FOOTNOTEStein2000108,_115-7"><a href="#cite_ref-FOOTNOTEStein2000108,_115_7-0">^</a> <a href="#CITEREFStein2000">Stein 2000</a>, pp. 108, 115.

</li>

<li id="cite_note-FOOTNOTEOppenheimVerghese201614-8"><a href="#cite_ref-FOOTNOTEOppenheimVerghese201614_8-0">^</a> <a href="#CITEREFOppenheimVerghese2016">Oppenheim & Verghese 2016</a>, p. 14.

</li>

<li id="cite_note-FOOTNOTEOppenheimVerghese2016422–423-10"><a href="#cite_ref-FOOTNOTEOppenheimVerghese2016422–423_10-0">^</a> <a href="#CITEREFOppenheimVerghese2016">Oppenheim & Verghese 2016</a>, pp. 422–423.

</li>

<li id="cite_note-FOOTNOTEMillerChilders2012429–431-11"><a href="#cite_ref-FOOTNOTEMillerChilders2012429–431_11-0">^</a> <a href="#CITEREFMillerChilders2012">Miller & Childers 2012</a>, pp. 429–431.

</li>

<li id="cite_note-FOOTNOTEMillerChilders2012433-13"><a href="#cite_ref-FOOTNOTEMillerChilders2012433_13-0">^</a> <a href="#CITEREFMillerChilders2012">Miller & Childers 2012</a>, p. 433.

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFDennis_Ward_Ricker2003" class="citation book cs1">Dennis Ward Ricker (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NF2Tmty9nugC&q=%22power+spectral+density%22+%22energy+spectral+density%22&pg=PA23">Echo Signal Processing</a>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4020-7395-3</bdi>.</cite>

</li>

<li id="cite_note-FOOTNOTEBrownHwang1997-15"><a href="#cite_ref-FOOTNOTEBrownHwang1997_15-0">^</a> <a href="#CITEREFBrownHwang1997">Brown & Hwang 1997</a>.

</li>

<li id="cite_note-FOOTNOTEMillerChilders2012431-16"><a href="#cite_ref-FOOTNOTEMillerChilders2012431_16-0">^</a> <a href="#CITEREFMillerChilders2012">Miller & Childers 2012</a>, p. 431.

</li>

<li id="cite_note-FOOTNOTEDavenportRoot1987-17"><a href="#cite_ref-FOOTNOTEDavenportRoot1987_17-0">^</a> <a href="#CITEREFDavenportRoot1987">Davenport & Root 1987</a>.

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite id="CITEREFWilliam_D_Penny2009" class="citation web cs1">William D Penny (2009). <a rel="nofollow" class="external text" href="http://www.fil.ion.ucl.ac.uk/~wpenny/course/course.html">"Signal Processing Course, chapter 7"</a>.</cite>

</li>

<li id="cite_note-FOOTNOTEIranmaneshRodriguez-Villegas2017-19"><a href="#cite_ref-FOOTNOTEIranmaneshRodriguez-Villegas2017_19-0">^</a> <a href="#CITEREFIranmaneshRodriguez-Villegas2017">Iranmanesh & Rodriguez-Villegas 2017</a>.

</li>

<li id="cite_note-FOOTNOTEImtiazRodriguez-Villegas2014-20"><a href="#cite_ref-FOOTNOTEImtiazRodriguez-Villegas2014_20-0">^</a> <a href="#CITEREFImtiazRodriguez-Villegas2014">Imtiaz & Rodriguez-Villegas 2014</a>.

</li>

<cite id="CITEREFMichael_CernaAudrey_F._Harvey2000" class="citation web cs1 cs1-prop-unfit">Michael Cerna & Audrey F. Harvey (2000). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120915030050/http://www.lumerink.com/courses/ece697/docs/Papers/The%20Fundamentals%20of%20FFT-Based%20Signal%20Analysis%20and%20Measurements.pdf">"The Fundamentals of FFT-Based Signal Analysis and Measurement"</a> (PDF). Archived from the original on September 15, 2012.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFBirolini2007" class="citation book cs1">Birolini, Alessandro (2007). Reliability Engineering. Berlin ; New York: Springer Science & Business Media. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-49388-4</bdi>.</cite></li>

<li><cite id="CITEREFBrownHwang1997" class="citation book cs1">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. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-12839-7</bdi>.</cite></li>

<li><cite id="CITEREFDavenportRoot1987" class="citation book cs1">Davenport, Wilbur B. (Jr); Root, William L. (1987). An Introduction to the Theory of Random Signals and Noise. New York: Wiley-IEEE Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-87942-235-6</bdi>.</cite></li>

<li><cite id="CITEREFImtiazRodriguez-Villegas2014" class="citation journal cs1">Imtiaz, Syed Anas; Rodriguez-Villegas, Esther (2014). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4204008">"A Low Computational Cost Algorithm for REM Sleep Detection Using Single Channel EEG"</a>. Annals of Biomedical Engineering. 42 (11): 2344–59. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10439-014-1085-6">10.1007/s10439-014-1085-6</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4204008">4204008</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/25113231">25113231</a>.</cite></li>

<li><cite id="CITEREFIranmaneshRodriguez-Villegas2017" class="citation journal cs1">Iranmanesh, Saam; <a href="Esther_Rodriguez-Villegas" title="Esther Rodriguez-Villegas">Rodriguez-Villegas, Esther</a> (2017). "An Ultralow-Power Sleep Spindle Detection System on Chip". IEEE Transactions on Biomedical Circuits and Systems. 11 (4): 858–866. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017ITBC...11..858I">2017ITBC...11..858I</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTBCAS.2017.2690908">10.1109/TBCAS.2017.2690908</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10044%2F1%2F46059">10044/1/46059</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/28541914">28541914</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:206608057">206608057</a>.</cite></li>

<li><cite id="CITEREFMaral2004" class="citation book cs1">Maral, Gerard (2004). VSAT Networks. West Sussex, England ; Hoboken, NJ: Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-470-86684-9</bdi>.</cite></li>

<li><cite id="CITEREFMillerChilders2012" class="citation book cs1">Miller, Scott; Childers, Donald (2012). Probability and Random Processes. Boston, MA: Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-386981-4</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/696092052">696092052</a>.</cite></li>

<li><cite id="CITEREFNortonKarczub2003" class="citation book cs1">Norton, M. P.; Karczub, D. G. (2003). Fundamentals of Noise and Vibration Analysis for Engineers. Cambridge: Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-49913-2</bdi>.</cite></li>

<li><cite id="CITEREFOppenheimVerghese2016" class="citation book cs1">Oppenheim, Alan V.; Verghese, George C. (2016). Signals, Systems & Inference. Boston: Pearson. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-394328-3</bdi>.</cite></li>

<li><cite id="CITEREFRiskenFrank1996" class="citation book cs1">Risken, Hannes; Frank, Till (1996). The Fokker-Planck Equation. New York: Springer Science & Business Media. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-61530-9</bdi>.</cite></li>

<li><cite id="CITEREFStein2000" class="citation book cs1">Stein, Jonathan Y. (2000). Digital Signal Processing. New York Weinheim: Wiley-Interscience. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-29546-4</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://vibrationdata.wordpress.com/category/power-spectral-density/">Power Spectral Density Matlab scripts</a></li></ul>

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</style><div id="Decibel_suffixes_(dB)140" style="font-size:114%;margin:0 4em"><a href="Decibel" title="Decibel">Decibel</a> <a href="Decibel#Suffixes" title="Decibel">suffixes</a> (dB)</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="DBZ_(meteorology)" title="DBZ (meteorology)">dBZ</a> (radar)</li>

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<dd><a href="Logarithmic_unit" class="mw-redirect" title="Logarithmic unit">logarithmic unit</a></dd>

<dd><a href="Link_budget" title="Link budget">link budget</a></dd>

<dd><a href="Signal_noise" class="mw-redirect" title="Signal noise">signal noise</a></dd>

<dd><a href="Telecommunications" title="Telecommunications">telecommunications</a></dd></dl>

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