Editing Spectral density (section)

{{Short description|Relative importance of certain frequencies in a composite signal}}
{{about|signal processing and relation of spectra to time-series|further applications in the physical sciences|Spectrum (physical sciences)}}
{{distinguish-redirect|Spectral power density|Spectral power}}
{{Use American English|date = March 2019}}
[[File:Fluorescent lighting spectrum peaks labelled.svg|thumb|right|The spectral density of a [[fluorescent light]] as a function of optical wavelength shows peaks at atomic transitions, indicated by the numbered arrows.]]
[[File:Voice waveform and spectrum.png|thumb|right|The voice waveform over time (left) has a broad audio power spectrum (right).]]{{Too technical|date=June 2024}}
In [[signal processing]], the power spectrum <math>S_{xx}(f)</math> of a [[continuous time]] [[signal]] <math>x(t)</math> describes the distribution of [[Power (physics)|power]] into frequency components <math>f</math> composing that signal.<ref name="P Stoica">{{cite web
 | url = http://user.it.uu.se/~ps/SAS-new.pdf
 | title = Spectral Analysis of Signals
 |author1=P Stoica
 |author-link=Peter Stoica
 |author2=R Moses
  |name-list-style=amp | year = 2005
 }}</ref>  According to [[Fourier analysis]], any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of any sort of signal (including [[Noise (electronics)|noise]]) as analyzed in terms of its frequency content, is called its [[spectrum]].

When the energy of the signal is concentrated around a finite time interval, especially if its total energy is finite, one may compute the '''energy spectral density'''. More commonly used is the '''power spectral density''' (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 energy distribution that would be found per unit time, since the total energy of such a signal over all time would generally be infinite. [[Summation]] 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>x^2(t)</math> over the time domain, as dictated by [[Parseval's theorem]].<ref name="P Stoica" />

The spectrum of a physical process <math>x(t)</math> often contains essential  information about the nature of <math>x</math>. For instance, the [[Pitch (music)|pitch]] and [[timbre]] of a musical instrument are immediately determined from a spectral analysis. The [[color]] of a light source is determined by the spectrum of the electromagnetic wave's electric field <math>E(t)</math> as it fluctuates at an extremely high frequency. Obtaining a spectrum from time series such as these involves the [[Fourier transform]], and generalizations based on Fourier analysis. In many cases the time domain is not specifically employed in practice, such as when a [[dispersive prism]] is used to obtain a spectrum of light in a [[spectrograph]], or when a sound is perceived through its effect on the auditory receptors of the inner ear, each of which is sensitive to a particular frequency.
 
However this article concentrates on situations in which the time series is known (at least in a statistical sense) or directly measured (such as by a microphone sampled by a computer). The power spectrum is important in [[statistical signal processing]] and in the statistical study of [[stochastic process]]es, as well as in many other branches of [[physics]] and [[engineering]]. Typically the process is a function of time, but one can similarly discuss data in the spatial domain being decomposed in terms of [[spatial frequency]].<ref name="P Stoica" />