Editing Periodogram

{{Short description|Estimate of the spectral density of a signal}}
In [[signal processing]], a '''periodogram''' is an estimate of the [[spectral density]] of a signal. The term was coined by [[Arthur Schuster]] in 1898.<ref name="Schuster"/> Today, the periodogram is a component of more sophisticated methods (see [[spectral estimation]]).  It is the most common tool for examining the amplitude vs frequency characteristics of [[FIR filter]]s and [[window functions]].  [[spectrum analyzer|FFT spectrum analyzers]] are also implemented as a time-sequence of periodograms.

==Definition==
There are at least two different definitions in use today.<ref name="Comparison"/>  One of them involves time-averaging,<ref name="Wolfram"/> and one does not.<ref name="Matlab"/> Time-averaging is also the purview of other articles ([[Bartlett's method]] and [[Welch's method]]).  This article is not about time-averaging.  The definition of interest here is that the power spectral density of a continuous function, <math>x(t),</math>&nbsp; is the [[Fourier transform]] of its auto-correlation function (see [[Fourier transform#Cross-correlation theorem|Cross-correlation theorem]], [[Spectral density#Power spectral density|Spectral density]], and [[Wiener–Khinchin theorem]]):
<math display="block">\mathcal{F}\{x(t)\circledast x^*(-t)\} = X(f)\cdot X^*(f) = \left| X(f) \right|^2.</math>

==Computation==

[[File:Periodogram.svg|thumb|400px|A power spectrum (magnitude-squared) of two sinusoidal basis functions, calculated by the periodogram method.]]
[[File:Periodogram windows.svg|thumb|400px|Two power spectra (magnitude-squared) (rectangular and Hamming [[Window function|window functions]] plus background noise), calculated by the periodogram method.]]

For sufficiently small values of parameter {{mvar|T,}} an arbitrarily-accurate approximation for {{math|''X''(''f'')}} can be observed in the region &nbsp;<math>-\tfrac{1}{2T} < f < \tfrac{1}{2T}</math>&nbsp; of the function:

<math display="block">X_{1/T}(f)\ \triangleq \sum_{k=-\infty}^{\infty} X\left(f - k/T\right),</math>

which is precisely determined by the samples {{math|''x''(''nT'')}} that span the non-zero duration of {{math|''x''(''t'')}} &nbsp;(see [[Discrete-time Fourier transform]]).

And for sufficiently large values of parameter {{mvar|N}}, &nbsp;<math>X_{1/T}(f)</math> can be evaluated at an arbitrarily close frequency by a summation of the form:

<math display="block">X_{1/T}\left(\tfrac{k}{NT}\right) = \sum_{n=-\infty}^\infty \underbrace{T\cdot x(nT)}_{x[n]}\cdot e^{-i 2\pi \frac{kn}{N}},</math>

where {{mvar|k}} is an integer.  The periodicity of &nbsp;<math>e^{-i 2\pi \frac{kn}{N}}</math>&nbsp; allows this to be written very simply in terms of a [[Discrete Fourier transform]]:

<math display="block">X_{1/T}\left(\tfrac{k}{NT}\right) = \underbrace{\sum_{n} x_{_N}[n]\cdot e^{-i 2\pi \frac{kn}{N}},}_\text{DFT} \quad \scriptstyle{\text{(sum over any }n\text{-sequence of length }N)},</math>

where <math>x_{_N}</math> is a periodic summation: &nbsp;<math>x_{_N}[n]\ \triangleq \sum_{m=-\infty}^{\infty} x[n - mN].</math>

When evaluated for all integers, {{mvar|k}}, between 0 and {{mvar|N}}-1, the array:
<math display="block">S\left(\tfrac{k}{NT}\right) = \left| \sum_{n} x_{_N}[n]\cdot e^{-i 2\pi \frac{kn}{N}}  \right|^2</math>
is a ''periodogram''.<ref name="Matlab"/><ref name=Oppenheim/><ref name=Rabiner/>

==Applications==

[[File:Periodogram for Proxima Centauri b.jpg|thumb|Periodogram for [[Proxima Centauri b]] is shown at the bottom.<ref>{{cite web|title=Do-it-yourself Science — is Proxima c hiding in this graph?|url=https://www.eso.org/public/images/potw1737a/|website=www.eso.org|access-date=11 September 2017}}</ref>]]

When a periodogram is used to examine the detailed characteristics of an [[FIR filter]] or [[window function]], the parameter {{mvar|N}}  is chosen to be several multiples of the non-zero duration of the {{math|''x''[''n'']}} sequence, which is called ''zero-padding'' (see {{slink|Discrete-time Fourier transform|Sampling the DTFT|nopage=y}}).{{
efn-ua|{{mvar|N}} is designated {{mvar|NFFT}} in the Matlab and Octave applications.
}}&nbsp; When it is used to implement a [[filter bank]], {{mvar|N}} is several sub-multiples of the non-zero duration of the {{math|''x''[''n'']}} sequence (see {{slink|Discrete-time Fourier transform|Sampling the DTFT|nopage=y}}).

One of the periodogram's deficiencies is that the variance at a given [[frequency]] does not decrease as the number of samples used in the computation increases.  It does not provide the averaging needed to analyze noiselike signals or even sinusoids at low signal-to-noise ratios.  Window functions and filter impulse responses are noiseless, but many other signals require more sophisticated methods of [[spectral estimation]].  Two of the alternatives use periodograms as part of the process:
*The ''method of averaged periodograms'',<ref name="Engelberg"/>&nbsp; more commonly known as [[Welch's method]],<ref name="Welch"/><ref name="Welch2"/>&nbsp; divides a long x[n] sequence into multiple shorter, and possibly overlapping, subsequences.  It computes a windowed periodogram of each one, and computes an array average, i.e. an array where each element is an average of the corresponding elements of all the periodograms.  For [[stationary process]]es, this reduces the noise variance of each element by approximately a factor equal to the reciprocal of the number of periodograms.
*[[Smoothing]] is an averaging technique in frequency, instead of time.  The smoothed periodogram is sometimes referred to as a ''spectral plot''.<ref name="smoothing"/><ref name="dataplot"/>

Periodogram-based techniques introduce small biases that are unacceptable in some applications.  Other techniques that do not rely on periodograms are presented in the [[spectral density estimation]] article.

== See also ==
*[[Matched filter]]
*[[Radon transform|Filtered backprojection]] (Radon transform)
*[[Welch's method]]
*[[Bartlett's method]]
*[[Discrete-time Fourier transform]]
*[[Least-squares spectral analysis]], for computing periodograms in data that is not equally spaced
*[[MUSIC_(algorithm)|MUltiple SIgnal Classification]] (MUSIC), a popular parametric [[Super-resolution imaging|superresolution]] method
*[[SAMV (algorithm)|SAMV]]

== Notes ==
{{notelist-ua}}

== References ==
{{reflist|1|refs=
<ref name="Engelberg">Engelberg, S. (2008), ''Digital Signal Processing: An Experimental Approach'',  Springer, Chap. 7 p. 56</ref>

<ref name="Schuster">{{cite journal |first=Arthur |last=Schuster |author-link=Arthur Schuster |title= On the investigation of hidden periodicities with application to a supposed 26 day period of meteorological phenomena | url= http://iranithenticate.ir/download/On%20the%20investigation%20of%20hidden%20periodicities%20with%20application%20to%20a%20supposed%2026%20day%20period%20of%20meteorological%20phenomena/iranithenticate-ir.pdf |date=January 1898 |journal=Terrestrial Magnetism |volume=3 |issue=1 |pages=13–41 |doi=10.1029/TM003i001p00013 |quote=It is convenient to have a word for some representation of a variable quantity which shall correspond to the ‘spectrum’ of a luminous radiation. I propose the word ''periodogram'', and define it more particularly in the following way.|bibcode=1898TeMag...3...13S }}</ref>

<ref name="Welch">{{cite journal |last=Welch |first=Peter D.|title=The use of Fast Fourier Transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms |journal=IEEE Transactions on Audio and Electroacoustics |volume=AU-15 |issue=2 |date=June 1967 |pages=70–73 |doi=10.1109/TAU.1967.1161901 |bibcode=1967ITAE...15...70W}}</ref>

<ref name="Welch2">{{cite web |url=https://www.mathworks.com/help/signal/ref/pwelch.html |title=Welch's power spectral density estimate - MATLAB pwelch}}</ref>

<ref name="Matlab">{{cite web|url=https://www.mathworks.com/help/signal/ref/periodogram.html |title=Periodogram power spectral density estimate - MATLAB periodogram }}</ref>

<ref name="Wolfram">{{cite web|url=https://reference.wolfram.com/language/ref/Periodogram.html|title=Periodogram—Wolfram Language Documentation}}</ref>

<ref name="smoothing">[http://www.itl.nist.gov/div898/handbook/eda/section3/eda33r.htm Spectral Plot], from the [[NIST]] Engineering Statistics Handbook.</ref>

<ref name="Comparison">
{{cite journal
 | last =McSweeney
 | first =Laura A.
 | title =Comparison of periodogram tests
 | journal =Journal of Statistical Computation and Simulation
 | volume =76
 | issue =4
 | pages =357–369
 | publisher =online ($50)
 | date =2004-05-14
 | doi =10.1080/10629360500107618
 | s2cid =120439605
 }}</ref>

<ref name="dataplot">
{{cite web
 | url =http://www.itl.nist.gov/div898/software/dataplot/refman1/ch2/spectral.pdf
 | title =DATAPLOT Reference Manual
 | date =1997-03-11
 | website =NIST.gov
 | publisher =National Institute of Standards and Technology (NIST)
 | access-date =2019-06-14
 | quote =The spectral plot is essentially a “smoothed” periodogram where the smoothing is done in the frequency domain.
}}</ref>

<ref name=Oppenheim>
{{Cite book |ref=Oppenheim |last1=Oppenheim |first1=Alan V. |author-link=Alan V. Oppenheim |last2=Schafer |first2=Ronald W. |author2-link=Ronald W. Schafer |last3=Buck |first3=John R. |title=Discrete-time signal processing |year=1999 |publisher=Prentice Hall |location=Upper Saddle River, N.J. |isbn=0-13-754920-2 |edition=2nd |page=732 (10.55) |url-access=registration |url=https://archive.org/details/discretetimesign00alan
}}
</ref>

<ref name=Rabiner>
{{Cite book
| author1=Rabiner, Lawrence R.
| author2=Gold, Bernard
| title=Theory and application of digital signal processing
| year=1975
| publisher=Prentice-Hall
| location=Englewood Cliffs, N.J.
| isbn=0-13-914101-4
| chapter=6.18
| pages=[https://archive.org/details/theoryapplicatio00rabi/page/415 415]
| chapter-url-access=registration
| chapter-url=https://archive.org/details/theoryapplicatio00rabi/page/415
}}
</ref>
}}

==Further reading==
{{refbegin|}}
*{{cite book|last1=Box|first1=George E. P.|last2=Jenkins|first2=Gwilym M.|title=Time series analysis: Forecasting and control|publisher=Holden-Day|date=1976|location=San Francisco|bibcode=1976tsaf.conf.....B }}
*{{cite journal|last=Scargle|first=J.D.|title=Studies in astronomical time series analysis. II - Statistical aspects of spectral analysis of unevenly spaced data|journal=Astrophysical Journal, Part 1|volume=263|pages=835–853|date=December 15, 1982|doi=10.1086/160554|bibcode=1982ApJ...263..835S}}
*{{cite journal|last1=Vaughan|first1=Simon|last2=Uttley|first2=Philip|title=Detecting X-ray QPOs in active galaxies|journal=Advances in Space Research|volume=38|issue=7|pages=1405–1408|date=2006|doi=10.1016/j.asr.2005.02.064|arxiv=astro-ph/0506456|bibcode=2006AdSpR..38.1405V|s2cid=21054467 }}
{{refend}}

[[Category:Frequency-domain analysis]]
[[Category:Fourier analysis]]