Editing Periodogram (section)

==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.