# Wolfram Language & System 11.0 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# PowerSpectralDensity

PowerSpectralDensity[data,ω]
estimates the power spectral density for data.

PowerSpectralDensity[data,ω,sspec]
estimates the power spectral density for data with smoothing specification sspec.

PowerSpectralDensity[tproc,ω]
represents the power spectral density of a time series process tproc.

## Details and OptionsDetails and Options

• PowerSpectralDensity is also known as the energy spectral density.
• PowerSpectralDensity[tproc,ω] is defined for weakly stationary time series processes as , where denotes CovarianceFunction[proc,h].
• The following smoothing specifications sspec can be given:
•  c use c as a cutoff w use a window function w {c,w} use both a cutoff and a window function
• For a window function w and positive integer c, PowerSpectralDensity[data,ω,{c,w}] is computed as , where is defined as CovarianceFunction[data,h].
• By default, the cutoff c is chosen to be , where is the length of data, and the window function is DirichletWindow.
• A window function is an even function such that , , for , including standard windows such as HammingWindow, ParzenWindow, etc.
• A window function can be given as a list of values {w0,}, where , and it will be applied symmetrically in the vector case.
• PowerSpectralDensity takes the FourierParameters option. Common settings for FourierParameters include:
•  {1,1} default setting {-1,1} often used for time series {a,b} general setting

## ExamplesExamplesopen allclose all

### Basic Examples  (3)Basic Examples  (3)

Estimate the power spectral density for some data:

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Calculate the power spectral density for a univariate time series:

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The sample power spectral density for a random sample from autoregressive time series:

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Calculate power spectral density with cutoff:

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