# 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 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

# Examples

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## Basic Examples(3)

Estimate the power spectral density for some data:

Calculate the power spectral density for a univariate time series:

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

Calculate power spectral density with cutoff:

## Scope(14)

### Empirical Estimates(4)

Estimate the power spectral density for a univariate time series:

Power spectral density for a vector time series:

Power spectral density for each component:

Cross power spectral density between components:

Estimate the power spectral density for an ensemble of paths:

Compare empirical and theoretical power spectral densities functions:

### Smoothing(5)

Obtain a smoothed estimate using a cutoff at 5:

Compare the smoothed spectrum to the original:

Compute the power spectral density using a NuttallWindow:

Compare the smoothed spectrum to the original:

Define a window using a pure function:

Compare the smoothed spectrum to the original:

Estimate the power spectral density using specified window function values:

Compare to power spectral density with explicit TukeyWindow:

Compare the smoothed spectrum to the original:

Compute the power spectral density, given a cutoff and a window function:

Compare the smoothed spectrum to the original:

### Random Processes(5)

Power spectral density for an ARProcess:

Vector ARProcess:

Cross spectral density:

Power spectral density for an MAProcess:

Vector MAProcess:

Cross spectral density:

Power spectral density for an ARMAProcess:

Vector ARMAProcess:

Cross spectral density:

Power spectral density for a fractionally integrated time series:

Vector FARIMAProcess:

Cross spectral density:

Power spectral density for a seasonal time series:

Vector SARMAProcess:

Cross spectral density:

## Options(2)

The default value of FourierParameters:

Change FourierParameters:

It is the default value scaled:

## Applications(1)

Use power spectral density for estimating time series processes:

Use a smoothing window:

## Properties & Relations(11)

Power spectral density of a time series is a transform of the CovarianceFunction:

Compare to the power spectrum:

For a vector time series:

Power spectral density of data is a transform of the sample CovarianceFunction:

Compare to SamplePowerSpectralDensity:

For a vector values time series:

Power spectrum of white noise:

Compare to special case of an MAProcess:

Integrate to find the variance:

Compare to the variance of the time series:

Integrate to find the sample second moment:

Compare to the sample second moment:

Power spectral density for harmonic frequencies is related to PeriodogramArray:

Compare with PeriodogramArray:

For zero frequency:

For nonzero frequencies:

Diagonal elements of the power spectral density for vector data:

Compare to univariate power spectral density for each data component:

Power spectral density of a vector process is conjugate symmetric about zero:

Power spectral density of a univariate process is symmetric about zero:

Power spectral density of a vector process is Hermitian:

Also non-negative definite:

The magnitude of the sample cross spectral density is given by each component:

The determinant of the sample power spectral density is constant equal to zero:

Use TransferFunctionModel to calculate PowerSpectralDensity of a time series:

Define transfer function:

Calculate spectral density:

Check:

## Neat Examples(1)

Plot a product of two power spectral densities in 3D:

Wolfram Research (2012), PowerSpectralDensity, Wolfram Language function, https://reference.wolfram.com/language/ref/PowerSpectralDensity.html.

#### Text

Wolfram Research (2012), PowerSpectralDensity, Wolfram Language function, https://reference.wolfram.com/language/ref/PowerSpectralDensity.html.

#### BibTeX

@misc{reference.wolfram_2021_powerspectraldensity, author="Wolfram Research", title="{PowerSpectralDensity}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/PowerSpectralDensity.html}", note=[Accessed: 01-August-2021 ]}

#### BibLaTeX

@online{reference.wolfram_2021_powerspectraldensity, organization={Wolfram Research}, title={PowerSpectralDensity}, year={2012}, url={https://reference.wolfram.com/language/ref/PowerSpectralDensity.html}, note=[Accessed: 01-August-2021 ]}

#### CMS

Wolfram Language. 2012. "PowerSpectralDensity." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PowerSpectralDensity.html.

#### APA

Wolfram Language. (2012). PowerSpectralDensity. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PowerSpectralDensity.html