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PowerSpectralDensity

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:

Use FourierSequenceTransform:

Compare to the power spectrum:

For a vector time series:

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

Apply ListFourierSequenceTransform:

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:

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