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:

The default value of FourierParameters :

Change FourierParameters :

It is the default value scaled:

Use power spectral density for estimating time series processes:

Use a smoothing window:

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:

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