# Periodograms - periodogram estimation

DSP.Periodograms.arraysplitFunction
arraysplit(s, n, m)

Split an array into arrays of length n with overlapping regions of length m. Iterating or indexing the returned AbstractVector always yields the same Vector with different contents.

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DSP.Periodograms.periodogramMethod
periodogram(s; onesided=eltype(s)<:Real, nfft=nextfastfft(n), fs=1, window=nothing)

Computes periodogram of a signal by FFT and returns a Periodogram object.

For real signals, the two-sided periodogram is symmetric and this function returns a one-sided (real only) periodogram by default. A two-sided periodogram can be obtained by setting onesided=false.

nfft specifies the number of points to use for the Fourier transform. If length(s) < nfft, then the input is padded with zeros. By default, nfft is the closest size for which the Fourier transform can be computed with maximal efficiency.

fs is the sample rate of the original signal, and window is an optional window function or vector to be applied to the original signal before computing the Fourier transform. The computed periodogram is normalized so that the area under the periodogram is equal to the uncentered variance (or average power) of the original signal.

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DSP.Periodograms.welch_pgramFunction
welch_pgram(s, n=div(length(s), 8), noverlap=div(n, 2); onesided=eltype(s)<:Real, nfft=nextfastfft(n), fs=1, window=nothing)

Computes the Welch periodogram of a signal s based on segments with n samples with overlap of noverlap samples, and returns a Periodogram object. For a Bartlett periodogram, set noverlap=0. See periodogram for description of optional keyword arguments.

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DSP.Periodograms.spectrogramFunction
spectrogram(s, n=div(length(s), 8), noverlap=div(n, 2); onesided=eltype(s)<:Real, nfft=nextfastfft(n), fs=1, window=nothing)

Computes the spectrogram of a signal s based on segments with n samples with overlap of noverlap samples, and returns a Spectrogram object. See periodogram for description of optional keyword arguments.

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DSP.Periodograms.stftFunction
stft(s, n=div(length(s), 8), noverlap=div(n, 2); onesided=eltype(s)<:Real, nfft=nextfastfft(n), fs=1, window=nothing)

Computes the STFT of a signal s based on segments with n samples with overlap of noverlap samples, and returns a matrix containing the STFT coefficients. See periodogram for description of optional keyword arguments.

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DSP.Periodograms.periodogramMethod
periodogram(s::AbstractMatrix; nfft=nextfastfft(size(s)), fs=1, radialsum=false, radialavg=false)

Computes periodogram of a 2-d signal by FFT and returns a Periodogram2 object.

Returns a 2-d periodogram by default. A radially summed or averaged periodogram is returned as a Periodogram object if radialsum or radialavg is true, respectively.

nfft specifies the number of points to use for the Fourier transform. If size(s) < nfft, then the input is padded with zeros. By default, nfft is the closest size for which the Fourier transform can be computed with maximal efficiency. fs is the sample rate of the original signal in both directions.

For radialsum=true the value of power[k] is proportional to $\frac{1}{N}\sum_{k\leq |k'|<k+1} |X[k']|^2$. For radialavg=true it is proportional to $\frac{1}{N \#\{k\leq |k'|<k+1\}} \sum_{k\leq |k'|<k+1} |X[k']|^2$. The computation of |k'| takes into account non-square signals by scaling the coordinates of the wavevector accordingly.

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DSP.Periodograms.freqFunction
freq(p)

Returns the frequency bin centers for a given Periodogram, Spectrogram, CrossPowerSpectra, or Coherence object.

Returns a tuple of frequency bin centers for a given Periodogram2 object.

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DSP.Periodograms.powerFunction
power(p)

For a Periodogram, returns the computed power at each frequency as a Vector.

For a Spectrogram, returns the computed power at each frequency and time bin as a Matrix. Dimensions are frequency × time.

For a CrossPowerSpectra, returns the pairwise power between each pair of channels at each frequency. Dimensions are channel x channel x frequency.

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DSP.Periodograms.coherenceFunction
coherence(c::Coherence)

Given an Coherence object, returns an n_channels x n_channels x length(freq(c)) array consisting of the pairwise coherences between each channel for each frequency.

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## Multitaper periodogram estimation

DSP.Periodograms.mt_pgramFunction
mt_pgram(s; onesided=eltype(s)<:Real, nfft=nextfastfft(n), fs=1, nw=4, ntapers=iceil(2nw)-1, window=dpss(length(s), nw, ntapers))
mt_pgram(signal::AbstractVector, config::MTConfig)

Computes the multitaper periodogram of a signal s.

If window is not specified, the signal is tapered with ntapers discrete prolate spheroidal sequences with time-bandwidth product nw. Each sequence is equally weighted; adaptive multitaper is not (yet) supported.

If window is specified, each column is applied as a taper. The sum of periodograms is normalized by the total sum of squares of window.

Returns a Periodogram.

See also mt_pgram! and MTConfig.

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DSP.Periodograms.mt_pgram!Function
mt_pgram!(output, s::AbstractVector{T}; onesided::Bool=eltype(s)<:Real,
nfft::Int=nextfastfft(length(s)), fs::Real=1,
nw::Real=4, ntapers::Int=ceil(Int, 2nw)-1,
window::Union{AbstractMatrix,Nothing}=nothing) where T<:Number
mt_pgram!(output::AbstractVector, signal::AbstractVector, config::MTConfig) -> Periodogram

Computes a multitapered periodogram, storing the output in output. Arguments:

• signal::AbstractVector: should be of length config.n_samples
• output::AbstractVector: should be of length length(config.freq)

Optionally pass an MTConfig object to preallocate temporary variables and choose configuration settings; otherwise, keyword arguments may be passed to choose those settings.

Returns a Periodogram.

See also mt_pgram and MTConfig.

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DSP.Periodograms.mt_spectrogramFunction
mt_spectrogram(signal::AbstractVector{T}, n::Int=length(s) >> 3,
n_overlap::Int=n >> 1; fs::Int=1,
onesided::Bool=T <: Real, kwargs...) where {T}
mt_spectrogram(signal::AbstractVector, config::MTSpectrogramConfig)

Compute a multitaper spectrogram, returning a Spectrogram object. Optionally pass a MTSpectrogramConfig object; otherwise, any additional keyword arguments accepted by MTConfig may be passed to configure the tapering.

Returns a Spectrogram.

See also mt_spectrogram!.

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DSP.Periodograms.mt_spectrogram!Function
mt_spectrogram!(output, signal::AbstractVector{T}, n::Int=length(signal) >> 3,
n_overlap::Int=n >> 1; fs::Int=1, onesided::Bool=T <: Real, kwargs...) where {T}
mt_spectrogram!(destination::AbstractMatrix, signal::AbstractVector, config::MTSpectrogramConfig)

Computes a multitaper spectrogram using the parameters specified in config. Arguments:

• destination: length(config.mt_config.freq) x length(config.time) matrix. This can be created by DSP.allocate_output(config).
• signal: vector of length config.n_samples
• config: optionally, pass an MTSpectrogramConfig object to hold temporary variables and configuration settings. Otherwise, settings arguments may be passed directly.

Returns a Spectrogram.

See also mt_spectrogram.

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DSP.Periodograms.mt_cross_power_spectraFunction
mt_cross_power_spectra(signal::AbstractMatrix{T}; fs=1, kwargs...) where {T}
mt_cross_power_spectra(signal::AbstractMatrix, config::MTCrossSpectraConfig)

Computes multitapered cross power spectra between channels of a signal. Arguments:

and choose configuration settings. Otherwise, any keyword arguments accepted by MTCrossSpectraConfig may be passed here.

Produces a CrossPowerSpectra object holding the n_channels x n_channels x n_frequencies output array (accessed by power) and the corresponding frequencies (accessed by freq).

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DSP.Periodograms.mt_cross_power_spectra!Function
mt_cross_power_spectra!(output, signal::AbstractMatrix; fs=1, kwargs...)
mt_cross_power_spectra!(output, signal::AbstractMatrix, config::MTCrossSpectraConfig)

Computes multitapered cross power spectra between channels of a signal. Arguments:

• output: n_channels x n_channels x length(config.freq). Can be created by DSP.allocate_output(config).
• signal: n_channels x n_samples
• config: MTCrossSpectraConfig{T}: optionally pass a MTCrossSpectraConfig to preallocate temporary and choose configuration settings. Otherwise, one may pass any keyword arguments accepted by this object.

Produces a CrossPowerSpectra object holding the n_channels x n_channels x n_frequencies output array and the corresponding frequencies (accessed by freq).

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DSP.Periodograms.mt_coherenceFunction
mt_coherence(signal::AbstractMatrix{T}; fs=1, freq_range = nothing, demean=false, kwargs...) where T
mt_coherence(signal::AbstractMatrix, config::MTCoherenceConfig)

Arguments:

• signal: n_channels x n_samples matrix
• Optionally pass an MTCoherenceConfig to pre-allocate temporary variables and choose configuration settings, otherwise, see MTCrossSpectraConfig for the meaning of the keyword arguments.

Returns a Coherence object.

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DSP.Periodograms.mt_coherence!Function
mt_coherence!(output, signal::AbstractMatrix; fs=1, freq_range=nothing, demean=false, kwargs...)
mt_coherence!(output, signal::AbstractMatrix, config::MTCoherenceConfig)

Computes the pairwise coherences between channels.

• output: n_channels x n_channels matrix
• signal: n_samples x n_channels matrix
• config: optional configuration object that pre-allocates temporary variables and choose settings.

Returns a Coherence object.

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### Configuration objects

DSP.Periodograms.MTConfigType
MTConfig{T}(n_samples; fs=1,
nfft = nextpow(2, n_samples),
window = nothing,
nw = 4,
ntapers = 2 * nw - 1,
taper_weights = fill(1/ntapers, ntapers),
onesided::Bool=T<:Real,
fft_flags = FFTW.MEASURE)

Creates a config object which holds the configuration state and temporary variables used in multitaper computations, e.g. mt_pgram!, mt_spectrogram, MTSpectrogramConfig, MTCrossSpectraConfig, and MTCoherenceConfig.

An MTConfig can be re-used between computations as long as none of the input arguments change.

• n_samples: the number of samples to be used as input when computing multitaper periodograms with this configuration. Used for pre-allocating temporary buffers.
• fs: the number of samples per second of the input signal
• nfft: length of input vector to the FFT; if nfft > n_samples, then the input signal will be zero-padded until it is of length nfft.
• window: window function to use for tapering. If left at the default of nothing, window will be set to dpss(n_samples, nw, ntapers).
• ntapers: the number of tapers to use.
• taper_weights = fill(1/ntapers, ntapers): how to weight the contribution of each taper. The default setting is to simply average them.
• onesided: whether or not to compute a "one-sided" FFT by using that real signal data yields conjugate-symmetry in Fourier space.
• fft_flags: flags to control how the FFT plan is generated.
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DSP.Periodograms.MTSpectrogramConfigType
MTSpectrogramConfig(n_samples, mt_config::MTConfig{T}, n_overlap_samples) where {T}
MTSpectrogramConfig{T}(n_samples, samples_per_window, n_overlap_samples; fs=1, kwargs...) where {T}

Creates a MTSpectrogramConfig which holds configuration and temporary variables for mt_spectrogram and mt_spectrogram!. Any keyword arguments accepted by MTConfig may be passed here, or an MTConfig object itself.

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DSP.Periodograms.MTCrossSpectraConfigType
MTCrossSpectraConfig{T}(n_channels, n_samples; fs=1, demean=false, freq_range=nothing,
ensure_aligned = T == Float32 || T == Complex{Float32}, kwargs...) where {T}
MTCrossSpectraConfig(n_channels, mt_config::MTConfig{T}; demean=false, freq_range=nothing,
ensure_aligned = T == Float32 || T == Complex{Float32})

Creates a configuration object used for mt_cross_power_spectra and mt_cross_power_spectra!.

• n_channels: the number of channels of the input.
• n_samples: the number of samples for each channel of the input.
• demean: if true, the channelwise mean will be subtracted from the input signals before the cross spectral powers are computed.
• freq_range: if nothing, all frequencies are retained. Otherwise, only frequencies between first(freq_range) and last(freq_range) are retained.
• ensure_aligned = T == Float32 || T == Complex{Float32}: perform an extra copy to ensure that the FFT output is memory-aligned
• Additionally, either pass an MTConfig object, or keyword arguments such as fs accepted by MTConfig.

Returns a CrossPowerSpectra object.

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DSP.Periodograms.MTCoherenceConfigType
MTCoherenceConfig{T}(n_channels, n_samples; fs=1, demean=false, freq_range=nothing, kwargs...) where T
MTCoherenceConfig(cs_config::MTCrossSpectraConfig{T}) where {T}
MTCoherenceConfig(n_channels, mt_config::MTConfig{T}; demean=false, freq_range=nothing,
ensure_aligned = T == Float32 || T == Complex{Float32}) where {T}

Creates a configuration object for coherences from a MTCrossSpectraConfig. Provides a helper method with the same arugments as MTCrossSpectraConfig to construct the MTCrossSpectraConfig object.

See also mt_coherence and mt_coherence!.

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