Function index

The multitaper spectrum is given as a MTSpectrum structure which holds

• frequency (f),
• spectrum (S),
• phase (optional),
• chosen values of the multitaper time bandwidth product etc of type MTParameters (params)
• eigencoefficients (coef, optional),
• Ftest values (Fpval, optional),
• jackknife output (jkvar, optional), and
• Tsquared test results (Tsq_pval, optional).

The multitaper coherence structure, MTCoherence, holds

• frequency (f),
• coherence (coh),
• phase (phase),
• eigencoefficients (coef, optional),
• jackknife output (jkvar, optional), and
• Tsquared test results (Tsq, optional).

The multitaper transfer function is given as a MTTransferFunction structure which holds

• frequency (f),
• transfer function (transf),
• phase (phase),
• MTParameters (params),
• eigencoefficients (EigenCoefficientfs, optional),
• jackknife output (jkvar, optional), and
• Tsquared test results (Tsq, optional).

The EigenCoefficient structure holds

• multitaper eigencoefficients (coef) and, optionally,
• adaptive weights (wts)

Multitaper parameters MTParameters struct contains

• time bandwidth (NW) as Float,
• number of tapers (K),
• number of samples (N),
• sampling rate (dt) in temporal units (e.g. seconds),
• padded length (M),
• number of segments (nsegments), and
• overlap if nsegments is greater than 1, nothing otherwise.

Multitaper complex demodulates are held in the Demodulate struct which contains

• magnitude (Mag),
• phase (Phase)

The multitaper autocorrelation function is given in the MTAutocorrelationFunction structure which holds

• lags (lags),
• autocorrelation function (acf),
• MTParameters (params)

The multitaper autocovariance function is given in the MTAutocovarianceFunction structure, which holds

• lags (lags),
• autocovariance function (acvf),
• MTParameters (params)

The multitaper cepstrum is given in the MTCepstrum structure which holds

• lags (lags),
• cepstrum (ceps),
• MTParameters (params)

The multitaper cross-covariance function is given in the MtCrossCovarianceFunction structure which holds

• lags (lags),
• cross-covariance function (acvf),
• MTParameters (params)

The multitaper cross-correlation function is given as a MTCrossCorrelationFunction structure which holds

• lags (lags),
• cross-correlation function (acf),
• MTParameters (params)

dpss_tapers(n,w,k,tap_or_egval)

Simply compute discrete prolate spheroidal sequence tapers, eigenvalues

...

Arguments

• n::Int64: Length of the taper

• nw::Float64: Time-bandwidth product

• k::Int64: Number of tapers

• tap_or_egval::Symbol = :tap: Either :tap, :egval, or :both

...

...

Outputs

• vv::Vector{Float64}: The matrix of eigenvalues, if taporegval is set to :tap

• dpss_eigval: Struct conaining the dpss tapers

...

mdslepian(w, k, t)

Generalized prolate spheroidal sequences for the 1D missing data problem

...

Arguments

Positional Arguments

• w::Float64: the bandwidth

• k::Int64: number of Slepian tapers, must be <=2bwlength(x)

• t::Vector{Int64}: vector containing the time indices

...

...

Outputs

• lambda,u::Tuple{Vector{Float64}, Vector{Float64}}: tuple containing the

concentrations and the tapers

...

See also: mdmultispec, gpss

gpss(w, k, t, f; <keyword arguments>)

Generalized prolate spheroidal sequences on an unequal grid

...

Arguments

Positional Arguments

• w::Float64: the bandwidth

• k::Int64: number of Slepian tapers, must be <=2bwlength(x)

• t::Vector{Int64}: vector containing the time indices

• f::Float64: frequency at which the tapers are to be computed

Keyword Arguments

• beta::Float64 = 0.5: analysis half-bandwidth (similar to Nyquist rate)

...

...

Outputs

• lambda::Vector{Float64} the concentrations of the generalized prolate spheroidal

sequences

• u::Matrix{Float64} the matrix containing the sequences themselves

• R the Cholesky factor for the generalized eigenvalue problem

...

See also: gpss_orth, mdmultispec, mdslepian

multispec(S1; <keyword arguments>)

Computes univariate multitaper spectra with a handful of extra gadgets.

...

Arguments

• S1::Vector{T} where T<:Float64: the vector containing the time series
• NW::Float64 = 4.0: time-bandwidth product of estimate
• K::Int64 = 6: number of slepian tapers, must be <= 2*NW
• dt::Float64: sampling rate in time units
• ctr::Bool: whether or not to remove the mean before computing the multitaper spectrum
• pad::Float64 = 1.0: factor by which to pad the series, i.e. spectrum length will be pad times length of the time series.
• dpVec::Union{Matrix{Float64},Nothing} = nothing: Matrix of dpss's, if they have been precomputed
• egval::Union{Vector{Float64},Nothing} = nothing: Vector of concentratins of said dpss's
• guts::Bool = false: whether or not to return the eigencoefficients in the output struct
• a_weight::Bool = true: whether or not to use adaptive weighting
• Ftest::Bool = true: Compute the F-test p-value
• highres::Bool = false: Whether to return a "high resolution" spectrum estimate
• jk::Bool = true: Compute jackknifed confidence intervals
• Tsq::Union{Vector{Int64},Vector{Vector{Int64}},Nothing} = nothing: which frequency indices to compute the T-squared test for multiple line components. Defaults to none.

...

...

Outputs

• MTSpectrum struct containing the spectrum

...

See also: dpss_tapers, MTSpectrum, mdmultispec, mdslepian

multispec(S1, S2; <keyword arguments>)

Computes multitaper cross-spectrum or coherence when given two time series with same sampling.

...

Arguments

• S1::Union{Vector{T},EigenCoefficient} where T<:Number: the vector containing the first time

series

• S2::Union{Vector{P},EigenCoefficient} where P<:Number: the vector containing the second

time series

• outp::Symbol: output can be either :coh for coherence, :spec for cross-spectrum,

or :transf for transfer function

• NW::Float64 = 4.0: time-bandwidth product of estimate

• K::Int64 = 6: number of slepian tapers, must be <= 2*NW

• offset::Union{Float64,Int64} = 0 set to nonzero value if offset coherence or

cross-spectrum is desired. If Float64 is used, this will be converted to nearest FFT bin.

• dt::Float64: sampling rate in time units

• ctr::Bool: whether or not to remove the mean before computing the multitaper

spectrum

• pad::Float64 = 1.0: factor by which to pad the series, i.e. spectrum length will

be pad times length of the time series.

• dpVec::Union{Matrix{Float64},Nothing} = nothing: Matrix of dpss's, if they have

been precomputed

• guts::Bool = false: whether or not to return the eigencoefficients in the output

struct

• jk::Bool = true: Compute jackknifed confidence intervals

• Tsq::Union{Vector{Int64},Vector{Vector{Int64}},Nothing} = nothing: which

frequency indices to compute the T-squared test for multiple line components. Defaults to none.

• alph::Float64 = 0.05: significance cutoff for the Tsquared test

...

...

Outputs

• MTSpectrum, MTCoherence, or MTTransferFunction struct containing the spectrum, coherence or

transfer function, depending on the selection of outp input.

...

See also: dpss_tapers, MTSpectrum, mdmultispec, mdslepian

multispec(S1; <keyword arguments>)

Multivariate version of the multispec call, data are in the columns of a matrix ...

Arguments

• S1::Matrix{T} where T<:Float64: the vector containing the first time series
• outp::Symbol: output can be either :coh for coherence, :justspeccs to compute just the spectra, or :cross for cross-spectra
• NW::Float64 = 4.0: time-bandwidth product of estimate
• K::Int64 = 6: number of slepian tapers, must be <= 2*NW
• dt::Float64: sampling rate in time units
• ctr::Bool: whether or not to remove the mean before computing the multitaper spectrum
• pad::Float64 = 1.0: factor by which to pad the series, i.e. spectrum length will be pad times length of the time series.
• dpVec::Union{Matrix{Float64},Nothing} = nothing: Matrix of dpss's, if they have been precomputed
• guts::Bool = false: whether or not to return the eigencoefficients in the output struct
• a_weight::Bool = true: whether or not to adaptively weight the spectra
• jk::Bool = false: Compute jackknifed confidence intervals
• Ftest:Bool = false: Compute F-test for line components
• Tsq::Union{Vector{Int64},Vector{Vector{Int64}},Nothing} = nothing: which frequency indices to compute the T-squared test for multiple line components. Defaults to none.
• alph::Float64 = 0.05: significance cutoff for the Tsquared test

...

...

Outputs

• Tuple{Vector{MTSpectrum},Vector{P},Union{Float64,Vector{Float64}}} where P = Union{MTCoherence,MTSpectrum}

struct containing the spectra, coherence or crossspectra, and Tsquared test p-values. Ouput of middle arg depends on the selection of outp input. ...

See also: dpss_tapers, MTSpectrum, mdmultispec, mdslepian

mdmultispec(tt, x; <keyword arguments>)

Multitaper power spectrum estimation for time series with missing data (gaps)

...

Arguments

Positional Arguments

• tt::Vector{T} where T<:Real: the vector containing the time indices

• x::Vector{P} where P<:Number: data vector

Keyword Arguments

• bw = 5/length(tt): bandwidth of estimate

• k::Int64 = 2*bw*length(x)-1: number of Slepian tapers, must be `<=

2bwlength(x)`

• dt::T = tt[2]-tt[1]: sampling rate in time units

• nz = 0.0: zero padding factor

• Ftest::Bool = true: Compute the F-test p-value

• jk::Bool = true: Compute jackknifed confidence intervals

• dof::Bool = false: Compute degrees of freedom for the adaptively weighted

spectrum estimate

• lambdau::Union{Tuple{Array{Float64,1},Array{Float64,2}},Nothing} = nothing:

Slepians, if precomputed ...

...

Outputs

• pkg::MTSpectrum struct containing the spectrum

• nu1::Vector{Float64} optional vector containing the degrees of freedom, given

if the dof kwarg is set to true.

...

See also: multispec, mdslepian

mdmultispec(tt, x, y; <keyword arguments>)

Multitaper coherence estimation for time series with missing data (gaps)

...

Arguments

Positional Arguments

• tt::Vector{T} where T<:Real: the vector containing the time indices

• x::Vector{P} where P<:Number: data vector 1

• y::Vector{Q} where Q<:Number: data vector 2

Keyword Arguments

• bw = 5/length(tt): bandwidth of estimate

• k::Int64 = 2*bw*length(x)-1: number of Slepian tapers, must be `<=

2bwlength(x)`

• dt = tt[2]-tt[1]: sampling rate in time units

• nz = 0.0: zero padding factor

• Ftest::Bool = true: Compute the F-test p-value

• jk::Bool = true: Compute jackknifed confidence intervals

• lambdau::Union{Tuple{Array{Float64,1},Array{Float64,2}},Nothing} = nothing:

Slepians, if precomputed

...

...

Outputs

• pkg::MTCoherence struct containing the coherence

...

See also: multispec, mdslepian

mdmultispec(tt, x; <keyword arguments>)

Multitaper coherence estimation for multiple time series with the same missing data (gaps)

...

Arguments

Keyword Arguments

• tt::Vector{T} where T<:Real: the vector containing the time indices

• x::Matrix{P} where P<:Number: time series in the columns of a matrix

Positional Arguments

• bw = 5/length(tt): bandwidth of estimate

• k::Int64 = 2*bw*length(x)-1: number of Slepian tapers, must be `<=

2bwlength(x)`

• dt = tt[2]-tt[1]: sampling rate in time units

• nz = 0.0: zero padding factor

• Ftest::Bool = false: Compute the F-test p-value

• jk::Bool = true: Compute jackknifed confidence intervals

• lambdau::Union{Tuple{Array{Float64,1},Array{Float64,2}},Nothing} = nothing:

Slepians, if precomputed

...

...

Outputs

• Tuple{Vector{MTSpectrum},Matrix{MTCoherence},Nothing} struct containing the spectra,

coherences, and T^2 test significances (currently set to return nothing)

...

See also: multispec, mdslepian

welch(S1, nsegments; <keyword arguments>)

Computes univariate multitaper Welch specrum starting with an input time series.

...

Arguments

• S1:Vector{T} where T<:Number: the vector containing the time series

• nsegments::Int64: the number of segments into which to divide the time series

• overlap::Float64 = 0.5: number between 0 and 1 which accounts for the amount of

overlap between the segments

• NW::Float64 = 4.0: time-bandwidth product of estimate

• K::Int64 = 6: number of slepian tapers, must be <= 2*NW

• dt::Float64: sampling rate in time units

• ctr::Bool: whether or not to remove the mean before computing the multitaper

spectrum

• pad::Float64 = 1.0: factor by which to pad the series, i.e. spectrum length will

be pad times length of the time series.

• dpVec::Union{Matrix{Float64},Nothing} = nothing: Matrix of dpss's, if they have

been precomputed

• egval::Union{Vector{Float64},Nothing} = nothing: Vector of concentratins of said

dpss's

• guts::Bool = false: whether or not to return the eigencoefficients in the output

struct

• a_weight::Bool = true: whether or not to use adaptive weighting

...

...

Outputs

• MTSpectrum struct, depending on the selection of outp above

• Float64 containing the effective bandwidth

...

See also: multispec

mt_acf(S)

Computes univariate multitaper autocorrelation function. Inputs a MTSpectrum struct.

...

Arguments

• S::MTSpectrum: the vector containing the result of an univariate call to multispec

...

...

Outputs

• MTAutocorrelationFunction struct containing the autocorrelation function

...

See also: multispec

mt_acf(S1; <keyword arguments>)

Computes univariate multitaper autocorrelation function starting with an input time series. ...

Arguments

• S1::Vector{T} where T<:Number: the vector containing the time series
• NW::Float64 = 4.0: time-bandwidth product of estimate
• K::Int64 = 6: number of slepian tapers, must be <= 2*NW
• dt::Float64: sampling rate in time units
• ctr::Bool: whether or not to remove the mean before computing the multitaper spectrum
• pad::Float64 = 1.0: factor by which to pad the series, i.e. spectrum length will be pad times length of the time series.
• dpVec::Union{Matrix{Float64},Nothing} = nothing: Matrix of dpss's, if they have been precomputed
• egval::Union{Vector{Float64},Nothing} = nothing: Vector of concentratins of said dpss's
• a_weight::Bool = true: whether or not to use adaptive weighting

...

...

Outputs

• MTAutocorrelationFunction struct containing the autocorrelation function

...

See also: multispec

mt_acvf(S)

Computes univariate multitaper autocovariance function. Inputs a MTSpectrum struct.

...

Arguments

• S::MTSpectrum: the vector containing the result of an univariate call to multispec

...

...

Outputs

• MTAutocovarianceFunction struct containing the autocovariance function.

...

See also: multispec

mt_acvf(S1; <keyword arguments>)

Computes univariate multitaper autocovariance function starting with an input time series. ...

Arguments

• S1::Vector{T} where T<:Number: the vector containing the time series
• NW::Float64 = 4.0: time-bandwidth product of estimate
• K::Int64 = 6: number of slepian tapers, must be <= 2*NW
• dt::Float64: sampling rate in time units
• ctr::Bool: whether or not to remove the mean before computing the multitaper spectrum
• pad::Float64 = 1.0: factor by which to pad the series, i.e. spectrum length will be pad times length of the time series.
• dpVec::Union{Matrix{Float64},Nothing} = nothing: Matrix of dpss's, if they have been precomputed
• egval::Union{Vector{Float64},Nothing} = nothing: Vector of concentratins of said dpss's
• a_weight::Bool = true: whether or not to use adaptive weighting

...

...

Outputs

• MTAutocovarianceFunction struct containing the autocovariance function

...

See also: multispec

mt_ccvf(S; <keyword arguments>)

Computes univariate multitaper cross-covariance/cross-correlation function. Inputs a MTSpectrum struct.

...

Arguments

• S::MTSpectrum: the vector containing the result of an multiivariate call to multispec
• typ::Symbol = :ccvf: whether to compute cross-correlation function (:ccf) or cross-covariance function (:ccvf)

...

...

Outputs

• MtCrossCovarianceFunction, MTCrossCorrelationFunction depending on the selection of typ input above.

...

See also: multispec

mt_ccvf(S1, S2; <keyword arguments>)

Computes bivariate multitaper cross-covariance/cross-correlation function from two time series

...

Arguments

• S1::Vector{T} where T<:Number: the vector containing the first time series
• S2::Vector{T} where T<:Number: the vector containing the second time series
• typ::Symbol: whether to compute cross-covariance function (:ccvf), or cross-correlation function (:ccf)
• NW::Float64 = 4.0: time-bandwidth product of estimate
• K::Int64 = 6: number of slepian tapers, must be <= 2*NW
• dt::Float64: sampling rate in time units
• ctr::Bool: whether or not to remove the mean before computing the multitaper spectrum
• pad::Float64 = 1.0: factor by which to pad the series, i.e. spectrum length will be pad times length of the time series.
• dpVec::Union{Matrix{Float64},Nothing} = nothing: Matrix of dpss's, if they have been precomputed

...

...

Outputs

• MtCrossCovarianceFunction struct, depending on the selection of typ input above.

...

See also: multispec

mt_cepstrum(S)

Computes multitaper cepstrum. Inputs a MTSpectrum struct.

...

Arguments

• S::MTSpectrum: the vector containing the result of an univariate call to multispec

...

...

Outputs

• MTCepstrum struct containing the cepstrum

...

See also: multispec

mt_cepstrum(S1; <keyword arguments>)

Computes multitaper cepstrum starting with an input time series. ...

Arguments

• S1::Vector{T} where T<:Number: the vector containing the time series
• NW::Float64 = 4.0: time-bandwidth product of estimate
• K::Int64 = 6: number of slepian tapers, must be <= 2*NW
• dt::Float64: sampling rate in time units
• ctr::Bool: whether or not to remove the mean before computing the multitaper spectrum
• pad::Float64 = 1.0: factor by which to pad the series, i.e. spectrum length will be pad times length of the time series.
• dpVec::Union{Matrix{Float64},Nothing} = nothing: Matrix of dpss's, if they have been precomputed
• egval::Union{Vector{Float64},Nothing} = nothing: Vector of concentratins of said dpss's
• a_weight::Bool = true: whether or not to use adaptive weighting

...

...

Outputs

• MTCepstrum struct containing the cepstrum

...

See also: multispec

demodulate(x, f0, NW, blockLen; <keyword arguments>)

Multitaper complex demodulation

...

Arguments

• x::Vector{T} where T<:Float64: the vector containing the time series
• f0::Float64: center frequency for the complex demodulate
• NW::Float64: the time bandwidth product
• blockLen::Int64: the length of the blocks to use
• wrapphase::Bool = true: whether or not to unwrap the phase
• dt::Float64 = 1.0: the sampling rate of the series, in time units
• basetime::Float64 = 0.0: the time at which the time series begins

...

...

Outputs

• A struct of type Demodulate

...

See also: Demodulate, and relevant plots recipe demodplot

tanhtrans(trcsq,ntf)

Inverse of the magnitude squared coherence variance stabilizing transform

...

Arguments

• trcsq::Float64: The transformed squared coherence
• ntf::Int64: the number of tapers use to compute the coherence

...

...

Outputs

• The output is a Float64 indicating the reverse-transformed MSC

...

Example

If the transformed coherence is 7.3 and 6 dof, the significance is

julia> invmscsig(tanhtrans(7.3,6),6)

If the significance is 0.9 and 6 dof the transformed msc is

julia> atanhtrans(sqrt(mscsig(0.9,6)),6)

See also: multispec

atanhtrans(c,ntf)

Magnitude squared coherence variance stabilizing transform

...

Arguments

• c::Float64: the value of the coherence
• ntf::Int64: the number of tapers use to compute the coherence

...

...

Outputs

• The output is a Float64 indicating the transformed MSC

...

See also: multispec

ejn(DoF)

Expected jackknife variance of an univariate multitaper spectrum estimate with DoF tapers

...

Arguments

• DoF::Int64: the number of tapers use to compute the spectrum

...

...

Outputs

• The output is a Float64 indicating the expected jackknife variance.

...

See also: multispec

unwrapphase(y, typ)

Unwrap the phase

...

Arguments

• y::Vector{Float64}: The vector of phases

• typ::Symbol: Whether to compute the unwrapped phase in degrees or radians

...

...

Outputs

• x::Vector{Float64}: The vector of unwrapped phases

...

Alternative phase unwrapper which takes a complex argument.

blockerr(lengt, nsegmentts; <keyword arguments>)

Blocker code to divide the data into segments

...

Arguments

• lengt::Int64: input length of a time series
• nsegments::Int64: number of segments
• overlap::Float64 = 0.0: fraction of overlap between segments, between 0.0 and 1.0

...

...

Outputs

• seq::Vector{Int64}: Beginning index for each segment
• seg_len::Int64: length of the segments
• ov::Float64: overlap that actually resulted (≈overlap, above)

...

Pgram_upcrossings(z, N)

Number of upcrossings of the level z of a periodogram spectrum per Rayleigh resolution ...

Arguments

• z::Float64: the level of the standardized spectrum
• N::Int64: sampling rate in time units

...

...

Outputs

• Float64: the number of upcrossings per Rayleigh

...

MT_Upcrossings(z, α, CR, N)

Number of upcrossings of the level z of a multitaper spectrum per Rayleigh resolution ...

Arguments

• z::Float64: the level of the standardized spectrum
• α::Float64: degrees of freedom of estimate
• `CR::Float64: time bandwidth product
• N::Int64: sampling rate in time units

...

...

Outputs

• Float64: the number of upcrossings per Rayleigh

...

Gives an upcrossing table for α = K, CR = NW (Float), optionally num_Ray has default 1e5, and signficiance levels that can be chosen, or set to default

bspec(times, dat, W, K, beta, nz, Ftest)

Computes the Bronez spectrum of an unequally-spaced time series

...

Positional Arguments

• times::Vector{T} where T<: Number: the vector containing the times
• dat::Vector{T} where T<:Number: the vector containing the time series
• W::Float64: bandwidth of estimate
• K::Int64: number of slepian tapers, must be <= 2*NW
• beta::Float64: estimated process bandwidth (aka Nyquist rate)
• nz::Union{Float64,Int64} = length(t): Number of frequencies
• Ftest::Bool = false: Whether to compute the F-test or not

...

...

Outputs

• MTSpectrum struct containing the Bronez spectrum

...

...

Example usage

N = 256
t = collect(1:N).^(1.05)
W = 0.008
K = 5
x = randn(N)
bet = 0.5 / (last(t) / (N-1))
M = 2*N
S = bspec(t, x, W, K, bet, nz, true)

...

See also: multispec, mdmultispec, mdslepian, gpss

bspec(time, dat1, dat2, W, K, bet, nz; <keyword arguments>)

Computes the Bronez coherence or cross-spectrum of two unequally-spaced time series

...

Positional Arguments

• time::Vector{T} where T<:Number: the vector containing the times
• dat1::Union{Vector{P}, EigenCoefficient} where P<:Number: the vector containing the first time series
• dat2::Union{Vector{P}, EigenCoefficient} where P<:Number: the vector containing the second time series
• W::Float64: bandwidth of estimate
• K::Int64: number of slepian tapers, must be <= 2*NW
• bet::Float64: Nyquist frequency
• nz::Union{Float64,Int64} = length(t): Number of frequencies

Keyword Arguments

• outp::Symb = :coh: Output, either :cross for cross spectrum or :coh (default)
• params::Union{MTParameters,Nothing} = nothing: parameters struct, important when x,y are EigenCoefficients
• Ftest::Bool = false: Whether to compute the F-test or not

...

...

Outputs

• MTCoherence containing the Bronez coherence

...

...

Example usage

N = 256
t = collect(1:N).^(1.05)
W = 0.008
K = 5
x = randn(N)
y = randn(N) # Incoherent
M = 2*N
beta = 0.5
S = bspec(t, x, y, W, K, M, beta)

...

See also: multispec, mdmultispec, mdslepian, gpss

bspec(time, dat, W, K, bet, nz; <keyword arguments>)

Computes the Bronez spectra and coherences of p unequally-spaced time series

...

Positional Arguments

• time::Vector{T} where T<:Number: the vector containing the times
• dat::Matrix{Vector{P}} where T<:Number: the matrix containing the time series in its columns
• W::Float64: bandwidth of estimate
• K::Int64: number of slepian tapers, must be <= 2*NW
• bet::Float64: Nyquist frequency
• nz::Union{Float64,Int64} = length(t): Number of frequencies

Keyword Arguments

• outp::Symb = :coh: Output, either :cross for cross spectrum or :coh (default)
• Ftest::Bool = false: Whether to compute the F-test or not

...

...

Outputs

• tuple(Vector{MTSpectrum},Matrix{MTCoherence},Nothing) containing the Bronez spectra and coherences

...

...

Example usage

N = 256
t = collect(1:N).^(1.05)
W = 0.008
K = 5
x = randn(N)
y = randn(N) # Incoherent
M = 2*N
beta = 0.5
S = bspec(t, hcat(x, y), W, K, beta, M, outp = :coh, Ftest = true)

...

See also: multispec, mdmultispec, mdslepian, gpss