This routine returns the multitaper coupling matrix for a given set of power spectra of arbitrary localization windows. This matrix relates the expectation of the localized multitaper spectrum to the expectation of the power spectrum of the global function.

## Usage

Mmt = SHMTCouplingMatrix (lmax, tapers_power, [lwin, k, taper_wt])

## Returns

Mmt : float, dimension (lmax+lwin+1, lmax+1)
The full multitaper coupling matrix that relates the expectation of the localized multitaper spectrum to the global power spectrum of the function.

## Parameters

lmax : integer
The spherical harmonic bandwidth of the global power spectrum.
tapers_power : float, dimension (lwinin+1, kin)
An array of power spectra of the k windowing functions, arranged in columns.
lwin : optional, integer, default = lwinin
The spherical harmonic bandwidth of the windowing functions in the array tapers.
k : optional, integer, default = kin
The number of tapers utilized in the multitaper spectral analysis.
taper_wt : optional, float, dimension (kin)
The weights used in calculating the multitaper spectral estimates. Optimal values of the weights (for a known global power spectrum) can be obtained from the routine SHMTVarOpt.

## Description

SHMTCouplingMatrix returns the multitaper coupling matrix that relates the expectation of the localized multitaper spectrum to the expectation of the global power spectrum of the function (assumed to be stationary). This is given by eqs 4.5 and 4.6 in Wieczorek and Simons (2007):

< S_{Phi Phi}^(mt) > = M^(mt) S_{ff}

where S_{Phi Phi} is a vector containing the lmax+lwin+1 localized multitaper power spectral estiamtes, S_{ff} is a vector of the global power spectrum up to degree lmax, and < ... > is the expectation operator. The coupling matrix is given explicitly by

M_{ij} = Sum_{l=0}^L Sum_{k=1}^K a_k S_{hh}^{k}(l) [ C_{l0j0}^{i0} ]^2

where a_k are the taper weights, S_{hh} is the power of the window, and C is a Clebsch-Gordon coefficient.

Note that this routine returns the “full” coupling matrix of dimension (lmax + lwin + 1, lmax + 1). When multiplied by a global input power spectrum with bandwidth lmax, it returns the output power spectrum with a bandwidth of lmax + lwin. In doing so, it is implicitly assumed that input power spectrum is exactly zero for all degrees greater than lmax. If this is not the case, the ouput power spectrum should be considered valid only for the degrees up to and including lmax - lwin.

Wieczorek, M. A. and F. J. Simons, Minimum-variance multitaper spectral estimation on the sphere, J. Fourier Anal. Appl., 13, 665-692, doi:10.1007/s00041-006-6904-1, 2007.