Calculate the (cross-)power spectrum expectation of a windowed function from its global spectrum.

## Usage

outcspectra = SHBias (shh, incspectra, [ldata, lwin, save_cg])

## Returns

outcspectra : float, dimension (ldata+lwin+1)
The expectation of the localized (cross-)power spectrum.

## Parameters

shh : float, dimension (lwinin+1)
The power spectrum of the localizing window.
incspectra : float, dimension (ldatain+1)
The global unwindowed (cross-)power spectrum.
ldata : optional, integer, default = ldatain
The maximum degree of the global unwindowed power spectrum.
lwin : optional, integer, default = lwinin
The spherical harmonic bandwidth of the localizing window.
save_cg : optional, integer, default = 0
If set equal to 1, the Clebsch-Gordon coefficients will be precomputed and saved for future use (if lwin or ldata change, this will be recomputed). To deallocate the saved memory, set this parameter equal to -1. If set equal to 0 (default), the Clebsch-Gordon coefficients will be recomputed for each call.

## Description

SHBias will calculate the (cross-)power spectrum expectation of a function multiplied by a localizing window. This is given by equation 35 of Wieczorek and Simons (2005) and equation 2.11 of Wieczorek and Simons (2007),

<SFG> = Sum_{j=0}^L Shh Sum_{i=|l-j|}^{|l+j|} Sfg (C_{j0i0}^{l0})^2

where <SFG> is the expectation of the localized (cross-)power spectrum, Shh is the power spectrum of the window bandlimited to degree L, Sfg is the global unwindowed (cross-)power spectrum, and C is a Clebsch-Gordan coefficient. The Clebsch-Gordan coefficients are calculated using a simple relationship to the Wigner 3-j symbols. It is implicitly assumed that the power spectrum of inspectrum is zero beyond degree ldata. If this is not the case, the ouput power spectrum should be considered valid only for the degrees up to and including ldata - lwin.

If this routine is to be called several times using the same values of lwin and ldata, then the Clebsch-Gordon coefficients can be precomputed and saved by setting the optional parameter save_cg equal to 1. To deallocate the saved memory, which is a matrix of size (lwin+ldata,lwin,2*lwin+ldata+1), set save_cg equal to -1.

## References

Wieczorek, M. A. and F. J. Simons, Localized spectral analysis on the sphere, Geophys. J. Int., 162, 655-675, doi:10.1111/j.1365-246X.2005.02687.x, 2005.

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