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shbias.doc
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shbias.doc
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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.