Calculate the expectation of the product of two functions, each multiplied by a different data taper, for a given spherical harmonic degree and two different angular orders.
Usage
value = SHSjkPG (incspectra, l, m, mprime, hj_real, hk_real, mj, mk, lwin, hkcc)
Parameters
value: output, complex(dp)- The expectation of the product of two functions, each multiplied by a different data taper, for a given spherical harmonic degree and two different angular orders.
incspectra: input, real(dp), dimension (l+lwin+1)- The global cross-power spectrum of
fandg. l: input, integer(int32)- The spherical harmonic degree for which to calculate the expectation.
m: input, integer(int32)- The angular order of the first localized function,
Phi. mprime: input, integer(int32)- The angular order of the second localized function,
Gamma. hj_real: input, real(dp), dimension (lwin+1)- The real spherical harmonic coefficients of angular order
mjused to localize the first functionf. These are obtained by a call toSHReturnTapers. hk_real: input, real(dp), dimension (lwin+1)- The real spherical harmonic coefficients of angular order
mkused to localize the second functiong. These are obtained by a call toSHReturnTapers. mj: input, integer(int32)- The angular order of the window coefficients
hj_real. mk: input, integer(int32)- The angular order of the window coefficients
hk_real. lwin: input, integer(int32)- the spherical harmonic bandwidth of the localizing windows
hj_realandhk_real. hkcc: input, integer(int32)- If 1, the function described in the
descriptionwill be calculated as is. If 2, the second localized functionGammawill not have its complex conjugate taken.
Description
SHSjkPG will calculate the expectation of two functions (f and g), each localized by a different data taper that is a solution of the spherical cap concentration problem, for a given spherical harmonic degree and two different angular orders. As described in Wieczorek and Simons (2007), this is the function
/ m(j) mprime(k)* \
| Phi Gamma |
\ l l /
The global cross-power spectrum of f and g is input as incspectra, and the real coefficients of the two data tapers of angular order mj and mk (obtained by a call to SHReturnTapers) are specified by hj_real and hk_real. If hkcc is set to 1, then the above function is calculated as is. However, if this is set to 2, then the complex conjugate of the second localized function is not taken.
References
Wieczorek, M. A. and F. J. Simons, Minimum-variance multitaper spectral estimation on the sphere, J. Fourier Anal. Appl., 13, doi:10.1007/s00041-006-6904-1, 665-692, 2007.