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#Xpol : a Power Spectrum estimator based on cross correlation between maps.
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*Xpol* is a method to estimate the polarized angular power spectra $$C_\ell$$s by computing the cross-power spectra between a collection of input maps coming either from multiple detectors of the same experiment or from different instruments. The `pseudo' cross-power spectra are explicitly corrected for incomplete sky coverage, beam smoothing, filtering and pixelization. Assuming no correlation between the noise contribution from two different maps, each of the corrected cross-power spectra is an unbiased estimate of the C'_l_'s. Analytical error bars are derived for each of them. The cross-power spectra, that do not include the classical ''auto''-power spectra, are then combined using a Gaussian approximation of the likelihood function. This result can be compared to the auto-spectra for which the noise bias in multipole domain is requested as input.
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# Xpol : a Power Spectrum estimator based on cross correlation between maps.
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The method computes an analytical estimate of the cross-correlation matrix directly from the data avoiding any Monte Carlo simulations. This allows to include naturally the mode coupling in this matrix. Further, this permits the computation of analytical error bars which are very compatible with those obtained from simulations (see [[#xspect | Tristram et al. 2005a]] for the temperature case). Because of the above, this method can be applied without modification to the estimation of the power spectrum of the correlated signal between a set of maps of the sky coming from multiple instruments with potentially different sky coverages. This method has been used on '''''Archeops''''' data to estimate the CMB angular power spectrum [[#archeops_cl2 | Tristram et al. 2005b]] and the polarized foreground emission at the sub-millimeter and millimeter wavelength in [[#archeops_polar | Ponthieu et al. 2005]].
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*Xpol* is a method to estimate the polarized angular power spectra C<sub>l</sub>s by computing the cross-power spectra between a collection of input maps coming either from multiple detectors of the same experiment or from different instruments. The `pseudo' cross-power spectra are explicitly corrected for incomplete sky coverage, beam smoothing, filtering and pixelization. Assuming no correlation between the noise contribution from two different maps, each of the corrected cross-power spectra is an unbiased estimate of the C<sub>l</sub>s. Analytical error bars are derived for each of them. The cross-power spectra, that do not include the classical ''auto''-power spectra, are then combined using a Gaussian approximation of the likelihood function. This result can be compared to the auto-spectra for which the noise bias in multipole domain is requested as input.
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The method computes an analytical estimate of the cross-correlation matrix directly from the data avoiding any Monte Carlo simulations. This allows to include naturally the mode coupling in this matrix. Further, this permits the computation of analytical error bars which are very compatible with those obtained from simulations (see [Tristram et al. 2005a](#xspect) for the temperature case). Because of the above, this method can be applied without modification to the estimation of the power spectrum of the correlated signal between a set of maps of the sky coming from multiple instruments with potentially different sky coverages. This method has been used on *Archeops* data to estimate the CMB angular power spectrum [[#archeops_cl2 | Tristram et al. 2005b]] and the polarized foreground emission at the sub-millimeter and millimeter wavelength in [[#archeops_polar | Ponthieu et al. 2005]].
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Generalizing MASTER equation for classical auto-spectra, we can compute the `pseudo' cross-power spectrum C'_l_''^X'_1_' Y'_2_'^' between any two detectors
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... | ... | @@ -81,7 +83,7 @@ and the final error bars are given by |
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* [[#master]] [Hivon et al. 2002] Hivon E., Gorski K. M., Netterfield C. B., Crill B. P., Prunet S., Hansen F., 2002, ApJ, 567, 2
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* [[#wmap_polar]] [Kogut et al. 2003] Kogut A. et al., 2003 ApJS 148 161
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* [[#archeops_polar]] [Ponthieu et al. 2005] Ponthieu N. et al., 2005 A&A 444 327
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* [[#xspect]] [Tristram et al. 2005a] Tristram M. et al., 2005 MNRAS 358 833
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* [xspect] \[Tristram et al. 2005a\] Tristram M. et al., 2005 MNRAS 358 833
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* [[#archeops_cl2]] [Tristram et al. 2005b] Tristram M. et al., 2005 A&A 436 785
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* [[#angularmomentum]] [Varshalovich et al. 1988] Varshalovich D. A., Moskalev A. N., Khersonoskii V. K., 1988, ''Quantum theory of Angular Momentum'', World Scientific, Singapore.
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