... | ... | @@ -16,11 +16,11 @@ Generalizing MASTER equation for classical auto-spectra, we can compute the `pse |
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%height=40px%Attach:image1.png
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where
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* B'_l_' is the beam transfer function describing the beam smoothing effect which can be computed, for example via models and/or Monte Carlo signal-only simulations;
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* p'_l_' is the transfer function of the pixelization scheme of the map describing the effect of smoothing due to the finite pixel size and geometry;
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* F'_l_' is an effective function that represents any filtering applied to the time ordered data that can also be computed via Monte Carlo;
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* <N'_l_'> is the noise power spectrum;
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* M'_ll'_' the coupling kernel matrix computed analytically from the weighting function as intensively described in [[#wmap_polar | Kogut et al. 2003]].
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* B<sub>l</sub> is the beam transfer function describing the beam smoothing effect which can be computed, for example via models and/or Monte Carlo signal-only simulations;
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* p<sub>l</sub> is the transfer function of the pixelization scheme of the map describing the effect of smoothing due to the finite pixel size and geometry;
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* F<sub>l</sub> is an effective function that represents any filtering applied to the time ordered data that can also be computed via Monte Carlo;
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* <N<sub>l</sub>> is the noise power spectrum;
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* M<sub>ll</sub> the coupling kernel matrix computed analytically from the weighting function as intensively described in [Kogut et al. 2003](#wmap_polar).
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\\
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... | ... | @@ -32,11 +32,11 @@ The main advantage of using cross-power spectra is that the noise is generally u |
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The system can thus be solved for large sky fraction, otherwise one has to bin multipoles into bandpowers.
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From @@N@@ input maps we can obtain @@N(N-1)/2@@ cross-power spectra for each polarized mode (@@TT@@, @@EE@@, @@BB@@, @@TE@@, @@TB@@ and @@EB@@) which are unbiased estimates of the angular power spectrum but which are obviously not independent. For each polarized power spectra independently, '''''Xpol''''' can estimate the cross-correlation matrix between cross-spectra and multipoles from which error bars and covariance matrix in multipole space can be deduced for each cross-power spectra. Each element of this matrix reads :
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From @@N@@ input maps we can obtain @@N(N-1)/2@@ cross-power spectra for each polarized mode (@@TT@@, @@EE@@, @@BB@@, @@TE@@, @@TB@@ and @@EB@@) which are unbiased estimates of the angular power spectrum but which are obviously not independent. For each polarized power spectra independently, *Xpol* can estimate the cross-correlation matrix between cross-spectra and multipoles from which error bars and covariance matrix in multipole space can be deduced for each cross-power spectra. Each element of this matrix reads :
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Considering the completeness relation for spherical harmonics [[#angularmomentum | Varshalovich et al.1988]] and in the limit of large sky coverage [[#efstathiou | Efstathiou 2004 & 2005]], @@Ξ@@ reads (see [[#xspect|Tristram et al. 2005]]) :
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Considering the completeness relation for spherical harmonics [Varshalovich et al.1988](#angularmomentum) and in the limit of large sky coverage [Efstathiou 2004 & 2005](#efstathiou), @@Ξ@@ reads (see [Tristram et al. 2005](#xspect)) :
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... | ... | @@ -45,11 +45,11 @@ with |
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%height=30px%Attach:image5.png
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where M'^(2)^' is the quadratic coupling kernel matrix and %height=15px%Attach:image6.png .
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where M^(2)^ is the quadratic coupling kernel matrix and %height=15px%Attach:image6.png .
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\\
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To combine the cross-power spectra and obtain the best estimate of the power spectrum C'_l_', we maximize the Gaussian approximated likelihood function
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To combine the cross-power spectra and obtain the best estimate of the power spectrum C<sub>l</sub>, we maximize the Gaussian approximated likelihood function
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[[#likelihood]]%height=30px%Attach:image7.png
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... | ... | @@ -59,7 +59,7 @@ Neglecting the correlation between adjacent multipoles, the estimate of the angu |
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the final covariance matrix can be obtained from [[#likelihood | Eq.]],
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the final covariance matrix can be obtained from [Eq.](#likelihood),
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%height=30px%Attach:image10.png
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... | ... | @@ -87,7 +87,3 @@ and the final error bars are given by |
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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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\\
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\\
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[[Xpol/Current work]] %red%restricted area%%. |