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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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* \<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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The main advantage of using cross-power spectra is that the noise is generally uncorrelated between different detectors so that
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<img name="image5" src="https://gitlab.in2p3.fr/tristram/Xpol/uploads/13b08d4211b8c26415f0711eb4ff227e/image5.png" width="500px">
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where M^(2)^ is the quadratic coupling kernel matrix and ![image6](https://gitlab.in2p3.fr/tristram/Xpol/uploads/d874dfcad1e84570f4d8a69741d7d483/image6.png).
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where M^\(2\) is the quadratic coupling kernel matrix and <img name="image6" src="https://gitlab.in2p3.fr/tristram/Xpol/uploads/d874dfcad1e84570f4d8a69741d7d483/image6.png" with="100px">.
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