One goal of CMB data analysis is to combine data at different frequencies , angular resolutions , and noise levels in order to best extract the component with a Plankian spectral behaviour .
A multi-frequency Wiener filtering method has been proposed in this context by Bouchet , Gispert and Puget ( 1995 ) and in parallel by Tegmark and Efstathiou ( 1996 ) .
As shown in Bouchet and Gispert ( 1998a ) , this linear method is also convenient to estimate a priori , given a sky model and an experimental description , the residual errors on the CMB power spectrum assuming the foregrounds have been removed with this method .
In this paper , we extend the method to the case when additional polarisation data is available .
In particular , we derive the errors on the power spectra involving polarisation and show numerical results for the specifications of the future CMB space missions MAP and Planck For current noise specifications and angular and frequency coverage of these experiments , see http : //map.gsfc.nasa.gov and http : //astro.estec.esa.nl/SA-general/Projects/Planck when it is assumed that the Galactic synchrotron and dust emission are respectively about 40 % and 10 % polarised .
We consider two underlying models for our study : we take a standard CDM model with \tau = 0.1 for the extraction of E -mode polarisation and ET cross-correlation ; for B -mode polarisation we consider a tilted CDM model with n _ { s } = 0.9 , n _ { T } = -0.1 and T / S = 0.7 .
We find that : ( 1 ) The resulting fractional errors on E mode polarisation and TE cross-correlation power spectra are \la 10 \hbox { - - } 30 \% for 50 \la \ell \la 1000 for Planck .
The fractional errors are between 50 \% to 150 \% for \ell \leq 50 , ( 2 ) The corresponding fractional errors for MAP are \geq 300 \% for most of the \ell range , ( 3 ) the Wiener filtering give extraction errors \leq 2 times the expected performance for the combined sensitivity of all the channels of Planck .
For MAP , the corresponding degradation is \simeq 4 .
( 4 ) if , instead of individual modes , one considers band-power estimates with a logarithmic interval \Delta \ell / \ell = 0.2 then the fractional error for MAP drops to \la 100 \% at the Doppler peak around \ell \simeq 300 for the ET signal , and ( 5 ) The fractional error for B -mode polarisation detection is \la 100 \% with Planck for \ell \leq 100 .
A band-power estimate with \Delta \ell / \ell = 0.2 reduces the fractional errors to \la 25 \% for 20 \leq \ell \leq 100 .