A Statistical Analysis of the "Internal Linear Combination" Method in Problems of Signal Separation as in CMB Observations

AIMS: The separation of foreground contamination from cosmic microwave background (CMB) observations is one of the most challenging and important problem of digital signal processing in Cosmology. In literature, various techniques have been presented, but no general consensus about their real performances and properties has been reached. This is due to the characteristics of these techniques that have been studied essentially through numerical simulations based on semi-empirical models of the CMB and the Galactic foregrounds. Such models often have different level of sophistication and/or are based on different physical assumptions (e.g., the number of the Galactic components and the level of the noise). Hence, a reliable comparison is difficult. What actually is missing is a statistical analysis of the properties of the proposed methodologies. Here, we consider the "Internal Linear Combination" method (ILC) which, among the separation techniques, requires the smallest number of "a priori" assumptions. This feature is of particular interest in the context of the CMB polarization measurements at small angular scales where the lack of knowledge of the polarized backgrounds represents a serious limit. METHODS: The statistical characteristics of ILC are examined through an analytical approach and the basic conditions are fixed in a way to work satisfactorily. RESULTS: ILC provides satisfactory results only under rather restrictive conditions. This is a critical fact to take into consideration in planning the future ground-based observations (e.g., with ALMA) where, contrary to the satellite experiments, there is the possibility to have a certain control of the experimental conditions.