A pixelised signal f(p) is the average within each pixel p (with surface
area
) of the underlying signal
However, complete analysis of a pixelised map with the exact wlm(p) defined above would be computationally intractable (because of azimutal variation of pixel shape over the polar caps of the HEALPix grid), and some simplifying asumptions have to be made. If the pixel is small compared to the signal correlation length (determined by the beam size), the exact structure of the pixel can be ignored in the subsequent analysis and we can assume
If we assume all the pixels to be identical, the power spectrum of the
pixelized map,
, is related to the hypothetical unpixelized
one,
, by
The pixel window functions are now available for both temperature and polarization.
For
, those window functions are computed exactly using
Eqs. (28) and (30). For
the
calculations are too costly to be done exactly at all l. The temperature
windows are
extrapolated from the case
assuming a scaling in l similar
to the one exhibited by the window of a tophat pixel. The polarization
windows are assumed to be proportional to those for temperature, with a
proportionality factor given by the exact calculation of wl at low
l.
Because of a change of the extrapolation scheme used, the temperature window
functions provided with HEALPix 1.2 and higher for
are slighty different from those
provided with HEALPix 1.1. For a given
, the relative difference
increases almost linearly with l, and is of the order of
at
and
at
.
Eric Hivon 2005-08-31