The effect of the finite pixel size on the measured power of a pixelized map
depends on the Spherical Harmonic Transforms of each pixel p, defined as
where wp is equal to one within the pixel, and equal to 0 outside.
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 the m-averaged window function
which is independent of the pixel location on the sky,
can be introduced.
If we assume all the pixels to be identical, the power spectrum of the
pixelized map,
Clpix, is related to the hypothetical unpixelized
one,
Clunpix, by
where the effective pixel window function wl is defined as
This function is provided with the HEALPix package for
l4Nside for each
resolution parameter
Nside.
The pixel window functions are now available for both temperature and polarization.
For
Nside128, those window functions are computed exactly using
Eqs. (24) and (26). For
Nside > 128 the
calculations are too costly to be done exactly at all l. The temperature
windows are
extrapolated from the case
Nside = 128 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 for
Nside > 128 are slighty different from those
provided with HEALPix 1.1. For a given
Nside, the relative difference
increases almost linearly with l, and is of the order of
w/w < 7 10-4 at
l = 2Nside and
w/w < 1.7 10-3 at
l = 4Nside.
Eric Hivon 2003-02-07