Pixel window functions

A pixelated signal $f(p)$ is the average within each pixel $p$ (with surface area $\Omega_{\mathrm{pix}}$) of the underlying signal

$$f(p) = \int d\u w_p(\u )f(\u )$$ (30)

where $w_p$ is equal to $1/\Omega_{\mathrm{pix}}$ within the pixel, and equal to 0 outside, so that $\int d\u w_p(\u ) = 1$. Eq. (4) then becomes

$$f(p) = \sum_{\ell =0}^{\ell_{\mathrm{max}}}\sum_{m}a_{\ell m}w_{\ell m}(p),$$ (31)

where

$$w_{\ell m}(p) = \int d\u w_p(\u ) Y_{\ell m}(\u ),$$ (32)

is the Spherical Harmonic Transform of the pixel $p$.

However, complete analysis of a pixelated map with the exact $w_{\ell m}(p)$ 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

$$w_{\ell m}(p) = w_\ell(p) Y_{\ell m}(p)$$ (33)

where we introduced the $m$-averaged window function

$$w_{\ell}(p) = \left(\frac{4 \pi}{2\ell+1}\sum_{m=-\ell}^{\ell} \left\vert w_{\ell m}(p)\right\vert^2\right)^{1/2},$$ (34)

which is independent of the pixel location on the sky.

If we assume all the pixels to be identical, the power spectrum of the pixelated map, $C_{\ell}^{\mathrm{pix}}$, is related to the hypothetical unpixelated one, $C_{\ell}^{\mathrm{unpix}}$, by

$$C_{\ell}^\mathrm{pix} = w^2_{\ell} C_{\ell}^\mathrm{unpix}$$ (35)

where the effective pixel window function $w_{\ell}$ is defined as

$$w_{\ell} = \left(\frac{1}{N_{\mathrm{pix}}}\sum_{p=0}^{N_{\mathrm{pix}}-1} w^2_{\ell}(p)\right)^{1/2}.$$ (36)

This function is provided with the HEALPix package for $\ell\le 4N_{\mathrm{side}}$ for each resolution parameter $N_{\mathrm{side}}$.

The pixel window functions are now available for both temperature and polarization.

For $N_{\mathrm{side}}\le 128$, those window functions are computed exactly using Eqs. (34) and (36). For $N_{\mathrm{side}}> 128$ the calculations are too costly to be done exactly at all $\ell$. The temperature windows are extrapolated from the case $N_{\mathrm{side}}= 128$ assuming a scaling in $\ell$ 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 $w_{\ell}$ at low $\ell$.

Because of a change of the extrapolation scheme used, the temperature window functions provided with HEALPix 1.2 and higher for $N_{\mathrm{side}}> 128$ are slighty different from those provided with HEALPix 1.1. For a given $N_{\mathrm{side}}$, the relative difference increases almost linearly with $\ell$, and is of the order of $\Delta w/w < 7\ 10^{-4}$ at $\ell=2N_{\mathrm{side}}$ and $\Delta w/w < 1.7\ 10^{-3}$ at $\ell=4N_{\mathrm{side}}$.