Max Tegmark's library: window.ps
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A METHOD FOR EXTRACTING MAXIMUM RESOLUTION POWER SPECTRA
FROM MICROWAVE SKY MAPS
Author: Max Tegmark
A method for extracting maximal resolution power spectra
from microwave sky maps is presented and applied to
the 2 year COBE data, yielding a power
spectrum that is consistent with a
standard n=1, Q=20 micro-Kelvin model.
By using weight functions that fall off smoothly near the
galactic cut, it is found that the spectral resolution \Delta l can be
more than doubled at l=15 and more than tripled at l=20
compared to simply using galaxy-cut spherical harmonics.
For a future high-resolution experiment with reasonable sky coverage,
the resolution around the CDM Doppler peaks
would be enhanced by a factor of about 100, down
to \Delta l\approx 1, thus allowing spectral features such as the locations
of the peaks to be determined with great accuracy.
The reason that the improvement is so large is basically that
functions with a sharp edge at the galaxy cut
exhibit considerable ``ringing" in the Fourier domain,
whereas smooth functions do not.
The method presented here is applicable to any survey geometry,
chopping strategy and
exposure pattern whatsoever.
The so called signal-to-noise eigenfunction
technique is found to be a special case, corresponding to
ignoring the width of the window functions.
Published in MNRAS, 280, 299-308 (1996)
This site also contains the latest versions of a paper that is
referenced in the text;
This is what the weight functions tailored to measure
the multipole l=20, m=0 look like, before
and after the optimization process:
Although the they look rather similar in real this
figure (in real space), the absence of sharp edges in the optimized
version makes them differ radically in the Fourier (multipole) domain,
as shown in Figure 3 at the top of this page.
Finally, here is what the power spectra obtained from the COBE DMR
2 year data look like when estimated with the naive method and with this
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This page was last modified July 1, 1998.