[mmaimcal] Matching of frequency profiles for ALMA and ACA correlators

[rlaing at eso.org]

The purpose of this e-mail is to start a discussion on a small issue which came 
up during the PDR of the ACA correlator. It concerns the differences in
frequency profile for the 64-station (hereafter ALMA) and ACA correlators. The
former is an XF design, and the response for a narrow spectral line is a sinc
function in the frequency domain. The latter is an FX design, and the response
is a sinc^2 function.  

See
http://edm.alma.cl/forums/alma/dispatch.cgi/revsactive/docProfile/100643/d20050510035512/No/2005-05-09%20CORL-62.00.00.00-006-A-REP.pdf 

(pp 45-46) and

http://edm.alma.cl/forums/alma/dispatch.cgi/revsactive/docProfile/101003/d20050611223211/No/3-2Compatibility-freq-ch_profile.pdf

for the material presented at the review.

If we combine ALMA and ACA data without doing something to equalize the
frequency profiles, there will be imaging errors. The ACA team have looked at
reproducing the ALMA frequency profile from a linear combination of ACA channels
(second document above). This gives fairly small differences except in the case
of a uniform weighting function at the highest spectral resolution (see second
document, above).  [It is, in any case, arguably the wrong way round, as the
sinc^2 response of the ACA (FX) correlator has lower sidelobes.]

In practice, it is likely that some weighting function other than uniform will
be used to reduce sidelobes at the cost of spectral resolution. For example: 
 - A triangular weighting function in the time (lag) domain for the ALMA
 correlator will produce exactly a sinc^2 response. This matches the form of the
 ACA correlator response, but for a lower resolution. 
 - Hanning weighting (which is actually quite close to this) will also suppress
 the sidelobes very effectively (and the ACA correlator can reproduce this
 profile to within 0.4% - see second document) by weighted frequency binning.

If the ALMA correlator is used at its highest resolution with uniform weighting,
this approach does not work, since the ACA correlator cannot then bin over
narrower channels.

The design issues are therefore as follows:

For anything other than the maximum frequency resolution, we should use 
appropriate weighting functions (Hanning or triangular) to match the profiles.
This should probably be the default option.
 
The match between the frequency responses of the two correlators at high
spectral resolution can be improved by increasing the length of the FFT in the
ACA correlator (to give narrower spectral channels) and then re-binning with
appropriate weights.  Is this justified by the need to match the frequency
profiles of the two correlators to better than the levels shown in the review
documentation? [There are cost and schedule implications.]

Do we need imaging simulations in order to answer this question, or are the
existing calculations adequate?

Regards

Robert