[alma-config]Re: Nyquist and more

[David Woody]

Some information on Nyquist sampling.

There is a generalization of the Nyquist
theorem that we all know and love that says
that all you really need is that the average
sampling over the time interval (or UV-plane)
be consistent with the band limited Nyquist
sampling theorem.  This is briefly discussed
in Bracewell "Fourier Transform and its
applications" 3rd edition, 2000, pp.232-235
and an article by D. A. Linden, "A discussion
of sampling theorems", Proc. IRE vol. 47, 
p. 1219, 1959.

The basic idea is that two samples can be
replace by one sample and a measure of the
slope obtained using a closely spaced nearby
sample.  This can be extended to higher derivatives.

The penalty is in robustness, i.e. noise becomes
progressively more important in deducing the
higher order derivatives.

The non-linear imaging algorithms could be 
intentionally or unintentionally exploiting this.

There is also a reference for linear random exponential
(sort of Guassian) sensor arrays
P. S. Naidu, "Sensor Array Signal Processing," CRC Press,
2001, p.99.  This looks like a very interesting
book, but I have not had time to read much of it yet.

*****************************************************
Carrying on the discussion of edge tapered 
UV distributions for the next meeting:

What John said in his previous E-mail is correct, but in
light of the above information about Nyquist sampling
I don't see the advantage of striving for complete
UV coverage at the expense of having a 7-10dB discontinuity
at the edge of the coverage.  Discontinuities don't 
seem correct in a continuous world.  Note that the 
natural maximum radius for a Gaussian distribution of 
4,000 points is at ~30dB edge taper.

John's discussion of the high signal-to-noise images being
able to afford apodizing to give better dynamic range is
correct.  But it will be very difficult to write an
observing proposal justification for a long track on
a bright object just to get complete UV coverage
so that you can apodize the data to get more dynamic
range.  Many, if not most, observations will have short 
tracks and then the difference between truncated Gaussians
and a smooth bell shaped distribution is just in degree,
i.e. the UV radius at which the snapshot image has
complete coverage and the interpolation distance between
samples beyond that radius.

Introducing an apodization option also complicates the
pipeline imaging.  (I don't mean to open this controversial
issue here.)  You have at least two paths, 1) full data 
naturally weighted and restored with the natural beam width 
(effectively extrapolating data beyond the edge), 2) apodized 
data restored with the slightly wider apodized beam width.

The analogy with antenna apertures is correct
for the S/N but antenna engineers only use 10-13dB edge
taper because an antenna's cost scales >D^5/2.  10-13dB
is a compromise between effective collecting area and
sidelobes.  Presumably the cost of longer baselines 
is <<BL^5/2 so you do not have to stop until you 
hit a natural or artificial boundary.  The collecting
area is fixed but we are pretty much free to distribute
it anyway we want within reasonable limits.

Are there any imaging studies that show whether a 10dB edge
taper is better than 30dB taper with the same natural angular
resolution?  If so, this would help resolve this issue.

*****************************************************
Minor side issue

When using earth rotation to complete or fill in the
UV coverage you not only have to look at the edge of the
UV coverage, but also at the central coverage.  The number
of UV cells added per hour increases as the UV radius times
the UV sample density.  I have examples of centrally condensed
UV distributions for the large configurations that look 
just fine, except that it takes much longer for all of the
central UV cells to be sampled than it does for the farthest
cells.  I think it would be a mistake to not have complete
coverage of the critical short spacing cells after a 6 hour
track in all but the extended 14km configuration.  This is
related to the question of the number of short spacings 
required and should be decided consciously.

*****************************************************
Concluding thoughts

Since the mathematics says it is only the noise that
limits us from recovering all of the information
given enough samples anywhere on the UV-plane we
need to test the proposed configurations 
using simulations, including noise, of the
kinds of images we expect.
This is what the "imagers" have been telling us all
along.  The answer must be that the best images will
come from configurations whose UV coverage mimics
the image visibilities, i.e. a lot of
UV samples where the visibilities are large and
few where they are small.  This leads to bell or
Gaussian shaped UV distributions.  These distributions
in turn have small near sidelobes in the PSF.  Is it
coincidence that the image visibility distributions
and the desire for small near sidelobes drives us
towards the same centrally condensed UV coverage?

Once we know the UV distribution we want, Fredrik Boone's
algorithm is excellent for finding a configuration
that gives a close fit to that distribution.
The next step would be to apply a sidelobe minimization
algorithm to reduce the far sidelobes.  This should
only perturb the antenna positions slightly and not
change the UV distribution or the excellent near
sidelobes that the bell or Gaussian distribution
produced.  The PSF then tells you what kind of images
you would get from linear processing and is a convenient
method for evaluating how well the UV coverage matches
the desired distribution.

Cheers
David
****************************************
| David Woody
| Owens Valley Radio Observatory
| P.O. Box 968, 100 Leighton Lane                         
| Big Pine, CA 93513, USA                                  
| Phone 760-938-2075ext111, FAX 760-938-2075
|dwoody at caltech.edu 
****************************************