[Pafgbt] save PAF cross-correlations rather than formed beam outputs

Mon May 20 18:02:02 EDT 2013

Another comment below.

Brian

On May 20, 2013, at 2:27 PM, Karl Warnick wrote:

One comment below..

Karl

On 5/20/2013 12:48 PM, Brian Jeffs wrote:
Comments below.

Brian

On May 20, 2013, at 10:28 AM, Karl Warnick wrote:

I agree that correlations are important. Correlated outputs will probably be the default mode for most PAF observations, and real time beams would only be used when phase information is important, such as for pulsar surveys (can pulsar searches be done with fine grained, narrowband correlations?)

I'm not sure whether the bandwidth issue is the real driver. Real time beams also have to be formed in subbands and then either processed for pulsar surveys or integrated for power spectra.
I don't think bandwidth is really what drives the difference between a B engine and an X engine. It's true that if one only forms a few PAF beams, then with a given disk storage data rate one could store the beam outputs over finer frequency subbands than correlations, but if fully sampled beams are formed, the data rate wouldn't be very different between beam outputs and correlations with the same integration time and frequency resolution. Am I missing something?

For the same integration time and frequency resolution, roughly the same number of beams as PAF elements (fully sampled beams and Nyquist array sampling), ignoring the N log N complexity growth of an FFT,  ignoring processor architecture constraints, and if you don't need time series data for downstream processing, then there is no relative advantage between an integrating B engine and an X engine as to data rate.  There are however some very real practical limitations.

If you have an ASKAP PAF you have many more elements than beams, so a B engine is more efficient.  With our PAF, which has about the same number of beams as elements, the calculus changes.

All frequency channelization must be performed before the correlations (X engine) stage, so if you need fine-grained frequency resolution you need a huge FFT to handle the 800 Msamp/s target sample rate of our back end.  We don't have a processor architecture that can do that FFT size (due to N log N growth).  You must do coarse FFTs first, then either real-time beamform or correlate.  If you correlate (e.g. and then beamform) you can never get to the fine frequency resolution.  If you real-time beamform, you can follow with another FFT stage to get fine frequency channels.  You could then correlate, or integrate squared spectra, or send to pulsar processing as suites the application.
If we were correlating outputs of an N channel PAF over the full bandwidth, wouldn't the cost of a fine FFT, while large, be negligible in comparison in relation to the correlator?

Can be, but not necessarily, and probably not for the arrays we are building.  The correlator grows as M^2, M= # antennas.  F engine grows as N log N, N= # frequency bins.  B engines grow linearly with both M and K= # of beams).  All grow linearly with total bandwidth (keeping N fixed).  The X and B engine computations are nearly independent of N.

For small arrays (like ours) the F and X engines are about the same size.   Our real-time correlator is an example of this, with F and X engines divided evenly in two topped out ROACHs with M = 32 and N = 2048.   This is for 20 MHz of analog bandwidth.  If you up that to 300 MHz, (e.g. using more than 2 ROACH IIs and GPUs) the 2048 FFT gives 800 kHz wide frequency channels.  If that is fine enough, we are home free and the F-X architecture works.

If you want a finer scale spectrum for an F-X architecture, Then the F engine outgrows the X engine.  On the other hand, for an F-B-F architecture, the first F engine is coarser, and you add an FFT after the beamformer, and can save computations.  Which approach is computationally more efficient? Depends on K, M, and N, and I guess I'm not sure how big N  can be before we have troubles with the FFT size in an F-X architecture.

If we are close to needed resolution with a N=2048 FFT, for 300 MHz BW, then my argument that we can't fit in the FFT is probably wrong for an F-X implementation.  If we need N >> 2048 I still worry that F-X is impractical, and we would need F-B-F.

When I say fine grained, narrowband correlations, I mean a fine FFT followed by X engine, with the integration time very small. The time scale certainly can be no smaller than N_FFT samples, but it could still be pretty small. That's certainly possible in principle, with a large enough back end. I wonder if there are applications that would be well served by this kind of architecture, with a fine F engine followed by a correlator.

There may well be an application that this serves.  In principle you could save out correlations with an integration time of L=1 FFT output and still beamform on these for power averaging.   But there is no data reduction (limited reduciton if small L > 1 is used) and no time domain processing is possible after this.  Once you have cross multiplied, you lose all phase relationship between successive FFT output sample.  So the issue really is whether the narrow frequency time series needs to be preserved for the application, such as later super fine spectral analysis.  I just don't know what is needed.

For pulsar work I don't know what the frequency resolution requirements are, so I can't judge whether you can beamform after correlation.  I don't think "fine grained, narrowband correlations" are possible since to be narrowband means correlations would necessarily be coarse grained in time.

There is another motivation for beamformers in the broader PAF community, and that is for synthesis arrays. The correlator for the dish array is expensive, so one would only want to correlate as small a number of beams from each PAF as possible. This raises a question - could correlations from each PAF be used to get correlations of the dish array somehow? Is there an efficient two-level correlator architecture, with a PAF correlator for each dish, followed by the synthesis array correlator? I suspect not, but I can't quite convince myself. In any case, it seems that for single dish telescopes, there's less motivation to use a beamformer back end instead of a correlator.

I agree with this.  That is why ASKAP uses a real-time beamformer ahead of the correlator.  Correlation across the dishes requires frequency channelized times series, not integrated data.  I have not seen proposals for a two-level correlator.  The first stage (PAF level) correlation removes all absolute time or phase reference information for one dish relative to another, which is the key signal component needed to form the 2nd level (dish to dish) correlation.  The other reason for doing real-time beamforming at the PAF level is to reduce data transport requirements (many more elements than beams).  Eigenbeams further reduce the data transport load.  Integrated correlations would of course be even lower data rate, but I don't see how to get the cross dish visibilities from PAF correlations.

For a single dish PAF the arguments are weaker for real-time beamforming unless actual time samples or superfine spectral resolution are needed.

Finally, there is probably a bit of analysis one could to to show how closely beams must be spaced in order for the information in the beams to be equivalent to the correlations. It's essentially the problem of recovering the matrix R from a set of inner products w'*R*w for many vectors w. I suspect that if one forms HPBW/2 spaced beams over the PAF FoV, the information in the beams is less than but on the order of the information in R in some quantiable sense. Information in the deep sidelobes is lost, but most of the large eigenvalues of R represent sources in the field of view. With finer beams, even just over the PAF field of view, all information even out in deep sidelobes could well be contained in the beam outputs, but that's a moot point, as one would not form that many beams in practice. There's also the idea of eigenbeams proposed by Cornwell et al., so that one can form very few PAF beams yet still have information over the full field of view.

I agree with this.

Karl

On 5/20/2013 9:25 AM, Brian Jeffs wrote:
Rick,

I agree that you have much more flexibility to try different beamformer designs, detection algorithms, interference mitigation techniques, superresolution, calibration correction, etc. if you store and operate on the accumulated cross products (correlation matrices).  However, you give up the ability to do fine resolution spectral processing.  You are stuck with the coarseness of the correlator's frequency channelization.  I don't know how problematic this is for some applications, such as pulsar searches, where fine spectral resolution may be needed.

Brian

On May 20, 2013, at 6:38 AM, Anish Roshi wrote:

Yes indeed. We can form images with beams with different optimization if the correlations are recorded.
Anish

On Sun, May 19, 2013 at 9:57 AM, Rick Fisher <rfisher at nrao.edu<mailto:rfisher at nrao.edu>> wrote:
Brian, Karl,

In trying to understand the ASKAP data processing architecture, I'm
beginning to understand the fundamental importance of saving the
cross-products between array element outputs in our own PAF data
processing.  In forming beams you throw away a lot of information in the
array's field of view that can be recovered only by forming many beams
with very close spacing (much closer than HPBW/2).  This has important
consequences for the sensitivity to point sources, as in the search for
pulsars.  Hence, I would suggest that the most important archived outputs
from your signal processor are the cross-products rather than formed
beams.  For a given data storage volume, there's more information in the
cross-products than in the formed beam outputs.  In some respects, the
"beam" concept is a holdover from a waveguide feed where there's only one
output, and most of the information in the focal plane is reflected back
into the sky.

Rick
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Karl F. Warnick
Department of Electrical and Computer Engineering
Brigham Young University
459 Clyde Building
Provo, UT 84602
(801) 422-1732

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Karl F. Warnick
Department of Electrical and Computer Engineering
Brigham Young University
459 Clyde Building
Provo, UT 84602
(801) 422-1732

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