[Pafgbt] save PAF cross-correlations rather than formed beam outputs
Anish Roshi
anish.roshi at gmail.com
Tue May 21 11:45:00 EDT 2013
Thanks Brian. I will read the paper.
Anish
On Tue, May 21, 2013 at 11:37 AM, Brian Jeffs <bjeffs at byu.edu> wrote:
> Anish,
>
> Yes, this is the way we do it to find the signal vector for calibration,
> with an eigen decomposition. I think we were the first to propose that
> approach. We have an improved method using a generalized eigenvector
> decomposition which also uses a dark sky noise covariance matrix estimate
> contemporary with the calibration observation. It is in:
>
> Elmer, M.; Jeffs, B.D.; Warnick, Karl F. ; Fisher, J.R. ; Norrod,
> "R.D.Beamformer Design Methods for Radio Astronomical Phased Array
> Feeds," Antennas and Propagation, IEEE Transactions on, Volume: 60, Issue:
> 2, Part: 2, Digital Object Identifier: 10.1109/TAP.2011.2173143, 2012 ,
> Page(s): 903- 914.
>
> Brian
>
> On May 21, 2013, at 7:14 AM, Anish Roshi wrote:
>
>
>
> Hi Rick,
>
> A clarification (see below) --
>
>
> On Mon, May 20, 2013 at 4:02 PM, Anish Roshi <anish.roshi at gmail.com>wrote:
>
>>
>>
>>
>>
>>
>>> Anish can correct me if I'm wrong, but I think he said that the ASKAP
>>> default plan is to cross-correlate all array PAF elements with all other
>>> PAF elements across all synthesis array antennas, which avoids the need to
>>> form voltage beams.
>>
>>
>> No they correlate between all PAF elements in an antenna to get the
>> weights for 30 beams. Then they will correlate between the 30 beams for
>> synthesis array. Eigen beam forming is not what they are going for in the
>> first place.
>>
>>
>
> Getting the weights for the beam has two steps -- (1) to measure the
> noise correlation matrix using all the PAF elements
> on a blank sky and (2) measuring the `signal vector'. They take the
> `signal vector' (say S) as the complex voltages obtained by cross
> correlating elements of the PAF in one antenna with a feed (it is a horn at
> present; but I think one element of PAF can also be used) in a second
> antenna.
>
> As I was discussing with Doug Hayman about the measurement of the signal
> vector -- I think S will not give the signal vector (Brain may be
> interested in this since I heard he will at ASKAP for a year). This is
> because of mutual coupling in PAF elements. One should form a matrix SS^H
> and diagonalize the matrix. The `signal vector' should then be the eigen
> vector corresponding to the dominant eigen value.
>
> Anish
>
>
> This is more expensive in signal processing power than forming
>>> modestly spaced voltage beams, but it retains all of the information in the
>>> focal plane at full sensitivity.
>>
>> Forming "eigen beams" may be a compromise between these to approaches.
>>>
>>
>> Yes that is right -- at least that is my understanding.
>>
>> Anish
>>
>>
>>> Good discussion regardless of the final outcome.
>>>
>>>
>>> Rick
>>>
>>> On Mon, 20 May 2013, 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.
>>>>
>>>> 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> 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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