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This test makes use of a 128 Mbps VLBA observation from June 2009, used to compare DiFX against the VLBA hardware correlator (VHWC) and to validate DiFX-2.0 vs DiFX-1.5. The test below used about 2 seconds of data on 4C39.25 (exactly 2 for SFXC and DiFX, and 2.09s for VHWC). Below, a summary of results from FRING in AIPS for DiFX vs VHWC vs SFXC (with various windowing and spectral averaging) is reported.
The columns FRING:X/Y in the table below refers to the S/N reported by FRING for antenna X, polarisation Y.
|Correlator run||FRING 1/1||FRING 3/1||FRING 1/2||FRING 3/2||AVERAGE|
|DiFX 128→32 channels||432||428||365||364||397|
|VHWC 128→32 channels||481||476||376||378||427|
|SFXC 256→32 channels via lag (zero pad+Hann)||544||548||440||433||491|
|SFXC 256→32 channels via lag (zero pad)||608||616||460||455||534|
|SFXC 256→32 channels via lag (zero pad+Rect)||595||604||447||442||522|
|SFXC 256→32 channels via freq boxcar (zero pad+Hann)||422||424||358||352||389|
|SFXC 32 channels (zero pad+Hann)||520||523||415||411||467|
|SFXC 128 channels (zero pad+Hann)||628||631||553||545||589|
|SFXC 128→32 channels via AIPS (zero pad+Hann)||432||433||367||362||398|
|w/4 channel boxcar smoothing in FRING|
|DiFX 128→32 channels||669||664||554||553||610|
|VHWC 128→32 channels||689||683||520||521||603|
|SFXC 256→32 channels via lag (zero pad+Hann)||658||666||511||504||584|
|SFXC 256→32 channels via lag (zero pad)||663||676||478||473||572|
|SFXC 256→32 channels via lag (zero pad+Rect)||649||662||462||458||557|
|SFXC 256→32 channels via freq boxcar (zero pad+Hann)||652||657||539||529||594|
|SFXC 32 channels (zero pad+Hann)||642||649||500||494||571|
|SFXC 128 channels (zero pad+Hann)||848||851||694||685||769|
|SFXC 128→32 channels via AIPS (zero pad+Hann)||649||653||535||527||591|
The bandpasses show that DiFX and VHWC are very similar, while SFXC is “smoother”. It looks like the channel response of SFXC is much wider, and hence that there is redundant information in adjacent channels, which is messing with the estimate of noise in FRING. Below are some plots that show delay/rate space for simulated data (very simple, no bandpass, no quantisation, zero relative delay, just pure white noise with a 0.05 correlation coefficient) that has been processed in a number of different ways:
This shows the result of a simple FFT, no zero padding, no windowing. The noise is flat everywhere.
This shows the result of a zero-padded FFT, no windowing. The peak is the same height, but the noise is marginally reduced.
This shows the result of windowed-overlapped FFT (Hamming window), no zero padding. The peak goes up but it is sitting on a plateau of higher noise. If you just estimate the noise globally, then the S/N seems higher, but if you use an estimate of the local noise in the vicinity of the peak, the S/N is the same.
This shows the result of windowed-overlapped FFT (Hamming window) with zero padding. Basically the same as the window-overlapped without zero padding.
But if you average the windowed/overlapped data, you're zooming in to a more representative area of the delay/rate space noise-wise, so the fringe is no longer sitting on a higher-noise plateau (relatively speaking).
So as you can see, the noise rolls off towards high lags when a Hamming window and overlapping is used rather than boxcar, as you expect. So if you estimate the noise globally, you get a lower value and hence higher S/N. But if you estimate the noise in the vicinity of the peak (in this case, near zero lag), it doesn't matter whether you zero pad or window-overlap, you get the same S/N as expected.
Here is a summary in table form (noise estimated by blanking the fringe and then taking the square root of the sum of the squared remaining points, global noise using all points, local noise using only +/- 10 lags):
|Type of correlation||signal / global noise||signal / local noise|
|No overlap, straight FFT||16.06||15.94|
|No overlapping, zero padding||18.55||16.33|
|Hanning window + overlap||21.58||15.87|
|Hanning window + overlap, zero padded||21.76||15.88|
|Hanning window + overlap, zero padded, averaged 4x in freq||17.37||15.94|
So as you can see, the S/N estimated locally is always pretty comparable, even though the S/N estimated globally is 35% higher with a windowed overlap. Zero padding does help a little, but at the level of 2-3%.