1. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 1 Cross Correlators Michael P. Rupen NRAO/Socorro

2. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 2 In an optical telescope… –a lens or a mirror collects the light & brings it to a focus –a spectrograph separates the different frequencies What is a Correlator?

3. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 3 In an interferometer, the correlator performs both these tasks, by correlating the signals from each telescope (antenna) pair:

4. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 4 The basic observables are the complex visibilities: amplitude & phase as functions of baseline, time, and frequency. The correlator takes in the signals from the individual telescopes, and writes out these visibilities.

5. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 5 Correlator Basics The cross-correlation of two real signals and is A simple (real) correlator.

6. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 6 Antenna 1:

7. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 7 Antenna 2:

8. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 8  =0:

9. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 9  =0.5:

10. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 10  =1:

11. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 11  =1.5:

12. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 12  =2:

13. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 13  Correlation:

14. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 14 Correlation of a Single Frequency For a monochromatic signal: and the correlation function is So we need only measure with

15. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 15  Correlation: xIxI xRxR

16. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 16 At a given frequency, all we can know about the signal is contained in two numbers: the real and the imaginary part, or the amplitude and the phase. A complex correlator.

17. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 17 Broad-band Continuum Correlators 1.The simple approach: use a filterbank to split the signal up into quasi- monochromatic signals hook each of these up to a different complex correlator, with the appropriate (different) delay: add up all the outputs 2.The clever approach: instead of sticking in a delay, put in a filter that shifts the phase for all frequencies by  /2

18. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 18

19. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 19 Spectral Line Correlators 1.The simple approach: use a filterbank to split the signal up into quasi- monochromatic signals hook each of these up to a different complex correlator, with the appropriate (different) delay: record all the outputs:

20. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 20 Fourier Transforms: a motivational exercise The frequency spectrum is the Fourier transform of the cross-correlation (lag) function. Short lags (small delays) high frequencies Long lags (large delays) low frequencies …so measuring a range of lags corresponds to measuring a range of frequencies!

21. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 21 Spectral Line Correlators (cont’d) 2.Clever approach #1: the FX correlator F: replace the filterbank with a Fourier transform X: use the simple (complex) correlator above to measure the cross-correlation at each frequency average over time, & record the results 3. Clever approach #2: the XF correlator X: measure the correlation function at a bunch of different lags (delays) average over time F: Fourier transform the resulting time (lag) series to obtain spectra record the results

22. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 22 FX vs. XF t t v1v1 v2v2 S 1 ( ) S 2 ( ) S( ) F Fourier transform F X multiply X

23. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 23 Fig. 4-6: FX correlator baseline processing. Fig. 4-1: Lag (XF) correlator baseline processing.

24. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 24 Details, Details Why digital? precise & repeatable lots of duplication accurate & stable delay lines …but there are some complications as well…

25. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 25 1.Sampling: v(t)  v(t k ), with t k =(0,1,2,…)  t –For signal v(t) limited to 0  , this is lossless if done at the Nyquist rate:  t  1/(2  ) –n.b. wider bandwidth  finer time samples! –limits accuracy of delays/lags 2.Quantization: v(t)  v(t) +  t –quantization noise –quantized signal is not band-limited  oversampling helps Digitization

26. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 26 Quantization & Quantization Losses

27. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 27 Michael’s Miniature Correlator 0.3 V1V1 V2V2 Signals come in…sampled…quantized…delayed…multiplied… integrated & normalized

28. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 28 We measure the cross-correlation of the digitized (rather than the original) signals. digitized CC is monotonic function of original CC 1-bit (2-level) quantization: – is average signal power level – NOT kept for 2-level quantization! –roughly linear for correlation coefficient For high correlation coefficients, requires non-linear correction: the Van Vleck correction Cross-Correlating a Digital Signal

29. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 29 Van Vleck Correction

30. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 30 XF correlator: limited number of lags N  ‘uniform’ coverage to max. lag N  t  Fourier transform gives spectral response - 22% sidelobes! - Hanning smoothing FX correlator: as XF, but Fourier transform before multiplication  spectral response is - 5% sidelobes Spectral Response; Gibbs Ringing

31. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 31 Spectral Response: XF Correlator

32. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 32 sinc( ) vs. sinc 2 ( )

33. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 33 n.b. radio frequency interference is spread across frequency by the spectral response Gibbs phenomenon: ‘ringing’ off the band edges

34. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 34 The size of a correlator (number of chips, speed, etc.) is generally set by the number of baselines and the maximum total bandwidth. [note also copper/connectivity costs…] Subarrays … trade antennas for channels Bandwidth -- cut  :  same number of lags/spectral points across a smaller  : N chan = constant  narrower channels:  …limited by filters How to Obtain Finer Frequency Resolution

35. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 35 -- recirculation: chips are generally running flat-out for max.  (e.g. EVLA/WIDAR uses a 256 MHz clock with  = 128 MHz/sub- band) For smaller , chips are sitting idle most of the time: e.g., pass 32 MHz to a chip capable of doing 128 M multiplies per second  add some memory, and send two copies of the data with different delays  N chan  1/      2 …limited by memory & data output rates

36. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 36 VLA Correlator: Bandwidths and Numbers of Channels

37. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 37 difficult to send the data to a central location in real time long baselines, unsynchronized clocks  relative phases and delays are poorly known So, record the data and correlate later Advantages of 2-level recording VLBI

38. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 38 Correlator Efficiency  c quantization noise overhead –don’t correlate all possible lags –blanking errors –incorrect quantization levels –incorrect delays

39. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 39 Choice of Architecture number of multiplies: FX wins as {N ant, N chan }  multiplies per second ~ N ant 2  N prod N chan number of logic gates: XF multiplies are much easier than FX; which wins, depends on current technology shuffling the data about: “copper” favors XF over FX for big correlators bright ideas help: hybrid correlators, nifty correlator chips, etc.

40. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 40 New Mexico Correlators VLA EVLA (WIDAR)VLBA ArchitectureXFfilter-XFFX Quantization3-level16/256-level2- or 4-level N ant 274020 Max.  0.2 GHz16 GHz0.256 GHz N chan 1-51216,384-262,144256-2048 Min.  381 Hz0.12 Hz61.0 Hz dt min 1.7 s0.01 s0.13 s Power req’t.50 kW135 kW10-15 kW Data rate3.3 x 10 3 vis/sec2.6 x 10 7 vis/sec3.3 x 10 6 vis/sec

41. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002 41 Current VLAEVLA/WIDAR