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Cross Correlators

Cross Correlators. Michael P. Rupen NRAO/Socorro. What is a Correlator?. In an optical telescope… a lens or a mirror collects the light & brings it to a focus a spectrograph separates the different frequencies.

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Cross Correlators

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

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

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

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

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

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

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

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

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

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

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

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

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

  17. 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 • The clever approach: • instead of sticking in a delay, put in a filter that shifts the phase for all frequencies by /2 Broad-band Continuum Correlators M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002

  18. 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: Spectral Line Correlators M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002

  19. 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! Fourier Transforms: a motivational exercise M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002

  20. Spectral Line Correlators (cont’d) • 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 M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002

  21. F v1 S1() v2 S2()  Fourier transform t X X multiply multiply Fourier transform S() t F  FX vs. XF M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002

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

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

  24. Digitization • Sampling:v(t)v(tk), with tk=(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 • Quantization:v(t)v(t) + t • quantization noise • quantized signal is not band-limited  oversampling helps M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002

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

  26. V1 0.3 V2 integrated & normalized Signals come in… sampled… quantized… delayed… multiplied… Michael’s Miniature Correlator M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002

  27. Cross-Correlating a Digital Signal • 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 M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002

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

  29. Spectral Response; Gibbs Ringing • 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 M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002

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

  31. sinc( ) vs. sinc2( ) M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002

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

  33. How to Obtain Finer Frequency Resolution • 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 : Nchan= constant •  narrower channels:  • …limited by filters M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002

  34. -- 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 • Nchan 1/ •   2 …limited by memory & data output rates M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002

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

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

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

  38. Choice of Architecture • number of multiplies: FX wins as {Nant, Nchan} multiplies per second ~ Nant2  Nprod Nchan • 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. M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002

  39. New Mexico Correlators M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002

  40. Current VLA EVLA/WIDAR M.P. Rupen, Synthesis Imaging Summer School, 18 June 2002

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