cross correlators - physics &...
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Astronomy 423 at UNM Radio Astronomy
Cross Correlators
Jayce Dowell/Greg Taylor University of New Mexico Spring 2017
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G. Taylor, Astr 423 at UNM
2 Outline
• Re-cap of interferometry – What is a correlator?
• The correlation function • Simple correlators • Spectral line correlators • Details – Sampling and quantization – Delay model
• The VLA and LWA correlators
This lecture is complementary to Chapter 4 of ASP 180 and is based on a lecture by Walter Brisken
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G. Taylor, Astr 423 at UNM
Re-cap of Interferometry
• What are we fundamentally trying to measure?
• How do we accomplish this in a traditional telescope? – Optical or radio
• What changes when we go to a interferometer? – A “sparse” telescope
• What do visibilities tell us about the sky?
3
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G. Taylor, Astr 423 at UNM
4 What is a Correlator?
• Visibilities are in general a function of – Frequency – Antenna pair – Time
• They are used for – Imaging – Spectroscopy / polarimetry – Astrometry
A correlator is a hardware or software device that combines sampled voltage time series from one or more antennas to produce sets of complex visibilities, .
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G. Taylor, Astr 423 at UNM
5 The Correlation Function
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G. Taylor, Astr 423 at UNM
6 Auto-Correlation and Convolution Functions
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G. Taylor, Astr 423 at UNM
7 Auto-Correlation and Convolution Functions
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G. Taylor, Astr 423 at UNM
8 The Correlation Function
• If it is an auto-correlation (AC). Otherwise it is a cross-correlation (CC).
• Useful for – Determining timescales (CC and AC) – Motion detection (2-D CC) – Optical character recognition (2-D CC) – Pulsar timing – Template matching (CC)
• Also called “matched filtering”
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G. Taylor, Astr 423 at UNM
9 A Real (valued) Cross Correlator
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G. Taylor, Astr 423 at UNM
10 Visibilities
What astronomers really want is the complex visibility where the real part of is the voltage measured by antenna . So what is the imaginary part of ? It is the same as the real part but with each frequency component phase lagged by 90 degrees.
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G. Taylor, Astr 423 at UNM
11 The Complex Correlator
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G. Taylor, Astr 423 at UNM
12 Spectral Line Correlators
• Chop up bandwidth for – Calibration
• Bandpass calibration • Fringe fitting
– Spectroscopy – Wide-field imaging – (Its all Spectral Line these days)
• Conceptual version – Build analog filter bank – Attach a complex correlator to each filter
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G. Taylor, Astr 423 at UNM
13 Practical Spectral Line Correlators
• Use a single filter / sampler – Easier to calibrate – Practical, up to a point
• The FX architecture – F : Replace filterbank with digital Fourier transform – X : Use a complex-correlator for each frequency channel – Then integrate
• The XF architecture – X : Measure correlation function at many lags – Integrate – F : Fourier transform
• Other architectures possible
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G. Taylor, Astr 423 at UNM
14 The XF Correlator
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G. Taylor, Astr 423 at UNM
15 XF Spectral Response
• XF correlators measure lags over a finite delay range
• Results in convolved visibility spectrum
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G. Taylor, Astr 423 at UNM
16 XF Spectral Response (2)
22% sidelobes!
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G. Taylor, Astr 423 at UNM
17 Hanning Smoothing
• Multiply lag spectrum by Hanning taper function
• This is equivalent to convolution of the spectrum by
● Note that sensitivity and spectral resolution are reduced.
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G. Taylor, Astr 423 at UNM
18 Hanning Smoothing (2)
2 chans wide
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G. Taylor, Astr 423 at UNM
19 FX Correlators
• Spectrum is available before integration – Can apply fractional sample delay per channel – Can apply pulsar gate per channel
• Most of the digital parts run N times slower than the sample rate
• Fewer computations (compared to XF)
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G. Taylor, Astr 423 at UNM
20 The FX correlator
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G. Taylor, Astr 423 at UNM
21 FX Spectral Response
• FX Correlators derive spectra from truncated time series
● Results in convolved visibility spectrum
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G. Taylor, Astr 423 at UNM
22 FX Spectral Response (2)
5% sidelobes
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G. Taylor, Astr 423 at UNM
23 Time Series, Sampling, and Quantization
• are real-valued time series sampled at “uniform” intervals, .
• The sampling theorem allows this to accurately reconstruct a bandwidth of .
• Sampling involves quantization of the signal – Quantization noise – Strong signals become non-linear – Sampling theorem violated!
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G. Taylor, Astr 423 at UNM
24 Quantization Noise
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G. Taylor, Astr 423 at UNM
25 Automatic Gain Control (AGC)
• Normally prior to sampling the amplitude level of each time series is adjusted so that quantization noise is minimized.
• This occurs on timescales very long compared to a sample interval.
• The magnitude of the amplitude is stored so that the true amplitudes can be reconstructed after correlation.
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G. Taylor, Astr 423 at UNM
26 The Correlation Coefficient
• The correlation coefficient, measures the likeness of two time series in an amplitude independent manner:
• Normally the correlation coefficient is much less than 1
• Because of AGC, the correlator actually measures the correlation coefficient. The visibility amplitude is restored by dividing by the AGC gain.
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G. Taylor, Astr 423 at UNM
27 Van Vleck Correction
• At low correlation, quantization increases correlation
• Quantization causes predictable non-linearity at high correlation
• Correction must be applied to the real and imaginary parts of separately – Thus the visibility phase is
affected as well as the amplitude
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G. Taylor, Astr 423 at UNM
28 The Delay Model
• is the difference between the geometric delays of antenna and antenna . It can be + or - .
• The delay center moves across the sky – is changing constantly
• Fringes at the delay center are stopped. – Long time integrations can be done – Wide bandwidths can be used
• Simple delay models incorporate: – Antenna locations – Source position – Earth orientation
• VLBI delay models must include much more!
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G. Taylor, Astr 423 at UNM
29 Pulsar Gating
• Pulsars emit regular pulses with small duty cycle • Period in range 1 ms to 8 s; • Blanking during off-pulse improves sensitivity • Propagation delay is frequency dependent
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G. Taylor, Astr 423 at UNM
30 The [old] VLBA Correlator
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G. Taylor, Astr 423 at UNM
31 VLBA Multiply Accumulate (MAC) Card
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G. Taylor, Astr 423 at UNM
32 [Old] VLA MAC Card
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G. Taylor, Astr 423 at UNM
33 BEE2-based Correlator
• BEE2: FPGA-based, scalable, modular, upgradeable signal processing system for radio astronomy developed at Berkeley
• ROACH2 boards at LWA-SV • Being used for several projects
– 300-station FX correlator for EOR telescope (HERA)
– 288-station correlator for LWA-OVRO • Modest hardware cost ($15k/ROACH2 +
switch) – LWA-SV uses 16 ROACH2 + 7 GPU
servers • Real effort is in the FPGA “software”
ROACH board
IBOB: Internet BreakOut Board
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G. Taylor, Astr 423 at UNM
34 The VLA WIDAR Correlator
• XF architecture duplicated 64 times, or “FXF” – Four 2GHz basebands per polarization (3 bit sampling) – Digital filterbank makes 16 subbands per baseband – 16,384 channels/baseline at full sensitivity – 4 million channels with less bandwidth!
• Initially will support 32 stations with plans for 48 • 2 stations at 25% bandwidth or 4 stations at 6.25%
bandwidth can replace 1 station input • Correlator efficiency is about 95%
– Compare to 81% for VLA • VLBI and LWA ready
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G. Taylor, Astr 423 at UNM
35 Basic Correlator Stages for the LWA
1. Correlate LWA1 beams with single dipoles at LWA1 and LWA-SV (partial success)
2. Digitize VLA dishes and correlate with LWA1 (works!)
3. Digitize VLA dishes and correlate with LWA1 and LWA-SV (soon)
4. Correlate ~10 LWA Phase II stations (the “LWIA”)
5. Correlate full LWA (up to 50 stations)
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G. Taylor, Astr 423 at UNM
36 Some Potential Correlator Options
• Software (up to ~10 stations?) – LWA Software Library (UNM)? – DifX software correlator? – GPU-based correlator?
• Hardware (for 10+) – GPU-based correlator? – CASPER-based (FPGA/GPU) correlator?
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G. Taylor, Astr 423 at UNM
37 Strawman LWA Correlator Plan
• Correlate LWA1 with a single dipole placed few hundred meters away. – Use LSL
• Correlate VLA + LWA1 + LWA-SV – Use LSL
• Correlate first 9 (or so) LWA stations – Use software correlator on a cluster ?
• Correlate full LWA – LEDA-style FPGA/GPU correlator?
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G. Taylor, Astr 423 at UNM
Further Reading
• http://www.nrao.edu/whatisra/mechanisms.shtml • http://www.nrao.edu/whatisra/ • www.nrao.edu
• Synthesis Imaging in Radio Astronomy • ASP Vol 180, eds Taylor, Carilli & Perley