Cross- correlators for Radio Astronomy

1. ATP-NSI Cross-correlators for Radio Astronomy Brent Carlson, NRAO Synthesis Imaging Summer School 29 May 2012

2. Outline • What is the purpose of the correlator? • Simplified signal flow • Basic correlator architectures • XF, FX, hybrid • Technology • How do the electronics “work” • The development process • JVLA WIDAR correlator • Now and the future

3. Purpose • To calculate the integrated cross-power response for each pair of antennas “X” and “Y” in the array over some integration time “T”. Statistically independent signals integrate down, whereas the common radio source signal establishes some power level and the uncertainty in that level integrates down.

5. Purpose • These are the visibilities—spatial Fourier components—for each baseline B in the u-v plane that are used to build the image. • The fun begins: • As N increases, number of pairs of antennas goes as ~N2 / 2. (e.g. JVLA has 27 antennas…378 baselines (including “auto” correlations). • As bandwidth and number of (spectral) channels increases, the speed of the electronics increases along with power and cost.

6. Purpose • Analog correlators, operating on analog (continuous time) voltage signals have been built and do work… • But nowadays the analog signal is quantized—in time and amplitude—as soon as possible for stability and to take advantage of “cheap” high-speed digital electronics. • Once the signal is digitized there are no more unknown/unquantifiable effects (well, unless something broke…)

7. Simplified signal flow • STEP #1: • Receive and amplify the signal from the antenna. • The LNA adds significant noise to the weak signal, and must be optimized to optimize sensitivity of the telescope. • Additive noise stability (or calibrated as such) here is important for stable correlator output.

8. Simplified signal flow • What does the signal “look” like? • Time domain (analog):

9. Simplified signal flow • STEP #2: • down-convert (mix) and filter the signal…ready for digital sampling. • Gain fluctuations here not so critical as long as additive noise is stable. • It is the ratio of radio source signal to system noise that is important.

10. Simplified signal flow • STEP #3: • quantize (digitize/sample) the signal. • One of the most critical+difficult steps in the processing chain because it is where the quiet analog world meets the noisy digital world. • Speed is important to maximize bandwidth.

11. “Flash” A-to-D converter diagram.

12. Simplified signal flow • What does the signal look like? • Time domain (digital):

13. Simplified signal flow • Sampling: • Nyquist sampling theorem: must sample at least 2X the highest frequency content to obtain all information about the signal. If less, leads to “aliasing” (confusion). • With noise input: • 1-bit results in ~35% sensitivity loss. • 2-bit results in ~12% sensitivity loss. • 3-bit: 3.5% -- JVLA wideband samplers • 4-bit: 1.5% -- JVLA correlator “internal samplers”

14. Simplified signal flow • Sampling: • adding more bits/sample produces diminishing sensitivity returns for noise input and integrated output. • When narrowband interference is present, need more bits so as not to contaminate the spectrum with saturated sampler-generated harmonics. Get ~6 dB per bit dynamic range for a pure tone. dB=10log(x); if x is a power value. • For real-time signals (music/video) need lots of bits to accurately represent the real-time waveform (e.g. CD ~16-bit sampling=216 = 65,536 levels)

15. Simplified signal flow • STEP #4: • Correct for antenna-dependent wavefront delay. • Two phases: • pure digital delay to +/-0.5 samples. Cheapest is to use memory. Get up to +/-90 deg phase changes at the upper edge of the band…severe decorrelation, therefore need: • sub-sample delay to << +/- 0.5 samples. Various methods, sometimes analog, often digital…more later.

16. Simplified signal flow • STEP #5: • Cross-correlate and accumulate. • Must also correct for “fringe phase” due to the fact that wavefront delay compensation occurs at a different frequency (baseband) than where it originally occurred (at RF in free space). • Various correlator methods to be discussed later…

17. Simplified signal flow • What does the signal look like? • Frequency domain (10e6 samples integrated):

18. Simplified signal flow • What does the signal look like? • Frequency domain…add some lines:

19. Simplified signal flow

20. Correlator architectures • There are two basic methods for correlation: • “XF”: Cross-correlate in the time (tau) domain, then Fourier transform (after integration) to the frequency domain. a.k.a. “lag correlator” • “FX”: Fourier transform in the (real) time domain, then multiply and integrate in the frequency domain. “Convolution in one domain is multiplication in the other domain”

21. Correlator architectures • Hybrid: • Combination of the two. JVLA WIDAR does this as does the ALMA correlator. • Coarse filter into sub-bands, XF each sub-band. • More details later.

22. Correlator architectures • XF: • traditionally simpler to understand+implement—especially for 1-bit or 2-bit correlators (e.g. 1-bit correlator multiplier is XOR gate). Important in “earlier days” because of speed and logic availability. • Data Nant2 distribution in correlator still uses the same few bits. • O(Nchan x sample rate) multiplies/sec…but, very simple operations (multiply-accumulate).

29. Correlator architectures • FX: • More complex, many-bit operations (FFT). (Has been) more difficult to implement. • O(log Nchan) x sample rate multiplies/sec…much more efficient…in principle. • Problems: • Have word-width expansion after FFT: (has been) 1 or 2-bit in, many bits out. • How to window the real-time data before FFT?

31. Correlator architectures • FX problems: • Word-width expansion: • Keep full word width: large Nant2 data distribution problem in correlator. • Reduced word width quasi-floating point. Original VLBA correlator did this. • Re-quantize to ~4 bits. Not much sensitivity loss. With strong interference just toss the bad channel. With strong source lines…measure power in each channel before re-quantization, correct levels afterwards. • Keep reasonable # bits to accurately represent the dynamic range of signals possible…no pre-requantizer power measurement required.

32. Correlator architectures • FX problems: • How to window the real-time data before the FFT? • Don’t…use “box-car” with no overlap…large ringing/sidelobes (sin(x)/x = “sinc” function) but if signal is just noise it is ok. Reduces SNR slightly since all of the data isn’t being correlated. • Triangular 50% overlap…reduces ringing/sidelobes (sinc2), no SNR loss since all data is used. Orig VLBA correlator did this. • Poly-phase FIR filter…excellent!

33. Correlator architectures • Poly-phase FX: • Put poly-phase FIR (Finite Impulse Response) filter in front of the FFT. • Acts like a window/taper function that is longer than the FFT…achieves superior channel-to-channel isolation and spectral dynamic range. ~10 taps per frequency channel is required. • This performance cannot fundamentally be obtained with an XF correlator: window function limited to size of FFT, and causes broadening of spectral features. • ~2X the amount of logic to implement over just the FFT. • Now not such a problem with modern high-performance programmable devices (FPGAs).

34. “Ringing” caused by truncated FFT. a.k.a. “Gibb’s phenomenon. Caused by convolving the true spectrum with a sinc() function. Ringing greatly reduced by tapered windowing before FFT but lines are broadened.

38. Correlator architectures • More on poly-phase FX: • If many channels ~>1k required, must cascade filterbanks to avoid excessive numbers of poly-phase FIR taps (e.g. 1 stage 1Mchannels ~10Mtaps(!); 2 stage ~2k taps). • Each channel out of the FFT will have some leakage (aliasing) from the adjacent channel since it is critically sampled. • Therefore, must oversample the output. Performed by “barrel-rolling” the input across phases such that the output after FFT is oversampled. The next stage poly-phase-FFT of a particular channel will therefore not contain aliasing in regions of interest so that all taken together, clean alias-free spectra are obtained…

39. Correlator architectures • See: • Harris, Dick, Rice, “Digital Receivers and Transmitters using PolyphaseFilterbanks for Wireless Communications”, IEEE transactions on microwave theory and techniques, Vol. 51, No. 4, April 2003. • …for more detail on poly-phase filterbanks (great paper!)

40. Correlator architectures • Hybrid FX-XF (a.k.a. “FXF”) • 1st stage: coarse filterbank (FIR or poly-phase FFT). • Useful as “digital BBC” for frequency-agile sub-band placement. • 2nd stage: XF. • Attractive as an efficient parallel processing method for wideband signals since no large multiplier operations are required (FIRs built with memory LUTs and adders). Has larger noise-power bandwidth in sub-band so can use fewer bits connecting to the ‘X’ part. • JVLA and ALMA correlators built this way (some slight differences in implementations). Probably the last of this breed!

41. Correlator architectures • The actual signal processing operations are just one piece of the puzzle when putting a system together. • Much of the logic and power in a system is consumed by transporting data around, synchronizing things, providing various modes of operation, error detection and recovery etc. • Let’s look “under the hood” of the electronics…

42. Technology • It all starts with fundamental physics…but moving up a level or two: • Transistor “switch”: the FET – Field Effect Transistor.

43. Technology • N-MOS: applying a voltage to the Gate opens a conduction channel between the Drain and Source. • P-MOS: applying a voltage to the Gate closes the conduction channel between the Source and the Drain.

44. MOS: Metal Oxide Semi-conductor. MO is the insulator between the gate and the conduction channel. Extremely sensitive to electro-static discharge (ESD). When the transistor is ON or OFF, no current flows from the gate to the conduction channel (unless it is blown…) Current (power) only flows when changing states, to charge/discharge the gate capacitance…faster state changes consumes more power.

45. CMOS: Complementary Metal Oxide Semiconductor. Output changes faster since it is being driven both high and low. Small amount of leakage current (power) when the conduction channel is switching states.

48. F.A. (Full Adder) A 4-bit 2’s complement multiplier

49. Technology • A logic gate “immediately” reflects changes on its input to its output. It can’t store a value. • A “bunch of gates doing something” is often referred to as “combinational logic”. • A “Flip-Flop” transfers “Data in” to the output “Data out” only on the edge of its clock: A Flip-flop can therefore store a value…a single bit. A Flip-flop is actually a set of cross-coupled gates with feedback implementing an “asynchronous Finite State Machine”.

50. Technology • In digital electronics, pretty much everything is “synchronous”. i.e. changes occur on the clock edge all “in step”. • It’s like a production line…the speed of the line is the clock speed and in each clock cycle each “worker” (bunch of gates doing some logic function) must get their step done before the next clock cycle starts. • As the clock speed increases, the logic “workers” must go faster to keep up.