Relevance of a Generic and efficient "E-field Parallel Imaging Correlator”(EPIC) for future radio telescopes Nithyanandan Thyagarajan (ASU, Tempe) Adam P. Beardsley (ASU, Tempe) Judd Bowman (ASU, Tempe) Miguel Morales (UW, Seattle)

Outline Synthesis imaging in radio interferometry Traditional Interferometry correlator architectures – FX – XF Direct Imaging Modular Optimal Fast Fourier Imaging EPIC implementation of MOFF in software – EPIC imaging in action – EPIC Calibration in action – Imaging performance of EPIC vs. FX EPIC on future large-N dense array layouts Testing GPU-based EPIC on HERA

Quick Refresher on Synthesis Imaging Van Cittert – Zernike equations for synthesis imaging Interferometers in the observer’s frame make Fourier plane measurements of spatial structures in the sky frame Each pair of antennas samples a spatial wave mode in the sky plane Signal sensitivity and angular resolution depend on collecting area and widest antenna spacings - thus large-N dense array layouts are desirable

Traditional Correlator Architectures FX Antenna based temporal F() (~N antennas) to get spectrum Cross-multiply spectra between antenna pairs (X) Efficient for large-N arrays Fractional sample delay correction Robust to RFI e.g., GMRT XF Antenna pair based temporal correlation X() (~N 2 antenna pairs) to get lags Temporal F() on lags on N 2 antenna pairs Modest requirements on data wordsize Simpler architecture e.g., EVLA

Motivation for Direct Imaging To gain angular resolution, longer baselines required To gain sensitivity, larger collecting area required between the longest baselines Cost of the correlator scales as N 2 which gets steep for next-generation antenna arrays which will have ~1000 antennas

Concept of Direct Imaging Antennas placed on a grid and perform spatial FFT of antenna voltages on grid to get complex voltage images Square the transformed image to obtain real- valued intensity images Using spatial FFT requires antenna voltages to be measured on a grid Current implementation: – 8x8 array in Japan (Daishido et al. 2000) – 4x8 BEST-2 array at Radiotelescopi de Medicina, Italy (Foster et al. 2014)

Limitations of Direct Imaging Uniformly arranged arrays have poor response in the sky frame in the form of grating lobes – thus not ideal for imaging Aliasing of objects from outside field of view Assumptions have to be made that antennas are identical and hence poor calibration Calibration still requires antenna correlations

Modular Optimal Frequency Fourier Imaging A generalized direct imaging algorithm – No requirement for antennas to be on a grid – No assumptions made that antennas have to be identical

Mathematical Framework Measurement Equation rearranged as: Optimal Image Estimate: The two are equivalent

E-field based imaging Remembering… Interferometer power pattern is the correlation of individual antenna voltage patterns Visibility measured by interferometer is the correlation of individual antenna voltages Measured antenna E-field is obtained by propagation of the E-field from the object Optimal Image estimate can be re-written as

Intuition behind E-field based Imaging Basis: Convolution – multiplication theorem of Fourier Transform applied in spatial domain Digitized E-field from each antenna is weighted by thermal noise Then gridded using antenna holographic gain pattern Spatially Fourier transformed to get E-field image E-field image is squared and averaged

Advantages over grid-based Direct Imaging Antennas can be placed in any pattern, not necessarily on a grid Gridding with voltage beams naturally places them on a grid besides ensuring optimal inversion Grid spacing can be chosen to match the field of view and the science case – no problems with aliasing Heterogeneous arrays can be easily absorbed since voltage gains for each antenna are used Output image has equivalent information as from FX/XF correlator visibilities Imaging effects such as w-projection, time-dependent wide-field refractive and scintillating atmospheric distortions can be accounted for

EPIC implementation of MOFF imaging Object Oriented Python codes Parallelization for efficiency and emulating real- life telescope arrays Implements generic antenna layouts Accounts for non-identical antenna shapes Calibrates using only measured antenna voltages Contains E-field simulator and FX/XF correlator/imaging pipelines for reference Publicly available on github –

EPIC Flowchart F t () A1A1 E 1 (t) E 1 (f) Calibrate K1K1 E 1 (f) F t () ANAN E N (t) E N (f) Calibrate KNKN E N (f) Grid() F r () E g (f) Square() Image() F t () A1A1 E 1 (t) E 1 (f) Calibrate K1K1 E 1 (f) F t () ANAN E N (t) E N (f) Calibrate KNKN E N (f) Grid() F r () E g (f) Square() Image() I s t 1 I s t M Average() I s (f) Propagated Sky E-field t1t1 tMtM Accumulated Image I s (f)

FX Flowchart F t () A1A1 E 1 (t) E 1 (f) F t () ANAN E N (t) E N (f) F t () A1A1 E 1 (t) E 1 (f) F t () ANAN E N (t) E N (f) Propagated Sky E-field t1t1 tMtM X t () V ij (f) Calibrate V ij (f) Average() V ij t m V ij t 1 V ij t M Image() F r () X t () V ij (f) Calibrate V ij (f) I s (f)

Imaging with EPIC vs. FX Toy Example: Nchan = 16 df = s f0 = 150 MHz MWA core layout inside 150 m (51 antennas) Square antenna kernels

EPIC on LWA Data

Imaging with EPIC vs. FX (zero spacing)

Imaging EPIC vs. FX (Rounding Noise in Gridding)

EPIC Calibration E a = uncalibrated antenna electric field (per frequency channel) from antenna A E ϑ = holographic image pixel at a particular direction of interest g a = gain of antenna A B(ϑ) = Antenna beam in direction ϑ. Assumes an average beam for all antennas. V ϑ ab = model “true” visibilities, phasing the array to position ϑ. Only needed once per calibration cycle, not a real-time requirement.

EPIC Calibration in Action (Single Calibrator)

EPIC Calibration in Action (Multiple Calibrators)

Scaling relations EPIC vs. FX

Implications from Scaling Relations EPIC Most expensive step – 2D spatial FFT at every ADC output cycle – O(N g log N g ) For a given N g, it does not depend on N a N g is insensitive to for increasing N a if densely packed. e.g. HERA Thus the array layout can get dense with no additional cost FX Most expensive step – FX operations on all antenna pairs at every ADC output cycle – O(N a 2 ) Accumulation in visibilities before imaging offers advantage for small-N arrays Advantage lost for large arrays which have short allowed accumulation times (due to rapid fringe rate, ionospheric changes, etc.)

Current and future telescopes in MOFF-FX parameter space Each aperture size has a MOFF “wedge” where MOFF wins over FX MOFF is favored by large number of antennas densely packed within a given maximum baseline length Top left of dotted line is where MOFF is more efficient than FX Solid line shows where expanded HERA will be Shaded area is where LWA will evolve to be MOFF will benefit most of future instruments

Science applications with EPIC Real-time transient science is our primary target – Can sample timescales down to inverse of channel width (~100 kHz) – Ideal for bright, fast (FRBs) and slow transients (planetary) with large-N dense arrays

Upcoming plans Design a GPU-based for HERA-19/37 HERA specs B = 100MHz (Could reduce bandwidth to ease the throughput) df = 97.7 kHz (1024 channels. Don’t necessarily need all) at 150 MHz, FoV ~ 10 degrees. So we should be able to grid at ~4 lambda N a =19 N chan = 1024 N g ~ 256 (16^2) – at an image every s (df ~ 100kHz), the FFT will require ~1 GFLOPS per channel – bandwidth into the FFT will be ~ few hundred MB/s per channel

Summary Efficient alternative to FX imaging techniques for large-N dense arrays Promising future for time-domain astronomy with large-N, dense arrays GPU-based implementation planned with HERA-37 Highly parallelized implementation with EPIC code publicly available Watch out for the paper coming out soon!