1. CALIM 2014 2-7 March 2014 Kiama, NSW Australia 1 PAF Beamformer Calibration Using Extended Sources Brian D. Jeffs Brigham Young University CSIRO - CASS

2. Acknowledgements 2 Thanks to the following people for their valuable contributions to this work: From CSIRO, CASS: Aaron Chippendale Aidan Hotan Maxim Voronkov From Brigham Young University Karl Warnick Michael Elmer

3. CAN EXTENDED SOURCES BE USED TO CALIBRATE A PAF BEAMFORMER WHEN GOOD BRIGHT POINT-SOURCE CALIBRATORS ARE UNAVAILABLE? 3 Summary problem statement:

4. Early PAF Experiments (2006) 19 element L band PAF on 3m dish Moving RFI (hand held) BYU campus 4

5. Early PAF RFI Experiments (2006) Moving FM sweep RFI, 10 second integration Subspace Projection and max SNR beamforming 5

6. Green Bank 20 Meter Telescope 19 element L band single and dual polarization, room temperature and cryo cooled PAFs, 2008-2012 6

7. Arecibo Telescope 7 BYU 19 element dual pol wideband room temp PAF, 2010 Cornell 19 element dual pol fully cryo cooled AO19 PAF, 2013

8. Green Bank Telescope 8 NRAO/BYU 19 element dual pol cryo cooled PAF, December 2013 Image credit, NRAO

9. ASKAP 9 94 element dual pol room temperature Image credit, CSIRO D = 12m f/D = 0.5

10. ESTIMATING ARRAY RESPONSE VECTORS IN DIRECTIONS OF INTEREST 10 Fundamentals of PAF Beamformer Calibration

11.  Signal Model: The Narrowband Beamformer Repeat for each frequency channel. w is (weakly) frequency dependent. 11  Beamformer weight vector

12.  Signal Model: The Narrowband Beamformer Repeat for each frequency channel. w is (weakly) frequency dependent. 12  Beamformer weight vector UNKNOWN!

13. Covariance and Array Response Estimation Calculating w relies critically on array covariance estimation.  Definitions:  is computed at the PAF digital receiver / beamformer / correlator (ACM processor for ASKAP). For calibration s(i) is a known bright point-source  Compute a new for each 2-D pointing relative to the calibration source.  Estimate array response vector for each pointing. 13

14. PAF Beamforming Calibration Procedure 14 Calibration vectors needed for:  Every beam mainlobe direction.  Every response constraint direction. We used a 31×31 raster grid of reflector pointing directions:  Centered on calibrator source e.g. Cas A, Cygnus A, Tau A, Virgo A. e.g. any 10+ Jy star for Arecibo PAF.  3-10 sec integration time per pointing.  Acquire array covariance matrices. One off-pointing per row to estimate (2-5 degrees away). Calibration grid calibration source

15. PAF Beamforming Calibration Procedure 15 Algorithm: Calibrations are stable for several weeks [Elmer 2012 Feb.] Calibration grid calibration source 1.The telescope is steered to angle θ k relative to the calibration source. 2.A signal-plus-noise covariance is obtained. 3.The telescope is steered several degrees in azimuth and an off-source, noise only is obtained 4.The calibration vector is computed as where is the dominant solution to:

16. Calculating Beamformer Weights Maximum SNR beamformer  Maximize signal to noise plus interference power ratio:  Point source case (e.g. calibrator) yields the MVDR solution: LCMV beamformer  Minimize total output power subject to linear constraints:  Direct control of response pattern at points specified by C. Equiripple or hybrid beamformers [Elmer 2012 Jan] 16

17. FOR SUFFICIENTLY BRIGHT CONTINUUM SOURCES, EXTENDED OBJECTS MAY NEED TO BE CONSIDERED 17 The Challenge: Find a Sizable Catalog of Suitable Calibrator Sources

18. Calibrator Requirements 18 High radio surface brightness  High SNR calibration produces low error beamformer weights Point-like compact structure Sources covering a variety of ‘RA and Dec locations  Convenient if at least one of the sources is usually up  Variation in Dec allows for pointing dependent calibration Continuum sources  A distinct w must be computed for every frequency channel.

19. Calibrator Requirements 19 High radio surface brightness  High SNR calibration produces low error beamformer weights Point-like compact structure Sources covering a variety of ‘RA and Dec locations  Convenient if at least one of the sources is usually up  Variation in Dec allows for pointing dependent calibration Continuum sources  A distinct w must be computed for every frequency channel. Few candidates exit for Southern Hemisphere observation with a small dish! (Cas A and Cyg A are not usable.)

20. Parkes ASKAP Testbed Beamformer Calibration 20 12m Patriot dish  CSIRO methods listed below were developed and used by: - Aaron Chippendale - Maxim Voronkov - Aidan Hotan Interferometric assist  A 64m aperture helps!  Allows use of much weaker sources that can’t be detected at calibration levels by the 12m dish alone.  Can multiple dishes at ASKAP site be phased up to use in this mode? D = 12m f/D = 0.4 D = 64m f/D = 0.428

21. Parkes ASKAP Testbed Beamformer Calibration 21 Successful single dish calibration using:  The Sun  Virgo A  A few other compact sources at lower SNR Other bright extended sources attempted:  Crab nebula  Orion nebula (M42)  Galactic center  None produced stable dominant calibration eigenvectors in all frequency channels  Consider also: the Moon, Centarus A (very wide), etc. D = 12m f/D = 0.4

22. WITH NO CORRECTION ALGORITHM 22 Calibration Performance Analysis with an Extended Source

23. Continuous Sun Intensity Profile Model Represents average 2-D extended source suface intensity function g(θ). 32.1 arc minute cross section, plus corona region. Arbitrary relative scale. Reference: A.D. Kuzmin, Radioastronomical Methods of Antenna Measurements, Academic Press, 1966. 23

24. Discrete Sample Model Continuous distribution g(θ) is modeled by a grid of independent point sources. Sample spacing varies with D and beamwidth. 16 points per HPBW. As seen in array covariance R, discrete model acts like a Riemann integral of g(θ). 24 Continuous distribution is modeled by a grid of independent point sources:

25. Extended Source Cal Performance Metrics 25 Correlation coefficient between true and extended source estimated boresight calibration vectors: Beampattern distortion comparison for matched filter beamformer weights (i.e. max SNR with R n = I): Respective a values are calculated with a detailed full-wave simulation of dish and PAF, including element patterns, mutual coupling, etc. Used single pol 19 element BYU PAF.

26. Correlation Coefficient vs. Dish size for Sun Cal 26 Reflector TypeDiameter, mf/Dρ, Correlation Coeff. ASKAP120.50.9994 Green Bank 20 Meter200.430.9951 VLA300.360.9758 Green Bank 140 ft430.50.9252 Green Bank 140 ft430.4280.9182 Generic 50 long f500.50.4911 Generic 50 short f500.42801572 Parkes 64m640.4280.0327

27. Correlation Coefficient vs. Dish size for Sun Cal 27

28. Beampatterns for Sun Calibration 28 12 m, f/D = 0.5 (~ASKAP) 43 m, f/D = 0.43 (~Green Bank 140’)

29. Beampatterns for Sun Calibration 29 50 m, f/D = 0.5 64 m, f/D = 0.43 (Parkes)

30. A WORK IN PROGRESS: OBSERVATIONS, APPROACHES, AND IDEAS 30 A Closed-Form Deconvolution Solution for Extended Source Beamformer Calibration

31. A Matrix-Vector Calibration Model 31 Represent the sampled source model in matrix form: Require that source sample points and cal pointing directions be on the same regular grid. Many columns of A j and A k, k≠ j, are repeated, though shifted into different positions. In this case we may write where contains the unknown set of all observed array response vectors, and is a known sparse column selection matrix. (1)

32. A Matrix-Vector Calibration Model (cont.) 32 Rewrite (1): Use Kronecker product form to isolate the unknowns Now stack all of these column vectors for each calibration grid pointing into a large matrix This solution must be studied to see if it is practical.

33. Conclusions For ASKAP PAF, the Sun and Moon are viable single dish calibrator sources without deconvolution or interferometry with a large dish reference. Performance drops off rapidly as cal source extend exceeds a beamwidth. More work is needed to develop a deconvolution method that can exploit truly extended sources for calibration. 33

34. Bibliography A.D. Kuzmin, Radioastronomical Methods of Antenna Measurements, Academic, 1966. J. R. Nagel, K. F. Warnick, B. D. Jeffs, J. R. Fisher, and R. Bradley, “Experimental verification of radio frequency interference mitigation with a focal plane array feed,” Radio Science, vol. 42, RS6013, doi 10.1029/2007RS003630, 2007. S. van der Tol and A.-J. van der Veen, “Application of robust Capon beamforming to radio astronomical Imaging,” Proceedings of ICASSP 2005, vol. iv, pp. 1089-1092, March 2005. M.J. Elmer, B.D. Jeffs, and K.F. Warnick, “Long-term Calibration Stability of a Radio Astronomical Phased Array Feed,” The Astronomical Journal, Vol. AJ 145, 24, Jan. 2013. M. Elmer*, B.D. Jeffs, K.F. Warnick, J.R. Fisher, and R. Norrod, “Beamformer Design Methods for Radio Astronomical Phased Array Feeds,” IEEE Transactions on Antennas and Propagation, vol. 60, no. 2, Feb. 2012. B.D. Jeffs, K.F. Warnick, J. Landon*, J. Waldron*, D. Jones*, J.R. Fisher, and R.D. Norrod, “Signal processing for phased array feeds in radio astronomical telescopes,” IEEE Journal of Selected Topics in Signal Processing, vol. 2, no. 5, Oct., 2008, pp. 635-646. 34

35. notes ASKAP beamwidth: 1.1 deg., GB 20 Meter: 0.64, VLA: 0.43, GB 43: 0.23, Parkes 0.20 @ 1.6 GHz Sun is 0.535 deg., apparent 0.665 deg w/ corona 35