Report

PAF Beamformer Calibration Using Extended Sources 1 Brian D. Jeffs Brigham Young University CSIRO - CASS CALIM 2014 2-7 March 2014 Kiama, NSW Australia 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 Summary problem statement: 3 C A N E X T E N D ED S O U R C E S B E USED TO CALIBRATE A PAF BEAMFORMER WHEN GOOD BRIGHT POINT-SOURCE CALIBRATORS ARE UNAVAILABLE? Early PAF Experiments (2006) 4 19 element L band PAF on 3m dish Moving RFI (hand held) BYU campus Early PAF RFI Experiments (2006) 5 Moving FM sweep RFI, 10 second integration Subspace Projection and max SNR beamforming Green Bank 20 Meter Telescope 6 19 element L band single and dual polarization, room temperature and cryo cooled PAFs, 2008-2012 Arecibo Telescope 7 BYU 19 element dual pol wideband room temp PAF, 2010 Cornell 19 element dual pol fully cryo cooled AO19 PAF, 2013 Green Bank Telescope 8 NRAO/BYU 19 element dual pol cryo cooled PAF, December 2013 Image credit, NRAO ASKAP 9 D = 12m f/D = 0.5 94 element dual pol room temperature Image credit, CSIRO Fundamentals of PAF Beamformer Calibration 10 ESTIMATING ARRAY RESPONSE VECTORS IN DIRECTIONS OF INTEREST The Narrowband Beamformer 11 Noise : n 1 (i) Space signal of interest y1 (i) w*1 s(i) q y3 (i) w* 3 z 1(i) nM (i) Repeat for each frequency channel. w is (weakly) frequency dependent. H Beamformer weight vector w = [w1, , wM ]T yM(i) Interference H b(i) = = w w y(i) y(i) b(i) + w* M Signal Model: D y(i) = a s(i) + å v d (i)zd (i) + n(i) d=1 The Narrowband Beamformer 12 Noise : n 1 (i) Space signal of interest y1 (i) w*1 s(i) q y3 (i) w* 3 z 1(i) nM (i) Repeat for each frequency channel. w is (weakly) frequency dependent. H Beamformer weight vector w = [w1, , wM ]T yM(i) Interference H b(i) = = w w y(i) y(i) b(i) + UNKNOWN! w* M Signal Model: D y(i) = a s(i) + å v d (i)zd (i) + n(i) d=1 Covariance and Array Response Estimation 13 Calculating w relies critically on array covariance estimation. Definitions: R = E{y(i)y H (i)} = Rs + R n + R z (k+1)N-1 1 H ˆ = R y(i)y (i) å k N i=kN ˆ is computed at the PAF digital receiver / beamformer / R k correlator (ACM processor for ASKAP). For calibration s(i) is a known bright point-source ˆ » R for each 2-D pointing q relative Compute a new R k S(q k ) k to the calibration source. ˆ k » a(qk )for each pointing. Estimate array response vector a PAF Beamforming Calibration Procedure 14 ˆ k needed for: Calibration vectors a Every beam mainlobe direction. Every response constraint direction. Calibration grid 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. ˆ . Acquire array covariance matrices R k One off-pointing per row to estimate (2-5 degrees away). calibration source Rn PAF Beamforming Calibration Procedure 15 Algorithm: 1. 2. 3. 4. The telescope is steered to angle θk relative to the calibration source. ˆ is A signal-plus-noise covariance R k obtained. The telescope is steered several degrees in azimuth and an off-source, ˆ is obtained noise only R n The calibration vector is computed as aˆ k = Rˆ n uk where u k is the dominant solution to: Rˆ k u k = lmax Rˆ n uk Calibration grid calibration source Calibrations are stable for several weeks [Elmer 2012 Feb.] Calculating Beamformer Weights 16 Maximum SNR beamformer Maximize signal to noise plus interference power ratio: w H Rs w ˆ w wsnr = argmax H ® Rs wsnr = lmax R n snr w w R w n Point source case (e.g. calibrator) yields the MVDR solution: Rs,k = s s2 a k a kH ˆ -1a ® wmvdr,k = R n k LCMV beamformer Minimize total output power subject to linear constraints: ˆ s.t. CH w = f ® w = R ˆ -1C[CH RC] ˆ -1 f wlcmv = argmin w H Rw lcmv w Direct control of response pattern at points specified by C. Equiripple or hybrid beamformers [Elmer 2012 Jan] The Challenge: Find a Sizable Catalog of Suitable Calibrator Sources 17 FOR SUFFICIENTLY BRIGHT CONTINUUM SOURCES, EXTENDED OBJECTS MAY NEED TO BE CONSIDERED 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. 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.) 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 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 Calibration Performance Analysis with an Extended Source 22 WITH NO CORRECTION ALGORITHM 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. 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). ò a(q ) f (q,i)dq R = E { òò a(q )a (f ) f (q , i) f (f,i)dq d f } = ò g(q )a(q )a (q )dq R » Då g(q )a distribution a , where g(q )=isEmodeled Continuous { f (q ) } by a ys (i) = H p p H * H p 2 p p grid of independent point sources: p Extended Source Cal Performance Metrics 25 Correlation coefficient between true and extended source estimated boresight calibration vectors: r= H aˆ extn a true aˆ extn a true , 0 £ r £1 Beampattern distortion comparison for matched filter beamformer weights (i.e. max SNR with Rn = I): b(qk ) = wH ak , wtrue = a0 , wextn = aˆ extn 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. Correlation Coefficient vs. Dish size for Sun Cal 26 Reflector Type Diameter, m f/D ρ, Correlation Coeff. ASKAP 12 0.5 0.9994 Green Bank 20 Meter 20 0.43 0.9951 VLA 30 0.36 0.9758 Green Bank 140 ft 43 0.5 0.9252 Green Bank 140 ft 43 0.428 0.9182 Generic 50 long f 50 0.5 0.4911 Generic 50 short f 50 0.428 01572 Parkes 64m 64 0.428 0.0327 Correlation Coefficient vs. Dish size for Sun Cal 27 /rho Correlation coeff between response vecs 1 0.8 0.6 0.4 0.2 0 10 20 30 40 Dish diameter, meters 50 60 70 Beampatterns for Sun Calibration 28 12 m, f/D = 0.5 (~ASKAP) 43 m, f/D = 0.43 (~Green Bank 140’) Beampatterns for Sun Calibration 29 50 m, f/D = 0.5 64 m, f/D = 0.43 (Parkes) A Closed-Form Deconvolution Solution for Extended Source Beamformer Calibration 30 A WORK IN PROGRESS: OBSERVATIONS, APPROACHES, AND IDEAS A Matrix-Vector Calibration Model 31 Represent the sampled source model in matrix form: R k = å g(q p )a p,k a Hp,k = A k GA kH , where (1) p A k = [a1,k , , a P,k ], G = Diag{[g(q1 ), , g(q P )]}, and a p,k is the response vector from the pth source sample point to the array during the kth calibration grid pointing. Require that source sample points q p and cal pointing directions q k be on the same regular grid. Many columns of Aj and Ak , k≠ j, are repeated, though shifted into different positions. In this case we may write Ak = ASk where A contains the unknown set of all observed array response vectors, and Sk is a known sparse column selection matrix. A Matrix-Vector Calibration Model (cont.) 32 Rewrite (1): Rk = Ak GAkH = ASk GSkH A H Use Kronecker product form to isolate the unknowns vec{Rk } = vec{ASk GSkH AH } = (A* Ä A)vec{Sk GSkH } Now stack all of these column vectors for each calibration grid pointing into a large matrix [ vec{R1}, , vec{R K }] = (A* Ä A)éëvec{S1G S1H }, , vec{SK G SKH }ùû (A Ä A) = [ vec{R1}, , vec{R K }] éëvec{S1G S1H }, , vec{SK G SKH }ùû where (- ^) indicates matrix pseudo inverse. * This solution must be studied to see if it is practical. (-^) Conclusions 33 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. Bibliography 34 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. notes 35 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