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CALIM 20142-7 March 2014

Kiama, NSW Austral ia

PAF Beamformer Calibration Using Extended Sources

Brian D. JeffsBrigham Young University

CSIRO - CASS

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Acknowledgements

Thanks to the following people for their valuable contributions to this work:

From CSIRO, CASS:Aaron ChippendaleAidan HotanMaxim Voronkov

From Brigham Young UniversityKarl WarnickMichael Elmer

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CAN EXTENDED SOURCES BE USED TO CALIBRATE A PAF BEAMFORMER WHEN GOOD

BRIGHT POINT-SOURCE CALIBRATORS ARE

UNAVAILABLE?

Summary problem statement:

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Early PAF Experiments (2006)

19 element L band PAF on 3m dishMoving RFI (hand held)BYU campus

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Early PAF RFI Experiments (2006)

Moving FM sweep RFI, 10 second integration

Subspace Projection and

max SNR beamforming

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Green Bank 20 Meter Telescope

19 element L band single and dual polarization, room temperature and cryo cooled PAFs, 2008-2012

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Arecibo Telescope

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

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Green Bank Telescope

NRAO/BYU 19 element dual pol cryo cooled PAF,December 2013 Image credit, NRAO

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ASKAP

94 element dual pol room temperatureImage credit, CSIRO

D = 12mf/D = 0.5

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ESTIMATING ARRAY RESPONSE VECTORS IN DIRECTIONS OF INTEREST

Fundamentals of PAF Beamformer Calibration

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Signal Model:

The Narrowband Beamformer

Repeat for each frequency channel. w is (weakly) frequency dependent.

Beamformer weight vector

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Signal Model:

The Narrowband Beamformer

Repeat for each frequency channel. w is (weakly) frequency dependent.

Beamformer weight vector

UNKNOWN!

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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.

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PAF Beamforming Calibration Procedure

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

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PAF Beamforming Calibration Procedure

Algorithm:

Calibrations are stable for several weeks [Elmer 2012 Feb.]

Calibration grid

calibration source

1. The telescope is steered to angle θk 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:

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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]

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FOR SUFFICIENTLY BRIGHT CONTINUUM SOURCES, EXTENDED OBJECTS MAY NEED

TO BE CONSIDERED

The Challenge: Find a Sizable Catalog of Suitable Calibrator

Sources

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Calibrator Requirements

High radio surface brightness High SNR calibration produces low error beamformer

weightsPoint-like compact structureSources 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.

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Calibrator Requirements

High radio surface brightness High SNR calibration produces low error beamformer

weightsPoint-like compact structureSources 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.)

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Parkes ASKAP Testbed Beamformer Calibration

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 = 12mf/D = 0.4

D = 64mf/D = 0.428

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Parkes ASKAP Testbed Beamformer Calibration

Successful single dish calibration using: The Sun Virgo A A few other compact

sources at lower SNROther 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 = 12mf/D = 0.4

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WITH NO CORRECTION ALGORITHM

Calibration Performance Analysis with an Extended

Source

23Continuous 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.

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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).

Continuous distribution is modeled by a grid of independent point sources:

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Extended Source Cal Performance Metrics

Correlation coefficient between true and extended source estimated boresight calibration vectors:

Beampattern distortion comparison for matched filter beamformer weights (i.e. max SNR with Rn = 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.

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Correlation Coefficient vs. Dish size for Sun Cal

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

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Correlation Coefficient vs. Dish size for Sun Cal

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Beampatterns for Sun Calibration

12 m, f/D = 0.5(~ASKAP)

43 m, f/D = 0.43(~Green Bank 140’)

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Beampatterns for Sun Calibration

50 m, f/D = 0.5

64 m, f/D = 0.43(Parkes)

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A WORK IN PROGRESS:OBSERVATIONS, APPROACHES, AND IDEAS

A Closed-Form Deconvolution Solution for Extended Source

Beamformer Calibration

A Matrix-Vector Calibration Model31

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 Aj and Ak , 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)

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A Matrix-Vector Calibration Model (cont.)

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.

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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.

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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.

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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