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