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Spatial Array Processing Methods
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SPATIAL ARRAY PROCESSING METHODS FOR RADIO ASTRONOMICAL RFI MITIGATION
Timely Technical Tutorials, URSI Boulder 2013, J2
Brian D. Jeffs and Karl F. Warnick Brigham Young University
Many array-based instruments suitable for spatial filtering are now in use, and new classes and forms are on the way
Applications in Radio Astronomy
Synthesis Imaging Array Spatial Filtering
Image credit: Dave Finely, National Radio Astronomy Observatory / Associated Universities, Inc. / National Science Foundation.
¨ Arrays such as the EVLA, WSRT, GMRT, etc. are well suited for adaptive array processing.
¨ The full array can be configured as a nulling beamformer. ¨ Array elements are the single dishes, or a station beam.
Synthesis Array Spatial Filtering
High gain main antenna arrayLow gain auxiliaryantennas
q
pI(p,q)
s(i)
A(p,q)
uwv
r1r2 rM
Interferingsatellite
Deep-spaceobject
y (i)m
y (i)a
Ground-basedtransmitter
z (i)1
z (i)2
m
¨ Post-correlation processing removes interference from visibilities.
¨ Large aperture means tracking speed is an issue. Need short integration dump times.
¨ Dish directivity helps and hurts: partially rejects RFI, but reduces INR needed to estimate interference subspace.
¨ Auxiliary antennas help.
Phased Array Feed Spatial Filtering
y(i)
s(i)z(i)
Radio telescope dish with a phased array feed
¨ Just now being deployed.
¨ Small array aperture helps with moving interferer tracking.
¨ Beamshape distortions may be a critical problem.
¨ Dish directivity helps and hurts cancelation.
¨ Must cancel sources outside the FOV in dish sidelobes.
Aperture Arrays
e.g. a LOFAR station aperture array with 96 antennas Images copyright ASTRON
Interior view showing four dual pol broadband dipole elements
Aperture Arrays
¨ Low frequency: LOFAR, LWA, MWA, PAPER, etc. ¨ Parabolic dish collecting area is replaced with a
“station” of closely packed simple antennas ¨ Wide fields of view for station elements make them
highly susceptible to RFI from horizon to horizon. ¨ Non SOI deep space objects must be treated as RFI
and cancelled (or peeled). ¨ Tracking is slow for station beams, fast for full
array.
These have always been promising, but have issues in the RA application
Basic Processing Structures
¨ Signal Model:
The Narrowband Beamformer
+
w*1
w*3
w*M
b(i) = w y(i) H
!
Space signalof interest
Noise :
Interference
y (i)1
y (i)3
y (i)M
n (i)1
n (i)M
s(i)
z (i)1
¨ Repeat for each frequency channel.
¨ w is (weakly) frequency dependent. y(i) = a s(i)+ vdd=1
D
∑ (i)zd (i)+n(i)
Time-dependent Covariance Estimation
¨ Calculating w for spatial nulling relies critically on array covariance estimation. ¤ Definitions:
¤ Must identify the interferer vector subspace portion Rz
¨ Must update frequently to track interferer motion ¤ Compute for all short term integrations (STI), k. ¤ STI windows are Nsti samples long, which depends on
motion rate and aperture size.
R = E{y(i)yH (i)} =Rs +Rn +Rz
Rk =1N
y(i)yH (i)i=kN
(k+1)N−1
∑
Rk
Classical Adaptive Canceling Beamformers
¨ Maximum SNR beamformer ¤ Maximize signal to noise plus interference power ratio:
¤ Point source case yields the MVDR solution:
¨ LCMV beamformer ¤ Minimize total variance subject to linear constraints:
¤ Direct control of response pattern at points specified by C. ¤ Can also constrain derivatives (slope) or eigen vectors.
wsnr = argmaxwwHRsw
wH (Rz +Rn )w→ Rswsnr = λmax (Rz + Rn )wsnr
wmvdr = (Rz + Rn )−1a
w lcmv = argminwwHRw s.t. CHw = f→ w lcmv = R
−1C[CHRC]−1f
−100 −80 −60 −40 −20 0 20 40 60 80 100−80
−70
−60
−50
−40
−30
−20
−10
0
10
Bearing in degrees
Res
pons
e in
dB
rela
tive
to p
eak
Conventional, Kaiser windowedMax SNR, no interferer in Rn model
Max SNR, interferer at −10 deg
An Example of MaxSNR canceling
¨ Very high INR case, +70 dB. ¨ SNR = +40 dB
¨ Max SNR output SINR = 50 dB
¨ 10 element ULA
¨ Exact covariances
¨ Output SIR is 139 dB!
Problems with the RA Signal Scenario
¨ SOI and interferer are well bellow the noise floor. ¤ SNR of -30 dB or worse is common. ¤ INR <0 dB can still severely corrupt SOI. ¤ Extremely hard to estimate Rz from R.
¨ Motion limits integration time, increases sample estimation error in Rz.
¨ Weak but troublesome interferers yield shallow nulls. ¨ Canceling distorts beam beampatterns.
¤ Raises confusion limit in sidelobes. ¤ Main beam may not have known shape.
Realistic Example of MaxSNR Canceling
¨ Low INR case, -10 dB. ¨ SNR = -40 dB
¨ Input SINR = -40 dB ¨ Max SNR output SINR = -30 dB
¨ 10 element ULA
¨ Exact covariances
¨ Output SIR = -5 dB
−100 −80 −60 −40 −20 0 20 40 60 80 100−80
−70
−60
−50
−40
−30
−20
−10
0
10
Bearing in degrees
Resp
onse
in d
B re
lativ
e to
pea
k
Conventional, Kaiser windowedMax SNR, no interferer in Rn model
Max SNR, interferer at −20 deg
Pattern Distortion with Moving RFI ¨ While adapting to null a moving interferer, sidelobe structure is
unpredictable. ¨ Becomes severe as null approaches the main lobe. ¨ Even sidelobe “rumble” increases confusion noise, hampers on – off
subtraction.
¨ Uniform line array.
¨ 2 moving interferers, starting at +33 and -35 degrees, then moving to the right.
¨ 0 dB INR
Achieving deeper cancelation nulls
Improved Methods for RA
The Good News
¨ Real-time correlators are already needed for all array types (imaging, aperture, PAF) ¤ Used to calculate visibilities or calibrate beams. ¤ Just need rapid dump times to handle motion ¤ Additional computational for spatial filtering is small
¨ Much progress has been made to address null depth limitations in the RA scenario.
Subspace Projection
¨ Well suited for synthesis arrays, aperture arrays, PAFs, and post correlation processing.
¨ Zero forcing, deeper nulls than with total variance minimization. ¨ Must assume interference is the dominant source. ¨ Use eigenvector decomposition to identify the interference subspace. ¨ Partition eigenspace. Largest eigenvalues(s) correspond to
interference.
¨ Form perpendicular subspace projection matrix:
¨ Compute weights and beamform:
€
ˆ R k[Uint | Usig+noise] = [Uint | Usig+noise]Λ
€
Pk = I−UintUintH
wSSP,k = Pkwnominal, b(i) =wSSP,kH y(i), k = i
N!
"!#
$#
Real-World PAF Subspace Projection
¨ 19 element L band PAF on Green Bank 20 Meter Telescope. ¨ Snapshot radio camera image, 21 by 21, 441 simultaneous beams. ¨ CW interference, hand held antenna and signal generator walking in
front of Jansky Lab.
Cross Elevation (Degrees)
Elev
atio
n (D
egre
es)
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1
-10
0
10
20
30
Cross Elevation (Degrees)
Elev
atio
n (D
egre
es)
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1
0
20
40
60
80
100
Cross Elevation (Degrees)
Elev
atio
n (D
egre
es)
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1
-10
0
10
20
30
W3OH, no RFI RFI corrupted image (moving function generator and antenna on the ground)
Adaptive spatial filtering Subspace projection algorithm
Subspace Bias Correction
¨ Projecting out interference subspace distorts the beampattern. ¨ Leshem and van der Veen proposed a correction for moving
interference over the N sample long term integration (LTI).
¨ Well suited for satellite RFI at large imaging arrays. ¨ Use directly as an imaging visibility matrix.
¨ RFI is removed from each STI, but in no subspace is missing.
Rk = PkRkPk
H
R = unvec B−1 vecRk
k=0
J−1
∑#
$%
&
'(
)*+
,+
-.+
/+
B = P k*⊗
k=0
J−1
∑ Pk, J = N / Nsti12 34
R j
R j
Auxiliary Antenna Methods
¨ Improve interference subspace estimate and increase null depth.
¨ Aux antennas return higher INR signal.
¨ Must track moving interferers.
¨ One antenna per interferer.
Rz
High gain main antenna arrayLow gain auxiliaryantennas
q
pI(p,q)
s(i)
A(p,q)
uwv
r1r2 rM
Interferingsatellite
Deep-spaceobject
y (i)m
y (i)a
Ground-basedtransmitter
z (i)1
z (i)2
m
Aux Antenna Performance Analysis
ï220 ï200 ï180 ï160 ï140 ï120 ï100 ï80 ï60ï60
ï40
ï20
0
20
40
60
Total Interference power in dBm at primary feed
Out
put S
igna
l to
Inte
rfere
nce
Ratio
(SIR
), dB
SPSPACSPMSCNo Processing
¨ Extend array vector to include auxiliaries.
¨ Compute “projection” matrix with SVD on
VLA simulation for two stationary interferers and two small auxiliary dish antennas. Source is 1 Jy OH emission. INR at primary feeds is 146 dB above the plotted dBm interferer power level.
Cross Subspace Projection
y(i) =ym(i)ya (i)
!
"##
$
%&&
R =Rmm Rma
Ram Raa
!
"##
$
%&&
Rma
Rma =UΣVH
Us = [uD+1,,uMm]
PCSP = UsUsH , 0Mm
"# $%
RCSP = PCSP RPCSPH
Bias Corrected Array PSD Estimation
¨ When averaging is performed for a beamformed PSD estimator, or total power observation, beampattern distortion can be completely corrected.
¨ Based on subspace projection beamforming.
Sy,c =αJ(wH ⊗wT )B−1 (Pk ⊗
k=0
J−1
∑ Pk*) (YkGF) (YkGF)
*( )
B = 1J
P k*⊗
k=0
J−1
∑ Pk, J = N / Nsti$% &'
Pk = I−UintUintH , Rk[Uint |Usig+noise ]= [Uint |Usig+noise ]Λ
Where F is the Fourier transform matrix, G is a diagonal window weighting matrix (e.g. Hamming window), α= 1/(diag{G}T diag{G}), and
Yk = y(k Nsti ), y(k Nsti +1),, y([k +1]Nsti −1)[ ]
Bias Corrected Array PSD Results
ï4 ï3 ï2 ï1 0 1 2 3 4ï20
ï15
ï10
ï5
0
5
10
15
Frequency in rad/sample
Pow
er/fr
eque
ncy
(dB
/rad/
sam
ple)
Power Spectral Density using all data via Welch’s method
Subspace ProjConventionalBias corrected subspace
2 interferers, weak SOI, J = 900, 512-point STIs
Bias Corrected Array PSD Results (cont.)
ï100 ï80 ï60 ï40 ï20 0 20 40 60 80 100ï80
ï70
ï60
ï50
ï40
ï30
ï20
ï10
0
10Effective beam response in PSD estimate
Bearing in degrees
Bea
m re
spon
se in
dB
Bias corrected proj.Subspace projectionConventional
¨ Effective beampatterns as seen by the PSD estimators (i.e. these are averaged over all STIs due to the PSD calculation.)
¨ Bias corrected PSD matches the conventional fixed beam response, even while canceling interference.
¨ Time varying adaptive beampattern.
Conventional SSP Limitations
101 102 103 1040
20
40
60
80
IRR
= P
int,i
n / P in
t,BF d
B (s
olid
line
s)
IRR & Residual Interference Power, motion = 2 °/sec
STI length (samples)
101 102 103 104 −160
−130
−100
−70
P int,B
F dBm
(dah
sed
lines
)
INR = 60dBINR = 40dBINR = 20dBINR = 0dBINR = −20dB
¨ Detailed simulation ¨ 19-element PAF on
20m reflector, 0.43f/D ¨ Correlated spillover
noise, mutual coupling, modeled 33K Ciao Wireless LNAs.
¨ Measured array element radiation patterns.
¨ Physical Optics, full 2D integration over reflector.
Subspace estimation error due to sample noise from short STI
Subspace smearing error due to motion, with no sample estimation error.
¨ Fit a series of STI covariances to a polynomial that can be evaluated at arbitrary timescale. ¤ Beamformer weights can be updated every time sample. ¤ Use entire data window to fit polynomial for less sample estimation error.
¨ Minimize the squared error
between STI sample covariances and the polynomial model CLS:
Low Order Parametric Models for SSP
CLS = argminC
Rk − Rint (tk,C)k=1
K
∑F
2
,
where tk=kNstiTs
105−180
−170
−160
−150
−140
−130
STI length (samples)
Res
idua
l Int
erfe
rer P
ower
(dB
m)
Residual Interferer Power, motion = 3°/sec
Conv−SPEIG−7−SPOPT−7−SP
Rk
Rint (t,C) = apoly (t,C)apolyH (t,C)
t=nTs
where C = [c0cr ]apoly (t,C) = c0 + c1t ++ crt
r
Real Data Cancelation Results
102 103 104 105
−160−155−150−145−140−135−130−125−120−115
STI length (samples)
Res
idua
l Int
erfe
rer U
pper
Lim
it
Tx near boresight, Slow MotionINR −12dB, Poly = 8, with Pre−Whitening
MaxSens on−offConv−SP on−offEIG−8−SP on−off
102 103 104 105
−160
−150
−140
−130
−120
STI length (samples)
Res
idua
l Int
erfe
rer U
pper
Lim
it
Tx in sidelobes, Slow MotionINR −12dB, Poly = 8, with Pre−Whitening
MaxSens on−offConv−SP on−offEIG−8−SP on−off
50 100 150 200
−160
−150
−140
−130
−120
−110
PSD bin
PS
D
Tx in sidelobes, Slow MotionINR −12dB, Poly = 8, with Pre−Whitening
MaxSens on−offNoise onlyConv−SP on−offEIG−8−SP on−off
50 100 150 200
−160
−155
−150
−145
−140
−135
−130
−125
−120
−115
PSD bin
PS
D
Tx near boresight, Slow MotionINR −12dB, Poly = 8, with Pre−Whitening
MaxSens on−offNoise onlyConv−SP on−offEIG−8−SP on−off
Real Data Cancelation Results
102 103 104 105
−160−155−150−145−140−135−130−125−120−115
STI length (samples)
Res
idua
l Int
erfe
rer U
pper
Lim
it
Tx near boresight, Slow MotionINR −12dB, Poly = 8, with Pre−Whitening
MaxSens on−offConv−SP on−offEIG−8−SP on−off
102 103 104 105
−160
−150
−140
−130
−120
STI length (samples)
Res
idua
l Int
erfe
rer U
pper
Lim
it
Tx in sidelobes, Slow MotionINR −12dB, Poly = 8, with Pre−Whitening
MaxSens on−offConv−SP on−offEIG−8−SP on−off
50 100 150 200
−160
−150
−140
−130
−120
−110
PSD bin
PS
D
Tx in sidelobes, Slow MotionINR −12dB, Poly = 8, with Pre−Whitening
MaxSens on−offNoise onlyConv−SP on−offEIG−8−SP on−off
50 100 150 200
−160
−155
−150
−145
−140
−135
−130
−125
−120
−115
PSD bin
PS
D
Tx near boresight, Slow MotionINR −12dB, Poly = 8, with Pre−Whitening
MaxSens on−offNoise onlyConv−SP on−offEIG−8−SP on−off
Final Observations
Additional Methods
¨ Robust beamforming ¤ Helpful if array calibration has errors. Keeps SOI from
being canceled. ¨ Spectral scooping correction
¤ Block mode covariance matrix processing for beamformer weight updates.
¤ Narrowband RFI. ¤ Surprising behavior: the time-varying spatial filter
places a spectral null in PSD estimates. ¤ The cause of this bias has been described analytically. ¤ A straightforward correction has been proposed.
Progress Towards Adoption
¨ Parks Telescope adaptive filter for digital TV may be the only current production system.
¨ Despite significant promise and potential, adoption has been slow.
¨ Perhaps astronomers are reluctant to move from the tried and true.
¨ We have not yet identified the critical science case. ¨ Computational and infrastructure costs are
incremental, but non-trivial. ¨ I hope presented progress in overcoming spatial
filtering limitations will spur adoption.
Conclusions
¨ RFI spatial Filtering for radio astronomy offers a new mode of mitigation beyond time blanking, frequency excision, and avoidance.
¨ It is challenging due to low INR and SNR. ¨ Algorithms common to wireless comm., radar, and sonar
do not drive deep enough nulls, and distort beampatterns.
¨ Several algorithm enhancements have been proposed which may solve these limitations.
¨ Widespread adoption of these promising methods may take some time for confidence to build.
¨ The “critical science case” which cannot be pursued without spatial filtering would spur deployment.
Bibliography
¨ A. Leshem, A.-J. van der Veen and A.-J. Boonstra, “Multichannel interference mitigation techniques in radio astronomy,” Astrophysical Journal Supplement, vol. 131, no. 1, 2000.
¨ A. Leshem and A.-J. van der Veen, “Radio-Astronomical Imaging in the Presence of Strong Radio Interference,” IEEE Jour. on Information Theory, vol. 46, no. 5, Aug. 2000.
¨ B.D. Jeffs, L. Li and K.F. Warnick, “Auxiliary antenna assisted interference mitigation for radio astronomy arrays,” IEEE Trans. on SP, vol. 53, No. 2, February, 2005.
¨ B.D. Jeffs and K.F. Warnick, “Bias corrected PSD estimation for an adaptive array with moving interference,” IEEE Trans. on SP, vol. 56, no. 7, July, 2008.
¨ J. Landon, B.D. Jeffs, and K.F. Warnick, “Model-Based Subspace Projection Beamforming for Deep Interference Nulling,” IEEE Trans. on SP, vol. 60, no. 3, March 2012.
¨ B.D. Jeffs and K.F. Warnick, “Spectral bias in adaptive beamforming with narrowband interference,” IEEE Trans. on SP, vol. 57, no. 4, Apr., 2009.
¨ 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.
¨ S. van der Tol and A.-J. van der Veen, “Performance analysis of spatial filtering of rf interference in radio astronomy,” IEEE Trans. on SP, vol. 53, no. 3, Mar. 2005.