C h i n a - U S W o r k s h o p , 1 3 - 1 6 O c t o b e r , 2 0 1 5 , S h a n g h a i , C h i n a

1

Calibration, Beamforming, and RFI Mitigation with Radio

Astronomical Phased Array Feeds

Brian D. Jeffs, K.F. Warnick, R. Black: - Brigham Young University

D. Campbell, G. Cortes, S. Parshley: - Cornell University

A. Chippendale, A. Hotan, G. Hellbourg: - CSIRO

Abstract 2

This presentation will discuss the current state of the art for calibrating the array response, computing beamformer weights, and canceling RFI sources using radio telescopes equipped with phased array feed (PAFs). The promise of PAFs for providing wide fields of view, fast survey speeds, multiple simultaneous beams with dense coverage, active beamshape control, high sensitivity, and the ability to mitigate RFI with new spatial filtering approaches, all require significant array signal processing capabilities. We will review these algorithms and present current “best practices” and examples of realistic performance capabilities.

PAF Telescope Pros and Cons 3

y(i)

s(i)z(i)

Radio telescope dish with a phased array feed

¨  Wide field of view. ¨  Fast surveys. ¨  Full FOV coverage, more

dense than multibeam. ¨  Explicit beampattern

control. ¨  Active RFI spatial

nulling. ¨  High computational

burden and system complexity

¨  Regular beamformer calibration required.

¨  Difficult to cool.

3

Some PAF Instruments and Projects 4

�  ASKAP – CSIRO, MRO, chequerboard, 188 el., 36 beams, interferometer, room temp., 380 MHz.

�  APERTIF – ASTRON, WSRT, Vivaldi, 112 el., ~25 beams, interferometer, room temp.

�  AO40 – Cornell & BYU, Arecibo, dipole, 160 el., 40 beams, single dish, full cryo, 300 MHz.

�  FLAG – NRAO & BYU, GBT, dipole, 38 el., 7-13 beams, cryo LNA, 150 MHz.

�  PHAMAS – UMASS, GBT, mm wave, horn grid, 64 el., 20 beams, full cryo.

�  AFAD – DRAO, Vivaldi, 41 el. room temp., very low noise.

�  Effelsberg – Max Planck Inst., ASKAP array on single 100m dish.

�  SKA – Future 3-band PAF interferometer.

Early BYU PAF Experiments (2006)

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

5

Early PAF RFI Experiments (2006)

Moving FM sweep RFI, 10 second integration

NAGEL, ET AL.: ARRAY FEED EXPERIMENTS 5

1610.9 1611 1611.1 1611.2 1611.3 1611.420

25

30

35

40

45

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55

60

Frequency (MHz)

PSD

(Arb

dB

/Hz) SIGNAL

Figure 6. PSD of the max-SINR beamformer output using in-terferer subspace partitioning for a moving interferer.

4.2.2. Performance Versus Interferer Power

0 5 10 15 20 25 30 35 40−5

0

5

10

15

20

25

30

35

40

Center Element INR (dB)

IRR

(dB

)

SINR−ISPSINR−TRNLCMVGAIN

Figure 7. Interference rejection ratio of the beamformers as afunction of interference to noise ratio at the center element. Thestraight line corresponds to an output interferer power spectraldensity equal to the center element noise floor.

The performance of a given beamformer dependsstrongly on the relative power levels of the signal andthe interferer. To quantify this effect, a series of datasets were captured with varying interferer power levels.A useful metric for beamformer performance is the in-terference rejection ratio (IRR), the interference to noiseratio (INR) at the center element divided by the INR ofthe beamformer output, so that

IRR =INRel1

INRx(7)

where the INR is defined for convenience in terms ofnoise and modulated interferer power spectral densities.Figure 7 summarizes the performance of several beam-former techniques as a function of interferer power level.The first curve (SINR-ISP) represents max-SINR us-

ing interferer subspace partitioning. The second curve(SINR-TRN) was obtained with a fixed max-SINR beam-former with interferer spatial statistics calculated oncefrom training data. The third beamformer is LCMV. Forreference, the fourth curve (GAIN) is a fixed maximum-gain beamformer calculated from training data whichdoes not suppress the interference.

4.3. Correlation Time and Nonstationarity

10−5 10−4 10−3 10−2 10−1 10012

14

16

18

20

22

24

26

28

Correlation Window (s)

IRR

(dB

)

SINR−ISPLCMV

Figure 8. Interference rejection ratio as a function of STI win-dow length for a stationary interferer.

There is a general trade-off with adaptive beamformingbetween STI window length and nonstationarity. Spatialfiltering relies on an accurate estimate of the covariance ofthe array response to the interferer. Assuming stationarysignal and noise statistics, covariance estimates improvewith longer STI lengths. If an interferer moves signifi-cantly over the STI window, however, smearing of the in-terferer spatial response leads a poor covariance estimate.For short STI lengths, the interferer response estimationerror is dominated by noise, and for long STI lengths, esti-mation error is dominated by nonstationarity. The formereffect can be seen in Fig. 8, which shows the IRR of twobeamformers as a function of correlation time for a sta-tionary interferer. For integration times longer than 5 ms,the IRR levels off and shows no improvement for longeraveraging windows. This may be due to mechanical vi-brations or other sources of nonstationarity that limit theinterferer null depth even with long correlation times.

4 NAGEL, ET AL.: ARRAY FEED EXPERIMENTS

dBm FM transmission centered at 1611.3 MHz, with 30kHz deviation and 1.0 kHz modulation rate.

Since the signal processing is narrowband, RFI miti-gation performance is essentially independent of the tem-poral signal characteristics (although the impact of resid-ual RFI on SOI detection certainly depends on the RFIspectral characteristics). The modulation was chosen forconvenience in displaying results.

Figure 3 shows the power spectral density (PSD) of thesignal at the center array element. This represents the con-trol signal that would be seen by a standard, single-feedreceiver in a radio telescope. The SOI is obscured by boththe variance of the noise floor and the interfering signal.Figure 4 shows the resulting PSD after 10 seconds of in-tegration. The noise floor variance decreases, but the FMinterferer remains and the SOI is not observable. Figure5 shows the beamformer output with the max-SINR-ISPalgorithm with an STI length of 4.9 ms, or 6125 real sam-ples. As can be seen, the FM interferer is suppressed andthe SOI is recovered. A small amount of residual RFI isvisible.

1610.9 1611 1611.1 1611.2 1611.3 1611.420

25

30

35

40

45

50

55

60

Frequency (MHz)

PSD

(Arb

dB

/Hz)

Figure 3. Short-time PSD as seen by the center element. A CWSOI is obscured both by variance of the noise floor and by anFM-modulated interferer.

1610.9 1611 1611.1 1611.2 1611.3 1611.420

25

30

35

40

45

50

55

60

Frequency (MHz)

PSD

(Arb

dB

/Hz)

Figure 4. PSD of the center element signal after 10 seconds ofintegration. The noise floor variance is reduced by integration,but the SOI is still obscured by interference.

1610.9 1611 1611.1 1611.2 1611.3 1611.420

25

30

35

40

45

50

55

60

Frequency (MHz)

PSD

(Arb

dB

/Hz) SIGNAL

Figure 5. PSD of the max-SINR beamformer output using in-terferer subspace partitioning for a stationary interferer. A smallamount of residual interference is visible after 10 seconds of in-tegration.

4.2.1. Nonstationary Interferer

To simulate a moving interferer, the RFI source wasmanually moved at a walking pace. As seen from thereceiver, the angular velocity was on the order of 0.1±/s,which is typical for a satellite in medium Earth orbit. Dur-ing this trial, the SOI was a CW transmission at°90 dBmwith a °10 dBm FM interferer overlapping in frequency.Figure 6 shows the beamformer output with max-SINR-ISP. The signal power is different from that of Fig. 5 be-cause the SOI source power was varied between data setsin order to test beamformer algorithms in a variety of SNRregimes.

Subspace Projection and

max SNR beamforming

6

2008 Radio Camera Image 7

Cross Elevation (Degrees)

Elev

atio

n (D

egre

es)

-1 0 1-1.5

-1

-0.5

0

0.5

1

1.5

0

5

10

15Source: 3C295 Flux density: 21 Jy at 1400 MHz Observation freq. 1612 MHz Integration time: 60 sec

Green Bank 20 Meter Telescope, 19 element single pol array

2008: Active RFI Mitigation 8

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

Green Bank 20 Meter Telescope 19 element single pol array

Green Bank Telescope: FLAG 9

�  NRAO/BYU 19 element dual pol cryo cooled PAF �  Real-time correlator / beamformer, 150 MHz BW

Image credit, NRAO

FLAG Archtecture for GBT 10

Cryo

stat

XB Engine: Correlator/Beamformer, Spectrometer

Array aperture, Antenna elements, LNAs, Cryo system, Down converters

•••

Ch. 1

ROACH II

FPGA

12 TB SATA RAID 0 Disk Array

Rack Mount PC

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System control and data storage (existing)

(× 5)

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Ch. 8

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CPU/GPU (Blade server +

2× nvidia GTX780)

LNA

Ant.

Back End (Jansky Lab) Front End (GBT)

LNA

Ant. Signal Transport: Optical fiber

•••

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(× 5)

CPU/GPU (Blade server +

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Ch. 33

4 X 10 GbE

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F Engine: DDL deserialization, boun-dary alignment, polyphase filter bank and 10 Gbe I/O

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NRAO DDL System BYU Correlator Beamformer

In Mezzanine I/F Slot

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I-Q mix, ADC, Serialize & Optical Xmit

I-Q mix, ADC, Serialize & Optical Xmit

LO

LO

Arecibo Telescope: AO19 11

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

ASKAP 12

�  94 element dual pol room temperature array �  3-axis mount Image credit, CSIRO

D = 12m f/D = 0.5

E S T I M A T I N G A R R A Y R E S P O N S E V E C T O R S I N D I R E C T I O N S O F I N T E R E S T

13

Fundamentals of PAF Beamformer Calibration

¨  Signal Model:

The Narrowband Beamformer

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

+

w*1

w*3

w*M

b(i) = w y(i) H

θ

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

y(i) = a s(i)+ vdd=1

D

∑ (i)zd (i)+n(i)

14

� � �

Beamformer weight vector w = [w1,,wM ]

T

b(i) =wHy(i)

¨  Signal Model:

The Narrowband Beamformer

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

+

w*1

w*3

w*M

b(i) = w y(i) H

θ

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

y(i) = a s(i)+ vdd=1

D

∑ (i)zd (i)+n(i)

15

� � �

Beamformer weight vector w = [w1,,wM ]

T

b(i) =wHy(i)

UNKNOWN!

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.

R = E{y(i)yH (i)} =Rs +Rn +Rv

Rk =1N

y(i)yH (i)i=kN

(k+1)N−1

∑

Rk ≈ RS(θk )

16

ak ≈ a(θk )

θk

Rk

PAF Beamforming Calibration Procedure 17

�  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).

Rn

Calibration grid

calibration source

ak

Rk

PAF Beamforming Calibration Procedure 18

�  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:

Rk

Rn

ak = Rnuk uk

Rkuk = λmaxRnuk

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]

wsnr = argmaxwwHRswwHRnw

→ Rswsnr = λmaxRnwsnr

Rs,k =σ s2akak

H → wmvdr,k = Rn−1ak

w lcmv = argminwwHRw s.t. CHw = f→ w lcmv = R

−1C[CHRC]−1f

19

Calibrator Requirements 20

�  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.

�  Calibration longevity ¡  Must repeat every few weeks for L band. ¡  Much more frequent for W band.

Parkes ASKAP Testbed Beamformer Calibration 21

�  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 22

�  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

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

−6

−4

−2

0

2

4

6

x 10−3

−6

−4

−2

0

2

4

6

x 10−30

0.5

1

1.5

2

2.5

Beampatterns for Sun Calibration 24

12  m,  f/D  =  0.5  (~ASKAP)  

43  m,  f/D  =  0.43  (~Green  Bank  140’)  

25

RFI Mitigation with a PAF Beamformer

26

Jacinta Delhaize: HI Spectral Stacking - RFI Challenges 26

�  RFI increased significantly at Parkes between 2008 -12.

�  RFI prevents interesting science at z > 0.1

�  Observed at Parkes and ATCA. �  With new nav. satellites, a large

contiguous band will be gone.

Auxiliary Antenna Methods

�  Improve interference subspace estimate and increase null depth.

�  Aux antennas return higher INR signal than 0 dBi dish sidelobe.

�  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

27

Reference Antenna at ASKAP Test Platform 28

Reference antenna tracks Galileo satellite, and signal is correlated with PAF array covariance matrix.

29

Take-away Points: 29

�  RFI is getting worse. �  Navigation satellites, COMPASS, Galileo,

GLONASS, and GPS are chewing up a large contiguous band below the HI line.

�  Some moderately to highly red shifted HI science is threatened or impossible.

�  Flagging is essential, but can be inadequate; too much data loss.

�  We are approaching some “critical cases” where important science will need spatial array processing to have any hope of success.

An Example of MaxSNR Canceling

�  Very high INR case, +70 dB. �  SNR = +40 dB

¨  Max SNR output SINR = 50 dB

30

¨  10 element ULA ¨  Exact

covariances ¨ 

−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!

31

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 Rv from R.

�  Motion limits integration time, increases sample estimation error in Rv.

�  Weak but troublesome interferers yield shallow nulls.

�  Canceling distorts beam beampatterns. ¡  Raises confusion limit in sidelobes. ¡  Main beam may not have known shape.

32

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 ¨ 

33

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

34

Cross Subspace Projection

�  Well suited for synthesis arrays, aperture arrays, PAFs, and post correlation processing.

�  Zero forcing, deeper nulls than with maxSNR.. �  Extend array vector to include auxiliaries.

�  Compute “projection” matrix with SVD on

�  Compute projection matirx and weights for kth STI, and beamform:

35

y(i) =ym(i)ya (i)

!

"##

$

%&&, R =

Rmm Rma

Ram Raa

!

"##

$

%&&

Rma

Rma =UΣVH , Us = [uD+1,!,uMm

], PCSP = UsUsH , 0Mm

"# $%

wCSP,k = PCSP,kwnominal, b(i) =wCSP,kH y(i), k = i

N!

"!#

$#

CSP Simulated Performance for ASKAP 36

−50 0 50−80

−60

−40

−20

0

20

40

60Comparison of Interference Mitigation Techniques − INR

INR at Feed (dB)

INR

at C

entra

l Cor

rela

tor O

utpu

t (dB

)

No MitigationSP−XSOP−XCSP−XCSOP−XX−SPX−CSPCSP−CSPCSOP−CSP

37

Back End Architectures

FLAG Archtecture for GBT (2) 38

Ethe

rnet

Sw

itch

Mel

anox

SX

1012

Chan

nel S

elec

tion

(k =

... )

Nvidia GTX780 Ti GPU, 1 of 2 per HPC

40 Gbe

Packets from 5 ROACH IIs: 50 out of 500 channels for all 40 input ports

Fine PFB 32 point FFT, 256 tap Filter

Lust

re D

isk

Arra

y St

orag

e

Rack-mount HPC, 1 of 5 (Data transfer & real time thread management from NIC to GPU via HASHPIPE)

All 50 Coarse Channels, 15 MHz total BW @303k samp/s

5 Selected Coarse Channels

160 Fine Channels, 1.51 MHz total BW @9.5k samp/s

To 4 other HPCs

xk[n]

Channel Selection (k = ... ) Coarse: For N = 30: pick 5 chan. for N ≥ 303: pick 50 chan. Fine: Pick all 160 fine channels

Coarse/ Fine

Correlator / Integrator (XGPU code) Coarse: N = 30 for 0.1 ms

N = 303 for 1.0 ms Fine: N = 4,734 for 500 ms

Rk =1N

xk[n] xkH [n]

n=0

N−1

∑ Rk

FITS

For

mat

ter

Real-time Beamformer

bk, j[n]=

wk, jH x k[n]

Integrator N = 30 for 0.1 ms

Sk,( j, j )c =

1N

bk, jbk, j*

n=0

N−1

∑

Sk,( j, j )c

Rk

AO40 Back End Architecture 39

•••

Xilinx Vertex

6 FPGA

(×20)

8 in

put,

800

Msp

s 2

x AD

C16-

250-

8

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5 x

10 G

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FP+

Xilinx Vertex

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8 in

put,

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•••

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From

ana

log

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nver

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To mass storage

Mercury GPU208 2U server HPC (#1)

ROACH II (#1)

ROACH II (#20)

Mercury GPU208 2U server HPC (#25)

Eth

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(1)

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x SX

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(5)

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•••

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GTX 980 TI GPU

GTX 980 TI GPU

GTX 980 TI GPU

GTX 980 TI GPU

1.1-

1.1

2.1-

1.1

3.1-

1.1

4.1-

1.1

20.1

-1.5

17.1

-1.5

18

.1-1

.5

19.1

-1.5

1.4-

4.1 1.3-

3.1

1.5-

5.1

2.5-

5.1

3.5-

5.1

4.5-

5.1

1.2-

2.1

20.5

-5.5

19

.5-5

.5

18.5

-5.5

17

.5-5

.5

20.4

-4.5

20

.3-3

.5

20.2

-2.5

(5-8).1 -1.2

(9-12).1 -1.3

(13-16).1 -1.4

(5-8).5- 5.2

(9-12).5 -5.2

(13-16).5 -5.4

1.6 -1.(1-4) 1.7-

2.(1-4) 1.8-

3.(1-4)

1.10- 5.(1-4)

1.9- 4.(1-4)

5.10- 25.(1-4)

5.9- 24.(1-4)

5.8- 23.(1-4)

5.6- 21.(1-4)

5.7- 22.(1-4)

1.1

1.2 1.3 1.4

25.1

25.2 25.3 25.4

To mass storage

1.11 1.12

5.12 5.11

AO40 GPU Processing 40

Fine PFB Polyphase filter bank (32 point FFT)

xk[n]

Correlator / Integrator Rk =

1N

xk[n] xkH [n]

n=0

N−1

∑Rk

Real-time Beamformer bk, j[n]=

wk, jH x k[n]

Fine Spectrometer Long-term Integrator

Sk,( j, j )f xk[n]

From

net

wor

k sw

itch:

16

coa

rse

chan

nels

from

eac

h of

the

160

ante

nnas

To m

ass s

tora

ge

Coarse Spectrometer Fast-dump Integrator

Sk,( j, j )c =

1N

bk, j[n]bk, j* [n]

n=0

N−1

∑

Sk,( j, j )c

bk, j[n]

!S !k,( j, j )f =

1N

!b!k, j[n] !b!k, j* [n]

n=0

N−1

∑

!b!k, j[n]

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.

�  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.

�  Richard Black, Digital Back End Development and Interference Mitigation Methods for Radio Telescopes with Phased-Array Feeds, masters thesis, Brigham Young University, Aug. 2014.

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