Examples of cyclostationary approaches for phased array radio
telescopes
Examples of cyclostationary approaches for phased array radio
telescopes R. Weber (Station de radioastronomie de Nançay / Université d’Orléans)
R. Feliachi (Université d’Orléans)
A.J. Boonstra (Astron)
1 Project PrepSKA (contract no 212243), ANR (contract ANR-09-BLAN-0225-04)
OutlineOutline
Examples of cyclostationary approaches for phased array radio telescopes, R.Weber et al., RFI 2010, Groningen, NL2
• The contextThe context• The cyclostationarity concept• First example : Cyclic detectors• Second example : Estimation and subtraction• Third example : cyclic spatial filtering• Conclusions and perspectives
Classic approachClassic approach
Examples of cyclostationary approaches for phased array radio telescopes, R.Weber et al., RFI 2010, Groningen, NL3
Hypothesis : noise
r nH
rrz
RR
IARAR 2
rfi
M-K1 last eigenvectors
dominant eigenvectors
NU
Signal subspace depending on RFI
Noise subspace
rS AU
EVD
zR
Spatial filtering
RFI detection
(see papers from A. Leshem, A.J. Boonstra, A.J van der Veen)
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x
y
initial map INR=10 BPSK
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Illustration:
Initial sky map filtered sky map
IR 2cosmicrfi , nnoisePP
rA
M a
nten
nas
K1 RFI
ka
General case issueGeneral case issue
Examples of cyclostationary approaches for phased array radio telescopes, R.Weber et al., RFI 2010, Groningen, NL4
IRRARAARARRR
2
noisecosmicrfi
nRFInHsss
Hrrr zR
zR
EVD
dominant eigenvectors
NUM-K1-K2 last eigenvectors
Signal subspace depending on cosmic
sources +RFI (+ noise)
Noise subspace
sr AAU ,S
Illustration with :
cosmicrfi PP
Initial sky map filtered sky map
IR 2nnoise
OutlineOutline
Examples of cyclostationary approaches for phased array radio telescopes, R.Weber et al., RFI 2010, Groningen, NL5
• The context• The cyclostationarity conceptThe cyclostationarity concept• First example : Cyclic detectors• Second example : Estimation and subtraction• Third example : cyclic spatial filtering• Conclusions and perspectives
Cyclostationarity conceptCyclostationarity concept
Examples of cyclostationary approaches for phased array radio telescopes, R.Weber et al., RFI 2010, Groningen, NL6
Stationary process Cyclostationary process
statistics are periodic statistics time-independent
)(),( RtR ),(),( tRTtR
T = hidden periodicity s(t) or
n(t) r(t)
Example : second order statistics
( )tFT ( )tFT
T
1
T
2
T
300 )(R
Cyclic correlation
Cyclic signature
Cyclicfrequency
Cyclicfrequency
)(R )(R
In practice :
)2exp()2/()2/()( * tjtztzRz
Gardner and Giannakis
The multidimensional caseThe multidimensional case
Examples of cyclostationary approaches for phased array radio telescopes, R.Weber et al., RFI 2010, Groningen, NL7
)()()(
)2exp()2
()2
(
nHsss
Hrrr
H tjttz
RARAARA
R
zz
)(RFIR 0 0
z
R
SVD
dominant eigenvectors
NUM-K1 last
eigenvectors
Signal subspace depending on RFI
Noise subspace
rS AU
Spatial filtering
RFI detection Estimation
& subtraction
Cyclic correlation matrix
OutlineOutline
Examples of cyclostationary approaches for phased array radio telescopes, R.Weber et al., RFI 2010, Groningen, NL8
• The context• The cyclostationarity concept• First example : Cyclic detectorsFirst example : Cyclic detectors• Second example : Estimation and subtraction• Third example : cyclic spatial filtering• Conclusions and perspectives
Cyclic detectorsCyclic detectors
Examples of cyclostationary approaches for phased array radio telescopes, R.Weber et al., RFI 2010, Groningen, NL9
t
)1( LR
)(lR
)0(R
0R
Ll
R
LL 1
R
FFTFFT
)( Rfrob
T
1 T
2 T
3
0
Several implementations:- Blind detector
-
-
2/ )(k
Tkfrob R
)max( k
Example at Westerbork telescope with GPS data
Cyclic detector performanceCyclic detector performance
Examples of cyclostationary approaches for phased array radio telescopes, R.Weber et al., RFI 2010, Groningen, NL10
AM RFI, no cosmic sourcesPower fluctuations between antennas = 20%
OutlineOutline
Examples of cyclostationary approaches for phased array radio telescopes, R.Weber et al., RFI 2010, Groningen, NL11
• The context• The cyclostationarity concept• First example : Cyclic detectors• Second example : Estimation and subtractionSecond example : Estimation and subtraction• Third example : cyclic spatial filtering• Conclusions and perspectives
Estimation & subtraction (1)Estimation & subtraction (1)
Examples of cyclostationary approaches for phased array radio telescopes, R.Weber et al., RFI 2010, Groningen, NL12
noisecosmic
2
RRaaR hrH
rrz RT
02
2 2..0
jR
T hrT
rrf
z aaR
Consider 1 BPSK RFI : k
jtfjk tkTthctr 002
0 )()(
2) SVD
1)
3)
OK for AM, BPSK + QAM if h(t) known
rfiR̂4)
rfiR̂RR zcleanz5)
Estimation & subtraction (2)Estimation & subtraction (2)
Examples of cyclostationary approaches for phased array radio telescopes, R.Weber et al., RFI 2010, Groningen, NL13
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x coordinate of the spatial signature
y c
oord
inat
e of
the
spa
tial s
igna
ture
initial map INR=0 dB, 1 BPSK
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x coordinate of the spatial signature
y co
ordi
nate
of
the
spat
ial s
igna
ture
cleaned map INR=0 dB, 1 BPSK
0
2
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-20
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50
Pow
er
Spectr
um
Magnitude (
dB
)
Frequency (normalized)
Real filtered spectrum + source
Estimated spectrum
Example of an AM RFI in the LOFAR band
Simulation with a BPSK RFI
AM Fake RFI
OutlineOutline
Examples of cyclostationary approaches for phased array radio telescopes, R.Weber et al., RFI 2010, Groningen, NL14
• The context• The cyclostationarity concept• First example : Cyclic detectors• Second example : Estimation and subtraction• Third example : cyclic spatial filteringThird example : cyclic spatial filtering• Conclusions and perspectives
Cyclic Spatial Filtering (1)Cyclic Spatial Filtering (1)
Examples of cyclostationary approaches for phased array radio telescopes, R.Weber et al., RFI 2010, Groningen, NL15
Hrr
Hrr AAAAIP 1)( Spatial filtering:
PRPR clean
1) projector
cleaning:2)
1) SVD :
2) Extract the RFI subspace :
3) Since ,
Projector
Estimation of : rAHVUΛR
rr span AU Hrr
Hrr UUUUIP 1)(
1 … Kr 4 5 p
eigenvalues
rU
rU noiseUU
eigenvectors
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initial map INR=0 BPSK
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cleaned map classic INR=0 BPSK
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cleaned map cyclic INR=0 BPSK
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Initial sky map filtered sky maps
classic cyclic
Simulationswith 3 BPSK
Cyclic Spatial Filtering (2)Cyclic Spatial Filtering (2)
Examples of cyclostationary approaches for phased array radio telescopes, R.Weber et al., RFI 2010, Groningen, NL16
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Frequency (MHz)
Pow
er S
pect
rum
Mag
nitu
de (
dB)
Spectrum of the data before and after Spatial Filtering
cyclic SF
classic SF
initial
16
RFI : pager
R
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65
Frequency (MHz)
Pow
er
Spectr
um
Magnitude (
dB
)
Spectrum of the data before and after Spatial Filtering
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Frequency
Pow
er S
pect
rum
Mag
nitu
de (
dB)
cyclic autocorreltion x*conj(x)
Example with real data from LOFAR
Conclusions & PerspectivesConclusions & Perspectives
Examples of cyclostationary approaches for phased array radio telescopes, R.Weber et al., RFI 2010, Groningen, NL17
- Robustness of the cyclic approach- Just replace R by R in the classical algorithms- For some modulations , it is possible to estimate Rrfi
performance analysis, impact on sky map exhaustive tests on real data (LOFAR,…) broadband issue. “virtual reference antenna” approach
Automatic cyclic beamformer
array
Clean signal+
-