Examples of cyclostationary approaches for phased array radio telescopes

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


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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-5

0

5

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15

Illustration:

Initial sky map filtered sky map

IR 2cosmicrfi , nnoisePP

rA

M a

nten

nas

K1 RFI

ka

General case issueGeneral case issue


IRRARAARARRR

2

noisecosmicrfi

nRFInHsss

Hrrr zR

zR

EVD


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


• 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


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


)()()(

)2exp()2

()2

(

nHsss

Hrrr

H tjttz

RARAARA

R

zz

)(RFIR 0 0

z

R

SVD


NUM-K1 last

eigenvectors

Signal subspace depending on RFI

Noise subspace

rS AU

Spatial filtering

RFI detection Estimation

& subtraction

Cyclic correlation matrix

OutlineOutline


• 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


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


AM RFI, no cosmic sourcesPower fluctuations between antennas = 20%

OutlineOutline


• 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)


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)


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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

2

4

6

8

10

12

14

16

18

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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

4

6

8

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16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-20

-10

0

10

20

30

40

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


• 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)


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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1

x

y

initial map INR=0 BPSK

2

4

6

8

10

12

14

16

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x

y

cleaned map classic INR=0 BPSK

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-5

0

5

10

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cleaned map cyclic INR=0 BPSK

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-5

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5

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15

Initial sky map filtered sky maps

classic cyclic

Simulationswith 3 BPSK

Cyclic Spatial Filtering (2)Cyclic Spatial Filtering (2)


160 170 180 190 200 210 220 230 240-10

0

10

20

30

40

50

60

70

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

168 168.5 169 169.5 170 170.5 171 171.5

20

25

30

35

40

45

50

55

60

65

Frequency (MHz)

Pow

er

Spectr

um

Magnitude (

dB

)

Spectrum of the data before and after Spatial Filtering

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.570

80

90

100

110

120

130

Frequency

Pow

er S

pect

rum

Mag

nitu

de (

dB)

cyclic autocorreltion x*conj(x)

Example with real data from LOFAR

Conclusions & PerspectivesConclusions & Perspectives


- 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+

-

Documents

Examples of cyclostationary approaches for phased array radio telescopes