spatial filtering of interfering signals at the initial ... · spatial filtering results (2)...

IUCAF RFI2004 workshop, 16-18 July 2004- 1 -A.J.Boonstra, S.van der Tol

Spatial filtering of interfering signals at the initial LOFAR phased array test station

Albert-Jan Boonstra ASTRON

Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands

Sebastiaan van der TolDelft University of Technology

Department of Electrical EngineeringMekelweg 4, 2628 CD Delft, The Netherlands


Contents

LOFAR RFI strategy

LOFAR initial test station (ITS)

Data model

Spatial filtering approach and results

Imaging of intermodulation products

Imaging and beamforming

Conclusions


LOFAR – RFI strategy (1)

Part ofthe radio spectrum in

East-DrentheThe Netherlands



LOFAR RFI strategy

• Select cleanest (order 100 kHz) subbands• Reduce RFI levels by RFI mitigation down to Cas.A level• Reduce the RFI further to levels close to or below noise in sky maps by

o Selfcal / peelingo Spectral dilution (cont. observations)o Spatial dilution (RFI ~ 1/N, noise ~1/sqrt(N))o RFI mitigation

Issues (a.o.): stationarity, stability


LOFAR Initial test station (1)

LOFAR ITS overview

• 60 sky noise limited inverse shaped V-dipoles (EW)

• Five arm spiral configuration• 10 – 40 MHz bandpass filter• Digitization: 12 bit• High speed optical connection to input

module, 2 GB menory• 16 Data acq. PC’s, 1 central processing

PC, 1Gb data network• Observation modes e.g.:

o Storage of 6.7 s data blockso Semi online correlation, beamforming

LOFAR ITS



Sky noise limited observations• Monitoring results• Antenna measurements with

matched load• Noise analysis, cf poser

S.Wijnholds SKA2004 conf.

Crosstalk• Crosstalk between cables etc.• Mutual coupling for close antenna spacings, |b10 – b20|= λ/2 at 30 MHz,

difficult to separate from the sky (large scale structures) • Method of moment simulations show low crosstalk levels:

At 40 MHz: -20 dB, at 30 MHz: -43 dB, at 20 MHz: -65 dB

ConslusionITS is sky noise dominated and has low crosstalk levels (< -30 dB)


Data model (1)

Received data model:

Astr. sources and one interferer: x(t) = v(t) + a(t)e(t) +n(t)

Astr. sources and multiple interferers: x(t) = v(t) + A(t)e(t) +n(t)v(t): astronomical sources vector, n(t): system noise vector e(t): interferere(t): interferer vector (q interferers)A(t) = [a1(t), ... aq(t)], direction signatures

Covariance model:Sample estimate:

Model, Rk=E{xk(t)xk(t)H}: Rk = Rv + Rn +AkBkAkH

Rn much smaller than Rv

Rv: astronomical visibilities, Rn: (diagonal) noise matrixBk: diagonal matrix containing RFI source fluxesAk: matrix containing spatial signature vectors


Spatial filtering approach (1)

Spatial filtering using projections

Data model with interference:

Projection matrix:

Applying projection:

Spatial filtering using subtraction



Distortion correctionBoth filter options: bias correction needed.For projections, use correction matrix C,(using vec(ABC) = (Ct⊗A)vec(B))

Recall:

then

where

and where

var(R0) = (1/N) C–1σ04 11t



Residual interference after spatial filteringDetector and filter combinedEffectiveness determined by estimaton accuracy of a (spatial signature vector)



Approaches how to find spatial sigature vector a:

Eigenvalue decompositon after whitening with (diag(R))-0.5

obtained in nearby frequency bins

Factor analysis followed by eigenvalue decomposition

Where: U,Us, Un: noise (sub) spacesΛ0 Λs: diagonal eigenvalue matrices, D: (diagonal) noise matrix


Spatial filtering results (1)

Multiple transmitters – eigenvalues


Spatial filtering results (2)

transmitter at horizon (26.75 MHz)

projection filtering (26.75 MHz)subtraction filtering (26.75 MHz)

no interference (26.89 MHz)

WSRT LOFAR - ITS


Imaging of intermodulation products (1)

Single antenna / telescopeConsider a second order model of (non)linear devic:e (e.g. LNA) :

Input x(t) consists of two cosines with aplitudes α1 and α2 :

Output y(t) with (non)linearity parameters β1 and β2 :



Antenna / telescope arrayInput of two cosines for array vector x(t) and amplitude vectors α1 and α2 :

Where:

Consider f12 = f1 +f2 in more detail:

Then:



Antenna / telescope array

Suppose there is a real source at f12 = f1+f2 , andsuppose a source direction s12:

Then this source will have the following phase:

The phase relation for the intermod was:

Conclusion: the intermod product will appear as a pointsource in the map.


Imaging and beamforming (1)

BeamformingArray weight vector w:Output signal y(t):

Output power P:

Classical (Capon) beamformer:

Multiple constraints beamformer:

k is a vector with constraints k=(1 0 0 0)t for 1 beam and 4 nulls

⇒So there is a certain equivalence in pre correlation filtering and postvorrrelation filtering



Beamforming

MVDR beamformer:

constraint:

Solution using Lagrange multipliers

better spatial resolution than classical beamformingbut senstive to calibration errors (gain “scaling” errors)use Robust Capon Beamforming



Measurement equation

Matrix formalism

Classical inverse fourier imaging

dirty image:

convolution:

point source model:

classical imaging after projections: use space-varying beam

Imaging via beamforming techniquesCLEAN is a generalized classical sequential beamformer:

MVDR

Robust MVDR versions exist (Stoica et al)



Beamforming: classical vs MVDR



Beamforming: robust capon beamfoming in comparison with classical beamforming and MVDR

intensity scalingdiffersMVDR: narrower beam, but scaling lost

due to calibration errors

Robustcapon beamforming: scaling recovered,implications fur use in imaging/calibration to be studied


Conclusions

• The LOFAR RFI strategy is based (a.o.) on selection of cleanest spectral (subband) regions and on suppression of RFI down to Cas.A. levels

• RFI at the Cas.A level at LOFAR stations will be reduced to levels below the system (sky) noise by spatial dilution. Assumption is that interferers are point sources.

• Interference popping up a 30 dB level above Cas.A could be reduced to levels below Cas.A using spatial filtering.

• Under certain stationarity/stability conditions, intermodulation products of RFI pointsources remain pointsources and can be filtered in the same way as direct interfering sources. They also will be spatially diluted.

• ITS is sky noise limited with low crosstalk levels.

• The observed spectrum occupancy, and the successful interferencemitigation tests support the expectation that LOFAR can be succesfully built and used in the Netherlands.

spatial filtering of interfering signals at the initial ... · spatial filtering results (2)...

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