SPATIAL FILTERING OF NEAR-FIELD RADIO FREQUENCY

SPATIAL FILTERING OF NEAR-FIELD RADIO FREQUENCY
SPATIAL FILTERING OF NEAR-FIELD RADIO FREQUENCY INTERFERENCE AT A
LOFAR LBA STATION
Stefan J. Wijnholds
Jan-Willem W. Steeb, David B. Davidson
∗
Department of Electrical and Electronic Engineering
Stellenbosch University
Stellenbosch, South Africa
[email protected]
ABSTRACT
In preparation for the SKA, many new RFI (radio frequency
interference) mitigation algorithms have been developed.
However, these algorithms usually assume that the RFI source
is in the far-field and that the array is calibrated. In this paper, the recovery of astronomical signals from uncalibrated
RFI-corrupted LOFAR visibility data using spatial filtering
methods are presented. For this demonstration, a near-field
continuous-wave RFI source was generated by a hexacopter
that was flown around one of the LOFAR LBA (low-band
antenna) arrays. Four spatial filtering methods were applied
to the RFI contaminated data: orthogonal projection, orthogonal projection with subspace bias correction, oblique
projection and subspace subtraction. Overall, orthogonal
projection with subspace bias correction performed the best,
however it requires that the RFI source moves relative to the
array and it is computationally expensive. Oblique projection performs similar to orthogonal projection with subspace
bias correction when point sources are to be recovered and
is furthermore considerably less computationally expensive.
Subspace subtraction is a suitable alternative if a large field
of view is to be recovered at a relatively low computational
cost.
R&D Department
ASTRON
Dwingeloo, The Netherlands
[email protected]
and Dept E&E Eng,
Stellenbosch University
1. INTRODUCTION
LOFAR is part of a new generation of radio telescope arrays
with large bandwidths, high sensitivity and resolution. To obtain high resolutions long baselines are required and therefore most RFI sources will be in the near-field. Consequently,
powerful near-field RFI presents a serious challenge. In this
paper the application of spatial RFI mitigation techniques to
uncalibrated data which has been corrupted with a near-field
source is presented. The experimental setup is explained, followed by a description of the mathematical model and applied
RFI mitigation techniques. Finally, the experimental results
are given.
2. EXPERIMENTAL SETUP
For this demonstration, a near-field continuous-wave RFI
source was generated by a hexacopter that was flown around
the LOFAR LBA (low-band antenna) array CS302. A signifcant feature of this test is that the hexacopter’s flight path
was within the array’s near-field. The Rayleigh distance for a
LOFAR station is approximately 1900 m for a given longest
baseline of approximately 85 m and a wavelength of 6.74 m
(the sub-band with centre frequency 44.5095 MHz was used).
Index Terms— RFI mitigation, LOFAR, near-field RFI,
spatial filtering.
3. MODEL
∗ This work is supported by SKA South Africa, the South African Research Chairs Initiative of the Department of Science and Technology, the
National Research Foundation, and a Marie Curie International Research
Staff Exchange Scheme Fellowship within the 7th European Community
Framework Programme MIDPREP, Grant Agreement PIRSES-GA-2013612599.
The authors would like to thank Millad Sardarabadi for the fruitful discussions and useful feedback.
The following general model (used in [1, 2, 3, 4, 5]) is considered for the output generated at time t by an antenna array that
consists of Ne elements, for one polarization and frequency
channel:
y(t) = g (xc (t) + xr (t)) + xn (t)
(1)
where
y(t)
g
xc (t)
is an Ne × 1 vector of measured array output signals,
is the vector of complex gains for each antenna,
Hadamard product,
is the vector where each element is the sum of Nc
delayed cosmic signals for a given antenna,
xr (t) is the vector where each element is the sum of Nr
delayed RFI signals for a given antenna,
xn (t) is the vector of instrumental noise for each antenna.
The gains g are unknown, since the array is assumed to be
uncalibrated.
The frequency channel is assumed to be sufficiently narrowband, so that the time delays τ can be represented as phase
delays. Therefore, a delayed signal can be approximated by
s(t − τ ) ≈ s(t)e−i2πν0 τ , where ν0 is the centre frequency
of the channel. This condition is satisfied for the array, if
2π∆ντmax 1, where ∆ν is the signal’s bandwidth and
τmax is the delay given by the longest baseline (greatest distance between any two antennas) [6].
The phase delays for the k th RFI source can be stacked
into a vector that is called the geometric delay vector
 −i2πν0 τ1r 
k
e


.
..
ark = 
(2)
.
e
−i2πν0 τN
e rk
and the same applies to Br and Rn if the RFI and noise signals
are, respectively, uncorrelated. Since the signals are spatially
and temporally stationary, the covariance matrix is estimated
by
Nt
X
b= 1
y(iTs )yH (iTs ),
R
Nt i=1
(6)
where
b
R
is the estimated covariance matrix,
Nt is the number of samples for which the signals
are stationary,
Ts is the sample time.
The covariance matrix has the following useful properties: R
is Hermitian and is positive semi-definite [8, p. 558].
4. SPATIAL RFI MITIGATION
4.1. Orthogonal Projection
If the columns of Ar are linearly independent, they form a basis for a vector space Vr . Therefore, an orthogonal projector
can be constructed [8, p. 430] which projects along Vr onto a
vector space orthogonal to Vr , namely
−1 H
P = I − Ar (AH
Ar ,
r Ar )
(7)
An Ne × Nr matrix can now be constructed from the geometric delay vectors, Ar = [ar1 . . . arNr ]T (the same applies
for the cosmic signals, Ac = [ac1 . . . acNc ]T ), and therefore
the model in equation 1 can be written as
such that PAr = 0. The projector is Hermitian and therefore
P = PH [8, p. 433]. Applying the projector to equation 5
yields (assuming G = I)
y(t) = g (Ac sc (t) + Ar sr (t)) + xn (t),
PRP =PRc P + PAr Br AH
r P + PRn P
(3)
where sc (t) and sr (t) are respectively, the vectors of the cosmic and RFI signals without delays.
The zero lag covariance matrix (the ij th element of the
matrix is the covariance of the ith and j th antenna [7, p. 501])
of the vectorised data model in equation 1 is given by
R = E{y(t)yH (t)},
(4)
where E is the expectation, H is the Hermitian transpose or
complex conjugate transpose and it is assumed that for a given
time period none of the signals change position. Therefore,
the covariance is constant over this time period as long as the
signals are themselves stationary.
Independence between the cosmic, RFI and noise sources
is assumed, therefore, when substituting equation 3 into equation 4 the expectation of any non-self multiplication terms is
zero and consequently the substitution yields
R = G(Rc + Rr )GH + Rn
H
H
= G(Ac Bc AH
c + Ar Br Ar )G + Rn ,
(5)
H
where Bc = E{sc (t)sH
c (t)}, Br = E{sr (t)sr (t)}, Rn =
H
E{xn (t)xn (t)} and G is the diagonal matrix of g. The matrices Bc will be diagonal if the cosmic signals are uncorrelated
=PRc P + PRn P
=PRcn P.
(8)
The RFI contribution to the covariance is completely nulled;
however, the noise and cosmic signals are biased.
4.2. Orthogonal Projection with Subspace Bias Correction
For any useful orthogonal projector P the kernel basis includes the zero vector and at least one non-zero vector, therefore, P has a column rank less than the number of columns in
P which consequently makes the matrix singular. The orthogonal projection method bias (see equation 8) can therefore not
be corrected by multiplying with the inverted orthogonal projector.
For the orthogonal projection correction scheme ([2, 3, 4])
the number of samples Nt is divided into NG equally sized
groups, where each group consists of Nst samples (st denotes
short term), that is, Nt = NG Nst . For a sampling time Ts
the overall integration time is Nt Ts , while Nst Ts is the short
term integration time for any of the NG groups. The following
assumptions must also hold:
• The cosmic signals are stationary for Nt Ts seconds.
4.3. Oblique Projection
• The RFI signals are stationary for Nst Ts seconds.
The oblique projection method projects along the RFI vector space Vr onto the cosmic vector space Vc . To construct
this oblique projector it is required that the column vectors in
[Ac Ar ] are independent (Vr ∩ Vc = {0}). The oblique projector is given by [1, p. 51]
• The RFI signals are not stationary for Nt Ts seconds.
The k th short term covariance matrix estimate is given by
bk = 1
R
Nst
kN
st
X
y(nTs )y(nTs )H
(9)
n=(k−1)Nst +1
where k ∈ {1, · · · , NG }. The covariance matrix estimate can
then be written as
NG
X
bk.
b= 1
R
R
NG
(10)
k=1
For each short term integration covariance matrix estimate
b k , an orthogonal RFI projector Pk , can be constructed, since
R
the RFI is assumed stationary over the short term integration
time (Nst Ts ). The averaged orthogonal projected covariance
matrix estimate is then
k=1
NG
1 X
bc + R
b k,r + R
b n )Pk
Pk (R
NG
k=1
where P⊥
Vr is an orthogonal projector which projects along
Vr onto a vector space that is orthogonal to Vr . When an
oblique projector is applied, the RFI is nulled and the cosmic
signal is recovered, however the noise is biased
Robl = EVr →Vc REVHr →Vc
= Rc + EVr →Vc Rn EVHr →Vc .
k=1
(11)
If the power and the geometric delay vectors of the RFI
sources are known, then the effect of the RFI sources can be
subtracted [5, p. 115]
k=1
Rcn = R −
Applying the matrix identity
vec(ABC) ≡ (CT ⊗ A)vec(B),
(16)
4.4. Subspace Subtraction
NG
1 X
bc + R
b n )Pk
=
Pk (R
NG
NG
1 X
b cn )Pk .
=
Pk (R
NG
(15)
For the case when Vr and Vc are orthogonal, the oblique
and orthogonal projectors are equivalent. The basis for Vc
can be constructed from either a skymap or choosing an area
of interest that does not contain the RFI.
NG
X
b orth = 1
b k Pk
R
Pk R
NG
=
⊥
−1 H ⊥
EVr →Vc = Ac (AH
Ac PVr ,
c PVr Ac )
Nr
X
σi2 ai aH
i .
(17)
i=1
(12)
where vec(·) indicates the stacking of column vectors of a
matrix and ⊗ the Kronecker product, to equation 11 yields
The power of the RFI source and a basis for the geometric
delay vectors can be estimated by using factor analysis, see
section 4.5.
N
G
X
b orth ) = 1
b cn )
vec(R
(PTk ⊗ Pk )vec(R
NG
k=1
(
)
NG
1 X
T
b cn )
=
(Pk ⊗ Pk ) vec(R
NG
4.5. RFI Subspace Estimation
k=1
b cn ).
= Cvec(R
(13)
The RFI is however assumed to be non-stationary over the
total integration time Nt Ts , therefore, Pk will vary between
the short term integration groups. The matrix C becomes nonsingular if NG is large enough and the orthogonal projectors
vary sufficiently. The corrected covariance matrix is then
b cn = unvec(C−1 vec(R
b orth )),
R
(14)
where the unvec(·) operator is the inverse of the vec(·) operator in equation 12.
Any basis of the RFI subspace Vr can be used to construct
the aforementioned projectors, not just Ar . The ability of the
projection and subtraction methods to null the contribution of
RFI is dependent on the accuracy of the estimate of a basis set
that spans the vector space Vr . When the direction of arrival
of the RFI is not known, Ar cannot be calculated. However,
an orthogonal set of eigenvectors can be found by applying
eigenvalue decomposition (EVD) to the covariance matrix,
because the covariance matrix is positive semi-definite [8, p.
517]. If it can be assumed that the cosmic signal contribution
can be ignored, that the noise is independently and identically
distributed and that the RFI signals are uncorrelated, then the
EVD of the covariance matrix yields [1, p. 64-65]
5. EXPERIMENTAL RESULTS
R ≈Rr + Dn
Dr
= Mr Kr
0
= Mr
= Mr
0 MH
r
+ σn2 I
0 KH
r
H
Mr
Kr Drn
KH
r
H
Dr + σn2 INr
0
Mr
Kr
, (18)
0
σn2 INe −Nr KH
r
where
Mr
is the eigenvectors that form the range of Vr ,
Kr
is the eigenvectors that form the kernel of Vr ,
Dr
is the matrix of eigenvalues (λr,j ) for Rr ,
Drn is the matrix of eigenvalues (λj ) for R.
The column vectors of Mr are orthogonal as well as those
of Kr . For the case where there are two or more RFI signals it
is unlikely that Mr will be equal to Ar . Therefore, the vector
space Vr is spanned differently and the eigenvalues λr,j will
2
not be equal to the RFI powers σr,j
(however the total power
will be the same). The noise only affects the eigenvalues of
Rr but not its eigenvectors, because the noise is identically
distributed [1, p. 65].
The orthogonal projector in equation 7 can now be constructed using Mr , which is identified by the larger eigenvalues in Drn . One simple method [9] to identify the RFI is
to count the eigenvalues which exceed three median absolute
deviations from the median
To show the effect of the spatial RFI mitigation methods, full
sky dirty images were created by classical delay beamforming [1, p. 36] on each pixel. In figure 1 the RFI source is
clearly seen at the top and its intensity is chosen as the 0 dB
point. Data was also saved when the hexacopter was switched
off and a ground truth image was created, see figure 2. In the
ground truth image Cassiopeia A (the brightest source) and
Cygnus A are clearly seen. When the orthogonal projector is
applied it is seen in figure 3 that the strong cosmic sources are
recovered, however, there is a null in the position where the
RFI source was. Orthogonal projection with subspace bias
correction recovers the information that was lost due to the
orthogonal projection, as seen in figure 4. For the oblique
projector a skymap was chosen that consists of Cassiopeia A
and Cygnus A, see figure 5. The oblique projector recovers
what was specified in the skymap and nulls everything else.
Subspace subtraction seems to perform similarly to orthogonal projection with bias correction, however the reliability
of the information recovered in the position of the RFI can
be questioned, since this method effectively replaces the null
with noise, see figure 6.
λj > 3 · median(|Drn − median(Drn )|) + median(Drn ),
(19)
where λj is the j th eigenvalue contained in Drn . Using the
median lessens the influence of outliers, that is, the values
affected by the RFI. Alternative methods are given in [10, 11,
12].
If the noise is not identically distributed, then adding the
2
2
noise covariance matrix Rn = diag(σn,1
, . . . , σn,n
) to the
e
1
RFI covariance matrix Rr causes the eigenvectors of the sum
to change [1, p. 64-65].
When the instrumental noise is not calibrated for an interferometer, factor analysis [13] can be used. Factor analysis is
a statistical method that decomposes a p × p covariance matrix, that is, R = ZZH + D, where Z is a p × q matrix and
D is a p × p diagonal matrix. Applying this decomposition
to an interferometer’s covariance (the influence of the cosmic
source is considered negligible) yields
R =ZZH + D
=Rr + Rn .
(20)
This method places a restriction on the number of factors (that
√
is interferers), namely q < (p − p) [4, 13].
1 However, if the power of the RFI is much larger than the noise power
2 σ 2 ), the effect of noise on the RFI covariance matrix’s eigenval(σr,i
n,i
ues will diminish.
Fig. 1: Full skymap with RFI source visible at the top right
in dB. All other sources are drowned in the sidelobe response
of the RFI source. The power of the cosmic sources are at
least 39 dB below that of the RFI source. The scale is set to
saturate at -15 dB so that the RFI source is clearly visible.
As a figure of merit the Mean Absolute Percentage Error
was chosen
MAPE = 100
Ne X
Ne b
X
b clean,i,j |
|Rproj,i,j − R
b clean,i,j |
|R
i
(21)
j
b proj,i,j is the ij th element of the covariance matriwhere R
b clean is the
ces recovered with a spatial filtering method and R
th
ij element of the covariance matrix estimated from the data
where the hexacopter is switched off. The results are given in
Fig. 2: Full skymap without RFI source in dB.
Fig. 5: Full skymap with RFI source removed using Oblique
Projection in dB. The scale is set to saturate at -43 dB so that
the recovered sources are clearly visible.
Fig. 3: Full skymap with RFI source removed using orthogonal projection in dB.
Fig. 6: Full skymap with RFI source removed using subspace
subtraction.
Fig. 4: Full skymap with RFI source removed using orthogonal projection with bias correction in dB.
a bar graph in figure 7. To make the comparison more meaningful the MAPE is also calculated between two different time
step covariance matrices for when the hexacopter is switched
off (this is labelled as clean). Any mitigation method that has
a MAPE close to the clean MAPE is considered to have recovered the ground truth successfully. The orthogonal projector with subspace bias correction performs the best, however
it is computationally the most expensive. The subspace subtraction method also performs well in recovering the ground
truth. The oblique projector performs the poorest, since it was
implemented only to recover the two bright cosmic sources.
To measure the ability of the mitigation methods to recover a source’s power, the power of Cassiopia A in the RFI
mitigated images is compared to that of the RFI free sky
image, see figure 8. The percentage error in power is also
calculated between two different time step images for when
the hexacopter is switched off (this is labelled as clean). The
oblique projector and orthogonal projection with subspace
bias correction methods performs the best. However, all of
the methods produced results with recovered power within
3.5% of the estimate of the source’s power.
7. REFERENCES
[1] G. Hellbourg, Radio Frequency Interference Spatial
Processing for Modern Radio Telescopes, Ph.D. thesis,
University of Orleans, 2014.
[2] J. Raza, A. Boonstra, and A. van der Veen, “Spatial
filtering of RF interference in radio astronomy,” IEEE
SIGNAL PROCESSING LETTERS, vol. 9, no. 2, pp. 64–
67, February 2002.
Fig. 7: MAPE of Spatial RFI Mitigation Techniques Covariance Matrices relative to RFI Free Covariance Matrix.
[3] S. van der Tol and A. van der Veen, “Performance analysis of spatial filtering of RF interference in radio astronomy,” IEEE TRANSACTIONS ON SIGNAL PROCESSING, vol. 53, no. 3, pp. 896–910, February 2005.
[4] A. van der Veen, A. Leshem, and A. Boonstra, “Signal
processing for radio astronomical arrays,” IEEE Sensor
Array and Multichannel Signal Processing Workshop,
pp. 1–10, July 2004.
[5] A. J. Boonstra, Radio Frequency Interference Mitigation in Radio Astronomy, Ph.D. thesis, Delft University
of Technology, 2005.
[6] M. Zatman, “How narrow is narrowband?,” IEEE
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[7] S. Kay, Fundamentals of Statistical Signal Processing,
Volume I: Estimation Theory, Prentice-Hall, 1993.
Fig. 8: Percentage error of power for Cassiopeia A.
[8] C. D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM: Society for Industrial and Applied Mathematics, 2001.
6. CONCLUSION
[9] J. Wagner, “C++ beamformer library with rfi mitigation,” Tech. Rep., Max Planck Institute for Radio Astronomy, 2011.
The hexacopter signal was 40 dB above the cosmic signals
and saturated the entire skymap. All of the projection methods that were implemented are able to remove the hexacopter
signal and approximately recover the ground truth. If it is
assumed that factor analysis is used to determine the RFI subspace then subspace subtraction has the lowest computational
cost (since no projector needs to be constructed) followed by
orthogonal projection. The oblique projector which includes
first calculating the orthogonal projector and then the oblique
projector has an increased computational cost. The orthogonal projector with subspace bias correction has the highest
computational cost because the correction matrix C must be
calculated and inverted. Orthogonal projection with bias correction performs the best in recovering the entire image (this
is especially useful when the RFI source is in the desired field
of view). The oblique projector performs well when a region
is to be recovered where the RFI source is not located.
[10] H. Akaike, “Information theory and an extension of the
maximum likelihood principle,” Proc. 2nd Int. Symp.
Inf. Theory, pp. 267–281, 1973.
[11] M. Wax and I. Ziskind, “Detection of the number of
coherent signals by the mdl principle,” IEEE Trans.
Acoust., Speech, Signal Process., vol. 37, no. 7, pp.
1190–1196, Aug. 1989.
[12] M. S. Bartlett, “Tests of significance in factor analysis,”
British J. Psych., vol. 3, pp. 77–85, 1950.
[13] A. M. Sardarabadi, Covariance Matching Techniques
for Radio Astronomy Calibration and Imaging, Ph.D.
thesis, Delft University of Technology, 2016.
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