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SELF-CALIBRATION OF RADIO ASTRONOMICAL ARRAYS WITH
NON-DIAGONAL NOISE COVARIANCE MATRIX
Stefan J. Wijnholds and Alle-Jan van der Veen
ASTRON
R&D Department
Oude Hoogeveensedijk 4, NL-7991 PD, Dwingeloo,
The Netherlands
phone: +31 521 595 261, email:
[email protected]
web: www.astron.nl
ABSTRACT
The radio astronomy community is currently building a
number of phased array telescopes. The calibration of
these telescopes is hampered by the fact that covariances
of signals from closely spaced antennas are sensitive to
noise coupling and to variations in sky brightness on
large spatial scales. These effects are difficult and computationally expensive to model. We propose to model
them phenomenologically using a non-diagonal noise covariance matrix. The parameters can be estimated using a weighted alternating least squares (WALS) algorithm iterating between the calibration parameters and
the additive nuisance parameters. We demonstrate the
effectiveness of our method using data from the low frequency array (LOFAR) prototype station.
1. INTRODUCTION
The radio astronomical community is currently constructing a number of large scale phased array telescopes such as the low-frequency array (LOFAR) [1] and
Murchison wide field array (MWA) [2]. These instruments need to be calibrated regularly to track variations in the electronics of the antennas and receivers,
as well as direction-dependent variations of the ionosphere [3]. E.g., LOFAR will consist of order 50 stations (distributed over an area of a hundred kilometers or more), where each station consists of 96 “low
band” dual-polarized dipole antennas (10–90 MHz) and
96 “high band” antennas (110–240 MHz). The latter
antennas are in turn composed of 16 beamformed dualpolarized droopy dipole antennas. Each station provides
a number of beamformed outputs, which in turn are correlated at a central location to form images and other
astronomy products.
In this paper we focus on the calibration of the station antennas. The general aim is to estimate the direction independent gains and phases of each sensor, as
well as the direction dependent gains corresponding to
each source. This is done for the 2–10 brightest sources
in the sky, assuming a point source model. The problem is complicated by the fact that the covariances of
signals from closely spaced antennas within a station
are sensitive to noise coupling (for the low frequencies,
This work was supported by the Netherlands Institute for
Radio Astronomy (ASTRON) and by NWO-STW under the VICI
programme (DTC.5893).
Delft University of Technology
Department of Electrical Engineering
Mekelweg 4, NL-2628 CD, Delft, The Netherlands
phone: +31 15 278 6240, email:
[email protected]
web: www.tudelft.nl
the antennas are spaced closer than half a wavelength).
Also, the point source model does not entirely hold because of bright emission from the plane of the galaxy,
extending over the entire sky. Fortunately, this emission
is spatially smooth, which implies that it is dominant on
the short spatial scales in the array aperture, i.e. on the
short baselines. In this paper we propose to model both
short baseline effects by an additive noise covariance matrix, which in this case is not diagonal and has unknown
entries for each short baseline. If we can estimate this
matrix, a simple point source model will be sufficient
to calibrate the array, which reduces the problem to a
problem for which solutions are readily available [4–6].
Direction finding problems for calibrated arrays in
the presence of unknown correlated noise have been
extensively studied in the 1990s. It was proven that
the general problem is not tractable without imposing
some appropriate constraints on the noise covariance
matrix or exploiting differences in temporal characteristics between source and noise signals [7]. Radio astronomical signals generally behave like noise, thus temporal techniques (instrumental variables) are not applicable. Instead, we should rely on an appropriately constrained parameterized model of the noise covariance
matrix. Starting with [8], a series of papers were published; see [9] for an overview. ML estimators for the
source and instrument parameters under a generalized
noise covariance parameterization is provided in [9, 10],
whereas nonlinear least squares estimators were studied in [10–12]. In either case, an analytic source and
instrument parameter dependent solution is derived for
the noise model parameters which is substituted back
into the cost function. This cost function then has to be
minimized using a generalized solving technique, such
as Newton iterations. This approach works well if the
number of instrument and source parameters is small.
For larger problems (we consider 100 antenna/source parameters and over 750 noise covariance parameters), it
is convenient to exploit suboptimal but closed-form analytic solutions, at least for initialization. We therefore
propose a weighted alternating least squares (WALS)
approach which iterates over noise, source and instrument parameters. The proposed method can thus be
regarded as an extension to the methods proposed in [6]
T
Notation: The transpose operator is denoted by
, the complex conjugate (Hermitian) transpose by H ,
complex conjugation by (·) and the pseudo-inverse by
†
c ⊗ denotes the
. An estimated value is denoted by (·).
Kronecker product and ◦ is used to denote the KhatriRao or column-wise Kronecker product of two matrices. vec(·) converts a matrix to a vector by stacking the
columns of the matrix.
2. DATA MODEL AND PROBLEM
STATEMENT
We consider an array of P antennas. The measured
P × P array covariance matrix can be modeled as
R = R0 (θ) + Σn
(1)
where R0 (θ) is the signal model for an ideal noise free
array, which depends on a number of unknown real valued parameters accumulated in a column vector θ, and
Σn is a P × P matrix describing the noise corruption.
Note that Σn must be Hermitian, since the array covariance matrix is a Hermitian matrix. In our application,
the data covariance model is
H H
R0 (θ) = G1 AG2 Σs GH
2 A G1
(2)
where Σs is the covariance of the point sources (assumed
to be known from tables), A contains the direction
vectors, G1 a (diagonal) instrumental gain/phase matrix, and G2 a (diagonal) direction dependent gain matrix [6]. If the direction dependent gains are unknown,
we can introduce Σ = G2 Σs GH
2 , which implies that we
should estimate the apparent source powers. The direction dependent gains follow directly from the apparent
source powers if the actual source powers are known,
e.g. from tables. If the source positions are unknown
or perturbed by propagation conditions, A may be parameterized [6]. The contents of the parameter vector
θ therefore strongly depend on the available knowledge
on the instrument, the propagation conditions and the
sources.
As in [8] and subsequent papers, the unknown noise
covariance matrix is modeled as a linear sum of known
matrices, in this case simple selection matrices Eij which
are zero everywhere except for a ’1’ in entry (i, j),
X
σij Eij .
Σn =
(i,j)∈S
The set S contains the index pairs of the short baselines, including the autocorrelation entries (i, i). The
unknown coefficients σij are the nuisance parameters.
In the absence of the noise corruptions, i.e., R =
R0 (θ), the estimation problem to find the parameter
vector θ is commonly formulated either as a ML problem, or as a generalized least squares estimation problem
w w2
b − R0 (θ) Ww
b = argmin w
(3)
θ
w ,
wW R
F
θ
b is the measured array covariance matrix. It is
where R
known that with W = R−1/2 this estimator is asymptotically unbiased and asymptotically efficient [10, 12].
Indeed, the simulations in [10] show only a small improvement of the ML solution as compared to the WLS
solution. For R0 (θ) given in (2), this problem, as well
as the case with an unknown diagonal noise covariance,
was studied by us in [6], in the present paper, we will
assume that a solution to this problem is available.
In the presence of correlated noise on the short baselines, the problem is extended to
w o
w2
n
w
b σ
b − R0 (θ) − Σn Ww
b n = argmin wW R
θ,
w ,
F
θ ,σ n
(4)
where σ n is a vector containing all unique real valued
parameters required to describe the nonzero entries of
Σn . This vector can be related to Σn using a selection
matrix Is such that vec (Σn ) = Is σ n . By choosing the
selection matrix appropriately, we can ensure that the
estimated Σn is Hermitian.
3. PARAMETER ESTIMATION
3.1 Weighted Alternating Least Squares
We propose to solve the problem in Eq. (4) by alternating between weighted least squares (WLS) estimation
of the desired parameters θ and WLS estimation of the
nuisance parameters σ n . The first WLS problem can be
formulated as
w w2
b = argmin w
b − Σn − R0 (θ) Ww
θ
w . (5)
wW R
F
θ
which is identical in form to Eq. (3) for which a solution
is assumed to be available.
The second WLS problem can be formulated as
w w2
w
b − R0 (θ) − Σn Ww
b n = argmin wW R
σ
w
F
σn
w
w
b − R0 (θ) −
= argminw W ⊗ W vec R
σn
w2
w
W ⊗ W Is σ n w .
(6)
F
The solution to this problem is given by
†
b − R0 (θ) .
b n = W ⊗ W Is
W ⊗ W vec R
σ
(7)
In section 2 it was mentioned that W = R−1/2 provides optimal weighting for the LS cost function. Since
b is used instead in many applications.
R is not known, R
It can be shown that this may lead to a bias in the esb n for a finite number of samples [6]. This
timate of σ
bias can be avoided by using the best available model
b
R (θ, σ n ) instead of of R.
Estimation of receiver noise powers is a special case
of the general problem treated here. In this case Σn is
a diagonal, and Is = I ◦ I where I is the P × P identity
matrix. This form of selection matrix simplifies Eq. (7)
considerably [6]. In some problems, for example in estimating the receiver based gains, it is then possible to
simply ignore the diagonal entries instead of including
nuisance parameters [5]. It can further be shown that
ignoring the corrupted entries instead of including them
using nuisance parameters does not change the CramèrRao bound of the parameters of interest [13]. This can
30
South ← y → North
be explained intuitively by regarding the matrix equation describing the WLS problem as a set of scalar equations. If a unique nuisance parameter is added to one of
those scalar equations, that equation is required to solve
for the nuisance parameter and can thus not be used to
solve any other parameters. This implies that this equation could have been ignored if one would only focus on
the parameters of interest. In practice, however, it may
be hard to develop an algorithm that ignores the corrupted entries in a statistically efficient way.
20
10
0
−10
3.2 Algorithm
−20
The resulting algorithm is as follows:
−30
An algorithm that alternatingly optimizes for distinct groups of parameters, in our case θ and σ n , can
be proven to converge if the value of the cost function
decreases in each iteration. We assume that a suitable
method is available to find θ. Since we propose to estimate σ n using the well known standard solution for
least squares estimation problems, the value of the cost
function will decrease in both steps, thus ensuring convergence. Although there is no guarantee that the algorithm will converge to the global optimum, practical
experience with LOFAR and results from Monte Carlo
simulations in this paper and in earlier papers [5, 6] indicate that the proposed method produces good results
for most reasonable initial estimates provided.
4. EXPERIMENTAL RESULTS
4.1 The LOFAR prototype station
The first full-scale LOFAR prototype station with realtime backend became operational in the second quarter of 2006 [14]. This station consisted of 48 dualpolarization antennas operating between 10 and 90 MHz
arranged in a randomized configuration based on rings
with exponentially increasing radii as shown in Fig. 1.
Each of the two signals from every antenna was filtered
and digitized using a 12-bit 200 MHz ADC. A real-time
FPGA based digital processing backend splits the 100
MHz Nyquist sampled base band in 512 subbands, each
195 kHz wide, using a polyphase filter. The backend
also provides a correlator which can correlate in realtime the data from the 96 input channels for a single
subband. This subband may be any of the 512 available
subbands and this choice may change every second.
−30
−20
−10
0
10
West ← x → East
20
30
Figure 1: Array configuration of the 48 antenna LOFAR
prototype station.
1
1
0.8
0.95
0.6
South ← m → North
1. Initialization Set the iteration counter i = 1 and inib [0]
tialize σ
n based on any prior information if availb [0]
able, otherwise initialize σ
n to zero. Initially use
b −1/2 .
W=R
b[i] by solving the WLS problem formu2. Estimate θ
b [i−1]
lated in Eq. (5) using σ
as prior knowledge.
n
[i]
b[i] as prior knowlb n using Eq. (7) using θ
3. Estimate σ
edge.
4. Update W = R−1/2 to avoid the bias mentioned in
the previous section.
5. Check for convergence, otherwise continue with step
2.
0.9
0.4
0.85
0.2
0
0.8
−0.2
0.75
−0.4
0.7
−0.6
−0.8
−1
−1
0.65
−0.5
0
West ← l → East
0.5
1
Figure 2: Calibrated all-sky map for a single polarization at 50 MHz from the 48-element LOFAR prototype
station. The image shows the sky projected on the horizon plane of the station.
4.2 Results
For the demonstration in this paper we used data from a
single 1 s snapshot observation in the 195 kHz subband
centered at 50 MHz. This observation was done on 14
February 2008 at 1:42:07 UTC. We will calibrate the
data using the method described in Sec. 3.2, where we
will model correlated noise terms on all baselines shorter
than four wavelengths. The complete WLS problem
thus implies estimation of the amplitudes (48 parameters) and phases (47 parameters) of the antenna based
complex gains, the source power ratio of the two brightest sources (1 parameter) and 764 real valued nuisance
parameters describing all non-zero entries of the noise
covariance matrix for a total of 860 free real valued parameters per polarization. In this experiment we show
that use of such nuisance parameters can reduce the
complex source structure on the sky to a simple model
with just two point sources.
1
1
0.8
0.8
0.25
0.6
0.8
0.4
0.2
0.75
0
−0.2
0.7
−0.4
−0.6
South ← m → North
South ← m → North
0.6
0.4
0.2
0.2
0.15
0
−0.2
0.1
−0.4
0.05
−0.6
0.65
−0.8
−1
−1
−0.8
−0.5
0
West ← l → East
0.5
1
Figure 3: Calibrated all-sky map of extended emission
observed at baselines shorter than four wavelengths for
a single polarization at 50 MHz.
Figure 2 shows a calibrated all sky map for a single polarization. There are two bright point sources
near the northeastern horizon. The image also shows
a lot of extended emission from the galactic plane (on
the northwestern horizon) and the north polar spur (on
the eastern horizon). This extended emission is hard
to model accurately, but only affects the short baselines
since short distances in the aperture plane of a phased
array correspond to low spatial frequencies, which describe the structure on large spatial scales.
It was found that most of this extended emission is
captured by the correlations on baselines shorter than
four wavelengths. This affects 358 crosscorrelations and
the 48 autocorrelations resulting in the aforementioned
764 real valued parameters to describe the non-zero entries of Σn . Using the procedure outlined in the preb
b n was estimated simultaneously with θ
vious section, σ
b
containing the other parameters. Σn can therefore be
interpreted as an estimate of the extended source structure, noise coupling and receiver noise powers. This
is nicely demonstrated in Fig. 3 which shows an image
b n after the calibration was completed.
based on Σ
Figure 4 shows the difference between the maps
shown in Figs. 2 and 3. This shows that Σn provides a
description of the extended emission that is sufficiently
b Σ
b n to an array covariance matrix
accurate to reduce R−
that can be described by a model consisting of only two
point sources. This thus reduces our original problem
to one that has been discussed extensively in the array
signal processing literature.
5. IMPROVING THE COMPUTATIONAL
EFFICIENCY
b n using Eq. 7 forms the most expenThe calculation of σ
sive part of the algorithm in terms of CPU and memory
usage due to the Kronecker products. These Kronecker
products can only be reduced to simpler Khatri-Rao or
Hadamard products in a number of special cases, such as
a diagonal noise covariance matrix treated in [6]. How-
0
−1
−1
−0.5
0
West ← l → East
0.5
1
Figure 4: Difference between the calibrated all-sky map
shown in Fig. 2 and the contribution of extended emission shown in Fig. 3 showing that the remainder can be
accurately modeled using a model with just two point
sources.
#
1
2
3
4
5
l
0.24651
-0.34346
-0.13125
-0.29941
0.39290
m
-0.71637
0.76883
-0.31463
-0.52339
0.58902
σq2
1.00000
0.88051
0.79079
0.74654
0.69781
Table 1: Source powers and source locations used in the
simulations
ever, the parameterization of Σn chosen here implies
that each entry of the array covariance matrix with a
contribution from Σn is affected by a unique additive
parameter. Intuitively, one would therefore expect that
the weighting in Eq. (7) would not make much difference. Omitting this would reduce the CPU and memory
requirements considerably, since the Kronecker products
and theinverse increase the numerical complexity from
o N P 2 to o N 3 + P 6 and the size of the largest matrix from P 2 × N to P 2 × P 2 , where N is the number
of noise parameters stacked in σ n .
This idea was therefore tested in Monte Carlo simulations. For these simulations, a five armed array was
defined, each arm being an eight-element, one wavelength spaced ULA. The first element of each arm
formed an equally spaced circular array with half wavelength spacing between the elements. The source model
is presented in Table 1. This source model was generated with a random number generator to verify that the
proposed approach works for arbitrary source models.
Figure 5 compares the variance on the estimates for
the omnidirectional complex gains of the receiving elements and the source powers obtained after 100 runs of a
Monte Carlo simulation with weighted LS estimation of
b n and the computationally more efficient unweighted
σ
b n . The complex receiving element
LS estimation of σ
gains and the source powers, stacked in θ, were esti-
−4
10
x 10
WLS Σ estimate
n
[2]
LS Σ estimate
9
n
8
variance
7
[3]
6
5
[4]
4
3
[5]
2
1
0
20
40
60
parameter index
80
100
Figure 5: Comparison of the variance on the direction independent complex gain and the source power
estimates obtained in Monte Carlo simulations with
b n and unweighted LS esweighted LS estimation of σ
b n.
timation of σ
mated using weighted least squares in both cases. Both
algorithms generally converged within three iterations
to relative error per parameter of ∼ 10−4 , i.e. well below
the Cramèr-Rao bound. The convergence rate in these
simulations was one digit per iteration down to the numerical accuracy provided by double precision floating
point numbers.
The results indicate that the variance on these estimates is the same in both cases within the accuracy
provided by the simulations. We therefore conclude
that it is viable to discard the weighting in Eq. (7).
With this modification all 860 free parameters in the
experiment described in the previous section could be
extracted from the actual data using Matlab running
on a standard dual core 2.4 GHz CPU in only 0.4 seconds. This implies that a single 2.4 GHz core can keep
up with the data from the correlator at the LOFAR
station, which real-time correlates the antenna signals
for a single subband with one second integration time.
This update rate is required to track variations in the
electronic gains and the ionosphere.
[6]
[7]
[8]
[9]
[10]
[11]
[12]
6. CONCLUSIONS
We have demonstrated using data from a LOFAR prototype station that the effects of noise coupling, receiver
noise powers and extended emission on a radio astronomical phased array can be phenomenologically described by a non-diagonal noise covariance matrix with
non-zero entries on short baselines. These entries can be
computationally efficient and accurately estimated by a
WALS algorithm alternating between estimation of the
correlated noise parameters and calibration parameters.
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