Radio Interferometer Array Point Spread Functions I

Radio Interferometer Array Point Spread Functions I
ALMA MEMO 389
Radio Interferometer Array Point Spread Functions
I. Theory and Statistics
David Woody1
Abstract— This paper relates the optical definition of the PSF
to radio interferometer arrays. The statistical properties of the
PSF including the effect of missing UV data are derived as a
function of the number of antennas and array magnification,
defined as the ratio of the primary beam width from an
individual element to the synthesized beam width. The effect of
earth rotation synthesis on the PSF is also calculated and the
merits of various configuration strategies are discussed in terms
of their PSFs.
The concept of a pseudo-random array is introduced as an
array whose large-scale average distribution matches an
idealized continuous antenna distribution. The small-scale
difference between the actual discrete distribution and the
idealized continuous distribution produces far sidelobes in the
PSF. It is shown that the statistical distribution of the sidelobes,
s, of pseudo-random arrays of N antennas with sparse UV
coverage is given by P(s) = Nexp(− Ns) . The average sidelobe
is 1/N and the standard deviation is also 1/N. Note that the
single antenna measurements are included in the formulation of
the PSF used in this work. The expected peak sidelobe for a
pseudo-random array with a magnification mag is
s max ≈ 2 ln(mag) / N and it is predicted that optimization can
reduce the peak sidelobe to s max,opt ≈ (2 ln(mag) - ln(N))/ N .
Pseudo-random arrays provide a benchmark against which
proposed configurations can be compared.
I.
INTRODUCTION
The point spread function, PSF, is very useful and
convenient for evaluating the performance of an imaging
system [1]. The PSF is the response of an imaging system to
a point source and the “raw” image produced by the system is
the true image convolved with the PSF. Thus the PSF is a
good measure of the errors and artifacts that will appear in
the raw image.
The PSF functions for optical instruments are usually of
sufficient quality that the raw images can be published with
little or no image processing. Radio interferometers do not
directly measure the image, but must reconstruct the image
from limited visibility measurements. The response to a
point source or PSF can still be calculated, but unfortunately
most existing interferometers have relatively poor PSFs and
useful images are produced only after applying non-linear
deconvolution imaging techniques such as CLEAN and
August, 2001
Owens Valley Radio Observatory, California Institute of Technology,
Big Pine, CA 93513, USA
1
MEM [2]. The ability of the imaging algorithms to produce
faithful representations of the true sky brightness distribution
in the presence of noise is limited by both the near and far
sidelobes of the PSF. The quality of the PSF in typical radio
interferometers is indicated by the fact that it is often called
the “dirty beam”. Respectable PSFs can be produced by
interferometers with a large number of antennas, such as
ALMA which will have ~64 antennas.
Most of the imaging quality and UV distribution
parameters can be identified with features in the PSF. The
dynamic range between a strong unresolved source and
distant noise in the “raw” or “dirty” image is determined by
the far sidelobes of the PSF. The near sidelobes determine
the fidelity for imaging extended objects.
Although
algorithms such as CLEAN or MEM can dramatically
decrease the effect of the sidelobes, their ability of
accomplish this in the presence of noise and imperfect
knowledge of the phase and amplitude of the aperture
illumination is limited. The PSF measures the magnitude of
the defects that must be corrected by any imaging algorithm
and arrays with smaller PSF sidelobes should produce more
accurate images.
Many millimeter and sub-millimeter
observations will be noise limited and an array with a PSF
that has sidelobes less than the signal to noise on the
strongest source will be able to use the raw image directly.
The task of determining the best array configurations for
ALMA involves many conflicting requirements ranging from
imaging performance to geographical limits on where
antenna can be economically placed. Although complex
imaging simulations can be used to evaluate many aspects of
the imaging performance, it is necessary to have a quick and
easy method for determining the effect of perturbing the
configurations. A simple evaluation metric is very useful
during both the initial design phase when many different
configurations need to be characterized and during the
detailed design phase when practical considerations may
require moving some of the pad locations. The PSF serves
this purpose very well.
The general discussion of the merits of ring type arrays
with their more uniform UV coverage versus uniform or
centrally condensed arrays with more Gaussian UV coverage
can be carried out quantitatively in terms of the sidelobe
distribution. This paper gives a quantitative basis for
Holdaway’s discussion of the tradeoffs ring-like and filled
array configurations [3].
2
WOODY
The next section develops the basic foundation for
calculating the PSF for a radio interferometer array while
adhering closely to the optical terminology.
The
formulations presented in this section are well known, but it
is useful to explicitly restate them here for use in later
sections. Section III evaluates the near sidelobes produced by
various large-scale antenna distributions.
Section IV
introduces the pseudo-random array whose large-scale
distribution matches an idealized continuous distribution but
has small-scale deviations. The effect of these small-scale
deviations is measured by the difference between the actual
PSF and the PSF of the idealized distribution. The statistics
of this difference PSF are derived in this section, including
the expected magnitude of the largest sidelobes. Various
other considerations, such as earth rotation synthesis and the
implications of obtaining nearly complete UV coverage, are
discussed in section V. The implications of these results and
conclusions are discussed in section VI.
A second paper titled “Radio Interferometer Array Point
Spread Functions II. Evaluation and Optimization" presents
the PSF for several sample configurations and optimized
versions of these configurations. The paper also shows
different methods for presenting the PSF that make it easy to
discern the differences between various configurations and
demonstrates the validity of the statistical distributions
derived in this paper.
II.
INTERFEROMETER POINT SPREAD FUNCTION
The definition of the PSF for interferometers should be
consistent with the PSF defined for optical telescopes to
allow similar interpretation of the resulting images.
A. Optical PSF
The PSF is widely used for characterizing the performance
of optical telescopes.
The PSF given for radio
interferometers should be consistent with this usage. The
PSF for optical telescopes is the image intensity distribution
of the focal plane image of a plane wave incident on the
aperture. This intensity is the square of the field magnitude
in the focal plane which is in turn the Fourier transform of the
complex field (amplitude and phase) across the aperture.
An array of N small apertures produces a focal plane
voltage field that can be written as the sum of the focal plane
fields from the individual apertures,
!
1
U PSF ( p) =
N
N
∑U
!
n ( p ) exp
(− i k p! ⋅ r!n ).
(1)
n =1
!
!
p is the position vector on the sky and U n ( p ) is the voltage
!
beam pattern for aperture k. rn is the location vector for the
center of the nth aperture. The PSF for an array of N identical
apertures is then given by
!
! 2
PSF ( p ) = U PSF ( p )
=
1
N2
N
N
∑∑ B( p) exp(−i k p ⋅ (r
!
! !
m
! .
− rn ) )
(2)
m =1 n =1
!
! 2
B( p) = U ( p) is the primary beam power pattern for a
single aperture in the array.
The Wiener-Khintchine relations can be used to write the
PSF as the Fourier transform of the autocorrelation of the
total aperture field pattern. This autocorrelation is called the
Optical Transfer Function, OTF, in optical systems [4] or the
UV coverage for radio interferometers. The OTF for an array
is given by
!
1
OTF (u ) = 2
N
N
N
!
∑∑ A(u − b
!
m,n )
,
(3)
m =1 n =1
!
!
where u is the vector position in the UV plane and bm,n is
the baseline vector connecting the mth and nth elements.
!
A(u ) is the autocorrelation of the field pattern, or OTF, for a
single element in the array. The double summation over both
indices explicitly includes the single aperture measurements
and ensures that the PSF is positive everywhere.
B. Radio interferometer formulation
Radio interferometers measure the visibility or complex
product of the voltages received by pairs of antennas and do
not directly produce images in a focal plane. The measured
visibility is the Fourier transform of the sky brightness times
!
B( p ) [5]. The same PSF formulation given in equ. 1 can be
arrived at for a radio interferometer by calculating the cross
coupling or orthogonality of the measurements of two point
!
sources. The measurement vector, M(p) , is the set of
measured visibilities plus the single antenna power
measurements.
The lth component corresponding to
!
!
!
measuring the visibility on baseline bl = (rm − rn ) of a point
source at
!
p is
!
!
! !
M ( p ) l = B( p) exp(−i k p ⋅ bl ) .
(4)
A snapshot with an N element array will produce N2
components when the Hermitian conjugates and single
element measurements are included. The cross coupling
!
between measurements of a point source at p and a
!
neighboring point at p + ∆p is given by the dot product of
the measurement vectors
2
! !
! ! * 1 N
! !
!
!!
M ( p + ∆p)•M ( p ) =
B ( p + ∆p) B( p) exp −i k ∆p ⋅bl .
2
N l =1
∑
[
]
(5)
!
A cross coupling of zero means that two point sources as p
!
!
and p + ∆p can be uniquely measured with no confusion,
while a large cross coupling means that the sources are
3
ALMA MEMO 389
31 AUGUST, 2001
difficult to distinguished. Equ. 5 reduces to equ. 2 when
!
p = 0 , i.e. a point source at field center.
Equ. 5 can be used to define the PSF for a point source at a
!
point p away from the field center as a function of the offset
!
distance ∆p to a neighboring point,
! !
! !
! !
!
PSF ( p, ∆p) = M ( p + ∆p ) • M ( p ) .
(6)
!
There will be a family of PSFs, a different PSF for each p .
The maximum sidelobe as a function of distance from the
point source is of interest in evaluating an array’s imaging
performance and is given by the upper envelope of the family
of PSFs. A single plot can show this maximum for all point
source positions by using a modified primary beam pattern
!
!
!
! !
B ′(∆p ) = max[B( p + ∆p ) B( p ), p ] .
0.5
0
0
0.2
0.4
(7)
The max function returns the maximum of the product as a
!
function of p . The modified beam pattern is a wider version
of the single antenna primary beam. A Gaussian primary
beam remains Gaussian but becomes
case PSF’ is given by
1
0.6
0.8
1
radius
Fig. 1. Radial slices through five different cylindrically symmetric largescale antenna distributions.
2 wider. The worst
1
2
N
! 1
!
! !
PSF ′(∆p )=
B ′(∆p ) exp( −i k ∆p ⋅ bl ) .
2
N l =1
∑
(8)
!
The corresponding OTF for a single element, A′(b ) , is the
!
Fourier transform of B ′( p) .
Data weighting or additional measurements can be
incorporated into this formulation of the PSF. Added data
from other configurations or arrays just increases the number
of vector components. Component coefficients can be used
in equ. 8 to account for different noise levels in each visibility
or to improve the sidelobes. This approach can also be used
to evaluate an array’s ability to distinguish or identify sources
that are not point like by appropriate formulation of the
measurement vectors.
0.5
0
0
0.2
0.4
0.6
0.8
baseline
1
1.2
1.4
Fig. 2. The UV distribution or OTF for the five distributions shown in fig.
1. The line style and colors are the same as for fig. 1.
1
III. LARGE-SCALE DISTRIBUTION
The central beam and near sidelobes for an array
consisting of a large number of antennas will be determined
by the large-scale distribution of antennas. These features of
the PSF can be investigated by evaluating the PSF for various
candidate continuous functions. A sample of continuous
antenna distributions is shown in fig. 1. The autocorrelations
corresponding to the UV coverage of the distributions are
shown in fig. 2 while their PSF’s are presented in fig. 3. The
distributions were scaled to produce beams with the same
FWHM.
As expected, the size of the near sidelobes decreases as the
antenna distributions become smoother and more bell shaped.
The tradeoff between sidelobe level and maximum baseline
length for a given resolution is also apparent from these
figures. The thin ring array requires the shortest maximum
baseline and produces the most uniform UV coverage, but at
the cost of sidelobes as large as 16%. The uniform antenna
0.1
0.01
10
3
0.1
1
angle
Fig. 3. The PSF for each of the five antenna distributions shown in fig. 1.
10
4
WOODY
distribution has a first sidelobe of ~1.6%. The cos 2 bell
shaped distribution has no sidelobes above 0.1%. The basic
large-scale distribution can be selected based upon the
acceptable sidelobe level, desired resolution, and available
real estate.
IV. SMALL-SCALE DISTRIBUTION AND STATISTICS OF THE
DIFFERENCE PSF
Matching a large-scale distribution with a finite number of
antennas will necessarily leave small-scale deviations from
the target distribution
!
!
!
D(u ) = S (u ) − I (u ) .
(9)
!
S (u ) is the actual UV sample distribution or OTF given by
!
equ. 3. I (u ) is the ideal target UV sample distribution.
Fourier transforming the UV functions in equ. 9 gives their
corresponding PSFs
!
!
!
D ( p) = S ( p) − I ( p) .
(10)
We will assume that the actual distribution closely matches
the large-scale structure of the ideal distribution and hence
the error or difference beam is close to zero near the center of
the beam and equal to the actual PSF in the far sidelobes.
Alternatively, the idealized continuous distribution can be
define as the actual distribution smoothed on a suitable largescale. Note that single dish measurements from the N
antennas as well as the Hermitian conjugate UV samples are
included in this formulation and the actual PSF function from
an array as well as the ideal PSF derived from a positive real
aperture field distribution is positive and real.
Average sidelobes
The integral of the actual far sidelobes is closely given by
the integral of the difference beam. This in turn is equal to
the difference between the idealized distribution and the
!
actual UV sampling at u = 0 ,
!
1
D ( p) ≈
.
N
(12)
The actual average for the PSF sidelobes can be altered by
varying the number of single antenna measurements
!
contributing to the u = 0 sample. Kogan shows that in the
absence of single antenna measurements the average sidelobe
is zero and the peak negative sidelobes have amplitude
≤ 1 /( n − 1) [6].
B. Missing UV samples and standard deviation of sidelobes
It should be possible to match the ideal large-scale
distribution reasonably well in regions where the UV samples
are separated by less than the antenna diameter d , especially
if the ideal UV distribution is derived from the
autocorrelation of a feasible antenna distribution. But for the
larger configurations there will be regions where the UV
samples are separated by more than d and the contrast
between the areas where there are no samples and the discrete
samples will produce relatively large deviations from the
ideal distribution.
!
The difference distribution, D(u ) , becomes a two valued
function in the regions of sparse coverage if we approximate
!
the antenna OTF, A(u ) , by a top hat of height A(0) and
!
!
diameter d ′=2 π A(0) . The two values of D(u ) are − I (u )
!
and ( A(0) N 2 − I (u )) . The assumption that the actual and
ideal distributions match on the large scale implies that the
UV samples cover a fraction of the UV plane given by
!
! N2
f (u ) = I (u )
.
A(0)
(13)
A.
∫∫
! !
D ( p)dp = S (0) − I (0)
1
=
A(0) − I (0)
N
.
(11)
This uses equ. 3 noting that N single antenna measurements
!
!
contribute to S (0) . I (u ) and A(u ) are normalized to have
an integrated volume of unity and hence I (0) and A(0) are
inversely proportional to the area encompassing the array and
the area of a single telescope respectively. The area
encompassing the array must necessarily be significantly
larger than the total antenna collecting area and the first term
in the second line of equ. 11 will dominate.
The average of the PSF sidelobes is obtained by dividing
the integrated sidelobes by the area of the primary beam,
which is equal to A(0) , conveniently yielding
The mean square value of a function that takes on the value
!
!
( A(0) N 2 − I (u )) over a fraction f (u ) of the area and the
!
value − I (u ) over the rest of the area is
2
!
! A (0)
! A(0) !
!
− 2 f (u ) 2 I (u ) + I 2 (u )
D 2 (u ) = f (u )
4
N
N
.
! A(0)
!
2
= I (u ) 2 − I (u )
N
(14)
The second term in the second line is negligible and can be
safely ignored. Parseval’s theorem tells us the integrated
square of the difference PSF is equal to the integrated square
of the difference OTF. A lower limit to the integrated square
of the difference PSF is obtained by integrating equ. 14 over
!
the regions where fractional coverage, f (u ) , is less than one.
As with the calculation of the average value of the PSF, the
average square of the difference PSF is obtained by dividing
by A(0) . Hence a lower limit to the standard deviation of the
difference PSF is
σD


! 1 !
I (u ) 2 du 
≥
 !

N
 f (b ) ≤1

∫∫
5
ALMA MEMO 389
31 AUGUST, 2001
0.1
1/ 2
.
(15)
σD ≥
1
.
N
(16)
Using the actual single antenna OTF instead of the assumed
top hat function complicates the derivation, but the results
shown in equ. 15 and 16 are still valid.
Evaluating equ. 15 for smaller configurations requires
knowing the idealized distribution. The standard deviations
of the difference PSF for 64 antenna arrays with bell shaped
and uniform UV distributions are plotted as a function of the
array magnification in fig. 4. The magnification is the ratio
of the primary beam to the synthesized beam. The lower
limit for the standard deviation for the bell shaped
distribution of N antennas increases linearly for
magnifications up to ~N and saturates at a value 1/N for
magnifications >2N. The uniform UV distribution has
complete coverage up to a magnification of ~N and hence no
lower limit to the standard deviation of the difference PSF
until the magnification exceeds this magnification.
C. Statistical distribution of sidelobe peaks
Any detectable pattern in the antenna distribution will
result in noticeable features in the OTF and in the PSF
sidelobes. A random antenna distribution that statistically
matches the desired ideal distribution is expected to produce
the minimum sidelobes over the full primary beam. The full
statistical distribution of sidelobes can be calculated for such
pseudo-random arrays operating at large magnification, i.e.
sparse UV coverage.
Equ. 1 shows that the complex voltage in the image plane
is the result of adding N complex numbers or steps of
!
magnitude E ( p ) N with different phases. The phase of the
!
! !
lth complex number for the PSF at p is k rl • p . Away from
the central peak in the PSF this phase is many turns and each
step has an essentially random phase relative to the other N-1
steps. Each point in the PSF is then essentially a 2-D random
walk of N steps. The variance of each component of the
individual steps is
σ l2, x
= σ l2, y
! 2
 E ( p) 
=
 .
 2N 
The variance for the sum of N of these components is
!
B( p)
!
!
σ x2 ( p) = σ 2y ( p) =
.
4N
(17)
(18)
Standard Deviation
This reduces to a particularly simple result for very large
!
configurations where f (u ) < 1 over the full UV plane
0.01
1 .10
3
10
100
Array Magnification
1 .10
Fig. 4. Approximate lower limit to the standard deviation of the difference
PSF as a function of the array magnification for an array of 64 antennas. The
red curve is for a bell shaped distribution, while the blue cure is for a uniform
UV coverage.
Although the components of each step do not have a
Gaussian distribution, the central limit theorem tells us that
the distribution of a sum of a large number of steps will
approach a Gaussian distribution. Thus we can apply the
results obtained for the noise from a cross correlator [7, 8]
and the magnitude of the resultant vectors follows a Rayleigh
distribution
(
)
g (v) = 2 Nv exp − Nv 2 .
(19)
where v is the magnitude of the sum of the N steps of length
1/N and random phase.
The PSF is the magnitude squared of the complex voltage
in the image plane. The distribution of the PSF sidelobes,
s = v 2 , is given by
 ds 
g ( s ) = g (v ) 
 dv 
−1
= N exp(− Ns ) .
(20)
This distribution yields the same the sidelobe average and
standard deviation derived in equs. 12 and 16 above.
The largest sidelobe in the PSF is an important parameter
since it represents the largest imaging artifact that a strong
point source can produce. The number of independent
sidelobes is roughly given by the square of the ratio of the
primary beam to the synthesized beam, i.e. the magnification
squared, mag 2 . A rough estimate of the largest sidelobe is
given by
s max such that
3
6
WOODY
and proceed to optimize the further out sidelobes without
grossly worsening the near sidelobes. Because the near
sidelobe correspond to effectively smaller magnification, a
fully optimized configuration should have a distribution of
sidelobe peaks versus distance from the center of the PSF that
looks like equ. 22 or 23. This is shown in fig. 5 using the
alternate definition of the horizontal scale. Note that this type
of distribution of sidelobe peaks is also expected for the
unoptimized pseudo-random array simply because the
number of independent PSF samples per unit radial distance
increases linearly with radius.
Array Beam max
0.15
0.1
0.05
V.
0
1
10
100
mag or mag*angle/PB
1 .10
3
Fig. 5. Plot of the expected peak sidelobe as a function of magnification
(or radial distance form beam center) for pseudo-random (solid) and
optimized pseudo-random arrays (dotted).
∞
∫ g (s)ds = exp(− Ns max ) = mag 2 .
1
(21)
smax
Hence we expect the peak sidelobe for a pseudo-random
array configuration to be
s max ≈
2
ln(mag ) .
N
(22)
This function is plotted fig. 5. The same result can also be
derived using the approach employed to determine VLBI
false fringe statistics [9] [1].
D. Optimization
The statistical distribution derived in the previous applies to a
pseudo-random configuration. There algorithms that can be
applied to further improve the PSF. Kogan has developed an
algorithm that explicitly minimizes the peak sidelobe over
selected areas in the PSF [10] and Boone has developed an
algorithm to optimize the match between the UV coverage
and a desired large scale distribution [11].
What
improvements in the peak sidelobe performance might one
expect after trying to optimize the configuration?
There are 2(N-1) degrees of freedom in placing N antennas
on a plane. Only a small fraction of these degrees of freedom
are used in placing the antennas so as to statistically match
the desired large scale distribution in a pseudo-random array.
It should be possible to fine tune the antenna positions to
decrease the ~N largest peaks. Following the same steps used
in equs. 21 and 22 above we arrive at
s max,opt ≈
1
[2 ln(mag ) − ln( N )],
N
(23)
and is also plotted in fig. 5.
The sidelobes at different distances from the center of the
PSF correspond to different spatial wavenumbers or scale
sizes. It should be possible to optimize the near in sidelobes
OTHER CONSIDERATIONS
A. Antennas out of service
The effect of taking an antenna out of service can be
conveniently calculated using the formulation of the PSF as
the magnitude squared of the voltage in the image plane
given in equ. 1. Each antenna contributes a vector step of
length 1/N to the voltage sum. The maximum affect on a
typical PSF sidelobe of amplitude 1/N of removing the
contribution from one antenna is
2
 1
1
1
2 
.
±  ≈ 1 ±
stypical ⇒ 

N
N
 N N
(23)
Thus the typical sidelobe in a 64 antenna array will change by
less than 25% with the removal of one antenna. The sidelobe
average increases by 1.6%.
Pseudo-random arrays are particularly resilient to removal
of an antenna, since there is no pattern that can be disrupted
and all antennas are equally important. The worst possible
increment to the maximum sidelobe is
 2
1
s max ⇒ 
ln(mag ) + 
 N
N 


2
≈ ln(mag )1 +

N

2

2

N ln(mag ) 
.
(24)
A 64 antenna array operating at a magnification of 100 could
have its peak sidelobe increased by as much as 8%.
B. Combining configurations
Many astronomical projects will observe a source for more
than a few minutes and may combine several UV data sets.
Additional observations add more coverage of the UV plane
and can improve the PSF. Adding UV data is a linear
operation and the PSF from the different observations are
also added linearly. If the sidelobes from the different
observations are uncorrelated, the standard deviation of the
resultant PSF decreases as the square root of number of
configurations.
Extended observational tracks present a special case
because the successive UV data sets are highly correlated and
the sidelobes do not necessarily decrease. This is especially
ALMA MEMO 389
Standard Deviation
31 AUGUST, 2001
0.1
measurements are acquired continuously during the track.
The near sidelobes caused by the large-scale idealized
distribution are typically complete elliptical rings and earth
rotation will offer only a modest reduction in their peaks.
Additionally the UV coverage resulting from earth rotation
does not correspond to a pseudo-random array and hence the
sidelobes will not necessarily follow the distribution given in
equ. 20.
0.01
C. Complete coverage
Complete coverage can provide unique images but the
accuracy of the images is still limited by the near-in sidelobes
caused by the sharp cutoff in the UV coverage at the
maximum baseline length and by the far sidelobes caused by
the small-scale variations in the UV coverage. The principal
advantage of complete coverage is that there are no gaps or
holes in the UV coverage and with proper weighting of the
data a very smooth effective distribution can be produced that
accurately matches the idealized distribution everywhere.
This would produce a PSF with negligible sidelobes and
direct Fourier transform imaging can be used. This approach
will produce high fidelity and dynamic range images without
the application of special imaging techniques.
Weighting of the UV samples to produce a smoother
distribution or to apply an edge taper to the UV coverage
comes at the cost of decreased point source sensitivity. The
paper by Boone presents an algorithm for building arrays that
give complete coverage while approaching a Gaussian UV
distribution [11]. Boone also quantifies the tradeoff between
array size, sidelobe level and sensitivity loss associated with
weighting the data to give a smooth bell shaped UV
distribution. As the array magnification increases the array
configurations become more ring-like and eventually circular.
A 5-fold symmetric circular array of 64 antennas can achieve
nearly complete coverage at a magnification of ~400 [12].
(13)
The discussion of complete UV coverage carries over to
incomplete but uniform coverage for observations of regions
where the sky brightness is confined to an area smaller than
the angular resolution corresponding to the typical separation
of the UV samples. Keto describes a method for constructing
such uniform coverage configurations based upon Reuleaux
triangles [13].
1 .10
3
10
100
Array Magnification
1 .10
3
Fig. 6. Plot of the lower limit to the standard deviation for bell shaped
(red curves) and uniform UV coverage (blue curves) for 64 antenna arrays
for snapshot (solid), 6 min (dashed) and 1 hr tracks (dotted).
true for the near sidelobes of the PSF. Long tracks can be
handled by scaling the effective number of UV samples and
!
replacing N 2 by g (b ) N 2 in equs. 13 and 14, where
!
θb
!
g (b ) = 1 +
(25)
d′
and
7
θ is the earth rotation angle. Equ. 15 becomes
2
σD
≥
! ∫∫ !
g (b ) f (b )≤1
!
I (b )
!
1
! 2 db .
g (b ) N
(26)
Earth rotation can significantly improve the UV coverage
and even short tracks will decrease the PSF sidelobes. Fig. 6
shows the improvement than can be obtained with only 6 min
and 1 hr tracks. This analysis assumes a circularly symmetric
array at the pole observing a polar source. The long baselines
sweep over a large area of the UV plane reducing the gaps in
the UV coverage and significantly decreasing the standard
deviation of the sidelobes. Interestingly, the increased
coverage for the higher magnification configurations more
than compensates for the reduced snapshot coverage and the
sidelobe standard deviation decreases as the configuration
size increases. Both the bell shaped and uniform coverage
arrays show similar behavior for large magnifications, but
there can be significant differences for the small
configurations. Because the added coverage is not randomly
distributed, the actual decrease in sidelobe standard deviation
will be much less than shown in fig. 6.
Earth rotation greatly reduces the standard deviation and
peaks of the intermediate and far sidelobes, but does not
change the average sidelobe level if the single antenna
VI. DISCUSSION AND SUMMARY
The basic result here is that the near-in sidelobes are
determined by the large-scale distribution of antennas, while
the far sidelobes depend mostly upon the number of antennas
and array magnification, assuming the configuration is
suitably randomized. For magnifications larger than N,
where N is the number of antennas, the standard deviation of
the far sidelobes is σ D ≈ 1 N and if the antennas are placed
pseudo-randomly the distribution of the sidelobe amplitudes
is g ( s ) = N exp(− Ns ) and the peak sidelobe should be
s max ≈ 2 N ln(mag ) . Placement of the antennas with a
detectable pattern is likely to worsen the PSF while
8
optimization might improve both the distribution and the
peak sidelobe level.
The expected standard deviation for the sidelobes is
proportional to the magnification for magnifications less than
N, σ D ∝ mag , and also decreases significantly when earth
rotation synthesis is applied.
Applying an edge taper to the UV can improve the near-in
sidelobes at the cost of a loss of sensitivity. The far sidelobes
can be improved by weighting of the data to produce a
smooth distribution if the fractional UV coverage exceeds
unity over most of the UV plane of interest, again at the cost
of losing point source sensitivity. But there is not much that
can be done to improve the far sidelobes for snapshot images
obtained
with
the
highest
magnification
sparse
configurations.
In general it is reasonable to select a large-scale
distribution with near-in sidelobes that are at least as small as
the far sidelobes to give the highest quality images without
having to weight the data and suffer the associated loss in
sensitivity. Thus filled or even centrally condensed array
configurations are more appropriate for large arrays such as
ALMA while arrays with a small number of antennas
operating in snapshot mode would favor ring-like
configurations, as has been pointed out by Holdaway [3].
Large arrays of 50 or more antennas require centrally
condensed or bell shaped antenna distributions to keep the
near-in sidelobes below the average far sidelobes. The
standard deviation of the far sidelobes also decrease below
1/N as the array magnification drops below N or when earth
rotation synthesis is used, further supporting the use of bell
shaped antenna distributions. Earth rotation synthesis can
also significantly reduce the far sidelobes for sparse arrays
and hence even very small arrays may benefit from using
bell shaped distributions.
A case for large arrays having complete UV coverage at
the highest practical magnification (and therefore nearly
uniform UV coverage) can be made for observation of very
bright and complex sources. Appropriate tapering and
weighting of the UV data can produce nearly perfect images
from complete UV coverage data while suffering some loss
in sensitivity. Unfortunately, given the current and projected
array sensitivities at millimeter and sub-millimeter
wavelengths, there are only a small number of sources bright
enough to benefit from this approach.
The formulas in this paper can aid in making the tradeoff
between the size and number of antennas during the initial
conceptual design of an interferometer array. Equations 12,
16, 20 and 22 give a pretty complete picture of PSF sidelobe
statistics and hence of the image quality as a function of the
number of antennas. These equations quantify the improved
image quality that an array of many small antennas will
produce over an array of fewer larger antennas with the same
total collecting area.
The results obtained are very general and can serve as a
basis for developing configuration strategies. The statistics
of the PSF for an actual or proposed configuration can be
compared against the above formulas to determine whether it
WOODY
performs better or worse than a pseudo-random array. A
companion paper will present the results for a few different
arrays and demonstrate the accuracy and utility of the results
given in this paper. The optimal configuration will depend
upon the science goals for an array but the concepts and
results derived here should make it easier to investigate the
many options.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
J. W. Hardy, Adaptive Optics for Astronomical Telescopes, Oxford
University Press, 1998, p. 116.
A.R. Thompson, J. M. Moran, and G. W. Swenson, Interferometry
and Synthesis in Radio Astronomy, John Wiley and Sons, 1986.
M. Holdaway, “Comments on Minimum Sidelobe Configurations,”
MMA Memo 172, May 7, 1997.
Ref [1] p. 109.
Thompson, Moran and Swenson "Radio Interferometery"
L. Kogan, “Level of Negative Sidelobes in an Array Beam,” Pub.
Astron. Soc. Pac. vol. 111, pp. 510-511, 1999.
Ref. [2] p. 165.
J. Bass, Principal of Probability, p. 230.
Ref [2] p. 265.
L. Kogan, “Optimizing a Large Array Configuration to Minimize the
Sidelobe,” IEEE Trans. Antennas Propagat., vol. 48, pp. 1075-1078,
July 2000.
F. Boone, “Interferometric array design: antenna positions optimized
for ideal distributions of visibilities,” Aston. & Astrophy, Feb. 2001.
D. Woody, “ALMA Configurations with complete UV coverage,”
MMA memo #270, July 1999.
E. Keto, “The Shapes of Cross-Correlation Interferometers,” Ap. J. vol.
475, pp. 843-852, Feb. 1997.
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