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NASSP Masters 5003F - Computational Astronomy - 2009
Lecture 16
Further with interferometry –• Digital correlation• Earth-rotation synthesis and non-planar
arrays• Resolution and the field of view;• Binning in frequency and time, and its
effects on the image;• Noise in cross-correlation;• Gridding and its pros and cons.
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NASSP Masters 5003F - Computational Astronomy - 2009
• The first thing necessary is to sample each continuous y at a number of times kΔt.
• Then R1,2(kΔt) is approximated by
• But, how many bits to use to store each yk value?
Digital correlation
N
j
tkjytjyN
tkR1
212,11~
y
yk
t
k
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NASSP Masters 5003F - Computational Astronomy - 2009
Digital correlation
1
• Surprisingly, 1 bit works pretty well!
• Multiplication becomes a boolean NOT(XOR).
• Allows us to use simple boolean logic circuits (cheap).
• SNR drops by about 2/π though.
• 2 or 3 bits improves the SNR without too much increase in circuit cost.
1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 10 0 0
y
yk>0
k
t
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NASSP Masters 5003F - Computational Astronomy - 2009
Earth-rotation synthesis
Apply appropriate delays: like measuring Vwith ‘virtual antennas’ in a plane normalto the direction of the phase centre.
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NASSP Masters 5003F - Computational Astronomy - 2009
Earth-rotation synthesis
Apply appropriate delays: like measuring Vwith ‘virtual antennas’ in a plane normalto the direction of the phase centre.
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NASSP Masters 5003F - Computational Astronomy - 2009
Earth-rotation synthesis
Apply appropriate delays: like measuring V
with ‘virtual antennas’ in a plane normal
to the direction of the phase centre.
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NASSP Masters 5003F - Computational Astronomy - 2009
Field of view and resolution.
Single dish:FOV and resolution are the same.
FOV ~ λ/d(d = dish diameter)
Resolution ~ λ/d
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NASSP Masters 5003F - Computational Astronomy - 2009
Field of view and resolution.
Aperture synthesis array:FOV is much larger than resolution.
FOV ~ λ/d Resolution ~ λ/D(D = longest baseline)
d
D
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NASSP Masters 5003F - Computational Astronomy - 2009
Field of view and resolution.
Phased array:Signals delayed then added.FOV again = resolution.
FOV ~ λ/D Resolution ~ λ/Dd
D
Good for spectroscopy,VLBI.
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NASSP Masters 5003F - Computational Astronomy - 2009
LOFAR – can see the whole sky at once.
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NASSP Masters 5003F - Computational Astronomy - 2009
Reconstructing the image.• The basic relation of aperture synthesis:
where all the (l,m) functions have been bundled into I´. We can easily recover the true brightness distribution from this.
• The inverse relationship is:
• But, we have seen, we don’t know V everywhere.
vmulimlIdmdlvuV 2exp,,
vmulivuVdvdumlI 2exp,,
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NASSP Masters 5003F - Computational Astronomy - 2009
Sampling function and dirty image• Instead, we have samples of V. Ie V is
multiplied by a sampling function S.
• Since the FT of a product is a convolution,
where the ‘dirty beam’ B is the FT of the sampling function:
ID is called the ‘dirty image’.
vmulivuSvuVdvdumlI 2D e,,,
mlBmlImlI ,,,D
vmulivuSdvdumlB 2e,,
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NASSP Masters 5003F - Computational Astronomy - 2009
Painting in V as the Earth rotates
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NASSP Masters 5003F - Computational Astronomy - 2009
Painting in V as the Earth rotates
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NASSP Masters 5003F - Computational Astronomy - 2009
But we must ‘bin up’ in ν and t.
This smears out the finer ripples.Fourier theory says: finer ripples come from distant sources.Therefore want small Δν, Δt for wide-field imaging. But: huge files.
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NASSP Masters 5003F - Computational Astronomy - 2009
We further pretend that these samples are points.
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NASSP Masters 5003F - Computational Astronomy - 2009
What’s the noise in these measurements?• Theory of noise in a cross-correlation is a little
involved... but if we assume the source flux S is weak compared to sky+system noise, then
• If antennas the same,
• Root 2 smaller SNR from single-dish of combined area (lecture 9).– Because autocorrelations not done information lost.
tT
AkS total
erms
2
tTT
AAkS total2total1
e2e1rms
2
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NASSP Masters 5003F - Computational Astronomy - 2009
Resulting noise in the image:Spatially uniform – but not ‘white’.
(Note: noise in real and imaginaryparts of the visibility is uncorrelated.)
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NASSP Masters 5003F - Computational Astronomy - 2009
Transforming to the image plane:• Can calculate the FT directly, by summing
sine and cosine terms.– Computationally expensive - particularly with
lots of samples.• MeerKAT: a day’s observing will generate about
80*79*17000*500=5.4e10 samples.
• FFT:– quicker, but requires data to be on a regular
grid.
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NASSP Masters 5003F - Computational Astronomy - 2009
How to regrid the samples?
Could simply add samples in each box.
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NASSP Masters 5003F - Computational Astronomy - 2009
But this can be expressed as a convolution.
Samples convolved with a square box.
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NASSP Masters 5003F - Computational Astronomy - 2009
Convolution gridding.
• ‘Square box’ convolver is
• Gives
• But the benefit of this formulation is that we are not restricted to a ‘square box’ convolver.– Reasons for selecting the convolver carefully will be
presented shortly.
vvuuGvuVdvduV kjkj ,,,
else. 0 ,5.0,5.0,for 1, vuvuG
vuVdvduV kj ,
5.0,5.0,
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NASSP Masters 5003F - Computational Astronomy - 2009
What does this do to the image?• Fourier theory:
– Convolution Multiplication.– Sampling onto a grid ‘aliasing’.
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NASSP Masters 5003F - Computational Astronomy - 2009
A 1-dimensional example ‘dirty image’ ID:V I via direct FT:
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NASSP Masters 5003F - Computational Astronomy - 2009
A 1-dimensional example ‘dirty image’ ID:Multiplied by the FT ofthe convolver:
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NASSP Masters 5003F - Computational Astronomy - 2009
A 1-dimensional example:The aliased resultis in green:
Image boundariesbecome cyclic.
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NASSP Masters 5003F - Computational Astronomy - 2009
A 1-dimensional example:Finally, dividingby the FT of theconvolver:
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NASSP Masters 5003F - Computational Astronomy - 2009
Effect on image noise:
Direct FT Gridded then FFT
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NASSP Masters 5003F - Computational Astronomy - 2009
Aliasing of sources – none in DT
This is a direct transform. The green box indicatesthe limits of a gridded image.
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NASSP Masters 5003F - Computational Astronomy - 2009
Aliasing of sources – FFT suffers from this.
The far 2 sources are now wrapped or ‘aliased’into the field – and imperfectly suppressed by thegridding convolver.