Further with interferometry

1
Lecture 13

• Further with interferometry
• 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.

2
Earth-rotation synthesis
Apply appropriate delays like measuring V with
virtual antennas in a plane normal to the
direction of the phase centre.
3
Earth-rotation synthesis
Apply appropriate delays like measuring V with
virtual antennas in a plane normal to the
direction of the phase centre.
4
Earth-rotation synthesis
Apply appropriate delays like measuring V with
virtual antennas in a plane normal to the
direction of the phase centre.
5
Field of view and resolution.
Single dish FOV and resolution are the same.
FOV ?/d (d dish diameter)
Resolution ?/d
6
Field of view and resolution.
Aperture synthesis array FOV is much larger than
resolution.
d
FOV ?/d
Resolution ?/D (D longest baseline)
D
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Field of view and resolution.
Phased array Signals delayed then added. FOV
again resolution.
Good for spectroscopy, VLBI.
d
FOV ?/D
Resolution ?/D
D
8
LOFAR can see the whole sky at once.
9
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 dont know V everywhere.

10
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.

11
Painting in V as the Earth rotates
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Painting in V as the Earth rotates
13
But we must bin up in ? and t.
This smears out the finer ripples. Fourier theory
says finer ripples come from distant
sources. Therefore want small ??, ?t for
wide-field imaging. But ? huge files.
14
We further pretend that these samples are points.
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Whats 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 skysystem noise, then
• If antennas the same,
• Root 2 smaller SNR from single-dish of combined
  area (lecture 9).
• Because autocorrelations not done ? information
  lost.

16
Resulting noise in the image
Spatially uniform but not white.
(Note noise in real and imaginary parts of the
visibility is uncorrelated.)
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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 days observing will generate about
  8079170005005.4e10 samples.
• FFT
• quicker, but requires data to be on a regular
  grid.

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How to regrid the samples?
Could simply add samples in each box.
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But this can be expressed as a convolution.
Samples convolved with a square box.
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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.

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What does this do to the image?

• Fourier theory
• Convolution ? Multiplication.
• Sampling onto a grid ? aliasing.

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A 1-dimensional example dirty image ID
V ? I via direct FT
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A 1-dimensional example dirty image ID
Multiplied by the FT of the convolver
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A 1-dimensional example
The aliased result is in green
Image boundaries become cyclic.
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A 1-dimensional example
Finally, dividing by the FT of the convolver
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Effect on image noise
Direct FT
Gridded then FFT
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Aliasing of sources none in DT
This is a direct transform. The green box
indicates the limits of a gridded image.
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Aliasing of sources FFT suffers from this.
The far 2 sources are now wrapped or
aliased into the field and imperfectly
suppressed by the gridding convolver.