nassp masters 5003f - computational astronomy - 2009 lecture 16 further with interferometry –...

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.


• 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


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


Earth-rotation synthesis

Apply appropriate delays: like measuring Vwith ‘virtual antennas’ in a plane normalto the direction of the phase centre.


Earth-rotation synthesis

Apply appropriate delays: like measuring V

with ‘virtual antennas’ in a plane normal

to the direction of the phase centre.


Field of view and resolution.

Single dish:FOV and resolution are the same.

FOV ~ λ/d(d = dish diameter)

Resolution ~ λ/d



Aperture synthesis array:FOV is much larger than resolution.

FOV ~ λ/d Resolution ~ λ/D(D = longest baseline)

d

D



Phased array:Signals delayed then added.FOV again = resolution.

FOV ~ λ/D Resolution ~ λ/Dd

D

Good for spectroscopy,VLBI.


LOFAR – can see the whole sky at once.


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,,


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,,


Painting in V as the Earth rotates


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.


We further pretend that these samples are points.


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


Resulting noise in the image:Spatially uniform – but not ‘white’.

(Note: noise in real and imaginaryparts of the visibility is uncorrelated.)


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.


How to regrid the samples?

Could simply add samples in each box.


But this can be expressed as a convolution.

Samples convolved with a square box.


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,


What does this do to the image?• Fourier theory:

– Convolution Multiplication.– Sampling onto a grid ‘aliasing’.


A 1-dimensional example ‘dirty image’ ID:V I via direct FT:


A 1-dimensional example ‘dirty image’ ID:Multiplied by the FT ofthe convolver:


A 1-dimensional example:The aliased resultis in green:

Image boundariesbecome cyclic.


A 1-dimensional example:Finally, dividingby the FT of theconvolver:


Effect on image noise:

Direct FT Gridded then FFT


Aliasing of sources – none in DT

This is a direct transform. The green box indicatesthe limits of a gridded image.


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.

nassp masters 5003f - computational astronomy - 2009 lecture 16 further with interferometry –...

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