Principles of Interferometry 3

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Principles of Interferometry 3 Practical
Applications John M. Dickey University of
Tasmania
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wavefronts (lines of constant phase)
extra path length
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direction to source
extra path length
voltages

phase centre
adding circuit
baseline to each antenna
output
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direction to source
extra path length
Relative to the phase centre, the extra path
length is so the extra phase is
voltages

phase centre
adding circuit
baseline to each antenna
output
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The voltages E1 and E2 are similar, but with
different path lengths, depending on the
direction to the source of the radiation. These
different path lengths change the phase of the
radiation (depending on the frequency or
wavelength) as

something x
For a point source, this is the (flux density)0.5.
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The sum of the two voltages from the two antennas
is
where A is the antenna response in direction
(x,y) at frequency n and T is the sky brightness
in direction (x,y) at frequency n The
expression in --
For a narrow band system i.e. observing a point
source i.e. we get power magnitude squared of
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In the notes the square aperture problem is
worked - try it yourself!
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The power response is
This can be rewritten as
where f (u,v) is the autocorrelation function of
e(u,v)
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So for the adding interferometer (or single dish,
or phased array) telescope, the power pattern
P(x,y) is the Fourier Transform of the
autocorrelation function of the aperture
illumination
E (x,y)
e (u,v)
f (u,v)
P (x,y)
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To make a map of the sky using a telescope with
beam power pattern P(x,y), we perform a
convolution of the sky brightness, T(x,y) with
P. The map or image of the sky is
f (u,v)
P (x,y)
T (x,y)
V(u,v)
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• The multiplying or correlation
• interferometer is very similar
• and very different!

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direction to source
extra path length
ti
delay to match path length difference
B is now the baseline between the two antennas
x
multiplying correlator
output
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If we can ignore the w term, then
Since the Fourier Transform of the product of two
functions is the convolution of the Transforms of
the two functions.
Aperture Synthesis means measuring for many
values of (u,v) using the rotation of the earth
to sweep each baseline into an ellipse.
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P (x,y)
A(u,v)
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Now the process of making an image is first
gridding, then a Fourier Transform, as in the red
box.
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So if the frequency response of the receiver is
then is
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The rest of this synthesis school will explain
how to make it all work with real data.
But, if the Fourier transforms were
non-obvious, or if the convolution concept was
unfamiliar, help is on the way tomorrow morning!
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direction to source
extra path length
delay to match path length difference

adding circuit
phase centre
baseline to each antenna
output
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extra path length
delay to match path length difference
x
multiplying correlator
output
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V(n)
dn
Dn
n
1
V(t)
dt
Dn
1
Dt
dn
t
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