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<p>Hi Eric,</p>
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<p>This is very helpful. Thanks! </p>
<p><br>
</p>
<p>Diana</p>
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<div id="x_divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif" color="#000000" style="font-size:11pt"><b>From:</b> Eric Greisen <egreisen@nrao.edu><br>
<b>Sent:</b> Thursday, June 13, 2019 4:02:08 PM<br>
<b>To:</b> Chen, Diana; daip@nrao.edu<br>
<b>Subject:</b> Re: Detailed questions</font>
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<div class="PlainText">On 06/13/2019 11:44 AM, Chen, Diana wrote:<br>
> Hi Eric,<br>
> <br>
> <br>
> I'm a bit ocd when it comes to understanding concepts so I have to ask <br>
> some more detailed questions about the image synthesis from multiple <br>
> telescopes. I apologize in advance if this takes too much time to <br>
> answer. I'm just frustrated that I'm still having a hard time <br>
> intuitively grasping what is actually happening.<br>
> <br>
> <br>
> Below are some questions I have regarding the mathematics of <br>
> synthesizing signals from the interferometer. I learn by intuition and <br>
> visual explanations, so sometimes chugging through the mathematics can <br>
> be difficult.<br>
> <br>
> For my questions, for clarification purposes, I’m going to go with the <br>
> simplest case, which is two telescope array with a relatively long <br>
> baseline.<br>
> <br>
> First, I wanted to double check my understanding of interferometry in <br>
> general.Since the diffraction limit is wavelength/Diameter, astronomers <br>
> uses multiple telescopes to create a very large diameter through having <br>
> long baselines. Then, the images from each telescope is synthesized <br>
> together. The synthesis of the images is where I am having trouble <br>
> understanding. What I know is that since the distance to each separate <br>
> element of the array is different, there is a time delay between <br>
> separate elements.<br>
> <br>
> 1. The signal from each telescope is a 2D image, right?<br>
<br>
NO. Assuming single-pixel receivers, each antenna produces <br>
a voltage stream that is the sum of signals received by the telescope.<br>
The correlator delays one stream so as to match the arrival times of the <br>
2 streams and then multiplies the signals. The result is a "visibility" <br>
which is the sum over the antenna beam areas of the emission from each <br>
part of the source. The emission from the presumed source center has <br>
zero phase, but emission from other directions arrives somewhat out of <br>
phase depending on the direction and the projected baseline between the <br>
2 antennas. This visibility is a sample of the Fourier transform of the <br>
sky emission. As the Earth rotates, the projected baseline length and <br>
position angle changes, so over time that baseline samples the Fourier <br>
transform of the sky emission along an ellipse in the Fourier space (we <br>
call the Fourier components u and v).<br>
<br>
> 2. The next step, as you stated, is to combine the image from each<br>
> telescope and find the spatial fourier transform. Why , if you are<br>
> finding the fourier spatial transform, are you integrating the<br>
> signal in time, as you stated in one of your emails.<br>
NO - this is not what I said.<br>
<br>
> 3. Also, because there is a time delay, for the 2 telescope case, are<br>
> you shifting the signal temporally from telescope 1 to match the<br>
> time at telescope 2?<br>
See above, we delay the one signal so that the times match <br>
up. The source is assumed to emit entirely incoherently. If we did not <br>
match the times then the result of the multiplication even from a single <br>
point source would be essentially zero.<br>
<br>
> 4. Also, why not just find the spatial fourier transform from one<br>
> signal? If the purpose of fourier transform is to break the signal<br>
> into its components, why use more than one?<br>
The spatial Fourier transform of the sky comes from the <br>
interference of the 2 signals. The single telescope simply measures the <br>
sum of the sky emission (scaled by the beam pattern of the single <br>
telescope). Single telescopes are also used in radio astronomy. They <br>
make images by changing their pointing position and simply measuring the <br>
input power as a function of position. Their spatial resolution is that <br>
of the single telescope (rather poor compared to an interferometer)<br>
<br>
<br>
> 5. I don’t quite understand why changing the orientation of the 2<br>
> telescopes help. I think(?) it’s because when you have the pair at 0<br>
> degrees(or any random orientation), the fourier transform of the<br>
> combined signals give you the spatial components that are also<br>
> orientated in a certain way. As you rotate the pair of telescopes,<br>
> this gives you other spatial components that orient another way.<br>
> Then, after you get a decent number of spatial components with<br>
> orientations, you combine them to the final image. Is this correct?<br>
> Still don’t understand why you need 2 telescopes for this. <br>
<br>
Since you did not understand the most basic point, it is not surprising <br>
that you do not understand the rest. It takes 2 antennas to sample a <br>
Fourier component and then you know that the Earth rotation changes the <br>
projection and allows many components to be sampled. Having 27 antennas <br>
in the VLA means we have 351 separate antenna pairs sampling the Fourier <br>
space. The image is then made by Fourier transforming the complete set <br>
of measurements.<br>
<br>
Try looking at wikipedia on such subjects as aperture synthesis, radio <br>
interferometry, and the like.<br>
<br>
Eric Greisen<br>
<br>
<br>
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