[daip] Detailed questions

[Eric Greisen]

On 06/13/2019 11:44 AM, Chen, Diana wrote:
> Hi Eric,
> 
> 
> I'm a bit ocd when it comes to understanding concepts so I have to ask 
> some more detailed questions about the image synthesis from multiple 
> telescopes. I apologize in advance if this takes too much time to 
> answer. I'm just frustrated that I'm still having a hard time 
> intuitively grasping what is actually happening.
> 
> 
> Below are some questions I have regarding the mathematics of 
> synthesizing signals from the interferometer. I learn by intuition and 
> visual explanations, so sometimes chugging through the mathematics can 
> be difficult.
> 
> For my questions, for clarification purposes, I’m going to go with the 
> simplest case, which is two telescope array with a relatively long 
> baseline.
> 
> First, I wanted to double check my understanding of interferometry in 
> general.Since the diffraction limit is wavelength/Diameter, astronomers 
> uses multiple telescopes to create a very large diameter through having 
> long baselines. Then, the images from each telescope is synthesized 
> together. The synthesis of the images is where I am having trouble 
> understanding. What I know is that since the distance to each separate 
> element of the array is different, there is a time delay between 
> separate elements.
> 
>  1. The signal from each telescope is a 2D image, right?

             NO.  Assuming single-pixel receivers, each antenna produces 
a voltage stream that is the sum of signals received by the telescope.
The correlator delays one stream so as to match the arrival times of the 
2 streams and then multiplies the signals.  The result is a "visibility" 
which is the sum over the antenna beam areas of the emission from each 
part of the source.  The emission from the presumed source center has 
zero phase, but emission from other directions arrives somewhat out of 
phase depending on the direction and the projected baseline between the 
2 antennas.  This visibility is a sample of the Fourier transform of the 
sky emission.  As the Earth rotates, the projected baseline length and 
position angle changes, so over time that baseline samples the Fourier 
transform of the sky emission along an ellipse in the Fourier space (we 
call the Fourier components u and v).

>  2. The next step, as you stated, is to combine the image from each
>     telescope and find the spatial fourier transform. Why , if you are
>     finding the fourier spatial transform, are you integrating the
>     signal in time, as you stated in one of your emails.
            NO - this is not what I said.

>  3. Also, because there is a time delay, for the 2 telescope case, are
>     you shifting the signal temporally from telescope 1 to match the
>     time at telescope 2?
            See above, we delay the one signal so that the times match 
up.  The source is assumed to emit entirely incoherently.  If we did not 
match the times then the result of the multiplication even from a single 
point source would be essentially zero.

>  4. Also, why not just find the spatial fourier transform from one
>     signal? If the purpose of fourier transform is to break the signal
>     into its components, why use more than one?
           The spatial Fourier transform of the sky comes from the 
interference of the 2 signals.  The single telescope simply measures the 
sum of the sky emission (scaled by the beam pattern of the single 
telescope).  Single telescopes are also used in radio astronomy.  They 
make images by changing their pointing position and simply measuring the 
input power as a function of position.  Their spatial resolution is that 
of the single telescope (rather poor compared to an interferometer)


>  5. I don’t quite understand why changing the orientation of the 2
>     telescopes help. I think(?) it’s because when you have the pair at 0
>     degrees(or any random orientation), the fourier transform of the
>     combined signals give you the spatial components that are also
>     orientated in a certain way. As you rotate the pair of telescopes,
>     this gives you other spatial components that orient another way.
>     Then, after you get a decent number of spatial components with
>     orientations, you combine them to the final image. Is this correct?
>     Still don’t understand why you need 2 telescopes for this. 

Since you did not understand the most basic point, it is not surprising 
that you do not understand the rest.  It takes 2 antennas to sample a 
Fourier component and then you know that the Earth rotation changes the 
projection and allows many components to be sampled.  Having 27 antennas 
in the VLA means we have 351 separate antenna pairs sampling the Fourier 
space.  The image is then made by Fourier transforming the complete set 
of measurements.

Try looking at wikipedia on such subjects as aperture synthesis, radio 
interferometry, and the like.

Eric Greisen