Suppose we are in a 1D universe where:
The source s lives on a 1D sky, the \bold{l}- axis, emitting monochromatic radio wave of wavelength \lambda .
The 2 stations are on a 1D Earth, the \bold{x}- axis , the \vec{b} formed is on the \bold{u}- axis, which is \bold{x}- axis scaled by \frac{1}{\lambda} .
The figure below shows such 1D universe, feel free to drag the yellow point source s around.
Figure :
source position on 1D sky: draggable {\color{yellow}\text{source } s}
As you drag the source around, you can observe that every distinct location of s has a corresponding distinct l value.
In this 1D universe:
Different stations pair results in different baseline \vec{b} , or, u .
Different s position results in a different l .
So the cosine correlator output v_{R_c} is a function of u and l :
v_{R_c}(\tau_g) &= v_{R_c} (u,l) = I(l) \cos(2\pi u l ) \\ &(u \text{ corresponds to baseline } \vec{b})\\ &(l \text{ corresponds to source } s)\\and the sine correlator outout v_{R_s} is also a function of u and l :
v_{R_s}(\tau_g) &= v_{R_s} (u,l) = I(l) \sin(2\pi u l ) \\ &(u \text{ corresponds to baseline } \vec{b})\\ &(l \text{ corresponds to source } s)\\1.3.1: Sky Intensity
Now let's fill the 1D sky line with \infty sources. In such setting:
Each source s_n line up one by one on the \bold{l}- axis, emitting monochromatic radio wave of frequency \omega but with different amplitudes M(s_n) . Suppose the amplitude of source s_n detected at antenna station 1 is a function of time shown as below: x_{s_n}(t) = M(s_n)\cos(\omega t + \phi_n)
Figure
But we know that the location of each s_n can be represented by a distinct l value, so with \infty of sources we can rewrite the source magnitude function above to:
x_{s_n} (t) = M(l)\cos(\omega t+ \phi_n)In radio astronomy, we are interested in the sky image, i.e., the magnitude of the sources, M(l) . With M(l) , we can create a sky magnitude plot like the figure below:
Figure
And with I = \frac{1}{2}M^2 , the ultimate goal is to recover the sky intensity plot like the figure below:
Figure
We will show you how to recover the Sky Intensity I(l) for the rest of this page.
1.3.2: Two or More Sources
Let's start with 2 sources, s_a and s_b :
Those two sources are spatially incoherent, meaning that their respective phase shift functions \phi_a and \phi_b are Independent and Identically Distributed (IID).
Station b_2 and b_1 form a baseline u .
Suppose station b_1 receive signal from sum of sources s_a and s_b as such:
\color{blue} x_1(t) \color{black} &= x_{s_a}(t) + x_{s_b}(t) \\ &= \large{M(l_a) \cos(\omega t +\phi_a{\scriptstyle(t)}) } \; \small{+} \; \large{M(l_b) \cos(\omega t +\phi_b{\scriptstyle(t)}) }\\station b_2 receive the same signal with delays \tau_a and \tau_b for s_a and s_b respectively:
{\color{green} x_2(t) } &= {\color{green} \Large[ } x_{s_a}(t-\tau_{a}) {\color{green} + } x_{s_b}(t-\tau_b) {\color{green} \Large] }\\ &={\color{green} \Large[ } M(l_a) \cos(\omega (t-\tau_a) +\phi_{a}{\scriptstyle(t-\tau_a)}) {\color{green} \Large +} M(l_b) \cos(\omega (t-\tau_b) +\phi_{b}{\scriptstyle(t-\tau_a)}) {\color{green} \Large] }\\ &(\text{assuming } \tau_a \ll \tau_c \;,\; \tau_b \ll \tau_c )\\ &= {\color{green} \Large[ } M(l_a) \cos(\omega (t-\tau_a) +\phi_{a}{\scriptstyle(t)}) {\color{green}\Large + } M(l_b) \cos(\omega (t-\tau_b) +\phi_{b}{\scriptstyle(t)}) {\color{green} \Large] } \\Next we calculate the Cosine Interferometer output v_{R_c}(\tau_a,\tau_b) :
v_{R_c}(\tau_a,\tau_b) = \langle {\color{green} x_2(t)} , {\color{blue}x_1(t) } \rangle1. Multiplication:
{\color{green}x_2(t)}{\color{blue}x_1(t)} =& {\color{green} \Large[ }M(l_a) \cos({\scriptstyle \omega (t-\tau_a) +\phi_{a}{\scriptstyle(t)}}) {\color{green}\Large + } M(l_b) \cos({\scriptstyle \omega (t-\tau_b) +\phi_{b}{\scriptstyle(t)}}) {\color{green} \Large] } \; {\color{blue} \Large[ }M(l_a) \cos({\scriptstyle \omega t +\phi_{a}{\scriptstyle(t)}}) {\color{blue} \Large+ } M(l_b) \cos({\scriptstyle \omega t +\phi_{b}{\scriptstyle(t)}}) {\color{blue} \Large] } \\ = & M^2(l_a) \cos( {\scriptstyle \omega (t-\tau_a) +\phi_{a}{\scriptstyle(t)}}) \cos({\scriptstyle \omega t +\phi_{a}{\scriptstyle(t)}}) \\ &+ M(l_a)M(l_b) \cos( {\scriptstyle \omega (t-\tau_a) +\phi_{a}{\scriptstyle(t)}}) \cos( {\scriptstyle \omega t+\phi_{b}{\scriptstyle(t)}}) \\ &+ M(l_b)M(l_a) \cos( {\scriptstyle \omega (t-\tau_b) +\phi_{b}{\scriptstyle(t)}}) \cos( {\scriptstyle \omega t+\phi_{a}{\scriptstyle(t)}}) \\ &+ M(^2l_b) \cos( {\scriptstyle \omega (t-\tau_b) +\phi_{b}{\scriptstyle(t)}}) \cos( {\scriptstyle \omega t+\phi_{b}{\scriptstyle(t)}}) \\2. Average:
v_{R_c}(t,\tau_a,\tau_b)|_T= & \langle{\color{green}x_2(t)},{\color{blue}x_1(t)} \rangle\\ = &\frac{1}{T} \int_{t-T/2}^{t+T/2} M^2(l_a) \cos( {\scriptstyle \omega (\psi-\tau_a) +\phi_{a}{\scriptstyle(\psi)}}) \cos({\scriptstyle \omega \psi +\phi_{a}{\scriptstyle(\psi)}}) d\psi \\ &+\frac{1}{T} \int_{t-T/2}^{t+T/2} M(l_a)M(l_b) \cos( {\scriptstyle \omega (\psi-\tau_a) +\phi_{a}{\scriptstyle(\psi)}}) \cos( {\scriptstyle \omega \psi+\phi_{b}{\scriptstyle(\psi)}})d\psi \\ &+\frac{1}{T} \int_{t-T/2}^{t+T/2} M(l_b)M(l_a) \cos( {\scriptstyle \omega (\psi-\tau_b) +\phi_{b}{\scriptstyle(\psi)}}) \cos( {\scriptstyle \omega \psi+\phi_{a}{\scriptstyle(\psi)}})d\psi \\ &+\frac{1}{T} \int_{t-T/2}^{t+T/2} M(^2l_b) \cos( {\scriptstyle \omega (\psi-\tau_b) +\phi_{b}{\scriptstyle(\psi)}}) \cos( {\scriptstyle \omega \psi+\phi_{b}{\scriptstyle(\psi)}}) d\psi \\ \\ = &\tfrac{1}{2}M^2(l_a) \frac{1}{T} \int_{t-T/2}^{t+T/2} \cos( {\scriptstyle \omega (\psi-\tau_a) +\phi_{a}{\scriptstyle(\psi)} +\omega \psi +\phi_{a}{\scriptstyle(\psi)} }) {\color{brown} + } \cos({\scriptstyle \omega (\psi-\tau_a) +\phi_{a}{\scriptstyle(\psi)} -\omega \psi -\phi_{a}{\scriptstyle(\psi)}}) d\psi \\ &+\tfrac{1}{2} M(l_a)M(l_b) \frac{1}{T} \int_{t-T/2}^{t+T/2} \cos( {\scriptstyle \omega (\psi-\tau_a) +\phi_{a}{\scriptstyle(\psi)} +\omega \psi+\phi_{b}{\scriptstyle(\psi)} }) {\color{brown} + } \cos( {\scriptstyle \omega (\psi-\tau_a) +\phi_{a}{\scriptstyle(\psi)} - \omega \psi-\phi_{b}{\scriptstyle(\psi)}}) d\psi \\ &+\tfrac{1}{2} M(l_b)M(l_a)\frac{1}{T} \int_{t-T/2}^{t+T/2} \cos( {\scriptstyle \omega (\psi-\tau_b) +\phi_{b}{\scriptstyle(\psi)}+ \omega \psi+\phi_{a}{\scriptstyle(\psi)}}){\color{brown} + } \cos( {\scriptstyle \omega (\psi-\tau_b) +\phi_{b}{\scriptstyle(\psi)}-\omega \psi-\phi_{a}{\scriptstyle(\psi)}}) d\psi \\ &+\tfrac{1}{2}M^2(l_b) \frac{1}{T} \int_{t-T/2}^{t+T/2} \cos( {\scriptstyle \omega (\psi-\tau_b) +\phi_{b}{\scriptstyle(\psi)} + \omega \psi+\phi_{b}{\scriptstyle(\psi)}}) {\color{brown} + } \cos( {\scriptstyle \omega (\psi-\tau_b) +\phi_{b}{\scriptstyle(\psi)} - \omega \psi-\phi_{b}{\scriptstyle(\psi)}}) d\psi \\ \\ = & \tfrac{1}{2}M^2(l_a) \frac{1}{T} \int_{t-T/2}^{t+T/2} \cos( {\scriptstyle \omega (\psi-\tau_a) +\phi_{a}{\scriptstyle(\psi)} +\omega \psi +\phi_{a}{\scriptstyle(\psi)} }) d\psi {\color{brown} + } \tfrac{1}{2}M^2(l_a) \frac{1}{T} \int_{t-T/2}^{t+T/2} \cos({\scriptstyle \omega (\psi-\tau_a) +\phi_{a}{\scriptstyle(\psi)} -\omega \psi -\phi_{a}{\scriptstyle(\psi)}}) d\psi \\ &+\tfrac{1}{2} M(l_a)M(l_b) \frac{1}{T} \int_{t-T/2}^{t+T/2} \cos( {\scriptstyle \omega (\psi-\tau_a) +\phi_{a}{\scriptstyle(\psi)} +\omega \psi+\phi_{b}{\scriptstyle(\psi)} }) d\psi {\color{brown} + }\tfrac{1}{2}M(l_a)M(l_b) \frac{1}{T} \int_{t-T/2}^{t+T/2} \cos( {\scriptstyle \omega (\psi-\tau_a) +\phi_{a}{\scriptstyle(\psi)} - \omega \psi-\phi_{b}{\scriptstyle(\psi)}}) d\psi \\ &+\tfrac{1}{2} M(l_b)M(l_a)\frac{1}{T} \int_{t-T/2}^{t+T/2} \cos( {\scriptstyle \omega (\psi-\tau_b) +\phi_{b}{\scriptstyle(\psi)}+ \omega \psi+\phi_{a}{\scriptstyle(\psi)}})d\psi {\color{brown} + } \tfrac{1}{2}M(l_b)M(l_a) \frac{1}{T} \int_{t-T/2}^{t+T/2} \cos( {\scriptstyle \omega (\psi-\tau_b) +\phi_{b}{\scriptstyle(\psi)}-\omega \psi-\phi_{a}{\scriptstyle(\psi)}}) d\psi \\ &+\tfrac{1}{2}M^2(l_b) \frac{1}{T} \int_{t-T/2}^{t+T/2} \cos( {\scriptstyle \omega (\psi-\tau_b) +\phi_{b}{\scriptstyle(\psi)} + \omega \psi+\phi_{b}{\scriptstyle(\psi)}}) d\psi {\color{brown} + } \tfrac{1}{2}M^2(l_b) \frac{1}{T} \int_{t-T/2}^{t+T/2} \cos( {\scriptstyle \omega (\psi-\tau_b) +\phi_{b}{\scriptstyle(\psi)} - \omega \psi-\phi_{b}{\scriptstyle(\psi)}}) d\psi \\ \\ \\ = & \tfrac{1}{2}M^2(l_a) \underbrace{ \frac{1}{T} \int_{t-T/2}^{t+T/2} \cos({\scriptstyle 2\omega\psi-\omega \tau_a+2 \phi_a(\psi) })d\psi }_{=0}\; {\color{brown} + } \; \tfrac{1}{2}M^2(l_a) \frac{1}{T} \int_{t-T/2}^{t+T/2} \cos({\scriptstyle -\omega \tau_a })d\psi \\ &+\tfrac{1}{2} M(l_a)M(l_b) \underbrace{ \frac{1}{T} \int_{t-T/2}^{t+T/2} \cos({\scriptstyle 2\omega\psi-\omega\tau_a+\phi_a(\psi)+\phi_b(\psi) })d\psi }_{=0}{\color{brown} + }\tfrac{1}{2}M(l_a)M(l_b) \underbrace{ \frac{1}{T} \int_{t-T/2}^{t+T/2} \cos({\scriptstyle -\omega \tau_a +\phi_{a}(\psi) -\phi_b(\psi) })d\psi }_{=0}\\ &+\tfrac{1}{2} M(l_b)M(l_a) \underbrace{ \frac{1}{T} \int_{t-T/2}^{t+T/2} \cos({\scriptstyle 2\omega\psi-\omega\tau_b+\phi_b(\psi)+\phi_a(\psi) }) d\psi}_{=0} {\color{brown} + }\tfrac{1}{2}M(l_a)M(l_b) \underbrace{ \frac{1}{T} \int_{t-T/2}^{t+T/2} \cos({\scriptstyle -\omega \tau_b +\phi_{b}(\psi) -\phi_a(\psi) })d\psi }_{=0}\\ &+ \tfrac{1}{2}M^2(l_b) \underbrace{ \frac{1}{T} \int_{t-T/2}^{t+T/2} \cos({\scriptstyle 2\omega\psi-\omega \tau_b+2 \phi_b(\psi) })d\psi}_{=0} \; {\color{brown} + } \; \tfrac{1}{2}M^2(l_b) \frac{1}{T} \int_{t-T/2}^{t+T/2} \cos({\scriptstyle -\omega \tau_b })d\psi \\ \\(the underlined integral terms = 0 are from Time average of cosine for fast varying random phase :\text{1-0-2} )
For N>2 sources, if you do the same arithmetic above, the result can be a written as the following summation:
v_{R_c}(u) = \sum_{n=1}^{N} I(l_n) \cos(2\pi u l_n)Same result can be obtained for Sine Interferometer:
v_{R_s}(u) = \sum_{n=1}^{N} I(l_n) \sin(2\pi u l_n)1.3.3: 1D FT Visibility
Now suppose we have N \to \infty sources in the sky with 2 stations, then the summation over l_n shown above turn into integrals:
v_{R_c}(u) &= \int_{l} I(l) \cos(2\pi u l) dl\\ v_{R_s}(u) &= \int_{l} I(l) \sin(2\pi u l) dl\\Now we put them together and create a new function V(u) , the Visibility Function:
\text{Visibility}: V(u) &= v_{R_c}(u) - j v_{R_s}(u) \\ &= \int_l I(l) \cos( 2\pi u l) dl - j \int_l I(l) \sin( 2\pi u l) dl \\ &= \int_l I(l) [ \cos( 2\pi u l) -j \sin( 2\pi u l)] dl \\ &= \int_l I(l) e^{-j 2\pi u l} dl \\With
\boxed{ V(u) = \int_l I(l) e^{-j 2\pi u l} dl}From the normal Fourier Transform's time v.s. frequency domain standpoints:
we can view l as time, which corresponds to source s .
we can view I(l) , the sky intensity, as the time-domain signal.
we can view u as frequency, which corresponds to baseline \vec{b} .
So the more V(u) we can get from different u (baseline), the better we can recover the I(l) , the sky intensity plot function.
1.3.4: Phase Center
We have been ignoring an important concept, the Phase Center s_0 . Now let's discuss what it is.
Phase center s_0 , is an abstract mathematical variable that allow us to decide which direction of the sky is the center.
So far all the calculation for cos/sin Interferometer is based on that the phase center s_0 to be vertically upward of the antennas , in other words, the phase center has been assigned to be at l_0=0 in the \bold{l}- axis (l_0 = \sin(0) = 0 ).
One trivial consequence is that if there is a source s located at l=\sin(0)=0 , the same location of the phase center, then:
v_{R_c}(u,l) &= I\cos(2\pi u 0) \\ &= I\\We can move the phase center s_0 to somewhere else. For example, Suppose we want to move the phase center s_0 to a new l_0\neq 0 location like the figure below, we can simply add an artificial delay \tau_0 as shown in the same figure below.
Figure
Next we solve for \tau_0 , which can be calculated using the values of \vec{b} and \hat{s_0} :
\tau_0 = \frac{\vec{b} \cdot \hat{s_0}}{\nu \lambda} \That's it , we plug in the \tau_0 and run the interferometer. With the added \tau_0 , Vola! we have a new phase center! Let's see what happen to v_{R_c} for some source s with a new phase center as shown in the figure below:
Figure
Now let's see the correlator output v_{R_c} :
v_{R_c}(t,u,l,l_0)|_T = \langle x_2(t) , x_1(t) \rangle ={\color{green} \frac{1}{T} \int_{\psi = t-T/2}^{\psi=t+T/2}}{\color{blue} x_2(\psi)x_1(\psi)} \, \color{green}d\psiStep 1: Multiplication:
\color{blue}x_2(\psi)x_1(\psi) =& M^2\cos\big(\omega( \psi-\tau_g ) + \phi_1{\scriptstyle(\psi)} \big)\cos(\omega (\psi-\tau_0) + \phi_1{\scriptstyle(\psi)}) \\ =& M^2\cos\big(\omega( \psi-\tau_g ) + \phi_1{\scriptstyle(\psi)} \big)\cos(\omega (\psi- \tau_0) + \phi_1{\scriptstyle(\psi)}) \\ =& \frac{1}{2}M^2\left[\cos\big(2 \omega \psi- \omega \tau_g + 2\phi_1{\scriptstyle(\psi)} - \omega \tau_0 \big)+\cos(-\omega \tau_g + \omega \tau_0 )\right] \\ \\ &(\text{let } I=\frac{1}{2} M^2 , \text{ where $I$ is referred as Intensity and } I \propto M^2)\\ \\ =& I\cos\big(2 \omega \psi- \omega \tau_g + 2\phi_1{\scriptstyle(\psi)} - \omega \tau_0 \big)+I\cos(-\omega \tau_g + \omega \tau_0 ) \\
Step 2: Average over integration time T :
(The first integral term = 0 is from Time average of cosine for fast varying random phase :\text{1-0-2} )
=& I\cos( -\omega\tau_g + \omega \tau_0 ) \\ =&I \cos(\omega \tau_g - \omega \tau_0 ) \\ =&I \cos\left( 2\pi \frac{\vec b \cdot \hat s}{\lambda} - 2\pi \frac{\vec b \cdot \hat s_0}{\lambda}\right) \\ =&I \cos\left( ul - ul_0\right) \\ =&I \cos\big( u(l - l_0) \big) \\ =& v_{R_c}(u,l,l_0)Once we decide the location of the phase center s_0 , l_0 is fixed, so we can omit the l_0 in v_{R_c}(u,l,l_0) and the equation v_{R_c}(u,l) implies that v_{R_c}(u,l) is shifted by the corresponding l_0 :
v_{R_c}(u,l) = I \cos\big( u(l - l_0) \big)so now if we have a source s that is at location l=l_0 , we see that:
v_{R_c}(u,l_0) &= I \cos\big( u(l_0 - l_0) \big) = I \\The interactive figure below should help you visualize how phase center work, feel free to drag around the blue triangle source \color{blue} s , the brown circle \color{brown} u , and the green triangle phase center \color{green} s_0 :
Figure :
Interactive Phase Center demonstration: draggable {\color{green}\text{phase center } s_0}, {\color{blue}\text{source } s}, {\color{brown}\text{baseline }u}
Same logic applies to Sine Interferometer:
v_{R_s}(u,l) = &I \sin\big( u(l - l_0) \big)In practice, when we change the phase center by adding an artificial delay \tau_0 , we usually also rotate the physical antenna to point at the direction of the new phase center s_0 for getting the best antenna gain around the phase center.
Expanding the phase center idea to the Visibility function V(u) , we have:
V(u) &= \int_l I(l) e^{-j 2\pi u (l-l_0) } dl \\ &=e^{j2\pi u l_0} \int_l I(l) e^{-j 2\pi u}dlSo the effect of changing the phase center on the Visibility function is just a constant phase shift.