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.
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 R_c is a function of u and l :
R_c &= 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 R_s is also a function of u and l :
R_s &= 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 wavelength \lambda but with different amplitudes M(s_n) : M_{s_n}(t) = M(s_n)\cos(\omega t + \phi_n)
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's magnitude function above to:
M_{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:
And with I = \frac{1}{2}M^2 , the ultimate goal is to recover the sky intensity plot like the figure below:
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).
M_{s_a}(t) &= M(l_a) \cos(\omega t +\phi_a{\scriptstyle(t)})\\ M_{s_b}(t) &= M(l_b) \cos(\omega t +\phi_b{\scriptstyle(t)})\\Station b_0 and b_0 & b_1 form a baseline u .
station b_1 receive signal as such :
{\color{blue} M_{b1}(t) } &= {\color{blue} \Large[ }M_{s_a}(t) {\color{blue} \Large +} M_{s_b}(t) {\color{blue} \Large] }\\ &= {\color{blue} \Large[ } M(l_a) \cos(\omega t +\phi_{a}{\scriptstyle(t)}) {\color{blue} \Large + } M(l_b) \cos(\omega t +\phi_{b}{\scriptstyle(t)}) {\color{blue} \Large] }station b_0 receive signal with delays \tau_a and \tau_b for s_a and s_b respectively, compare to b_1 , remember the delays can be negative:
{\color{green} M_{b0}(t) } &= {\color{green} \Large[ } M_{s_a}(t-\tau_{a}) {\color{green} + } M_{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 R_c(u) :
1. Multiplication:
{\color{green}M_{b0}(t)}{\color{blue}M_{b1}(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] } \\ = {\color{red} \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)}}) {\color{red} \Large] }\\2. Average:
R_c(u)=& {\color{red} E\Large[ }M_{b0}(t)M_{b1}(t) {\color{red}\Large]}\\ = &{\color{red} E\Large[ }M^2(l_a) \cos( {\scriptstyle \omega (t-\tau_a) +\phi_{a}{\scriptstyle(t)}}) \cos({\scriptstyle \omega t +\phi_{a}{\scriptstyle(t)}}) {\color{red}\Large]}\\ &+{\color{red} E\Large[ } M(l_a)M(l_b) \cos( {\scriptstyle \omega (t-\tau_a) +\phi_{a}{\scriptstyle(t)}}) \cos( {\scriptstyle \omega t+\phi_{b}{\scriptstyle(t)}}){\color{red}\Large]}\\ &+{\color{red} E\Large[ } M(l_b)M(l_a) \cos( {\scriptstyle \omega (t-\tau_b) +\phi_{b}{\scriptstyle(t)}}) \cos( {\scriptstyle \omega t+\phi_{a}{\scriptstyle(t)}}){\color{red}\Large]}\\ &+{\color{red} E\Large[ } M(^2l_b) \cos( {\scriptstyle \omega (t-\tau_b) +\phi_{b}{\scriptstyle(t)}}) \cos( {\scriptstyle \omega t+\phi_{b}{\scriptstyle(t)}}) {\color{red}\Large]}\\ \\ = &\tfrac{1}{2}M^2(l_a) {\color{red} E\Large[ } \cos( {\scriptstyle \omega (t-\tau_a) +\phi_{a}{\scriptstyle(t)} +\omega t +\phi_{a}{\scriptstyle(t)} }) {\color{brown} + } \cos({\scriptstyle \omega (t-\tau_a) +\phi_{a}{\scriptstyle(t)} -\omega t -\phi_{a}{\scriptstyle(t)}}) {\color{red}\Large]}\\ &+\tfrac{1}{2} M(l_a)M(l_b) {\color{red} E\Large[ } \cos( {\scriptstyle \omega (t-\tau_a) +\phi_{a}{\scriptstyle(t)} +\omega t+\phi_{b}{\scriptstyle(t)} }) {\color{brown} + } \cos( {\scriptstyle \omega (t-\tau_a) +\phi_{a}{\scriptstyle(t)} - \omega t-\phi_{b}{\scriptstyle(t)}}){\color{red}\Large]}\\ &+\tfrac{1}{2} M(l_b)M(l_a){\color{red} E\Large[ } \cos( {\scriptstyle \omega (t-\tau_b) +\phi_{b}{\scriptstyle(t)}+ \omega t+\phi_{a}{\scriptstyle(t)}}){\color{brown} + } \cos( {\scriptstyle \omega (t-\tau_b) +\phi_{b}{\scriptstyle(t)}-\omega t-\phi_{a}{\scriptstyle(t)}}) {\color{red}\Large]}\\ &+\tfrac{1}{2}M^2(l_b) {\color{red} E\Large[ } \cos( {\scriptstyle \omega (t-\tau_b) +\phi_{b}{\scriptstyle(t)} + \omega t+\phi_{b}{\scriptstyle(t)}}) {\color{brown} + } \cos( {\scriptstyle \omega (t-\tau_b) +\phi_{b}{\scriptstyle(t)} - \omega t-\phi_{b}{\scriptstyle(t)}}) {\color{red}\Large]}\\ \\ = &\tfrac{1}{2}M^2(l_a) {\color{red} E\Large[ } \cos( {\scriptstyle \omega (t-\tau_a) +\phi_{a}{\scriptstyle(t)} +\omega t +\phi_{a}{\scriptstyle(t)} }) {\color{brown} + } \cos({\scriptstyle -\omega \tau_a}) {\color{red}\Large]}\\ &+\tfrac{1}{2} M(l_a)M(l_b) {\color{red} E\Large[ } \cos( {\scriptstyle \omega (t-\tau_a) +\phi_{a}{\scriptstyle(t)} +\omega t+\phi_{b}{\scriptstyle(t)} }){\color{brown} + } \cos( {\scriptstyle -\omega \tau_a +\phi_{a}{\scriptstyle(t)} -\phi_{b}{\scriptstyle(t)}}){\color{red}\Large]}\\ &+\tfrac{1}{2} M(l_b)M(l_a){\color{red} E\Large[ } \cos( {\scriptstyle \omega (t-\tau_b) +\phi_{b}{\scriptstyle(t)}+ \omega t+\phi_{a}{\scriptstyle(t)}}){\color{brown} + } \cos( {\scriptstyle - \omega \tau_b +\phi_{b}{\scriptstyle(t)}-\phi_{a}{\scriptstyle(t)}}) {\color{red}\Large]}\\ &+\tfrac{1}{2}M^2(l_b) {\color{red} E\Large[ } \cos( {\scriptstyle \omega (t-\tau_b) +\phi_{b}{\scriptstyle(t)} + \omega t+\phi_{b}{\scriptstyle(t)}}) {\color{brown} + } \cos( {\scriptstyle -\omega \tau_b }) {\color{red}\Large]}\\ \\ = &\tfrac{1}{2}M^2(l_a) {\color{red} E\Large[ } \xcancel{ \cos( {\scriptstyle \omega (t-\tau_a) +\phi_{a}{\scriptstyle(t)} +\omega t +\phi_{a}{\scriptstyle(t)} }) } {\color{brown} + } \cos({\scriptstyle -\omega \tau_a}) {\color{red}\Large]}\\ &+\tfrac{1}{2} M(l_a)M(l_b) \xcancel{ {\color{red} E\Large[ } \cos( {\scriptstyle \omega (t-\tau_a) +\phi_{a}{\scriptstyle(t)} +\omega t+\phi_{b}{\scriptstyle(t)} }){\color{brown} + } \cos( {\scriptstyle -\omega \tau_a +\phi_{a}{\scriptstyle(t)} -\phi_{b}{\scriptstyle(t)}}){\color{red}\Large]} }\\ &+\tfrac{1}{2} M(l_b)M(l_a) \xcancel{ {\color{red} E\Large[ } \cos( {\scriptstyle \omega (t-\tau_b) +\phi_{b}{\scriptstyle(t)}+ \omega t+\phi_{a}{\scriptstyle(t)}}){\color{brown} + } \cos( {\scriptstyle - \omega \tau_b +\phi_{b}{\scriptstyle(t)}-\phi_{a}{\scriptstyle(t)}}) {\color{red}\Large]} }\\ &+\tfrac{1}{2}M^2(l_b) {\color{red} E\Large[ } \xcancel{ \cos( {\scriptstyle \omega (t-\tau_b) +\phi_{b}{\scriptstyle(t)} + \omega t+\phi_{b}{\scriptstyle(t)}}) } {\color{brown} + } \cos( {\scriptstyle -\omega \tau_b }) {\color{red}\Large]}\\ \\ &(\text{with } \omega \tau_g = 2\pi u l) \\ \\ &= \tfrac{1}{2}M^2(l_a) \cos(\omega \tau_a) + \tfrac{1}{2}M^2(l_b) \cos(\omega \tau_b) \\ &= I(l_a) \cos(2\pi u l_a) + I(l_b) \cos(2\pi u l_b) \\For N>2 sources, if you do the same arithmetic above, the result can be a written as the following summation:
R_c(u) = \sum_{n=1}^{N} I(l_n) \cos(2\pi u l_n)Same result can be obtained for Sine Interferometer:
R_s(u) = \sum_{n=1}^{N} I(l_n) \sin(2\pi u l_n)1.3.3: 1D Fourier Transform
Now suppose we have \infty sources in the sky all beaming their signals to the 2 stations, then the summations above turn into integrals:
R_c(u) &= \int_{l} I(l) \cos(2\pi u l) dl\\ 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) &= R_c(u) - j 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:
R_c &= 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.
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 R_c for some source s with a new phase center as shown in the figure below:
The calculation above shows that a new phase center s_0 shifts R_c(u,l) by the corresponding l_0 :
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:
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 :
Same logic applies to Sine Interferometer:
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.