The analysis on monochromatic interferometry is nice and all but is not practical. In practice, due to the limitation of physics, hardware, and DSP algorithm, the observation is always made over a narrow bandwidth 2\Delta\omega centered at \omega_c instead of a single frequency. \omega_c \pm \Delta\omega

So with that, let's redo the calculation for cosine interferometry output (sine interferometer can be easily obtained, so we will skip the derivation for sine):

First the received signal at a station x_1(t) is always a bandpass-filtered signal of the sky.

Then we use the assumptions at page 63 from the book :

Assuming that the receiving system's bandpass filter is a rectangular shape with passband \omega_c \pm \Delta\omega , then we can write our x_1(t) as

x_1(t) &= a(t)\cos(\omega_c t + \phi(t)) \\ &= \mathcal{Re}\{ a(t) e^{j\phi(t)} e^{j\omega t} \} \\ &= \mathcal{Re}\{ A(t) e^{j\omega_c t} \}, \; \text{where } A(t) =a(t) e^{j\phi(t)} \\

A(t) is a band limited (2\Delta\omega ) wide-sense stationary the complex baseband envelope with constant PSD, S_{A}(\omega) = M^2 .

Before we proceed, we insert an artificial delay \tau_0 at station b_1 (known as phase center) shown in the figure below. You will see why adding that \tau_0 is necessary.

so:

Now let's see the correlation calculation:

\large v_{R_c}(\bar t,\tau_g,\tau_0)|_T =& \large \frac{1}{2T} \int_{\bar t-T}^{\bar t+T} x_1(t-\tau_0) x_2^*(t) dt \\ =& \large \frac{1}{2T} \int_{\bar t-T}^{\bar t+T} \mathcal{Re} \left\{ A(t-\tau_0)e^{j\omega_c(t-\tau_0) } A^*(t-\tau_g) e^{-j\omega_c(t-\tau_g)} \right\} dt \\ =& \large \frac{1}{2T} \mathcal{Re}\left\{ \int_{\bar t-T}^{\bar t+T} e^{j\omega_c(\tau_g-\tau_0) } A(t-\tau_0) A^*(t-\tau_g) dt \right\} \\ =& \large \mathcal{Re}\left\{ e^{j\omega_c(\tau_g-\tau_0) } {\color{blue} \frac{1}{2T} \int_{\bar t-T}^{\bar t+T} A(t-\tau_0) A^*(t-\tau_g) dt } \right\} \\ \color{blue} \frac{1}{2T} \int_{\bar t-T}^{\bar t+T} A(t-\tau_0) A^*(t-\tau_g) dt &= \text{auto-correlation of $A(t)$ at $\Delta \tau$} = r_{nn}(\Delta \tau), \text{where } \Delta \tau = \tau_0-\tau_g\\ \color{blue} \text{using Wiener-Khinchin theorem } &: F^{-1}\{ PSD_A(\omega) \}(\tau) = r_{nn}(\tau)\\ =& \large \mathcal{Re}\left\{ e^{j\omega_c \Delta \tau } {\color{blue} F^{-1}\{ PSD_A(\omega) \}(\Delta \tau) } \right\} \\ =& \large \mathcal{Re}\left\{ e^{j\omega_c \Delta \tau } {\color{blue} (\frac{1}{2\pi} \int_{-\Delta \omega}^{\Delta \omega} M^2 e^{j\omega \Delta \tau } dw) }\right \} \\ =& \large \frac{1}{2\pi} \mathcal{Re} \{ e^{j\omega_c \Delta \tau }M^2 2\Delta \omega \text{sinc} (\Delta \omega \Delta \tau) \}\\ =& \large \frac{M^2}{\pi} \Delta \omega \cos(\omega_c \Delta \tau) \text{sinc} (\Delta \omega \Delta \tau) \\ &(\text{set }I = \frac{M^2}{\pi} )\\ =& \large I \Delta \omega \cos(\omega_c \Delta \tau) \text{sinc} (\Delta \omega \Delta \tau) \\

As you see in the final result, \bar t is irrelevant

The final quasi-monochromatic cosine interferometer visibility is:

v_{R_c}(\tau_g,\tau_0)|_T= &I \Delta\omega \cos(\omega_c (\tau_g-\tau_0) ) \text{sinc}\left( \Delta\omega (\tau_g-\tau_0) \right) \\

Here now you can see that the artificial delay \tau_0 is necessary and should try to be as close to the geometric delay \tau_g as possible to achieve \text{sinc}(\Delta\omega (\tau_g-\tau_0) ) = \text{sinc}(0) = 1 . This artificial delay \tau_0 is referred as Phase Center, we will go over more about it later.

Next we apply the geometric tricks we have shown you in 1.1 $\cos$ Interferometer - 1.1.4: Putting It All Together :

so:

v_{R_c}(\tau_g,\tau_0) &=I \Delta\omega \;\cos(\omega_c (\tau_g-\tau_0) ) \text{sinc}\left( \Delta\omega (\tau_g-\tau_0) \right)\\ \\ &=I \Delta\omega \;\cos(2\pi u (l_g-l_0) ) \text{sinc}\left( \Delta\omega\frac{ |\vec{b}|}{c} (l_g-l_0) \right) \\ &= v_{R_c}(l_g,l_0)

Compare to the visibility of monochromatic interferometer with phase center: v_{R_c(mono)}(\tau_g,\tau_0) &= I \;\cos(\omega_c (\tau_g-\tau_0) )\\ &= I \;\cos(2\pi u (l_g-l_0) )\\ &= v_{R_c(mono)}(l_g,l_0)

The visibility of quasi-monochromatic interferometer has this extra term of \Delta\omega \text{sinc}\left( \Delta\omega (\tau_g-\tau_0) \right)

In real VLBI observation, you insert the \tau_0 by delaying station b_1 's signal when calculating the correlation.

The interactive figure below shows that with the introduction of \text{sinc}\left( \Delta\omega (\tau_g-\tau_0) \right) , you can see that if the phase center \color{green}s_0 is not close to source \color{blue}s , the Visibility value v_{R_c} (the blue dot \color{blue}\cdot\text{ } ) is easily squashed to 0.

Now let's see the case for 2 sources s_x,s_y for station b_1,b_2 :

b_1 signal:

\large x_1(t) = \mathcal{Re}\{ A_x(t) e^{j\omega_c t} + A_y(t) e^{j\omega_c t} \}, \; \text{where } A_x(t) =a_x(t) e^{j\phi_x(t)}, A_y(t) =b_y(t) e^{j\phi_y(t)} \\

b_2 signal:

\large x_2(t) = \mathcal{Re}\{ A_x(t-\tau_x) e^{j\omega_c (t-\tau_x)} + A_y(t-\tau_y) e^{j\omega_c( t-\tau_y)} \}

Now let's calculate the correlation with the phase center \tau_0 :

v_{R_c}(\tau_x, \tau_y, \tau_0) =& \langle {\color{red} x_1(t-\tau_0)} , {\color{brown} x_2(t)} \rangle \\ =& \large \frac{1}{2T} \int_{-T}^{T} \mathcal{Re} \left\{ ({\color{red} A_x(t-\tau_0)e^{j\omega_c(t-\tau_0) } + A_y(t-\tau_0) e^{j\omega_c(t-\tau_0)} } ) ({\color{brown} A_x^*(t-\tau_x)e^{-j\omega_c(t-\tau_x) } + A_y^*(t-\tau_y) e^{-j\omega_c(t-\tau_y)} } ) \right\} dt \\ =& \large \frac{1}{2T} \int_{-T}^{T} \mathcal{Re} \left\{ A_x(t-\tau_0)e^{j\omega_c(t-\tau_0) } A_x^*(t-\tau_x)e^{-j\omega_c(t-\tau_x) } + A_y(t-\tau_0) e^{j\omega_c(t-\tau_0)} A_x^*(t-\tau_x)e^{-j\omega_c(t-\tau_x) } + A_x(t-\tau_0)e^{j\omega_c(t-\tau_0) }A_y^*(t-\tau_y) e^{-j\omega_c(t-\tau_y)} + A_y(t-\tau_0) e^{j\omega_c(t-\tau_0)} A_y^*(t-\tau_y) e^{-j\omega_c(t-\tau_y)} \right\} dt \\ =& \large \frac{1}{2T} \int_{-T}^{T} \mathcal{Re} \left\{ A_x(t-\tau_0) A_x^*(t-\tau_x) e^{j\omega_c(\tau_x-\tau_0) } + A_y(t-\tau_0) A_x^*(t-\tau_x) e^{j\omega_c(\tau_x-\tau_0) } + A_x(t-\tau_0) A_y^*(t-\tau_y) e^{j\omega_c(\tau_y-\tau_0)} + A_y(t-\tau_0)A_y^*(t-\tau_y) e^{j\omega_c(\tau_y-\tau_0)} \right\} dt \\ =& \mathcal{Re} \left\{ e^{j\omega_c(\tau_x-\tau_0) } \frac{1}{2T} \int_{-T}^{T} A_x(t-\tau_0) A_x^*(t-\tau_x) dt \right\} + \mathcal{Re} \left\{e^{j\omega_c(\tau_x-\tau_0) } \underbrace{ \frac{1}{2T}\int_{-T}^{T} A_y(t-\tau_0) A_x^*(t-\tau_x) dt}_{ = 0 \; \because \; A_x , A_y \text{ are uncorrelated} } \right\} \\ & + \mathcal{Re} \left\{ e^{j\omega_c(\tau_y-\tau_0)} \underbrace{ \frac{1}{2T}\int_{-T}^{T} A_x(t-\tau_0) A_y^*(t-\tau_y) dt }_{ = 0 \; \because \; A_x , A_y \text{ are uncorrelated} } \right\} + \mathcal{Re} \left\{ e^{j\omega_c(\tau_y-\tau_0)} \frac{1}{2T} \int_{-T}^{T} A_y(t-\tau_0)A_y^*(t-\tau_y)dt \right\} \\ \\ &\text{by following the steps for single source $\cos$ interferometer calculation in 2.0.1 , we have:}\\ \\ =& \large I_x \Delta \omega \cos(\omega_c (\tau_x-\tau_0)) \text{sinc} (\Delta \omega (\tau_x-\tau_0)) + \large I_y \Delta \omega \cos(\omega_c (\tau_y-\tau_0)) \text{sinc} (\Delta \omega (\tau_y-\tau_0)) \\

So For N>2 sources, you can just simply add them up:

V_{R_c}(\tau_0)|_T =& \sum_{n=1}^{N} I_n\Delta\omega\cos(\omega_c (\tau_{n}-\tau_0)) \text{sinc}\left( \Delta\omega (\tau_{n}-\tau_0) \right)\\ \\ &(\text{with $l$ notation} )\\ \\ =& \sum_{n=1}^{N} I_n \Delta\omega \;\cos(2\pi u (l_{n}-l_0) ) \text{sinc}\left( \Delta\omega\frac{ |\vec{b}|}{c} (l_{n}-l_0) \right) \\ \\ &(\text{define $l_n$ as : } l_n = l_{n}-l_0 )\\ \\ =& \sum_{n=1}^{N} I_n \Delta\omega \;\cos(2\pi u l_{n} ) \text{sinc}\left( \Delta\omega\frac{ |\vec{b}|}{c} l_n \right) \\

Now suppose we have N \to \infty sources in the sky with 2 stations, then then the summation over l_n above turn into integrals:

v_{R_c}(u) &=\Delta\omega \int_{l} I(l) \cos(2\pi u l) \text{sinc}\left( \Delta\omega \frac{|\vec{b}|}{c} l\right) dl\\ v_{R_s}(u) &= \Delta\omega \int_{l} I(l) \sin(2\pi u l) \text{sinc}\left( \Delta\omega \frac{|\vec{b}|}{c} l\right) dl\\

Suppose that:

• we are looking only at the line segment that is close to the phase center thus l\approx 0
• bandwidth \Delta\omega is small

Then we can do the following approximation:

\text{sinc}\left( \frac{\Delta\omega }{2}\frac{|\vec{b}|}{c} l\right) \approx 1

so with the 2 assumptions above, we have: v_{R_c}(u) &\approx \Delta\omega \int_{l} I(l) \cos(2\pi u l) dl\\ v_{R_s}(u) &\approx \Delta\omega \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) \\ &= \Delta\omega \int_l I(l) \cos( 2\pi u l) dl - j \Delta\omega \int_l I(l) \sin( 2\pi u l) dl \\ &= \Delta\omega \int_l I(l) [ \cos( 2\pi u l) -j \sin( 2\pi u l)] dl \\ &= \Delta\omega \int_l I(l) e^{-j 2\pi u l} dl \\

With

\boxed{ V(u) = \Delta\omega \int_l I(l) e^{-j 2\pi u l} dl}