For this section, we will use analytic signal for cleaner mathematical presentation.

Given two stations that form a baseline vector \vec{b}=(u,v) . Now suppose the bandpassed RF signal arriving at the uv -plane with earth center as origin (0,0,0) is:

E(t) &= A(t) e^{j \omega_{rf}(t) } \\

Keep in mind that A(t) is a band-limited ergodic stationary process, and the N points FFT of A(t) is:

A(t) \overset{\mathcal{F}}{\rarr} \xi_{\nu}[k]

And the visibility output from the correlator for this baseline \vec{b}=(u,v) is:

\large \color{navy} V_{\nu}(u,v)

Notice that A(t) is a band-limited ergodic stationary random process.

The dataflow below shows the simple FX correlator operation on data from station i,j that form baseline vector (u,v) :

&\boxed{\textcircled{1}_i \; \text{RF signal arriving at station $i$ with negative delay } -\tau_i } \\ &\Downarrow \\ E_i(t) &= A(t+\tau_i) e^{j \omega_{rf}(t+\tau_i) } \\ &\Downarrow \\ &\boxed{\textcircled{2}_i \; {\color{blue} \omega_{lo} } \text{ mixer } } \\ &\Downarrow \\ E_{ilo}(t) &= A(t+\tau_i) e^{j \omega_{rf}(t+\tau_i) } {\color{blue} e^{-j\omega_{lo}t} } \\ &= A(t+\tau_i) e^{j \omega_{rf}\tau_i} e^{j(\omega_{rf}-\omega_{lo})t} \\ &\Downarrow \\ &\boxed{\textcircled{3}_i \; \text{Baseband conversion}} \\ &\Downarrow \\ E_{ib}(t) &= A(t+\tau_i) e^{j \omega_{rf}\tau_i} e^{j(\omega_{rf}-\omega_{lo})t} {\color{blue} e^{-j(\omega_{rf}-\omega_{lo})t} }\\ &= A(t+\tau_i) e^{j \omega_{rf}\tau_i}\\ &\Downarrow \\ &\boxed{\textcircled{4}_i \; \text{Delay Compensation } \color{blue} \tau_i} \\ &\Downarrow \\ E_{\tau_i}(t) &= A(t+\tau_i {\color{blue} -\tau_i } ) e^{j \omega_{rf}\tau_i}\\ &= A(t ) e^{j \omega_{rf}\tau_i}\\ &\boxed{\textcircled{5}_i \; \text{Phase Compensation (Fringe Stopping) } \color{blue} \theta_i = \omega_{rf} \tau_i} \\ &\Downarrow \\ E_{\theta_i}(t) &=A(t ) e^{j \omega_{rf}\tau_i} \color{blue} e^{-j \omega_{rf}\tau_i}\\ &= A(t )\\ &\Downarrow \\ &\boxed{\textcircled{6}_i \; \text{F Step: } N\text{ points }FFT \text{ on } E_{\theta_i}(t) } \\ &\large{\Downarrow_{k=0} } \;\;\;\;\; \cdot \cdot\cdot \cdot \; \;\;\;\; \Downarrow \;\;\;\; \Downarrow \;\;\;\; \Downarrow \;\; \cdot \cdot \cdot \;\;\Downarrow_{k=N-1} \\ &\large \xi_{i\nu}[k] \\ &\Large \Downarrow \\ \boxed{ \textcircled{7}_{ij} \; \text{X Step: Correlation} } \; & \Large \otimes \Rightarrow V[k] = \langle \xi_{i\nu}[k] , \xi_{j\nu}[k] \rangle = \color{navy} V_{\nu_k+\nu_{rf}}(u,v)\\ &\Large\Uparrow \\ &\large \xi_{j\nu}[k] \\ &\large{\Uparrow_{k=0} } \;\;\;\;\; \cdot \cdot\cdot \cdot \; \;\;\;\; \Uparrow \;\;\;\; \Uparrow \;\;\;\; \Uparrow \;\; \cdot \cdot \cdot \;\; \Uparrow_{k=N-1} \\ &\boxed{\textcircled{6}_j \; \text{F Step: } N\text{ points }FFT \text{ on } E_{\theta_j}(t) } \\ &\Uparrow \\ &\; \large\vdots\\ &\Uparrow \\ E_j(t) &= A(t+\tau_j) e^{j \omega_{rf}(t+\tau_j) } \\ &\Uparrow \\ &\boxed{\textcircled{1}_j \; \text{RF signal arriving at station $j$ with negative delay } -\tau_j } \\ &\Uparrow \\ E(t) &= A(t) e^{j \omega_{rf}t }\\

As you see, after the \boxed{ \textcircled{7}_{ij} \; \text{X Step: Correlation} } step, you will have N frequencies of V_{\nu-\nu_{rf} } (u,v)

Usually, after \boxed{\textcircled{3}_i \; \text{Baseband conversion}} , we are in the digital domain, so the station based delay \tau_i has to be represented as sum of integer delay and fractional delay: \color{blue} \tau_i = n_{\tau_i} + \delta_i . The fractional delay can be treated as phase shift using Fourier time shifting property which can be mitigated after the FFT. See the data flow below.

&\boxed{\textcircled{1}_i \; \text{RF signal arriving at station $i$ with negative delay } -\tau_i } \\ &\Downarrow \\ E_i(t) &= A(t+\tau_i) e^{j \omega_{rf}(t+\tau_i) } \\ &\Downarrow \\ &\boxed{\textcircled{2}_i \; {\color{blue} \omega_{lo} } \text{ mixer } } \\ &\Downarrow \\ E_{ilo}(t) &= A(t+\tau_i) e^{j \omega_{rf}(t+\tau_i) } {\color{blue} e^{-j\omega_{lo}t} } \\ &= A(t+\tau_i) e^{j \omega_{rf}\tau_i} e^{j(\omega_{rf}-\omega_{lo})t} \\ &\Downarrow \\ &\boxed{\textcircled{3}_i \; \text{Baseband conversion + } \bold{Digitizing} } \\ &\Downarrow \\ E_{ib}[n] &= A[n+(n_{\tau_i}+\delta_i)] e^{j \omega_{rf}\tau_i} e^{j(\omega_{rf}-\omega_{lo})n} {\color{blue} e^{-j(\omega_{rf}-\omega_{lo})n} }\\ &= A[n+(n_{\tau_i}+\delta_i)] e^{j \omega_{rf}\tau_i}\\ &\Downarrow \\ &\boxed{\textcircled{4}_i \; \text{Discrete-time Delay Compensation } \color{blue} n_{\tau_i}} \\ &\Downarrow \\ E_{n_i}[n] &= A[n+(n_{\tau_i}+\delta_i) {\color{blue} -n_{\tau_i}} ] e^{j \omega_{rf}\tau_i } \\ &= A[n {\color{green} +\delta_i} ] e^{j \omega_{rf}\tau_i } \\ &\Downarrow \\ &\boxed{\textcircled{5}_i \; \text{Phase Compensation (Fringe Stopping) } \color{blue} \theta_i = \omega_{rf} \tau_i} \\ &\Downarrow \\ E_{\theta_i}[n] &=A[n {\color{green} +\delta_i}]e^{j \omega_{rf}\tau_i} \color{blue} e^{-j \omega_{rf}\tau_i}\\ &= A[n {\color{green} +\delta_i}]\\ &\Downarrow \\ &\boxed{\textcircled{6}_i \; \text{F Step: } N\text{ points }FFT \text{ on } E_{\theta_i}[n] } \\ &\large{\Downarrow_{k=0} } \;\;\;\;\; \cdot \cdot\cdot \cdot \; \;\;\;\; \Downarrow \;\;\;\; \Downarrow \;\;\;\; \Downarrow \;\; \cdot \cdot \cdot \;\;\Downarrow_{k=N-1} \\ &\large \xi_{i\nu}[k] {\color{green} e^{j { 2\pi \over N } k \delta_i} }\\ &\Large \Downarrow \\ & \Large \otimes \large \leftarrow {\color{green} e^{-j { 2\pi \over N } k \delta_i} } \normalsize \;\;\boxed{ \textcircled{6a}_i \;\text{Fractional Delay Compensation} } \\ &\Large \Downarrow \\ &\large \xi_{i\nu}[k]\\ &\Large \Downarrow \\ \boxed{ \textcircled{7}_{ij} \; \text{X Step: Correlation} } \; & \Large \otimes \Rightarrow V[k] = \langle \xi_{i\nu}[k] , \xi_{j\nu}[k] \rangle = \color{navy} V_{\nu_k+\nu_{rf}}(u,v)\\ &\Large\Uparrow \\ &\large \xi_{j\nu}[k] \\ &\Large \Uparrow \\ & \Large \otimes \large \leftarrow {\color{black} e^{-j { 2\pi \over N } k \delta_j} }\normalsize \;\;\boxed{ \textcircled{6a}_j \;\text{Fractional Delay Compensation} } \\ &\Large \Uparrow \\ &\large \xi_{j\nu}[k] {\color{black} e^{j { 2\pi \over N } k \delta_j} }\\ &\large{\Uparrow_{k=0} } \;\;\;\;\; \cdot \cdot\cdot \cdot \; \;\;\;\; \Uparrow \;\;\;\; \Uparrow \;\;\;\; \Uparrow \;\; \cdot \cdot \cdot \;\; \Uparrow_{k=N-1} \\ &\boxed{\textcircled{6}_j \; \text{F Step: } N\text{ points }FFT \text{ on } E_{\theta_j}[n] } \\ &\Uparrow \\ &\; \large\vdots\\ &\Uparrow \\ E_j(t) &= A(t+\tau_j) e^{-j \omega_{rf}(t+\tau_j) } \\ &\Uparrow \\ &\boxed{\textcircled{1}_j \; \text{RF signal arriving at station $j$ with negative delay } -\tau_j } \\ &\Uparrow \\ E(t) &= A(t) e^{-j \omega_{rf}t }\\

Let's see how FX correlator works.

The figure below shows how to obtain the visibility of one cycle of FX operation.

After {\color{red}M} integration time, we then have a visibility grid, going from V[k,{\color{red}0}] to V[k,{\color{red}M-1}] : \text{Visibility Grid : }& V[k,{\color{red}m}] \\ \\ & k \text{ is the frequency index }, k \in [0,N-1] \\ & {\color{red}m} \text{ is the time index }, {\color{red}m} \in [{\color{red}0},{\color{red}M-1}] \\

The figure below shows a visual representation of the V[k,{\color{red}m}] grid.

When we calculate the phase center delay for station i as n_{\tau_i} + \delta_i as close to the true delay \tau_i as we can, unfortunately, there is always a residual error \color{red} \epsilon_i such that:

{\color{red}\epsilon_i }= \tau_i - (n_{\tau_i} + \delta_i )

With this error value \color{red}\epsilon_i , let's see how it affects the FX correlator dataflow below:

& \text{station }i\\ &\; \; \vdots\\ &\Downarrow \\ &\boxed{\textcircled{5}_i \; \text{Phase Compensation (Fringe Stopping) } \color{blue} \theta_i = \omega_{rf} (n_{\tau_i} + \delta_i)} \\ &\Downarrow \\ E_{\theta_i}[n] &=A[n {\color{green} +\delta_i } {\color{red}+\epsilon_i } ]e^{j \omega_{rf}\tau_i} \color{blue} e^{-j \omega_{rf}(n_{\tau_i} + \delta_i)}\\ &= A[n {\color{green} +\delta_i} {\color{red}+\epsilon_i } ] \color{brown}e^{j \omega_{rf}\epsilon_i} \\ &\Downarrow \\ &\boxed{\textcircled{6}_i \; \text{F Step: } N\text{ points }FFT \text{ on } E_{\theta_i}[n] } \\ &\large{\Downarrow_{k=0} } \;\;\;\;\; \cdot \cdot\cdot \cdot \; \;\;\;\; \Downarrow \;\;\;\; \Downarrow \;\;\;\; \Downarrow \;\; \cdot \cdot \cdot \;\;\Downarrow_{k=N-1} \\ &\large \xi_{i\nu}[k] {\color{green} e^{j { 2\pi \over N } k (\delta_i {\color{red} + \epsilon_i) } } } \color{brown}e^{j \omega_{rf}\epsilon_i} \\ &\Large \Downarrow \\ & \Large \otimes \large \leftarrow {\color{green} e^{-j { 2\pi \over N } k \delta_i} } \normalsize \;\;\boxed{ \textcircled{6a}_i \;\text{Fractional Delay Compensation} } \\ &\Large \Downarrow \\ &\large \xi_{i\nu}[k] {\color{red} e^{j { 2\pi \over N } k \epsilon_i } } \color{brown}e^{j \omega_{rf}\epsilon_i} \\ &\Large \Downarrow \\ \boxed{ \textcircled{7}_{ij} \; \text{X Step: Correlation} } \; & \large \begin{matrix} \otimes \Rightarrow V[k] &=& \langle \xi_{i\nu}[k] {\color{red} e^{j { 2\pi \over N } k \epsilon_i } }{\color{brown}e^{j \omega_{rf}\epsilon_i}} , \xi_{j\nu}[k] {\color{red} e^{j { 2\pi \over N } k \epsilon_j } } {\color{brown}e^{j \omega_{rf}\epsilon_j}} \rangle \\ &=& \braket{ \xi_{i\nu}[k] \xi_{j\nu}^*[k] {\color{red} e^{j { 2\pi \over N } k (\epsilon_i-\epsilon_j) } }{\color{brown}e^{j \omega_{rf}(\epsilon_i-\epsilon_j)}} } \end{matrix} \\ &\Large\Uparrow \\ &\large \xi_{j\nu}[k] {\color{red} e^{j { 2\pi \over N } k \epsilon_j} } { \color{brown}e^{j \omega_{rf}\epsilon_j} } \\ &\Large\Uparrow \\ &\; \;\vdots\\ &\text{station }j

Looking at the correlation output V[k] :

\large V[k] = \braket{ \xi_{i\nu}[k] \xi_{j\nu}^*[k] {\color{red} e^{j { 2\pi \over N } k (\epsilon_i-\epsilon_j) } } { \color{brown}e^{j \omega_{rf}(\epsilon_i -\epsilon_j)} } }

First we simplify the errors to just one term: \Delta \epsilon = \epsilon_i-\epsilon_j :

\large V[k] &= \braket{ \xi_{i\nu}[k] \xi_{j\nu}^*[k] {\color{red} e^{j { 2\pi \over N } k \Delta \epsilon } }{ \color{brown}e^{j \omega_{rf}\Delta\epsilon} } }

Next we realize that \xi_{i\nu}[k] , \xi_{j\nu}[k] refer to the same random process: A(t) \overset{\mathcal{F}}{\rarr} \xi_{\nu}[k] , so we can rewrite to:

V[k] &= \large \braket{ \xi_{\nu}[k] \xi_{\nu}^*[k] {\color{red} e^{j { 2\pi \over N } k \Delta \epsilon } }{ \color{brown}e^{j \omega_{rf}\Delta \epsilon} } } \\ &=\large \braket{ |\xi_{\nu}[k]|^2 {\color{red} e^{j { 2\pi \over N } k \Delta \epsilon } }{ \color{brown}e^{j \omega_{rf}\Delta \epsilon} }}

And since earth rotate, so \Delta \epsilon is actually a function of time, which we can approximate as a linear function of time:

\Delta \epsilon \to \epsilon(t) &= \epsilon_0 + \dot \epsilon t \\ &(\text{considering discrete-time})\\ \rightsquigarrow \; & \epsilon(m\Delta t) = \epsilon[m] = \epsilon_0 + \dot \epsilon \; m \Delta t \\

Also remember that the FX correlator output a grid V[k,m] where k is the frequency index and m is the time index.

So we can rewrite the expression for V[k] so far as:

V[k,m] \large &=\large \braket{ |\xi_{\nu}[k,m]|^2 {\color{red} e^{j { 2\pi \over N } k \; \epsilon(m \Delta t) } }{ \color{brown}e^{j \omega_{rf} \; \epsilon(m \Delta t) } }}\\ &=\large \braket{ |\xi_{\nu}[k,m]|^2 {\color{black} e^{j \phi(k,m) }}}

where \phi(k,m) is:

\phi(k,m) &= {\color{red}\frac{2\pi}{N} k \;\; \epsilon(\Delta t\; m)}+{\color{brown} \omega_{rf} \;\; \epsilon(\Delta t \; m)}\\ &= \frac{2\pi}{N} k (\epsilon_0 + \dot \epsilon \; \Delta t \; m)+ 2\pi \nu_{rf} (\epsilon_0 + \dot \epsilon \; \Delta t \; m)\\ &= \frac{2\pi}{N} k \epsilon_0 + \frac{2\pi}{N} k \dot \epsilon \; \Delta t \; m+ 2\pi \nu_{rf} \epsilon_0 + 2\pi \nu_{rf} \dot \epsilon \; \Delta t \; m\\ &= (\frac{2\pi}{N} \epsilon_0 ) k + \underbrace{ \bcancel{\frac{2\pi}{N} k \dot \epsilon \; \Delta t \; m}}_{ \dot \epsilon \ll 1 , \text{can ignore}} + 2\pi \nu_{rf} \epsilon_0 +\underbrace{ (2\pi \nu_{rf} \dot \epsilon \; \Delta t )\; m }_{ \dot \epsilon \ll 1, \text{but } \nu_{rf} \gg 1}\\ &\approx 2\pi \nu_{rf} \epsilon_0 + (\frac{2\pi}{N} \epsilon_0 ) k + (2\pi \nu_{rf} \dot \epsilon \; \Delta t )\; m \\

Comparing the above function with the general 1^{st} order Taylor series approximation for the phase function \phi(k,m) :

\phi(k,m) &= \phi(k_0,m_0) + \frac{\delta \phi}{\delta k} (k-k_0) + \frac{\delta \phi}{\delta m} (m-m_0) \\ &(\text{with } k_0 = 0, m_0 = 0) \\ &= \phi(0,0) + \frac{\delta \phi}{\delta k} k + \frac{\delta \phi}{\delta m} m \\

We can see the following matches:

2\pi \nu_{rf} \epsilon_0 \to& \;\phi(0,0) \\ \\ \frac{2\pi}{N} \epsilon_0 \to& \frac{\delta \phi}{\delta k} := \text{delay} \\ \\ 2\pi \nu_{rf} \dot \epsilon \; \Delta t \to& \frac{\delta \phi}{\delta m} := \text{delay rate} \\

I think \frac{\delta \phi}{\delta k} is referred as delay, and \frac{\delta \phi}{\delta m} is referred as delay rate is because:

\text{phase shift } \phi &\lrarr 2\pi \nu \tau \\ &\lrarr 2\pi (k/N) m \\ &\text{so:} \\ \frac{\delta \phi}{\delta k} &\to \frac{ 2\pi (k/N) m}{k} \to 2\pi(1/N)m \to \text{unit is time} \\ \frac{\delta \phi}{\delta m} &\to \frac{ 2\pi (k/N) m}{m} \to 2\pi(k/N) \to \text{unit is frequency which is rate}

Anyway now we have V[k,m] as:

V[k,m] &= \braket{ |\xi_{\nu}[k,m]|^2 {\color{black} e^{j \phi(k,m) }}} \\ &= \braket{ |\xi_\nu[k,m]|^2 \large e^{j ( 2\pi \nu_{rf} \epsilon_0 + (\frac{2\pi}{N} \epsilon_0 ) k + (2\pi \nu_{rf} \dot \epsilon \; \Delta t )\; m )} }

Now we take 2D-FFT on V[k,m] :

F[u,v] &= \sum_{k=\braket{K}}\sum_{m=\braket{M}} V[k,m] \;\;\large e^{-j 2\pi (\frac{u}{K}k + \frac{v}{M}m)}\\ &= \sum_{k=\braket{K}}\sum_{m=\braket{M}} \braket{ |\xi_\nu[k,m]|^2 \large e^{j ( 2\pi \nu_{rf} \epsilon_0 + (\frac{2\pi}{N} \epsilon_0 ) k + (2\pi \nu_{rf} \dot \epsilon \; \Delta t ) m )} } \;\;\large e^{-j 2\pi (\frac{u}{K}k + \frac{v}{M}m)}\\ &= e^{j 2\pi \nu_{rf} \epsilon_0} \sum_{k=\braket{K}}\sum_{m=\braket{M}} \braket{ |\xi_\nu[k,m]|^2 \large {\color{navy} e^{j(\frac{2\pi}{N} \epsilon_0 ) k }} {\color{olive} e^{j(2\pi \nu_{rf} \dot \epsilon \; \Delta t )m }} } \;\;\large {\color{navy}e^{-j 2\pi \frac{u}{K}k }} {\color{olive} e^{-j 2\pi \frac{v}{M}m}} \\ &= e^{j 2\pi \nu_{rf} \epsilon_0} \sum_{k=\braket{K}}\sum_{m=\braket{M}} \braket{ |\xi_\nu[k,m]|^2 \large {\color{navy} e^{j 2\pi(\frac{\epsilon_0 }{N} - \frac{u}{K}) k }} {\color{olive} e^{j2\pi( \nu_{rf} \dot \epsilon \; \Delta t - \frac{v}{M})m }} }

So suppose

\argmax_{(u,v)} F[u,v] = (u',v')

That implies:

\frac{\epsilon_0 }{N} - \frac{u'}{K} \approx & 0 \\ \nu_{rf} \dot \epsilon \; \Delta t - \frac{v'}{M} \approx & 0 \\

Since we the only unknowns in the above equations are \epsilon_0 \text{ and } \dot \epsilon , we can solve for their approximations and recover the residual delay function:

\epsilon(t) = \epsilon_0 + \dot \epsilon \; t

So now we can use the solved \epsilon(t) to adjust the raw result of the FX-correlator:

\begin{matrix} V[k] &=& \langle \xi_{i\nu}[k] {\color{red} e^{j { 2\pi \over N } k \epsilon_i } }{\color{brown}e^{j \omega_{rf}\epsilon_i}} , \xi_{j\nu}[k] {\color{red} e^{j { 2\pi \over N } k \epsilon_j } } {\color{brown}e^{j \omega_{rf}\epsilon_j}} \rangle \\ &=& \braket{ \xi_{i\nu}[k] \xi_{j\nu}^*[k] {\color{red} e^{j { 2\pi \over N } k (\epsilon_i-\epsilon_j) } }{\color{brown}e^{j \omega_{rf}(\epsilon_i-\epsilon_j)}} } \\ &=& \braket{ \xi_{i\nu}[k] \xi_{j\nu}^*[k] {\color{red} e^{j { 2\pi \over N } k \; \Delta \epsilon} }{\color{brown}e^{j \omega_{rf} \; \Delta \epsilon }} } \\ &=& \braket{ \xi_{i\nu}[k] \xi_{j\nu}^*[k] {\color{red} e^{j { 2\pi \over N } k \; \epsilon(t) } }{\color{brown}e^{j \omega_{rf} \;\epsilon(t)}} } \end{matrix} \\

Side note: Because u,v is integer, to get more accuracy, we can apply zero-padding or interpolation during 2D-FFT.

Let's see the effect of the signal's spectrum when we use FFT.

At step \boxed{\textcircled{6}_i \; \text{F Step: } N\text{ points }FFT \text{ on } E_{\theta_i}[n] } , to take FFT, we take a section of the continuous baseband signal A_i(t) , the cut out signal can be expressed as:

A_i'(t) &= A_i(t) \cdot \sqcap\left({t \over T}\right) \\ &\text{where } \\ &\sqcap\left({t \over T}\right) = \begin{cases} 1 &, |t/T| < 1/2 \\ 0 &, \text{else} \end{cases}

So basically we only preserve the value of A_i(t) for \frac{-T}{2} < t < \frac{T}{2} and 0 for everywhere else. And we can also shift around where t=0 is as we desire.

now using the convolution property, and knowing that rectangle function in time is sinc function in frequency, The Fourier Transform of A_i(t) is:

A_{i\nu}'(\nu) = A_{i\nu}(\nu) \ast T \text{sinc}(T\nu)

And after applying correlation(multiplication) with another station, we have:

A_{ij\nu}'(\nu) = A_{ij\nu}(\nu) \ast T^2 \text{sinc}^2(T\nu)