Suppose we can operate at the sky frequency without doing any down conversion.
There are 2 ways we can represent the same correlator output v(t,\tau_0) |_T :
Given that x_1 is closer to the source than x_2 :
Option 1: Delay x_1 by \tau_0 :
v(t,\tau_0) |_T = \frac{1}{T} \int_{\psi = t-T/2}^{\psi=t+T/2} x_2(\psi)x_1(\psi-\tau_0) d\psiOption 2: Advance x_2 by \tau_0 :
v(t,\tau_0) |_T = \frac{1}{T} \int_{\psi = t-T/2}^{\psi=t+T/2} x_2(\psi+\tau_0)x_1(\psi) d\psiWe will show you the correlator output operating at sky frequency for both each option:
Option 1: Delay x_1 by \tau_0 :
During real VLBI observation, we don't operate with integral since we have discrete signal.
instead, in VLBI software, such as DiFX, use FFT/DFT to compute visibility.
Given the below visibility function with phase center:
v(t,\tau_0) |_T = \frac{1}{T} \int_{\psi = t-T/2}^{\psi=t+T/2} x_2(\psi)x_1(\psi-\tau_0) d\psiLet's discretize this v(t) by first changing the variables:
t &\to r \\ \tau_0 &\to n \\ T &\to N \\ \psi &\to p \\ x_2(\psi) &\to x_2[p] \\ x_1(\psi-\tau_0) &\to x_1[p-n] \\we assume x_2[p] \;\&\; x_1[p] is periodic with period N :
v(t,\tau_0) |_T \to v[r,n]|_N &\\ v[r,n]|_N &= \frac{1}{N} \sum_{p=r-N/2}^{r+N/2-1} x_2[p]x_1[p-n] \\ \\ (\because & \; x_2[p] \; \& \; x_1[p] \text{ has period } N) \\ \\ &= \frac{1}{N} \sum_{p=0}^{N-1} x_2[p]x_1[p-n] \\ &= v[n]Given the standard DFT formula:
X[k] = \sum_{n=0}^{N-1} x[n] \large e^{ -j\frac{2\pi kn}{N}}Let's find the DFT of v[n] :
\mathcal{F} \{ v[n]\} =& V[k] \\ =& \sum_{n=0}^{N-1} v[n] \large e^{ -j\frac{2\pi kn}{N}} \\ =& \sum_{n=0}^{N-1} \left( \frac{1}{N} \sum_{p=0}^{N-1} x_2[p]x_1[p-n] \right) \large e^{ -j\frac{2\pi kn}{N}} \\ =& \frac{1}{N} \sum_{n=0}^{N-1} \left( \sum_{p=0}^{N-1} x_2[p]x_1[p-n] \right) \large e^{ -j\frac{2\pi kn}{N}} \\ =& \frac{1}{N} \sum_{p=0}^{N-1} \left( \sum_{n=0}^{N-1} x_2[p]x_1[p-n] \large e^{ -j\frac{2\pi kn}{N}} \right) \\ =& \frac{1}{N} \sum_{p=0}^{N-1} x_2[p] \left( \sum_{n=0}^{N-1} x_1[p-n] \large e^{ -j\frac{2\pi kn}{N}} \right) \\ \\ &\begin{pmatrix} \text{set }& q =p-n \\ &\to n=p-q \end{pmatrix}\\ \\ =& \frac{1}{N} \sum_{p=0}^{N-1} x_2[p] \left( \sum_{q=p}^{p-(N-1)} x_1[q] \large e^{ -j\frac{2\pi k(p-q)}{N}} \right) \\ =& \frac{1}{N} \sum_{p=0}^{N-1} x_2[p] \left( \sum_{q=0}^{N-1} x_1[q] \large e^{ -j\frac{2\pi kp}{N}} \large e^{ j\frac{2\pi kq}{N}} \right) \\ =& \frac{1}{N} \sum_{p=0}^{N-1} x_2[p] {\large e^{ -j\frac{2\pi kp}{N}} } \left( \sum_{q=0}^{N-1} x_1[q] \large e^{ j\frac{2\pi kq}{N}} \right) \\ =& \frac{1}{N} X_2[k] \left( \sum_{q=0}^{N-1} x_1[q] \large e^{ j\frac{2\pi kq}{N}} \right) \\ =& \frac{1}{N} X_2[k] \left( \sum_{q=0}^{N-1} x_1[q] \large e^{ j\frac{2\pi kq}{N}} \right)^{**} \\ =& \frac{1}{N} X_2[k] \left( \sum_{q=0}^{N-1} x_1^*[q] \large e^{ -j\frac{2\pi kq}{N}} \right)^{*} \\ \\ &\begin{pmatrix} \text{given that $x_1,x_2$ are real} \\ \end{pmatrix}\\ \\ =& \frac{1}{N} X_2[k] \left( \sum_{q=0}^{N-1} x_1[q] \large e^{ -j\frac{2\pi kq}{N}} \right)^{*} \\ =& \frac{1}{N} X_2[k] \left(X_1[k]\right)^{*} \\ =& \frac{1}{N} X_1^*[k] X_2[k] \\The result is:
\mathcal{F} \{ v[n]\} =& V[k] \\ =& \frac{1}{N} X_1^*[k] X_2[k] \\Let's go over how FX correlator is implemented:
- As baseband signal coming in as input, We record arrays of size N voltage data from both stations: x_1[p], x_2[p]
- Calculate the phase center delay in terms of sample number n_{\tau_0} based on the baseline's geometric parameters. n_{\tau_0} is most likely not an integer: n_{\tau_0} = n_0 + n_d\; , \; n_d <1
- Now instead of directly calculate v[n_{\tau_0}] , We delay x_1[p] by n_{\tau_0} samples: x_{1\_n_{\tau_0}}[p] = x_1[p-n_{\tau_0}] such that v_{n_{\tau_0}}[0] = v[n_{\tau_0}]
- But as pointed out above, n_{\tau_0} is most likely not an integer. So we deal with the integer n_0 and fractional n_d separately.
- First we delay x_1[p] by the integer n_0 : x_{1\_n_{0}}[p] = x_1[p-n_0]
- Use FFT to calculate: X_{1\_n_{0}}[k], X_2[k]
- Now we apply fractional delay n_d to X_{1\_n_{0}}[k] as phase shift (based on time shifting property): X_{1\_n_{0}+n_d}[k] = X_{1\_n_{0}}[k]\large e^{-j\frac{2\pi k n_d}{N}} , \; {\small 0 \le k \le N-1}
- Calculate the complex visibility V_{n_0+n_d}[k] : V_{n_0+n_d}[k] =& \frac{1}{N} X_2[k] X^*_{1\_n_{0}+n_d}[k] \\ =& \frac{1}{N} X_2[k] X_{1\_n_{0}}^*[k]\large e^{j\frac{2\pi k n_d}{N}} , \; {\small 0 \le k \le N-1}
- Now with the FX correlator output, assuming no timing error, we can calculate the final visibility v_{n_0+n_d}[0] with IDFT/IFFT: v_{n_0+n_d}[0] &= \frac{1}{N}\sum_{k=0}^{N-1} \left(\frac{1}{N} X_2[k] X_{1\_n_{0}}^*[k] \large e^{j\frac{2\pi k n_d}{N}} \right) \large \underbrace{ e^{ j\frac{2\pi k \cdot 0 }{N}} }_{=1}\\ &= \frac{1}{N}\sum_{k=0}^{N-1}\frac{1}{N} X_2[k] X_{1\_n_{0}}^*[k] \large e^{j\frac{2\pi k n_d}{N}} \\ &= \frac{1}{N}\sum_{k=0}^{N-1}\frac{1}{N}X_{1\_n_{0}}^*[k] X_2[k] \large e^{j\frac{2\pi k n_d}{N}} \\
Option 2: Advance x_2 by \tau_0 :
During real VLBI observation, we don't operate with integral since we have discrete signal.
instead, in VLBI software, such as DiFX, use FFT/DFT to compute visibility.
Given the below visibility function with phase center:
v(t,\tau_0) |_T = \frac{1}{T} \int_{\psi = t-T/2}^{\psi=t+T/2} x_2(\psi+\tau_0)x_1(\psi) d\psiLet's discretize this v(t) by first changing the variables:
t &\to r \\ \tau_0 &\to n \\ T &\to N \\ \psi &\to p \\ x_2(\psi+\tau_0) &\to x_2[p+n] \\ x_1(\psi) &\to x_1[p] \\we assume x_2[p] \;\&\; x_1[p] is periodic with period N :
v(t,\tau_0) |_T \to v[r,n]|_N &\\ v[r,n]|_N &= \frac{1}{N} \sum_{p=r-N/2}^{r+N/2-1} x_2[p+n]x_1[p] \\ \\ (\because & \; x_2[p] \; \& \; x_1[p] \text{ has period } N) \\ \\ &= \frac{1}{N} \sum_{p=0}^{N-1} x_2[p+n]x_1[p] \\ &= v[n]Given the standard DFT formula:
X[k] = \sum_{n=0}^{N-1} x[n] \large e^{ -j\frac{2\pi kn}{N}}Let's find the DFT of v[n] :
\mathcal{F} \{ v[n]\} =& V[k] \\ =& \sum_{n=0}^{N-1} v[n] \large e^{ -j\frac{2\pi kn}{N}} \\ =& \sum_{n=0}^{N-1} \left( \frac{1}{N} \sum_{p=0}^{N-1} x_2[p+n]x_1[p] \right) \large e^{ -j\frac{2\pi kn}{N}} \\ =& \frac{1}{N} \sum_{n=0}^{N-1} \left( \sum_{p=0}^{N-1} x_2[p+n]x_1[p] \right) \large e^{ -j\frac{2\pi kn}{N}} \\ =& \frac{1}{N} \sum_{p=0}^{N-1} \left( \sum_{n=0}^{N-1} x_2[p+n]x_1[p] \large e^{ -j\frac{2\pi kn}{N}} \right) \\ =& \frac{1}{N} \sum_{p=0}^{N-1} x_1[p] \left( \sum_{n=0}^{N-1} x_2[p+n] \large e^{ -j\frac{2\pi kn}{N}} \right) \\ \\ &\begin{pmatrix} \text{set }& q =p+n \\ &\to n=q-p \end{pmatrix}\\ \\ =& \frac{1}{N} \sum_{p=0}^{N-1} x_1[p] \left( \sum_{q=p}^{p+(N-1)} x_2[q] \large e^{ -j\frac{2\pi k(q-p)}{N}} \right) \\ =& \frac{1}{N} \sum_{p=0}^{N-1} x_1[p] \left( \sum_{q=0}^{N-1} x_2[q] \large e^{ -j\frac{2\pi kq}{N}} \large e^{ j\frac{2\pi kp}{N}} \right) \\ =& \frac{1}{N} \sum_{p=0}^{N-1} x_1[p] {\large e^{ j\frac{2\pi kp}{N}} } \left( \sum_{q=0}^{N-1} x_2[q] \large e^{ -j\frac{2\pi kq}{N}} \right) \\ =& \frac{1}{N} \left( \sum_{p=0}^{N-1} x_1[p] \large e^{ j\frac{2\pi kp}{N}} \right) X_2[k] \\ =& \frac{1}{N} \left( \sum_{p=0}^{N-1} x_1[p] \large e^{ j\frac{2\pi kp}{N}} \right) ^{**} X_2[k] \\ =& \frac{1}{N}\left( \sum_{p=0}^{N-1} x_1^*[p] \large e^{ -j\frac{2\pi kp}{N}} \right)^{*} X_2[k] \\ \\ &\begin{pmatrix} \text{given that $x_1,x_2$ are real} \\ \end{pmatrix}\\ \\ =& \frac{1}{N} \left( \sum_{p=0}^{N-1} x_1[p] \large e^{ -j\frac{2\pi kp}{N}} \right)^{*}X_2[k] \\ =& \frac{1}{N} X_1^{*}[k] X_2[k] \\The result is:
\mathcal{F} \{ v[n]\} =& V[k] \\ =& \frac{1}{N} X_1^*[k] X_2[k] \\Let's go over how FX correlator is implemented:
- As baseband signal coming in as input, We record arrays of size N voltage data from both stations: x_1[p], x_2[p]
- Calculate the phase center delay in terms of sample number n_{\tau_0} based on the baseline's geometric parameters. n_{\tau_0} is most likely not an integer: n_{\tau_0} = n_0 + n_d\; , \; n_d <1
- Now instead of directly calculate v[n_{\tau_0}] , We advance x_2[p] by n_{\tau_0} samples: x_{2\_n_{\tau_0}}[p] = x_2[p+n_{\tau_0}] such that v_{n_{\tau_0}}[0] = v[n_{\tau_0}]
- But as pointed out above, n_{\tau_0} is most likely not an integer. So we deal with the integer n_0 and fractional n_d separately.
- First we advance x_2[p] by the integer n_0 : x_{2\_n_{0}}[p] = x_2[p+n_0]
- Use FFT to calculate: X_{2\_n_{0}}[k], X_1[k]
- Now we apply fractional advancement n_d to X_{2\_n_{0}}[k] as phase shift (based on time shifting property): X_{2\_n_{0}+n_d}[k] = X_{2\_n_{0}}[k]\large e^{j\frac{2\pi k n_d}{N}} , \; {\small 0 \le k \le N-1}
- Calculate the complex visibility V_{n_0+n_d}[k] : V_{n_0+n_d}[k] = & \frac{1}{N} X_1^*[k] X_{2\_n_{0}+n_d}[k] \\ =& \frac{1}{N} X_1^*[k] X_{2\_n_{0}}[k]\large e^{j\frac{2\pi k n_d}{N}} , \; {\small 0 \le k \le N-1}
- Now with the FX correlator output, assuming no timing error, we can calculate the final visibility v_{n_0+n_d}[0] with IDFT/IFFT: v_{n_0+n_d}[0] &= \frac{1}{N} \sum_{k=0}^{N-1} \left(\frac{1}{N} X_1^*[k] X_{2\_n_{0}}[k] \large e^{j\frac{2\pi k n_d}{N}} \right) \large \underbrace{ e^{ j\frac{2\pi k \cdot 0 }{N}} }_{=1}\\ &= \frac{1}{N} \sum_{k=0}^{N-1}\frac{1}{N} X_1^*[k] X_{2\_n_{0}}[k] \large e^{j\frac{2\pi k n_d}{N}}
2.1.1: Sky to Baseband
Let's see the simplified signal data flow of a real VLBI observation:
Figure
Figure
Figure
Figure
Figure
Let's see some details in \boxed{\textcircled{4} \text{: Downconversion}} on how convert intermediate frequency signal x_{if}(t) to baseband signal/digital baseband signal,x_{b}(t)/x_b[n] :
Suppose the intermediate frequency signal, x_{if}(t) is:
x_{if}(t) = a(t) \cos(\omega_{if} t +\theta(t))where \omega_{if} is the center frequency.
because x_{if}(t) is real, its frequency magnitude spectrum (|X_{if}(\omega)| ) is symmetric around 0 due to conjugate symmetry property :
\text{suppose } x(t)& \text{ is some real signal , and } X(\omega) =m(\omega) e^{j\theta(\omega)} \\ X^*(\omega)=&X(-\omega) \text{ (conjugate symmetry property)}\\ \to X^*(\omega) = & m(\omega) e^{-j\theta(\omega)} = X(-\omega) = m(-\omega) e^{j\theta(-\omega)} \\ \to m(\omega) = & m(-\omega) , -\theta(\omega) = \theta(\omega)Figure
let's create an associate complex signal from x_{if}(t) with the help of Hilbert Transform:
x_c(t) =& a(t) \cos(\omega_{if} t +\theta(t)) + j [a(t) \sin(\omega_{if} t +\theta(t))]\\ =& a(t)e^{j( \omega_{if} t + \theta(t) )} \\where:
\mathcal{Re} \{ x_c(t) \} = x_{if}(t)and the magnitude spectrum is one-sided:
Figure
Using shifting property to bring the x_c(t) to baseband:
\mathcal{F} \{ x_c(t) e^{-j\omega_{if} t}\} = X_c(\omega+\omega_{if})Notice that with X_c(\omega + \omega_{if}) , the intermediate frequency signal is now brought down to baseband.
Which means x_c(t) e^{-j\omega_c t} is our the baseband signal x_{b}(t) :
Figure
and we can express x_{b}(t) as :
x_b(t) =& x_c(t) e^{-j\omega_{if} t} \\ =& [a(t)e^{j( \omega_{if} t + \theta(t) )}] e^{-j\omega_{if} t} \\ =& a(t)e^{j \theta(t) } \\Notice that x_{b}(t) is complex, but we can also represent it the combination of 2 real signals:
x_b(t) =& a(t)e^{j \theta(t) } \\ =& I(t) + j Q(t) \\ \\ \to I(t) =& a(t) \cos\theta(t) \\ \to Q(t) =& a(t) \sin\theta(t) \2.1.2: Fringe Rotation
Now let's see what is fringe rotation and how to fix it.
Basically, fringe rotation happens because of time delay and \omega_{lo} mixing. Let's see in detail for a baseline:
In this example baseline, Antenna 1 is closer to the target source than Antenna 2. So Antenna 2 experiences geometric delay \tau_g :
Starts at the output of \boxed{\textcircled{1} \text{: Bandpass Filter}} , the bandpass signal for Antenna 1 is:
x_{1p} (t)= a(t) \cos(\omega_{rf} t +\theta(t))Goes through \boxed{\textcircled{2} \text{: LO Mixer }\omega_{lo}} to get x_{1lo}(t) :
x_{1lo} (t) =& a(t) \cos(\omega_{rf} t +\theta(t)) \cos(\omega_{lo}t )\\ =& \frac{1}{2} a(t) [ \cos( (\omega_{rf}-\omega_{lo} ) t +\theta(t)) +\cos( (\omega_{rf}+\omega_{lo}) t +\theta(t)) ] \\Goes through \boxed{\textcircled{3} \text{: Lowpass Filter}} to get x_{1if}(t) :
x_{1if} (t) =& \frac{1}{2}a(t) \cos(\omega_{if} t +\theta(t)), \; \omega_{if} = \omega_{rf} - \omega_{lo} \\After \boxed{\textcircled{4} \text{: Downconversion}} , the baseband signal for Antenna 1 is:
x_{1b}(t) =\frac{1}{2} a(t)e^{j \theta(t) }Now let's analysis the signals for Antenna 2 that incurs geometric delay \tau_g , and the phase center delay \tau_0 is applied to Antenna 2:
Starts at the output of \boxed{\textcircled{1} \text{: Bandpass Filter}} , the bandpass signal for Antenna 2 is:
x_{2p} (t)= a(t-\tau_g) \cos(\omega_{rf} (t-\tau_g) +\theta(t-\tau_g))Goes through \boxed{\textcircled{2} \text{: LO Mixer }\omega_{lo}} to get x_{2lo}(t) :
x_{2lo} (t) =& a(t-\tau_g) \cos(\omega_{rf} (t-\tau_g) +\theta(t-\tau_g)) \cos(\omega_{if}t )\\ =& \frac{1}{2} a(t-\tau_g) [ \cos( (\omega_{rf}-\omega_{if} ) t - \omega_{rf}\tau_g +\theta(t-\tau_g)) +\cos( (\omega_{rf}+\omega_{if}) t- \omega_{rf}\tau_g +\theta(t-\tau_g)) ] \\Goes through \boxed{\textcircled{3} \text{: Lowpass Filter}} to get x_{2if}(t) :
x_{2if} (t) =& \frac{1}{2}a(t-\tau_g) \cos(\omega_{if} t - \omega_{rf}\tau_g +\theta(t-\tau_g)), \; \omega_{if} = \omega_{rf} - \omega_{lo} \\Even though the phase center delay is applied in the FX correlator, to see the effect of the mixing with \omega_{lo} , we apply the phase center delay here.
In the previous section, we show the math for applying \tau_0 as time delay to Antenna 1 which is closer to the target source, but we can achieve the way result if we apply \tau_0 as time advancement to Antenna 2:
x_{2if} (t+\tau_0) =& \frac{1}{2}a(t+\tau_0-\tau_g) \cos(\omega_{if} (t+\tau_0) - \omega_{rf}\tau_g +\theta(t+\tau_0-\tau_g)), \; \omega_{if} = \omega_{rf} - \omega_{lo} \\\tau_g, \tau_0 are actually both function of time (thus the term fringe rotation), but for the sake of simplified explanation of fringe rotation , we will treat them as constant.
We also assume that we calculate \tau_0 with perfect accuracy such that \tau_g= \tau_0 :
x_{2if} (t+\tau_0) =& \frac{1}{2}a(t) \cos(\omega_{if} (t+\tau_0) - \omega_{rf}\tau_g +\theta(t)), \; \omega_{if} = \omega_{rf} - \omega_{lo} \\ =& \frac{1}{2}a(t) \cos((\omega_{rf} - \omega_{lo}) (t+\tau_0) - \omega_{rf}\tau_g +\theta(t)) \\ =& \frac{1}{2}a(t) \cos([\omega_{rf} t - \omega_{lo} t +\omega_{rf} \tau_0 - \omega_{lo} \tau_0 ] - \omega_{rf}\tau_g +\theta(t)) \\ =& \frac{1}{2}a(t) \cos((\omega_{rf} - \omega_{lo}) t + \omega_{rf}\underset{\approx 0}{\bcancel{(\tau_0 - \tau_g)}}- \omega_{lo} \tau_0 +\theta(t)) \\ =& \frac{1}{2}a(t) \cos(\omega_{if}t - \omega_{lo} \tau_0 +\theta(t)) , \; \omega_{if} = \omega_{rf} - \omega_{lo} \\ =& \frac{1}{2}a(t) \cos(\omega_{if}t +\theta'(t)) \\ &\begin{matrix} \text{where } & \omega_{if} = \omega_{rf} - \omega_{lo} \\ \text{ } & \theta'(t)= \theta(t)- \omega_{lo} \tau_0 \\ \end{matrix}After \boxed{\textcircled{4} \text{: Downconversion}} , the baseband signal for Antenna 2 is:
x_{2b}(t+\tau_0) =&\frac{1}{2} a(t)e^{j \theta'(t) } \\ =& \frac{1}{2} a(t)e^{j (\theta(t)- \omega_{lo} \tau_0 ) } \\ =& \frac{1}{2} a(t)e^{j \theta(t) } \color{red} e^{-j \omega_{lo} \tau_0 } \\With this result, you see that even if we calculate perfectly that \tau_0 = \tau_g the baseband signal will have an extra fringe rotation phase that need to be corrected:
\color{red} e^{-j \omega_{lo} \tau_0 }To fix this is easy, the idea is basically multiply the baseband signal with e^{j \omega_{lo} \tau_0 } : x_{2b}(t) {\color{green} e^{j \omega_{lo} \tau_0 }} = \frac{1}{2} a(t)e^{j \theta(t) } = x_{1b}(t)
Now let's see how fringe rotation correction can be applied in a FX correlator:
There are 2 ways:
1. Pre-FFT:
Inside the FX correlator, for every discrete signal of Antenna 2, multiple the time domain signal with e^{j \omega_{lo} \tau_0 } :
x_{2b\_{\color{green} \text{fringe fix}}}[n] = x_{2b}[n] \color{green} e^{j \omega_{lo} \tau_0 }then proceed with x_{2b\_{\color{green} \text{fringe fix}}}[n]
This way is required more computations but more accurate because \tau_0 technically changes every instance as earth rotates and travels.
2. Post-FFT:
Inside the FX correlator, after taking FFT of Antenna 2 signal, and at the step of adjusting for the fractional advancement, add in the term for fringe rotation phase correction: X_{2\_n_0+{\color{blue}n_d}+{\color{green}\text{fringe fix} }}[k] = X_{2\_n_0}[k] \large { {\color{blue}e^{j\frac{2\pi k n_d}{N}}} } {\color{green} e^{j \omega_{lo} \tau_0 }}, \; {\small 0 \le k \le N-1}
It requires less computing compared to Pref-FFT, but introduces error known as "time smearing/fringe washing."
Now roll back a bit and starts at x_{2if}(t) , but this time with practical \tau_0 where \tau_0-\tau_g \neq 0 :
x_{2if} (t+\tau_0) =& \frac{1}{2}a(t+\tau_0-\tau_g) \cos(\omega_{if} (t+\tau_0) - \omega_{rf}\tau_g +\theta(t+\tau_0-\tau_g)), \; \omega_{if} = \omega_{rf} - \omega_{lo} \\ &(\text{set } \tau_e = \tau_0-\tau_g) \\ =& \frac{1}{2}a(t+\tau_e) \cos((\omega_{rf}-\omega_{lo}) (t+\tau_0) - \omega_{rf}\tau_g +\theta(t+\tau_e)) \\ =& \frac{1}{2}a(t+\tau_e) \cos(\omega_{rf}t +\omega_{rf} \tau_0-\omega_{lo}t-\omega_{lo}\tau_0- \omega_{rf}\tau_g +\theta(t+\tau_e)) \\ =& \frac{1}{2}a(t+\tau_e) \cos(\omega_{if}t + \underbrace{\omega_{rf} (\tau_0- \tau_g)-\omega_{lo}\tau_0 +\theta(t+\tau_e)}_{\theta'(t+\tau_e)}) \\ =& \frac{1}{2}a(t+\tau_e) \cos(\omega_{if}t +\theta'(t+\tau_e)) \\ &\begin{matrix} \text{where } & \omega_{if} = \omega_{rf} - \omega_{lo} \\ \text{ } & \tau_e = \tau_0- \tau_g \\ \text{ } & \theta'(t+\tau_e)= \theta(t+\tau_e)+\omega_{rf}\tau_e-\omega_{lo}\tau_0 \\ \end{matrix}Now if we bring this signal to baseband, we will get:
x_{2b}(t+\tau_0) =&\frac{1}{2} a(t+\tau_e)\large e^{j \theta'(t+\tau_e) } \\ =& \frac{1}{2} a(t+\tau_e)\large e^{j (\theta(t+\tau_e)+\omega_{rf}\tau_e-\omega_{lo}\tau_0 ) } \\ =& \frac{1}{2} a(t+\tau_e)\large e^{j \theta(t+\tau_e) } \color{red} e^{-j \omega_{lo} \tau_0 } \color{brown}e^{j\omega_{rf}\tau_e } \\So notice that even if we remove the fringe rotation term \large \color{red} e^{-j \omega_{lo} \tau_0 } in the FX correlator, in practice we will still be left with extra term \large \color{brown}e^{j\omega_{rf}\tau_e }:
x_{2b\text{-fx}}(t+\tau_0) =& \frac{1}{2} a(t+\tau_e)\large e^{j \theta(t+\tau_e) } \color{brown}e^{j\omega_{rf}\tau_e } \\2.1.3: FX Correlator Iteration
Let's see how FX correlator works.
The figure below shows how to obtain the visibility of one cycle of FX operation.
Figure :
Breakdown on FX operation
Procedure:
- Iteration 0 :During the first integration time period (0 \le t \le T ):
After producing finite amount of V_{\Delta t}[k] and reaching time t=T , we normalize/average the accumulated V_{\Delta t}[k] and produce V[k,0] .
- Iteration 1 :During the second integration time period (T \le t \le 2T ):
After producing finite amount of V_{\Delta t}[k] and reaching t=2T , we normalize/average the accumulated V_{\Delta t}[k] and produce V[k,1] .
\vdots
- Iteration M-1 :During the second integration time period ((M-1)T \le t \le MT ):
After producing finite amount of V_{\Delta t}[k] and reaching t=MT , we normalize/average the accumulated V_{\Delta t}[k] and produce V[k,M-1] .
After M integration time, we then have a visibility grid, going from V[k,0] to V[k,M-1] : \text{Visibility Grid : }& V[k,m] \\ \\ & k \text{ is the frequency index }, k \in [0,N-1] \\ & m \text{ is the time index }, m \in [0,M-1] \\
The figure below shows a visual representation of the V[k,m] grid.
Figure :
V_{}[k,m] Grid