2.2.0: Constant Residual Delay
When we calculate the phase center delay \tau_0 to minimize the geometric delay between 2 stations, there will always be an error compare to the true delay \tau_g :
\tau_e = \tau_g-\tau_0 \neq 0If we treat this residual delay \tau_e as constant, let's see its effect and how to find its value:
Now suppose we are inside the FX correlator.
The F step:Start from the baseband signal for station 1:
x_1(t) &=\frac{1}{2} a(t)e^{j \theta(t) }\\and Fourier transform of x_1(t) is:
\mathcal{F}\{x_1(t) \} = X_1(\omega)As for station 2's signal. After inserting calculated phase center time advancement \tau_0 and removing the fringe rotation, we have:
x_2(t+\tau_0) &=\left[\frac{1}{2} a(t+\tau_e) \large e^{j \theta(t+\tau_e) }\right]e^{j\omega_{rf}\tau_e} \\&= x_1(t+\tau_e)e^{j\omega_{rf}\tau_e} \\ \text{where } &\tau_0 = \text{calculated phase center delay} \\ &\tau_e = \tau_g-\tau_0\\ &\omega_{rf} = \text{center of sky frequency band, a constant}\\Now we make a new signal: y_2(t) = & x_2(t+\tau_0) \\=& x_1(t+\tau_e) \large e^{j\omega_{rf}\tau_e}
And we take the Fourier transform of y_2(t) :
\mathcal{F}\{y_2(t) \} = Y_2(\omega)Assuming \tau_e is constant and using the time-shifting property:
Y_2(\omega) = X_1(\omega) \large e^{j\omega \tau_e}e^{j\omega_{rf}\tau_e}Now multiply the 2 Fourier transformed signals:
V(\omega) =& Y_2(\omega) X_1^*(\omega)\\ =& X_1(\omega) X_1^*(\omega)\large e^{j\omega \tau_e}e^{j\omega_{rf}\tau_e}\\ &(\text{suppose } |X_1(\omega)|=A) \\ =& A^2 \large e^{j\omega \tau_e}e^{j\omega_{rf}\tau_e} \\ =& A^2 \large e^{j\omega_{rf}\tau_e} \color{red} e^{j\omega \tau_e}\\Looking at the output of the correlator, we have a linear phase term :
\color{red} e^{j\omega \tau_e}that will mess up the desired result. so we need to find and remove this \tau_e
Continuous-time V(\omega) \to Discrete-time V[k] :
The visibility V(\omega) is in continuous-time but FX operates and output discrete time V[k] , let's find out what is V[k] :
Suppose the sampling rate is \omega_s , and we are doing N points FFT, then the visibility is V[k] with \{ k \in \Z \; \vert \; 0 \leq k \leq N-1 \}
Now for a particular k :
its center frequency \omega_k is:
\omega_k = k\frac{\omega_s}{N}and its channel bandwidth is:
\Delta \omega_k = \frac{\omega_s}{N}and V[k] is:
V[k] = V(\omega_k)\bigg|_{-\Delta\omega_k /2}^{\Delta\omega_k /2} & = \int_{\omega_k -\Delta \omega_k/2}^{\omega_k + \Delta \omega_k/2} V(\omega) d\omega \\ & = \int_{\omega_k -\Delta \omega_k/2}^{\omega_k + \Delta \omega_k/2} A^2 \large e^{j\omega_{rf}\tau_e} e^{j\omega \tau_e} d\omega \\ & = A^2 \large e^{j\omega_{rf}\tau_e} \int_{\omega_k -\Delta \omega_k/2}^{\omega_k + \Delta \omega_k/2} e^{j\omega \tau_e} d\omega \\ \\ &\left(\begin{aligned} \text{set } u&=\omega-\omega_k\\ \text{ }d u&=d\omega\\ \end{aligned}\right) \\ & = A^2 \large e^{j\omega_{rf}\tau_e} \int_{-\Delta \omega_k/2}^{\Delta \omega_k/2} e^{j(u+\omega_k) \tau_e} du \\ & = A^2 \large e^{j\omega_{rf}\tau_e} \int_{-\Delta \omega_k/2}^{\Delta \omega_k/2} e^{j u \tau_e}e^{j\omega_k \tau_e }du \\ & = A^2 \large e^{j\omega_{rf}\tau_e} e^{j\omega_{k}\tau_e}\left[ \frac{e^{ju\tau_e}}{j \tau_e} \right]_{-\Delta \omega_k/2}^{\Delta \omega_k/2} \\ & = A^2 \large e^{j\omega_{rf}\tau_e} e^{j\omega_{k}\tau_e}\left[ \frac{ e^{j (\Delta \omega_k/2) \tau_e } -e^{-j(\Delta \omega_k/2 )\tau_e}}{j \tau_e} \right] \\ & = A^2 \large e^{j\omega_{rf}\tau_e} e^{j\omega_{k}\tau_e}\left[\frac{2}{\tau_e} \frac{ e^{j (\Delta \omega_k/2) \tau_e } -e^{-j(\Delta \omega_k/2 )\tau_e}}{2j} \right] \\ & = A^2 \large e^{j\omega_{rf}\tau_e} e^{j\omega_{k}\tau_e} \frac{2}{\tau_e} \sin ( \frac{\Delta \omega_k}{2}\tau_e) \\ & = A^2 \large e^{j\omega_{rf}\tau_e} e^{j\omega_{k}\tau_e} \frac{\Delta \omega_k}{\Delta \omega_k\tau_e/2} \sin ( \frac{\Delta \omega_k}{2}\tau_e) \\ & = A^2 \large e^{j\omega_{rf}\tau_e} e^{j\omega_{k}\tau_e} \Delta \omega_k \; \text{sinc} ( \frac{\Delta \omega_k}{2}\tau_e) \\As you can see because of the sinc function, if your \tau_e is big, V[k] can be squashed to 0.
Finding \tau_e :
We take FT of V(\omega) , think of the FT here as a weighted sum.
\mathcal{F}\{V(\omega) \} = R(\tau) =& \int_{-\infty}^{\infty} (V(\omega) )e^{-j\omega \tau}d\omega \\ =& \int_{\frac{-\Delta \omega }{2}}^{\frac{\Delta \omega }{2}} (A^2 \large e^{j\omega_{rf}\tau_e} {\color{red} e^{j\omega \tau_e}} )e^{-j\omega \tau}d\omega \\ =& A^2 \large e^{j\omega_{rf}\tau_e} \int_{\frac{-\Delta \omega }{2}}^{\frac{\Delta \omega }{2}} ( {\color{red} e^{j\omega \tau_e}} )e^{-j\omega \tau}d\omega \\ =& A^2 \large e^{j\omega_{rf}\tau_e} \int_{\frac{-\Delta \omega }{2}}^{\frac{\Delta \omega }{2}} ( {\color{red} e^{j\omega \tau_e}} )e^{-j\omega \tau}d\omega \\ =& A^2 \large e^{j\omega_{rf}\tau_e} \int_{\frac{-\Delta \omega }{2}}^{\frac{\Delta \omega }{2}} e^{-j\omega (\tau-\tau_e)} d\omega \\ =& A^2 \large e^{j\omega_{rf}\tau_e} \left[ \frac{e^{-j\omega (\tau-\tau_e)}}{-j(\tau-\tau_e)}\right]_{\frac{-\Delta \omega }{2}}^{\frac{\Delta \omega }{2}} \\ =& A^2 \large e^{j\omega_{rf}\tau_e} \left[ \frac{e^{j\frac{\Delta \omega }{2} (\tau-\tau_e)}- e^{-j\frac{\Delta \omega }{2} (\tau-\tau_e)}}{j(\tau-\tau_e)}\right] \\ =& A^2 \large e^{j\omega_{rf}\tau_e} \frac{2}{\tau-\tau_e}\left[ \frac{e^{j\frac{\Delta \omega }{2} (\tau-\tau_e)}- e^{-j\frac{\Delta \omega }{2} (\tau-\tau_e)}}{2j}\right] \\ =& \large A^2 e^{j\omega_{rf}\tau_e} \frac{2}{\tau-\tau_e} \sin\left(\frac{\Delta \omega }{2} (\tau-\tau_e)\right)\\ =& \large A^2 e^{j\omega_{rf}\tau_e} \frac{2}{\tau-\tau_e}\frac{\frac{\Delta \omega }{2}}{\frac{\Delta \omega }{2}} \sin\left(\frac{\Delta \omega }{2} (\tau-\tau_e)\right)\\ =& \large A^2 e^{j\omega_{rf}\tau_e} \Delta \omega \frac{ \sin\left(\frac{\Delta \omega }{2} (\tau-\tau_e)\right)}{ \frac{\Delta \omega }{2} (\tau-\tau_e) }\\ =& \large A^2 e^{j\omega_{rf}\tau_e} \Delta \omega \; \text{sinc}\left(\frac{\Delta \omega }{2} (\tau-\tau_e)\right)\\With sinc function, we can easily find \tau_e by finding peak value of |R(\tau)| :
\argmax_\tau |R(\tau)| = \tau_eWe take FFT of V[k] , think of the FFT here as a weighted sum.
R[\tau] &= \large \sum_{k=0}^{N-1} (V[k] )e^{-j \frac{2\pi k}{N} \tau} \\ &= \large \sum_{k=0}^{N-1} \left( A^2 e^{j\omega_{rf}\tau_e} e^{j\omega_k \tau_e } \Delta \omega_k \; \text{sinc} \left( \tfrac{\Delta \omega_k}{2} \tau_e \right) \right) e^{-j \frac{2\pi }{N}k \tau} \\ &= \large A^2 e^{j\omega_{rf}\tau_e} \sum_{k=0}^{N-1} \left( e^{j\omega_k \tau_e } \Delta \omega_k \; \text{sinc} \left( \tfrac{\Delta \omega_k}{2} \tau_e \right) \right) e^{-j \frac{2\pi }{N}k \tau} \\ \\ & (\omega_k \to \frac{2 \pi}{N} k \; , \; \; \Delta \omega_k \to \frac{2\pi}{N} ) \\ \\ &= \large A^2 e^{j\omega_{rf}\tau_e} \sum_{k=0}^{N-1} \left( e^{j\frac{2 \pi}{N} k \tau_e } \tfrac{2\pi}{N} \; \text{sinc} \left( \tfrac{\pi}{N} \tau_e \right) \right) e^{-j \frac{2\pi }{N}k \tau} \\ &= \large \frac{A^2 2\pi}{N} e^{j\omega_{rf}\tau_e} \text{sinc} \left( \tfrac{\pi}{N} \tau_e \right) \sum_{k=0}^{N-1} \left( e^{j\frac{2 \pi}{N} k \tau_e } \right) e^{-j \frac{2\pi }{N}k \tau} \\ &= \large \frac{A^2 2\pi}{N} e^{j\omega_{rf}\tau_e} \text{sinc} \left( \tfrac{\pi}{N} \tau_e \right) \sum_{k=0}^{N-1} e^{j\frac{2 \pi}{N} k (\tau_e -\tau) } \\ \\ &\text{using geometric sum }=\frac{a(1-r^N)}{1-r} , \text{where } a=1, r= e^{j\frac{2\pi}{N}(\tau_e-\tau)}\\ \\ &= \large \frac{A^2 2\pi}{N} e^{j\omega_{rf}\tau_e} \text{sinc} \left( \tfrac{\pi}{N} \tau_e \right) \frac{1-e^{j2\pi(\tau_e -\tau)}}{1-e^{j2\pi(\tau_e -\tau)/N}} \\ &= \large \frac{A^2 2\pi}{N} e^{j\omega_{rf}\tau_e} \text{sinc} \left( \tfrac{\pi}{N} \tau_e \right) \frac{ e^{j\pi(\tau_e-\tau)} ( e^{-j\pi(\tau_e-\tau)}-e^{j\pi(\tau_e -\tau)} )} {e^{j\pi(\tau_e-\tau)/N}( e^{-j\pi(\tau_e-\tau)/N}-e^{j\pi(\tau_e -\tau)/N})} \\ &= \large \frac{A^2 2\pi}{N} e^{j\omega_{rf}\tau_e} \text{sinc} \left( \tfrac{\pi}{N} \tau_e \right) \frac{ -e^{j\pi(\tau_e-\tau)} ( e^{j\pi(\tau_e -\tau)} -e^{-j\pi(\tau_e-\tau)} ) \frac{2j}{2j}} {-e^{j\pi(\tau_e-\tau)/N}( e^{j\pi(\tau_e -\tau)/N} -e^{-j\pi(\tau_e-\tau)/N}) \frac{2j}{2j}} \\ &= \large \frac{A^2 2\pi}{N} e^{j\omega_{rf}\tau_e} \text{sinc} \left( \tfrac{\pi}{N} \tau_e \right) \frac{ e^{j\pi(\tau_e-\tau)} \sin (\pi (\tau_e-\tau)) } {e^{j\pi(\tau_e-\tau)/N} \sin (\pi (\tau_e-\tau)/N) } \\ &= \large \frac{A^2 2\pi}{N} e^{j\omega_{rf}\tau_e} \text{sinc} \left( \tfrac{\pi}{N} \tau_e \right) e^{j[\pi(\tau_e-\tau)-\pi(\tau_e -\tau)/N]} \frac{ \sin (\pi (\tau_e-\tau)) } {\sin (\pi (\tau_e-\tau)/N) } \\ &= \large \frac{A^2 2\pi}{N} e^{j\omega_{rf}\tau_e} \text{sinc} \left( \tfrac{\pi}{N} \tau_e \right) e^{j \frac{N-1}{N}\pi(\tau_e-\tau)} \frac{ \sin (\pi (\tau_e-\tau)) } {\sin (\pi (\tau_e-\tau)/N) } \\ &(\text{usually } N \gg 1 \text{ in VLBI observation}) \\ &\approx \large \frac{A^2 2\pi}{N} e^{j\omega_{rf}\tau_e} \text{sinc} \left( \tfrac{\pi}{N} \tau_e \right) e^{j \pi(\tau_e-\tau)} \frac{ \sin (\pi (\tau_e-\tau)) } {\sin (\pi (\tau_e-\tau)/N) } \\ &(\text{using small angle approximation} ) \\ &\approx \large \frac{A^2 2\pi}{N} e^{j\omega_{rf}\tau_e} \text{sinc} \left( \tfrac{\pi}{N} \tau_e \right) e^{j \pi(\tau_e-\tau)} \frac{ \sin (\pi (\tau_e-\tau)) } {\pi (\tau_e-\tau)/N } \\ &= \large \frac{A^2 2\pi}{N} e^{j\omega_{rf}\tau_e} \text{sinc} \left( \tfrac{\pi}{N} \tau_e \right) e^{j \pi(\tau_e-\tau)} N \frac{ \sin (\pi (\tau_e-\tau)) } {\pi (\tau_e-\tau)} \\ &= \large \frac{A^2 2\pi}{N} e^{j\omega_{rf}\tau_e} \text{sinc} \left( \tfrac{\pi}{N} \tau_e \right) N e^{j \pi(\tau_e-\tau)} \; \text{sinc} (\pi (\tau_e-\tau)) \\Again with sinc function, we can easily find \tau_e by finding peak value of |R[\tau]| :
\argmax_\tau |R[\tau]| = \tau_eFX correlator output a grid : V[k,m] , so we will take a total of m FFTs that integrate over frequency index k and produce m numbers of R[\tau] . Meaning we will have a \tau_e for each row of V[k,m] .
2.2.1: Linear Residual Delay
Previously when trying to find residual delay, we assume \tau_e is constant, which is false because of earth's rotation. We can approximate \tau_e as a linear function of time:
\tau_e \to \tau_e (t) = \tau_{eo} + \dot \tau_{eo} tso x_2(t+\tau_0) becomes:
x_2(t+\tau_0) &=\frac{1}{2} a\big(t+\tau_e(t)\big) \large e^{j \theta\big(t+\tau_e(t)\big) }e^{j\omega_{rf}\tau_e(t)}\\ \text{where } &\tau_0 = \text{calculated phase center delay} \\ &\tau_e(t) = \tau_{eo} + \dot \tau_{eo} t\\ &\omega_{rf} = \text{center of sky frequency band, a constant}\\plug in \tau_e(t) for y_2(t) : y_2(t) =x_1\big(t+\tau_e(t)\big)\large e^{j\omega_{rf}\tau_e(t)}
Now we take the Fourier transform of y_2(t) .
Remember that the fourier transform we are taking is time dependent. i.e., we take fourier transform over short time intervals \Delta t centered at time t , this technique is referred as Short-Time Fourier Transform.
\mathcal{STFT}\{ y_2(t) \} &= Y_2(\omega,t) \\ &= \int_{-\Delta t/2}^{\Delta t /2} y_2(t+u)\large e^{-j\omega u}du \\ &= \int_{-\Delta t/2}^{\Delta t /2} x_1\big(t+u+\tau_e(t+u)\big)\large e^{j\omega_{rf}\tau_e(t+u)} e^{-j\omega u}du \\ \\ &\left(\begin{aligned} \text{claim: } &\tau_e(t+u) \approx \tau_e(t) \\ \text{proof: } &\\ \tau_e(t+u) &= \tau_{eo} + \dot \tau_{eo} (t+u) \\ & = \tau_{eo} + \dot \tau_{eo} t + \dot \tau_{eo} u \\ & (\dot \tau_{eo} \ll 1) \\ & \approx\tau_{eo} + \dot \tau_{eo} t \\ & =\tau_e(t) \\ \end{aligned}\right) \\ Y_2(\omega,t)&\approx \int_{-\Delta t/2}^{\Delta t /2} x_1\big(t+u+\tau_e(t)\big)\large e^{j\omega_{rf}\tau_e(t)} e^{-j\omega u}du \\ \\ &\left(\begin{aligned} \text{set }\bar u&=u+\tau_e(t)\\ \text{ }d\bar u&=du\\ \end{aligned}\right) \\ &= \large e^{j\omega_{rf}\tau_e(t)}\int_{-\Delta t/2+\tau_e(t)}^{\Delta t /2+\tau_e(t)} x_1(t+\bar u) e^{-j\omega (\bar u - \tau_e(t))}d\bar u \\ &= \large e^{j\omega_{rf}\tau_e(t)}\int_{-\Delta t/2+\tau_e(t)}^{\Delta t /2+\tau_e(t)} x_1(t+\bar u) e^{-j\omega \bar u}e^{j\omega \tau_e(t)}d\bar u \\ &= \large e^{j\omega_{rf}\tau_e(t)}e^{j\omega \tau_e(t)}\int_{-\Delta t/2+\tau_e(t)}^{\Delta t /2+\tau_e(t)} x_1(t+\bar u) e^{-j\omega \bar u}d\bar u \\ \\ &\left(\begin{aligned} \text{looking at the bounds: }&\\ \text{ because }& \Delta t /2 \gg \tau_e(t) \\ \text{ so } &\Delta t /2 + \tau_e(t) \approx \Delta t /2 \\ \text{ } &-\Delta t /2 + \tau_e(t) \approx -\Delta t /2 \\ \end{aligned}\right) \\ &\approx \large e^{j\omega_{rf}\tau_e(t)}e^{j\omega \tau_e(t)}\int_{-\Delta t/2}^{\Delta t /2} x_1(t+\bar u) e^{-j\omega \bar u}d\bar u \\ &= \large e^{j\omega_{rf}\tau_e(t)}e^{j\omega \tau_e(t)} \underbrace{ \int_{-\Delta t/2}^{\Delta t /2} x_1(t+ u) e^{-j\omega u}d u }_{=X_1(\omega,t) \text{ by definition}}\\ & = X_1(\omega,t) \large e^{j\omega \tau_e(t)}e^{j\omega_{rf}\tau_e(t)} \\ & = X_1(\omega,t) \large e^{j\omega ( \tau_{eo} + \dot \tau_{eo} t)}e^{j\omega_{rf}( \tau_{eo} + \dot \tau_{eo} t)} \\ & = X_1(\omega,t) \large e^{j\omega \tau_{eo} }{\color{red}e^{j\omega \dot \tau_{eo} t}}e^{j\omega_{rf} \tau_{eo} }{\color{red}e^{j\omega_{rf} \dot \tau_{eo} t} }\\ & = X_1(\omega,t) \large e^{j\omega \tau_{eo} }e^{j\omega_{rf} \tau_{eo} }{\color{red}e^{j(\omega +\omega_{rf} )\dot \tau_{eo}t}}\\ &\large ( \because \omega_{rf} \gg \omega \; \therefore {\color{red}e^{j(\omega +\omega_{rf} )\dot \tau_{eo}t}} \approx \;{\color{brown} e^{j\omega_{rf} \dot \tau_{eo}t }})\\ & \approx X_1(\omega,t) \large e^{j\omega \tau_{eo} }e^{j\omega_{rf} \tau_{eo} } \color{brown} e^{j\omega_{rf} \dot \tau_{eo} t}The X step:
So now the V(\omega,t) is:
V(\omega,t) =& Y_2(\omega,t) X_1^*(\omega,t)\\ \approx & [ X_1(\omega) \large e^{j\omega \tau_{eo} }e^{j\omega_{rf} \tau_{eo} }e^{j\omega_{rf} \dot \tau_{eo} t} ] X_1^*(\omega,t)\\ &(\text{suppose } |X_1(\omega,t)|=A) \\ =& A^2 \large \large e^{j\omega \tau_{eo} }e^{j\omega_{rf} \tau_{eo} }e^{j\omega_{rf} \dot \tau_{eo} t} \\ =& A^2 \large \large e^{j\omega_{rf} \tau_{eo} } \color{purple} e^{j\omega \tau_{eo} } \color{brown}e^{j\omega_{rf} \dot \tau_{eo} t} \\Continuous-time V(\omega,t) \to Discrete-time V[k,m] :
The visibility V(\omega,t) is in continuous-time but FX operates and output discrete time V[k,m] , let's find out what is V[k,m] :
Suppose the sampling rate is \omega_s , and we are doing N points FFT, then the visibility is V[k,m] with \{ k,m \in \Z \; \vert \; 0 \leq k \leq N-1 ,\vert \; 0 \leq m \leq M-1 \}
Now for a particular k and m :
its center frequency \omega_k is:
\omega_k = k\frac{\omega_s}{N}and its channel bandwidth is:
\Delta \omega_k = \frac{\omega_s}{N}and its center time is:
t_m = mT+\tfrac{T}{2}, \; \; \text{where $T=$ integration time}and V[k,m] is:
V[k,m] & \approx \int_{t_m -\Delta t/2}^{t_m + \Delta t/2} \int_{\omega_k -\Delta \omega_k/2}^{\omega_k + \Delta \omega_k/2} V(\omega,t) d\omega dt \\ & = A^2 \large \large e^{j\omega_{rf} \tau_{eo} } \int_{t_m -\Delta t/2}^{t_m + \Delta t/2} \int_{\omega_k -\Delta \omega_k/2}^{\omega_k + \Delta \omega_k/2}e^{j\omega \tau_{eo} }e^{j\omega_{rf} \dot \tau_{eo} t} d\omega dt \\ & = A^2 \large \large e^{j\omega_{rf} \tau_{eo} } {\color{green} \int_{t_m -\Delta t/2}^{t_m + \Delta t/2} e^{j\omega_{rf} \dot \tau_{eo} t} dt }{\color{blue} \int_{\omega_k -\Delta \omega_k/2}^{\omega_k + \Delta \omega_k/2}e^{j\omega \tau_{eo} }d\omega } \\ & = A^2 \large \large e^{j\omega_{rf} \tau_{eo} } {\color{green} \left[ \frac{e^{j\omega_{rf}\dot \tau_{eo}t}}{j\omega_{rf}\dot \tau_{eo}}\right]_{t_m -\Delta t/2}^{t_m + \Delta t/2} } {\color{blue}\left[ \frac{e^{j\omega \tau_{eo} }}{j \tau_{eo}}\right]_{\omega_k -\Delta \omega_k/2}^{\omega_k + \Delta \omega_k/2}} \\ & = A^2 \large \large e^{j\omega_{rf} \tau_{eo} } {\color{green}e^{j \omega_{rf}\dot \tau_{eo} t_m }\left[ \frac{e^{j\omega_{rf}\dot \tau_{eo} \frac{\Delta t}{2} }- e^{-j\omega_{rf}\dot \tau_{eo} \frac{\Delta t}{2} } }{j\omega_{rf}\dot \tau_{eo}}\right]} {\color{blue} e^{j \omega_{k} \tau_{eo} } \left[ \frac{e^{j\frac{\Delta \omega_k}{2} \tau_{eo} }-e^{-j\frac{\Delta \omega_k}{2} \tau_{eo} }}{j \tau_{eo}}\right]} \\ & = A^2 \large \large e^{j\omega_{rf} \tau_{eo} } {\color{green}e^{j \omega_{rf}\dot \tau_{eo} t_m } \frac{2}{\omega_{rf} \dot \tau_{eo}} \left[ \frac{e^{j\omega_{rf}\dot \tau_{eo} \frac{\Delta t}{2} }- e^{-j\omega_{rf}\dot \tau_{eo} \frac{\Delta t}{2} } }{2j}\right]} {\color{blue} e^{j \omega_{k} \tau_{eo} }\frac{2}{\tau_{eo}} \left[ \frac{e^{j\frac{\Delta \omega_k}{2} \tau_{eo} }-e^{-j\frac{\Delta \omega_k}{2} \tau_{eo} }}{2j }\right]} \\ & = A^2 \large \large e^{j\omega_{rf} \tau_{eo} } {\color{green}e^{j \omega_{rf}\dot \tau_{eo} t_m } \frac{2}{\omega_{rf} \dot \tau_{eo}} \sin \left( \frac{\Delta t }{2} \omega_{rf} \dot \tau_{eo} \right)} {\color{blue} e^{j \omega_{k} \tau_{eo} }\frac{2}{\tau_{eo}} \sin \left( \frac{\Delta \omega_k }{2} \tau_{eo} \right)} \\ & = A^2 \large \large e^{j\omega_{rf} \tau_{eo} } {\color{green}e^{j \omega_{rf}\dot \tau_{eo} t_m } \frac{\Delta t}{ \frac{\Delta t}{2} \omega_{rf} \dot \tau_{eo}} \sin \left( \frac{\Delta t }{2} \omega_{rf} \dot \tau_{eo} \right)} {\color{blue} e^{j \omega_{k} \tau_{eo} }\frac{\Delta \omega_k}{ \frac{\Delta \omega_k}{2} \tau_{eo}} \sin \left( \frac{\Delta \omega_k }{2} \tau_{eo} \right)} \\ & = A^2 \large \large e^{j\omega_{rf} \tau_{eo} } {\color{green}e^{j \omega_{rf}\dot \tau_{eo} t_m } \Delta t \; \text{sinc} \left( \frac{\Delta t }{2} \omega_{rf} \dot \tau_{eo} \right)} {\color{blue} e^{j \omega_{k} \tau_{eo} } \Delta \omega_k \; \text{sinc} \left( \frac{\Delta \omega_k }{2} \tau_{eo} \right)} \\As you can see because of the sinc function, if your \tau_{eo} or \omega_{rf} \dot \tau_{eo} is big, V[k,m] can be squashed to 0.
Finding \tau_{eo} , \dot \tau_{eo} :
We take 2D-FT of V(\omega,t) , again, think of the FTs here as weighted sum.
R(\tau,\dot \tau) =& {\color{brown}\int_{-T/2}^{T/2}} {\color{purple}\int_{-\Delta \omega/2}^{\Delta \omega/2}} \large \underbrace{ \left[A^2 e^{j\omega_{rf} \tau_{eo} } {\color{purple}e^{j\omega \tau_{eo} }} {\color{brown}e^{j(\omega_{rf} \dot \tau_{eo}) t} }\right] }_{V(\omega,t) } {\color{purple}e^{-j\omega \tau }} {\color{brown}e^{-j(\omega_{rf} \dot \tau) t }} {\color{purple}d\omega} {\color{brown}dt} \\ =& A^2 e^{j\omega_{rf} \tau_{eo} } \large {\color{brown}\int_{-T/2}^{T/2}} {\color{brown}e^{j(\omega_{rf} \dot \tau_{eo}) t} } {\color{brown}e^{-j(\omega_{rf} \dot \tau) t }} {\color{brown}dt} {\color{purple}\int_{-\Delta \omega/2}^{\Delta \omega/2}} {\color{purple}e^{j\omega \tau_{eo} }} {\color{purple}e^{-j\omega \tau }} {\color{purple}d\omega} \\ =& A^2 e^{j\omega_{rf} \tau_{eo} } \large {\color{brown}\int_{-T/2}^{T/2}} {\color{brown}e^{-j\omega_{rf} (\dot \tau -\dot \tau_{eo}) t} } {\color{brown}dt} {\color{purple}\int_{-\Delta \omega/2}^{\Delta \omega/2}} {\color{purple}e^{-j\omega (\tau- \tau_{eo}) }} {\color{purple}d\omega} \\ =& A^2 e^{j\omega_{rf} \tau_{eo} } \large {\color{brown} \left[ \frac{e^{-j\omega_{rf} (\dot \tau - \dot \tau_{eo}) t}}{-j\omega_{rf} (\dot \tau - \dot \tau_{eo})} \right]_{-T/2}^{T/2} } {\color{purple} \left[ \frac{e^{-j\omega (\tau - \tau_{eo})}}{-j (\tau - \tau_{eo})} \right]_{-\Delta \omega/2}^{\Delta \omega/2} } \\ =& A^2 e^{j\omega_{rf} \tau_{eo} } \large {\color{brown} \left( \frac{e^{-j\omega_{rf} (\dot \tau - \dot \tau_{eo}) \frac{T}{2}} - e^{j\omega_{rf} (\dot \tau - \dot \tau_{eo}) \frac{T}{2}}}{-j\omega_{rf} (\dot \tau - \dot \tau_{eo})} \right) } {\color{purple} \left( \frac{e^{-j\frac{\Delta \omega}{2} (\tau - \tau_{eo})} - e^{j\frac{\Delta \omega}{2} (\tau - \tau_{eo})}}{-j (\tau - \tau_{eo})} \right) } \\ =& A^2 e^{j\omega_{rf} \tau_{eo} } \large {\color{brown} \left( \frac{2 \sin\left(\omega_{rf} (\dot \tau - \dot \tau_{eo}) \frac{T}{2}\right)}{\omega_{rf} (\dot \tau - \dot \tau_{eo})} \right) } {\color{purple} \left( \frac{2 \sin\left(\frac{\Delta \omega}{2} (\tau - \tau_{eo})\right)}{(\tau - \tau_{eo})} \right) } \\ =& A^2 e^{j\omega_{rf} \tau_{eo} } \large {\color{brown} T \left( \frac{\sin\left( \frac{T}{2} \omega_{rf} (\dot \tau - \dot \tau_{eo}) \right)}{ \frac{T}{2} \omega_{rf} (\dot \tau - \dot \tau_{eo}) } \right) } {\color{purple} \Delta \omega \left( \frac{\sin\left( \frac{\Delta \omega}{2} (\tau - \tau_{eo}) \right)}{ \frac{\Delta \omega}{2} (\tau - \tau_{eo}) } \right) } \\ =& A^2 e^{j\omega_{rf} \tau_{eo} } \large \color{brown} T \; \text{sinc}\left( \frac{T}{2} \omega_{rf} (\dot \tau - \dot \tau_{eo})\right) \color{purple} \Delta \omega \; \text{sinc}\left( \frac{\Delta \omega }{2} (\tau- \tau_{eo})\right)With 2 sinc functions, we can easily find \tau_{eo} , \dot \tau_{eo} by finding peak value of |R(\tau,\dot \tau)| :
\argmax_{\tau,\dot \tau} |R(\tau,\dot \tau)| = \tau_{eo} , \dot \tau_{eo}Continuous-time R(\tau,\dot \tau) \to Discrete-time R[\tau,\dot \tau] :
We take 2D-FFT of V[k,m] , think of the FFT here as a weighted sum.
R[\tau,\dot \tau] &= \large \sum_{m=0}^{M-1} \sum_{k=0}^{N-1} V[k,m] e^{-j \frac{2\pi}{N}k \tau} e^{-j \frac{2\pi}{M}m \omega_{rf}\dot \tau} \\ &= \large \sum_{m=0}^{M-1} \sum_{k=0}^{N-1} A^2 \large \large e^{j\omega_{rf} \tau_{eo} } {\color{green}e^{j \omega_{rf}\dot \tau_{eo} t_m } \Delta t \; \text{sinc} \left( \frac{\Delta t }{2} \omega_{rf} \dot \tau_{eo} \right)} {\color{blue} e^{j \omega_{k} \tau_{eo} } \Delta \omega_k \; \text{sinc} \left( \frac{\Delta \omega_k }{2} \tau_{eo} \right)} e^{-j \frac{2\pi}{N}k \tau} e^{-j \frac{2\pi}{M}m \omega_{rf}\dot \tau} \\ &= \large A^2 \large \large e^{j\omega_{rf} \tau_{eo} } \sum_{m=0}^{M-1} \sum_{k=0}^{N-1} {\color{green}e^{j \omega_{rf}\dot \tau_{eo} t_m } \Delta t \; \text{sinc} \left( \frac{\Delta t }{2} \omega_{rf} \dot \tau_{eo} \right)} {\color{blue} e^{j \omega_{k} \tau_{eo} } \Delta \omega_k \; \text{sinc} \left( \frac{\Delta \omega_k }{2} \tau_{eo} \right)} e^{-j \frac{2\pi}{N}k \tau} e^{-j \frac{2\pi}{M}m \omega_{rf}\dot \tau} \\ & ( \color{green} t_m \to \frac{2\pi}{M}m , \; \Delta t \to \frac{2\pi}{M} ,\; \; \color{blue} \omega_k \to \frac{2\pi}{N}k , \; \Delta \omega_k \to \frac{2\pi}{N} \color{black} )\\ &= \large A^2 \large \large e^{j\omega_{rf} \tau_{eo} } \sum_{m=0}^{M-1} \sum_{k=0}^{N-1} {\color{green}e^{j \omega_{rf}\dot \tau_{eo} \frac{2\pi}{M}m } \frac{2\pi}{M} \; \text{sinc} \left( \frac{\tfrac{2\pi}{M} }{2} \omega_{rf} \dot \tau_{eo} \right)} {\color{blue} e^{j \frac{2\pi}{N}k \tau_{eo} } \frac{2\pi}{N} \; \text{sinc} \left( \frac{\frac{2\pi}{N} }{2} \tau_{eo} \right)} e^{-j \frac{2\pi}{N}k \tau} e^{-j \frac{2\pi}{M}m \omega_{rf}\dot \tau} \\ &= \large A^2 \large \large e^{j\omega_{rf} \tau_{eo} } \sum_{m=0}^{M-1} \sum_{k=0}^{N-1} {\color{green}e^{j \omega_{rf}\dot \tau_{eo} \frac{2\pi}{M}m } \frac{2\pi}{M} \; \text{sinc} \left( \tfrac{\pi}{M} \omega_{rf} \dot \tau_{eo} \right)} {\color{blue} e^{j \frac{2\pi}{N}k \tau_{eo} } \frac{2\pi}{N} \; \text{sinc} \left( \tfrac{\pi}{N} \tau_{eo} \right)} e^{-j \frac{2\pi}{N}k \tau} e^{-j \frac{2\pi}{M}m \omega_{rf}\dot \tau} \\ &= \large A^2 \large \large e^{j\omega_{rf} \tau_{eo} }\frac{2\pi}{M} \; \text{sinc} \left( \tfrac{\pi}{M} \omega_{rf} \dot \tau_{eo} \right) \frac{2\pi}{N} \; \text{sinc} \left( \tfrac{\pi}{N} \tau_{eo} \right) \sum_{m=0}^{M-1} \sum_{k=0}^{N-1} {\color{green}e^{j \omega_{rf}\dot \tau_{eo} \frac{2\pi}{M}m } } {\color{blue} e^{j \frac{2\pi}{N}k \tau_{eo} } } e^{-j \frac{2\pi}{N}k \tau} e^{-j \frac{2\pi}{M}m \omega_{rf}\dot \tau} \\ &= \large A^2 \large \large e^{j\omega_{rf} \tau_{eo} }\frac{2\pi}{M} \; \text{sinc} \left( \tfrac{\pi}{M} \omega_{rf} \dot \tau_{eo} \right) \frac{2\pi}{N} \; \text{sinc} \left( \tfrac{\pi}{N} \tau_{eo} \right) \left( \sum_{m=0}^{M-1} {\color{green}e^{j \frac{2\pi}{M}m \omega_{rf}\dot \tau_{eo} } } e^{-j \frac{2\pi}{M}m \omega_{rf}\dot \tau} \right) \left( \sum_{k=0}^{N-1} {\color{blue} e^{j \frac{2\pi}{N}k \tau_{eo} } } e^{-j \frac{2\pi}{N}k \tau} \right) \\ &= \large A^2 \large \large e^{j\omega_{rf} \tau_{eo} }\frac{2\pi}{M} \; \text{sinc} \left( \tfrac{\pi}{M} \omega_{rf} \dot \tau_{eo} \right) \frac{2\pi}{N} \; \text{sinc} \left( \tfrac{\pi}{N} \tau_{eo} \right) \left( \sum_{m=0}^{M-1} {\color{green}e^{j \omega_{rf} \frac{2\pi}{M}m (\dot \tau_{eo}-\dot \tau) } } \right) \left( \sum_{k=0}^{N-1} {\color{blue} e^{j \frac{2\pi}{N}k (\tau_{eo}-\tau) } } \right) \\ &\text{using geometric sum =}\frac{a(1-r^N)}{1-r} :\\ &\large \begin{pmatrix} {\color{green} a_m = 1 } & {\color{blue} a_k = 1 } \\ {\color{green} r_m = e^{j \omega_{rf} \frac{2\pi}{M} (\dot \tau_{eo}-\dot \tau) } } & {\color{blue} r_k = e^{j \frac{2\pi}{N} (\tau_{eo}-\tau) } } \\ \end{pmatrix} \\ \\ &= \large A^2 \large \large e^{j\omega_{rf} \tau_{eo} }\frac{2\pi}{M} \; \text{sinc} \left( \tfrac{\pi}{M} \omega_{rf} \dot \tau_{eo} \right) \frac{2\pi}{N} \; \text{sinc} \left( \tfrac{\pi}{N} \tau_{eo} \right) {\color{green}\left( \frac{1-e^{j \omega_{rf} 2\pi (\dot \tau_{eo}-\dot \tau) } }{1-e^{j \omega_{rf} \frac{2\pi}{M} (\dot \tau_{eo}-\dot \tau) } } \right) } {\color{blue} \left( \frac{ 1- e^{j 2\pi (\tau_{eo}-\tau) } }{1-e^{j \frac{2\pi}{N} (\tau_{eo}-\tau) } } \right) } \\ &( \text{set } {\color{green} \mu = \frac{\omega_{rf} \pi (\dot \tau_{eo}-\dot \tau) }{M} } \text{ and } {\color{blue} \phi = \frac{ \pi ( \tau_{eo}- \tau) }{N} } ) \\ &= \large A^2 \large \large e^{j\omega_{rf} \tau_{eo} }\frac{2\pi}{M} \; \text{sinc} \left( \tfrac{\pi}{M} \omega_{rf} \dot \tau_{eo} \right) \frac{2\pi}{N} \; \text{sinc} \left( \tfrac{\pi}{N} \tau_{eo} \right) {\color{green}\left( \frac{1-e^{j 2M\mu } }{1-e^{j 2\mu} } \right) } {\color{blue} \left( \frac{1-e^{j 2N\phi } }{1-e^{j 2\phi} } \right) } \\ &= \large A^2 \large \large e^{j\omega_{rf} \tau_{eo} }\frac{2\pi}{M} \; \text{sinc} \left( \tfrac{\pi}{M} \omega_{rf} \dot \tau_{eo} \right) \frac{2\pi}{N} \; \text{sinc} \left( \tfrac{\pi}{N} \tau_{eo} \right) {\color{green}\left( \frac{-e^{jM\mu}(e^{jM\mu}-e^{-jM\mu})/2j}{ -e^{j\mu}(e^{j\mu}-e^{-j\mu})/2j } \right) } {\color{blue} \left( \frac{-e^{jN\phi}(e^{jN\phi}-e^{-jN\phi})/2j}{ -e^{j\phi}(e^{j\phi}-e^{-j\phi})/2j } \right) } \\ &= \large A^2 \large \large e^{j\omega_{rf} \tau_{eo} }\frac{2\pi}{M} \; \text{sinc} \left( \tfrac{\pi}{M} \omega_{rf} \dot \tau_{eo} \right) \frac{2\pi}{N} \; \text{sinc} \left( \tfrac{\pi}{N} \tau_{eo} \right) {\color{green}\left( e^{j(M-1)\mu}\frac{\sin(M\mu)}{ \sin(\mu) } \right) } {\color{blue} \left( e^{j(N-1)\phi} \frac{\sin(N\phi)}{ \sin(\phi) } \right) } \\ &\text{using small angle approximation} :\\ & \approx\large A^2 \large \large e^{j\omega_{rf} \tau_{eo} }\frac{2\pi}{M} \; \text{sinc} \left( \tfrac{\pi}{M} \omega_{rf} \dot \tau_{eo} \right) \frac{2\pi}{N} \; \text{sinc} \left( \tfrac{\pi}{N} \tau_{eo} \right) {\color{green}\left( e^{j(M-1)\mu}\frac{\sin(M\mu)}{ \mu } \right) } {\color{blue} \left( e^{j(N-1)\phi} \frac{\sin(N\phi)}{ \phi } \right) } \\ & =\large A^2 \large \large e^{j\omega_{rf} \tau_{eo} } 2\pi \; \text{sinc} \left( \tfrac{\pi}{M} \omega_{rf} \dot \tau_{eo} \right) 2\pi \; \text{sinc} \left( \tfrac{\pi}{N} \tau_{eo} \right) {\color{green}\left( e^{j(M-1)\mu}\frac{\sin(M\mu)}{ M \mu } \right) } {\color{blue} \left( e^{j(N-1)\phi} \frac{\sin(N\phi)}{ N\phi } \right) } \\ & =\large A^2 \large \large e^{j\omega_{rf} \tau_{eo} } 2\pi \; \text{sinc} \left( \tfrac{\pi}{M} \omega_{rf} \dot \tau_{eo} \right) 2\pi \; \text{sinc} \left( \tfrac{\pi}{N} \tau_{eo} \right) {\color{green}\left( e^{j(M-1)\mu}\text{sinc} (M\mu) \right) } {\color{blue} \left( e^{j(N-1)\phi} \text{sinc} (N\phi) \right) } \\ & =\large A^2 \large \large e^{j\omega_{rf} \tau_{eo} } 2\pi \; \text{sinc} \left( \tfrac{\pi}{M} \omega_{rf} \dot \tau_{eo} \right) 2\pi \; \text{sinc} \left( \tfrac{\pi}{N} \tau_{eo} \right) {\color{green}\left( e^{j\frac{ M-1}{M} \omega_{rf} \pi (\dot \tau_{eo}-\dot \tau)}\text{sinc} (\omega_{rf} \pi (\dot \tau_{eo}-\dot \tau)) \right) } {\color{blue} \left( e^{j\frac{ N-1}{N} \pi ( \tau_{eo}- \tau) } \text{sinc} (\pi ( \tau_{eo}- \tau)) \right) } \\Using the properties of sinc function, we can easily find \tau_{eo}, \dot \tau_{eo} by finding peak value of |R[\tau,\dot \tau]| :
\argmax_{\{\tau, \dot \tau\}} |R[\tau, \dot \tau]| =\{ \tau_{eo}, \dot \tau_{eo} \}FX correlator output a grid : V[k,m] , so we will take a total of M+N FFTs. And in this 2D-FFT grid R[\tau,\dot \tau] , the \{\tau, \dot \tau\} pair that gives the maximum value will be our approximated \tau_{eo}, \dot \tau_{eo} values.