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Matt Pitkin added We_can_define_the_Fourier__.tex
over 8 years ago
Commit id: a529f451035e0b7966f816319cd7ea3bf95d53ee
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We can define the Fourier transform of our signal, which has complex amplitude at time $t_k$ of $y_k$ as
\begin{align}
H_k(f) \approx & y_k e^{i\phi_k-i\pi f \Delta t} \int_{-\Delta t/2}^{\Delta t/2} \exp{\left[2\pi i \left((f_k-f)t + \frac{t^2}{2}\dot{f}_k + \frac{t^3}{6}\ddot{f}_k \right) \right]} \nonumber \\
& + y_k^{*}e^{-i\phi_k-i\pi f \Delta t} \int_{-\Delta t/2}^{\Delta t/2} \exp{\left[-2\pi i \left((f_k+f)t - \frac{t^2}{2}\dot{f}_k - \frac{t^3}{6}\ddot{f}_k \right) \right]}.
\end{align}
