added reverberation equations to press

press/reverberation/reverb_equations.tex

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+\documentclass[11pt,letterpaper]{article}
+
+\usepackage{natbib}
+\usepackage{graphicx}
+\usepackage[margin=1.in,centering]{geometry}
+
+\begin{document}
+
+Consider two lightcurves $x(t)$ and $y(t)$, where $x(t)$ is the driving lightcurve and $y(t)$ is the reprocessed lightcurve.  If they are related by a linear impulse response, $g(\tau)$, then: 
+
+\begin{equation}
+y(t) = \int_{-\infty}^{\infty} g(\tau) x(t-\tau)  {\rm d}\tau
+\end{equation} 
+
+So, $y(t)$ is a delayed and blurred version of $x(t)$, with the amount of delay and blurring encoded in $g(\tau)$.
+
+The power spectral density (PSD) of $x(t)$ is calculated from the Fourier transform of $x(t)$, which we denote $X(\nu)$.  The PSD is $|X(\nu)|^2 = X^*(\nu)X(\nu)$, where the $^*$ denotes the complex conjugate.  From the convolution theorem of Fourier transforms we can write:
+
+\begin{equation}
+Y(\nu) = G(\nu) X(\nu)
+\end{equation}
+
+This means it is easy to relate the PSD of the reprocessed lightcurve to the PSD of the driving lightcurve and the impulse response function:
+
+\begin{equation}
+|Y(\nu)|^2 = |G(\nu)|^2 |X(\nu)|^2
+\end{equation}
+
+The cross spectrum is defined as
+\begin{equation}
+C(\nu) = X^*(\nu) Y(\nu)
+\end{equation}
+the phase, $\phi$, of which gives the phase lag between X and Y at each Fourier frequency, $\nu$.  This can be converted to a time lag through: 
+\begin{equation}
+\tau(\nu) = \frac{\phi(\nu)}{2\pi\nu}
+\end{equation}
+Since $Y(\nu) = G(\nu) X(\nu)$,  the cross spectrum can be written as:
+\begin{equation}
+C(\nu) = X^*(\nu) G(\nu) X(\nu) =  G(\nu) |X(\nu)|^2 
+\end{equation}
+thus, for a given impulse response function, one can trivially predict the time lags as a function of frequency, $\tau(\nu)$, by calculating the phase of $G(\nu)$, and the frequency dependence of the lags directly relates to the shape of the response function.
+
+\end{document}