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a/press/freqdomain/reverb_equations.pdf b/press/freqdomain/reverb_equations.pdf new file mode 100644 index 0000000..89873fe Binary files /dev/null and b/press/freqdomain/reverb_equations.pdf differ diff --git a/press/freqdomain/reverb_equations.tex b/press/freqdomain/reverb_equations.tex new file mode 100644 index 0000000..afe59e4 --- /dev/null +++ b/press/freqdomain/reverb_equations.tex @@ -0,0 +1,43 @@ +\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} \ No newline at end of file