mirror of
https://asciireactor.com/otho/psdlag-agn.git
synced 2024-11-23 17:05:05 +00:00
43 lines
1.8 KiB
TeX
43 lines
1.8 KiB
TeX
|
\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}
|