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Details of Normalization

The Fourier transform convensions of Numerical Recipes are used here. In particular, suppose that the time series $a(t)$ is sampled at intervals of $\Delta t=1/\texttt{srate}$ and these samples are stored in the array array[0..n-1] where n is the number of samples taken (thus, the total observation time is $T=\texttt{n}\Delta t$. Then, the Fourier transform $\tilde{a}(f)=\int dt\,e^{2\pi ift}a(t)$ is related to the FFT of array by

\begin{displaymath}
\tilde{a}(f_j) = \Delta t \times
( \texttt{atilde[2j]} + i\,\texttt{atilde[2j+1]} )
\end{displaymath} (13.2.306)

where atilde[0..n-1] is produced from array[0..n-1] by realft(...,1). Note that the frequency $f_j=j\Delta f$ where $\Delta f=(\texttt{n}\Delta t)^{-1}=\texttt{srate}/\texttt{n}$. Define the one-sided mean power spectrum of $a(t)$ by
\begin{displaymath}
S_a(\vert f\vert) = \frac{2}{T}\,\langle\vert\tilde{a}(f)\vert^2\rangle
\end{displaymath} (13.2.307)

and similarly define
\begin{displaymath}
\texttt{power[j]} = 2 \langle (\texttt{atilde[2j]})^2
+ (\texttt{atilde[2j+1]})^2 \rangle .
\end{displaymath} (13.2.308)

Notice that $S_a(\vert f\vert)$ has dimensions of time while power[j] is, of course, dimensionless. These two power spectra are related by
\begin{displaymath}
S_a(f_j) = \frac{1}{\texttt{n}\times\texttt{srate}}\,\texttt{power[j]}.
\end{displaymath} (13.2.309)

A well known ``feature'' of the inverse FFT produced by realft(...,-1) is that

\begin{displaymath}
\texttt{realft(atilde,n,-1)} \Rightarrow
\frac{\texttt{n}}{2}\times\texttt{array[0..n-1]}.
\end{displaymath} (13.2.310)

We can express the correlation between two time series, $a(t)$ and $b(t)$, weighted by twice the inverse power spectrum, as
$\displaystyle c(t_k)$ $\textstyle =$ $\displaystyle \frac{1}{2}\int_{-\infty}^{\infty} df\,e^{-2\pi ift_k}
\tilde{a}(f)\tilde{b}^\ast(f)\,\frac{2}{S(\vert f\vert)}$  
  $\textstyle \simeq$ $\displaystyle \frac{1}{2}\frac{\texttt{srate}}{\texttt{n}} \biggl\{
\sum_{\text...
...\scriptsize n}/2-1}
e^{-2\pi i\texttt{\scriptsize {jk}}/\texttt{\scriptsize n}}$  
    $\displaystyle \qquad
\times\frac{\texttt{atilde[2j]}+i\,\texttt{atilde[2j+1]}}{\texttt{srate}}$  
    $\displaystyle \qquad
\times\frac{\texttt{btilde[2j]}-i\,\texttt{btilde[2j+1]}}{\texttt{srate}}$  
    $\displaystyle \qquad
\times\frac{2\times\texttt{n}\times\texttt{srate}}{\texttt{power[j]}}
\biggr\} + {\mathrm{cc}}$  
  $\textstyle \Leftarrow$ $\displaystyle \texttt{correlate(c,atilde,btilde,weight,n)}$ (13.2.311)

[cf. equation ([*])] where $\texttt{weight[j]}=2/\texttt{power[j]}$ and $c(t_k)=\texttt{c[k]}$. Notice that all the factors of 2, n, and srate are accounted for. However, the correlation defined in the first line is one half of the correlation defined in equation ([*]).


next up previous contents
Next: Function: strain_spec() Up: Binary Inspiral Search on Previous: Reception of a Signal   Contents
Bruce Allen 2000-11-19