For a given frequency row $l$, the cross-correlation coefficient $X$ measures the sum of $\rho_1\rho_2$ when the $\rho_1$ time bins are shifted by $m-N_\tau/2$ bins with respect to those of $\rho_2$. The coefficient $X$ can be seen as a function of the time shift between detector 1 and detector 2:

For a given frequency row $l$, the cross-correlation coefficient $\xi$ measures the sum of $\rho_1\rho_2$ when the $\rho_1$ time bins are shifted by $m-N_\tau/2$ bins with respect to those of $\rho_2$. The coefficient $\xi$ can be seen as a function of the time shift between detector 1 and detector 2:

@@ -68,11 +68,11 @@ We note that, in Eq.~\ref{eq:xcorr}, the index $N_\tau/2-m+m^{\prime}$ can be ou

\begin{figure}

\center

\includegraphics[width=16cm]{./figures/xcorr.pdf}

\caption{Representation of three cross-correlation coefficients for a given frquency row $l$: $X[0][l]$ (left), $X[N_\tau/2][l]$ (center), and $X[N_\tau-1][l]$. The detector 1 (blue) time bins are shifted by $\delta t$ with respect to those of detector 2 (green). The resulting cross-correlation coefficient is represented by the red square. The dashed-line squares indicate the circular periodicity used to compute the cross-correlation.}

\caption{Representation of three cross-correlation coefficients for a given frquency row $l$: $\xi[0][l]$ (left), $\xi[N_\tau/2][l]$ (center), and $\xi[N_\tau-1][l]$. The detector 1 (blue) time bins are shifted by $\delta t$ with respect to those of detector 2 (green). The resulting cross-correlation coefficient is represented by the red square. The dashed-line squares indicate the circular periodicity used to compute the cross-correlation.}

\label{fig:xcorr}

\end{figure}

To optimize the computation of $X[m][l]$, the cross-correlation is calculated in the Fourier domain. We define the Fourier transform of a discrete time series $d$ of size $N_\tau$ as:

To optimize the computation of $\xi[m][l]$, the cross-correlation is calculated in the Fourier domain. We define the Fourier transform of a discrete time series $d$ of size $N_\tau$ as:

\begin{equation}

\tilde{d}[k] = \frac{1}{N_\tau}\sum_{m=0}^{(N_\tau-1)} d[m] e^{-2i\pi km / N_\tau}.

\label{eq:dft}

...

...

@@ -84,8 +84,22 @@ The inverse Fourier transform is given by:

\end{equation}

Using the definition of Eq.~\ref{eq:dft}, we write Eq.~\ref{eq:xcorr} in the Fourier domain:

\begin{equation}

\tilde{X}[k][l] = \frac{1}{N_\tau}\sum_{m=0}^{(N_\tau-1)}\sum_{m^{\prime}=0}^{(N_\tau-1)}\rho_1[m^{\prime}][l] \times\rho_2[N_\tau/2-m+m^{\prime}][l] \times e^{+2i\pi km / N_\tau}.

\tilde{\xi}[k][l] = \frac{1}{N_\tau}\sum_{m=0}^{(N_\tau-1)}\sum_{m^{\prime}=0}^{(N_\tau-1)}\rho_1[m^{\prime}][l] \times\rho_2[N_\tau/2-m+m^{\prime}][l] \times e^{-2i\pi km / N_\tau}.

\end{equation}

We substitute $n$ for $N_\tau/2-m+m^{\prime}$ and we get:

In Eq.~\ref{eq:xcorr_theorem}, the first sum is the Fourier transform of $\rho_1$. For the second sum, $\rho_2$ and the exponential are $N_\tau$-periodic functions of so that we can apply a shift of $m^{\prime}-N_\tau/2+1$ without changing the result of the sum:

where $\rho_2^*$ is the complex conjugate of $\rho_2$. Equation~\ref{eq:xcorr_ft} is a variant of the well-known cross-correlation theorem: there is an extra phase of $\pi$ due to the fact that we chose a $N_\tau/2$ shift in Eq.~\ref{eq:xcorr}.