next up previous contents
Next: Reception of a Signal Up: The Statistical Theory of Previous: A Receiver for a   Contents

Reception of a Signal with Unknown Arrival Time

In general, the expected signals will be much shorter than the observation time, and we do not know when the signal will occur. We wish not only to detect these signals but also to measure their arrival time. To do this, we construct the time series

\begin{displaymath}
x(t) = \int_{-\infty}^\infty df\,e^{-2\pi ift}
\frac{\tild...
...{s}_0(f)+\tilde{h}(f)\tilde{s}_0^\ast(f)}
{S_h(\vert f\vert)}
\end{displaymath} (13.1.300)

and
\begin{displaymath}
y(t) = \int_{-\infty}^\infty df\,e^{-2\pi ift}
\frac{\tild...
...{s}_1(f)+\tilde{h}(f)\tilde{s}_1^\ast(f)}
{S_h(\vert f\vert)}
\end{displaymath} (13.1.301)

for the case of an unknown phase, or just the quantity $x(t)$ with $s_0(t)=s(t)$ if the signal is known completely (up to the arrival time). For each observation period, we compute the mode statistic,
\begin{displaymath}
\rho = \sigma^{-1}\max_t\left\{
\begin{array}{ll}
\vert x...
...rt{x^2(t)+y^2(t)} & \mbox{unknown phase},
\end{array} \right.
\end{displaymath} (13.1.302)

and set some threshold $\rho_\ast$ for this statistic.

In general, it is difficult to compute the false alarm probability for a given threshold because the values of $x(t)$ and $y(t)$ are correlated. However, in the limit of large thresholds (small false alarm probability for a large observation time), each sample $x(t_i)$ and $y(t_i)$ becomes effectively independent in the sense that if the threshold is exceeded at a time $t_i$, then it is unlikely that it will be exceeded again at any time within the correlation timescale of time $t_i$. The false alarm probability is then the probability computed for a single time receiver (above) multiplied by the total number of samples in the observation time. Stated another way, the rate of false alarms is

\begin{displaymath}
\nu(\mbox{false alarm}) \simeq \Delta^{-1} \left\{
\begin{...
...box{unknown phase}
\end{array} \right. \qquad (\rho_\ast\gg1)
\end{displaymath} (13.1.303)

where $\Delta^{-1}$ is the sampling rate. The probability of a false alarm in the observation time $T$ is $P(\mbox{false alarm})=T\times\nu(\mbox{false alarm})$. Although this false alarm rate is good only in the limit of high thresholds, it provides an overestimate for more modest thresholds.

The true detection probability is even more difficult to compute than the false alarm probability. However, a quick estimate would be to assume that the detection will always be made at the correct time, and then the correct detection probability will be the same as was computed in the previous two sections. This will provide an underestimate of the correct detection probability.


next up previous contents
Next: Reception of a Signal Up: The Statistical Theory of Previous: A Receiver for a   Contents
Bruce Allen 2000-11-19