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

(13.1.301) |

(13.1.302) |

In general, it is difficult to compute the false alarm probability for
a given threshold because the values of and are correlated.
However, in the limit of large thresholds (small false alarm probability
for a large observation time), each sample and becomes
effectively independent in the sense that if the threshold is exceeded
at a time , then it is unlikely that it will be exceeded again at
any time within the correlation timescale of time . 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

(13.1.303) |

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.