In the case that the signal waveform possesses parameters, other than the time of arrival that we wish to measure, then we must consider a bank of filters, (and if there is also an unknown phase), corresponding to a correlation of the output with all of the possible signals . Here, the set of parameters is . A set of statistics is then created according to the methods given in the previous section. The largest of these statistics is compared to the threshold to determine if a signal is present or not. If it is decided that a signal is present, then the estimate of the parameters of the signal are those parameters corresponding to the largest statistic .
The filters in the filter bank will typically be densely packed into
the parameter space in order to accurately match any expected signal.
Thus, these filters will likely be highly correlated with one another.
To estimate the rate of false alarms, we appeal to the high threshold
limit in which each filter is effectively independent. Then, the
rate of false alarms is