next up previous contents
Next: Details of Normalization Up: The Statistical Theory of Previous: Reception of a Signal   Contents

Reception of a Signal with Additional Unknown Parameters

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, $\{x_i(t;\lambda_i)\}$ (and  $\{y_i(t;\lambda_i)\}$ if there is also an unknown phase), corresponding to a correlation of the output $h(t)$ with all of the possible signals  $s(t;\lambda_i)$. Here, the set of $N_\lambda$ parameters is $\{\lambda_i\}$. A set of statistics $\{\rho_i\}$ 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 $\lambda_i$ corresponding to the largest statistic $\rho_i$.

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

\begin{displaymath}
\nu(\mbox{false alarm}) \simeq N_\lambda\Delta^{-1} \left\{...
...ox{unknown phase}
\end{array} \right. \qquad (\rho_\ast\gg1).
\end{displaymath} (13.1.304)

Similarly, we assume that if a signal is detected, then it will be detected with the correct parameters. The probability of true detection under this assumption is
\begin{displaymath}
P(\mbox{true detection}) \simeq \left\{
\begin{array}{ll}
...
...
Q(A\sigma,\rho) & \mbox{unknown phase}.
\end{array} \right.
\end{displaymath} (13.1.305)

These expressions provide an overestimate of the false alarm rate and an underestimate of the probability of true detection; thus, they may be used to set a threshold for an upper limit to the desired false alarm rate, and the false dismissal probability should be no greater than the value computed from that threshold.


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