4.6 DETECTION AND FALSE-ALARM PROBABILITIES

A threshold decision may be the right way to separate signal from noise, but how do we choose the threshold? For any threshold Y, four types of events can occur. These are:

• Target present; y(t) > Y; correct detection.

• Target present; y(t) < Y; missed detection.

• Target not present; y(t) > Y; false alarm.

• Target not present; y(t) < Y; no action.

The frequency of occurrence of these events can be described using two probabilities. The first is the probability of deciding that a signal was present when in fact there was only noise. This is the probability of a false alarm, given by

The second is the probability of a correct detection, given by

The subscripts indicate the PDFs of noise (n) and signal plus noise (sn). Using the PDFs of Fig. 4.7, the areas defining P _fa and P _d are shown in Fig. 4.8.

Figure 4.8: The areas in the PDFs of (a) noise and (b) signal plus noise corresponding to the false-alarm and detection probabilities for a threshold Y.

It is clear that, whatever the value of Y, increasing it will cause both P _fa and P _d to decrease. This illustrates that there is always a trade-off between improving detections and reducing false alarms. In practice, most radars are operated at a threshold level giving very low false-alarm rates (and hence long times between false alarms),...