Transient signal detection and Page's test

Date of Completion

January 1996


Engineering, Electronics and Electrical|Engineering, System Science




A transient signal, or burst, can be thought of as a two-sided change: at some unknown time $n\sb{s}$, the observations process $\{U\sb{n}\}$ switches from being governed by probability law $f\sb0$ to being governed by $f\sb1$; and at a later time $n\sb{e}$, returns to $f\sb0$. Hence, while the signal-absent hypothesis is simple, the signal-present hypothesis is composite with $n\sb{s}$ and $n\sb{e}$ being the nuisance parameters. Such problems can be found in sonar, radar and fault detection. If $n\sb{e}$ approaches infinity, meaning that the return to $f\sb0$ does not happen, then we have a permanent change in distribution.^ Page's test, also known as the quickest detector, is often used to detect a permanent change. It has lately been also used for transient signal detection. The performance of Page's test for the detection of a permanent change in distribution is reasonably well-understood in the sense of average delay-to-detection versus average time between false alarms. However, there are few parallel results on its application to the detection of a transient change. It is desired to calculate or approximate the probability of detection of a transient signal given the threshold determined by the specifications on the false alarm rate.^ In this dissertation, the probability of detection of a discrete-valued Page's test is evaluated exactly and that of the continuous-valued test is bounded and approximated. Subsequently, the Fourier transform of the threshold-crossing time for a regulated random walk is derived.^ A min-max test is also developed. This is optimal for detecting a transient signal in the sense of minimizing the maximum probability of a miss, subject to a false alarm rate constraint. The performance of this min-max test is compared with that of Page's test under alternative hypotheses with different nuisance parameters $n\sb{s}$ and $n\sb{e}$.^ Finally, the sequential probability ratio test is investigated. A generalised sequential probability ratio test is proposed, which, when designed properly, can give constant probability of type I and II errors conditioned on the time of decision. ^