4.18 Partially known signals and noise

Rather than assuming that aspects of the signal, such as its amplitude are beyond any set of justifiable assumptions and arethus "unknown," we may have a situation where these signal aspects are "uncertain." For example, the amplitude may be known to bewithin ten percent of a nominal value. If the case, we would expect better performance characteristics from a detectionstrategy exploiting this partial knowledge from one that doesn't. To derive detectors that use partial information aboutsignal and noise models, we apply the approach used in robust model evaluation: find the worst-case combination of signal andnoise consistent with the partial information, then derive the detection strategy that best copes with it. We have seen that theoptimal detection strategy is found from the likelihood ratio: no matter what the signal and noise model are, the likelihood ratioyields the best decision rule. When applied to additive Gaussian noise problems, the performance of the likelihood ratio testincreases with the signal-to-noise ratio of the difference between the two hypothesized signals. Since we focus on deciding whether aparticular signal is present or not, performance is determined by that signal's SNR and the worst-case situation occurs when thissignal-to-noise ratio is smallest. The results from robust model evaluation taught us to design the detector to the worst-casesituation, which in our case roughly means employing matched filters based on the worst-case signal. Employing this approachresults in what are known as robust detectors .