Worst-case signal spectra for broadband detection

As a class, broadband detection problems are difficult to analyze because, even for a fixed noise spectrum and fixed broadband signal-to-noise ratio (SNR), a continuum of different signal spectrum choices exists, each resulting in a different receiver operating characteristic (ROC) point. This paper derives the asymptotically worst-case signal spectrum corresponding to a given noise spectrum and SNR. The resulting signal spectrum asymptotically minimizes the log-likelihood ratio as the number of samples K increases without bound and the SNR approaches zero. We then develop ROC performance equations that characterize the worst-case detection performance in a given broadband noise background, and provide some insights into the nature of broadband detection. To illustrate the theoretical results, we derive the worst-case signal spectra for three noise spectrum cases and present a performance analysis of their ROC behavior. Comparisons among these cases reveals some surprising conclusions.