Probability of Error for Quadratic Detectors

A procedure is presented for evaluating the performance of a general class of digital detectors which square or multiply signal waves contaminated by gaussian noise. In addition to simplifying and unifying the treatment of a number of previously solved problems and some hitherto unsolved ones, the method achieves a considerable advance toward a complete evaluation of postdetection filtering. In contrast to most of the earlier related work, which is typically restricted to filters described as accepting low-frequency difference products and rejecting high-frequency sum products, the present analysis offers a tractable inclusion of filters which do significant selective processing of the detected low-frequency signal and noise components. A principal goal sought is the asymptotic form approached by the error probability expressed as a function of the signal-to-noise ratio when the latter is large. This is a primary region of interest in digital data transmission over the telephone network, and the applicable results give a basis for comparing performance of different systems. The mathematical problem is one of calculating the probability that a quadratic form in a set of gaussian variables with arbitrary means and variances will assume values critically far removed from that obtained when each variable is at its mean. The mean values represent signal contributions unperturbed by noise and for good performance should dominate over the noise except at the tails of the distribution. Concentration of attention on the infrequent large noise peaks calls for an approach inherently different from the conventional series expansions appropriate near the center of the distribution. The results are of importance not only in detection theory but also in general statistical analysis of rare events.