Single-Threshold Detection of a Random Signal in Noise with Multiple Independent Observations, Part 2: Continuous Case

A single-threshold detector is derived for a wide class of classical binary decision problems involving the likelihood-ratio detection of a signal embedded in noise. The class of problems considered encompasses the case of multiple independent (but not necessarily identically distributed) observations of a nonnegative (or nonpositive) signal embedded in additive and independent noise, where the range of the signal and noise is continuous. It is shown that a comparison of the sum of the observations with a unique threshold comprises an optimum detector if a weak condition on the noise is satisfied independent of the signal. Examples of noise densities that satisfy and that violate this condition are presented. A sufficient condition on the likelihood ratio which implies that the sum of the observations is also a sufficient statistic is considered.