On Optimal Detectors, Asymptotic Efficiency, and Some CDMA Problems

Necessary and sufficient conditions for a linear detector to be asymptotically optimal are given. In particular, it is shown that finding the asymptotically best linear detector and the largest asymptotic efficiency is a standard problem of convex analysis in Euclidean space, namely, finding the distance from a point to a convex set. As examples, decorrelating and conventional detectors are considered. In the case of randomly chosen CDMA signals, we show that, under certain conditions, the decorrelating detector is with high probability asymptotically optimal. This allows us to find the largest asymptotic efficiency of linear detectors for randomly chosen CDMA signals.