The Application of Extreme-Value Theory to Error-Free Cornrnunication

Abstract

This article gives an application of Gumbel's extreme-value theory to the problem of the estimation of low error probabilities in certain types of communications receivers, for the purpose of rejecting receivers having unacceptably high error rates. The systems considered all involve threshold detectors in a binary system. Thus, the receiver calls a received symbol a “0” if and only if a certain detected voltage falls below a certain value; otherwise, the symbol is called a “1”. The maximum of a run of a large number of successive independent samples of voltage is taken, and these maxima have approximately a Gumbel distribution. A large number of these maxima themselves are taken, and the two parameters of the Gumbel distribution are thereby estimated using large sample theory. These two parameters in turn allow the estimation of error probability. The variance of the estimate is found, as well as a confidence interval. In addition, the savings to be obtained using the Gumbel method, as opposed to the sim...