Normal Approximations to the Error Rate for Hard-Limited Correlators

Abstract

The approximation to tile normal law of the error transition probability for a correlation receiver preceded by a wideband hard limiter is reviewed. As is well known, the correlator output is asymptotically normal. This circumstance, however, is in general no justification for computing an error probability with the use of the complimentary error function. A binary message alphabet is assumed and the probability that the correlator output is positive (negative), given the transmission of a negative (positive) message bit, is examined with Chernoff bounding techniques as refined by Shannon and Gallagher. This method avoids the usual objections raised to the Edgeworth series when used to compute an error probability. Subject to restrictions of weak carrier-to-noise ratios at the limiter input and sufficient correlator processing gain that the error rate will be less than 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> , the conclusion reached is that the error probability can indeed be computed, to within the accuracy of the mathematical idealization of the real channel, with the use of the complimentary error function; that is to say as if the correlator output were indeed normal.