Error probabilities for equicorrelated<tex>m</tex>-ary signals under phase-coherent and phase-incoherent reception

Abstract

Formulas for the error probabilities of equicorrelated <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> -ary signals under optimum phase-coherent and phase-incoherent reception are derived in the form of previously untabulated single and double integrals. These integrals are amenable to computer evaluation for arbitrary <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> . Two modes of reception are considered. In the first, one of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> equal energy equiprobable signals is known to be transmitted during a symbol interval of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</tex> seconds through a nonfading channel with additive white Gaussian noise. The receiver is assumed to be synchronized in time and frequency with the incoming signal, and reception is on a per-symbol basis. Furthermore, the cross-correlation coefficients between all the signals are equal. The probability of correct decision in both phase-coherent and phase-incoherent reception is derived exactly, as a function of the signal-to-noise ratio, the common cross-correlation coefficient, and the size of the signal set <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> . In the second mode of reception, the only difference is that a threshold is incorporated in the receiver. The probability of false detection and the probability of detection and correct decision are derived exactly for both phase-coherent and phase-incoherent reception as a function of the threshold level, the signal-to-noise ratio, the common cross-correlation coefficient, and the size of the signal set <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> . The method of reduction of multiple integrals presented here can be generalized, and may find application in other statistical studies in which the Gaussian density form is encountered under an integral.