Error probability for incoherent diversity reception

Abstract

The problem of computing error probabilities for multi-channel communications using an incoherently terminated receiver is analyzed. The signaling alphabet is composed of two equal-energy, equiprobable, correlated waveforms and the multichannel model is presumed to be of the slowly fading "Rician" type, i.e., each subchannel is presumed to be composed of a fixed or specular component and a scatter-like or Rayleigh fading component. The main result of this paper is a generalization (21) of an earlier result derived by Helstrom. Novel by-products of this generalization include, as special cases, results derived by Turin, Pierce, Price, and Lindsey. Also closed-form solutions to very general integrals (22) (heretofore seemingly unknown) involving Bessel function products are presented as part of the main result. These integrals are known to arise in the analysis of multichannel adaptive communication systems. Numerical computations for the error probabilities are given for special values of the signal cross-correlation coefficient <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\lambda</tex> and multi-channel order. These results graphically illustrate that the optimum set of equal-energy binary signals which minimize the error rate for the Rayleigh fading multichannel are orthogonal. Specifically, to maintain the same error probability in two systems, one employing nonorthogonal signals having correlation coefficient <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\lambda</tex> , the other employing orthogonal signals, the transmitter power must be increased in the former. In fact, for large SNR's, the graphical data indicate that the required increase in transmitter power is approximately <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">10 \log_{10}(1 - \lambda^{2})^{-1}</tex> dB.