Prony and Polynomial Approximations for Evaluation of the Average Probability of Error Over Slow-Fading Channels

A novel and simple semianalytical method for evaluating the average probability of transmission error for digital communication systems that operate over slow-fading channels is presented. The proposed method applies a sum of exponentials fit known as the Prony approximation to the conditional probability of error. Hence, knowledge of the moment-generating function of the instantaneous signal-to-noise ratio (SNR) at the detector input can be used to obtain the average probability of error. Numerical results show that knowledge of the conditional probability of error at only a small number of points and the sum of only two exponentials are sufficient to achieve very high accuracy; the relative approximation error of the exact average probability of error is less than 6% in most of the cases considered. Furthermore, a piecewise polynomial approximation of the conditional probability of error is investigated as an alternative to the sum of exponentials fit. In this case, knowledge of the partial moments of the instantaneous SNR at the detector input can be used to obtain the average probability of error. Numerical results indicate that, to achieve good accuracy, the method based on the polynomial approximation requires that the product of the polynomial degree and the number of approximation subintervals be larger than 10.

I. Introduction

The Probability of detection error is often considered the most important performance measure in the receiver design for a communication system. An error event occurs when the receiver erroneously decides about the transmitted bit, symbol, or sequence of symbols. These error events will be referred to as (transmission) errors, and their relative frequency of occurrence will be termed bit-error rate (BER), symbol-error rate (SER), and frame-error rate (FER). Hence, the probability of error and the error rate are defined for stationary and ergodic channels. In some cases, the probability of error equals the error rate; for example, the BER can be defined as the arithmetic average of the probabilities of error for all the bits in a transmitted sequence. The probability of error on an additive white Gaussian noise (AWGN) channel is conditioned on the channel (instantaneous) signal-to-noise ratio (SNR). On fading (time-varying) channels, the probability of error is a random variable, and its first moment is typically assumed to be a sufficient performance measure, provided that the observations are long enough to consider the channel as ergodic. The first moment (average) of the probability of error is obtained by averaging the conditional probability of error over the channel SNR distribution at the input to the detector. Numerous methods have been developed to accomplish averaging of the conditional probability of error over the SNR distribution. In particular, the moment-generating function (MGF) [1] and the characteristic function (CHF) [2] methods are appealing, because the Laplace or Fourier transform of the SNR distribution is usually known, even for multichannel reception with correlated and nonidentically distributed branches, whereas, in general, the SNR distribution is often unknown or difficult to obtain. The CHF can be also obtained using the statistical moments of the SNR [3]. The SNR distribution is estimated from known moments in [4]. The method of residues [5], Gauss–Chebyshev quadrature (GCQ) [6], saddle-point integration [7], and the Beaulieu series [8]–[10] can be used to invert the CHF and obtain the SNR distribution. Overviews and further description of these methods can be found, for example, in [7] and in [11].