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].