Nonergodic Error Rate Analysis of Finite-Length Received Sequences

The Shannon coding theorems assume transmission of an infinite number of possibly infinite-length codewords. Previous works then considered more realistic scenarios with transmission of an infinite number of strictly finite codewords. In this paper, a pragmatic approach is used to analyze a novel scenario with the transmission of a finite number of possibly infinite codewords or, equivalently, to analyze the transmission of a finite number of independent and identically distributed (i.i.d.) bits over a stationary binary symmetric channel. Since the bit error rate (BER) of finite-length received sequences is a random variable, the corresponding performance analysis is nonergodic. The BER is then well described by the Bayesian credible intervals rather than by the expectation. In addition, a degree-of-ergodicity measure is introduced to quantify the level of ergodicity of the received sequence with respect to its instantaneous BER and to describe the transition of the data detector from the nonergodic to the ergodic zone of operation with an increasing number of received bits. The nonergodic analysis developed in this paper can be used to optimize transmission parameters to guarantee, with a given probability, the worst case instantaneous BER performance when processing the received sequence of bits.