Universal decoding for channels with memory

Abstract

A universal decoder for a parametric family of channels is a decoder whose structure depends on the family but not on the individual channel over which transmission takes place, and it yet attains the same random-coding error exponent as the maximum-likelihood receiver tuned to the channel in use. The existence and structure of such decoders is demonstrated under relatively mild conditions of continuity of the channel law with respect to the parameter indexing the family. It is further shown that under somewhat stronger conditions on the family of channels, the convergence of the performance of the universal decoder to that of the optimal decoder is uniform over the set of channels. Examples of families for which universal decoding is demonstrated include the family of finite-state channels and the family of Gaussian intersymbol interference channels. Index Terms-Compound channel, error exponent, finite-state channel, Gilbert-Elliott channel, intersymbol interference, random coding, universal decoding.

A training based scheme for communicating over unknown channels with feedback

“…This family is used in [5] and [7] to prove negative results about training based schemes. We use this family to show that training based schemes with multiple training phases can achieve error exponents that have the same form as Burnashev's exponent.…”

“…First, lets reconsider the arguments of [5] on why training based schemes perform badly for fixed length communication over an unknown channel. In that scenario, the probability of decoding error is similar to expression (10) for Pr(D (t,1) ): a sum of two exponential terms.…”

“…(BSC(p) denotes a binary symmetric channel with crossover probability p). With fixed length communication, training based schemes do no not achieve universal error exponent [5]. With variable length communication with noiseless feedback, training based scheme that have a single training phase will have error exponent with zero slope near capacity.…”

“…The second point puts the result of [5] and [7] in perspective. In both [5] and [7], a simple family W p = {BSC(p), BSC(1 − p)}, with 0 ≤ p < 1 2 was used to prove negative results about training based schemes.…”

“…In both [5] and [7], a simple family W p = {BSC(p), BSC(1 − p)}, with 0 ≤ p < 1 2 was used to prove negative results about training based schemes. (BSC(p) denotes a binary symmetric channel with crossover probability p).…”