A note on the Poor-Verdu upper bound for the channel reliability function

Abstract

Abstract-In an earlier work, Poor and Verdú established an upper bound for the reliability function of arbitrary single-user discrete-time channels with memory. They also conjectured that their bound is tight for all coding rates. In this note, we demonstrate via a counterexample involving memoryless binary erasure channels (BECs) that the Poor-Verdú upper bound is not tight at low rates. We conclude by examining possible improvements to this bound.Index Terms-Arbitrary channels with memory, binary erasure channels (BECs), channel coding, channel reliability function, information spectrum, probability of error.

The Asymptotic Generalized Poor-Verdu Bound Achieves the BSC Error Exponent at Zero Rate

“…We derived a closed-form formula for the asymptotic generalized Poor-Verdú error bound to the multihypothesis testing error probability and proved that, unlike the case for the BEC [12], it achieves the zero-rate error coding exponent of the BSC.…”

“…Indeed for the BEC, the bound in (1) is unchanged for every θ ≥ 1 (including when θ → ∞) and is hence identical to the original Poor-Verdú bound. The latter bound was shown in [12] not to achieve the BEC's error exponent at low rates.…”

A Generalized Poor-Verdú Error Bound for Multihypothesis Testing

“…(14)] and they conjectured that the bound is tight. In [1], this bound was shown not to be tight at low rates for memoryless BECs.…”

“…We next compute the new bound in (4) for , and for different values of and plot it in Fig. 1, along with Fano's original bound (referred to as "Fano" in the figure) given by (1). As can be seen from the figure, bound (4) for and 100 improves upon (1) and both Fano bounds and approaches the exact probability of error as is increased without bound (e.g., for and , the bound is quite close to ).…”

“…I N [3], Poor and Verdú establish a lower bound to the minimum error probability of multihypothesis testing. Specifically, given two random variables and with joint distribution , taking values in a finite or countably-infinite alphabet and taking values in an arbitrary alphabet , they show that the optimal maximum-a-posteriori (MAP) estimation of given results in the following lower bound on the probability of estimation error : (1) for each…”