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IEEE Information Theory Workshop (ITW)

Joint channel intrinsic randomness and channel resolvability

Alexandre J. Pierrot, Matthieu R. Bloch

Published Sep 1, 2013• 7 citations
Publisher

Abstract

This paper investigates the separation of channel intrinsic randomness and channel resolvability. We derive joint exponents, which are compared to the tandem exponents obtained with a separate approach. We prove at once, in a simple manner, achievability results for channel intrinsic randomness, random number generation, and channel resolvability. We also provide converse results in different special settings.

DOI
10.1109/itw.2013.6691334

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This paper investigates the separation of channel intrinsic randomness and channel resolvability. We derive joint exponents, which are compared to the tandem exponents obtained with a separate approach. We prove at once, in a simple manner, achievability results for channel intrinsic randomness, random number generation, and channel resolvability. We also provide converse results in different special settings.
The authors have introduced the new notion of resolvability of a channel, as the dual of the capacity, which is defined as the minimum complexity per input letter needed to generate an input process whose output distribution via the channel arbitrarily accurately approximates any prescribed achievable output distribution. The resolvability thus introduced has revealed a deep relationship between the minimum achievable rate for source coding, the channel capacity, the identification capacity and the problem of random number generation. However, the validity of the proof of the converse for the resolvability formula established by Han and Verdu (see IEEE Trans. on Inform. Theory, vol.39, no.3, p.379, 1993) hinged heavily on the assumption that the input alphabet of the channel is finite. Our main purpose in this paper is to show that we can relax this restriction and to show that the resolvability formula of Han and Verdu continues to hold also for a wide class of channels with continuous input alphabet, including as a special case, additive white Gaussian noise (AWGN) channels with power constraint.>
Given a channel and an input process we study the minimum randomness of those input processes whose output statistics approximate the original output statistics with arbitrary accuracy. We introduce the notion of resolvability of a channel, defined as the number of random bits required per channel use in order to generate an input that achieves arbitrarily accurate approximation of the output statistics for any given input process. We obtain a general formula for resolvability which holds regardless of the channel memory structure. We show that, for most channels, resolvability is equal to Shannon capacity. By-products of our analysis are a general formula for the minimum achievable (fixed-length) source coding rate of any finite-alphabet source, and a strong converse of the identification coding theorem, which holds for any channel that satisfies the strong converse of the channel coding theorem.
We study the problem of generating an approximately i.i.d. string at the output of a discrete memoryless channel using a limited amount of randomness at its input in presence of causal noiseless feedback. Feedback does not decrease the channel resolution, the minimum entropy rate required to achieve an accurate approximation of an i.i.d. output string. However, we show that, at least over a binary symmetric channel, a significantly larger resolvability exponent (the exponential decay rate of the divergence between the output distribution and product measure), compared to the best known achievable resolvability exponent in a system without feedback, is possible. We show that by employing a variable-length resolvability scheme and using an average number of R coin-flips per channel use, the average divergence between the distribution of the output sequence and product measure decays exponentially fast in the average length of output sequence with an exponent equal to [R - I(U; V)]+where I(U; V) is the mutual information developed across the channel.
We study the problem of generating an approximately i.i.d. string at the output of a discrete memoryless channel using a limited amount of randomness at its input in presence of causal noiseless feedback. Feedback does not decrease the channel resolution, the minimum entropy rate required to achieve an accurate approximation of an i.i.d. output string. However, we show that, at least over a binary symmetric channel, a significantly larger resolvability exponent (the exponential decay rate of the divergence between the output distribution and product measure), compared to the best known achievable resolvability exponent in a system without feedback, is possible. We show that by employing a variable-length resolvability scheme and using an average number of coin-flips per channel use, the average divergence between the distribution of the output sequence and product measure decays exponentially fast in the average length of output sequence with an exponent equal to [R−I(U;V)]+[R-I(U;V)]^+[R−I(U;V)]+ where I(U;V)I(U;V)I(U;V) is the mutual information developed across the channel.