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IEEE Transactions on Information Theory

On identification capacity of infinite alphabets or continuous-time channels

M. V. Burnashev

Published Jan 1, 2000• 23 citations
Publisher

Abstract

Two kinds of channels are considered: (1) discrete-time channel with additive noise, and (2) Poisson and white Gaussian (i.e., continuous-time) channels. For the type (1) channel there are given some sufficient conditions when the Shannon and identification capacities coincide. It is shown that the identification capacity of Poisson and Gaussian channels without bandwidth constraint is infinite. Conversely, for a white Gaussian channel with bandwidth constraint, the identification capacity coincides with the Shannon capacity.

DOI
10.1109/18.887853

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Two kinds of channels are considered: (1) discrete-time channel with additive noise, and (2) Poisson and white Gaussian (i.e., continuous-time) channels. For the type (1) channel there are given some sufficient conditions when the Shannon and identification capacities coincide. It is shown that the identification capacity of Poisson and Gaussian channels without bandwidth constraint is infinite. Conversely, for a white Gaussian channel with bandwidth constraint, the identification capacity coincides with the Shannon capacity.
Given a general channel, we first formulate the idetification capacity problem as well as the resolvability problem with input cost constraint in as the general form as possible, along with relevant fundamental theorems. Next, we establish some mild sufficient condition for the key lemma linking the identification capacity with the resolvability to hold for the continuous input alphabet case with input cost constraint. Under this mild condition, it is shown that we can reach the {\em continuous}-input fundamental theorem of the same form as that for the fundamental theorem with {\em finite} input alphabet. Finally, as important examples of this continuous-input fundamental theorem, we show that the identification capacity as well as the resolvability coincides with the channel capacity for stationary additive white (and also non-white) Gaussian noise channels.
Deterministic identification (DI) for the discrete-time Poisson channel, subject to an average and a peak power constraint, is considered. It is established that the code size scales as 2(nlogn)R, where n and R are the block length and coding rate, respectively. The authors have recently shown a similar property for Gaussian channels [1]. Lower and upper bounds on the DI capacity of the Poisson channel are developed in this scale. Those imply that the DI capacity is infinite in the exponential scale, regardless of the dark current, i.e., the channel noise parameter.
Identification capacity is developed without randomization at neither the encoder nor the decoder. In particular, full characterization is established for the deterministic identification (DI) capacity for the Gaussian channel and for the general discrete memoryless channel (DMC) with and without constraints. Originally, Ahlswede and Dueck established the identification capacity with local randomness given at the encoder, resulting in a double exponential number of messages. In the deterministic setup, the number of messages scales exponentially, as in Shannon’s transmission paradigm, but the achievable identification rates can be significantly higher than those of transmission. Ahlswede and Dueck further stated a capacity result for the deterministic setting of a DMC, but did not provide an explicit proof. In this paper, a detailed proof is given for both the Gaussian channel and the general DMC. The DI capacity of a Gaussian channel is infinite regardless of the noise.
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.>
We consider a problem of communication over a continuous-time additive Gaussian noise channel. A covertness constraint is imposed on the communication protocol, which says that the channel output must statistically resemble pure noise. When there is no bandwidth constraint on the input, we argue that the covert communication capacity of this channel is positive, in contrast to the band-limited case where covertness requires that the amount of transmission grow at most like the square root of total communication time. This claim is formally proven in the case where the Gaussian noise is white with respect to the bandwidth used by the transmitter.
In this paper, we establish the second-order randomized identification capacity (RID capacity) of the Additive White Gaussian Noise Channel (AWGNC). On the one hand, we obtain a refined version of Hayashi's theorem to prove the achievability part. On the other, we investigate the relationship between identification and channel resolvability, then we propose a finer quantization method to prove the converse part. Consequently, the second-order RID capacity of the AWGNC has the same form as the second-order transmission capacity. The only difference is that the maximum number of messages in RID scales double exponentially in the block length.