To feed or not to feed back

Abstract

We study the communication over Finite State Channels (FSCs), where the encoder and the decoder can control the availability or the quality of the noise-free feedback. Specifically, the instantaneous feedback is a function of an action taken by the encoder, an action taken by the decoder, and the channel output. Encoder and decoder actions take values in finite alphabets, and may be subject to average cost constraints.We prove capacity results for such a setting by constructing a sequence of achievable rates, using a simple scheme based on 'code tree' generation, that generates channel input symbols along with encoder and decoder actions. We prove that the limit of this sequence exists. For a given block length N and probability of error, , we give an upper bound on the maximum achievable rate. Our upper and lower bounds coincide and hence yield the capacity for the case where the probability of initial state is positive for all states. Further, for stationary indecomposable channels without intersymbol interference (ISI), the capacity is given as the limit of normalized directed information between the input and output sequence, maximized over an appropriate set of causally conditioned distributions. As an important special case, we consider the framework of 'to feed or not to feed back' where either the encoder or the decoder takes binary actions, which determine whether current channel output will be fed back to the encoder, with a constraint on the fraction of channel outputs that are fed back. As another special case of our framework, we characterize the capacity of 'coding on the backward link' in FSCs, i.e. when the decoder sends limited-rate instantaneous coded noise-free feedback on the backward link. Finally, we propose an extension of the Blahut-Arimoto algorithm for evaluating the capacity when actions can be cost constrained, and demonstrate its application on a few examples.

Capacity of a POST channel with and without feedback

“…The simple POST channel is similar to the Ising channel introduced by Berger and Bonomi [9], but rather than the previous input being the state of the channel, here the state of the channel is the previous output. This channel arose in the investigation of controlled feedback in the setting of "to feed or not to feed back" [10]. The POST channel can also be useful in modeling memory affected by past channel outputs, as is the case in flash memory and other storage devices.…”

“…De MM 0 0 0 0 amenable to analysis under the "to feed or not to feed" framework of constrained feedback in [10], while exhibiting non-trivial dependence of its capacity on the extent to which the feedback is constrained. However, numerical results based on the computational algorithm devised in [10] suggested that feedback does not increase the capacity of the simple POST channel.…”

“…However, numerical results based on the computational algorithm devised in [10] suggested that feedback does not increase the capacity of the simple POST channel. This paper stemmed from our attempts to make sense of these surprising observations.…”

Multiple access channel with partial and controlled cribbing encoders

“…Recently, several problems on "action" in information theory have been considered in [8]- [11]. In these problems, the side information is not freely available, but it depends on a cost-constrained action taken by the encoder or the decoder.…”

Information embedding on actions

“…In [2][3], a related channel coding problem is studied in which the encoder in a point-to-point channel can take actions to affect, or probe, the state of a channel. Generalizations of these works include models with additional design constraints [4], [5], with adaptive actions [6], with memory [7], [5] and with multiple terminals [8][9] [10]. To illustrate the problem of interest in this paper, consider the example of a link in which the encoder takes actions to probe the channel by sending a training packet.…”