Capacity Region of the Finite-State Multiple Access Channel with and without Feedback

📝 Abstract

The capacity region of the Finite-State Multiple Access Channel (FS-MAC) with feedback that may be an arbitrary time-invariant function of the channel output samples is considered. We characterize both an inner and an outer bound for this region, using Masseys’s directed information. These bounds are shown to coincide, and hence yield the capacity region, of FS-MACs where the state process is stationary and ergodic and not affected by the inputs. Though multi-letter' in general, our results yield explicit conclusions when applied to specific scenarios of interest. E.g., our results allow us to: - Identify a large class of FS-MACs, that includes the additive mod-2 noise MAC where the noise may have memory, for which feedback does not enlarge the capacity region. - Deduce that, for a general FS-MAC with states that are not affected by the input, if the capacity (region) without feedback is zero, then so is the capacity (region) with feedback. - Deduce that the capacity region of a MAC that can be decomposed into a multiplexer’ concatenated by a point-to-point channel (with, without, or with partial feedback), the capacity region is given by $\sum_{m} R_m \leq C $, where C is the capacity of the point to point channel and m indexes the encoders. Moreover, we show that for this family of channels source-channel coding separation holds.

💡 Analysis

The capacity region of the Finite-State Multiple Access Channel (FS-MAC) with feedback that may be an arbitrary time-invariant function of the channel output samples is considered. We characterize both an inner and an outer bound for this region, using Masseys’s directed information. These bounds are shown to coincide, and hence yield the capacity region, of FS-MACs where the state process is stationary and ergodic and not affected by the inputs. Though multi-letter' in general, our results yield explicit conclusions when applied to specific scenarios of interest. E.g., our results allow us to: - Identify a large class of FS-MACs, that includes the additive mod-2 noise MAC where the noise may have memory, for which feedback does not enlarge the capacity region. - Deduce that, for a general FS-MAC with states that are not affected by the input, if the capacity (region) without feedback is zero, then so is the capacity (region) with feedback. - Deduce that the capacity region of a MAC that can be decomposed into a multiplexer’ concatenated by a point-to-point channel (with, without, or with partial feedback), the capacity region is given by $\sum_{m} R_m \leq C $, where C is the capacity of the point to point channel and m indexes the encoders. Moreover, we show that for this family of channels source-channel coding separation holds.

📄 Content

the achievable region given by Cover and Leung for a memoryless channel with feedback is optimal for a class of channels where one of the inputs is a deterministic function of the output and the other input. More recently Bross and Lapidoth [4] improved Cover and Leung’s region, and Wu et. al. [5] have extended Cover and Leung’s region for the case that non-causal state information is available at both encoders.

Ozarow derived the capacity of a memoryless Gaussian MAC with feedback in [6], and showed it to be achievable via a modification of the Schalkwijk-Kailath scheme [7]. In general, the capacity in the presence of noisy feedback is an open question for the point-to-point channel and a fortiori for the MAC. Lapidoth and Wigger [8] presented an achievable region for the case of the Gaussian MAC with noisy feedback and showed that it converges to Ozarow’s noiseless-feedback sum-rate capacity as the feedback-noise variance tends to zero. Other recent variations on the Schalkwijk-Kailath scheme of relevance to the themes of our work include the case of quantization noise in the feedback link [9] and the case of interference known non-causally at the transmitter [10].

Verdú characterized the capacity region of a Multi-Access channel of the form P (y i |x i 1 , x i 2 , y i-1 ) = P (y i |x i 1,i-m , x i 2,i-m ) without feedback in [11]. Verdú further showed in that work that in the absence of frame synchronism between the two users, i.e., there is a random shift between the users, only stationary input distributions need be considered. Cheng and Verdú built on the capacity result from [11] in [12] to show that for a Gaussian MAC there exists a water-filling solution that generalizes the point-to-point Gaussian channel.

In [13] [14], Kramer derived several capacity results for discrete memoryless networks with feedback. By using the idea of code-trees instead of code-words, Kramer derived a ‘mulit-letter’ expression for the capacity of the discrete memoryless MAC. One of the main results we develop in the present paper extends Kramer’s capacity result to the case of a stationary and ergodic Markov Finite-State MAC (FS-MAC), to be formally defined below.

In [15] [16], Han used the information-spectrum method in order to derive the capacity of a general MAC without feedback, when the channel transition probabilities are arbitrary for every n symbols. Han also considered the additive mod-q MAC, which we shall use here to illustrate the way in which our general results characterize special cases of interest. In particular, our results will imply that feedback does not increase the capacity region of the additive mod-q MAC. In this work, we consider the capacity region of the Finite-State Multiple Access Channel (FS-MAC), with feedback that may be an arbitrary time-invariant function of the channel output samples. We characterize both an inner and an outer bound for this region. We further show that these bounds coincide, and hence yield the capacity region, for the important subfamily of FS-MACs with states that evolve independently of the channel inputs. Our derivation of the capacity region is rooted in the derivation of the capacity of finite-state channels in Gallager’s book [17, ch 4,5]. More recently, Lapidoth and Telatar [18] have used it in order to derive the capacity of a compound channel without feedback, where the compound channel consists of a family of finite-state channels. In particular, they have introduced into Gallager’s proof the idea of concatenating codewords, which we extend here to concatenating code-trees.

Though ‘multi-letter’ in general, our results yield explicit conclusions when applied to more specific families of MACs. For example, we find that feedback does not increase the capacity of the mod-q additive noise MAC (where q is the size of the common alphabet of the input, output and noise), regardless of the memory in the noise. This result is in sharp contrast with the finding of Gaarder and Wolf in [19] that feedback can increase the capacity even of a memoryless MAC due to cooperation between senders that it can create. Our result should also be considered in light of Alajaji’s work [20], where it was shown that feedback does not increase the capacity of discrete point-to-point channels with mod-q additive noise. Thus, this part of our contribution can be considered a multi-terminal extension of Alajaji’s result. Our results will in fact allow us to identify a class of MACs larger than that of the mod-q additive noise MAC for which feedback does not enlarge the capacity region.

Further specialization of the results will allow us to deduce that, for a general FS-MAC with states that are not affected by the input, if the capacity (region) without feedback is zero, then so is the capacity (region) with feedback. It will also allow us to identify a large class of FS-MACs for which source-channel coding separation holds.

The remainder of this paper is organized as follows. We c

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