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Feedback Does Not Increase the Capacity of Approximately Memoryless Surjective POST Channels

Xiaojing Zhang, Jun Chen, Guanghui Wang

Abstract

We study a class of finite-state channels, known as POST channels, in which the previous channel output serves as the current state. A POST channel is deemed approximately memoryless when the state-dependent transition matrices are sufficiently close to one another. For this family of channels, under a surjectivity condition on the associated memoryless reference channel, we show that the feedback capacity coincides with the non-feedback capacity. Consequently, for almost all approximately memoryless POST channels whose input alphabet size is no smaller than the output alphabet size, feedback provides no capacity gain. This result extends Shannon's classical theorem on discrete memoryless channels and demonstrates that the phenomenon holds well beyond the strictly memoryless case.

Feedback Does Not Increase the Capacity of Approximately Memoryless Surjective POST Channels

Abstract

We study a class of finite-state channels, known as POST channels, in which the previous channel output serves as the current state. A POST channel is deemed approximately memoryless when the state-dependent transition matrices are sufficiently close to one another. For this family of channels, under a surjectivity condition on the associated memoryless reference channel, we show that the feedback capacity coincides with the non-feedback capacity. Consequently, for almost all approximately memoryless POST channels whose input alphabet size is no smaller than the output alphabet size, feedback provides no capacity gain. This result extends Shannon's classical theorem on discrete memoryless channels and demonstrates that the phenomenon holds well beyond the strictly memoryless case.
Paper Structure (13 sections, 7 theorems, 139 equations, 2 figures)

This paper contains 13 sections, 7 theorems, 139 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Non-Feedback Capacity vs. Feedback Capacity
  3. Non-Feedback Capacity
  4. Feedback Capacity
  5. Main Results
  6. Proof of the Main Result
  7. $|\mathcal{X}|=|\mathcal{Y}|$
  8. $|\mathcal{X}|>|\mathcal{Y}|$
  9. Discussion
  10. Conclusion
  11. Proof of Lemma \ref{['lem:indecomposable_connected']}
  12. Proof of Lemma \ref{['lem:uniform_convergence']}
  13. Proof of Theorem \ref{['thm:necessary_sufficient']}

Key Result

Lemma 1

A $W$-centered $\delta$-approximately memoryless POST channel $Q$ is indecomposable whenever and connected whenever

Figures (2)

  • Figure 1: In the provided visualization, the dashed triangle, the solid triangle, and the asterisk represent the affine span of the columns of $\mathbb{Q}^{(n)}_{y_0}$, their convex hull, and the output probability vector $\mathbbm{q}^{(n)}_{y_0}$ induced by the optimal feedback strategy, respectively. Under the memoryless setting where $\delta=0$, the vector $\mathbbm{q}^{(n)}_{y_0}$ is guaranteed to reside within the convex hull. When a small perturbation $\delta$ is introduced---representing an approximately memoryless regime---neither $\mathbbm{q}^{(n)}_{y_0}$ nor the convex hull undergoes significant deformation, maintaining their containment relationship. However, as the magnitude of the perturbation $\delta$ increases, the divergence of their relative trajectories may eventually cause $\mathbbm{q}^{(n)}_{y_0}$ to fall outside the boundaries of the convex hull.
  • Figure 2: In this visualization, the dashed triangle, the solid line segment, and the asterisk represent the ambient output space, the convex hull of the columns of $\mathbb{Q}^{(n)}_{y_0}$, and the output probability vector $\mathbbm{q}^{(n)}_{y_0}$ induced by the optimal feedback strategy, respectively. Under the memoryless setting ($\delta=0$), $\mathbbm{q}^{(n)}_{y_0}$ is guaranteed to reside within the convex hull. However, as the perturbation $\delta$ deviates even slightly from zero, the availability of an extra dimension in the ambient output space allows $\mathbbm{q}^{(n)}_{y_0}$ to move beyond the linear span of the columns of $\mathbb{Q}^{(n)}_{y_0}$, breaking the containment relationship.

Theorems & Definitions (9)

  • Definition 1: Non-Feedback Capacity
  • Definition 2: Feedback Capacity
  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 2
  • Lemma 5