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Feedback Capacity of the Continuous-Time ARMA(1,1) Gaussian Channel

Jun Su, Guangyue Han, Shlomo Shamai

TL;DR

The paper derives a closed-form characterization of the feedback capacity for the continuous-time ARMA(1,1) Gaussian channel under an average power constraint. Using a continuous-time Schalkwijk–Kailath coding framework and discrete-time approximations, it shows that C_fb(P) equals the unique positive root of P(x+κ)^2 = 2x(x+|κ+λ|)^2 when -2κ<λ<0, and equals P/2 otherwise, revealing that color noise can either boost or leave unchanged capacity under feedback. It also determines when feedback matches non-feedback capacity and when it can reach the two-times limit, and it disproves continuous-time analogues of the half-bit and Cover’s 2P conjectures. The results bridge continuous- and discrete-time Gaussian channel analyses and provide a foundation for analyzing feedback effects in broader continuous-time ACGN models.

Abstract

We consider the continuous-time ARMA(1,1) Gaussian channel and derive its feedback capacity in closed form. More specifically, the channel is given by $\boldsymbol{y}(t) =\boldsymbol{x}(t) +\boldsymbol{z}(t)$, where the channel input $\{\boldsymbol{x}(t) \}$ satisfies average power constraint $P$ and the noise $\{\boldsymbol{z}(t)\}$ is a first-order {\em autoregressive moving average} (ARMA(1,1)) Gaussian process satisfying $$ \boldsymbol{z}^\prime(t)+κ\boldsymbol{z}(t)=(κ+λ)\boldsymbol{w}(t)+\boldsymbol{w}^\prime(t), $$ where $κ>0,~λ\in\mathbb{R}$ and $\{\boldsymbol{w}(t) \}$ is a white Gaussian process with unit double-sided spectral density. We show that the feedback capacity of this channel is equal to the unique positive root of the equation $$ P(x+κ)^2 = 2x(x+\vert κ+λ\vert)^2 $$ when $-2κ<λ<0$ and is equal to $P/2$ otherwise. Among many others, this result shows that, as opposed to a discrete-time additive Gaussian channel, feedback may not increase the capacity of a continuous-time additive Gaussian channel even if the noise process is colored. The formula enables us to conduct a thorough analysis of the effect of feedback on the capacity for such a channel. We characterize when the feedback capacity equals or doubles the non-feedback capacity; moreover, we disprove continuous-time analogues of the half-bit bound and Cover's $2P$ conjecture for discrete-time additive Gaussian channels.

Feedback Capacity of the Continuous-Time ARMA(1,1) Gaussian Channel

TL;DR

The paper derives a closed-form characterization of the feedback capacity for the continuous-time ARMA(1,1) Gaussian channel under an average power constraint. Using a continuous-time Schalkwijk–Kailath coding framework and discrete-time approximations, it shows that C_fb(P) equals the unique positive root of P(x+κ)^2 = 2x(x+|κ+λ|)^2 when -2κ<λ<0, and equals P/2 otherwise, revealing that color noise can either boost or leave unchanged capacity under feedback. It also determines when feedback matches non-feedback capacity and when it can reach the two-times limit, and it disproves continuous-time analogues of the half-bit and Cover’s 2P conjectures. The results bridge continuous- and discrete-time Gaussian channel analyses and provide a foundation for analyzing feedback effects in broader continuous-time ACGN models.

Abstract

We consider the continuous-time ARMA(1,1) Gaussian channel and derive its feedback capacity in closed form. More specifically, the channel is given by , where the channel input satisfies average power constraint and the noise is a first-order {\em autoregressive moving average} (ARMA(1,1)) Gaussian process satisfying where and is a white Gaussian process with unit double-sided spectral density. We show that the feedback capacity of this channel is equal to the unique positive root of the equation when and is equal to otherwise. Among many others, this result shows that, as opposed to a discrete-time additive Gaussian channel, feedback may not increase the capacity of a continuous-time additive Gaussian channel even if the noise process is colored. The formula enables us to conduct a thorough analysis of the effect of feedback on the capacity for such a channel. We characterize when the feedback capacity equals or doubles the non-feedback capacity; moreover, we disprove continuous-time analogues of the half-bit bound and Cover's conjecture for discrete-time additive Gaussian channels.
Paper Structure (12 sections, 14 theorems, 216 equations, 1 figure)

This paper contains 12 sections, 14 theorems, 216 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and Terminologies
  3. Continuous-Time SK Coding
  4. Mutual Information Rate
  5. Feedback Capacity
  6. Proof of the Converse Part
  7. Proof of the Achievability Part
  8. Effect of Feedback
  9. Concluding Remarks
  10. Proof of Theorem \ref{['detailed-structure']}
  11. Proof of Lemma \ref{['exist-limit-ode']}
  12. Proofs of $(a)-(f)$

Key Result

Theorem 3.1

Assume that If $R< C_{fb,\infty}(P)$ and $C_{fb,\infty}(P)$ is continuous in $P$, then the rate $R$ is achievable. Conversely, if a rate $R$ is achievable, then $R\le C_{fb,\infty}(P)$.

Figures (1)

  • Figure 1: The continuous-time ARMA$(1,1)$ Gaussian channel with feedback. Note that the OU process $\{\boldsymbol{u}(t)\}$ can be interpreted as the white Gaussian noise $\{\boldsymbol{w}(t) \}$ filtered by a Lorentzian filter (see, e.g., bibbona2008ornstein).

Theorems & Definitions (35)

  • Theorem 3.1: Ihara1992channelcodingthm
  • Lemma 3.2: ihara1990capacity
  • Lemma 3.3
  • Theorem 3.4: ihara1980capacity Reformulated
  • Lemma 4.1
  • Theorem 4.2
  • proof
  • Remark 4.3
  • Example 4.4
  • Example 4.5
  • ...and 25 more