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Generalized Schalkwijk-Kailath Coding for Autoregressive Gaussian Channels

Jun Su, Guangyue Han, Shlomo Shamai

TL;DR

This work extends the classical SK coding framework to stationary AR($p$) Gaussian channels with feedback by introducing the SK(2) coding scheme, a Gaussian random coding strategy with a second-order deterministic recursion for the message process. It derives a closed-form lower bound $\bar{I}_{\mathrm{SK2}}(P)$ on the feedback capacity, and shows that SK(2) reduces to SK(1) when the second parameter vanishes, recovering the known AWGN and AR(1) capacity results. Crucially, SK(2) can strictly outperform SK(1) for AR($p$) channels with higher order, notably AR(2), thereby disproving Butman\'s conjecture of universal optimality for SK(1). The results provide a broader constructive approach to achieving higher rates with feedback in colored Gaussian noise, clarifying the limits of SK-based schemes and highlighting new opportunities for capacity gains via higher-order recursion structures.

Abstract

We propose a Gaussian random coding scheme for AR($p$) Gaussian channels that generalizes the celebrated Schalkwijk-Kailath (SK) coding scheme. This constructive coding scheme, termed the SK(2) coding scheme, yields a closed-form characterization for the corresponding achievable rate. Among many others, this result shows that the celebrated SK coding scheme is not universally optimal, and therefore, disprove the conjecture proposed by Butman in \cite{butman1976linear}.

Generalized Schalkwijk-Kailath Coding for Autoregressive Gaussian Channels

TL;DR

This work extends the classical SK coding framework to stationary AR() Gaussian channels with feedback by introducing the SK(2) coding scheme, a Gaussian random coding strategy with a second-order deterministic recursion for the message process. It derives a closed-form lower bound on the feedback capacity, and shows that SK(2) reduces to SK(1) when the second parameter vanishes, recovering the known AWGN and AR(1) capacity results. Crucially, SK(2) can strictly outperform SK(1) for AR() channels with higher order, notably AR(2), thereby disproving Butman\'s conjecture of universal optimality for SK(1). The results provide a broader constructive approach to achieving higher rates with feedback in colored Gaussian noise, clarifying the limits of SK-based schemes and highlighting new opportunities for capacity gains via higher-order recursion structures.

Abstract

We propose a Gaussian random coding scheme for AR() Gaussian channels that generalizes the celebrated Schalkwijk-Kailath (SK) coding scheme. This constructive coding scheme, termed the SK(2) coding scheme, yields a closed-form characterization for the corresponding achievable rate. Among many others, this result shows that the celebrated SK coding scheme is not universally optimal, and therefore, disprove the conjecture proposed by Butman in \cite{butman1976linear}.
Paper Structure (7 sections, 6 theorems, 116 equations, 2 figures)

This paper contains 7 sections, 6 theorems, 116 equations, 2 figures.

Table of Contents

  1. Introduction and Summary
  2. Notation and Terminologies
  3. The Schalkwijk--Kailath Coding
  4. The SK(2) Coding Scheme
  5. Proof of Lemma \ref{['lemma1']}
  6. Proof of Theorem \ref{['ISK1-thm']}
  7. Proof of (\ref{['power-eqn-equivalence-case-c']})

Key Result

Lemma 3.1

Let $\{u_k;k=1,2,...,m\}\subset \mathbb{C}^{n}$ for $n,m \ge 1$. Then, it holds that where $U = $ and $A = I_m +U^\ast U$ and $M_{ij}$ is the $(i, j)$-minor of $A$, i.e., the determinant of the $(m-1) \times (m-1)$ matrix that results from deleting row $i$ and column $j$ of $A$. In particular, if $m=1$, then we have

Figures (2)

  • Figure 1: Comparison of $\bar{I}_\mathrm{SK1}(P)$ and $\bar{I}_\mathrm{SK2}(P)$ for stationary AR(1) Gaussian channels with $P=1,5$.
  • Figure 2: Comparison of $\bar{I}_\mathrm{SK1}(P)$ and $\bar{I}_\mathrm{SK2}(P)$ for stationary AR(2) Gaussian channels with $\beta_2=0.5,~0.9$ and $P=1$.

Theorems & Definitions (14)

  • Lemma 3.1
  • Theorem 3.2
  • Remark 3.3
  • proof
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • proof
  • Corollary 4.3
  • proof
  • ...and 4 more