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Feedback Capacity of Stationary Gaussian Channels: An Optimal Schalkwijk-Kailath Scheme

David Fay, Oron Sabag

arXiv:2609.12069v1cs.ITmath.IT

TL;DR

A gap in the proof that a feedback-independent Gaussian component can be removed left the stated optimality of the generalized SK scheme and dependent results unestablished. The paper proves component removal for stationary Gaussian noise with rational PSD using a state-space feedback-capacity optimization, then derives an explicit SK scheme from an optimizer. The scheme achieves every rate below feedback capacity with doubly-exponentially decaying maximal error probability.

  • Problem

    An omitted cross term left Kim’s proof of component removal—and the dependent generalized SK optimality claim—unestablished.

    Source: A. Feedback capacity of the ACGN channel · p. 2

  • Method

    The paper proves the zero-component property through a state-space feedback-capacity optimization and uses an optimizer to determine an explicit generalized SK scheme.

    Source: §III · p. 5

  • Results

    For rational-PSD noise, every optimizer removes the feedback-independent component, and the explicit SK scheme achieves every rate below feedback capacity with doubly-exponentially decaying maximal error probability.

    Source: §III · p. 5, §V · p. 14

  • Takeaways & Limitations

    For rational-PSD noise, the result supplies the missing component-removal premise for Kim’s reduction and subsequent arguments that rely on it.

    Source: §III · p. 5

    The scope beyond rational-PSD stationary noise remains open, including MIMO channels, nonstationary noise, and arbitrary stationary spectra without finite-dimensional state-space representations.

    Source: §V · p. 14

Abstract

from arXiv · show

We consider channels with additive colored Gaussian noise and noiseless feedback. Kim's seminal work derived a stationary variational characterization of feedback capacity and further asserted that the capacity-achieving stationary input need not contain a feedback-independent Gaussian component. These results led to the construction of a simple coding scheme, based on the Schalkwijk--Kailath (SK) refinement principle, which was shown to be capacity-achieving. A recent note identified a gap in the proof of the feedback-independent component-removal assertion, thereby leaving the optimality of the SK scheme and subsequent results that rely on it incomplete. In this paper, we prove the component-removal assertion for channels with stationary Gaussian noise that has a rational power spectral density. Our proof uses a perturbation analysis of a convex optimization formulation of feedback capacity and, indeed, shows that every optimizer assigns zero power to the feedback-independent component. Using this stronger property, we construct from any optimizer an explicit SK coding scheme that achieves every rate below feedback capacity with doubly-exponentially decaying maximal error probability.

I. INTRODUCTION

For additive colored Gaussian noise channels, feedback capacity and the optimality of associated coding schemes depend on whether the stationary input may omit a feedback-independent component. A proof gap in Kim’s component-removal argument left the stated optimality of the generalized SK scheme and related results unestablished.

I. INTRODUCTION

  • For colored Gaussian noise, feedback capacity can exceed no-feedback capacity, unlike for AWGN, where feedback does not increase capacity.The ACGN feedback-capacity literature includes Cover and Pombra’s treatment of arbitrary Gaussian noise.Source: §I · p. 1, A. Feedback capacity of the ACGN channel · p. 1

A. Feedback capacity of the ACGN channel

  • Kim showed that stationary Gaussian inputs suffice for the Cover–Pombra limit and asserted that the feedback-independent component can be removed without loss of optimality.The reduction supported results including a generalized SK coding scheme claimed to achieve feedback capacity [1, Lem. 6.1, Thm. 6.1].Source: A. Feedback capacity of the ACGN channel · p. 2
  • Later work used the reduced formulation for ARMA feedback-capacity characterizations and SK-type schemes.Liu and Han used polynomial equations; Li and Elia used a Youla parametrization.Source: A. Feedback capacity of the ACGN channel · p. 2
  • The earlier reduction proof omitted a cross term that need not vanish, so it did not establish component removal or the dependent generalized SK optimality claim.The gap also affects later results whose proofs rely on the reduction.Source: A. Feedback capacity of the ACGN channel · p. 2

B. Contribution

The paper establishes component removal for stationary Gaussian noise with rational PSD through a finite-dimensional state-space framework. It introduces a state-space noise model and its steady-state observer form for the feedback-capacity analysis.

B. Contribution

  • For stationary Gaussian noise with rational PSD, the component-removal result applies to general finite-dimensional state-space realizations and every optimizer assigns zero power to the feedback-independent component.The result covers rational PSDs, including finite-order ARMA processes.Source: B. Contribution · p. 2

A. State-space noise model

  • The analysis models noise with a finite-dimensional state and Gaussian disturbances, under conditions that yield a stationary nonwhite output process.The stated conditions include Schur stability of F and a nonwhite stationary limiting output.Source: §II · p. 3, A. State-space noise model · p. 3

II. STATE-SPACE MODELS AND FEEDBACK-CAPACITY OPTIMIZATION

  • The paper’s state-space framework and convex feedback-capacity characterization provide the basis for the component-removal proof.The paper introduces the model, observer form, capacity characterization, and rational-PSD realization in sequence.Source: B. Contribution · p. 2, §II · p. 3

B. Kalman filter and observer form

  • The observer form uses the encoder’s MMSE estimate of the hidden noise state, with a Kalman filter yielding steady-state parameters.A sublinear zero-input prefix lets encoder and decoder share an estimate and brings the covariance close to its steady-state value.Source: B. Kalman filter and observer form · pp. 3–4

C. Capacity optimization and optimal input law

A convex state-space optimization parametrizes a capacity-achieving stationary input law, whose feedback-independent Gaussian component is represented by M. Rational noise spectra admit finite-dimensional state-space realizations, connecting the optimization to the paper’s rational-PSD scope.

C. Capacity optimization and optimal input law

  • An optimizer of the convex feedback-capacity problem parametrizes a capacity-achieving stationary input law with a Gaussian component independent of feedback.The component has covariance M, defined from the power constraint and optimizer parameters.Source: C. Capacity optimization and optimal… · p. 4
  • When M = 0, the stationary input is a strictly causal linear function of past noise, so its stationary spectral representation has SV ≡ 0.For a Riccati-tight optimizer, stability on the relevant range makes the state-estimate difference a mean-square limit of linear combinations of past noise innovations.Source: C. Capacity optimization and optimal… · p. 4

D. Rational spectra as SSM

  • A rational nonwhite PSD admits a finite-dimensional realization with a stable state matrix, yielding a stationary Gaussian noise process.The realization’s output transfer function is the rational spectral factor Φ(z).Source: D. Rational spectra as SSM · p. 5

III. MAIN RESULT

The paper proves that every optimizer of the feedback-capacity optimization assigns zero power to the feedback-independent Gaussian component. For rational-spectrum noise, this establishes Kim’s reduction and yields an optimizer-derived SK refinement gain.

III. MAIN RESULT

  • Every optimal solution to the feedback-capacity optimization assigns zero power to the feedback-independent Gaussian component.The result establishes Kim’s reduction for the rational-spectrum realization and supplies a missing premise for later arguments that rely on it.Source: §III · p. 5
  • Every optimizer induces a capacity-achieving stationary input law of the form specified by the theorem.The supplied passage does not include the formula for that form.Source: §III · p. 5
  • The optimizer determines the refinement gain ΓΣ̂† used to construct an explicit generalized SK scheme that achieves capacity.The section identifies this as the operational consequence of the structural result.Source: §III · p. 5

A. Capacity-achieving SK scheme

The scheme embeds a PAM message by shaping the mismatch between encoder and decoder state estimates, then refines that mismatch using a stationary input law. Its parameters come from a capacity optimizer, and it achieves every rate below feedback capacity for sufficiently large blocklength.

A. Capacity-achieving SK scheme

  • The message is represented by one-dimensional PAM symbols and embedded during the first d channel uses in the mismatch between encoder and decoder state estimates.Using the same gain Kp for both state-estimate recursions makes this mismatch a deterministic linear function of the embedded PAM vector.Source: A. Capacity-achieving SK scheme · p. 6
  • During refinement, the encoder transmits using the stationary law and both terminals update their state estimates before nearest-neighbor decoding.The encoder’s input is xi = ΓΣ̂†(ŝi − ŝ̂i), and the decoder estimates the embedded PAM symbols after n channel uses.Source: A. Capacity-achieving SK scheme · pp. 6–7
  • The scheme’s parameters are derived from any optimizer (Γ, Σ̂), with a Riccati solution substituted when needed while preserving optimality.Rates are selected with 0 ≤ Rj < log ρj and sum to R; each sub-message is mapped to an equally spaced PAM constellation.Source: A. Capacity-achieving SK scheme · p. 6
  • For any R < CFB(P), Algorithm 1 achieves rate R for all sufficiently large blocklengths.The proposition states that the scheme also has an error-probability bound, but the supplied passage omits its form.Source: A. Capacity-achieving SK scheme · p. 7

IV. PROOFS

The proof section establishes Theorem 1 and Proposition 1.

IV. PROOFS

  • The section proves Theorem 1 and Proposition 1.The supplied passage gives no further proof details.Source: §IV · p. 7

A. Notation

The notation section introduces a single-variable formulation of the optimization and relates it to the original capacity formulation.

A. Notation

  • The reformulation uses Q as its single decision variable and introduces a power constraint.The supplied passages do not include the full constraint expression.Source: A. Notation · p. 7
  • The corresponding gain is KY(Q) := (FΓ⊤ + FΣ̂H⊤ + KpΨ)ΨY(Q)−1.This expression defines the gain in terms of Q and the listed system parameters.Source: A. Notation · p. 7
  • The reformulation is equivalent to the original optimization, and every optimizer uses the full power budget p = P.Increasing p to P preserves feasibility and strictly increases ΨY(Q).Source: A. Notation · p. 7

B. Proof of Theorem 1

The proof shows that every optimal solution has zero Schur complement for Q, eliminating the feedback-independent component. It obtains this result by moving to an optimizer with a tight Riccati constraint and ruling out a positive Schur complement by perturbation.

Lemma 2

  • Lemma 2 constructs a feasible Q⋆⪰Q with the same p and Γ and Schur(L(Q⋆))=0, allowing the proof to start from a tight solution.Because Q⋆⪰Q does not decrease the objective, it is also optimal.Source: B. Proof of Theorem 1 · pp. 8–9

Proof of Theorem 1

  • Every optimal Q satisfies Schur(Q)=0, so the feedback-independent component M is zero.The proof first selects an optimal Q with Schur(L(Q))=0, then applies Lemma 1 and identifies Schur(Q) with M.Source: B. Proof of Theorem 1 · p. 8

Lemma 1

  • If an optimal Q has Schur(L(Q))=0 but Schur(Q)>0, a small feasible perturbation strictly increases the objective, contradicting optimality.The perturbation preserves the power constraint and the positive-semidefinite constraints for sufficiently small ε.Source: B. Proof of Theorem 1 · pp. 8–9

C. Technical lemmas for Theorem 1

The technical lemmas establish that the perturbation used to prove component removal both improves the objective and remains feasible. They rely on a Lyapunov equation and kernel properties of the optimizer.

Lemma 3

  • The Lyapunov equation for Ξ has a unique positive-semidefinite solution, and the nonwhite-noise assumption implies HΞH⊤>0.The proof establishes that HF^kKp is nonzero for some k; otherwise the stationary noise would be white.Source: C. Technical lemmas for Theorem 1 · pp. 9–10

Lemma 4

  • For a feasible Q with Schur(Q)>0 and Schur(L(Q))=0, Lemma 4 constrains vectors in ker(Q) so the perturbation preserves that kernel.The proof characterizes ker(Q) using the zero Schur complement and the kernel of Σ̂.Source: C. Technical lemmas for Theorem 1 · p. 10

D. Proof of Proposition 1

The proposition constructs an SK refinement scheme from an optimizer using its active modes, message embedding, and the decoder’s effective channel. For every rate below feedback capacity, the scheme has doubly-exponentially decaying error probability.

Lemma 5 (Active modes)

  • The optimizer’s active modes satisfy FpW=WJ, J is antistable, JY is Schur, and CFB(P)=log|det J|.These properties connect the active dynamics used for refinement to the feedback-capacity expression.Source: D. Proof of Proposition 1 · p. 10

Lemma 6 (Message embedding)

  • For every message θ, the embedding inputs xd=−C†Wa(θ) produce the initial mismatch Δd+1=Wa(θ).The proof verifies that Range(W)⊆Range(C), ensuring CC†W=W.Source: D. Proof of Proposition 1 · p. 12

Lemma 7 (Effective channel)

  • Conditioned on θ, the decoder statistic is Gaussian, with mean determined by JY and covariance bounded by W†Σ̂(W†)⊤.During refinement, wi=a(θ)−J−(i−d−1)W†Δi, and JY is Schur.Source: D. Proof of Proposition 1 · pp. 10, 12, 13

Error analysis

  • When each Rj<log ρj, the growing PAM spacing relative to the estimation error makes each sub-message error probability decay doubly exponentially.The proof bounds the estimation error using the Gaussian tail bound and applies a union bound over the message coordinates.Source: D. Proof of Proposition 1 · pp. 13–14

Proposition 1

  • For every R<CFB(P), the constructed scheme achieves rate R with doubly-exponentially decaying error probability.The proof chooses coordinate rates Rj<log ρj with their sum equal to R; the sum of log ρj equals CFB(P).Source: D. Proof of Proposition 1 · pp. 10, 14

Power constraint

  • The refinement input has power at most P plus a summable correction, and bounded embedding energy yields total energy at most (n−d)P+K.Finite zero-padding makes the scheme satisfy the power constraint.Source: D. Proof of Proposition 1 · p. 14

V. CONCLUSIONS

For stationary Gaussian noise with rational PSDs, the paper proves component removal and uses it to construct a generalized SK scheme. The scope of component removal beyond this setting remains open.

Conclusions

  • For stationary Gaussian noise with rational PSDs, a capacity-achieving stationary input is generated by strictly causal filtering of noise reconstructed from feedback.A finite message-dependent initialization realizes this input structure as a generalized SK scheme, with refinement parameters determined by the convex optimizer.Source: §V · p. 14

Scope

  • Whether component removal holds for MIMO channels, nonstationary noise, or arbitrary stationary spectra remains to be characterized.For arbitrary spectra, the open question concerns validity of SV≡0 without a finite-dimensional state-space representation.Source: §V · p. 14
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