The Capacity Gains of Gaussian Channels with Unstable Versus Stable Autoregressive Noise
Bibliographic record
Abstract
In this paper, we consider Cover's and Pombra's formulation of feedback capacity of additive Gaussian noise (AGN) channels, with jointly Gaussian nonstationary and nonergodic noise. We derive closed-form feedback capacity formulas, using Karush-Kuhn-Tucker (KKT) conditions and convergence properties of difference Riccati equations to limiting algebraic Riccati equations of filtering theory, for unstable and stable autoregressive (AR) noise. Surprisingly, the capacity formulas depend on the parameters of the AR noise, its pole c∈(-∞,∞) and noise variance KW∈(0,∞), and the total transmit power κ∈[0,∞), indicating substantial gains for the unstable noise region ∀c2∈(1,∞),∀κ>κmin=▵KW1+4c2-32c2-12 compared to its complement region. In particular, feedback capacity is distinguished by three regimes, as follows. Regime 1, ∀c2∈(1,∞),∀κ>κmin: the optimal channel input includes an innovations part, the capacity increases as |c|>1 increases, while κmin and the allocated transmit power decrease. Regime 2, ∀c2∈(1,∞),∀κ≤κmin, Regime 3, ∀c∈[-1,1],∀κ∈[0,∞) (complement of Regime 1): the innovations part of the optimal channel is asymptotically zero and the capacity is fundamentally different compared to Regime 1. The differences of capacity formulas for Regimes 1, 2 and 3 are directly related to their operational meaning: (i) Regime 1 is an ergodic capacity while Regimes 2 and 3 are nonergodic capacities; (ii) Regime 1 is achieved by an asymptotically stationary channel input with a non-zero innovations part, while Regimes 2 and 3 are achieved by an asymptotically zero innovations part. The gains of capacity for Regime 1 are attributed to the high correlation of noise samples compared to stable noise and the use of an informative innovations part by the optimal channel input, which make possible the prediction of future noise samples from past samples, unlike memoryless noise. Our results provide answers to certain open questions regarding the validity of capacity formulas of stable noise that appeared in the literature.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".