Approximation of Achievable Rates in Additive Gaussian Mixture Noise Channels
Bibliographic record
Abstract
In this paper, we detail effective methods to approximate the achievable rates of channels with additive Gaussian mixture (GM) noise for both real and complex channels to achieve any desired level of accuracy. Attention is paid to a Gaussian input, a discrete real input, and a complex input with discrete amplitude and independent uniform phase. Such discrete inputs represent a wide range of input distributions and they include the capacity-achieving inputs as special cases. At first, we propose a simple technique to accurately calculate the noise entropy. Specifically, when the noise level is high, a lower bound on the integrand of the entropy is established and the noise entropy can be estimated using a closed-form solution. In the low noise region, the piecewise-linear curve fitting (PWLCF) method is applied. We then extend this result to calculate the achievable rate when the input is Gaussian distributed, which is shown to be asymptotically optimal. Next, we propose a simple PWLCF-based method to approximate the output entropy for a real GM channel when the input is discrete, and for a complex GM channel when the input is discrete in amplitude with independent uniform phase. In particular, for the real channel, the output entropy is evaluated by examining the output in high and low regions of amplitude using a lower bound on the integrand of the output entropy and PWLCF, respectively. For the complex channel, the output entropy is approximated a similar manner but using polar coordinates and the Kernel function. It is demonstrated that the output entropy, and consequently, the achievable rates, can be computed to achieve any given accuracy level.
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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 | 0.004 | 0.016 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
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".