Closed-Form CRLBs for the SNR Estimates from Turbo-Coded PAM- and Rectangular-QAM-Modulated Signals
Bibliographic record
Abstract
In this paper, we consider the problem of signal-to-noise ratio (SNR) estimation from turbo-coded (TC) PAM and rectangular-QAM (RQAM) modulated signals over flat-fading channels. We derive for the first time closed-form expressions for the Cramer-Rao lower bounds (CRLBs) of the SNR estimates. We exploit the structure of the binary-reflected-Gray-code (BRGC) for bits-to-symbols mapping, so that the likelihood function (LF) becomes factorized into a sum of two simple terms. This factorization allows the linearization of the log-likelihood function (LLF), which allowed us to carry the derivation of the Fisher Information Matrix (FIM) elementsanalytically, and derive the code aided (CA) SNR CRLB in closed-form (CF). These new CF expressions corroborate the CA bounds established previously in the particular case of square-QAM constellations. They finally tackle the new problem never addressed until now of CA SNR estimation from PAM/RQAM signals. In the low-to-medium SNR level, the new CRLBs for the CA estimates of the SNR range between their respective CRLBs in the non-data-aided (NDA) and data-aided (DA) scenarios, thereby highlighting and quantifying the advantage of CA estimation against the NDA. In high SNR, they coincide with the DA CRLB. These bounds also confirm the increase in estimation accuracy achievable by decreasing the coding rate.
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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 | 0.035 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.002 |
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".