Computation of complex number theoretic transforms using quadratic residue number systems
Bibliographic record
Abstract
Very recently, the Quadratic Residue Number System (QRNS) has been introduced [4,5]. The QRNS is obtained from a mapping of Gaussian integers over a finite ring to a ring of conjugate elements. The conjugate ring has the remarkable property that both addition and multiplication are performed component-wise, therefore complex multiplication only requires two base field multiplications and zero additions. The operations are performed over sub-rings, isomorphic to the conjugate ring via the Chinese Remainder Theorem isomorphism. The primary restriction is the limited form of the moduli set for RNS computations. The QRNS has since been generalized for any type of moduli set with an increase in multiplications from 2 to 3 and the resulting number system has been termed the Modified Quadratic Residue Number System (MQRNS) [1,2]. The direct FIR filter architecture and bit-slice architecture for FIR and recursive digital filters have, been presented using the QRNS and MQRNS [4]. In this paper, the computation of the Complex Number Theoretic Transform(CNTT) and the hardware implementation of a radix-2 butterfly structure, using high-density ROM arrays, are presented. This paper shows that both theQRNS and MQRNS require almost the same amount of hardware for the implementation of the butterfly structure. The computation of Cyclic Convolution in both the QRNS and MQRNS is also discussed.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.004 | 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".