Application of the Mittag–Leffler expansion to sampling discontinuous signals
Bibliographic record
Abstract
In applying Shannon's sampling theorem, evaluation of the sampled signal Fourier spectrum is based on the fact that sampling the continuous‐time signal is the result of multiplying the signal by distributions. If the signal has discontinuities, a multiplication of distributions – an undefined operation – is encountered. Such undefined operation has led to errors in the literature which to date accompany the formulation of sampling of signals containing discontinuities. This paper presents an approach to evaluating the product of distributions as a means of sampling discontinuous signals, eliminating such errors. It is shown that the value of the product of distributions may be found by invoking the Mittag–Leffler expansion. As an illustration of errors that have existed for decades and still exist in the digital signal processing literature whenever discontinuous signals are sampled the approach of impulse invariance provides a case in point. It was already noted that this approach has an inherent error. Yet, impulse invariance is still considered as one of the two main approaches for converting analogue to digital filters. In this study, the true spectra of sampled discontinuous signals are evaluated, and a new approach to the transformations between continuous‐time and discrete‐time systems eliminating the error, is proposed.
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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 | 0.002 |
| 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 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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".