Complete characterization of systems for simultaneous Lagrangian upsampling and fractional-sample delaying
Bibliographic record
Abstract
We present a complete formulation and an exact solution to the problem of designing systems for simultaneous sampling rate increase and fractional-sample delay in the Lagrangian sense. The problem may be regarded as that of a linear transformation, i.e., scaling, and/or shifting, of the uniform sampling grid of a discrete-time signal having a Newton series representation. It is proved that the solution forms a three-parameter family of maximally flat finite impulse response digital filters with a variable group-delay at the zero frequency. Various properties of the solution, including Nyquist properties and conditions for a linear phase response are analyzed. The solution, obtained in the closed form, is exact for polynomial inputs. We show that it is also suited for processing discrete-time versions of certain continuous-time bandlimited signals and signals having a rational Laplace transform. We then derive a generalization of the solution by augmenting the family with a fourth parameter that controls the number of multiple zeros at the roots of unity. This four-parameter family contains various types of maximally flat filters including those due to Herrmann and Baher. We list specific conditions on the four parameters to obtain many of the maximally flat filters reported in the literature. A significant part of the family of systems characterized by the solutions has been hitherto unknown. Examples are provided to elucidate this part as well.
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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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 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".