Bibliographic record
Abstract
Digital filters arising in the B-spline signal processing are characterized in a unified manner in this paper. The transfer functions of these filters are the z-transforms of the uniformly sampled central B-splines shifted by an arbitrary value. The transfer functions are cascades of an FIR kernel filter and a simple moving average FIR filter. For certain values of the shift parameter, the filters are identical to those referred to as the B-spline digital filters in the literature. The filters thus form a general family of B-spline digital filters. The kernel part of the B-spline filters may be used for transforming a discrete-time signal to a representation based on the B-spline coefficients. The B-spline filters may also be used to convert a sequence of B-spline coefficients to a discrete-time spline signal. The contributions of the paper are as follows. A unifying recurrence relation enabling the computation of the impulse response coefficients of the B-spline kernel filters is derived. An accompanying recurrence relation is also obtained for the entire transfer function of the kernel filters. The recurrences are valid for arbitrary values of the shift parameter. It is proved that the roots of the transfer functions of the kernel filters are distinct, negative and real. We also prove that the roots of the kernel filters of successive orders interlace. The results regarding the location of the zeros are also valid for arbitrary values of the shift parameter. The relation of the kernel filters to the Eulerian polynomials is discussed. It is shown that for certain choices of the parameters the kernel filters are equivalent to the classical Eulerian polynomials that frequently arise in combinatorics. An alternative closed-form expression for the kernel filters in the Bernstein form is also derived. Besides their importance in unifying the existing results on B-spline filters, the generalized family of B-spline filters studied in this paper find applications in fractional delay of B-spline signals.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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".