spectral op• Data kinds: signal → table
• Call: import fullseye as fs; fs.ledger.hilbert_analytic(x, fs: 'float' = 1.0) -> 'dict' (to call the implementation directly, import mathspectral; mathspectral.hilbert_analytic(x, fs: 'float' = 1.0) -> 'dict'; from the registry, opsmath.get("hilbert_analytic"))
The analytic signal z = x + i·H[x] (negative frequencies removed via FFT), and the amplitude, instantaneous phase and instantaneous frequency read from it.
Args:
x: real 1-D signal.
fs: sampling frequency (sets the unit of the instantaneous frequency).
Returns:
analytic complex analytic signal, amplitude |z| (envelope), phase unwrapped phase [rad],
frequency instantaneous frequency = time derivative of the phase / 2π (central differences, same length as x).
Gate: cos → sin (definition of the Hilbert transform), the instantaneous frequency of a linear chirp is f0 + k·t, Bedrosian's theorem (if the envelope is band-limited below the carrier, the amplitude of an AM signal comes back as is). The ends are disturbed by the periodicity assumption (tapering the signal with a window reduces this).
Every mathops op validates its input before computing (nothing slips through silently):
• complex input raises ValueError — coercing to float64 silently discards the imaginary part (numpy only emits a ComplexWarning and returns a plausible-looking wrong real number). State .real/.imag/abs() explicitly, or use complexops, which handles complex data.
• masked arrays with masked elements raise ValueError — the implicit conversion that peels off the mask and uses the raw values underneath is refused. Say explicitly whether to fill or to drop.
• NaN/Inf raises ValueError on every input (refused with the count stated — it propagates through the whole result).
• Shapes are strict: 1-D and 2-D are never implicitly promoted or broadcast (a matrix in a vector slot, or a vector in a matrix slot, raises ValueError; reshape explicitly).
• Size cap: ops that take a matrix, and the stat_histogram bins, raise ValueError beyond mathops.MAX_ELEMENTS (2^26 ≈ 67 million elements).
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
• The canonical algorithm (author, year) and its uses are named in the family usage guide above.
• spectral_tour — py -3.11 examples/spectral_tour.py
table as input)dynsys_poincare_section · dwt_inverse · chebyshev_eval_nd
spectral)lomb_scargle · ula_snapshots · music_doa · esprit_doa · n_sources_mdl
*Provenance: mathspectral.py — MATH operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.