Full Transform Functions

Object methods are documented in API Overview. All four full functions select window="meyer" by default. Gaussian accepts nu and support; WDM inverse methods pass their retained metadata automatically. Bare coefficient arrays carry no metadata, so low-level inverse callers must pass the matching originating configuration.

from_time_to_wdm

from_time_to_wdm(data: Any, *, nt: int, nf: int, a: float, d: float, dt: float, backend: str | Backend | None = None, window: Literal['meyer', 'gaussian'] = 'meyer', nu: float | None = None, support: int | None = None) -> Any

Compute Wilson coefficients with Meyer or modified-Gaussian windows.

Gaussian nu defaults to 0.5 and support to min(16, nf) half-band spacings. Its default a=1/3, d=1 values are ignored. Meyer rejects explicit nu or support.

from_freq_to_wdm

from_freq_to_wdm(data: Any, *, nt: int, nf: int, a: float, d: float, dt: float, backend: str | Backend | None = None, window: Literal['meyer', 'gaussian'] = 'meyer', nu: float | None = None, support: int | None = None) -> Any

Compute Wilson coefficients from full complex Fourier-domain samples.

The spectrum is projected to the Fourier transform of its real inverse. Window options follow :func:from_time_to_wdm; an rFFT half-spectrum is not a full Fourier grid.

from_wdm_to_time

from_wdm_to_time(coeffs: Any, *, a: float, d: float, dt: float, backend: str | Backend | None = None, window: Literal['meyer', 'gaussian'] = 'meyer', nu: float | None = None, support: int | None = None) -> Any

Reconstruct a time-domain signal with the originating window settings.

Bare arrays carry no window metadata: the caller must supply the family and Gaussian settings used for analysis. :class:WDM does this for you.

from_wdm_to_freq

from_wdm_to_freq(coeffs: Any, *, dt: float, a: float, d: float, backend: str | Backend | None = None, window: Literal['meyer', 'gaussian'] = 'meyer', nu: float | None = None, support: int | None = None) -> Any

Reconstruct a full Fourier grid with the originating window settings.

Compact Subbands

The compact functions and aliases below support both window families. Meyer retains its existing span convention; Gaussian expands spans according to its finite frequency support. These APIs use a one-sided Fourier grid and treat omitted bins or coefficient channels as zero. Bare coefficient arrays carry no window metadata: pass the originating settings to inverse synthesis.

from_freq_to_wdm_subband

from_freq_to_wdm_subband(data: Any, *, df: float, nfreqs_fourier: int, kmin: int, nfreqs_wdm: int, ntimes_wdm: int, a: float = 1.0 / 3.0, d: float = 1.0, backend: str | Backend | None = None, window: Literal['meyer', 'gaussian'] = 'meyer', nu: float | None = None, support: int | None = None) -> tuple[Any, int]

Compute the overlapping WDM sub-band from a compact Fourier span.

Use this when the input is already cropped in the one-sided Fourier domain. The function evaluates only the WDM frequency channels whose window support intersects the supplied Fourier span, rather than materializing the full WDM grid.

Parameters:

• data (Any) –

  One-dimensional one-sided Fourier samples for the compact span [kmin, kmin + len(data)).

• df (float) –

  Frequency spacing of the full one-sided Fourier grid.

• nfreqs_fourier (int) –

  Number of samples in the full one-sided Fourier grid, including DC and Nyquist.

• kmin (int) –

  Starting Fourier-bin index of data in the full one-sided grid.

• nfreqs_wdm (int) –

  Number of positive WDM frequency intervals in the full transform. A full WDM coefficient grid has nfreqs_wdm + 1 frequency channels.

• ntimes_wdm (int) –

  Number of WDM time bins.

• a (float, default: 1.0 / 3.0 ) –

  Window roll-off parameter.

• d (float, default: 1.0 ) –

  Window order parameter.

• backend (str | Backend | None, default: None ) –

  Backend name or object. If omitted, the default backend is used.

• window (Literal['meyer', 'gaussian'], default: 'meyer' ) –

  Meyer (default) or orthogonalized modified Gaussian.

• nu (float | None, default: None ) –

  Gaussian shape parameter, default 0.5; omitted for Meyer.

• support (int | None, default: None ) –

  Gaussian half-band spacings retained on each side, default min(16, nfreqs_wdm); omitted for Meyer.

Returns:

• Any –

  A tuple (coeffs, mmin). coeffs has shape

• int –

  (ntimes_wdm, nf_sub_wdm) and stores the touched WDM channels.

• tuple[Any, int] –

  mmin is the first WDM frequency-channel index represented by

• tuple[Any, int] –

  coeffs[:, 0].

Notes

Fourier bins outside the supplied crop are zero. Retain window, nu and support for the inverse; the returned array carries no window metadata.

from_freq_to_wdm_band

from_freq_to_wdm_band(data: Any, *, df: float, nfreqs_fourier: int, kmin: int, nfreqs_wdm: int, ntimes_wdm: int, mmin: int, nf_sub_wdm: int, a: float = 1.0 / 3.0, d: float = 1.0, backend: str | Backend | None = None, window: Literal['meyer', 'gaussian'] = 'meyer', nu: float | None = None, support: int | None = None) -> Any

Compute a requested WDM channel block from a compact Fourier span.

This is the study-facing variant of the subband API: it first computes the minimal touched WDM span, then returns only the requested compact block mmin : mmin + nf_sub_wdm. The requested block must lie inside the touched span of the supplied Fourier support.

Parameters:

• data (Any) –

  One-dimensional one-sided Fourier samples for the compact span [kmin, kmin + len(data)).

• df (float) –

  Frequency spacing of the full one-sided Fourier grid.

• nfreqs_fourier (int) –

  Number of samples in the full one-sided Fourier grid, including DC and Nyquist.

• kmin (int) –

  Starting Fourier-bin index of data in the full one-sided grid.

• nfreqs_wdm (int) –

  Number of positive WDM frequency intervals in the full transform. A full WDM coefficient grid has nfreqs_wdm + 1 frequency channels.

• ntimes_wdm (int) –

  Number of WDM time bins.

• mmin (int) –

  First WDM frequency-channel index to return.

• nf_sub_wdm (int) –

  Number of WDM frequency channels to return.

• a (float, default: 1.0 / 3.0 ) –

  Window roll-off parameter.

• d (float, default: 1.0 ) –

  Window order parameter.

• backend (str | Backend | None, default: None ) –

  Backend name or object. If omitted, the default backend is used.

• window (Literal['meyer', 'gaussian'], default: 'meyer' ) –

  Meyer (default) or orthogonalized modified Gaussian.

• nu (float | None, default: None ) –

  Gaussian shape parameter, default 0.5; omitted for Meyer.

• support (int | None, default: None ) –

  Gaussian support, default min(16, nfreqs_wdm).

Returns:

• Any –

  The requested coefficient block with shape

• Any –

  (ntimes_wdm, nf_sub_wdm).

from_wdm_to_freq_subband

from_wdm_to_freq_subband(coeffs: Any, *, df: float, nfreqs_fourier: int, mmin: int, nfreqs_wdm: int, ntimes_wdm: int, a: float = 1.0 / 3.0, d: float = 1.0, backend: str | Backend | None = None, window: Literal['meyer', 'gaussian'] = 'meyer', nu: float | None = None, support: int | None = None) -> tuple[Any, int]

Reconstruct the touched Fourier span from a compact WDM block.

This is the inverse of the compact sub-band transform. It expands a block of adjacent WDM frequency channels into the one-sided Fourier bins touched by those channels' window support.

Parameters:

• coeffs (Any) –

  Two-dimensional WDM coefficient block with shape (ntimes_wdm, nf_sub_wdm).

• df (float) –

  Frequency spacing of the full one-sided Fourier grid.

• nfreqs_fourier (int) –

  Number of samples in the full one-sided Fourier grid, including DC and Nyquist.

• mmin (int) –

  First WDM frequency-channel index represented by coeffs[:, 0].

• nfreqs_wdm (int) –

  Number of positive WDM frequency intervals in the full transform. A full WDM coefficient grid has nfreqs_wdm + 1 frequency channels.

• ntimes_wdm (int) –

  Number of WDM time bins.

• a (float, default: 1.0 / 3.0 ) –

  Window roll-off parameter.

• d (float, default: 1.0 ) –

  Window order parameter.

• backend (str | Backend | None, default: None ) –

  Backend name or object. If omitted, the default backend is used.

• window (Literal['meyer', 'gaussian'], default: 'meyer' ) –

  Window family used for the supplied coefficients.

• nu (float | None, default: None ) –

  Originating Gaussian shape parameter, default 0.5.

• support (int | None, default: None ) –

  Originating Gaussian support, default min(16, nfreqs_wdm).

Returns:

• Any –

  A tuple (spectrum, kmin). spectrum contains the reconstructed

• int –

  one-sided Fourier span, and kmin is the starting Fourier-bin index

• tuple[Any, int] –

  of spectrum[0] in the full one-sided grid.

Notes

WDM channels outside the supplied block are zero. Supply the same window, nu and support used to create the coefficients.

wdm_span_from_fourier_span

wdm_span_from_fourier_span(*, nfreqs_fourier: int, nfreqs_wdm: int, ntimes_wdm: int, kmin: int, lendata: int, window: str = 'meyer', nu: float | None = None, support: int | None = None) -> tuple[int, int]

Map a compact one-sided Fourier span onto the touched WDM channels.

Parameters:

• nfreqs_fourier (int) –

  Number of samples in the full one-sided Fourier grid, including DC and Nyquist.

• nfreqs_wdm (int) –

  Number of positive WDM frequency intervals in the full transform. A full WDM coefficient grid has nfreqs_wdm + 1 frequency channels.

• ntimes_wdm (int) –

  Number of WDM time bins.

• kmin (int) –

  Starting Fourier-bin index of the compact Fourier span.

• lendata (int) –

  Number of Fourier bins in the compact span.

• window (str, default: 'meyer' ) –

  Meyer or modified Gaussian; default Meyer preserves historical spans.

• nu (float | None, default: None ) –

  Gaussian shape parameter; it does not alter the geometric tap span.

• support (int | None, default: None ) –

  Gaussian spectral support in channel spacings on either side.

Returns:

• int –

  A tuple (mmin, nf_sub_wdm). mmin is the first touched WDM

• int –

  frequency-channel index, and nf_sub_wdm is the number of adjacent

• tuple[int, int] –

  WDM channels touched by the Fourier span.

fourier_span_from_wdm_span

fourier_span_from_wdm_span(*, nfreqs_fourier: int, nfreqs_wdm: int, ntimes_wdm: int, mmin: int, nf_sub_wdm: int, window: str = 'meyer', nu: float | None = None, support: int | None = None) -> tuple[int, int]

Map a compact WDM channel span onto the touched one-sided Fourier bins.

Parameters:

• nfreqs_fourier (int) –

  Number of samples in the full one-sided Fourier grid, including DC and Nyquist.

• nfreqs_wdm (int) –

  Number of positive WDM frequency intervals in the full transform. A full WDM coefficient grid has nfreqs_wdm + 1 frequency channels.

• ntimes_wdm (int) –

  Number of WDM time bins.

• mmin (int) –

  First WDM frequency-channel index in the compact WDM span.

• nf_sub_wdm (int) –

  Number of adjacent WDM frequency channels in the compact span.

• window (str, default: 'meyer' ) –

  Meyer or modified Gaussian; default Meyer preserves historical spans.

• nu (float | None, default: None ) –

  Gaussian shape parameter; it does not alter the geometric tap span.

• support (int | None, default: None ) –

  Gaussian spectral support in channel spacings on either side.

Returns:

• int –

  A tuple (kmin, lendata). kmin is the first touched one-sided

• int –

  Fourier-bin index, and lendata is the number of touched Fourier

• tuple[int, int] –

  bins.