signals.inversion

Regularized spectral inversion with frequency-dependent regularization.

Inverting a measured transfer function — to equalize a measurement loudspeaker, flatten a microphone response or post-process an acquired impulse response — cannot be a plain reciprocal $1/H(f)$: wherever the system radiates little energy (outside its passband, in deep notches) the reciprocal explodes and the inverse filter amplifies nothing but noise. Mueller & Massarani (“Transfer-Function Measurement with Sweeps”, JAES 49(6), 2001, Secs. 3.1 and 4.5) therefore confine the inversion to the transmission band: unity equalization in-band and a controlled, bounded gain outside.

This module implements that behaviour with the frequency-dependent Tikhonov regularization of Kirkeby & Nelson (“Digital Filter Design for Inversion Problems in Sound Reproduction”, JAES 47(7/8), 1999, Eq. (17)):

where the regularization profile $ϵ(f)$ is small inside the band $[f_1,f_2]$ (the filter equalizes to unity within an analytic bound) and large outside (the out-of-band gain is capped at $1/(2\sqrt{ϵ})$, the maximum of $x/(x^2+ϵ)$), with a smooth geometric cross-fade over a transition zone bordering the band edges. A modeling delay makes the mixed-phase inverse causal (Kirkeby & Nelson Sec. 2.4: the inverse of a non-minimum-phase response is anticausal and must be delayed to be realisable).

The closed forms above are the module’s oracles: in-band the equalized magnitude $|H{H}_{\mathrm{inv}}|$ deviates from unity by exactly $ϵ/(|H{|}^2+ϵ)$, and out-of-band the filter gain never exceeds the $1/(2\sqrt{ϵ})$ bound.

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InverseFilterResult(
    inverse: np.ndarray,
    frequencies: np.ndarray,
    spectrum: np.ndarray,
    response_spectrum: np.ndarray,
    regularization: np.ndarray,
    f_range: tuple[float, float],
    delay: int,
    fs: float,
    flatness_db: float,
    max_gain_db: float,
)

A regularized inverse filter with its achieved equalization.

Returned by regularized_inverse_filter. The causal filter samples live in inverse (the equalized response arrives delay samples late; apply compensates it). spectrum is the complex inverse spectrum including the modeling delay; regularization is the $ϵ(f)$ profile actually used, and flatness_db reports how flat the equalized magnitude $|H(f)H_{\mathrm{inv}}(f)|$ actually is across the requested band.

Attributes

InverseFilterResult.apply(x: list[float] | np.ndarray) -> np.ndarray

Equalize a signal with the inverse filter.

Convolves x with the filter and removes the modeling delay, so the output is time-aligned with the input and has the same length. Feeding the response the filter was designed from returns (in-band) a band-limited unit impulse at sample 0.

Parameters

Returns: The equalized, delay-compensated signal, len(x) samples.

InverseFilterResult.plot(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Plot the measured, inverse and equalized magnitudes.

One panel: $|H|$, $|H_{\mathrm{inv}}|$ and the equalized product $|H{H}_{\mathrm{inv}}|$ in dB over log-frequency, with the equalized band shaded. Requires matplotlib (pip install phonometry[plot]).

Parameters

property

Number of samples in the inverse filter.

regularized_inverse_filter(
    response: SignalInput | TimedResponse,
    fs: float | None = None,
    *,
    f_range: tuple[float, float],
    regularization_inside: float = 1e-06,
    regularization_outside: float = 1.0,
    transition_octaves: float = 0.3333333333333333,
    n_fft: int | None = None,
    delay: int | None = None,
) -> InverseFilterResult

Design a regularized inverse filter for a measured impulse response.

Computes the frequency-dependent Tikhonov inverse of Kirkeby & Nelson (JAES 47(7/8), 1999, Eq. (17)),

with $ϵ(f)$ small across $[f_1,f_2]$ and large outside, so the filter equalizes the response to unity in-band while the out-of-band gain stays bounded by $1/(2\sqrt{{ϵ}_{\mathrm{outside}}})$ — the behaviour Mueller & Massarani (JAES 49(6), 2001, Secs. 3.1/4.5) obtain by band-passing the plain inverse. A modeling delay of delay samples (default half the FFT block) shifts the generally anticausal inverse of a mixed-phase response into a causal filter (Kirkeby & Nelson, Sec. 2.4).

Both regularization levels are relative to the peak of $|H{|}^2$ (like the scalar regularization of phonometry.room.impulse_response, which this generalises): in-band the equalized magnitude deviates from unity by at most regularization_inside * max|H|**2 / min|H|**2 — the analytic residue $ϵ/(|H{|}^2+ϵ)$ — and the achieved figure is reported as InverseFilterResult.flatness_db.

Use the result’s InverseFilterResult.apply to equalize recordings (or the excitation, for pre-emphasis) and read InverseFilterResult.spectrum to apply it spectrally.

Parameters

Returns: An InverseFilterResult.