electroacoustics.intermodulation

Intermodulation distortion of audio equipment (IEC 60268-3 14.12.7-10).

Where the harmonic metrics of phonometry.electroacoustics.distortion drive the equipment with a single tone, these drive it with two (or with a sine against a square wave) and read the products that a non-linearity puts between the excitation frequencies — the components a harmonic measurement cannot see and the ear finds least forgiving, because they are inharmonic with the programme:

Modulation distortion d_m,2/d_m,3 (14.12.7): a large low tone $f_{1}$ modulating a small high tone $f_{2}$ , read on the sidebands at $f_{2} \pm (n - 1) f_{1}$ .
Difference-frequency distortion d_d,2/d_d,3 (14.12.8) from two equal-amplitude tones, and the total difference-frequency distortion (14.12.10) of the standard 8 kHz / 11,95 kHz pair.
Dynamic intermodulation distortion DIM (14.12.9) from the 15 kHz sine / 3,15 kHz square-wave test signal.

The per-order definitions are the IEC ones (arithmetic sums of the product amplitudes, referenced as each clause prescribes), with the SMPTE combined-RMS convention reported alongside where analyzers use it. As with the harmonic metrics, every quantity has an exact analytic oracle: a signal synthesised with known product amplitudes reproduces the closed-form ratio, and the tones are assumed to fall on (or very near) FFT bins.

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difference_frequency_distortion

difference_frequency_distortion(
    signal: NDArray[np.float64] | list[float],
    fs: float,
    f1: float,
    f2: float,
    *,
    order: int = 2,
    window: str = 'hann',
) -> float

Difference-frequency distortion of the nth order (IEC 60268-3 14.12.8).

Two equal-amplitude tones $f_{1} < f_{2}$ are applied. Per 14.12.8.1 the reference voltage is $U_{2, ref} = 2 U_{2, f_{2}}$ — realised here as the sum of both measured tone amplitudes, identical for the standard equal-amplitude tones — and

d_{d, 2} = a_{f_{2} - f_{1}} / (a_{f_{1}} + a_{f_{2}})

d_{d, 3} = (a_{2 f_{2} - f_{1}} + a_{2 f_{1} - f_{2}}) / (a_{f_{1}} + a_{f_{2}})

with the third order an arithmetic sum of the two products. Products that fall outside (0, Nyquist) or that cannot be separated from a primary tone or DC read zero.

Parameters

Name	Description
`signal`	Captured signal (1-D).
`fs`	Sample rate, in Hz.
`f1`	Lower tone, in Hz.
`f2`	Upper tone, in Hz.
`order`	Product order (2 or 3).
`window`	FFT window (default `'hann'`).

Returns: nth-order difference-frequency distortion, as a ratio.

Raises

Exception	When
ValueError	If `order` is not 2 or 3 or the inputs are invalid.

dynamic_intermodulation_distortion

dynamic_intermodulation_distortion(
    signal: NDArray[np.float64] | list[float],
    fs: float,
    *,
    f_sine: float = 15000.0,
    f_square: float = 3150.0,
    window: str = 'hann',
) -> float

Dynamic intermodulation distortion DIM (IEC 60268-3 14.12.9).

From the standard test signal — a f_sine = 15 kHz sine plus a low-pass-filtered f_square = 3.15 kHz square wave in a 1:4 peak ratio — the DIM is the RMS of the intermodulation products $∣ k f_{square} \pm f_{sine} ∣$ that fall below f_sine (IEC 60268-3 Table 2), relative to the 15 kHz sine amplitude.

Parameters

Name	Description
`signal`	Captured signal (1-D).
`fs`	Sample rate, in Hz.
`f_sine`	High sine frequency, in Hz (default 15 kHz).
`f_square`	Square-wave fundamental, in Hz (default 3.15 kHz).
`window`	FFT window (default `'hann'`).

Returns: Dynamic intermodulation distortion, as a ratio.

Raises

Exception	When
ValueError	If the inputs are invalid.

modulation_distortion

modulation_distortion(
    signal: NDArray[np.float64] | list[float],
    fs: float,
    f_low: float,
    f_high: float,
    *,
    window: str = 'hann',
) -> ModulationDistortionResult

Modulation distortion of the nth order (IEC 60268-3 14.12.7).

A low-frequency tone $f_{1}$ (f_low, large) and a high-frequency tone $f_{2}$ (f_high; small, amplitude ratio preferably 4:1) are applied; the nth-order distortion shows up as modulation sidebands at $f_{2} \pm (n - 1) f_{1}$ . Per 14.12.7.2 g)-h) the per-order values use the arithmetic sum of the two sideband amplitudes, referenced to the output voltage at f2:

d_{m, 2} = (a_{f_{2} + f_{1}} + a_{f_{2} - f_{1}}) / a_{f_{2}}

d_{m, 3} = (a_{f_{2} + 2 f_{1}} + a_{f_{2} - 2 f_{1}}) / a_{f_{2}}

(The alternative presentation $d_{m, n}^{'} = 5 d_{m, n}$ references the 4:1 reference output voltage $U_{2, ref} = 5 U_{2, f_{2}}$ instead.) The combined root-sum-square that SMPTE-type analyzers report is returned alongside as smpte.

Parameters

Name	Description
`signal`	Captured signal (1-D).
`fs`	Sample rate, in Hz.
`f_low`	Low modulating tone `f1`, in Hz (e.g. 60 Hz).
`f_high`	High carrier tone `f2`, in Hz (e.g. 7 kHz).
`window`	FFT window (default `'hann'`).

Returns: A ModulationDistortionResult with d2, d3 and the smpte combined RMS.

Raises

Exception	When
ValueError	If the inputs are invalid.

ModulationDistortionResult

ModulationDistortionResult(
    d2: float,
    d3: float,
    smpte: float,
    f_low: float | None = None,
    f_high: float | None = None,
    carrier_amplitude: float | None = None,
    sideband_frequencies: NDArray[np.float64] | None = None,
    sideband_amplitudes: NDArray[np.float64] | None = None,
)

Modulation (intermodulation) distortion (IEC 60268-3 14.12.7).

Attributes

Name	Description
`d2`	Second-order modulation distortion $d_{m, 2}$ (14.12.7.2 g): the arithmetic sum of the sideband amplitudes at $f_{2} \pm f_{1}$ relative to the output amplitude at `f2`.
`d3`	Third-order modulation distortion $d_{m, 3}$ (14.12.7.2 h): the arithmetic sum of the sidebands at $f_{2} \pm 2 f_{1}$ relative to the output amplitude at `f2`.
`smpte`	Combined-RMS convention of SMPTE-type analyzers (not an IEC 60268-3 quantity): $\sum a_{s}^{2} / a_{f_{2}}$ over all four sidebands.
`f_low`	Low modulating tone `f1`, in Hz.
`f_high`	High carrier tone `f2`, in Hz.
`carrier_amplitude`	Measured output amplitude at `f2` (the reference of the per-order ratios).
`sideband_frequencies`	The four intermodulation product frequencies in ascending order: $f_{2} - 2 f_{1}$ , $f_{2} - f_{1}$ , $f_{2} + f_{1}$ and $f_{2} + 2 f_{1}$ , in Hz.
`sideband_amplitudes`	Measured peak amplitudes at `sideband_frequencies` (zero for a product that falls outside the analysis band or cannot be separated from a primary tone).

ModulationDistortionResult.plot()

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

Plot the carrier and its modulation sidebands, with d2/d3 annotated.

Draws the output amplitude at f2 (the 0 dB reference) and the four intermodulation sidebands at $f_{2} \pm f_{1}$ and $f_{2} \pm 2 f_{1}$ as a stem-style spectrum in dB relative to the carrier, the modulation counterpart of plot.

Parameters

Name	Description
`ax`	Existing axes, or `None` to create a figure.
`language`	Label language, `"en"` (default) or `"es"`.
`kwargs`	Forwarded to the marker `plot` call.

Returns: The axes.

Raises

Exception	When
ValueError	If the result carries no sideband spectrum data (a result constructed by hand without the spectral fields).

total_difference_frequency_distortion

total_difference_frequency_distortion(
    signal: NDArray[np.float64] | list[float],
    fs: float,
    f1: float = 8000.0,
    f2: float = 11950.0,
    *,
    window: str = 'hann',
) -> float

Total difference-frequency distortion (IEC 60268-3 14.12.10).

A specific two-tone test with $f_{1} = 2 f_{0}$ and $f_{2} = 3 f_{0} - δ$ (the standard values, kept as defaults, are $f_{1} = 8$ kHz, $f_{2} = 11.95$ kHz, so $f_{0} = 4$ kHz and $δ = 50$ Hz). Only the two in-band products at $f_{0} \mp δ$ enter — the second-order product at $f_{2} - f_{1}$ and the third-order product at $2 f_{1} - f_{2}$ — combined in RMS over the arithmetic sum of the two tone output amplitudes (14.12.10.2 g):

d_{TDFD} = a_{f_{2} - f_{1}}^{2} + a_{2 f_{1} - f_{2}}^{2} / (a_{f_{1}} + a_{f_{2}})

(The out-of-band product at $2 f_{2} - f_{1}$ is explicitly not part of it.)

Parameters

Name	Description
`signal`	Captured signal (1-D).
`fs`	Sample rate, in Hz.
`f1`	Lower tone, in Hz (default 8 kHz, per 14.12.10.2 b).
`f2`	Upper tone, in Hz (default 11.95 kHz, per 14.12.10.2 b).
`window`	FFT window (default `'hann'`).

Returns: Total difference-frequency distortion, as a ratio.

Raises

Exception	When
ValueError	If the inputs are invalid.

Created and maintained by José Manuel Requena Plens ORCID All projects