electroacoustics.sound_reinforcement

Gain before feedback of a sound-reinforcement system.

A public-address system is a closed loop: the loudspeaker feeds the audience, but it also feeds the microphone that drives it. Long (Architectural Acoustics 2nd ed., Chapter 18, Equations (18.13) to (18.24)) writes the loop in terms of two decibel gains,

• the open-loop system gain $Z_{\mathrm{S}}$ (Equation (18.17)), the level the loudspeaker produces at an average listener minus the level the talker produces at the microphone,

  $$Z_{\mathrm{S}}=L_{\mathrm{H}\text{-}\mathrm{L}}-L_{\mathrm{T}\text{-}\mathrm{M}}$$

  so $Z_{\mathrm{S}}=-6$ dB (a typical auditorium or church) means the amplified sound at the listener sits 6 dB below what the talker delivers to the microphone, i.e. a comfortable conversational level at twice the talker-to-microphone distance;

• the feedback-loop gain $G_{\mathrm{S}}$ (Equation (18.18)), the part of that output that returns to the microphone,

  $$G_{\mathrm{S}}=L_{\mathrm{H}\text{-}\mathrm{M}}-L_{\mathrm{H}\text{-}\mathrm{L}}+D_{\mathrm{M}}(\theta )$$

  with $D_{\mathrm{M}}(\theta )$ the directivity index of the microphone toward the loudspeaker relative to the talker (zero for an omnidirectional microphone, about -2 to -3 dB for a cardioid pointed at the talker).

Summing the infinite series of round trips (Equation (18.14)) makes the system oscillate when the loop gain reaches unity, that is (Equation (18.16))

$$Z_{\mathrm{S}}+G_{\mathrm{S}}=0$$

Long takes a feedback stability margin of 10 dB for an equalised system (other authors quote 12 dB unequalised and 6 dB carefully equalised): 6 dB of it covers a tone that adds in phase with a reflection from a hard surface, the remaining 4 dB is safety. With several microphones open at once the returned signals add at the mixer, which is accounted for by the number of open microphones correction (Equation (18.23)) $\Delta L_{\text{nom}}=10{\log }_{10}{N}_{\mathrm{m}}$. The stability criterion is then Equation (18.24),

$$Z_{\mathrm{S}}+L_{\mathrm{H}\text{-}\mathrm{M}}+\Delta L_{\text{nom}}\le L_{\mathrm{H}\text{-}\mathrm{L}}-D_{\mathrm{M}}(\theta )-10$$

Note

Long prints Equation (18.24) with $+D_{\mathrm{M}}(\theta )$ on the right-hand side, which contradicts Equations (18.20) to (18.22) it generalises (and would flip the benefit of a directional microphone into a penalty; see docs/ERRATA.md). This module implements the sign of Equation (18.20), so that $N_{\mathrm{m}}=1$ reproduces Long’s own special cases: with $Z_{\mathrm{S}}=-6$ dB the criterion collapses to $L_{\mathrm{H}\text{-}\mathrm{M}}\le L_{\mathrm{H}\text{-}\mathrm{L}}-D_{\mathrm{M}}(\theta )-4$ (Equation (18.21)), which for an omnidirectional microphone puts the loudspeaker level at the microphone 4 dB below the average level in the audience, and for a cardioid ($D_{\mathrm{M}}=-2$ dB) 2 dB below it (Equation (18.22)).

Note what the criterion does not contain: the loudspeaker type, its number or its power. Only the sound field it produces at two points and the type and orientation of the microphone matter. Long also excludes the reverberant field deliberately, as it is uniform and therefore does not depend on where the microphone is or where it points; the direct-to-reverberant ratio enters the design separately, through the intelligibility table of his Table 18.3 (better than -3 dB excellent, -3 to -6 very good, -6 to -9 good, -9 to -12 fair, -12 to -15 poor, below -15 very poor).

This module is a closed-form design calculator anchored on those equations. Long gives no fully numeric worked case for them, so the tests anchor on the closed forms and on the special cases he states in words.

Auto-generated from the source docstrings by scripts/generate_api_docs.py (make api-docs). Do not edit by hand.

CARDIOID_RELATIVE_DIRECTIVITY

Constant (float).

CARDIOID_RELATIVE_DIRECTIVITY = -2.0

DEFAULT_STABILITY_MARGIN

Constant (float).

DEFAULT_STABILITY_MARGIN = 10.0

feedback_loop_gain

feedback_loop_gain(
    level_loudspeaker_at_microphone: float,
    level_loudspeaker_at_listener: float,
    *,
    microphone_directivity: float = 0.0,
) -> float

Feedback-loop gain $G_{\mathrm{S}}$ (Long Equation (18.18)).

$G_{\mathrm{S}}=L_{\mathrm{H}\text{-}\mathrm{M}}-L_{\mathrm{H}\text{-}\mathrm{L}}+D_{\mathrm{M}}(\theta )$.

Parameters

Name	Description
level_loudspeaker_at_microphone	Direct-field level $L_{\mathrm{H}\text{-}\mathrm{M}}$ the loudspeaker produces at the microphone, dB.
level_loudspeaker_at_listener	Direct-field level $L_{\mathrm{H}\text{-}\mathrm{L}}$ the loudspeaker produces at an average listener, dB.
microphone_directivity	Directivity index $D_{\mathrm{M}}(\theta )$ of the microphone toward the loudspeaker relative to the talker, dB (0 for an omnidirectional microphone, about CARDIOID_RELATIVE_DIRECTIVITY for a cardioid).

Returns: The feedback-loop gain $G_{\mathrm{S}}$, dB.

feedback_stability

feedback_stability(
    open_loop_gain: float,
    level_loudspeaker_at_microphone: float,
    level_loudspeaker_at_listener: float,
    *,
    microphone_directivity: float = 0.0,
    open_microphones: int = 1,
    stability_margin: float = 10.0,
) -> FeedbackStabilityResult

Stability of a reinforcement loop (Long Equations (18.16) to (18.24)).

The loop gain $Z_{\mathrm{S}}+G_{\mathrm{S}}+\Delta L_{\text{nom}}$ is compared with the oscillation threshold of Equation (18.16) reduced by the stability margin, that is Equation (18.24) written with the sign of Equation (18.20) (see the module docstring).

Parameters

Name	Description
open_loop_gain	Open-loop system gain $Z_{\mathrm{S}}=L_{\mathrm{H}\text{-}\mathrm{L}}-L_{\mathrm{T}\text{-}\mathrm{M}}$, dB; about -6 dB for a typical auditorium or church.
level_loudspeaker_at_microphone	Direct-field level $L_{\mathrm{H}\text{-}\mathrm{M}}$ produced by the loudspeaker system at the microphone, dB.
level_loudspeaker_at_listener	Direct-field level $L_{\mathrm{H}\text{-}\mathrm{L}}$ produced by the loudspeaker system at an average listener, dB.
microphone_directivity	Directivity index $D_{\mathrm{M}}(\theta )$ of the microphone toward the loudspeaker relative to the talker, dB.
open_microphones	Number $N_{\mathrm{m}}$ of microphones open at once.
stability_margin	Required margin below oscillation, dB (default DEFAULT_STABILITY_MARGIN, Long’s equalised-system value).

Returns: A FeedbackStabilityResult.

FeedbackStabilityResult

FeedbackStabilityResult(
    open_loop_gain: float,
    feedback_loop_gain: float,
    nom_correction: float,
    loop_gain: float,
    stability_margin: float,
    margin: float,
    headroom: float,
    is_stable: bool,
    maximum_open_loop_gain: float,
    maximum_level_at_microphone: float,
    level_loudspeaker_at_microphone: float,
    level_loudspeaker_at_listener: float,
    microphone_directivity: float,
    open_microphones: int,
)

Gain structure and stability verdict of a reinforcement loop.

Attributes

Name	Description
open_loop_gain	Open-loop system gain $Z_{\mathrm{S}}$, dB (Equation (18.17)).
feedback_loop_gain	Feedback-loop gain $G_{\mathrm{S}}$, dB (Equation (18.18)).
nom_correction	Number-of-open-microphones correction $\Delta L_{\text{nom}}$, dB (Equation (18.23)).
loop_gain	Total loop gain $Z_{\mathrm{S}}+G_{\mathrm{S}}+\Delta L_{\text{nom}}$, dB. The system oscillates at 0 dB (Equation (18.16)).
stability_margin	Required margin below oscillation, dB.
margin	Margin actually available, -loop_gain, dB.
headroom	Gain that may still be added before the required margin is used up, -stability_margin - loop_gain, dB; negative when the criterion of Equation (18.24) is already violated.
is_stable	Whether the criterion of Equation (18.24) holds.
maximum_open_loop_gain	Largest $Z_{\mathrm{S}}$ the loop tolerates, dB.
maximum_level_at_microphone	Largest $L_{\mathrm{H}\text{-}\mathrm{M}}$ the loop tolerates, dB, for the given $Z_{\mathrm{S}}$ (Equations (18.20) to (18.22)).
level_loudspeaker_at_microphone	Input $L_{\mathrm{H}\text{-}\mathrm{M}}$, dB.
level_loudspeaker_at_listener	Input $L_{\mathrm{H}\text{-}\mathrm{L}}$, dB.
microphone_directivity	Input $D_{\mathrm{M}}(\theta )$, dB.
open_microphones	Input $N_{\mathrm{m}}$.

FeedbackStabilityResult.plot()

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

Plot the gain structure against the oscillation and margin lines.

Bars for $Z_{\mathrm{S}}$, $G_{\mathrm{S}}$ and $\Delta L_{\text{nom}}$ accumulate into the total loop gain, with the 0 dB oscillation threshold of Equation (18.16) and the required stability margin marked. Requires matplotlib (pip install phonometry[plot]).

open_microphone_correction

open_microphone_correction(open_microphones: int) -> float

Number-of-open-microphones correction (Long Equation (18.23)).

$\Delta L_{\text{nom}}=10{\log }_{10}{N}_{\mathrm{m}}$: doubling the number of simultaneously open microphones costs about 3 dB of gain before feedback.

Parameters

Name	Description
open_microphones	Number $N_{\mathrm{m}}$ of microphones open at once (>= 1).

Returns: The correction $\Delta L_{\text{nom}}$, dB.

plot_sound_reinforcement_geometry

plot_sound_reinforcement_geometry(
    talker_distance: float,
    microphone_distance: float,
    listener_distance: float,
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw the four points of a reinforcement feedback loop.

A schematic in the layout of Long’s Figure 18.15, with each path annotated with its own length: the talker T close in front of the microphone M, the flown loudspeaker H above and downstage of it, and the average listener L out in the audience. The signal path T -> M and H -> L is solid, the feedback path H -> M dashed. The two loudspeaker paths are the direct-field levels L_H-M and L_H-L that drive phonometry.electroacoustics.feedback_stability; the drawing is deliberately not to scale, because a talker 0.3 m from the microphone and a listener 20 m from the loudspeaker cannot share one usable scale.

Parameters

Name	Description
talker_distance	Talker-to-microphone distance, m.
microphone_distance	Loudspeaker-to-microphone distance, m.
listener_distance	Loudspeaker-to-listener distance, m.
ax	Existing axes, or None to create a figure.
language	Label language, "en" (default) or "es".
kwargs	Forwarded to the feedback-path Axes.plot.

Returns: The axes.

Raises

Exception	When
ValueError	If any distance is not positive and finite.