Operator-qualified active acoustic inference

This note formalizes Fixture F-009 from the acoustics, hearing, and auditory-scene analysis audit. It supplies a hostile comparison boundary for Candidate 002, Candidate 006, Candidate 007, Candidate 009, Candidate 012, and Candidate 014.

Episode, operator, and action identity

For receiver $r$, source $s$, microphone or ear channel $m$, and episode $e$, preserve

where:

• $X_e$ is geometry, medium, boundaries, temperature in kelvins, relative humidity as a dimensionless fraction, flow in metres per second, occupancy, and time-varying room/material state;
• $S_e$ is every target and interfering source identity, waveform, position in metres, velocity in metres per second, directivity, emission time in seconds, and source-level record;
• $E_e$ is commanded and realized active emission: waveform, spectrum, duration in seconds, acoustic energy in joules, aim in degrees, repetition, and dose;
• $R_e$ is receiver/body/array position and orientation, aperture in metres, morphology, feasible motion, transducer response, gain, health, and saturation;
• $H_e$ is prior acoustic exposure, room/source history, adaptation, training, previous emissions and motions, and feedback with timestamps;
• $O_e$ is the observation operator: impulse responses, transfer functions, spatial/time support, sample rate in samples per second, quantization in bits, clock, latency in seconds, preprocessing, missingness, and selection;
• $C_e$ is calibration identity, uncertainty, reference pressure and distance, instrument class, traceability, and data vintage;
• $T_e$ is the target construct, literal outcome, deadline in seconds, and loss or utility in declared units;
• $U_e$ is the independent unit: waveform, frame, event, source, room, receiver, body, array, device, site, or population; and
• $B_e$ is the complete ceiling in samples, events, bytes, seconds, person-hours, joules, emissions, exposure, unsafe events, sensors, actuators, replacements, and opportunity.

For method $q$ and literal outcome $k$, the estimand is

where $Y_k$ uses the registered unit for outcome $k$. A contrast against baseline $b$ does not isolate $q$ if aperture, calibration, room response, source distribution, observation receipt time, emission or motion authority, training scenes, or any binding resource differs without intervention.

Pressure, level, exposure, and clock contract

For acoustic pressure $p(t)$ in pascals over interval $T$ in seconds,

where $p_{\mathrm{rms}}$ is RMS pressure in pascals, $p_0=20\,\mu\mathrm{Pa}$ is the reference pressure in air, and $L_p$ is in decibels relative to $20\,\mu\mathrm{Pa}$. Frequency weighting, time weighting, band, location, orientation, and calibration are fields, not implied defaults.

Sound exposure and its level are

where $E_p$ is in pascal-squared seconds, $L_E$ is in decibels relative to $E_0$, and reference duration $t_0=1\,\mathrm{s}$. Exposure is not acoustic emission energy, electrical energy, or a universal risk model.

For channel clock $c$, correct a device timestamp by

where device time $t^{\mathrm{dev}}_{c,n}$, corrected time $\bar t_{c,n}$, and version-$v$ offset $\delta_{c,v}$ are in seconds; $c$ is the channel-clock index, $n$ is the dimensionless sample index, and $v$ is the dimensionless calibration version. Retain residual clock uncertainty $\sigma_{c,v}$ in seconds, drift in seconds per second, capture time, receipt time, synchronization method, and calibration interval. At decision time $t$, only observations with receipt time no later than $t$ are available.

Propagation, rooms, and time-varying mixing

At channel $m$, use the time-varying mixture

where received waveform $y_m(t)$, source pressure $x_s(t)$, and noise $n_m(t)$ are in pascals; $S$ is dimensionless source count; $t$ and delay $\tau$ are in seconds; and transfer kernel $h_{m,s}(t,\tau)$ is in reciprocal seconds. A time-invariant room convolution is a special case. Source/receiver motion, changing boundaries, temperature, flow, occupancy, and adaptive emitters make the operator time-dependent.

For an ideal free-field point source with unchanged directivity and negligible absorption,

where distances $r_1,r_2$ are in metres and $\Delta L_p$ is in decibels. The comparison must expose directionality, near-field terms, boundaries, atmospheric absorption, scattering, and receiver orientation when present.

For a diffuse-field Sabine approximation,

where $T_{60}$ is decay time in seconds, room volume $V$ is in cubic metres, equivalent absorption area $A$ is in square metres, and $0.161$ has units seconds per metre. Report measured impulse responses and uncertainty by band, source, receiver, and support; this approximation is not an operator identity.

Masking, filterbank, compression, and events

A real gammatone-like channel is

where centre frequency $f_j$ and bandwidth $b_j$ are in hertz, filter order $n_j$ is dimensionless, phase $\phi_j$ is in radians, time $t$ is in seconds, and $a_j$ has units $\mathrm{s}^{-n_j}$ so $g_j$ is in reciprocal seconds. Fixed FFT, wavelet, mel, ERB, gammatone/gammachirp, modulation, and learned filterbanks remain competing representations.

A local normalized compression fit is

where magnitudes $x,x_0$ share one input unit, $z,z_0$ share one output unit, and exponent $\gamma$ is dimensionless. Register frequency, level, state, attack/release time in seconds, saturation, and distortion; one exponent does not describe an adaptive cochlea or compressor globally.

For yes/no target detection,

where both $P$ terms are dimensionless probabilities, $\Phi^{-1}$ is the inverse standard-normal cumulative distribution, and sensitivity $d'$ is dimensionless. Keep decision criterion, energetic overlap, informational uncertainty, spatial release, source grouping, and task history separate.

For event encoder threshold $\theta_j$ in the unit of channel response $u_j(t)$,

where $j$ is channel identity, event time $t_{j,n}$ is in seconds, $n$ is the dimensionless event index, $e_{j,n}$ is the retained event record, $t_{j,n-1}$ is the previous retained-event time in seconds, and $v_{\theta},v_c$ are dimensionless threshold and clock versions. Event rate is in events per second. Report missed sustained signals, false events, timestamp error, decoder cost, bytes, task information, and measured joules; event sparsity is not an energy unit.

ITD, ILD, aperture, correlation, and beamforming

For sound speed $c_{\mathrm{snd}}$ in metres per second and path-length difference $\Delta r$ in metres,

where interaural or interchannel time difference $\Delta t$ is in seconds. Clock offset, phase ambiguity, multipath, source extent, and head/array transfer functions are part of its uncertainty.

For left/right RMS pressures $p_L,p_R$ in pascals,

where interaural level difference is in decibels and is band-, time-, source-, and orientation-qualified. ITD, ILD, spectral cues, and head motion remain separate intervention axes.

For far-field array aperture $D$ in metres, frequency $f$ in hertz, and wavelength $\lambda=c_{\mathrm{snd}}/f$ in metres,

where angular resolution $\Delta\theta$ is in radians. Geometry, beampattern, SNR, estimator, bandwidth, and resolution definition determine the coefficient; no learned estimator restores spatial frequencies excluded by the physical support without additional prior information.

For signals $y_1(t),y_2(t)$ in pascals, define generalized cross-correlation

where $Y_1,Y_2$ are Fourier transforms in pascal-seconds, frequency $f$ is in hertz, delay $\tau$ and estimate $\widehat\tau$ are in seconds, $\mathcal T$ is the feasible delay set in seconds, superscript $*$ is complex conjugation, and weighting $\Psi(f)$ has reciprocal pascal-squared-second-squared units so $R_{12}^{(\Psi)}$ is in hertz after integration. The peak must carry a calibrated multimodal delay distribution under reverberation, not only an argmax.

For array snapshot $\mathbf y\in\mathbb C^M$ in pascals, dimensionless steering vector $\mathbf a$, and noise covariance $\mathbf R_n=\mathbb E[\mathbf n\mathbf n^H]$ in pascal-squared units,

where $M$ is microphone count, superscript $H$ is conjugate transpose, $\mathbf w_{\mathrm{MVDR}}$ is dimensionless, and output $z$ is in pascals. Steering mismatch, covariance error, short sample support, correlated sources, motion, and room change receive sealed interventions.

Reverberation, separation, and source identity

For direct component $d_m(t)$, early-reflection component $r_m^{\mathrm{early}}(t)$, late component $r_m^{\mathrm{late}}(t)$, and noise $n_m(t)$, all in pascals,

The early/late boundary is a registered time in seconds relative to the direct arrival. Waveform dereverberation, speech compensation, perceived distance, localization, and downstream intelligibility are distinct outcomes.

For estimated source $\widehat{\mathbf s}$ and reference source $\mathbf s\in\mathbb R^N$ with the same amplitude unit and sample count $N$,

where $\langle\cdot,\cdot\rangle$ is the waveform inner product and SI-SDR is in decibels. Report unknown source count, permutation, causal identity, localization, intelligibility, calibration, perceptual/task quality, and residual mixture separately.

For predicted scene outcome $z_n$ and causally available acoustic observation $o_n$, proper log loss is

where $N_e$ is independent event count and $L_q$ is in bits per event. Use it for detection, source count, assignment, localization bins, range bins, and abstention under held-out operators; do not pool incompatible outcomes.

Active emission, dose, and closed-loop action

For monostatic emission at $t_{\mathrm{emit}}$ and echo receipt at $t_{\mathrm{recv}}$, both in seconds,

where estimated range $\widehat r$ is in metres and sound speed $c_{\mathrm{snd}}$ is in metres per second. Clock uncertainty, target motion, refraction, multipath, transducer ringing, waveform ambiguity, association, and detector threshold must propagate to range uncertainty.

For acoustic output power $P_{\mathrm{ac}}(t)$ in watts over emission duration $T_e$ in seconds,

where $E_{\mathrm{emit}}$ is acoustic joules. Electrical input, transduction loss, body/sensor motion, sensing, compute, cooling, detectability, interference, and exposure remain separate axes.

At decision time $t$, a complete acoustic policy is

where $a_t^{\mathrm{emit}}$ contains waveform, level, spectrum, aim, and timing; $a_t^{\mathrm{body}}$ contains pose or head/body motion in metres, radians, and seconds; $a_t^{\mathrm{sensor}}$ contains aperture, sample rate, and channel selection; $a_t^{\mathrm{gain}}$ is dimensionless or in declared decibels; $a_t^{\mathrm{task}}$ is the downstream command in its native unit; $\mathcal H_t$ is causally received history; $\widehat X_t$ is scene state; $\widehat O_t$ is operator/calibration state; $\widehat U_t$ is uncertainty; $\mathcal A_t^{\mathrm{safe}}$ is the feasible action envelope; and $B_t$ is remaining budget in matched units. Attention- or efferent-like gain receives causal credit only when intervening on it changes its registered endpoint.

Hardware, lifecycle energy, and equal budgets

Lifecycle energy is

where every term is in joules over one service interval and denotes data acquisition, training, acoustic emission, sensing, inference, physical motion, communication, storage, calibration, maintenance, and amortized embodied energy. DSP, FPGA, ASIC, neuromorphic, CPU, GPU, and analog front ends are measured at identical physical boundaries and workload deadlines.

Human effort is

where every $H$ term is in person-hours and roles are reported separately.

The complete cost vector is

where the five $N$ terms count samples, emitted/received events, optimization or environment steps, queries, and bytes; $T_{\mathrm{wall}}$ is seconds; $H_q^{\mathrm{human}}$ is person-hours; $E_q^{\mathrm{life}}$ is joules; $E_p$ is pascal-squared-second exposure; $N_{\mathrm{unsafe}}$ counts unsafe events; $D_{\mathrm{harm}}$ uses a registered physical or severity unit; and $C_{\mathrm{opp}}$ is opportunity cost in task exposures or person-hours.

Method $q$ is feasible only when

where $\mathbf B$ is the preregistered componentwise ceiling with identical units. Over-budget runs are infeasible; removed ablation resources stay unused.

Confirmatory contrast and retirement

Let $b^*(j)$ be the strongest mature baseline for track $j$, frozen on development data. Orient every protected endpoint as a preregistered benefit, so larger $Q$ is better and a non-inferiority margin may be negative. For protected outcomes $k\in\mathcal P_j$, retain a residual only when

where $\delta_{j,k}$ is an improvement or non-inferiority margin in the unit of outcome $k$, and $\alpha_j$ is the dimensionless error budget. The result must survive held-out rooms, sources, arrays, bodies, operators, scenes, source counts, interference policies, model families, sites, and hardware classes at equal complete cost. Otherwise retire the architectural residual while keeping the operator/action/measurement contract.