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Echo Detection in Spatial Room Impulse Responses Measured with Spherical Microphone Arrays Using the Herglotz Wavefunction

Pierre Massé, Anthony Gallien, Wolfgang Kreuzer, Markus Noisternig

arXiv:2609.28068v1eess.AS

TL;DR

Early reflections in SMA-measured SRIRs are impulsive and temporally limited, complicating their localization. The paper estimates a Herglotz-based directional function and fits Gaussian components to estimate reflection directions and count within a frame. In a denser measured sequence, it reports a 70% detection rate versus 31–62% for SRP, maxWDI, and MUSIC.

  • Problem

    Early reflections have very limited temporal support, posing a challenge for their localization in SMA-measured SRIRs.

    Source: A. Related Work · p. 1, §I · p. 1

  • Method

    The method estimates a Herglotz-based directional function from SMA measurements, then fits radial Gaussian components and uses an information criterion to estimate reflection directions and count.

    Source: B. Overview of the Proposed Approach · p. 2, §V · p. 10

  • Results

    Across isolated and denser conditions, the Herglotz localization function shows an advantage as echo density increases; in a denser measured sequence, its detection rate was 70% against 31–62% for SRP, maxWDI, and MUSIC.

    Source: 2) Denser Reflection Sequence · p. 10

  • Takeaways & Limitations

    The results suggest the formulation is well suited to analyzing early SRIR portions where multiple broadband impulsive arrivals coexist within one time frame.

    Source: §V · p. 10

    Detection performance is limited by the dynamic range between echoes, and estimated model complexity depends on the information criterion's balance between fitting accuracy and parsimony.

    Source: C. Echo Detection · p. 8

Abstract

from arXiv · show

Early reflections in spatial room impulse responses (SRIRs) measured using spherical microphone arrays (SMAs) play an important role in spatial audio analysis, rendering, and reverberation modeling. Their accurate localization and characterization can facilitate the analysis, processing, and manipulation of measured reverberation fields. This paper proposes a method for detecting and characterizing early reflections based on the Herglotz wavefunction formalism. The measured sound field is represented as a continuous superposition of incident plane waves, from which a localization function is derived in the spherical harmonic domain. An adaptive radial Gaussian fitting procedure is then used to estimate both the directions of arrival (DoAs) and the number of incident reflections within a single analysis frame. The proposed framework is evaluated using simulated and measured SRIRs and compared with conventional steered response power (SRP) and multiple signal classification (MUSIC) localization methods. The results demonstrate improved localization accuracy and more reliable detection of multiple simultaneous reflections, highlighting the potential of the proposed Herglotz-based formulation for the analysis of early reflections in SMA-measured SRIRs.

I. Introduction

The paper targets localization and characterization of early reflections in SMA-measured SRIRs, focusing on DoAs within a single analysis frame. It situates this task within a broader effort to analyze and manipulate measured reverberation fields.

I. Introduction

  • Mapping dominant early reflections could support reflection filtering, redistribution, editing, and higher-level objective or perceptual controls.The broader framework also aims to facilitate spatial reverberation reproduction.Source: §I · p. 1

A. Related Work

  • SIRR and SDM typically estimate at most one dominant DoA per frame, while HO-SIRR allows multiple DoAs but relies on relatively long-window pseudo-intensity estimation.SRP angular sharpening likewise depends on spherical-sector pseudo-intensity estimation.Source: A. Related Work · p. 2
  • MUSIC and ESPRIT localize multiple sources but require covariance estimates from sufficiently long windows and generally assume sources are uncorrelated or weakly correlated.These assumptions are challenging for early reflections, which are impulsive and highly correlated with the direct sound and one another.Source: A. Related Work · pp. 1–2
  • Frequency-smoothed MUSIC uses a full-length SRIR snapshot, but spherical-array response limits usable bandwidth and the formulation's detectable reflections are limited by maximum SH order.Time-smoothed and modal-smoothed MUSIC adaptations also do not provide a complete framework for estimating both reflection count and DoAs; modal smoothing requires an SLA, restricting it to MIMO settings.Source: A. Related Work · p. 2
  • A phase-aligned spatial-correlation method demonstrated blind multi-reflection localization, whereas this work formulates localization directly in the SH domain using a Herglotz representation.The cited method does not require prior knowledge of reflection structure.Source: A. Related Work · p. 2

B. Overview of the Proposed Approach

The proposed framework estimates a directional localization function from SMA measurements, then fits spherical Gaussian components and adaptively selects their number to localize multiple reflections in one frame. Its measurement model and analysis are bounded by early/late temporal segmentation and the finite SH representation of the array.

B. Overview of the Proposed Approach

  • The method transforms SMA measurements into a Herglotz localization function, fits spherical Gaussian components to its peaks, and uses an adaptive information criterion to estimate reflection count and directions.The number of echoes is not assumed known in advance, and the target is multiple reflections within a single frame.Source: B. Overview of the Proposed Approach · p. 2

A. SRIR Time Segmentation

  • The assumed early SRIR region contains predominantly direct sound and discrete reflections before a stochastic, exponentially decaying late tail, with their transition called the mixing time.A complete analysis over t ∈ [t0, tmix] also requires frame selection, false-detection rejection, and associating repeat detections, which are beyond this work's scope.Source: A. SRIR Time Segmentation · p. 3

B. Plane-Wave Field Measurement with an SMA

  • The SMA measurement model represents the total field outside a rigid sphere as incident and scattered contributions, with the rigid-sphere mode strength relating to the SH representation of a plane wave.The mode strength is also called the holographic function.Source: 1) SH representation of a single plane… · p. 3

1) SH representation of a single plane wave:

  • SMA transducers sample the field on the sphere, and practical SH representations truncate at order L, which depends on quadrature accuracy and spatial sampling density.The EM32 supports accurate representations up to L = 4; the EM64 is practically limited to approximately L ≈ 6 because of poorer encoding-matrix conditioning. Encoding filters compensate mode strengths to make the effective modal response as flat and geometry-independent as possible over the usable band.Source: 1) SH representation of a single plane… · p. 3, 2) Compensation of sphere radius · p. 3

2) Compensation of sphere radius:

The method corrects the sphere-dependent response of HOA coefficients and uses the resulting Herglotz representation to infer incident directions from SMA measurements. Its directional analysis is tied to frequency-dependent array response and a discretized estimate of the Herglotz kernel.

2) Compensation of sphere radius:

  • Encoding filters compensate the mode strength, but the effective modal response is not perfectly flat and remains frequency-dependent.For the EM32 example, the maximally directive band is bounded above by the spatial aliasing frequency and below by the frequency where directivity remains at least as high as at that aliasing limit.Source: 2) Compensation of sphere radius · p. 3

A. Herglotz Wavefunction

  • The Herglotz representation expresses the sound field as a continuous superposition of plane waves, with the direction-dependent kernel weighting each component.The kernel is assumed square-integrable over the unit sphere.Source: A. Herglotz Wavefunction · p. 4
  • Estimating the kernel from measurements on the SMA surface forms an inverse problem, with dominant directions identified from its principal maxima.The sampled kernel provides a directional interpolation of the incident field, where a single echo appears as a localized contribution.Source: A. Herglotz Wavefunction · p. 4
  • The kernel integral is discretized over the unit sphere for numerical quadrature, and its sampled values are related to modified SH coefficients through a matrix representation.The quadrature nodes and weights determine the spherical discretization; this differs from beamforming's fixed grid of scanning directions.Source: A. Herglotz Wavefunction · pp. 4–5

B. Echo Localization Function

The method models overlapping early echoes through Herglotz kernels and estimates a localization function from spherical-harmonic coefficients. That function contains directional information for identifying echo DoAs, while its spatial resolution is limited by the maximum spherical-harmonic order.

B. Echo Localization Function

  • Each early analysis frame is modeled as overlapping impulsive echoes, each represented by a Herglotz wavefunction with a frequency-invariant kernel and frequency-dependent amplitude.The method therefore does not treat all echoes in a frame as independent plane-wave components.Source: B. Echo Localization Function · p. 5
  • The localization function m(f) = Ga(f) combines the directional contributions of all echoes and contains the information needed to identify their DoAs.The number of echoes is encoded implicitly and need not be known a priori.Source: B. Echo Localization Function · p. 5
  • Tikhonov regularization stabilizes estimation of the localization function, but its spatial smoothing leaves achievable resolution fundamentally limited by the maximum SH order.A finer sampling scheme can represent the solution but does not remove that resolution limit.Source: B. Echo Localization Function · p. 5
  • Estimation is restricted to the maximally directive frequency band, from fminMaxDI to falias, where DI(fk) ≥ DI(falias).Directional information becomes unreliable above the spatial aliasing frequency.Source: B. Echo Localization Function · p. 5

C. Echo Detection

Echo detection combines directional information across frequency and fits spherical Gaussian peaks to the resulting localization function. The fitted peak positions estimate DoAs, while peak amplitudes provide relative echo strengths.

C. Echo Detection

  • The detection procedure combines frequency information in the magnitude of the complex localization function, independently of any particular frequency bin.The frequency averaging is performed over the maximal directive frequency range.Source: C. Echo Detection · pp. 5–6
  • The localization function is modeled as a sum of spherical Gaussian radial basis functions, with peak positions giving echo DoAs and amplitudes measuring relative strengths.Each peak’s amplitude coefficient represents relative presence, and its angular concentration parameter controls lobe sharpness.Source: C. Echo Detection · p. 6
  • A weighted least-squares objective gives greater influence to errors at strong localization peaks than to errors in low-energy regions.Gradient descent uses an adaptive step size estimated with the Barzilai–Borwein method.Source: C. Echo Detection · p. 6

D. Estimation of the Number of Incident Echoes

Because the number of echoes is unknown in blind analysis, the method estimates the number of Gaussian components iteratively using an information criterion inspired by AIC. It stops adding components when the criterion rises, selecting the preceding model.

D. Estimation of the Number of Incident Echoes

  • The proposed AIC-inspired information criterion balances fitting error against model complexity to estimate the unknown number of echo components.Extra components may fit spatial fluctuations or noise rather than physically meaningful reflections.Source: D. Estimation of the Number of… · p. 6
  • The criterion was selected empirically using preliminary simulations; systematic comparison with alternatives such as BIC is beyond this work’s scope.Its influence on detection performance is examined briefly in Section IV-C.Source: D. Estimation of the Number of… · p. 6
  • Starting from one Gaussian, the procedure adds components and re-estimates the model until the criterion increases, then selects the preceding model.The criterion initially decreases as fit improves, then rises when the fit improvement no longer justifies added complexity.Source: D. Estimation of the Number of… · p. 6

IV. Evaluation

The evaluation uses simulated and measured data to compare localization maps and assess the full echo-detection framework. The same peak-extraction procedure is applied across localization methods so comparisons focus primarily on the underlying localization functions.

IV. Evaluation

  • The evaluation compares the proposed Herglotz localization function with conventional SRP and MUSIC localization maps using simulated and measured data.The comparison uses a shared peak-detection procedure across the localization maps.Source: §IV · p. 7
  • The complete framework is evaluated for estimating directions, relative amplitudes, and the number of incident echoes.These are the stated evaluation targets for the full detection framework.Source: §IV · p. 7
  • Every localization map uses the same radial-Gaussian fitting, gradient-based parameter estimation, and AIC-based peak-count estimation.The comparisons therefore focus primarily on localization-function quality rather than differences in peak extraction.Source: §IV · p. 7

A. Simulation Setup

The simulations model short early-SRIR frames containing prescribed impulsive reflections, then encode the resulting Eigenmike measurements as spherical-harmonic coefficients.

A. Simulation Setup

  • Each 256-sample frame represents 5.33 ms at 48 kHz and contains a prescribed number of impulsive reflections.The frames represent the early SRIR portion, from t0 to tmix.Source: A. Simulation Setup · p. 7
  • Echoes are modeled as discrete impulses bandpass-filtered from 125 Hz to 16 kHz to avoid temporal aliasing and preserve broadband reflection characteristics.The lower cutoff addresses long low-frequency responses in the finite analysis window; the bandwidth exceeds the EM32's maximally directive band.Source: A. Simulation Setup · p. 7
  • Eigenmike measurements are simulated and encoded into HOA coefficients, with Lsim = 20 and spatial aliasing arising from the array's 32 capsules and L = 4 truncation.The simulations use the EM32 unless otherwise stated; EM64 results are also presented later.Source: A. Simulation Setup · p. 7

B. Localization Function Estimation

The section compares Herglotz-based and conventional localization functions, then evaluates radial-Gaussian fitting for detecting and characterizing simulated echoes. The Herglotz function gives the strongest overall accuracy in the four-echo example, with detection subject to model-selection and dynamic-range limits.

B. Localization Function Estimation

  • The comparison plots Herglotz, PWD-SRP, maxWDI-SRP, and MUSIC localization functions; MUSIC's signal-subspace analysis estimated two sources.The plotted functions are estimated for the same test signals.Source: B. Localization Function Estimation · p. 7

C. Echo Detection

  • Radial-Gaussian fitting and model selection recover four echoes in the example, with accuracy assessed by angular and relative-presence errors.Presence errors compare estimated factors with reflection levels normalized to the strongest echo.Source: C. Echo Detection · p. 7
  • Detection depends on the information criterion's fit–parsimony balance, while larger level gaps can make weaker echoes indistinguishable from spatial noise or aliasing.Among the compared maps, Herglotz most closely represents the four simulated directions.Source: C. Echo Detection · p. 8
  • Herglotz yields the lowest angular errors for three of four echoes and the smallest relative-presence errors throughout; maxWDI wins angular error only on the −6 dB reflection.The comparison concerns a simulated frame with four discrete echoes.Source: C. Echo Detection · p. 8, Table I · p. 9

D. Validation with a Higher-Order SMA

The higher-order SMA and large-scale simulations compare Herglotz localization with SRP, maxWDI, and MUSIC, while measured SRIRs are evaluated in complementary frame settings.

D. Validation with a Higher-Order SMA

  • With six unequal-level echoes on EM64, Herglotz produces sharper, better-separated peaks and achieves an 83% detection rate, versus 50% for MUSIC.It has the lowest angular error for four of five detected reflections and the smallest relative presence errors overall; maxWDI's detection rate exceeds 100%.Source: D. Validation with a Higher-Order SMA · p. 9

E. Large-Scale Statistical Simulation

  • Across 250 configurations per reflection count, Herglotz detects more echoes than SRP and MUSIC, particularly as counts rise, and preserves twelve distinct peaks in an equal-level example.The simulations span one to twelve reflections, with a minimum 35° separation between echoes.Source: E. Large-Scale Statistical Simulation · p. 9, Figure 5 · p. 10

F. Measurement Validation

1) Direct Sound and First Reflection:

The methods perform comparably for isolated early arrivals, while Herglotz has a consistent advantage in denser reflection conditions. The work estimates reflections frame by frame, leaving full spatio-temporal cartography for future work.

1) Direct Sound and First Reflection:

  • For the direct sound and first reflection, all four methods detect and localize both arrivals; Herglotz has the lowest presence error for each.MUSIC has the lowest angular error for the direct sound, while Herglotz has the lowest for the first reflection.Source: 1) Direct Sound and First Reflection · p. 10

2) Denser Reflection Sequence:

  • In the 13-reflection measured sequence, Herglotz achieves a 70% detection rate, compared with 31–62% for SRP, maxWDI, and MUSIC.The comparison is for a denser frame excluding the direct sound and first two reflections.Source: 2) Denser Reflection Sequence · p. 10, Table IV · p. 11

V. Conclusion

  • Across simulations and measurements, Herglotz improves direction estimates, relative presence errors, and echo-count estimation, particularly with several closely spaced reflections in a frame.The authors conclude that it is well suited to early SRIRs where multiple broadband impulsive arrivals coexist.Source: §V · p. 10, 2) Denser Reflection Sequence · p. 10
  • The present work localizes reflections within individual frames; frame-to-frame tracking and complete early-reflection cartographies remain future work.Further studies are also proposed for the information criterion, reflection dynamic range, and maximum supported spherical-harmonic order.Source: §V · p. 11

Appendix A Wide Comparison Plots

The appendix plots show localization functions for simulated and measured SRIR frames, including both direct and reflected sound.

Appendix A Wide Comparison Plots

  • A simulated six-echo frame displays the localization function, with white stars marking true DoAs and echo levels ranging from 0 to −16.5 dB relative to the first.The echoes are placed at the listed directions.Source: Figure A.1 · p. 13
  • A simulated frame of twelve randomly placed, equal-level echoes shows the localization function, with a minimum 35° separation between echoes.Source: Figure A.2 · p. 13
  • A reconstructed localization function for a measured EM64 frame covers 13 later reflections predicted by second-order ISM, excluding the direct sound and first two reflections.White stars mark the ISM-predicted DoAs.Source: Figure A.3 · p. 14
  • A measured EM64 frame includes both direct sound and the first reflection over a 10 ms interval.The displayed localization function is calculated following Eq. (11).Source: Figure A.4 · p. 15
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