PaperlayerView exact revision on arXiv

Source-linked AI summary

Broadband Acoustic Intensity Direction Estimation with Tight-Frame Cardioid Arrays

Akira Omoto

arXiv:2609.03490v1eess.AS

TL;DR

This study experimentally tests whether tight-frame cardioid arrays can estimate source direction from acoustic intensity, addressing the lack of full-process validation in earlier numerical work. It evaluates measured-impulse-response arrays under single-source and coherent-interference conditions, finding broadband estimates from 50 Hz to 20 kHz and particularly low tracking errors for TF24 at the shorter spacing.

  • Problem

    Earlier numerical evaluation found accurate single-wave estimates but did not experimentally validate reproduction, propagation, spatial sampling, and impulse-response acquisition.

    Source: §1 · p. 2

  • Method

    The study synthesizes TF6 and TF24 arrays from anechoic-chamber impulse responses and evaluates source direction under single-source and coherent-arrival conditions.

    Source: §1 · p. 2, §2.1 · p. 2

  • Results

    TF24 with d = 20 mm kept median tracking error within approximately 2.5° across the frequency range at LJ/S = −20 and −30 dB, while direction estimation extended from 50 Hz to 20 kHz with reasonable accuracy.

    Source: §4 · p. 6

  • Takeaways & Limitations

    The combination of the C-C method and a tight-frame arrangement provided stable broadband direction estimation, with TF24 and shorter spacing generally yielding smaller errors.

    Source: §3.3 · p. 5, §4 · p. 6

    The study synthesized array data from sequential single-microphone measurements, leaving microphone variability and synchronization errors in simultaneous multimicrophone implementations for future evaluation.

    Source: §4 · p. 6

Abstract

from arXiv · show

This study experimentally investigates source-direction estimation from acoustic intensity using paired cardioid microphones, with particular emphasis on well-balanced arrangements known as tight frames. Impulse responses were measured at 6 or 24 microphone positions, and two opposed-pair spacings were examined. The resulting arrays were evaluated for a single source, coherent interference between waves arriving from two orthogonal directions, and directional tracking in the presence of interfering waves from multiple directions. The results demonstrate broadband acoustic-intensity direction estimation from 50 Hz to 20 kHz, with the 24-microphone arrangement and the shorter spacing generally providing smaller errors.

1. Introduction

The study addresses the gap between numerical evidence for tight-frame cardioid-cardioid arrays and experimental validation of the complete measurement process. It tests whether these arrays can estimate source direction from acoustic intensity under measured impulse-response conditions.

1. Introduction

  • Tight frames are highly symmetric arrays proposed for measuring acoustic intensity and diffuseness (Omoto, 2026b); the cardioid-cardioid method may support higher usable frequencies than pressure-pressure probes.The pressure-pressure probes’ upper usable frequency is constrained by finite-difference approximation and microphone spacing.Source: §1 · p. 2
  • A previous numerical study found accurate single-wave direction estimates up to high frequencies but did not validate reproduction, propagation, spatial sampling, and impulse-response acquisition (Omoto, 2026a).Source: §1 · p. 2
  • The first measured-impulse-response validation compares TF24 and TF6 at two opposed-pair spacings and assesses tracking with multiple coherent arrivals.The study uses 26-channel impulse responses measured in an anechoic chamber and constructs controlled multiple-arrival conditions by channel permutation and coherent superposition.Source: §1 · p. 2

2. Experimental examination

The experiments use synthesized TF6 and TF24 arrays, process measured impulse responses into bandwise intensity estimates, and evaluate two-wave and multiple-arrival conditions. The tracking benchmark compares estimated azimuth with the ideal plane-wave resultant for the same sources, amplitudes, and phases.

2.1 Microphone arrays

  • TF6 uses three orthogonal opposed pairs, while TF24 combines 12 directional intensities; both are synthesized from sequentially measured impulse responses at two spacings.A single cardioid microphone was rotated to measure the responses; the arrays were tested at d = 20 and 100 mm in an anechoic chamber.Source: §2.1 · p. 2, Figure 1 · p. 3

2.2 Calculation of time-integrated active acoustic intensity

  • The processing windows a common direct-sound interval, filters responses into one-third-octave bands from 50 Hz to 20 kHz, and integrates bandwise intensity before tight-frame reconstruction.The complete filtered response, including filter tails, is used in the time-domain C-C calculation; directional intensities are combined by weighting them with their unit vectors.Source: §2.2 · pp. 2–3

2.3 Coherent addition of two waves

  • The two-wave test coherently combines frontal and 90° responses with zero relative phase and varies their amplitudes to test estimated arrival direction.The resulting mixed response is used to calculate opposed-pair intensity, reconstruct the three-dimensional intensity components, and estimate azimuth; a frequency-dependent frontal offset is subtracted in later analyses.Source: §2.3 · pp. 3–4

2.4 Monte Carlo evaluation of directional tracking

  • The Monte Carlo benchmark varies jammer count from one to seven, holds total jammer energy constant, and compares estimated azimuth with the ideal resultant for matching source conditions.It uses 1,000 trials per condition, with randomized jammer phases; tracking error is defined against the independently calculated ideal plane-wave resultant intensity.Source: §2.4 · p. 4

3. Results and discussion

Across the tests, direction estimates followed single-source and ideal coherent-intensity directions, with errors varying by frequency, array configuration, spacing, and interference conditions. In multi-arrival tracking, TF24 and shorter spacing generally had smaller errors.

3.1 Direction estimation for a single source

  • Azimuth error was about −2° at 50 Hz and −1° above 200 Hz across arrays and spacings, while elevation varied with frequency and array configuration.At higher frequencies, elevation estimates converged to about 0° for TF6 and −2° to −3° for TF24; the systematic offset's cause remains unknown.Source: §3.1 · p. 4

3.2 Coherent addition of two orthogonal waves

  • For orthogonal arrivals, estimates closely followed theoretical values through 500 Hz; from 2 kHz, d = 100 mm deviated, most for TF6, and TF24 deviations reached about 6°.The test varied the relative amplitudes of two waves arriving from directions separated by 90°.Source: §3.2 · p. 5

3.3 Coherent addition of multiple arrivals

  • Across multi-arrival tests, shorter spacing and TF24 reduced error; at LJ/S = −20 and −30 dB, TF24 with d = 20 mm kept median tracking error within 2.5°.At LJ/S = −10 dB, d = 100 mm produced 10°–15° errors; TF24 with d = 20 mm stayed within 5° to about 5 kHz, reaching about 7.5° at higher frequencies.Source: §3.3 · p. 5
  • For d = 100 mm, error peaked near 3 and 4 kHz rather than rising monotonically, and its physical origin remains unresolved.The corresponding wavelengths were approximately 114 and 86 mm, suggesting a relationship with microphone spacing.Source: §3.3 · p. 6

4. Concluding Remarks

Both arrays estimated a single frontal source over a broad frequency range using measured impulse responses. TF24 at 20 mm had the strongest reported tracking result, while variability and synchronization in simultaneous-microphone measurements remain future work.

4. Concluding Remarks

  • Both arrays estimated the direction of a single frontal source over a broad frequency range using measured impulse responses.Source: §4 · p. 6
  • Under two-wave coherent interference, agreement with theory was good at low frequencies, but errors increased for d = 100 mm at higher frequencies.Source: §4 · p. 6
  • Tracking error showed a spacing- and wavelength-dependent peak followed by improved performance at higher frequencies.Source: §4 · p. 6
  • The arrays provided direction estimation from 50 Hz to 20 kHz with reasonable accuracy despite errors of several degrees under some conditions.At high frequencies, microphone spacing covered multiple wavelengths.Source: §4 · p. 6
  • Array data were synthesized from sequential single-microphone measurements; microphone variability and synchronization effects in simultaneous measurements remain for future work.Source: §4 · p. 6
  • Median tracking error for TF24 at d = 20 mm stayed within approximately 2.5° across the frequency range at LJ/S = −20 and −30 dB.Figure 4 compares median tracking error across array configurations and spacings.Source: §4 · p. 6, Figure 4 · p. 7
Loading 2609.03490v1…