^{1}, Jean-Hugh Thomas

^{1,a)}and Jean-Claude Pascal

^{1}

### Abstract

Near-field acoustic holography is a measuring process for locating and characterizing stationary sound sources from measurements made by a microphone array in the near-field of the acoustic source plane. A technique called real-time near-field acoustic holography (RT-NAH) has been introduced to extend this method in the case of nonstationary sources. This technique is based on a formulation which describes the propagation of time-dependent sound pressure signals on a forward plane using a convolution product with an impulse response in the time-wavenumber domain. Thus the backward propagation of the pressure field is obtained by deconvolution. Taking the evanescent waves into account in RT-NAH improves the spatial resolution of the solution but makes the deconvolution problem “ill-posed” and often yields inappropriate solutions. The purpose of this paper is to focus on solving this deconvolution problem. Two deconvolution methods are compared: one uses a singular value decomposition and a standard Tikhonov regularization and the other one is based on optimum Wiener filtering. A simulation involving monopoles driven by nonstationary signals demonstrates, by means of objective indicators, the accuracy of the time-dependent reconstructed sound field. The results highlight the advantage of using regularization and particularly in the presence of measurementnoise.

I. INTRODUCTION

II. REAL-TIME NEARFIELD ACOUSTIC HOLOGRAPHY

A. Forward propagation

B. Backward propagation

III. SOLVING THE BACKWARD PROPAGATION

A. The impulse response

B. Regularization method

C. Inverse filtering method

IV. NUMERICAL SIMULATIONS

A. Setup

B. Indicators for comparison

5. Results

A. Time-space comparisons

B. Spatial comparisons

C. Noise influence

VI. CONCLUSION

### Key Topics

- Sound pressure
- 19.0
- Acoustic holography
- 15.0
- Acoustic noise
- 10.0
- Deconvolution
- 9.0
- Fourier transforms
- 9.0

## Figures

Geometry of interest. Forward and backward propagation in real-time near-field acoustic holography. The distances between the measurement plane and the calculation plane in both configurations are the same.

Geometry of interest. Forward and backward propagation in real-time near-field acoustic holography. The distances between the measurement plane and the calculation plane in both configurations are the same.

Impulse response .

Impulse response .

Modulus and phase of the theoretical Fourier transform of the inverse impulse response with , where *f _{s} * is the sampling frequency.

Modulus and phase of the theoretical Fourier transform of the inverse impulse response with , where *f _{s} * is the sampling frequency.

(Color online) Reconstructed time signals obtained by inverse filtering using singular value decomposition coupled with regularization (a)–(c), or Wiener approach (d)–(f), vs reference signals (dotted line) on locations *R* _{2} [(a), (d)], *R* _{3} [(b), (e)], and *R* _{4} [(c), (f)] (see Fig. 1). The impulse response was processed using Chebyshev low-pass filtering.

(Color online) Reconstructed time signals obtained by inverse filtering using singular value decomposition coupled with regularization (a)–(c), or Wiener approach (d)–(f), vs reference signals (dotted line) on locations *R* _{2} [(a), (d)], *R* _{3} [(b), (e)], and *R* _{4} [(c), (f)] (see Fig. 1). The impulse response was processed using Chebyshev low-pass filtering.

(Color online) Spatial maps for indicator *T* _{1} in the case of regularization (a) and inverse filtering (b) with a contour line at the value 0.95. The locations of *R* _{1}(+), *R* _{2}(+), *R* _{3}(+), and *R* _{4}(*) are marked.

(Color online) Spatial maps for indicator *T* _{1} in the case of regularization (a) and inverse filtering (b) with a contour line at the value 0.95. The locations of *R* _{1}(+), *R* _{2}(+), *R* _{3}(+), and *R* _{4}(*) are marked.

Spatial maps for indicator *T* _{2} in the case of regularization (a) and inverse filtering (b). The areas in gray correspond to values of *T* _{2} below 0.05. The locations of *R* _{1}(+), *R* _{2}(+), *R* _{3}(+), and *R* _{4}(*) are marked.

Spatial maps for indicator *T* _{2} in the case of regularization (a) and inverse filtering (b). The areas in gray correspond to values of *T* _{2} below 0.05. The locations of *R* _{1}(+), *R* _{2}(+), *R* _{3}(+), and *R* _{4}(*) are marked.

(Color online) Time-dependent spatial errors *E _{x} _{,} _{y} * [see Eq. (34)] and[see Eq. (35)] in the case of regularization (a), (c), and inverse filtering (b), (d) for three processing methods applied to the impulse response (direct sampling, Kaiser and Chebyshev filtering).

(Color online) Time-dependent spatial errors *E _{x} _{,} _{y} * [see Eq. (34)] and[see Eq. (35)] in the case of regularization (a), (c), and inverse filtering (b), (d) for three processing methods applied to the impulse response (direct sampling, Kaiser and Chebyshev filtering).

(Color online) Comparison of the modulus in Pa of spatial sound pressure fields at time *t* = 6.2 ms: The back-propagated spatial sound pressure fields using regularization with (b) or without measurement noise (a), Wiener inverse filtering with (d) or without measurement noise (c), the reference field (e). Inversion is achieved from an impulse response preprocessed by Chebyshev low-pass filtering. In the case of measurement noise, the signal-to-noise ratio (SNR) is 3 dB.

(Color online) Comparison of the modulus in Pa of spatial sound pressure fields at time *t* = 6.2 ms: The back-propagated spatial sound pressure fields using regularization with (b) or without measurement noise (a), Wiener inverse filtering with (d) or without measurement noise (c), the reference field (e). Inversion is achieved from an impulse response preprocessed by Chebyshev low-pass filtering. In the case of measurement noise, the signal-to-noise ratio (SNR) is 3 dB.

(Color online) Comparison between reference signals (dotted line) and back-propagated signals using regularization method (a)–(c), and inverse Wiener filtering method (d)–(f), associated with Chebyshev low-pass filtering on locations *R* _{2}, *R* _{3}, and *R* _{4} with a signal to measurement noise ratio SNR = 3 dB.

(Color online) Comparison between reference signals (dotted line) and back-propagated signals using regularization method (a)–(c), and inverse Wiener filtering method (d)–(f), associated with Chebyshev low-pass filtering on locations *R* _{2}, *R* _{3}, and *R* _{4} with a signal to measurement noise ratio SNR = 3 dB.

(Color online) Noise influence on the spatial errors *E _{x} _{,} _{y} * and in the case of regularization (a), (c) and inverse filtering (b), (d) when Chebyshev filtering is applied to the impulse response. The vertical line indicates the time chosen (

*t*= 6.2 ms) for the spatial field representation in Fig. 8.

(Color online) Noise influence on the spatial errors *E _{x} _{,} _{y} * and in the case of regularization (a), (c) and inverse filtering (b), (d) when Chebyshev filtering is applied to the impulse response. The vertical line indicates the time chosen (

*t*= 6.2 ms) for the spatial field representation in Fig. 8.

## Tables

Indicator *T* _{1} [see Eq. (30)] computed from reference signals and pressure signals back-propagated to the plane *z* = *z _{c} * in locations

*R*

_{1},

*R*

_{2},

*R*

_{3}, and

*R*

_{4}(see Fig. 1) using the inverse impulse responses obtained by Wiener filtering or regularization from preprocessed impulse responses by the direct method with

*f*= 16 000 Hz, the average method, Chebyshev and Kaiser filtering.

_{e}Indicator *T* _{1} [see Eq. (30)] computed from reference signals and pressure signals back-propagated to the plane *z* = *z _{c} * in locations

*R*

_{1},

*R*

_{2},

*R*

_{3}, and

*R*

_{4}(see Fig. 1) using the inverse impulse responses obtained by Wiener filtering or regularization from preprocessed impulse responses by the direct method with

*f*= 16 000 Hz, the average method, Chebyshev and Kaiser filtering.

_{e}Indicator *T* _{2} [see Eq. (31)] computed from reference signals and pressure signals back-propagated to the plane *z* = *z _{c} * in locations

*R*

_{1},

*R*

_{2},

*R*

_{3}, and

*R*

_{4}(see Fig. 1) using the inverse impulse responses obtained by Wiener filtering or regularization from pre-processed impulse responses by the direct method with

*f*= 16 000 Hz, the average method, Chebyshev and Kaiser filtering.

_{e}Indicator *T* _{2} [see Eq. (31)] computed from reference signals and pressure signals back-propagated to the plane *z* = *z _{c} * in locations

*R*

_{1},

*R*

_{2},

*R*

_{3}, and

*R*

_{4}(see Fig. 1) using the inverse impulse responses obtained by Wiener filtering or regularization from pre-processed impulse responses by the direct method with

*f*= 16 000 Hz, the average method, Chebyshev and Kaiser filtering.

_{e}Indicators *T* _{1} and *T* _{2} [see Eqs. (30) and (31)] computed from reference signals and pressure signals back-propagated to the plane *z* = *z _{c} * in locations

*R*

_{1},

*R*

_{2},

*R*

_{3}, and

*R*

_{4}(see Fig. 1) using the inverse impulse responses obtained by Wiener filtering or regularization in the case of measurement noise with a signal to noise ratio SNR = 3 dB.

Indicators *T* _{1} and *T* _{2} [see Eqs. (30) and (31)] computed from reference signals and pressure signals back-propagated to the plane *z* = *z _{c} * in locations

*R*

_{1},

*R*

_{2},

*R*

_{3}, and

*R*

_{4}(see Fig. 1) using the inverse impulse responses obtained by Wiener filtering or regularization in the case of measurement noise with a signal to noise ratio SNR = 3 dB.

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