Monopole time reversal is an advanced noise source localization technique, but its sidelobe level in localizing sound sources is greatly affected by noise, leading to a decrease in the accuracy of sound source localization under a low signal-to-noise ratio. To address this issue, this paper proposes a time-domain monopole time reversal technique that excludes autocorrelation to reduce the impact of noise on sound source localization results. Numerical simulation and experimental results show that, compared to conventional monopole time reversal techniques, the proposed method can effectively reduce the sidelobes in sound source localization results and achieve accurate sound source localization under −20 dB low signal-to-noise ratio conditions.

Topics
Microphone array, Acoustical properties, Acoustic holography, Acoustic noise control, Acoustic signal processing, Image processing, Signal-to-noise ratio, Computer simulation, Integral equations, Geometrical optics

Noise source localization is crucial for understanding the mechanisms of noise generation and achieving noise control. In recent years, various techniques based on microphone array measurements have been developed to locate noise sources, such as Near-field Acoustic Holography (NAH),1–3 Delay-and-Sum (DAS) beamforming,4–6 and Time Reversal (TR).7–9 Among these, NAH is an inverse method with a strong ill-posed solution, while beamforming techniques are mainly suitable for locating mid-to-high frequency sound sources.

Time Reversal (TR), an acoustic frontier technology with powerful noise source localization capabilities, was developed at the end of the 1980s.7–9 The complete TR technique is derived from the Helmholtz–Kirchhoff integral equation and requires simultaneous measurement of sound pressure and its gradient as input, which not only increases the workload but also the hardware cost.10,11 Monopole Time Reversal (MTR) is a simplification of the complete TR technique that only requires sound pressure measurement as input, thus reducing workload and hardware costs and making it easier to apply in practical engineering.12 However, the noise source localization effectiveness of MTR is greatly affected by noise, mainly manifested in high sidelobe levels in the localization results under low signal-to-noise ratios.12 To solve this problem, Solimene et al.13 mixed the localization results obtained at different frequencies, which effectively reduced the sidelobes and removed spurious targets. Yu et al.14 proposed a vector TR method that uses the correlation of coherent signals and the difference in isotropic noise correlation to further improve the output signal-to-noise ratio through joint processing of sound pressure and particle velocity, thereby reducing the sidelobes of the localization results. However, this method requires simultaneous measurement of sound pressure and particle velocity, resulting in higher measurement costs. Considering that noise components have a larger proportion in the high wavenumber region, Bi et al.12,15 introduced a low-pass filter to directly remove some of the high wavenumber information from the sound field, effectively reducing the sidelobes of the sound source localization results. However, this method reduces resolution while lowering sidelobes due to the removal of some high wavenumber information. Li et al.16 used the characteristic that noise signals from each channel generally only exist in the autospectrum of the channel’s signals to eliminate the interference of channel noise signals using the cross-spectral algorithm, effectively reducing the sidelobes, even when only measuring sound pressure. However, the above methods are calculated in the frequency domain and are usually only applicable to single-frequency or narrowband sound sources, while actual sound source signals are often non-stationary with a wide bandwidth, which would increase the workload if calculated in the frequency domain.

This paper proposes a time-domain monopole time reversal method with excluding autocorrelation to achieve low sidelobe localization of sound sources. Based on conventional MTR techniques, this method considers the impact of channel self-noise during cross-correlation processing of time-domain measurement signals, removing the autocorrelation elements from the cross-correlation matrix to further reduce the sidelobes in the sound source localization results. This paper is organized as follows. Section II introduces the theory of the proposed method. Section III presents the numerical simulations to validate the effectiveness of the proposed method, and the experimental validation is performed in Sec. IV. Finally, conclusions are drawn in Sec. V.

In an ideal fluid medium, the radiation sound field of a point source satisfies the wave equation with a source,17 
(1)
In the equation, c represents the propagation speed of sound waves in the medium, p(r, t) is the sound pressure at point r, s(t) is the source signal, r0 is the position of the sound source, and δ is the Dirac delta function. From Eq. (1), we can solve for
(2)
where ⊗ represents the convolution operation and g(r, r0, t) is the impulse response function from the sound source to point r. In the free field, g(r,r0,t)=14πr−r0δ(t−r−r0c)⁠.
Suppose the measurement array is composed of N microphones, located at positions ri (i = 1, 2, …,N). Considering that in actual measurements, it is impossible to measure signals for an infinite amount of time, measurements are usually only taken within a finite time interval [0, T]. Therefore, the sound pressure generated at the receiving point R after the time reversal of the signal is
(3)
where ai=14πR−ri and τi=R−ric⁠.
Taking the time average of Eq. (3), we get the final output for the MTR method as
(4)
where ⋅ represents the time average operator.

In Eq. (4), when the receiving point is located at the position of the sound source, that is R = r0, all terms in Φ(R) have the same phase. When summed, they will output a maximum value, forming the main lobe. However, when R ≠ r0, the phases of the terms in Φ(R) are different. When summed, they will cancel each other out, causing the output value to decrease and forming side lobes. Therefore, the focal point where the maximum value appears in the time-reversed acoustic field is the location of the sound source. Moreover, the greater the dynamic range of the side lobe peak relative to the main lobe peak, the more accurate the noise source localization will be.

In the aforementioned time domain MTR method, when the signal-to-noise ratio of the measurement signal is low, the sidelobe of localization result is usually large, and may even obscure the main lobe, affecting the accurate localization of the sound source. To reduce the impact of noise, Eq. (4) can be written as
(5)
In Eq. (5), the first term represents the cross-correlation of signals from different measurement points, and the second term represents the autocorrelation of signals at each measurement point. Since the noise in the signals from each measurement point is generally uncorrelated, noise interference usually only exists in the autocorrelation of each measurement signal. Therefore, the second term can be removed in Eq. (5) to reduce the impact of noise signals. Then, the output becomes
(6)
Combining Eqs. (4) and (6), the output of the time-domain MTR method with excluding autocorrelation can be expressed as
(7)

To verify the effectiveness of the method proposed in this paper, numerical simulation studies will be conducted below. Figure 1 shows a schematic diagram of the simulation model, where point O is the origin of the coordinates. In the simulation, two monopole sources are used as target sound sources, located at (−0.1, 0, 0 m) and (0.1, 0, 0 m), respectively. The sound source signal uses white noise, which is filtered through a 1/3 octave band filter with a center frequency of 3000 Hz. The measurement surface contains 7 × 7 measurement points, with a spacing of 0.1 m in both x and y directions between each adjacent measurement point. The distance between the measurement surface and the sound source plane is Δz = 1 m.

FIG. 1.
The schematic diagram of the simulation.
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The schematic diagram of the simulation.

FIG. 1.
The schematic diagram of the simulation.
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The schematic diagram of the simulation.

Close modal
To verify the effectiveness of the proposed time-domain MTR method with excluding autocorrelation, Fig. 2 presents the sound source localization results using the conventional MTR method and autocorrelation-excluded MTR method at a signal-to-noise ratio (SNR) of 0 dB. The SNR is defined as
(8)
where Es denotes the signal power and En is the power of noise.
FIG. 2.
Source localization results at 0 dB. (a) Conventional MTR method. (b) Autocorrelation-excluded MTR method. Two “+” indicate the true position of the sound sources.
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Source localization results at 0 dB. (a) Conventional MTR method. (b) Autocorrelation-excluded MTR method. Two “+” indicate the true position of the sound sources.

FIG. 2.
Source localization results at 0 dB. (a) Conventional MTR method. (b) Autocorrelation-excluded MTR method. Two “+” indicate the true position of the sound sources.
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Source localization results at 0 dB. (a) Conventional MTR method. (b) Autocorrelation-excluded MTR method. Two “+” indicate the true position of the sound sources.

Close modal

From the figure, it can be observed that although the conventional MTR method can locate two sound sources, its sidelobes are relatively high. In contrast, the autocorrelation-excluded MTR method, by eliminating the influence of self-noise, can significantly reduce the sidelobe level, thus validating the effectiveness of the proposed method.

Figure 3 presents the sound source localization results of the conventional MTR method and autocorrelation-excluded MTR method when the signal-to-noise ratio is −20 dB. As can be seen from the figure, due to the high sidelobe level, the conventional MTR method is unable to identify the two target sound sources at this time; however, the autocorrelation-excluded MTR method can still effectively reduce the sidelobes and accurately distinguish the positions of the two target sound sources. This proves that the proposed method can still identify two target sound sources under the low signal-to-noise ratio condition of −20 dB.

FIG. 3.
Source localization results at −20 dB. (a) Conventional MTR method. (b) Autocorrelation-excluded MTR method. Two “+” indicate the true position of the sound sources.
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Source localization results at −20 dB. (a) Conventional MTR method. (b) Autocorrelation-excluded MTR method. Two “+” indicate the true position of the sound sources.

FIG. 3.
Source localization results at −20 dB. (a) Conventional MTR method. (b) Autocorrelation-excluded MTR method. Two “+” indicate the true position of the sound sources.
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Source localization results at −20 dB. (a) Conventional MTR method. (b) Autocorrelation-excluded MTR method. Two “+” indicate the true position of the sound sources.

Close modal

To further examine the feasibility of the proposed method, an experiment was arranged in a semi-anechoic chamber, as shown in Fig. 4. The measurement array consisted of a circular array containing 64 microphones, as depicted in Fig. 4(a). The sound source was a conical point source, located as indicated by the red circles in Figs. 4(b) and 4(c), where the distances between the sound source plane and the measurement plane were 1 and 2 m, respectively. The measurement signal also passed through a 1/3 octave band filter with a center frequency of 3000 Hz.

FIG. 4.
Experimental setup. (a) Measurement array. (b) Target source at 1 m. (c) Target source at 2 m.
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Experimental setup. (a) Measurement array. (b) Target source at 1 m. (c) Target source at 2 m.

FIG. 4.
Experimental setup. (a) Measurement array. (b) Target source at 1 m. (c) Target source at 2 m.
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Experimental setup. (a) Measurement array. (b) Target source at 1 m. (c) Target source at 2 m.

Close modal

Figure 5 presents the sound source localization results of two MTR methods. It can be seen that although the conventional MTR method can accurately locate the sound source, the side lobes of its sound source localization result are relatively high. However, the autocorrelation-excluded MTR method proposed in this paper can effectively reduce the side lobes of sound source localization, proving the superiority of the proposed method, and its conclusion is also consistent with the simulation.

FIG. 5.
Experimental localization result.
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Experimental localization result.

FIG. 5.
Experimental localization result.
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Experimental localization result.

Close modal

This paper proposes a time-domain MTR method with excluding autocorrelation, which reduces the sidelobes of the sound source localization results by removing the channel self-noise in the time domain. Simulations and experiment show that the autocorrelation-excluded MTR method can effectively reduce the sidelobes of sound source localization, and can accurately locate the sound source even under a signal-to-noise ratio of −20 dB.

This work was financially supported by the Science and Technology Project of State Grid Shaanxi Electric Power Company Limited (Grant No. 5226KY230009).

The authors have no conflicts to disclose.

Mingxin Geng: Conceptualization (equal); Methodology (equal); Validation (equal); Writing – original draft (equal). Chen Shen: Methodology (equal); Validation (equal); Writing – original draft (equal). Jiangang Ma: Conceptualization (equal); Methodology (equal); Writing – review & editing (equal). Xiaolong Wei: Validation (equal); Writing – review & editing (equal). Lv Wang: Validation (equal); Writing – review & editing (equal). Yalin Zhao: Validation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and can be used for the sake of comparison by other researchers. However, the specific computer code used for the computation of the results would remain confidential and would not be shared.

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