Localization of low-frequency coherent sound sources with compressive beamforming-based passive synthetic aperture

Localizing low-frequency sound sources is important in fault detection and diagnosis for fields that involve rotating mechanisms, power transformers, and other scenarios. Using a microphone array and conventional beamforming (CBF) is the most common method to robustly obtain the location of sound sources. However, obtaining a high resolution with CBF for an array with a small aperture is difficult when the sources emit low-frequency signals. Other methods to improve resolution, such as Capon's method and the multiple signal classification (MUSIC) algorithm, have been developed. However, the performance of these methods significantly degrades under a low signal-to-noise ratio (SNR) environment with strongly coherent sources and insufficient snapshots.^1,2 Another way to overcome the resolution limit of a single array is to form a virtually large array through a passive synthetic aperture (PSA).^3,4 The PSA enables detection of a low frequency that has an explicit mismatch with the operational frequency of the original array, but this aperture entails strict moving speed control and heavy computational burden. In recent years, the compressive sensing method^5,6 has attracted considerable attention in the field of signal processing. Compressive sensing-based beamforming (CS beamforming)^2,7,8 takes advantage of the prior information on the sparsity of sources in space domains or arriving angle domains, and it can achieve a higher resolution than traditional methods even in a noisy and coherent environment with limited samples.

In this letter, a compressive beamforming-based passive synthetic aperture (CS-PSA) is proposed to localize low-frequency coherent sound sources. Unlike the classic PSA, the proposed CS-PSA uses a fixed sensor to correct the phase of the moved array. Comparisons among CBF, CS beamforming, PSA, and CS-PSA are made to demonstrate how CS-PSA can achieve better performance than the other methods even in a low-frequency scenario.

For an arbitrary planar array with M microphones of which the ith microphone is located on $ri=(xi,yi,zi)$⁠, its array manifold matrix $A$ under the plane wave assumption is

where $λ$ is the wavelength of the input signal, $a(θu,ϕw)$ is the array manifold vector of the hypothetic direction $(θu,ϕw)$⁠, and $θu$ and $ϕw$ are the elevation angle and azimuth angle, respectively. At a specific time, the received signal can be written as $s=Aα+n$⁠, where $α$ denotes the vector of the emitted signals, and $n$ is the noise.

It should be noted that the wavelengths of the sources are both 1.36 m and the distance (R) between the microphone array plane and the sound sources plane is only 5.5 m in the subsequent experiment in this letter. So the array is in the near field of sound sources. As R is known, the scanning process is carried out in hypothetic position grids. The array manifold matrix $A$ in the near field can be rewritten by changing the array manifold vector $a$⁠. Given the distance R, $a$ can be written as

where $di(Xp,Yq)=(xi−Xp)2+(yi−Yq)2+R2$⁠, $(Xp,Yq)$ is the intersection of the qth horizontal direction and the pth vertical direction in the scanning plane.

Passive synthetic aperture techniques can be used by towed arrays to improve bearing resolution. The correlation of data received at different sensors in the same position is commonly used to calibrate the phase of synthetic data.³ Unlike these methods, which need prior knowledge of the array moving speed and signal frequency, in our proposed method, we set a reference receiver Re at a fixed position as shown in Fig. 1(a), which corrects the phases of signals when the array is moved to different positions. Similar concept has been used in spatially referenced towed array concept,⁹ which utilizes the frequency and location of a known transmitted source to provide smaller source location estimation errors. We assume that the frequency and amplitude of the signal emitted by a sound source are $f$ and $S$⁠, respectively. The range vector between the microphones of the array in the first position [Ra in Fig. 1(a)] and the sound source at time $ta$ is $ra$⁠. The received signals of the array can be expressed as s_ra = S exp[j2πft_a + jφ_a]exp(−j2πfra/c), where $φa$ denotes the initial phase of the sound source at time $ta$⁠, and $c$ denotes the sound speed in air. The received signal of the reference receiver is s_rea = S exp[j2πft_a + jφ_a]exp(−j2πfre/c). When the array is moved to the next position $rb$ [Rb in Fig. 1(a)] at time $tb$⁠, the received signals are s_rb = S exp[j2πft_b + jφ_b]exp(−j2πfrb/c) and s_reb = S exp[j2πft_b + jφ_b]exp(−j2πfre/c), where $φb$ denotes the initial phase at time $tb$⁠. The array phases of the signals in different positions can be calibrated by division of their reference signals as s_a = s_ra/s_rea = exp[−j2πfra/c]exp(j2πfre/c) and s_b = s_rb/s_reb = exp[−j2πfrb/c]exp(j2πfre/c). The initial phases $φa$ and $φb$ are eliminated, and the calibrated signals $sa$ and $sb$ are considered the synchronized samples at a specific time. With the preceding steps, the aperture of the array can be virtually expanded.

For acoustic localization problems, suppose the number of the sources is K. When an M element array is used and N directions are scanned in the beam domain, the sources are sparse when $K<M<N$⁠, which is common in cases of fault detection and underwater target localization, among others. In compressive sensing theory,⁵ the sparsity can be exploited to reconstruct the signal from a small number of measurements. If a signal of length N is K-sparse, it can be exactly recovered from $O(K log N)$ measurements with a high probability. Many existing techniques¹⁰ can be used to solve the reconstruction problem, such as matching pursuit, orthogonal matching pursuit, basis pursuit, and basis pursuit denoising. Compressive beamforming is a direction of arrival (DOA) estimation method that takes advantage of the prior information of sparsity and has been proved to achieve a higher resolution than most popular methods, such as conventional beamforming, Capon's method, and MUSIC method.² Furthermore the ability to resolve correlated sources makes compressive beamforming suitable for low-frequency coherent sources localization. A commonly used CS beamforming framework^2,7 is

where $Φ$ is the random measurement matrix, $Ψ$ is the dictionary that corresponds to the manifold matrix of the array (⁠$Ψ=A$⁠), $b=Φs$⁠, $s$ is the signal vector received by the array, $α̂$ is the sparse solution of the source location, and $ε$ is the upper bound of the Gaussian noise norm $‖n‖2$⁠. Equation (4) is well known as a basis pursuit denoising (BPDN)¹¹ problem which is often used in noisy environment. To get the solution of the BPDN problem, a lot of methods have been developed. These methods can be divided to several classes such as active-set/pivoting methods, interior-point methods, gradient-descent methods. Spectral project gradient for L1 minimization (SPGL1)⁸ is an extension of gradient methods and is reported suitable for large scale problems in the complex domain. It gets the solution through finding the root of a single variable nonlinear equation that determines the Pareto curve, and the Pareto curve indicates the relation between $||ΦΨα−b||2$ and $‖α‖1$⁠. In this letter, SPGL1 algorithm was used to solve Eq. (4).

Compressive beamforming can be combined with the passive synthetic aperture framework by replacement of the components $Ψ$⁠, $α$⁠, and $b$ in Eq. (4) with their synthetic expressions

where $Ai$ is the manifold matrix of the ith position of the array, and the passive synthetic measurements are taken at $L$ positions. $Φsyn$ is the new random measurement matrix.

A spiral array was used to localize two low-frequency sources. The array is 2 m in diameter and has 63 microphones distributed in seven arms, as shown in Fig. 1(b). The array is set to be movable on a slide guide in the horizontal direction and has a stretcher in the vertical direction, so the synthetic aperture processing can be conducted on a specific plane. The experiment was conducted in a semi-anechoic chamber. Two loudspeakers were placed in front of the spiral array. The distance between the loudspeakers was 3 m in the horizontal direction, and they were both 2.4 m in height. The distance between the spiral array plane and the loudspeakers plane was 5.5 m. A microphone was used as the reference point for the passive synthetic aperture and was kept in the same position during the entire experiment. The initial height of the central point of the spiral array was 2 m. During the experiment, the two sources emitted two similar tonal signals of 250 Hz. Figure 2(a) illustrates the experiment scenario.

Measurements were taken in the grid space, which was expanded by the horizontal and vertical positions set. All spacings of the adjacent horizontal grid and vertical grid were 1 m. The horizontal moving direction from H1 to H3 was from left to right, and the vertical moving direction from V1 to V2 was from low to high. Six measurements were made when the array was moved to six different positions. Sources A and B were positioned 2.04 m to the left and 0.96 m to the right of the central point of the array, respectively, and they were at a height of 0.4 m when position V1_H2 was chosen as the origin point for the localization results. Figure 2(b) shows the virtual array when all of the six measurements were conducted.

Location scanning was performed from −5 to 5 m in the horizontal direction and −5 to 5 m in the vertical direction relative to the origin point. The scanning step was 0.1 m in both directions. $α$ was a $10 201×1$ vector in the frequency domain after the scanning grids were reshaped, whereas $Ψsyn$ was a $378×10 201$ matrix under the synthetic aperture. $Φsyn$ was a $60×378$ Gaussian random matrix with the computing burden considered. Signal $si$ was obtained in the frequency domain and was a $63×1$ vector in the ith position. The synthetic signal $bsyn$ is a $60×1$ vector. Because the environment in the semi-anechoic chamber ensures a high SNR condition, the upper bound of the noise norm $ε$ in the experiment was set to 0.1, which was very close to 0.

The CBF result is shown in Fig. 3(a), in which the two extreme values are located on (−3.4,0.3) and (2.3,0.9) and have a large bias with the true positions (−2.04,0.4) and (0.96,0.4). The semi-anechoic chamber cannot eliminate the reflection from the floor, so a large error is produced when the aperture of the array is relatively small. The result of CS beamforming is shown in Fig. 3(b). The estimates of the locations are (−2.7,0) and (1.9,1.1). Although this result is inaccurate, we find that another source has been detected on (2.2,−4.1), which can be treated as the image source that is a consequence of the reflection from the floor. Figure 3(c) presents the result of the PSA. The localization results for the two sources are (−2.3,0.6) and (1.4,0.8). Because of the expanding aperture, the localization accuracy is improved, and the width of the main lobe is narrower than that of the CBF result. Figure 3(d) shows the result of CS-PSA. The localization results for the two sources are (−2.2,0.3) and (1.0,0.3), respectively. The horizontal biases of the results are 0.16 and 0.04 m, respectively, and the vertical biases are both 0.1 m. Although the two upper locations are nearly the same as those in Fig. 3(c), the width of the main lobe is narrower. Two image sources can be detected in (−2.2,−4) and (1.2,−4.3), and the results are close to the positions of the theoretical image sources, (−2.2,−4.4) and (0.96,−4.4). Compared with the results of CBF, CS beamforming, and PSA, that of the CS-PSA has the narrowest main lobe width and the best detection ability. It is worth noting that the intensity of the source on the right in Fig. 3(d) is significantly reduced. It is most likely due to the selection of $ε$⁠. The main purpose of this experiment is to validate the localization method, so the $ε$ was chosen empirically. In the future research, the optimum value of $ε$ will be investigated especially when sources are coherent.

In this letter, a compressive sensing-based passive synthetic aperture is proposed. Compared with classical passive synthetic aperture methods, the proposed approach uses a fixed reference sensor to calibrate the phase and does not need prior information about the array moving speed parameters. The localization results of two low-frequency coherent sources show that the CS-PSA has a narrower main lobe width and more accurate location estimation than CBF, CS beamforming, and PSA. Our approach has potential to localize low-frequency coherent sources with a small aperture array.

This work was supported by the National Natural Science Foundation of China (Grant No. 11174235), the Doctorate Foundation of Northwestern Polytechnical University, China (Grant No. CX201226), and the Fundamental Research Funds for the central Universities (Grant No. 3102014JC02010301).