Modified Wideband Acoustical Holography for Improving the Identification of Acoustic Sources at Low Frequencies

Source identification and sound field reconstruction using near-field acoustical holography (NAH) is particularly challenging when the measurement data are relatively fewer than the sources to reconstruct. Fast wideband acoustical holography (WBH) is an inverse solution method based on the compressed sensing theory, which enables the NAH with such sparse measurement and extends its applicable frequency range. WBH usually works well at high frequencies, but it is often unsatisfactory when dealing with low-frequency coherent sources. This work proposes a modified WBH method to improve low-frequency and high-frequency performance substantially. It is developed by combining the conventional WBH and the steepest descent methods in computing the initial source strength values. It contrasts the WBH method using specific initial values for the equivalent sources. The iteration is repeated to ensure that these source strengths are not involved in the threshold filtering process during the iteration process. The simulation and experiment show that the proposed modified WBH substantially improves the reconstruction accuracy of coherent sound sources at low frequencies without degrading the reconstruction performance at mid to high-frequencies.


I. INTRODUCTION
The reconstruction of acoustic parameters and localization for sound sources in the sound field can help engineers to enable the investigation of vibro-acoustic energy flow, noise control, and acoustic tuning. Near-field acoustical holography (NAH) [1], [2], [3], [4] has been widely used for sound field reconstruction and identification with good low-frequency resolution. The inverse equivalent source method (ESM) [5], [6], [7] is one of the acoustic imaging techniques in the NAH with great potential. In the ESM, the acoustic pressure at any field point can be expressed The associate editor coordinating the review of this manuscript and approving it for publication was Lin Wang . by the superposition of sound fields generated by a series of equivalent point sources distributed inside or outside the actual surface of the structure. Often, only monopole sources are employed as equivalent sources for simplicity, and computational efficiency, owing to their ability to reconstruct non-separable geometries [8], [9]. When the number of microphones is larger than that of the equivalent sources, the well-established L 2 -norm regularization, e.g., the Tikhonov regularization [10], [11] is usually used to obtain the inverse solution optimally and stably. The resulting equivalent source strengths are suitable for sound field reconstruction at low frequencies but less satisfying for high frequencies. When the number of microphones is smaller than that of the equivalent sources, the transfer matrix becomes underdetermined. It imposes a significant limitation at high frequencies [3] due to the violation of the Nyquist sampling theorem, apart from the instability problem. The compressed sensing technique using the L 1 -norm resolves the issue [12], [13] that results in good reconstruction performance.
The wideband acoustical holography method (WBH) [14], [15] is proposed based on the compressed sensing theory. It uses the steepest descent method to solve the inverse problem of underdetermined systems with the gradual elimination of small source strength in the iteration process, which enforces sparsity in the coefficient vector. In such an underdetermined situation, it is demonstrated that the WBH method yields a high reconstruction accuracy in the middle and high frequencies. However, its accuracy was unsatisfactory when dealing with low-frequency coherent sources. It happens because the iteration begins with zero source strength assigned to all sources, which means that no information about the sound field is assumed to be known. When the initial value for the WBH is obtained from the calculated results of the L 1 -norm, better results were obtained. However, the weighting parameter in the L 1 regularization is problematic because it is usually chosen manually based on multiple experiments or reconstruction comparisons and the expected results [16].
In the present work, a new method, called the modified wideband acoustical holography, hereafter called mWBH, is proposed to realize accurate sound source identification and reconstruction for a wide band spanning from low to high-frequency ranges. First, the inverse problem is solved by the steepest descent method (SDM). Then, instead of beginning the WBH iteration with zero source strengths, the source strength distribution resulting from the SDM is used as the initial solution for the WBH process. This way, the WBH method searches for the solution starting from a relatively reasonable guess rather than a zero-source strength guess without any ground. In addition, the threshold filtering condition in WBH is reset for each iteration, so the new threshold filtering condition can retain the preliminary sound field information obtained from the previous step to prevent them from being filtered out during the subsequent iteration processes.

II. WIDEBAND ACOUSTICAL HOLOGRAPHY METHOD A. WBH METHOD USING THE ESM
In the simplest form of the equivalent source method employing only monopole sources, the acoustic pressure at any field point can be approximately expressed by the superposition of sound fields, which are generated by many monopoles distributed over a surface near the actual surface structure. The distribution of the equivalent sources is generally arranged according to the dimension of the sound source plane. Assume that there are M measured points on the measurement plane, or hologram plane, in other words, the sound field and N equivalent sources are distributed on the equivalent source plane. The m-th measured data pressure P h m on the measurement or hologram plane, h, can be written as P h = N n=1 g(r h m , r n )q(r n ). Here, q(r n ) is the strength of equivalent sources on the equivalent source plane , g(r h m , r n ) denotes the free-field Green's function between the measured pressure and the equivalent sources, it can be represented as , ω is the angular frequency of the harmonic sound (∼ e jωt ), and −→ r m,n denotes the distance from the measurement points to the equivalent source points. One can rewrite the pressure vector in matrix-vector notation as Here, P h means the pressure vector on the measurement plane with the size of M × 1, G h p is the pressure transfer matrix between the measured pressure and the equivalent sources with the size of M × N , and Q denote the source strengths vector with the size of N × 1. Once the equivalent source strengths Q is obtained by solving the inverse problem of (1), the sound field can be reconstructed by multiplying the corresponding transfer function, G f p , connecting the equivalent sources to any point P f in the sound field.
In applying the inverse ESM based on NAH, many reconstruction points need to be arranged to obtain a high resolution. Otherwise, when the analysis frequency is relatively high, the number of equivalent source points should be arranged enough to describe the complex sound field. However, in either case, it will lead to a highly underdetermined condition of the transfer matrix, G h p , connecting the source points to the measurement points. In such an underdetermined system, the inverse problem for (1) can be solved well using the WBH method. Essential steps are the steepest decent iteration and a thresholding process in each iteration. The step is introduced as follows [14]. First, define the residual vector r and the quadratic residual function F as When the i-th iteration step is denoted as q (i) , it minimizes the residual function F(q) in the steepest descent direction as q (i) = s (i) w (i) , where the vector w (i) is the negative gradient vector given by The step length in the vector direction is defined as where the vector g (i) is given by g (i) = G h p w (i) . Then, the solution vectorq (i+1) after one iteration is obtained as  where α is a possible relaxation factor with values typically between 0.5 and 1.0. It should be noted that using the source strength vector obtained from (5) may introduce 'ghost' sources in the reconstruction. It is because these source strengths are assigned to all elements of the initial source model, possibly, even to the components posted at source-free locations. To eliminate the ghost sources, a threshold value, T (i) , is set by assigning the component inq (i+1) less than T (i) to be zero. Considering the effective signal magnitude, T (i) value is recommended as T (i) = 10 −D (i)/20 q (i) max , where D (i) (dB) implies the dynamic range is less than the amplitude of the largest elementq (i) max inq (i+1) . Thus, the elements q One can call the process the threshold filtering condition. The dynamic range D (i) is to be increased after each iteration so that more sources will be added to the model with each iteration; thus, it can be expressed as D (i+1) = D (i) + D. Finally, the iteration will be terminated when it satisfies the condition, D (i+1) > D max , where D max denotes an upper limit on D (i) . In the previous work [14], the following initial values are recommended in the WBH: To compare with the proposed mWBH in the present work, these data will be used for the conventional WBH computation.

B. PERFORMANCE SIMULATION
In the simulation, two pulsating sources are taken as actual sources to reconstruct using Tikhonov regularization (TR) [10], [11] and WBH methods. The radius of the pulsating sphere is a = 0.01m, and its velocity amplitude u a = 0.01m/s. The locations of two coherent pulsating spheres are (−0.16, 0, 0) m and (0.16, 0, 0) m, respectively. The measurement, reconstruction, and equivalent source planes are located at 0.3 m and 0.05 m above the plate and 0.001 m below the pulsating sphere, as seen in Fig. 1(a). The spacing of data points in the reconstruction plane is 0.02 m over a 0.40 × 0.40 m 2 area. The data point grid in the equivalent source plane is the same as that of the reconstruction plane. There are 36 microphones distributed on the measurement plane, of which the locations are shown in Fig. 1(b). In this test, the 441 equivalent source strengths are reconstructed using the 36 measured signals; thus, the entire transfer system is underdetermined. The analysis frequency range is from 100 Hz to 3000 Hz with a 100 Hz step. Uniform random noise with a signal-to-noise ratio of 40 dB is added to the pressure data, reflecting an actual measurement situation with noise contamination. The reconstruction error is presented in the root-mean-squared error norm, e rms , in %, given by where P s andP s denote the actual and the reconstruction pressure, respectively. The parameters setting of the WBH followed the recommendation as given in (7). Figure 2 compares the reconstruction errors between the conventional WBH and TR methods over the 100-3000 Hz frequency range. One can observe that the reconstruction error of the conventional WBH is higher than the TR method when the analysis frequency is below 1500 Hz, which reaches up to 40%. For mid-high frequencies, the reconstruction error of the conventional WBH can be kept at a low level, around 10% overall. Conversely, the reconstruction error of TR is lower at low frequencies and increases significantly at medium and high frequencies. Figure 3(a) shows the reconstruction results of 441 data points on the reconstruction plane using the conventional WBH method and the TR method for two representative frequencies in Fig. 2: 500 Hz and 2500 Hz. Correspondingly, the comparison of the reconstructed sound pressure levels using TR and WBH with the actual value along the middle row of the reconstruction plane (21 points) is shown in Fig. 3(b). One can see that, in the low-frequency case, i.e., 500 Hz, the reconstruction results of the TR method are close to the actual map. However, the conventional WBH does not identify the two sources distinctly, but it reconstructs a significant source located in the middle of the two real sources. Due to the superposition of the phases, the maximum sound pressure level is higher than the actual level. A similar phenomenon can also be found in the other previous works [14], [15], [16]. When the frequency is 2500 Hz, TR and WBH clearly identify the location of a given source. However, the source maps obtained with WBH have better agreement with the theoretical source map with respect to the size of the hot spot compared to the TR, which shows a slight difference.

III. MODIFIED WBH METHOD (mWBH)
The preceding section shows that the conventional WBH can not identify low-frequency coherent sources. The reconstruction accuracy for the low-frequency sound field is significantly reduced compared with the high-frequency case. Such limitation of the WBH stems from the zero source strength assigned to all sources in the starting phase of the iteration, i.e., q (0) = 0. It is reported [16] that there is a potential to improve the reconstruction results if proper assumptions are given to the initial sound field. The other point to note for the modification is the threshold filtering process. Equation (6) sets a portion of the source strength less than the threshold value to be zero in each iteration to eliminate ghost sources. However, it can also bring uncertainty that the finite source strength corresponding to the real sources may be set to zero in the iteration process. Although the zero source strengths will be changed to a nonzero value in the subsequent iterations, it can still cause some biased results.

A. CHANGE OF INITIAL SOURCE IN THE WBH
Three kinds of source strength values are considered as their initial data in the WBH to examine the effect of initial source strength on the WBH results as follows: (1) the zero vector, which is used in the conventional WBH; (2) the result of Tikhonov regularization method indicated as TR-WBH, for which the regularization parameter can be determined by the generalized cross-validation technique (GCV) [17]; (3) the result of the steepest descent method (SDM-WBH). The test model is the same as described in Sec. II-B. The One can see from Fig. 4 that the reconstruction accuracy at lower frequencies is improved using the initial source strength used for the beginning of the WBH by the TR method or the steepest descent method. One can also find that the satisfactory error condition is maintained at high frequencies similar to the conventional WBH using zero initial source strengths. Actually, a little bit of improvement is also made. It again demonstrates the previous conclusion that the initial source strength influences the reconstructed results of the WBH method. In Fig. 4, it is found that the reconstruction error using the calculated results of the SDM as the initial strength value of the WBH method is better than the TR method. One can recall that the transfer system connecting the sources to the measurement points is underdetermined, i.e., 441 sources and 36 measurement signals. In this underdetermined situation, it is not wise to use the TR method because there may exist too many solutions for underdetermined linear systems. It may lead to a bias in the initial source strength describing the acoustic model. Therefore, in the proposed method, the results of the SDM-WBH, q 0 = q sdm , are chosen as the initial source strength instead of the zero initial value, q 0 = 0. In addition, the steepest descent method does not need to determine the relevant parameters, i. e., the regularization parameter in the TR method or the weighting parameter in the L1-norm, which controls how much noise is included in the initial solution. This weighting parameter is difficult to quantify using a unified formula and is often chosen manually based on multiple experiments and comparisons of the reconstruction result and the expected result [16].

B. CHANGE OF THE THRESHOLD FILTERING CONDITION
When the non-zero initial source strengths, i. e., q sdm , are used at the beginning of the iterations for the WBH; some may be automatically set to zero because the magnitudes are less than the threshold value. Thus, the threshold filtering condition in (6) needs to be modified to ensure q sdm values are not changed by threshold filtering during the iterative process.
Combining with the preceding SDM-WBH method, the procedure of using the modified WBH, called the mWBH method, can be summarized as follows.
(1) Assume that there are M measurement points and N equivalent sources. The relationship between the measured sound pressure and the equivalent source strength is obtained according to (1) as P h = G h p Q , where G h p is an underdetermined matrix with a size of M × N.
(2) Compute the equivalent source strengths with the steepest descent method to achieve q sdm,N ×1 . where m is defined as the number of measured signals, M. (4) Reset the initial source strength vector in the WBH iterative algorithm, q 0 = q sdm,N ×1 . (5) Compute the equivalent source strengths using WBH with q 0 = q sdm,N ×1 , where the threshold filtering condition in the iterative process of the WBH is modified as

IV. SIMULATION
The present simulation aims to investigate the reconstruction abilities of the mWBH, mainly focusing on low frequencies because we already know that the high-frequency performance is nearly the same or a little better than the conventional WBH. The sound field at 0.05 m above the two pulsating spheres will be reconstructed using the mWBH and the conventional WBH methods. The parameters of the simulation object and the three planes, such as the measurement plane, reconstruction plane, and equivalent source plane, are entirely the same as in Sec. II-B. Similarly, the reconstruction error of the sound field on the reconstruction plane is used as the evaluation criterion for the algorithm reconstruction accuracy. The reconstruction error is obtained according to (8).

A. LOCALIZATION OF TWO SOURCES
In this section, the example used for Fig. 3 is again reconstructed using the mWBH to check its identification ability. The results are shown in Fig. 5. One can see from Fig. 5(a) that the mWBH successfully identifies the two sources distinctively to the correct positions at a low frequency (500 Hz). At a high frequency (2500 Hz), the source maps obtained with mWBH have better agreement with the theoretical source map in terms of the size of the hot spot than the WBH. Additionally, one can find from Fig. 5(b) that the reconstructed value of the mWBH has a better match with the actual value both at 500 Hz and 2500 Hz than the TR and WBH results in Fig. 3(b). At 500 Hz, the peak of the reconstructed value differs from the actual by only 0.2 dB, and at 2500 Hz, the peak differs from the actual one by only 0.1 dB.

B. EFFECT OF THE MEASURING DISTANCE
In the NAH technique, the accuracy of sound field reconstruction is very sensitive to the measuring distance. It is widely believed that measuring at relatively close locations to the sound source can yield good reconstruction accuracy. However, in practical applications, long measuring distances are often desired for some test environments, such as high temperature and vibration, which significantly limits the application of NAH. Thus, the analysis of the measuring distance is an important parameter to evaluate the performance of the reconstructed method. In the present simulation, the reconstruction ability of the WBH, mWBH, and TR methods are tested for the four different measuring distances, which are 0.1 m, 0.2 m, 0.3 m, and 0.4 m, respectively. The rest simulation parameters are set up entirely as in Sec II-B, except for the measuring distance. Figure 6 compares the reconstruction results with frequency variation by the WBH, mWBH, and TR methods varying the measuring distances: d = 0.1 m, 0.2 m, 0.3 m, 0.4 m. The first thing we can notice from either error plot of the measuring distance case is that the reconstruction accuracy of the WBH at high frequencies is still acceptable but inferior at low frequencies, and the reconstruction results of the TR are just the opposite. Compared to the WBH results, the mWBH result at low frequencies bears relatively more accurate error percentages, which is a dramatic improvement. Additionally, at least a slight improvement at high frequencies can be observed. We can also note that the reconstruction results of the mWBH and TR are comparable when the measuring distance is 0.1 m and 0.2 m, however, TR is slightly better than mWBH when the measuring distance is 0.3 m and 0.5 m. In addition, it is found that with the decrease of SNR further, mWBH has more accurate accuracy than TR at low frequencies, which can be observed in the next Sec. IV C. Integrating the error plots for these four measuring distance cases, one can find that the WBH and TR methods are relatively influenced strongly by the measuring distance; when the spatial separation becomes farther, the overall reconstruction of the error increases substantially at high frequencies. Compared with WBH and TR methods, mWBH is less affected by the change in measuring distance, and the overall reconstruction error of the mWBH is lower than 10%, except for individual low-frequency cases. It is worth noting that mWBH is still more effective than WBH when the analysis frequency decreases, i.e., below 100 Hz. However, the mWBH may not provide an advantage compared to the TR method, especially when the measuring distance becomes more extended. It can be further compensated by optimizing the array appropriately. Figure 7 shows the contour maps of relative error using the mWBH for different measuring distances (from 0.1 m to 0.5 m with a 0.05 m step) and frequencies (from 100 Hz to 3000 Hz with a 100 Hz step). It can be seen from the figure that for a single frequency, the reconstruction accuracy with the increase of the measuring distance. In near-field acoustic holography, one can recall that less information is captured with longer measuring distances, which means fewer evanescent waves are included. It implies that using smaller measurement information is challenging for source identification. In addition, longer measuring distances may cause the true source to be less ''visible'' in the measurement data, causing fewer problems when fitting the acoustic model to the data. In general, the reconstruction accuracy of the mWBH is good at all the measuring distances, and the reconstruction error is less than 13%. When the frequency is below 500 Hz, and the measuring distance is longer than 0.30 m, the reconstruction error is relatively higher. It can also be seen that the influence of the measuring distance on the reconstruction results of the mWBH is minor at the higher analysis frequency.

C. EFFECT OF THE SIGNAL-TO-NOISE RATIO (SNR)
In practice, all measured signals contain noise components, even within an anechoic chamber, which can eventually affect the reconstruction performance of the algorithm. In the present simulation, the reconstruction abilities of the WBH and the mWBH methods are tested for four different SNRs, which are 20 dB, 25 dB, 30 dB, and 35 dB, respectively. Except for the SNR, the other simulation parameters are set up completely the same as before. Figure 8 compares the reconstruction results with frequency variation by the WBH and the mWBH under different SNR conditions. To ensure the credibility of the results, the error corresponding to each SNR condition is calculated five times, and its average is used for the presentation. Figure 8 exhibits that the overall reconstruction error increases with the decrease of SNR. Compared with the WBH and TR, one can find that the mWBH performs well at the selected four SNR conditions, and the reconstruction accuracy at low frequencies is greatly improved compared to the WBH. The reconstruction errors are around 5% except for the lower SNR conditions, while they are about 10% for SNR=20 and 25 dB. The additional advantage is that the mWBH method's error is more uniform than the WBH and TR in the whole frequency range. Additionally, it is worth noting that, recalling the simulation results of Fig. 6(c) in Sec IVB, which corresponds to SNR= 40 dB and a measuring distance = 0.3 m, the TR is slightly better than mWBH at low frequencies. In comparison to Fig. 6(c), the SNR is further decreased in this section, it can be observed that mWBH outperforms the TR method at low frequencies when the signal is subjected to stronger interference.

D. EFFECT OF THE SOUND SOURCE DISTANCE
One can recall that the distance between the two coherent pulsating spheres is 0.32 m in the previous simulation object. When the analysis frequency is 500 Hz, the wavelength (λ= 0.68 m) is much larger than the distance between the two sources (0.32 m), causing trouble in identifying the slight difference clearly between them. This simulation examines the reconstruction ability of the mWBH method for two coherent sources with different distances to determine whether the mWBH can identify the neighboring sound sources within a compact range. The distance between the two coherent sources is set from 0.2 m to 0.3 m, with a step size of 0.01 m and the analysis frequency from 100 Hz to 3000 Hz. The other simulation parameters are set up completely the same as in Secs II-B. Figure 9 shows the contour maps of relative error using the mWBH and WBH methods for different source distances and frequencies.
One can see from Fig. 9 that the source distance influences the reconstruction accuracy of the WBH severely for all considered source distances when the frequency is relatively low. It is particularly true when the distance between sources is small, i.e., 0.2 m and 0.22 m, which exhibits the maximum error reaching up to 40%. It means that the conventional WBH method cannot identify the densely populated sources within a compact geometric range at low frequencies.
As depicted in Fig. 3, its result shows a reconstruction of a single large source located between two actual sources (WBH at 500 Hz).
Compared with the WBH, the reconstruction accuracy of the mWBH is significantly improved at the low-frequency range, where the reconstruction error is less than 13% for all the considered source distance cases. In addition, the overall reconstruction of the mWBH is maintained in a stable state. Therefore, the mWBH method can well reconstruct the sound field with multiple sources nearby. Such a finding suggests that the mWBH method would help deal with the extended vibro-acoustic sources.
It is worth noting that all the above simulations are performed using MATLAB on a PC with an INTEL i7-7700K chipset. As an example, the time required to calculate Fig. 6(c) takes the 20 s for the mWBH and 3 s for the WBH, which is because mWBH has an additional process for calculating the initial source strength than WBH and more iterations, i.e., a new threshold filtering condition.

V. EXPERIMENT
An experiment with dual loudspeakers with a 0.30 m distance was conducted in a semi-anechoic chamber to test the reconstruction performance of the mWBH method. The two loudspeakers were placed 0.3 m in front of a 36-channel uniform array (B&K type 4957), and the data was collected by the signal analyzer (B&K Pulse). The speakers produced pure tones of 500 Hz and 2500 Hz during the experiment. The sampling frequency was 16.384 kHz, and the sampling time was 5 s. Four separate measurements were conducted in front of the loudspeaker; each non-overlapping measurement covered the top left, bottom left, top right, and bottom right of the loudspeaker. Overall, 441 data were VOLUME 11, 2023  obtained from the different positions. Then, the transfer function between each microphone and the input signal to the loudspeakers using the averaged periodogram method [18] was obtained using the measured data. The method allowed sufficient simultaneous measurement data to be constructed by multiplying all transfer functions by the same input signal. Finally, 36 data points were selected among the preceding 144 data according to the position of the measurement points in the simulation, as exhibited in Fig. 1 (b). Note that, in the selected process of 36 measurement points, efforts were made to choose the microphone position that is consistent or very close to the measurement points' positions in the simulation; however, the possibility of including minor positioning errors could exist. The equivalent source and the reconstruction planes are located 0.001 m behind and 0.02 m in front of the loudspeakers, respectively. Uniformly distributed 441 source points are positioned on the equivalent source and reconstruction planes with 0.6 × 0.6 m 2 in size: 441 parameters must be estimated based on 36 measurements, thereby establishing the underdetermined system. Based on the measurement signals, the sound sources are visualized using the WBH and mWBH methods, as shown in Fig. 10(a). Figure 10(b) shows the comparison between the sound pressure levels reconstructed using mWBH and WBH and the actual measured values (21 points) along the middle row of the reconstruction plane.
From the result of 2500 Hz in Fig. 10, one can find that the WBH can roughly reconstruct the positions of the two coherent sources. Still, the identification results reveal that the high-order elements are distributed near the primary sources producing 'ghost' sources, which may lead to misinterpretation. Conversely, the source strength reconstructed by mWBH is highly concentrated at two small zones, and the positions of the identified sources are distinguishable. So, it eliminates the 'ghost sources' happening in the WBH results. In the 500 Hz result of Fig. 10, the WBH identifies a strong, seemingly a 'primary' source in the middle of the two actual sources: the result is consistent with the finding in the preceding numerical simulation. However, the mWBH can still successfully identify both coherent sources distinctly. Additionally, it can be seen from Fig. 10(b) that the reconstructed results of the mWBH have a better agreement with the actual measured values than WBH in both frequency cases, where the difference with the actual values is mainly caused by the deviation of the microphone position during the measurement and slightly noise. Based on the preceding results, one can assure that the mWBH method can reconstruct coherent sources well for a wide spectral range from low to high frequencies. The ability to restore the source field for a wide frequency band at once is a handy and practical feature.

VI. CONCLUSION
The motivation of this work is to drastically improve the accuracy of source localization and reconstruction using a small number of microphones based on conventional ESMbased NAH. The transfer system connecting the sources to the field points is set as underdetermined for the practical application to big-size extended sources. Such an underdetermined problem is usually troublesome in achieving the inverse solution, but the WBH has suggested a practical algorithm for this kind of problem. Although the WBH works well at high frequencies, it often fails in dealing with coherent sound sources at low frequencies: the identification results are often biased due to the inherent problem of the algorithm itself. In the present work, a new method, called the mWBH method, is proposed to substantially improve the WBH for accurate source identification and reconstruction for a wide spectral range covering low to high frequencies. In this method, the source strengths are initially solved with the fastest descent method, and it is used as a priori information of the sound field to replace the zero initial source strengths used in the WBH. Further, the threshold filtering condition is adjusted to prevent these source strengths from being set to zero during the iterative processes.
Simulation and experimental results of coherent sources show that the proposed mWBH can significantly improve the reconstruction of low-frequency sources compared with the WBH. It can also accurately identify coherent sources avoiding the bias encountered in the WBH method. For coherent sources at the middle and high frequencies, the mWBH method still maintains a high reconstruction performance. It is shown that the mWBH method can well reconstruct the sound field with multiple sources nearby. Such a finding suggests that the mWBH would have good potential in dealing with extended vibro-acoustic sources in general. In addition, the mWBH can provide a more balanced estimate of the source strengths and is very stable in a wide frequency range than the WBH. Moreover, in this work, the mWBH method has been tested with two experiments based on another randomly distributed and a uniformly distributed microphone array. The results of these experiments have a good agreement, which validates that the proposed method is practical and reliable. It is worth noting that in practical applications, when the shape of the sources is irregular, the results may be biased when using planar arrays for reconstruction. In this case, as the source-to-sensor distance increases, the information containing the detailed source activity is transmitted to the sensors on the hologram plane in a diminished way, which leads to a significant increase in the illness of the transfer matrix connecting the source to the sensor. One may further study this matter combined with the mWBH starting with [19], which employs optimized sensor placement to further improve the reconstruction of the sound field for irregular sources.