## 4 Simulation  
### Simulation Experiment Setup  
There are two main difficulties for the acoustic imaging problem in far-field measurements due to the limitations of microphone array aperture and microphone spacing [52]. On the one hand, the spatial resolution of the low-frequency sources is poor, and it is challenging to realize the correct identification and separation of multiple sources with close location distribution. On the other hand, due to long-distance propagation, the SNR is reduced, and side lobes or false sources are prone to exist in the cloud image, reducing the dynamic range. The above problems will seriously affect the accurate identification of real sound sources. In this section, the issue of sound source identification under complex working conditions at long measurement distances, low source frequencies, and close source spacing is discussed to compare the performance differences of various methods.  
In this section, the source localization process is simulated through a series of simulation experiments. To achieve high-precision identification of far-field low-frequency sound sources, in terms of the layout of the microphone array, an array evaluation function is constructed with spatial resolution and dynamic range as the goals, and an adaptive optimization design method for microphone arrays is designed. This method derives the response analytical expression and resolution constraint function of the annular microphone array, constructs the performance evaluation function of the microphone array composed of multi-circle annular arrays, and establishes an array optimization algorithm based on the annular array, which can achieve high-resolution and high-dynamic-range measurement of wideband noise sources. The microphone array used in the simulation is optimized in this way. It is a circular array consisting of 193 array elements [53], and the distribution of array element positions is shown in Figure 3a. The array aperture is 8 m, and the distance from the array to the sound source plane is 7 m. The sound source plane is set to be a rectangular plane of 8 \(\times\) 6 m, uniformly discretized into 81 \(\times\) 61 grid points (the spacing of adjacent discretized points in both directions is 0.1 m). The schematic diagram of the microphone array and sound source arrangement for the simulation experiment is shown in Figure 3b. Three monopole sources are set up, as shown in Figure 3c, with Source A, Source B, and Source C located at (0.36, \(-\)1.24) m, (0.54, \(-\)1.48) m, and (0.86, 1.6) m, respectively, and the amplitudes and phases of the sources are randomly assigned to ensure the generality of the methods. Complex Gaussian noise with a certain SNR is added linearly to the target signal to simulate the actual test environment.  
Figure 3: Simulation experiment setup. (**a**) Distribution of microphones; (**b**) Measurement schematic. (**c**) The locations of the three sound sources.  
The signal generation method is shown in Equation (4). First, generate the complex source strength snapshot \(\mathbf{Q}\) that obeys Gaussian distribution, and then multiply it by Green's function \(\mathbf{G}\) to obtain the snapshot matrix under noise-free conditions. On this basis, add complex interference noise \(\mathbf{E}\). It is known that the signal-to-noise ratio (SNR) is defined as:  
\[\text{SNR}=10\log_{10}\frac{\mathbf{P}_{s}}{\mathbf{P}_{e}} \tag{26}\]  
where \(\mathbf{P}_{s}\) is the power of the sound source signal and \(\mathbf{P}_{e}\) is the power of the noise signal. Therefore, according to the definition of SNR, the noise component \(\mathbf{E}\) under a given SNR can be obtained as follows [54]:  
\[\mathbf{E}=\frac{\mathbf{P}_{s}^{(1/2)}}{10^{(\text{SNR}/10)}}e^{i\theta} \tag{27}\]  
where \(\frac{\mathbf{P}_{s}^{(1/2)}}{10^{(\text{SNR}/10)}}\) denotes the amplitude of the \(\mathbf{E}\) and \(\theta\) is the phase of \(\mathbf{E}\).