## 3 Simulated Measurements
### Simply supported plate  
In this simulation, a 50 cm \(\times\) 50 cm \(\times\) 3 mm finite simply supported steel plate was used to test the performance of the proposed method. The plate was excited by point force of 1 N at the center. The radiated sound field was calculated by using the Rayleigh's first integral. The measurement plane and reconstruction plane were meshed by 21 \(\times\) 21 grids with a uniform interspacing of 0.025 m. The two planes were located 5 cm (measurement plane) and 2 cm (reconstruction plane) above the plate. There were 64 measurement points which were randomly selected from 441 points located on 21 \(\times\) 21 grids (with a 0.025 m interval) to simulate a random array, as shown in Figure 2. The equivalent sources were distributed on a plane 0.001 m from the plate, with a grid of 21 \(\times\) 21 and a uniform interspacing of 0.025 m. Random noise with an SNR of 20 dB was added to the pressures to simulate the actual measurement. Figure 6 shows the reconstruction errors of the four methods over the frequency range of 50-3000 Hz. In a relatively wide frequency range, ERDCESM provided a better accuracy than the other three methods. The CESM performed well only at low frequencies (e.g., at 100 Hz, 200 Hz), and the error gradually became unacceptable with the increased frequency. The main reason resulting in this phenomenon was that the sparse basis of CESM disagreed with the spatially extended sound source, and the equivalent point source strengths could not be well-compressed. The difference between ERDCESM and CMESM was that ERDCESM possessed a complex-valued dictionary which contains more phase information, and consequently yielded better reconstruction effects for the same number of measurement points. Sound source maps of simply supported plate at 500 Hz and 2000 Hz are presented in Figure 7. It can be seen that the reconstruction accuracy obtained by ERDCESM and CMESM match the theoretical one better than those obtained by TRESM and CESM. CESM provided less accurate reconstruction at these two frequencies. This was because the radiated sound field of simply supported plate was complex which lead to a worse spatial sparsity, and the sparse basis of CESM was not suitable. Therefore, to accurately reconstruct the sound field, the number of measurement points required by CESM was more than required by CMESM and ERDCESM. Moreover, in terms of sources distribution, ERDCESM had a more accurate reconstruction, compared with CMESM. Figure 5: Pressure distributions of **(a)** the sound pressure at 500 Hz, **(b)** ERDCESM with \(\
u=0\) at 500 Hz, **(c)** ERDCESM with \(\
u=1\) at 500 Hz, **(d)** ERDCESM with \(\
u=3\) at 500 Hz, **(e)** the sound pressure at 2000 Hz, **(f)** ERDCESM with \(\
u=0\) at 2000 Hz, **(g)** ERDCESM with \(\
u=1\) at 2000 Hz and **(h)** ERDCESM with \(\
u=3\) at 2000 Hz for a single source. Figure 6: Reconstructed error for a simply supported plate. Frequency interval is 50 Hz. Figure 7: The calculated sound maps of (**a**) Theoretical map at 500 Hz, (**b**) TRESH map at 500 Hz, (**c**) CESM map at 500 Hz, (**d**) CMESM map at 500 Hz, (**e**) ERDCESM map at 500 Hz, (**f**) Theoretical map at 2000 Hz, (**g**) TRESH map at 2000 Hz, (**h**) CESM map at 2000 Hz, (**i**) CMESM map at 2000 Hz and (**j**) ERDCESM map at 2000 Hz for a simply supported plate. Using ERDCESM-\(\
u\), the comparison of sound pressure distributions with different orders at 500 Hz is shown in Figure 8. It can be seen that when \(\
u=1\), the identified sound pressure distribution was similar to that of the theoretical one (see the Figures 8b and 7a). With the order \(\
u\) increasing, the coverage of the identified acoustic center was narrowed gradually. Therefore, the maximum area of sound pressure distribution was located by increasing order \(\
u\), which provided direction for the optimization and improvement of structural panels. However, it was worth noting that some important sound pressure information could be lost (e.g., Figure 8d, \(\
u=5\)) if the order \(\
u\) was too large.