## 5 Conclusions  
CS is an effective technique rooted on the idea that signal often has a sparse or nearly sparse representation in some bases or dictionaries. However, in many practical applications of acoustics, the characteristics of the source are not clear. It is difficult to find a suitable basis or dictionary for the source. To solve this problem and improve the performance of sound field reconstruction for different types of sources, a compressed ESM based on equivalent redundant dictionary was proposed. In the present work, the following key findings were acquired.  
(1) An equivalent redundant dictionary, which combined a series of orthogonal basis \(\mathbf{U}\) obtained by the SVD method with the equivalent dictionary \(\mathbf{\phi}\) used in CESM, was constructed, and it made the proposed method applicable both for sparse and even spatially extended sources.  
(2) A reformative method taking advantage of FB, named ERDCESM-\(\upsilon\), was developed in this paper. This improved the dynamic range when detecting the source location.  
Numerical simulations were performed to compare the reconstructed results of TRESH, CESM, CMESM, and ERDCESM. The results showed that whether the source was distributed sparsely or not sparsely, the reconstruction effect of TRESH was unsatisfactory due to the insufficient measurement points. When the source was distributed sparsely, CESM achieved the best reconstruction effect, and the accuracy of ERDCESM was almost as good as that of CESM. When the source was not distributed sparsely (e.g., the spatially extended source), both CMESM and ERDCESM had higher reconstruction accuracy than TRESH and CESM. However, compared with CMESM, ERDCESM could provide more accurate results since the dictionary of ERDCESM was complex-valued and contained more phase information. Therefore, ERDCESM produced a satisfactory reconstruction result, which indicated that  
Figure 13: Pressure distributions under experimental conditions of **(a)** ERDCESM with \(\upsilon=0\), **(b)** ERDCESM with \(\upsilon=1\), **(c)** ERDCESM with \(\upsilon=3\) and **(d)** ERDCESM with \(\upsilon=5\) for a plate sound source at 500 Hz.  
it was more flexible for different types of sources. By combining the advantages of FB, ERDCESM-\(v\) could provide more a accurate source position (for the spatially sparse source) or locate the strongest radiated sound pressure area (for the spatially extended source).  
The proposed method was also tested by experimental applications. The results indicated that the proposed method was practical and reliable for different types of sound sources.  
Investigation, Y.H.; Writing--original draft, Y. H.; Methodology, Y.H and L.C.; Software. L.C.; Data curation, L.C.; Writing--review and editing, Y.H. and L.C.; Project administration, Z.X. and Z.Z.; Resources, Z.X. and Z.Z.  
This research received no external funding.  
This work was supported by the Natural Science Foundation of China (Grant No. 11874096) and the Fundamental Research Funds for the Central Universities (Project No. 106112016 CDJXZ338825). These projects covered all the costs of this series of research. The authors appreciate their support in this research.  
The authors declare no conflict of interest.