## 1 Introduction  
The equivalent source method (ESM) based on Nearfield acoustic holography (NAH) is a powerful tool for identifying sound sources and visualizing sound fields [1; 2; 3]. The method is based on the fundamental idea that the actual sound field can be approximated as the superposition of waves radiated by a series of discrete simple sources. In the ESM, an equation relating the measured pressures to the equivalent sources is established, and the source strengths vector can be determined by matching the boundary conditions on the source or on the hologram plane. Recently, the ESM is appealing in the field of NAH for its simplicity, computational efficiency, and the adaptability to arbitrarily shaped sources [4; 5; 6]. The ESM, as well as other holography methods [7; 8; 9; 10], generally leads to underdetermined problems, since the number of microphones is limited in most acoustic studies. The classical way of solving the problem is in a least-squares (LS) sense, by means of Tikhonov regularized Moore-Penrose pseudoinverse with regularization parameter \(\lambda\). A combined criterion for selecting the parameter \(\lambda\) is described by [PERSON] in Ref. [11]. However, the above conventional ESM with Tikhonov regularization (TRESM) only gives the minimum-norm solution, which tends to unduly spread errors in the solution space. In recent years, some compressive sensing (CS) methods have been widely used in the field of signal/image processing [12; 13; 14]. CS is an effective signal acquisition and processing technique based on the idea that data often have a sparse representation in a suitable basis or dictionary, i.e., they can be described only with few non-zero coefficients [15]. In source identification, [PERSON] et al. [16] first introduced the concept of CS to NAH and constructed a sparsity framework for NAH. After that, [PERSON] et al. applied the CS theory to a wave expansion method based on measurements with a spherical microphone array [17]. [PERSON] has presented a detailed study of the compressive ESM (CESM) [18]. When the source is distributed sparsely, the CS theory can be directly applied to NAH to reduce the number of measurement points or to extend the frequency range, which has been intensively studied in many literatures [19; 20; 21]. When the source is not distributed sparsely (e.g., the spatially extended source), it is necessary to construct an appropriate basis or dictionary to sparsely decompose the sound field so that the sparsity condition required by the CS theory can be fulfilled. In this case, two typical methods, named the total generalized variation regularization (TGV) [22] and the compressed modal equivalent point source method (CMESM) [23], are introduced. However, in most of the practical applications, the characteristics of the source are not clear. Therefore, it is difficult to choose a suitable basis or dictionary for the sparse representation of source distribution. In this paper, a compressed ESM based on an equivalent redundant dictionary (ERDCESM) is proposed, which is applicable both for sparse and even spatially extended sources. The method aims at enhancing the performance of source field reconstruction for different types of sources. In this method, the constructed equivalent redundant dictionary contains two core modules: one is a series of orthogonal basis of sound field obtained by employing the singular value decomposition (SVD), and the other one is the equivalent dictionary used in the CS-based ESMs under the sparse assumption. Compared with CMESM and CESM, ERDCESM is more flexible for different types of sources. Moreover, combining the advantages of functional beamforming (FB) [24; 25], ERDCESM with order \(v\) (ERDCESM-\(v\)) is applied to make further improvement of the dynamic range when detecting the source location. The remainder of this paper is organized as follows. In Section 2, the basic theory of ESM-based NAH is provided first, and then a theoretical derivation of the proposed method is described.