###### Abstract  
A spherical antenna array (SAA) is an array-designed arrangement capable of scanning in almost all the radiation sphere with constant directivity. It finds recent applications in aerospace, spacecraft, vehicular and satellite communications. Therefore, estimation of the direction-of-arrival (DoA) of electromagnetic (EM) waves that impinge on an SAA with unknown mutual coupling called for research attention. This paper proposed a spherical harmonic atomic norm minimization (SHANM) for DoA estimation using an SAA configuration. The gridless sparse signal recovery problem is considered in the spherical harmonic (SH) domain in conjunction with the atomic norm minimization (ANM). Because of the unavailability of the Vandermonde structure in the SH domain, the theorem of Vandermonde decomposition that is the mathematical basis of the traditional ANM methods finds no application in SH. Addressing this challenge, a low-dimensional semidefinite programming (SDP) approach implementing the SHANM method is developed. This approach is independent of Vandermonde decomposition, and directly recovers the atomic decomposition in SH. The numerical experimental results show the superior performance of the proposed method against the previous methods. In addition, accounting for the impacts of mutual coupling, an experimental measured data, which is the generally accepted ground of testing any method, is employed to illustrate the efficacy and robustness of the proposed methods. Finally, for achieving DoA estimation with sufficient localization accuracy using a SAA, the proposed SHANM-based method is a better option. a pacs: 12.30.-p, 12.30.-d, 12.30. Hence, estimation of the DoA of electromagnetic waves (EM) impinging on antenna arrays remains an important and crucial task. The spherical antenna array (SAA) (depicted in Figure 1) satisfies isotropic requirements [6]. The SAA configuration exhibits the capability to receive signals in all directions with the same strength regardless of the DoA and the polarization [2; 5]. In order to explore the biggest degree of freedom, the SAA must have the ability to determine the polarization and DoA of the receiving signals impinging on the unit sphere. There are various postulates that describe the SAA in literature [1; 3; 5; 7; 8; 9; 10; 11; 12; 13; 14; 15], but correct theories with deeper intuition of the signal features of the SAA still beg for research curiosity and attention. Various methods or algorithms have been developed for DoA estimation in the spherical harmonic (SH) domain [16; 17; 18; 19; 20; 21; 22]. A good number of these methods calculate metrics over a dense azimuth-elevation grid in advance of the identification of peaks as the DoAs. Such algorithms are those that deal with the computation of the steered response power, because of beamformers that steer in all the source directions, and those that calculate the spatial spectrum via subspace techniques based on MUSIC (multiple signal classification) [5]. There are widely used methods, such as, the MUSIC method [1], ESPRIT (estimation of signal parameters through rotational invariance technique) method [5], beamforming, and ML (maximum likelihood) approaches [4; 15]. The above techniques have been investigated and employed for antenna arrays having arbitrary configurations and produce correct estimation of DoA. There are more approaches, such as MUSIC group delay [4], the SRP-PHAT (steered response power with phase transform) approach [5], GCC (generalized cross-correlation) technique [1; 5] adaptive eigenvalue decomposition method [13], 1-D MUSIC [11] method, and order aware algorithm [8]. In near field, the MUSIC method employed for spherical configurations [1; 2; 3; 5] is called SH-MUSIC. Because of the sensitivity of SH-MUSIC to distortions in the multipath case, [PERSON] in [13] developed another approach named direct path dominance (DPD). In the near field, the mode strength depends on a source range that is not known a priori.