# Critical Review of Basic Methods on DoA Estimation of EM Waves Impinging a Spherical Antenna Array
## 1 Introduction  
Direction-of-arrival estimation is a popular topic in various electromagnetic-related fields of study, finding applications in military surveillance, radar, sonar, and mobile communication systems [1, 2, 3, 4, 5, 6, 7, 8, 9]. Understanding the directions-of-arrival (DoAs) of incoming signals on receiving antenna can be effectively used to localize the positions of the corresponding sources. It facilitates adaptive beamforming of the receiving pattern to improve the sensitivity of the system towards the desired directions of signals and mitigates undesirable interferences. Invariably, it ensures that the array provides the maximum beamtowards the desired users and nulls in the directions of interferences, and it improves the performance of mobile and base stations. Therefore, DoA estimation of electromagnetic (EM) waves impinging on an antenna array is very important. A spherical antenna array (SAA) configuration is useful in obtaining an array with isotropic characteristics. There is also the spherical phased antenna array, which is widely used in spacecraft and satellite communication. Electronic beam scanning by the phased antenna array remains a better choice because it provides a hemispherical scan and almost uniformly distributed gain [3; 10]. Another major advantage of the SAA configuration is its three-dimensional symmetry, which is an advantage in the spatial analysis of signals. The geometry and size usually provide the directivity and degree of beam scanning in the elevation and azimuth needed from the array. A linear array, which scans in one axis, and a planar array, which scans in the x- and y-axes, are acceptable for a bounded scan demand of 60-70\({}^{\circ}\)[10]. The projected apertures of these arrays drop by the cosine factor of the scan angle. The array directivity is reduced for a large scan angle. For larger steerability, a multifold planar configuration or, in a general form, an SAA is preferred [10; 11; 12]. The directivity of an SAA remains almost the same when an active source is moved over the surface of the SAA. Since signals measured with an SAA can be coherent and highly correlated, it becomes important to investigate the effectiveness of the available DoA estimation methods with an SAA. Various DoA estimation methods have been reported in the literature. The most commonly used methods are: the multiple signal classification (MUSIC) algorithm [13; 14], estimation of signal parameters through rotational invariance technique (ESPRIT) [15; 16], beamforming, and the maximum likelihood (ML) DoA estimator [17; 18]. They have been deployed and applied to antenna arrays with arbitrary geometries and provide an appropriate estimate of DoA. Recently, the use of phased-array-based sub-Nyquist sampling [19] was proposed. The temporal properties of the impinging signals were explored to improve the detection capabilities. To date, linear and planar antenna array configurations have been considered in most DoA estimations. However, recent publications emphasize the importance and rapid development of SAAs, particularly in satellite communication [3; 10]. Multipath signals can be coherent or highly correlated in a narrow band of frequency. Coherent signals can cause spectral-based methods to fail in DoA estimation in linear or planar arrays [1; 11], and hence, these methods are generally employed with spherical arrays [11; 20; 21]. The performance of MUSIC and ESPRIT for the estimation of correlated signals can be improved by using spatial smoothing techniques [22; 23; 24]. In addition, different smoothing techniques have been proposed for general SAAs [21; 25]. The smoothing techniques preprocess the array output covariance matrix over time [26], space [21; 24], or frequency [25] before the estimation of DoA. In SAA processing, the array is usually divided into subarrays by transforming the SAA into a uniform linear array [21; 27]. Another spatial smoothing approach is the formation of a subarray using eigenbeam space rotation [28; 29].