# Critical Review of Basic Methods on DoA Estimation of EM Waves Impinging a Spherical Antenna Array
## 1 Introduction
The free space response may be different over time (i.e., DoA, amplitude, and phase) due to refractivity and absorption. The frequency resolution is reduced by averaging over the frequency, while averaging over time reduces the temporal resolution of the estimate. The spatial resolution is reduced by averaging over space because the number of elements for each subarray is less than the number of elements in the entire array. Contrary to MUSIC and other similar methods, ML methods [30; 31] with no smoothing can handle coherent signals. The coherent signal model can be U-dimensional (where U is the number of signals due to multipath) when each signal is differently modeled. If a spatial spectrum is computed with a grid size D, where the correct size and sampling are user-defined, then the DoA estimation using the ML function has a dimensionality of U \(\times\) D, but spectral-based techniques have 1 \(\times\) D dimensionality. Therefore, the search space quickly becomes very large as the number of signals increases. For ML methods [31], high dimensionality gives rise to the deployment of nonlinear optimization algorithms for reduced computational time. The general limitation of all methods is that the estimable number of signals must be less than the number of elements I, i.e., I \(>\) U [30]. Moreover, in practice, the free space response is always time- and frequency-dependent. This paper studies the DoA estimation of EM signals in free space trapped with an SAA. Specifically, we consider the case where more than one signal (i.e., in a multipath scenario) is present in a single analysis window. The number of signal points or responses due to multipath at the receiving antenna within a short time window rises with the square of time t. This impact is usually termed the density of reflection [20]. Therefore, with time, it results in a situation where a single analysis window encompasses multiple signals. There is a complete overlap of signals if the distance between the generating source and the receiving antenna is the same for one or more signals. This usually occurs for first-order signals in a symmetric source-receiver. We examined the ML DoA estimation methods within the context of the SAA's spatial response and large sample approximation (LSA), known as weighted subspace fitting (WSF). LSA assumes that the available number of measurements is large. Because of the aforementioned characteristics, the conditions in which one or more signals are present in one analysis window for narrowband and wideband frequencies are considered in this work. The key innovations and contributions of this study are as follow:  
1. Five DoA estimation techniques are presented and investigated in this study. MUSIC, plane-wave decomposition (PWD), weighted subspace fitting (WSF), and both deterministic (D) and stochastic (S) ML techniques were applied in the analysis of EM waves using SAA processing for the first time in this study. WSF is the large sample approximation of SML. 2. We adopted ML techniques in the spatial signal analysis with SAA and confirmed their suitability in DoA estimation using an SAA. It is numerically demonstrated how ML techniques outweighed MUSIC and beamforming methods and how WSF produces the best DoA estimation among the ML techniques. 3. The results obtained from ML techniques are compared to beamforming and MUSIC methods throughout the simulation and analysis section. The ML-based DoA estimation is more appropriate, less biased, and more robust against noise than beamforming and MUSIC techniques. In addition, the WSF technique exhibits lower computational complexity than DML and SML. 4. We explain how DoA estimation performance with a rigid sphere is expressed as a function of frequency. In addition, the radius of the sphere affects the estimation accuracy at a particular frequency. For instance, for a given SAA with a radius of 8.4 cm, the most accurate performance was achieved at 16 GHz. If similar performance were required, for example, at about 8 GHz, then a larger radius of SAA would be required.