## 1 Introduction
These approaches have been applied to the localization of reflected signals, but not for DoAs sources. There are other different methods for the estimation of DoA in the spherical realm that employed sources' directional sparsity according to the literature [29; 30; 31; 32]. For instance, [PERSON] et al. [29] reported an independent component analysis of signals in the spherical realm and the DoAs were estimated via comparison between the unmixing matrix columns and signal steering vectors in each direction. Furthermore, the intensity-dependent methods of the DoA estimation [33; 34; 35; 36] are different from the previously discussed methods, because with the direct computation of energy flow, the computation of spatial cost function is not needed. As such, there is a potential and significant saving in terms of computation. In specific directions, the intensity component was computed via two intensity probes [37]. The first probe operates using the velocity of particle approximation, using the differences that exist between the two firmly distanced omnidirectional pressure elements, and the second probe directly deals with the velocity of the particle [38]. The former technique is common but severely affected by noise and phase mismatch. The array with intensity probes generates an intensity vector in two or three dimensions in which the DoA is estimated [39; 40]. Methods that can be used for the estimation of the DoA with SAA were developed in [33], where many elements were employed for the transformation of signals in the spherical domain in which the velocity of the particle was approximated. The resulting vectors were named pseudointensity vectors (PVs). The first set of results shows the efficiency of the DoA estimation technique with a single source under a high-SNR scenario. Furthermore, the DoA estimation with multiple sources was computed using PVs k-means clustering in [34]. The formulation review and function of PVs are presented in [33]. For an understanding of the basics and some mathematical formulations, readers are referred to References [33; 41]. Compared to the previous works in the literature, the main innovation and contributions in this work are given as follows. We present the formulation of PV from an SH domain signal representation in Cartesian and propose an extended analysis of PVs in practical scenarios. Furthermore, a subspace pseudointensity vector (SPV) is proposed, which shows a higher level of robustness to noise and is more computationally efficient than PV. The main innovation of this work compared to [41] is that the PV methods are extended to a spherical antenna array, supported by measurements from experiments in EM vis-a-vis an SAA, which is a generally acceptable basis of testing and evaluating any procedure. The remaining parts of this article are structured as follows. In Section 2, the pseudointensity vector is mathematically formulated; the signal intensity and pseudointensity are briefly reviewed, and the PV system model and the subspace PV are mathematically presented. Section 3 presents the numerical experiments, results, and discussion. An estimation of DoA from PVs and SPVs, comparison of the results with baseline approaches using simulation and experimental data, and computational complexity analysis are given in Section 3. Section 4 concludes the article and suggests some future research directions.