# A Deep Learning Method for DOA Estimation with Covariance Matrices in Reverberant Environments
## 1 Introduction  
Direction-of-arrival (DOA) estimation has received more attention in the field of signal processing because it has a wide range of application. It can be applied to wireless communication, speech source separation, video conferencing and so on [1, 2, 3, 4, 5, 6]. Over the years, many conventional methods have been successfully developed for the estimation task. The subspace-based methods are more popular among traditional methods for the outstanding performances. The multiple signal classification (MUSIC) [7] and the estimation of signal parameter via rotational invariance (ESPRIT) [8] are relatively eminent representative methods in subspace-based algorithms. The noise subspace is used to generate a spectrum in the MUSIC algorithm. The position where the spectral peak appears is the corresponding estimated DOA. Although it can guarantee relatively higher accuracy, the peak search is very time-consuming. Conversely, the ESPRIT method utilizes the signal sub-space for DOA estimation. The ESPRIT is time-saving at the expense of accuracy. I the beamforming-based approach, such as the minimum variance distortionless response (MVDR) [9], is another traditional method. However, the Rayleigh limit and low resolution are the shortcomings of this method. Generalized cross-correlation phase transform (GCC-PHAT) is also a commonly used traditional method, which requires the time difference of arrival (TDOA) between microphone pairs [10, 11]. There is also a conventional called Maximum likelihood method [12]. It needs a multi-dimensional search, even though they can provide high estimation accuracy. Even worse, the global convergence can not be guaranteed. All these methods mentioned above can be usedin an anechoic condition. However, they generally suffer from problems such as high computational cost or performance degradation in noisy and reverberant environments. In recent years, the development of deep learning has brought new inspiration to the DOA estimation. In order to resist reverberation and noise effectively, many deep neural networks (DNNs) have been proposed to estimate DOA. The most regular way to deal with the DOA task is to treat it as a regression task or a classification task. In terms of the regression, features extracted from the observation signals, which is related to location information, are directly mapped to the source position by deep learning [13; 14; 15; 16; 17]. For the classification, the position space will be divided into different regions. Each region corresponds to a class. Neural networks learn the relationship between input features and classes and then identify the class that a signal belongs to. In many studies and simulations, it is shown that when DOA is acted as a classification task, it can be estimated more accurately [18; 19; 20; 21]. In the classification task, a vital step is to find proper input features for neural networks to learn the relationship between the features and labeled regions. Many different features have been proposed for DOA estimation, such as phases of signals [22], both phases and magnitudes of signals [23], real part and imaginary part of the signal [24], intensity vectors [25] and so on. [PERSON] used signal phases as the inputs for a uniform linear array (ULA) [22]. However, elevations cannot be estimated simultaneously. It is not enough to locate the signal exactly with the ULA. [PERSON] combined phase and amplitude spectra as inputs [23] and mapped them to the outputs. The input features are first mapped into space pseudo-spectra (SPS) and the DOAs in the two-dimensional polarized coordinates are estimated. [PERSON] utilized the intensity vectors of first-order ambisonic (FOA) signals as the inputs [25] to jointly estimate DOAs. However, the accuracy is relatively lower. As shown before, the spherical microphone array has not received much attention. However, it has many advantages for array signal processing. In particular, this array does not cause angular blurring due to the perfect symmetry of the sphere.