2.3.1. Empirical Covariance Estimation

Earlier spatiotemporal beamformer studies [21,22,28,29] use the empirical covariance and inverse covariance, calculated as follows:

$$
\hat{\mathbf{C}}\_{\text{emp}} = \frac{1}{n-1} \sum\_{i=1}^{n} \mathbf{x}\_i \mathbf{x}\_i^{\mathsf{T}} \tag{6}
$$

$$
\overline{\hat{\mathsf{C}}^{-1}}{\mathsf{C}^{-1}}\_{\mathsf{emp}} = \hat{\mathsf{C}}^{+}\_{\mathsf{emp}} \tag{7}
$$

The Moore–Penrose pseudoinverse <sup>+</sup> ensures that a solution exists when *C*ˆ emp is singular. Figure 1a,b show examples of the empirical estimators of the covariance and the inverse covariance matrices, respectively. The empirical estimator suffers from performance and stability issues if the number of epochs *n* used for estimation is not much larger than the number of features *cs* [30,31].
