# Interference Suppression in EEG Dipole Source Localization through Reduced-Rank Beamforming
## 2 Materials and Methods
### Proposed Reduced-Rank Beamforming Scheme  
In many practical beamforming applications, the available training data is insufficient to obtain a full-rank estimate of the covariance matrix of interference and noise, additionally the interference is typically of low rank. In [15], we proposed a reduced-rank beamforming approach using the cross-spectral metrics (CSM), since they allow the dimensionality of the filter to be reduced below the dimension of the noise eigenstructure without significant loss of performance in the signal-to-interference-plus-noise ratio (SINR). In addition, the crosspectral reduced-dimension filter can outperform the full-dimension one when the noise covariance is unknown (see [22] and references therein). Under those conditions, we re-evaluate the activity index first proposed in [15]:  
\[\iota_{\mathrm{RE}_{1}}(\mathbf{\theta})=\mathrm{tr}\bigg{\{}\Big{[}\widetilde{W} _{0}^{T}\widehat{R}^{-1}(\widetilde{W}_{0}-W_{h})-W_{h}^{T}\widehat{R}^{-1} \widetilde{W}_{0}\Big{]}^{-1}\bigg{\}}. \tag{13}\]  
This index applies a rank-reduction to \(W_{0}\) (indicated as \(\widetilde{W}_{0}\)) according to the \(J=M-\mathrm{rank}(C_{\perp})\) largest CSM as defined in [23]. Hence, \(\widetilde{W}_{0}\) is given by  
\[\widetilde{W}_{0}=\sum_{j=1}^{J}\frac{\mathbf{u}_{j}\mathbf{u}_{j}^{T}R_{X_{0}Q_{h}}}{ \lambda_{j}}, \tag{14}\]  
where \(\lambda_{j}\) are the eigenvalues and \(\mathbf{u}_{j}\) are the orthonormal eigenvectors of \(R_{X_{0}}\) with largest CSM, and \(R_{X_{0}Q_{h}}=\)E\([X_{0}Q_{h}^{T}]\). Note that \(\eta_{\mathrm{res}_{\mathrm{res}_{\mathrm{res}_{\mathrm{res}_{\mathrm{res}_{ \mathrm{res}_{\mathrm{res}_{\mathrm{res}_{\mathrm{res}_{\mathrm{res}_{ \mathrm{res}_{\mathrm{res}_{\mathrm{res}_{\mathrm{res}_{\mathrm{res}_{ \mathrm{res}}}}}}}}}}}}}}}}\) differs from the one in [15] because we are now considering \(C_{\perp}\) as an ABM whose rank will determine the rank-reduction of \(W_{0}\).  
In (13), all the weights of \(W_{\mathrm{csc}}\) are considered, yet the rank-reduction is only applied to \(W_{0}\). For this reason, we introduce an alternative activity index that only considers the weights of the orthogonal subspace of the noise and interference components:  
\[\iota_{\mathrm{RE}_{2}}(\mathbf{\theta})=\mathrm{tr}\bigg{\{}\Big{[}\widetilde{W} _{0}^{T}\widehat{R}^{-1}\widetilde{W}_{0}\Big{]}^{-1}\bigg{\}}. \tag{15}\]  
Figure 1: Structure of the generalized sidelobe canceler.  
Under those conditions, our proposed brain source localization based on the reduced-rank activity indices \(t_{\text{ss}_{1}}\) and \(t_{\text{ss}_{2}}\) is given by  
\[\widehat{\mathbf{\theta}}=\min_{\mathbf{\theta}}t_{\text{ss}_{1}}(\mathbf{\theta}),\text{for }\mathbf{\theta}\in\Omega. \tag{16}\]  
Note that in (13) and (15) a negative sign was omitted for convenience, hence the minimization in (16). In addition, note that their structure is inspired in the one of (6) but with the array response matrix \(A\) replaced by a reduced-rank version of the weights \(W_{\text{GSC}}\) and \(W_{0}\) in \(t_{\text{RR}_{1}}\) and \(t_{\text{RR}_{2}}\), respectively. This approach is used to \"force\" a desired response in a similar way as in array interpolation. There, a virtual or desired array response having all properties necessary for posterior processing is used to provide the array with the necessary characteristics to estimate the direction of correlated sources [24].