# Interference Suppression in EEG Dipole Source Localization through Reduced-Rank Beamforming
## 2 Materials and Methods
### Data Model  
EEG data is assumed to be generated by cortical sources that can be approximated by \(l=1,2,\ldots,L\) equivalent current dipoles (ECDs), whose magnitude is given by \(\mathbf{q}_{l}=[q_{l,x}(t),q_{l,y}(t),q_{l,z}(t)]^{T}\) (assuming a Cartesian coordinate system), for \(t=1,2,\ldots,N\) time samples, and located within the brain. The dipoles are allowed to change in time but remain at the same position \(\mathbf{r}_{l}\) during the measurements period. Note that the ECD approximation is valid in practice for evoked response and event-related experiments [18]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Variable/Notation** & **Description** \\ \hline \(W\) & spatial filter (beamformer) \\ \(I\) & identity matrix \\ \(\mathbf{0}\) & matrix full of zeros \\ \(\widehat{Q}\) & indicates an approximation of \(Q\) \\ \(\widehat{Q}_{k}\) & approximated dipoles at the \(k\)th trial \\ \(A(\mathbf{r})\) & lead field matrix as a function of \(\mathbf{r}\) \\ \(W(\mathbf{r})\) & beamformer as a function of \(\mathbf{r}\) \\ \(W_{\textsc{LCMV}}\) & linearly constrained minimum variance (LCMV) beamformer \\ \(R\) & data covariance matrix \\ \(P\) & noise covariance matrix \\ \(u_{\textsc{LCMV}}(\mathbf{r})\) & LCMV-based neural activity index (NAI) as a function of \(\mathbf{r}\) \\ \(u_{\textsc{MAI}}(\mathbf{\theta})\) & multi-source activity index (MAI) as a function of \(\mathbf{\theta}\) \\ \(G(\theta)\) & reciprocal of the noise power as a function of \(\mathbf{\theta}\) \\ \(H(\theta)\) & reciprocal of the sources’ power as a function of \(\mathbf{\theta}\) \\ \(W_{GSC}\) & generalized sidelobe canceler (GSC) \\ \(W_{h}\) & quiescent component of the GSC \\ \(W_{0}\) & noise-plus-interference components of the GSC \\ \(C_{\perp}\) & blocking matrix \\ \(P_{A}^{\perp}\) & projection matrix of \(A\) \\ \(X_{0}\) & undesired measurement components \\ \(R_{X_{0}}\) & autocorrelation matrix of the undesired signals \\ \(\mathbf{b}\) & Wiener filter that minimizes the mean-squares of \(X_{0}\) \\ \(Q_{0}\) & matrix containing all undesired signals \\ \(u_{\textsc{RR}_{1}}\) & first proposed reduced-rank (RR) NAI a function of \(\mathbf{\theta}\) \\ \(\widehat{W}_{0}\) & RR approximation of \(W_{0}\) \\ \(\lambda_{j}\) & eigenvalues of \(R_{X_{0}}\) \\ \(\mathbf{u}_{j}\) & orthonormal eigenvectors of \(R_{X_{0}}\) \\ \(t_{\textsc{RR}_{2}}\) & second proposed RR-NAI as a function of \(\mathbf{\theta}\) \\ \(R_{X_{0}Q_{k}}\) & cross-correlation of \(X_{0}\) and \(Q_{h}\) \\ \(\mathbf{\theta}_{A}\) & matrix with all positions of interference sources \\ \(A(\mathbf{\theta}_{A})\) & lead field matrix as a function of \(\mathbf{\theta}_{A}\) \\ \(Q_{\Delta}\) & matrix containing all interference dipole sources \\ \(\sigma_{\Delta}^{2}\) & variance of biological noise \\ \(\eta_{m}\) & signal-to-measurement-noise ratio \\ \(\eta_{b}\) & signal-to-biological-noise ratio \\ \(\mathbf{\theta}_{\textsc{cam}}\) & matrix containing candidate dipole’s positions \\ \(b_{l,k}\) & bias of the estimate of \(\mathbf{r}_{l}\) at the \(k\)th trial \\ \(\textsc{SS}_{k}\) & sum-of-squares of \(b_{l,k}\) at the \(k\)th trial \\ \(b_{\textsc{MAX},k}\) & maximum bias at the \(k\)th trial \\ \(\overline{\textsc{SS}}\) & average sum-of-squares \\ \(\sigma_{\textsc{MAX}}\) & standard deviation of the maximum bias \\ \(P(W_{0})_{\textsc{min}}\) & minimum power of \(W_{0}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(Cont.\)Under those conditions, we can define a spatio-temporal representation of our EEG measurements through a matrix \(Y_{k}\) of size \(M\times N\) at the \(k\)th trial, for the case of \(k=1,2,\ldots,K\) independent experiments (trials), such that  
\[Y_{k}=\left[ \begin{array}{cccc}y_{1}(1)&y_{1}(2)&\cdots&y_{1}(N)\\ y_{2}(1)&y_{2}(2)&\cdots&y_{2}(N)\\ \vdots&\vdots&&\vdots\\ y_{M}(1)&y_{M}(2)&\cdots&y_{M}(N)\end{array} \right], \tag{1}\]  
where \(y_{m}(t)\) is the time-varying EEG signal at the \(m\)th sensor, for \(m=1,2,\ldots,M\).