# Low-Cost SDR for GNSS Interference Mitigation Using Spatial Diversity Techniques +
## 3 Exploiting Spatial Diversity for Interference Mitigation
### Prewhitening Beamformer  
The fact that GNSS signals are buried below the noise floor when they first arrive at the receiver can be exploited to obtain an interference free signal at the output of the beamformer. This approach, known as Prewhitening beamformer (PREW), is the one followed in the technique presented in [17], where the autocorrelation matrix is decomposed and manipulated to obtain an interference-free subspace in which to project the incoming samples. Due to the uncorrelation between the interference and the LOSS, the projection of the samples contained in \(\mathbf{x}[n]\) onto this interference free subspace results in output samples where the undesired contributions have been suppressed. The essence of this algorithm lies, therefore, in finding a projection matrix that allows the receiver to eradicate the interference. To achieve this, the first step involves decomposing the autocorrelation matrix into its eigenvectors and eigenvalues, considering the signal model defined in Equation (3), where the presence of the LOSS can be omitted since it is imperceptible. This simplified but effective version of the decomposition is depicted in Equation (7). \[\mathbf{R}_{\mathbf{x}}\approx\mathbf{U}\mathbf{\Lambda}\mathbf{U}^{H}=\mathbf{U}_{i}\Big{(}\mathbf{ \Lambda}_{i}+\sigma_{n}^{2}\mathbf{I}\Big{)}\,\mathbf{U}_{i}^{H}+\sigma_{n}^{2}\mathbf{U} _{n}\mathbf{U}_{n}^{H} \tag{7}\]  
From the previous expression, \(\mathbf{U}_{i}\in\mathbb{C}^{L\times I}\) is a matrix containing the eigenvectors of the interference subspace that are associated with the eigenvalues \(\lambda_{i}\) enclosed in the diagonal matrix \(\mathbf{\Lambda}_{i}\). Similarly, \(\mathbf{U}_{n}\in\mathbb{C}^{L\times(L-I)}\) holds the eigenvectors that span the noise subspace, and \(\sigma_{n}^{2}\) represents the noise power. With the expression in Equation (7), a whitening matrix \(\mathbf{W}\) can be found through the Mahalanobis whitening procedure. Applying said matrix to the incoming signal allows the decorrelation of the contributions present in \(\mathbf{x}[n]\), ensuring the orthogonality between the interference and the LOSS. Therefore, the resulting whitening matrix \(\mathbf{W}\) has the form presented in Equation (8). \[\mathbf{W}=\mathbf{R}_{\mathbf{x}}^{-\frac{1}{2}}=\mathbf{U}_{t}\Big{(}\mathbf{\Lambda}_{i}+\sigma_{ n}^{2}\mathbf{I}\Big{)}^{-\frac{1}{2}}\mathbf{U}_{i}^{H}+\frac{1}{\sqrt{\sigma_{n}^{2}}} \mathbf{U}_{n}\mathbf{U}_{n}^{H} \tag{8}\]  
It can be fairly presumed that the power of the interference will be much higher than that of the noise, so the eigenvalues of the interference will be larger than the noise power, i.e., \(\lambda_{i}\gg\sigma_{n}^{2}\); thus, the whitening matrix can be approximated as stated in Equation (9). \[\mathbf{W}=\mathbf{R}_{\mathbf{x}}^{-\frac{1}{2}}\approx\frac{1}{\sqrt{\sigma_{n}^{2}}}\bm {U}_{n}\mathbf{U}_{n}^{H} \tag{9}\]  
Note now that the resulting whitening matrix \(\mathbf{W}\approx\frac{1}{\sigma_{n}}\mathbf{U}_{n}\mathbf{U}_{n}^{H}\) is the projection matrix onto the interference free subspace, and so when applied to \(\mathbf{x}[n]\), as depicted in Equation (10), the result will be samples free of the undesired contributions, denoted by \(\widetilde{\mathbf{x}}[n]\). \[\widetilde{\mathbf{x}}[n]=\mathbf{W}\mathbf{x}[n] \tag{10}\]  
Lastly, once the received signal is clean from interference, the weights can be easily selected as the eigenvector associated with the largest eigenvalue of the new autocorrelation matrix obtained from the modified samples, i.e., \(\widetilde{\mathbf{R}}_{\mathbf{x}}=\frac{1}{N}\mathbb{I}_{n=1}^{N}\widetilde{\mathbf{x}} [n]\widetilde{\mathbf{x}}^{H}[n]\), obtaining the final coefficients as depicted in Equation (11), where \(\mathcal{P}\{\cdot\}\) represents the operator that retrieves the eigenvector associated with the largest eigenvalue.