## 3 Interference Suppression through Beamforming Techniques
### Adaptive Beamforming (ABF)  
The second proposal to improve targets' detectability by decreasing the power of interference sources is adaptive beamforming. From the multiple techniques proposed in the literature for ABF, for this application, Sample Matrix Inversion (SMI) in combination with a Minimum Variance Distortionless (MVDR) optimum beamformer is used [28]. SMI means that the matrix containing the information about the interference for each channel--that is, the covariance matrix--is estimated from sample training data and must be inverted. MVDR is adequate in this case because post-Doppler covariance matrix estimation is performed. MVDR will optimise the weights needed to minimise the power of the sources contained in the sample data. Because of this, a good selection of RD cells for estimation will allow the interferences power to be minimised without cancelling any targets if they are not included in the sample data. The optimum MVDR beamformer weights are calculated from Equation (5) where \(v_{s}\) states for the steering vector and \(\hat{S}_{n}\) is the estimated interference covariance matrix [30]. \[w^{H}_{mvdr,semi}=\frac{v_{s}^{H}\hat{S}_{n}^{-1}}{v_{s}^{H}\hat{S}_{n}^{-1}v _{s}} \tag{5}\]  
The key point in this technique is the selection of training data from RD maps of all the channels to estimate the covariance matrix. The selected data must contain just interference plus noise. Targets must be excluded from training data to avoid their rejection. Two methods are proposed:  
* The first one takes advantage of the fact that the interference sources expand from zero Doppler and employs the cells surrounding zero Doppler for all ranges, excluding the nearest ones. This is because near ranges contain high-level peaks due to nearby clutter that comes from almost all directions. This proposal has been evaluated by selecting the Doppler rows of \(\pm 2\) Hz and \(\pm 4\) Hz. This proposal can include low-velocity targets, or some that, due to geometry, have a low Doppler shift, and therefore cancel it. * The second proposal tries to employ all the Doppler information of the interferences, so the range of the main interferences is calculated and the columns corresponding to these ranges, and the neighbouring ones, for all Doppler shifts are selected for covariance matrix estimation. This proposal has the advantage that if a target appears in the same range as one form of interference, it will be only cancelled in one or a few CPIs, as it will change range due to movement. The SNR loss due to covariance matrix estimation, instead of perfect knowledge of interferences, depends on the number of data samples employed for estimation. In the first proposal, the number of samples employed is equal to \(N_{s}=9840\); that is, 4 Doppler cells \(\times\) (\(2500-40\)) range cells. In the second proposal, \(N_{s}\) depends on the number of dominant interferences found in each CPI RD map. The total number of samples is \(N_{s}=N_{I}\times 500\), with \(N_{I}\) interferers or peaks selected (i.e., one range cell is selected for each interferer), and 500 is the number of Doppler cells of an RD map. The number of range cells employed is, in most cases, between 15 and 40 so, in the worst case scenario (minimum value) the number of samples is \(N_{s}=7500\). In [28] it is proved that for \(N_{s}=1000\), the SINR loss converges to 0 dB, so both proposals have an adequate number of samples needed to achieve a valid covariance matrix estimation. Both proposals of MVDR ABF achieve a similar level of interference suppression to Deterministic Null (DN) generation (Figure 12): most interference sources are suppressed except those in range bins 350 and 600. In the ABF case, the background level is highly decreased, and non-suppressed interferences seem to be stronger. This phenomenon is even more important in the second proposal for \(\hat{S}_{n}\) estimation. Quite the opposite of the DN case; in ABF cases, the targets' level is not decreased, as it can be checked in the RD map of cumulative maxima (Figure 13).