## 4 New Method, SMI-LU
### The Common Part of Both Algorithms  
The previous section presented Formula (1), describing the echo of the received signal. Graphically, the received echo is represented by a radar data cube. To determine the vector of the weights given by Formula (16), it is necessary to find the clutter covariance matrix (12) using the SMI method. To do this, the SMI algorithm operates on the raw data by using a given cross-section of the radar cube at a fixed distance bin. Hypotheses about the presence or absence of a target in a given distance bin are then tested. For this purpose, a filter is created, characterized by high amplification of the useful signal from the target and, at the same time, high attenuation of all other signals (the clutter and jammer). More generally, STAP aims to filter out echoes coming from sources of interference and preserve the signal coming from the target of interest.  
Figure 4 shows the estimation of the clutter covariance matrix using neighboring distance bins (the SMI method). In the figure, the test bin is marked, and guard bins are marked on either side of it. The covariance matrix used to calculate the optimal filter should not contain target data. For this reason, the distance bin data, in which the target location is expected, is not used. Therefore, guard bins are used to prevent the use of the data contained in the distance bin under test. The set of distance bins around the target test bin is called training bins. Finally, after selecting the correct training bins, the covariance matrix is determined. The clutter covariance matrix is then inverted.  
Figure 3: STAP.  
The following is an algorithm for determining the set of training bins based on the known number of distance bins and the number of guard bins, the required number of training bins, and the known number of the distance bin under test. For this purpose, the following designations were adopted:  
* \(K\)--number of distance bins;
* \(K_{T}\)--number of training distance bins;
* \(K_{O}\)--number of guard bins;
* \(K_{TARGET}\)--distance bin under test;
* \(I_{KOP}\)--lower index of guard bins;
* \(I_{KOF}\)--upper index of guard bins;
* \(K_{\text{BEFORE}}\)--the remaining number of training bins before \(K_{\text{TARGET}}\);
* \(K_{\text{after}}\)--the remaining number of training bins behind \(K_{\text{TARGET}}\);
* \(K_{\text{before}}\)--number of training bins before \(K_{\text{TARGET}}\);
* \(K_{\text{after}}\)--number of training bins behind \(K_{\text{TARGET}}\);
* \(I_{\text{before}}\)--distance training bin indexes before \(K_{\text{TARGET}}\);
* \(I_{\text{after}}\)--distance training bin indexes behind \(K_{\text{TARGET}}\);  
Let the initial value [1; 2; 3] be:  
\[I_{KOP}=\ K_{TARGET}-\frac{K_{O}}{2} \tag{17}\]  
\[I_{KOZ}=\ K_{TARGET}+\frac{K_{O}}{2} \tag{18}\]  
\[K_{BEFORE}=\max(I_{KOP}-1,\ 0) \tag{19}\]  
\[K_{AFTER}=\max(K-I_{KOZ},\ 0) \tag{20}\]  
If \(K_{\text{BEFORE}}<K_{AFTER}\), let the initial value:  
\[K_{\text{Tbefore}}=\ \min(\frac{K_{T}}{2},K_{\text{BEFORE}}\ \bigg{)} \tag{21}\]  
\[K_{\text{Tafter}}=\ \min(K_{T}-K_{\text{Tbefore}},K_{AFTER}\ \bigg{)} \tag{22}\]  
Figure 4: Radar data cube—SMI training strategy.  
If \(K_{BEFORE}>K_{AFTER}\), let the initial value:  
\[K_{Tafter}=\min\biggl{(}\frac{K_{T}}{2},K_{AFTER}\ \biggr{)} \tag{23}\]  
\[K_{Tbefore}=\ \min\Bigl{(}K_{T}-K_{Tafter},K_{BEFORE}\ \biggr{)} \tag{24}\]  
Finally, the training data matrix should be completed considering the data from the radar data cube with range bins indexes equal to  
\[I_{Kbefore}=I_{KOP}+\Bigl{(}-K_{Tbefore}:-1\ \Bigr{)} \tag{25}\]  
\[I_{Kafter}=I_{KOZ}+\Bigl{(}1:K_{Tafter}\Bigr{)} \tag{26}\]