# Removing False Targets for Cyclic Prefixed OFDM Sensing with Extended Ranging +
## 5 Designing Methods to Remove False Targets
### Removing False Targets by Nullifying CP Signals  
From Section 4, we see that the false targets are caused by \(\mathbf{w}\) and \(\left\langle\mathbf{w}\right\rangle_{N}\) in \(\mathbf{e}\). Recall that \(\mathbf{e}\) is the first column of \(\tilde{\mathbf{X}}\), as obtained in Step 2 of Algorithm 1. Similar to \(\mathbf{e}\), other columns of \(\tilde{\mathbf{X}}\) can also be decomposed, as performed in (9), and there will be \(\mathbf{w}\)-like vectors leading to false targets. From (9), we see that the nonzero entries in \(\mathbf{w}\) are actually CPs of OFDM signals. Therefore, a simple way of preventing false targets from being generated is to remove CPs, such as \(\mathbf{w}\), and the signals used for generating CPs, such as \(\left(\mathbf{w}\right)_{N}\), in \(\tilde{\mathbf{X}}\). The feasibility of modifying \(\tilde{\mathbf{X}}\) is ensured by the fact that \(\tilde{\mathbf{X}}\) is constructed at the sensing receiver using the copy of communication-transmitted signals in the time domain. In other words, whatever changes made to \(\tilde{\mathbf{X}}\) will not affect communications. Due to the reshaping performed in Step 2 of Algorithm 1, CPs will fall in different locations over the columns of \(\tilde{\mathbf{X}}\). However, they are traceable by referring to the indexes of the entries in \(\mathbf{x}\) that form a column of \(\tilde{\mathbf{X}}\). Specifically, according to Definition 1, we know that the \(m\)-th column of \(\tilde{\mathbf{X}}\) consists of entries in \(\mathbf{x}\) with indexes given by  
\[\mathcal{I}_{m}=\left\{(m-1)(\bar{N}-\bar{Q})+\left[1,2,\cdots,\bar{N}\right] \right\},\ m=1,\cdots,\left\lfloor\frac{I-\bar{Q}}{\bar{N}-\bar{Q}}\right\rfloor, \tag{14}\]  
where \(I(=M(N+Q))\) is the total number of samples in \(\mathbf{x}\). The index of original CPs in \(\mathbf{x}\) can be given by  
\[\mathcal{I}_{\text{CP}}=\{\mathcal{I}_{\text{CP}}^{m},\ m=0,1,\cdots,M-1\}, \tag{15}\]  
where \(\mathcal{I}_{\text{CP}}^{m}\) is the set of CP indexes in the \(m\)-th OFDM symbol. It can be given by  
\[\mathcal{I}_{\text{CP}}^{m}=\{(1,2,\cdots,Q,N+1,\cdots,N+Q)+m(N+Q)\}. \tag{16}\]  
Note that not only CPs but also the samples at the tail of each OFDM symbol that are used as CPs contribute to the presence of false targets. To nullify signals in \(\tilde{\mathbf{X}}\) that are related to CPs in the original OFDM signals, we introduce an indicating matrix \(\mathbf{W}\) of the same size as \(\tilde{\mathbf{X}}\) and initiate it with all ones. Then, for \(m=0,1,\cdots,\left\lfloor\frac{I-\bar{Q}}{\bar{N}-\bar{Q}}\right\rfloor\), we identify the entries in \(\mathcal{I}_{m}\) that are also in \(\mathcal{I}_{\text{CP}}\). At the indexes of those entries, we set the entries of \(\mathbf{W}\) in the \(m\)-th column as zeros. After enumerating all values of \(m\), we set \(\tilde{\mathbf{X}}\) in Algorithm 1 by \(\tilde{\mathbf{X}}\odot\mathbf{W}\) before performing Steps 3-7 to prevent false targets from being generated. The above illustration is summarised in Algorithm 3. ```
1. Run Steps 1-2 of Algorithm 1, obtaining \(\mathbf{Z}\) and \(\tilde{\mathbf{X}}\);
2. Set \(\mathbf{W}=\mathbf{1}_{\bar{N}\times\bar{M}}\), with the same dimension as \(\tilde{\mathbf{X}}\);
3. For \(m=1,\cdots,\left\lfloor\frac{I-\bar{Q}}{\bar{N}-\bar{Q}}\right\rfloor\) (a) Find entries in \(\mathcal{I}_{m}\), as given in (14), that also exist in \(\mathcal{I}_{\text{CP}}\), as given in (15), and collect their indexes in \(\mathcal{I}_{i}\); (b) Set \([\mathbf{W}]_{\mathcal{I},m}=0\), where \([\mathbf{W}]_{\mathcal{I},m}\) denotes the \(m\)-th column of \(\mathbf{W}\) with rows indexed by entries in \(\mathcal{I}_{i}\);
4. Set \(\tilde{\mathbf{X}}=\tilde{\mathbf{X}}\odot\mathbf{W}\);
5. Run Steps 3-7 of Algorithm 1, yielding an RDM without false targets;
```  
**Algorithm 3** Modified USF (MUSF). Nullifying CPs prevents the false targets from being generated. However, it will also reduce the processing gain on the true target. This can be seen from (6).