# Removing False Targets for Cyclic Prefixed OFDM Sensing with Extended Ranging +
## 4 Investigating Reasons for False Targets
### Analysing False Targets and Their General Features  
In light of (4), we propose to decompose \(\mathbf{e}\) as follows,  
\[\mathbf{e}=\mathbf{w}+\langle\mathbf{w}\rangle_{N}+\mathbf{r},\text{ s.t. }\mathbf{w}=\bigg{[}\mathbf{v}_{0};\mathbf{0}_{N\times 1}; \cdots;\mathbf{v}_{\left\lfloor\frac{N}{N+Q}\right\rfloor-1}\mathbf{:0}_{N \times 1};\mathbf{0}_{\left(\tilde{N}\right)_{N+Q}\times 1}\bigg{]}, \tag{9}\]where all the \(v\)-vectors have the dimension of \(Q\), \(\tilde{N}\) is the length of each segmentation at the sensing receiver (see Algorithm 1), and \(\big{(}\tilde{N}\big{)}_{N+Q}\) denotes the remainder of \(\tilde{N}\) divided by \((N+Q)\). Note that the first integer multiples of \((N+Q)\) samples of \(\mathbf{w}\) have a fixed pattern: only the first \(Q\) samples of each OFDM symbol (with CP) are kept. The following properties of the generalised \(\mathbf{e}\) will be useful for further analysis. Note that Lemma 1 can be proved using the definition of \(\mathbf{e}\) given in (4). Lemma 2 can be confirmed based on the decomposition defined in (9). **Lemma 1**. : _The decomposition of \(\mathbf{e}\) given in (9) is always valid, provided \(\tilde{N}\geq N+Q\)._  
**Lemma 2**. : _The three components \(\mathbf{w},\langle\mathbf{w}\rangle_{N}\), and \(\mathbf{r}\) have mutually exclusive positions for nonzero entries._  
Based on (9), the vector \(\mathbf{f}\), as obtained in Step 3 of Algorithm 1, can be written as  
\[\mathbf{f}\approx\langle\mathbf{e}\rangle_{L}=\langle\mathbf{w}\rangle_{L}+ \langle\mathbf{w}\rangle_{N+L}+\langle\mathbf{r}\rangle_{L}, \tag{10}\]  
where \(L\) denotes the target delay and \(\mu=0\) is continued to be assumed. As seen in Figure 1, false targets only appear in the range domain, so the Doppler information is irrelevant in the following analysis. Moreover, for illustration convenience, we continue assuming that \(\alpha=1\) and unit-amplitude signals are carried by sub-carriers. Furthermore, the approximation in (10) is made because the interference term from adding the virtual CP has negligible impacts on CCC. This can be inferred from (5) and (6). As illustrated in (6), the range cut of the RDM (note that RDM is a two-dimensional matrix. The row dimension spans the range domain and the column dimension spans the Doppler domain. When we take one row out of an RDM, we obtain a so-called range cut or profile [39], as if we cut a slice of the RDM at a specific range bin. Similarly, when we take one column, we obtain a so-called Doppler cut or profile. It depicts the sensing results over range bins at a certain Doppler bin) obtained by Algorithm 1 is the CCC between \(\mathbf{e}\) and \(\mathbf{f}\). Thus, the \(i\)-th sample of the range cut can be written as  
\[[\mathbf{g}]_{i}\approx(\langle\mathbf{e}\rangle_{i})^{\mathrm{H}}\langle \mathbf{e}\rangle_{L}, \tag{11}\]  
where the result in (10) has been used. As expected, \([\mathbf{g}]_{i}\) is maximised at \(i=L\), where the signals involved in the CCC are fully overlapped. Moreover, using (9) and (10), we can validate that  
\[ \begin{split}&[\mathbf{g}]_{L+N}=\langle\mathbf{e}\rangle_{L+N}^{ \mathrm{H}}\mathbf{f}\\ &=\langle\mathbf{w}\rangle_{L+N}^{\mathrm{H}}\langle\mathbf{w} \rangle_{L+N}^{\mathrm{H}}\langle\mathbf{w}\rangle_{N+L}+\langle\mathbf{w} \rangle_{L+N}^{\mathrm{H}}\langle\mathbf{r}\rangle_{L+N}^{\mathrm{H}}\langle \mathbf{r}\rangle_{L+N}^{\mathrm{H}}\langle\mathbf{w}\rangle_{L+2N}\langle \mathbf{w}\rangle_{N+L}+\langle\mathbf{w}\rangle_{L+2N}^{\mathrm{H}}\langle \mathbf{r}\rangle_{L}\\ &=\langle\mathbf{w}\rangle_{L+N}^{\mathrm{H}}\langle\mathbf{w} \rangle_{N+L}+\epsilon_{L+N}=\left\lfloor\frac{\tilde{N}}{N+Q}\right\rfloor Q +\epsilon_{L+N},\end{split}  \tag{12}\]  
where the other terms are zero based on Lemma 2.