# Removing False Targets for Cyclic Prefixed OFDM Sensing with Extended Ranging +
## 5 Designing Methods to Remove False Targets
### Sequential Target Detection and Removal  
Unlike the previous method preventing false targets from appearing, this method handles the false targets using the properties unveiled in Proposition 1. In particular, as stated therein, false targets actually have known locations relative to that of the true target. Moreover, the relation between the strengths of true and a false target is also known. Utilising these features, we develop a sequential target detection and removal method, based on the RDM \(\hat{\mathbf{Z}}\) obtained in Step 7 of Algorithm 1. We start with searching for the strongest peak in \(\hat{\mathbf{Z}}\):  
\[(n_{t},m_{t}):\ \arg\max_{n,m}\bigl{|}|\hat{\mathbf{Z}}|_{n,m}\bigr{|},\ n=1, \cdots,\tilde{N},\ m=1,\cdots,\tilde{M}, \tag{17}\]  
where \((n_{t},m_{t})\) is the two-dimensional index in the range-Doppler-domain, and the subscript \(t\) denotes the target index. Treating the peak at \((n_{t},m_{t})\) as a true target, we can deduce the features of the accompanying false targets. According to Proposition 1, the locations of false targets are  
\[\bigl{(}n_{t}^{\pm},m_{t}\bigr{)}=((n_{t}\pm N-1)_{\tilde{N}}+1,m_{t}), \tag{18}\]  
where '\(-1\)' and '\(1\)' are because the index starts from one here. Moreover, the amplitudes at the local peaks of the two false targets are the scaled version of the peak amplitude at the true target, specifically,  
\[\mathsf{P}_{\bigl{(}n_{t}^{\pm},m_{t}\bigr{)}}=\mathsf{P}_{t}k,\ \ \text{s.t.}\ k=\biggl{\lfloor}\frac{\tilde{N}}{N+Q}\biggr{\rfloor}\frac{Q}{ \tilde{N}},\ \mathsf{P}_{t}=[\dot{\mathbf{Z}}]_{n_{t},m_{t}}, \tag{19}\]  
where \(\mathsf{P}_{t}\) is the amplitude of the true target. Note that \(\Bigl{\lfloor}\frac{\tilde{N}}{N+Q}\Bigr{\rfloor}\frac{Q}{\tilde{N}}\) is the ratio depicting the amplitude relation between true and false targets, as derived in Proposition 1. If the target range and Doppler frequencies are integer multiples of their respective resolutions, we can reconstruct an RDM related to the strongest target and remove it from the \(\hat{\mathbf{Z}}\). However, it is likely that the target range and Doppler frequencies are noninteger multiples of their respective resolutions. To account for this case, it can be better to also include the neighbour bins surrounding the peaks of true and false targets when reconstructing the RDM for target removal. Let \(N_{\text{r}}\) (\(N_{\text{d}}\)) denote the number of the single-side range (Doppler) bins to be included for target reconstruction. Then, the set of indexes for the range-Doppler bins, where the true target is located, can be given by  
\[\mathcal{I}_{t}=\{(n_{t}+i,m_{t}+j),\ i=-N_{\text{r}},\cdots,0,\cdots,N_{\text {r}},\ j=-N_{\text{d}},\cdots,0,\cdots,N_{\text{d}}\}. \tag{20}\]  
The set of indexes for the range-Doppler bins, where the false targets are located, can be given by  
\[\mathcal{I}_{t}^{\pm}=\{((n_{t}\pm N-1)_{\tilde{N}}+1+i,m_{t}+j),\ i=-N_{\text {r}},\cdots,0,N_{\text{r}},\ j=-N_{\text{d}},\cdots,0,N_{\text{d}}\}. \tag{21}\]  
The RDM of the target can be reconstructed as  
\[[\mathbf{Z}_{t}]_{n,m}=\left\{ \begin{array}{ll}[\tilde{\mathbf{Z}}]_{n,m},& \text{if }(n,m)\in\{\mathcal{I}_{t}\}\\ [\tilde{\mathbf{Z}}]_{n^{\prime},m^{\prime}}k,&\text{if }(n,m)\in\{\mathcal{I}_{t}^{\pm} \}\\ 0,&\text{otherwise}\end{array} \right., \tag{22}\]  
where \((n,m)=((n^{\prime}\pm N-1)_{\tilde{N}}+1,m^{\prime})\) for the second case, and \(k\) is given in (19). Note that the amplitudes for the false targets are estimated based on the their properties derived in Proposition 1. To suppress the impact of the false targets associated with the strongest target on the detection of other targets, we can update \(\tilde{\mathbf{Z}}\) as  
\[\tilde{\mathbf{Z}}=\tilde{\mathbf{Z}}-\mathbf{Z}_{t}. \tag{23}\]  
Then, we detect the next strongest target, reconstruct it, and remove it, as illustrated above. The sequential target detection and removal method is summarised in Algorithm 4. Input: the number of targets \(T\), \(N_{r}\), \(N_{\text{d}}\)  
1.