# Removing False Targets for Cyclic Prefixed OFDM Sensing with Extended Ranging +
## 4 Investigating Reasons for False Targets
### A Special Case for Looking into False Targets  
From Figure 1a, we see that the false targets exist over the range dimension solely. This indicates that they are generated by some column operations in Algorithm 1. Let us consider a special case with \(L=N\), \(\bar{Q}=N+Q\), \(\bar{N}=2\bar{Q}\), \(\bar{Q}=0\), \(\alpha=1\), and \(\mu=0\). From (3), we see that \(\mu=0\) leads to \(\mathbf{d}=\mathbf{1}\). Based on these settings, the first column of \(\mathbf{\hat{X}}\), as obtained in Step 2 of Algorithm 1, can be given by  
\[\mathbf{e}=[\mathbf{v}_{0};\mathbf{u}_{0};\mathbf{v}_{0};\mathbf{v}_{1}; \mathbf{u}_{1};\mathbf{v}_{1}], \tag{4}\]  
where \"\(;\)\" concatenate column vectors. As given in (1), the dimensions of the \(u\)-vector and \(v\)-vector are \((N-Q)\) and \(Q\), respectively. For ease of illustration, the noise term will be ignored in this section, as the false target issue exists regardless of noises. Given \(L=N\) and the expression of \(\mathbf{y}\) in (3), the first column of \(\mathbf{Z}\) obtained in Step 1 of Algorithm 1 can be given by  
\[[\mathbf{0}_{N\times 1};\mathbf{v}_{0};\mathbf{u}_{0};\mathbf{v}_{0};\mathbf{v}_{ 1};\mathbf{u}_{1};\mathbf{v}_{1};\mathbf{v}_{2}],\]  
Figure 2: Doppler (**a**) and range (**b**) cuts of the RDM obtained by COS, as plotted in Figure 1b. Doppler (**c**) and range (**d**) cuts of the RDM obtained by USF, as plotted in Figure 1a. Please note that the symbol “\(\cdot\)” before all numbers, as automatically generated by MATLAB during plotting the figure, denotes the negative sign. where \(\mathbf{0}_{N\times 1}\) denotes an \(N\times 1\) zero vector and the zero Doppler is assumed. The above vector, going through Step 3 of Algorithm 1, becomes  
\[\mathbf{f}=\underbrace{[\mathbf{u}_{1};\mathbf{v}_{1};\mathbf{v}_{0};\mathbf{u} _{0};\mathbf{v}_{0};\mathbf{v}_{1}]}_{\langle\mathbf{e}\rangle_{N}}+\left[ \mathbf{0}_{N\times 1};\mathbf{v}_{2};\mathbf{0}_{(N+Q)\times 1}\right], \tag{5}\]  
where \(\langle\mathbf{e}\rangle_{N}\) indicates that the vector \(\mathbf{e}\) is circularly shifted by \(N\) entries. Note that the second vector on the right-hand side of (5) is an interference term. It is a price paid by USF to relieve the constraint \(L\leq Q\), as suffered by COS, given in Algorithm 2. The calculations in Steps 5-7 of Algorithm 1 make the first column of \(\tilde{\mathbf{Z}}\) (the RDM) become the cyclic cross-correlation (CCC) between \(\mathbf{e}\) and \(\mathbf{f}\). Let \(\mathbf{g}\) denote the first column of \(\tilde{\mathbf{Z}}\). Its \(i\)-th entry satisfies  
\[[\mathbf{g}]_{i}=(\langle\mathbf{e}\rangle_{i})^{\mathrm{H}}\mathbf{f}\approx( \langle\mathbf{e}\rangle_{i})^{\mathrm{H}}\langle\mathbf{e}\rangle_{N}. \tag{6}\]  
**Remark 1**: _We remark on the features of \([\mathbf{v}_{m}]\)\((\forall m)\), as given in (1). In particular, the entries of \([\overline{\mathbf{v}}_{m}]\) approximately converge in distribution to a complex Gaussian random process and it is independent over \(m\), provided that the communication symbols carried by sub-carriers are randomly and uniformly drawn from some constellations. The proof can be readily established based on Remark 2 in [41] and is hence suppressed here for brevity. Applying the above feature, we have that \(\mathbf{v}_{2}\) in \(\mathbf{f}\) is independent from any consecutive \(Q\) entries in \(\mathbf{e}\). This leads to approximation in (6)._  
From (4) and (5), we see that \([\mathbf{g}]_{i}\) is maximised at \(i=N\), leading to a peak at the true sample delay of the target.