# Removing False Targets for Cyclic Prefixed OFDM Sensing with Extended Ranging +
## 4 Investigating Reasons for False Targets
### Analysing False Targets and Their General Features
Note that \(\epsilon_{L+N}\) is a residual term, as given by  
\[\epsilon_{L+N}=\langle\mathbf{w}\rangle_{L+2N}^{\mathrm{H}}\langle\mathbf{w} \rangle_{L}+\langle\mathbf{w}\rangle_{L+2N}^{\mathrm{H}}\langle\mathbf{r} \rangle_{L}+\langle\mathbf{r}\rangle_{L+N}^{\mathrm{H}}\langle\mathbf{r} \rangle_{L}.\]  
Similar to \([\mathbf{g}]_{L+N}\), we can obtain  
\[[\mathbf{g}]_{L-N}=\langle\mathbf{w}\rangle_{L}^{\mathrm{H}}\langle\mathbf{w} \rangle_{L}+\epsilon_{L-N}=\left\lfloor\frac{\tilde{N}}{N+Q}\right\rfloor Q+ \epsilon_{L-N}, \tag{13}\]  
where \(\epsilon_{L-N}\) is a residual term similar to \(\epsilon_{L+N}\). Although \(\epsilon_{L-N}\) and \(\epsilon_{L+N}\) cannot be analytically expressed, we can safely assume that their values are small, due to their noise-like and (partially) independent entries. Similar to (12) and (13), one can easily calculate \([\mathbf{g}]_{i}\) at other \(i\)s. Given the noise-like signals in \(\mathbf{e}\) (see Remark 1) and Lemmas 1 and 2, we can further validate that have the largest amplitudes, only next to the global peak \([\mathbf{g}]_{L}\), among all CCC results. We summarise the above analysis formally into the following proposition. Proposition 1: _Given \(\tilde{N}\geq(N+Q)\), USF can present two strong false targets accompanying a true target in the range domain. They locate at \([(L+N-1)_{\tilde{N}}+1]\)-th and the \([(L-N-1)_{\tilde{N}}+1]\)-th range bins, where \(L\) is the true target delay, and \"\(+1\)\" and \"\(-1\)\" are because the starting index is one. The magnitudes of the two false targets are approximately identical. The value approximates \(\left\lfloor\frac{\tilde{N}}{N+Q}\right\rfloor\frac{Q}{N}\), when normalised by the peak magnitude of the true target._  
Proof: The analysis in Section 4.2 proves the results in the proposition for the case with \(\alpha=1\) and unit-amplitude communication signals over sub-carriers. Here, we mainly show that the proposition holds without the two assumptions. From (12) and (13), we see that the locations of the false targets mainly depend on the locations of nonzero entries in \(\mathbf{w}\). Thus, we only need to confirm the amplitude relation between the false targets and the true one. There can be two cases when the communication signals over OFDM sub-carriers have nonconstant amplitudes. First, constellations such as quadrature amplitude modulation (QAM) are used. Second, the single-carrier OFDM is performed. It means that the signals over sub-carriers are from the discrete Fourier transform of signals drawn from some constellation. In either case, the mean of the communication signals over sub-carriers is zero. Thus, the squared sum of the signals, normalised by the number of samples, approximates the variance of some statistic. This further leads to \([\mathbf{g}]_{L}\approx\alpha\tilde{N}\sigma^{2}\), where \(\sigma^{2}\) denotes the variance of the signals over sub-carriers. Similarly, based on (12) and (13), we have \([\mathbf{g}]_{L\pm N}\approx\alpha\left\lfloor\frac{\tilde{N}}{N+Q}\right\rfloor Q \sigma^{2}\), where the \(\epsilon\)-terms are suppressed here. Therefore, the ratio between \([\mathbf{g}]_{L\pm N}\) and \([\mathbf{g}]_{L}\) is \(\left\lfloor\frac{\tilde{N}}{N+Q}\right\rfloor\frac{Q}{N}\), as stated in Proposition 1.