# Removing False Targets for Cyclic Prefixed OFDM Sensing with Extended Ranging +
## 3 Problem Statement
As proved in our previous work [32], PWP theoretically achieves higher signal-to-interference-plus-noise ratio (SINR) than PWD, given high to medium noise power; the threshold for determining the noise level is also derived therein. This makes PWP more desirable in practice. Hence, we mainly consider PWP here. Step 7 of Algorithm 1 performs a two-dimensional Fourier transform of the PWP result, leading to the so-called range-Doppler map (RDM). Here, the discrete Fourier transform (DFT) along the symbol domain turns a Doppler steering vector into a discrete Doppler spectrum. The inverse DFT (IDFT) over sub-carriers turns an amplitude-weighted range steering vector into a discrete range spectrum. As a result of PWP in Step 6, the amplitude weights are the squared amplitudes of communications signals over sub-carriers. If a constant-modulus constellation, such as phase shift keying (PSK), is used, the amplitude weights become all ones. The discrete range spectrum can be depicted using a discrete sinc function. In contrast, if signals carried by sub-carriers have nonconstant amplitudes, the amplitude weighting on the range steering vector can make the discrete range spectrum different from a sinc function in shape. However, the impact of the weighting is not deterministic, as signals on sub-carriers are generally random in actual communications. It is also this randomness that prevents the sidelobe level in the RDM being constructively  
Figure 2 plots the range and Doppler cuts of the RDMs obtained in Figure 1. Given the sample delay \(L=N+Q\), the range bin (in the sensing receiver, each OFDM symbol is uniformly sampled. Each of the sampling times represents a different range increment and is often referred to as a range bin [39]) index for USF is \(L\) (starting from one here to comply with MATLAB). For COS, since the overall number of samples along the range bin is \(N\), smaller than \(L\), a modulo-\(N\) of \(L\), i.e., \((L-N)\) here, is taken as the range bin index for COS. The Doppler bin (similar to the range bin, the DFT of OFDM symbols at the same range bin leads to the Doppler domain. Each discretisation grid in the Doppler domain is termed as a Doppler bin) index can be calculated as \(\lfloor\,\mu T\rceil+1\), where \(\lfloor\,\cdot\,\rceil\) rounds the enclosed term, \"\(+1\)\" is because the index starts from one, \(T=M(N+Q)T_{8}\) for COS, and \(T=\tilde{M}\tilde{N}T_{8}\). Here, \(\tilde{M}\) is the number of columns of \(\mathbf{Z}\) obtained in Step 1 of Algorithm 1. According to Definition 1, we have \(\tilde{M}=\left\lceil\frac{(N+Q)M}{N}\right\rceil\). From Figure 2, we see strong peaks in the range and Doppler cuts of the USF-RDM. In contrast, we see noise-like signals over the whole range and Doppler bins in the cuts of the COS-RDM. Given the high peak-to-sidelobe ratio at the true target location, the target can be readily detected employing common detecting algorithms, e.g., the constant false alarm rate (CFAR) detector [40]. However, from Figure 2, we also see two false targets around the true one along the range dimension. This problem was noticed in [32], yet is unsolved. Below, we first investigate the reasons for the false targets and then develop new methods to solve the problem. \begin{table}
\begin{tabular}{c c c} \hline
**Variable** & **Description** & **Value** \\ \hline \(f_{\text{c}}\) & Carrier frequency & 2.4 GHz \\ \hline \(B\) & Bandwidth & 3.84 MHz \\ \hline \(M\) & No. of OFDM symbols & 128 \\ \hline \(Q\) & CP length & 64 \\ \hline \(N\) & No. of sub-carriers & 256 \\ \hline \(L\) & Target sample delay & \(320(=N+Q)\) \\ \hline \(\mu\) & Doppler frequency & 480 Hz \\ \hline \(|a|\) & Signal amplitude in \(\mathbf{y}\); see (3) & 0 dB \\ \hline – & Noise power in \(\mathbf{y}\) & 30 dB \\ \hline \([\tilde{Q},\tilde{N},\tilde{Q}]\) & Parameters of USF & \([L,2\tilde{Q},0]\) \\ \hline \end{tabular}
\end{table} 
Table 1: Simulation P]parameters. Figure 1: Illustrating the RDMs of UFS (**a**) and COS (**b**).