# Removing False Targets for Cyclic Prefixed OFDM Sensing with Extended Ranging +
## 3 Problem Statement  
In this section, we briefly review USF [32] and then illustrate the false target issue. Unlike in the original work using the scalar form for developing the algorithm, we introduce a more concise matrix form here. The new matrix form will also benefit the analysis to be performed in Section 4. To start with, we define the following operator, reshaping a vector into a matrix. Definition 1: The operator \(\underset{x,y}{\text{mat}}\{\mathbf{c}\}\) reshapes the vector \(\mathbf{c}\) into a matrix of size \(x\times\left\lfloor\frac{\mathbf{c}-y}{x-y}\right\rfloor\), where each column has \(x\) number of entries and starts from the \((i(x-y)+1)\)-th \((i=0,1,\cdots,\left\lfloor\frac{\mathbf{c}-y}{x-y}\right\rfloor-1)\) entry in \(\mathbf{c}\). Here, \(\left\lfloor\cdot\right\rfloor\) takes flooring, \(C\) denotes the dimension of \(\mathbf{c}\), \(x\) is a positive integer (PI), and \(y\) can be a PI or zero. Note that, if \(y\) is nonzero, adjacent columns of the resulted matrix have overlapping entries. To help understand the operator, let us look at an example. Given \(\mathbf{c}=[1,2,3,\cdots,9]^{\mathsf{T}}\)\(\underset{4,1}{\text{\rm mat}}\{\text{\rm c}\}\) and \(\underset{4,2}{\text{\rm mat}}\{\text{\rm c}\}\), respectively, result in  
\[ \begin{bmatrix}1&4\\ 2&5\\ 3&6\\ 4&7\end{bmatrix} \text{ and } \begin{bmatrix}1&3&5\\ 2&4&6\\ 3&5&7\\ 4&6&8\end{bmatrix} \]  
Based on the signal model given in Section 2, USF [32] is summarised in Algorithm 1. As performed in Step 1, the echo signal vector \(\mathbf{y}\) is reshaped into a matrix by \(\underset{\tilde{N}+\tilde{Q},\tilde{Q}+\tilde{Q}}{\text{\rm mat}}\{\mathbf{ y}\}\), where each symbol has \((\tilde{N}+\tilde{Q})\) samples and adjacent symbols are overlapped by \((\tilde{Q}+\tilde{Q})\) samples. Note that \((\tilde{N}+\tilde{Q})\) here can be different from \((N+\tilde{Q})\), the length of the original OFDM symbol (with CP). In Step 2, the communication-transmitted signal, \(\mathbf{x}\) given in (2), is reshaped into a matrix by \(\underset{\tilde{N},\tilde{Q}}{\text{\rm mat}}\{\mathbf{x}\}\). We refer interested readers to Figure 5 in [12] for an intuitive illustration of the reshaping performed above. As a result of Steps 1 and 2, each column of \(\mathbf{Z}\), as obtained in Step 3, contains a cyclically shifted version of the corresponding column in \(\tilde{\mathbf{X}}\), as obtained in Step 2. The shifting amount is linked with the target delay. Therefore, each column of \(\tilde{\mathbf{Z}}\), as obtained in Step 4, is the point-wise product (PWP) between the same column of \(\tilde{\mathbf{X}}\), as obtained in Step 5, and the range steering vector of the target. ```
1. Reshape \(\mathbf{y}\) by \(\underset{\tilde{N}+\tilde{Q}+\tilde{Q}}{\text{\rm mat}}\{\mathbf{y}\}\), leading to \(\mathbf{Z}\);
2. Reshape \(\mathbf{x}\) by \(\underset{\tilde{N},\tilde{Q}}{\text{\rm mat}}\{\mathbf{x}\}\), leading to \(\tilde{\mathbf{X}}\);
3. Add the last \(\tilde{Q}\) rows of \(\mathbf{Z}\) onto its first ones and remove the last \(\tilde{Q}\) rows, leading to \(\tilde{\mathbf{Z}}\);
4. Take the DFT of the columns of \(\tilde{\mathbf{Z}}\), giving \(\tilde{\mathbf{Z}}\);
5. Take the DFT of the columns of \(\tilde{\mathbf{X}}\), giving \(\tilde{\mathbf{X}}\);
6. Calculate the point-wise product (PWP) between \(\tilde{\mathbf{Z}}\) and \(\tilde{\mathbf{X}}^{*}\), yielding \(\hat{\mathbf{Z}}\), where \((\cdot)^{*}\) denotes conjugate;
7. Take the IDFT of the columns of \(\hat{\mathbf{Z}}\) and then the DFT of the rows, yielding an RDM \(\hat{\mathbf{Z}}\). ```  
**Algorithm 1** Unified sensing framework (USF) [32]. A point-wise division (PWD) between \(\hat{\mathbf{Z}}\) and \(\tilde{\mathbf{X}}\) can remove the communications information, making the extraction of sensing target parameters independent of communications information. However, when the entries of \(\tilde{\mathbf{X}}\) approach zero, the division can lead to the noise enhancement issue. To relieve the issue, it was proposed in [31] to replace PWD with PWP.