# Removing False Targets for Cyclic Prefixed OFDM Sensing with Extended Ranging +
## 2 Signal Model  
This section presents the signal model of OFDM-ISAC. Consider that a communication node is turned into an ISAC node by adding a sensing receiver. Sensing is performed using the echo signals resulted from the communication-transmitted signals. Being co-located, the communication transmitter and sensing receiver are assumed to be fully synchronised. It is also reasonable to assume that the sensing receiver has a copy to the time-domaincommunication-transmitted signals. It is further assumed that some proper full-duplex techniques are applied such that self-interference, i.e., the signal leaking from the transmitter to receiver, can be ignored. As for waveform, we employ OFDM with regular CP (i.e., one per symbol), which is widely used in modern communication systems.  
We consider \(M\) consecutive OFDM symbols, each having \(N\) sub-carriers. Let \(Q\) denote the CP length. After CP is added, the time-domain signal matrix can be given by [34]  
\[\hat{\mathbf{X}}=\begin{bmatrix}\mathbf{V}\\ \mathbf{X}\end{bmatrix},\text{ s.t. }\mathbf{X}=\begin{bmatrix}\mathbf{u}_{0}& \mathbf{u}_{1}&\cdots&\mathbf{u}_{M-1}\\ \mathbf{v}_{0}&\mathbf{v}_{1}&\cdots&\mathbf{v}_{M-1}\end{bmatrix}_{N\times M} \tag{1}\] \[\mathbf{V}=\begin{bmatrix}\mathbf{v}_{0}&\mathbf{v}_{1}&\cdots& \mathbf{v}_{M-1}\end{bmatrix}_{Q\times M}\]  
where each column of \(\hat{\mathbf{X}}\) represents an OFDM symbol, \(\mathbf{v}_{m}\) (\(\forall m\)) denotes CP, and \(\begin{bmatrix}\mathbf{u}_{m}\\ \mathbf{v}_{m}\end{bmatrix}\) (\(\forall m\)) is often referred to as the essential signal part of each OFDM symbol.  
Reshaping \(\hat{\mathbf{X}}\) into a vector by stacking its columns, we obtain  
\[\mathbf{x}=\text{vec}\{\hat{\mathbf{X}}\}. \tag{2}\]  
At the communication transmitter, \(\mathbf{x}\) is transmitted after being processed by a radio frequency (RF) chain. The RF signal will hit some targets and partially return to the sensing receiver. After some RF processing, the digital signal at the receiver can be given by [11]  
\[\mathbf{y}=\alpha\mathbf{S}\text{vec}\{\hat{\mathbf{X}}\}\odot\mathbf{d}+ \mathbf{z},\text{ s.t. }\mathbf{S}=\begin{bmatrix}\mathbf{0}_{L\times I}\\ \mathbf{I}_{(I-L)\times I}\end{bmatrix},\ [\mathbf{d}]_{i}=e^{\text{j2}\pi\mu iT_{ \text{s}}}, \tag{3}\]  
where \(\alpha\) denotes the scattering coefficient, \(\odot\) denotes the point-wise product, and \(\mathbf{z}\) denotes a noise vector with independent entries conforming to the same centred complex Gaussian distribution. Moreover, \(L\) denotes the sample delay, \(I=(N+Q)M\) is the total number of samples in a block, \(\mathbf{I}_{(I-L)\times I}\) denotes the first \((I-L)\) rows of the \(I\)-dimensional identity matrix, \(\mathbf{d}\) accounts for the Doppler impact, \([\cdot]_{i}\) takes the \(i\)-th entry of the enclosed vector, \(\mu\) denotes the Doppler frequency, and \(T_{\text{s}}\) the sampling time. A single target is modelled here for clarity. The multitarget case will be considered in later algorithm development and simulation.  
We emphasise that, in the considered ISAC, sensing is performed based on the echo signal at the receiver, i.e., \(\mathbf{y}\), and the known copy of the communication-transmitted signals, i.e., \(\hat{\mathbf{X}}\). That is, the ISAC scheme discussed here does not make any changes to the communication waveform. Moreover, we perform ISAC based on the predominant OFDM communication system that is widely under use in WiFi, LTE, and 5G. ISAC employing more advancing OFDM systems, such as the real-signal DHT-OFDM [35], NOMA-TDS-OFDM [36; 37], and the precoded IM-OFDM-SS [38], can be an interesting future work.