# A W-Band Communication and Sensing Convergence System Enabled by Single OFDM Waveform
## 2 Communication and Sensing Convergence Using OFDM Waveforms  
Motivated by the OFDM-based sensing presented in [26; 34], a reference system model for the convergence of communication and sensing is presented here. An OFDM waveform for communication purposes consists of several OFDM symbols, each with orthogonal subcarriers modulated by data symbols and cyclically extended by appending the last part of the signal at the beginning called cyclic prefix (CP). If \(\Delta f\) represents the subcarriers spacing, \(N\) the number of orthogonal subcarriers, \(T\) the OFDM symbol duration, \(T_{\text{cp}}\) as the CPI, \(T_{\rm S}=T_{\rm cp}+T\) the effective duration of the OFDM symbol, and \(M\) the number of OFDM symbols, then the analytical expression of the transmitted OFDM waveform is [34],  
\[s(t)=\sum_{\mu=0}^{M-1}\sum_{n=0}^{N-1}S(\mu N+n)e^{(2\tau n\Delta f(t-\mu T_{ \rm S}))}e^{(-j2\tau n\Delta f/T_{\rm cp})}{\rm rect}\bigg{(}\frac{t-\mu T_{\rm S }}{T_{\rm S}}\bigg{)}, \tag{1}\]  
where \(S(\mu N+n)\) is the data symbol at \(n\)th subcarrier of \(\mu\)th OFDM symbol. The \({\rm rect}(t)\) in (1) is the rectangular pulse shape, such that \({\rm rect}(t)=1\) for \(t\in[0\quad 1]\) and 0 otherwise. The term \(\exp\bigl{(}-j2\pi n\Delta fT_{\rm cp}\bigr{)}\) appears due to the cyclic extension of OFDM symbols by the CP. In order to fulfill the orthogonality among subcarriers, over the interval \(T\), the following condition must be held:  
\[\Delta f=\frac{1}{T}, \tag{2}\]  
and \(T_{\rm cp}\) should accommodate the maximum expected delay caused by the radio channel. The baseband signal \(s(t)\) is up-converted by a carrier frequency \(f_{\rm c}\) to form \(\widetilde{s}(t)\) for transmission,  
\[\widetilde{s}(t)=s(t)e^{j2\pi f_{\rm c}t}. \tag{3}\]  
The received signal \(\widetilde{a}(t)\) at the sensing receiver is the sum of echoes from different targets. Using point-target channel model for \(L\) number of targets,  
\[\widetilde{a}(t)=\sum_{l=1}^{L}b_{l}\widetilde{s}(t-\tau_{l}), \tag{4}\]  
where \(\tau_{l}\) and \(b_{l}\) represent delay and attenuation related to the \(l\)th target, respectively. If \(l\)th target is located at a distance \(R_{l}\) and moving with a speed of \(v_{l}\), delay \(\tau_{l}\) in the received echo can be expressed as  
\[\tau_{l}=\frac{2(R_{l}-v_{l}t)}{c_{0}}, \tag{5}\]  
and \(b_{l}\)[34],  
\[b_{l}=\sqrt{\frac{c_{0}^{2}G_{\rm Tx}G_{\rm Rx}\sigma_{\rm RCS_{l}}}{(4\pi)^{3 }R_{l}^{4}f_{\rm c}^{2}}}, \tag{6}\]  
where in (6), \(c_{0}\) is the speed of light in free space; \(\sigma_{\rm RCS_{l}}\) is the radar cross-section of the \(l\)th target; and \(G_{\rm Tx}\), \(G_{\rm Rx}\) represent transmitting and receiving antenna gain, respectively. For the communication link, the signal attenuation \(b_{\rm com}\) is  
\[b_{\rm com}=\sqrt{\frac{c_{0}^{2}G_{\rm Tx}G_{\rm Rx}}{(4\pi)^{2}R_{\rm com}^{ 2}f_{\rm c}^{2}}}, \tag{7}\]  
where \(R_{\rm com}\) indicates the distance of the communication link. For sensing processing, a single target is sufficient for mathematical derivations due to the linear operation in (1). The analytical expression for the received echo from a target, located at a distance \(R\), moving with the speed of \(v\), and attenuated by \(\tilde{b}\) (assuming constant attenuation factor for frequencies within the bandwidth) is obtained by using delay \(\tau=(2R-2 vt)/c_{0}\) in (1), i.e.,  
\[ \begin{split}\widetilde{a}(t)&=\sum_{\mu=0}^{M-1} \sum_{n=0}^{N-1}\tilde{b}S(\mu N+n)e^{\left(j2\pi n\Delta f(t-\frac{(2R-2 vt)} {c_{0}}-\mu T_{\rm S}-T_{\rm cp})\right)}\\ &\quad\cdot e^{\left(j2\pi f_{\rm c}(t-\frac{(2R-2 vt)}{c_{0}}) \right)}{\rm rect}\bigg{(}\frac{t-(\frac{2R-2 vt}{c_{0}})-\mu T_{\rm S}}{T_{\rm S }}\bigg{)}+\hat{z}(t),\end{split}  \tag{8}\]where \(\hat{z}(t)\) is to account for the additive white Gaussian noise (AWGN).