# A W-Band Communication and Sensing Convergence System Enabled by Single OFDM Waveform
## 2 Communication and Sensing Convergence Using OFDM Waveforms
Since \(f_{\rm c}\) is usually very high compared to the bandwidth of the signal, in particular in the millimeter-wave band, the Doppler shift \((n\Delta f2v)/c_{0}\) is negligible for the subcarriers and the overall Doppler shift appears only caused by \((2 vf_{\rm c})/c_{0}\). The received signal is down-converted to baseband, which is equivalent to  
\[a(t)=\sum_{\mu=0}^{M-1}\sum_{n=0}^{N-1}bS(\mu N+n)e^{\left(\left/2\pi n\Delta f (t-\frac{(2R)}{c_{0}}-\mu T_{\rm S}-T_{\rm cp})\right.\right)}e^{\left(2\pi \frac{2\pi T_{\rm c}}{c_{0}}t\right)}\text{rect}\left(\frac{t-(\frac{2R}{c_{0} })-\mu T_{\rm S}}{T_{\rm S}}\right)+z(t), \tag{9}\]  
where \(b\) and \(z(t)\) represent \(\hat{b}\text{exp}(-j2\pi f_{\rm c}2R/c_{0})\) and \(\hat{z}(t)\text{exp}(-j2\pi f_{\rm c}t)\), respectively. Finally, the signal \(a(t)\) is sampled, the CP part is removed before it is converted into the frequency domain by using FFT operation,  
\[ \begin{array}{rl}A(\mu N+n)=&bS(\mu N+n)e^{\left(-j2\pi n\Delta f\frac{2R}{ c_{0}}\right)}e^{\left(j2\pi\frac{2\pi f_{\rm c}}{c_{0}}\mu T_{\rm S}\right)}+Z( \mu N+n),\\ &n\in[0,N-1],\quad\mu\in[0,M-1]\end{array}  \tag{10}\]  
where \(A(\mu N+n)\) and \(Z(\mu N+n)\) are frequency domain equivalents of \(a(t)\) and \(z(t)\). Once the received signal is translated back into the frequency domain, the element-wise division of the received OFDM symbol by the respective transmitted OFDM symbol is performed to construct the channel matrix \(\mathbf{H}\), i.e.,  
\[H(\mu N+n)=be^{\left(-j2\pi n\Delta f\frac{2R}{c_{0}}\mu T_{\rm S}\right)}e^{ \left(j2\pi\frac{2\pi f_{\rm c}}{c_{0}}\mu T_{\rm S}\right)}+\frac{Z(\mu N+n)} {S(\mu N+n)}, \tag{11}\]  
where \(H(\mu N+n)\) represents the \(\mu\)th column and \(n\)th row of the channel matrix \(\mathbf{H}\), and \(Z(\mu N+n)/S(\mu N+n)\) defines the noise floor that depends on the digital modulation, e.g., a 16-QAM mapping affects the noise floor by approx 2.7 dB [36]. The IFFT of \(\mathbf{H}\), along subcarriers, provides the range information,  
\[ \begin{array}{rl}r(d)=e^{\left(2\pi\frac{2c_{0}}{c_{0}}\mu T_{\rm S}\right)} \frac{b}{N}\sum_{n=0}^{N-1}e^{\left(-j2\pi n\Delta f\frac{2R}{c_{0}}\right)}e ^{\left(j\frac{2R}{N}nd\right)}+\hat{z}(d),\\ &d\in[0,N-1]\end{array}  \tag{12}\]  
where \(\hat{z}(d)\) represents the noise part. The \(|r(d)|\) shows a peak value under the following condition:  
\[d=\left\lfloor\frac{2R\Delta fN}{c_{0}}\right\rfloor, \tag{13}\]  
i.e., the value of \(d\) corresponding to the maximum of \(|r(d)|\) holds the information of the target range, and the range resolution \(\Delta R\) (minimum distinguishable distance between the two targets) is defined as  
\[\Delta R=\frac{c_{0}}{2\Delta fN}. \tag{14}\]  
Similarly, the FFT operation over different OFDM symbols in \(\mathbf{H}\) (over \(\mu\) domain in (11)) provides the information about the speed of the target and can be recognized by using  
\[p=\left\lfloor\frac{2 vf_{\rm c}T_{\rm S}M}{c_{0}}\right\rfloor,\quad p\in[0,M -1] \tag{15}\]  
whereas the speed resolution \(\Delta v\) can be calculated by setting \(p=1\),  
\[\Delta v=\frac{c_{0}}{2f_{\rm c}T_{\rm S}M}. \tag{16}\]The IFFT/FFT operations on \(\mathbf{H}\) provide processing gain due to coherent addition of signals, and the overall processing gain is  
\[G=NM. \tag{17}\]  
If there are \(\hat{N}\) guardband subcarriers on each side of the OFDM symbol, then the processing gain reduces to \((N-2\hat{N})M\). Here, it is important to note that the guardband subcarriers reduce the bandwidth of the OFDM waveform, and consequently the value of \(\Delta R\) increases in (14) when \(N\) is replaced by \((N-2\hat{N})\). Although \(\Delta R\) increases due to guardband subcarriers, improvement in the range accuracy (resolution of IFFT) is linked to the size of IFFT [37]. Figure 1 highlights the sensing processing using \(\mathbf{H}\). Figure 1a shows the real part of the channel matrix \(\mathbf{H}\), which has sinusoidal variations due to the range and speed of a single target.