# A W-Band Communication and Sensing Convergence System Enabled by Single OFDM Waveform
## 3 Proposed Method for Range Extension  
In OFDM-based sensing, the maximum range is limited by the CPI [28; 35] as echoes falling outside the CPI cause ISI and suffer in processing gain. We propose a zero-delay shift method to compensate for delay \(\tau\) in an echo to maintain its processing gain \(G\) during IFFT operation for OFDM-based sensing. Using sampling intervals \(\Delta T=1/F_{\mathrm{s}}\) and \(\Delta f=1/(N\Delta T)\), the sampled version of (9) is  
\[ \begin{split} a(k\Delta T)&=\sum_{\mu=0}^{M-1}\sum_ {n=0}^{N-1}bS(\mu N+n)e^{\left(\frac{2m\pi}{(\mathrm{NAT})}\frac{1}{(k\Delta T -\frac{2\mathrm{R}}{c_{0}}-\mu\frac{T_{\mathrm{s}}}{\Delta T}-\frac{T_{ \mathrm{cp}}}{\Delta T})}\right)}\\ &\cdot e^{\left(2\pi\frac{2\gamma_{c}}{\gamma_{0}}k\Delta T \right)}\mathrm{rect}\left(\frac{k\Delta T-\frac{2\gamma_{c}}{c_{0}}-\mu\frac{ T_{\mathrm{s}}}{\Delta T}}{\frac{T_{\mathrm{s}}}{\Delta T}}\right)+z(k\Delta T), \end{split}  \tag{18}\]  
Figure 1: OFDM–based sensing from the channel matrix \(\mathbf{H}\). (**a**) IFFT operation over subcarriers. (**b**) FFT operation over OFDM symbols. (**c**) 3–D plot of the delay–Doppler profile. (**d**) Radar image, indicating the delay associated to the range and the Doppler frequency related to the speed of the target. where \(k\) is the sampled time index. Using \(F_{\text{s}}/\Delta f=T/\Delta T=N\), \(T_{\text{s}}/\Delta T=N_{\text{s}}\), \(T_{\text{cp}}/\Delta T=N_{\text{cp}}\), \(m=\lfloor(2R)/(c_{0}\Delta T)\rfloor\), and \(a(k)\), \(z(k)\) to represent \(a(k\Delta T)\) and \(z(k\Delta T)\) respectively,  
\[ \begin{split} a(k)=&\sum_{\mu=0}^{M-1}\sum_{n=0}^{N -1}bS(\mu N+n)e^{\left(\beta 2\pi\frac{R}{N}(k-m-\mu N_{\text{s}}-N_{\text{cp}}) \right)}\\ &\cdot e^{\left(\beta 2\pi\frac{2\gamma_{\text{s}}}{\varepsilon}k \Delta T\right)}\text{rect}\left(\frac{k-m-\mu N_{\text{s}}}{N_{\text{s}}} \right)+z(k).\end{split}  \tag{19}\]  
A delay-shift in \(k\) by \(m\) samples shifts the target at zero on the delay axis. Since \(m\) is unknown, sequentially increasing the delay-shift in \(k\) identifies \(m\) when a peak appears at delay zero. This process can identify echoes with delay longer than the CP, provided they arrive with detectable signal strength. If we extend the sensing range up to \(Q\) number of OFDM symbols, the proposed method can be described in following steps:  
1. \(N\) samples are selected (window of length \(N\)) from the received OFDM waveform to perform an \(N\)-point FFT operation;
2. Received data symbols are divided by the transmitted data symbols in the current and previous \(Q-1\) OFDM symbols;
3. \(N\)-point IFFT operation is performed on results obtained in step 2, which provides first columns (in the delay domain) for \(Q\) number of matrices;
4. Steps 1-3 are repeated for \(N_{\text{s}}\) number of delay-shifts in the selected window. Completion of this step provides \(Q\) number of matrices each of size \(N\times N_{\text{s}}\) in delay and delay-shift domains;
5. Delay-zero rows (\(d=0\)) of each of the \(Q\) number of matrices, generated in step 4, are concatenated to form a row of another matrix, which will be used for range/speed processing;
6. At this point, the selected window of \(N\) samples has been shifted by \(N_{\text{s}}\) samples; now, OFDM symbols, which are used for division in step 2, are replaced by next OFDM symbol, e.g., \([S_{q},S_{q-1},\ldots,S_{q-(Q-1)}]\) are replaced by \([S_{q+1},S_{q},\ldots,S_{q-(Q-2)}]\), where \(S_{q}\) denotes \(q\)th OFDM symbol;
7. Steps 1-6 are repeated \(M\) times. Using the above process, a matrix (in the delay-shift and time domains) of size \(QN_{\text{s}}\times M\) is constructed, which requires an \(M\)-point FFT operation to complete the range/speed plot. Figure 2 shows different steps in the proposed method to detect two targets separated by more than one OFDM symbol duration; \(Q=2\) is used, and the current OFDM symbol number is \(q\).