# A W-Band Communication and Sensing Convergence System Enabled by Single OFDM Waveform
## 3 Proposed Method for Range Extension
Figure 2a,b are obtained using steps 1-4 of the proposed method; Figure 2c is the plot of step 5 and using \(M=12\) for step 6; Figure 2d is the final range/speed plot, where the range is extended up to two OFDM symbols. A schematic diagram of the proposed method is presented in Figure 3. Delay shifts are used in the sampled version of the incoming OFDM waveform to get the frequency domain signal \(Y_{q}\). Element-wise division of \(Y_{q}\) is performed with the current OFDM symbol \(S_{q}\) and previous OFDM symbols for sensing and combining. We use periodogram to compare the performance of the proposed method with conventional OFDM-based sensing. Periodogram of the conventional OFDM-based sensing is defined as [31],  
\[ \begin{split} D_{\text{conv}(d,p)}=&\left|\sum_{ \mu=0}^{M-1}r(d)e^{\frac{-2\pi\mu p}{M}}\right|^{2},\\ & d\in[0,N-1]\quad p\in[0,M-1]\end{split}  \tag{20}\]  
where \(r(d)\) is defined in (12). A target is detected if the peak in \(D_{\text{conv}(d,p)}\) is above a threshold level (usually defined by the minimum detectable signal strength). For the range/speed plot, \(\mathbf{D}\) is often transformed to normalized power, and in dB scale using \(10\log_{10}\left(\mathbf{D}/max[\mathbf{D}]\right)\), where \(max[\mathbf{D}]\) represents the maximum value of \(\mathbf{D}\). Similarly, the periodogram of the proposed method is  
\[D_{\text{pro}(d,p)}=\left|\sum_{\mu=0}^{M-1}\hat{r}_{d}(0,\mu)e^{\frac{-2\pi p}{ M}}\right|^{2},\quad\hat{d}\in[0,(Q-1)N_{\text{s}}-1]\quad p\in[0,M-1] \tag{21}\]  
where \(\hat{d}\) represents the delay-shift domain and \(\hat{r}_{d}(0,\mu)\) is obtained by concatenation of \(Q\) segments as defined in step 5 of the proposed method, i.e.,  
\[\hat{r}_{d}(0,\mu)=[r_{d}(0,\mu),r_{d}(0,\mu-1),...,r_{d}(0,\mu-(Q-1))],\qquad d \in[0,N_{\text{s}}-1] \tag{22}\]  
Figure 3: Schematic of the proposed range extension method to detect targets beyond the range limit in the conventional OFDM–based sensing. Figure 2: Explanation of the proposed method when range extension is up to two OFDM symbols. (**a**,**b**) Steps 1–4 of the proposed method provide two matrices. (**c**) Delay–zero rows of the matrices in (**a**,**b**) are concatenated according to the step 5. (**d**) Range/speed plot is completed using \(M\)–point FFT operation over time domain. where \(r_{\hat{d}}(0,\mu)\) is obtained by using \(d=0\) in \(r_{\hat{d}}(d,\mu)\),  
\[r_{\hat{d}}(d,\mu)=\frac{1}{N}\sum_{n=0}^{N-1}H_{\hat{d}}(\mu N+n)e^{\frac{2\pi nd }{N}},\qquad d\in[0,N_{\rm s}-1] \tag{23}\]  
i.e.,  
\[r_{\hat{d}}(0,\mu)=\frac{1}{N}\sum_{n=0}^{N-1}H_{\hat{d}}(\mu N+n). \tag{24}\]  
If we represent \(m=gN_{\rm s}+\hat{m}\) where \(g\in[0,Q-2]\) and \(\hat{m}\in[0,N_{\rm cp}-1]\), received \(q\)th OFDM symbol and transmitted OFDM symbols provide  
\[{\bf H}_{\hat{d},(\hat{q}-i)}=\frac{{\bf Y}_{\hat{d},\hat{q}}}{{\bf S}_{(q-i)}}, \tag{25}\]  
where \({\bf H}_{\hat{d},(q-i)}\) is taken as simplified notation for \(H_{\hat{d}}((q-i)N+n)\), \(i\in[0,Q-1]\), and \({\bf Y}_{\hat{d},\hat{q}}={\rm FFT}(A_{\hat{d}}(k))\), where \(A_{\hat{d}}(k)\) is the delay-shift of \(A(k)\) by \(d\) and \(k\in[\hat{d}+(q)N_{\rm s}-gN_{\rm s}-\hat{m}+,\hat{d}+(q+1)N_{\rm s}-gN_{\rm s} -\hat{m}-N_{\rm cp}]\) (interval of the N-samples of the waveform is selected at the initial step of the proposed method). It is clear that at \(\hat{d}=\hat{d}_{0}=\hat{m}+N_{\rm cp}\), \(A_{\hat{d}_{0}}(k)\) represents \((q-g)\)th OFDM symbol without \(N_{\rm cp}\); hence, (25) changes to  
\[{\bf H}_{\hat{d}_{0},(q-i)}=\frac{b{\bf S}_{(q-g)}+{\bf Z}_{(q-g)}}{{\bf S}_{( q-i)}}, \tag{26}\]  
where \({\bf Z}_{(q-g)}\) represents noise part in the \((q-g)\)th OFDM symbol.