## 3 Modified Hybrid Integration Processing
### Communication Compensation of Receiver Signal  
The signal in (7) includes the communication information and parameters. In the receiver, the communication information will interfere with the range and velocity estimations. Hence, in order to obtain the range and velocity estimations, the communication information is compensated for first. Let \(\beta_{R,J}=e^{-\beta\pi f_{c}2R_{l}/c}\), \(\tilde{\beta}_{v,l}(n)=e^{-\beta 2\pi f_{c}2v_{l}nT_{c}/c}\), \(\widetilde{\beta}_{v,l}(p)=e^{-\beta 2\pi f_{c}2v_{l}pT_{p}/c}\), and (8) can be rewritten as  
\[ \begin{split} z_{n,p}(k)&=\sum_{l=0}^{L-1}\sum_{s=0 }^{S-1}\psi_{l}\beta_{R,l}\beta_{v,l}\tilde{\beta}_{v,l}(n)\widehat{\beta}_{v, l}(p)a_{s,n,p}\\ &\cdot e^{-\beta 2\pi s(1+2v_{l}/c)k/S}e^{-\beta 2\pi s\Delta f2R_{l}/c}e^{-\beta 2\pi s\Delta f2v_{l}T_{cp}/c}\\ &\cdot e^{\beta 2\pi s\Delta f2v_{l}nT_{c}/c}e^{\beta 2\pi s\Delta f2v_{l}T_{p}/c}e^{\beta 2\pi s\Delta f2v_{l}k \Delta t/c}\\ &+n_{n,p}(k)\end{split}  \tag{9}\]  
Stack the \(z_{n,p}(k)\) into a vector \(\mathbf{z}_{n,p}=\big{[}z_{n,p}(0),z_{n,p}(1),\cdots,z_{n,p}(S-1)\big{]}^{T}\), and let \(\mathbf{n}_{n,p}(k)=\big{[}n_{n,p}(0),n_{n,p}(1),\cdots,n_{n,p}(S-1)\big{]}^{T}\), \(\mathbf{a}_{n,p}=\big{[}a_{0,n,p},a_{1,n,p},\cdots,a_{S-1,n,p}\big{]}^{T}\), \(\rho_{l}=1+2v_{l}/c\),  
\[F(\rho_{l})= \begin{bmatrix}1&1&\cdots&1\\ 1&e^{\beta 2\pi\rho_{l}/S}&\cdots&e^{\beta 2\pi(S-1)\rho_{l}/S}\\ \vdots&\vdots&\ddots&\vdots\\ 1&e^{\beta 2\pi\rho_{l}(S-1)/S}&\cdots&e^{\beta 2\pi(S-1)\rho_{l}(S-1)/S} \end{bmatrix} , \tag{10}\]  
and \(\mathbf{\tilde{u}}(R_{l})=\big{[}1,e^{-\beta 2\pi\Delta f2R_{l}/c},\cdots,e^{- \beta 2\pi(S-1)\Delta f2R_{l}/c}\big{]}^{T}\), \(\mathbf{u}_{n,p}(v_{l})=[1,e^{\beta 2\pi\Delta f2v_{l}(T_{cp}+nT_{l}+pT_{p})/c}\] \[\cdots e^{\beta 2\pi(S-1)\Delta f2(T_{cp}+nT_{l}+pT_{p})/c}]^{T}\), \(\widetilde{\mathbf{u}}(v_{l})=\big{[}1,e^{\beta 2\pi f_{c}2v_{l}\Delta t/c},\cdots,e^{\beta 2\pi(S-1)f_{c}2v_{l}(S-1) \Delta t/c}\big{]}^{T}\). Stack \(S\) time samples of \(n\)th OFDM symbol and \(p\)th pulse into a column vector as follows:  
\[ \begin{split}&\mathbf{z}_{n,p}=\sum_{l=0}^{L-1}\psi_{l}\beta_{R, l}\beta_{v,l}\tilde{\beta}_{v,l}(n)\widehat{\beta}_{v,l}(p)\mathbf{D}\Big{(} \widetilde{\mathbf{u}}(v_{l})\Big{)}\\ &\cdot\mathbf{D}\big{(}\mathbf{u}_{n,p}(v_{l})\big{)}\mathbf{F}( \rho_{l})\mathbf{D}\big{(}\mathbf{c}_{n,p}\big{)}\widetilde{\mathbf{u}}(R_{l} )+\mathbf{n}_{n,p},\end{split}  \tag{11}\]  
where \(\mathbf{D}(\cdot)=diag\{\centerdot\center\}\) denotes a diagonal matrix with the diagonal elements. Note that \(e^{\beta 2\pi(S-1)\Delta f2(T_{cp}+nT_{s}+pT_{p})/c}\) and \(e^{\beta 2\pi(S-1)f_{c}2v_{l}(S-1)\Delta t/c}\) are approximately 1, since \(f_{c}\) is usually as large as \(c\) when \(v_{i}(S-1)\Delta t\) is usually much less than 1, and \((S-1)\Delta f\) is usually much less than \(c\) when \(2v_{i}(T_{cp}+pT_{p}+nT_{s})\) is usually much less than 1. Therefore, \(diag\big{\{}\widetilde{\mathbf{u}}(v_{l})\big{\}}\) is approximately \(I_{S}\) and \(diag\big{\{}\mathbf{u}_{n,p}(v_{l})\big{\}}\) is approximately \(I_{S}\). Moreover, since \(2v_{l}/c\ll 1\), \(\rho_{l}\approx 1\), \(\mathbf{F}(\rho_{1})\approx\mathbf{F}(1)=\mathbf{F}\), \(\mathbf{F}\) is the \(S\times S\) inverse fast Fourier transform (IFFT) matrix. Meanwhile, (11) can be rewritten as  
\[\mathbf{z}_{n,p}=\sum_{l=0}^{L-1}\psi_{l}\beta_{R,l}\beta_{v,l}\tilde{\beta}_{v,l}(n)\widehat{\beta}_{v,l}(p)\mathbf{F}(\rho_{l})\mathbf{D}\big{(}\mathbf{c} _{n,p}\big{)}\widetilde{\mathbf{u}}(R_{l})+\mathbf{n}_{n,p} \tag{12}\]  
The frequency domain signal can be obtained by multiplying FFT matrix \(\mathbf{F}^{-1}\) and matrix \(\mathbf{D}^{-1}\big{(}\mathbf{c}_{n,p}\big{)}\) by both sides in (12) as follows:  
\[\mathbf{F}^{-1}\mathbf{D}^{-1}\big{(}\mathbf{c}_{n,p}\big{)}\mathbf{z}_{n,p}= \beta_{R,l}\beta_{v,l}\tilde{\beta}_{v,l}(n)\widehat{\beta}_{v,l}(p)\widetilde{ \mathbf{u}}(R_{l})+\widetilde{\mathbf{n}}_{n,p}, \tag{13}\]  
where \(\widetilde{\mathbf{n}}_{n,p}=\mathbf{F}^{-1}\mathbf{D}^{-1}\big{(}\mathbf{c}_{n,p}\big{)}\mathbf{n}_{n,p}\).