## 2 RadCom System Model and Processing
### Radar Processing  
As we demonstrated in Section 2.2, the channel equation fully contains information about distance and velocity. Hence, estimating these two parameters for radar is equivalent to the channel estimation for the communication system. The main difference is the use of the entire transmitted grid in radar instead of only a few pilots for communication. On the basis of known transmitted symbols, raw channel estimation was first performed. For OFDM signals, it is expressed as  
\[\hat{H}_{o}=X^{-1}Y_{o} \tag{6}\]  
Figure 2: RadCom system model. where \(\hat{H}_{o}\) denotes the OFDM estimated channel, \(Y_{o}\) the received signal, and \(X^{-1}\) is the transmitted pilots for communication and the grid for radar. For UFMC, on the other hand, raw channel estimation at each subcarrier can be written as  
\[\hat{H}_{u}=(F_{s}X_{s})^{-1}Y_{u}, \tag{7}\]  
where \(\hat{H}_{u}\) denotes the UFMC estimated channel, \(Y_{u}\) the received signal, and \((. )_{s}\) means the subcarrier belonging to the \(s\)-th sub-band. \(F_{s}\) is the known filter frequency response at this subcarrier. From Equations (6) and (7), it is clear that UFMC channel estimation only differs from the OFDM one by the filter response. Hence, in the system model diagram (Figure 2), we propose to add a filter equalization block to account for the filter impact on the UFMC-received signal. This operation afterwards makes the UFMC signal equivalent to OFDM. Hence, below, we drop the \((. )_{u}\) and \((. )_{o}\) subscripts. This first step of channel estimation is simply a spectral division of the received signal by the transmitted one; hence, in the system model diagram, we present it by its name: spectral division. After spectral division (and filter equalization for UFMC), the estimated frequency-domain channel transfer function \(\hat{H}_{r,q}\) at \(q\)-th subcarrier for the \(r\)-th symbol (ignoring the stretching and compression effects) can be given by  
\[\hat{H}_{r,q}=\sum_{j=0}^{L_{q,q}-1}e^{-j\pi q\Delta f/\tau_{j}}e^{\frac{j\pi \tau_{j}\tau_{j}N_{o}}{N_{c}\Delta f}}+Z_{r,q}, \tag{8}\]  
where \(\Delta_{f}\) describes the subcarrier spacing, \(Z_{r,q}\) denotes the additive white Gaussian noise of the \(q\)-th subcarrier for the \(r\)-th symbol; \(N_{o}=N_{c}+N_{ep}\) is the OFDM symbol length, where \(N_{c}\) denotes Fast Fourier transform (FFT) length, and \(N_{ep}\) is CP length. Equation (8) shows that the delay (hence, range) can be estimated using inverse FFT (IFFT) over the frequency axis. The Doppler shift (hence, velocity) can be evaluated using FFT over the time axis. This is equivalent to applying a 2D periodogram, as depicted in Figure 3. The main advantage in this approach is that the delay and Doppler estimations are independent. At first, this operation estimates frequency \(\hat{q}\) and symbol \(\hat{r}\) indices, which should then be translated to distance and velocity through the following equations:  
\[\hat{d}_{j}=\hat{q}_{j}\frac{c}{2N\Delta f} \tag{9}\]  
\[\hat{v}_{j}=\hat{r}_{j}\frac{c}{2f_{c}MT_{o}\Delta f} \tag{10}\]  
\(N\geqslant N_{c}\) and \(M\geqslant N_{symb}\), where \(N_{symb}\) denotes the total number of the transmitted symbols, and \(N\) and \(M\) are the FFT and IFFT length of the 2D periodogram, respectively. \(T_{o}\) is the total duration of the OFDM symbol. Figure 3: Two-dimensional (2D) periodogram processing. Deciding if a peak in the periodogram is a true target is basically a statistical detection problem that boils down to binary hypothetical testing. Hypotheses can be formulated as in Equation (11), where in \(H_{0}\), the received signal is noise, and in \(H_{1}\), the received signal is the target-reflected one, propagating through the channel plus the noise. \[y[n]= \begin{cases}z[n]&H_{0}\quad noise\ only\\ x[n]\ \otimes h[n]+z[n]&H_{1}\quad signal\ plus\ noise\end{cases}