## 2 Preliminary  
### Signal Model  
Suppose that a quasi-monostatic MF-PBR system contains a reference channel (RC) and a surveillance channel (SC), as shown in Figure 1. The reference signal of the \(m\)-th frequency channel, i.e., \(S_{ref,m}(t)\), is expressed as  
\[S_{ref,m}(t)=b_{m}s_{m}(t)\exp(j2\pi f_{m}t)+n_{ref,m}(t),m=1,2,\ldots,M, \tag{1}\]  
where \(b_{m}\) denotes the signal amplitude of the transmitted signal in the \(m\)-th RC, \(f_{m}\) is the carrier frequency at the \(m\)-th frequency band, and \(s_{m}(t)\) is the normalized communication signal emitted by IO in the \(m\)-th RC. Generally, \(s_{i}(t)\) and \(s_{k}(t)\) are uncorrelated for \(i,k\in m\), \(i\
eq k\). \(n_{ref,m}(t)\) is the noise in the \(m\)-th RC and it is assumed to be circularly symmetric and complex Gaussian noise with zero mean and variance of \(\sigma_{r,m}^{2}\), i.e., \(n_{ref,m}(t)\sim\mathcal{CN}(0,\sigma_{r,m}^{2})\). \(M\) is the number of the frequency channel. The surveillance signal in the \(m\)-th SC is expressed as  
\[ \begin{split} S_{surv,m}(t)=c_{m}s_{m}(t)&\exp(j2 \pi f_{m}t)+a_{m}s_{m}(t-\tau_{m})\exp(j2\pi f_{m}t)\exp(j2\pi f_{d,m}t)+\\ & n_{surv,m}(t),\quad m=1,2,\ldots,M,\end{split}  \tag{2}\]  
where \(c_{m}\) represents the signal amplitude for the propagation effect of the direct path interference (DPI) to the receiving antenna of the \(m\)-th SC. The symbols \(a_{m}\), \(\tau_{m}\), and \(f_{d,m}\), respectively, denote the echo reflectivity, target delay, and Doppler frequency of the \(m\)-th SC. Without loss of generality, the reflected component in the \(m\)-th SC can be considered from the same target of interest. \(n_{surv,m}(t)\) is the noise in the \(m\)-th SC and it is assumed to be circularly symmetric and complex Gaussian noise with zero mean and variance of \(\sigma_{s,m}^{2}\), i.e., \(n_{surv,m}(t)\sim\mathcal{CN}(0,\sigma_{s,m}^{2})\). Generally, we assume that the noise in SC and that in RC are independently and identically distributed (IID). Through a digital down-conversion (DDC) operation, the signals of RC and SC in the radio frequency (RF) band are converted to the baseband, and the problem of detecting the existence of a target is then formulated as a binary hypothesis, as follows:  
\[\left\{\begin{array}{l}H_{0}:\left\{ \begin{array}{l}S_{ref,m}(t)=b_{m}s_{m}(t) +n_{ref,m}(t),\\ S_{surv,m}(t)=c_{m}s_{m}(t)+n_{surv,m}(t),\\ S_{ref,m}(t)=b_{m}s_{m}(t)+n_{ref,m}(t),\\ S_{surv,m}(t)=c_{m}s_{m}(t)+\alpha_{m}s_{m}(t-\tau_{m})e^{j2\pi f_{d,m}t}+n_{ surv,m}(t),\end{array} \right. \tag{3}\]  
where \(H_{0}\) and \(H_{1}\) represent the absence and presence of the target of interest, respectively. With the removal of DPI via the extensive cancellation algorithm (ECA) [36], the \(c_{m}s_{m}(t)\) can be ignored in the following. As a result, the target problem in (3) is then modified as follows:  
\[\left\{\begin{array}{l}H_{0}:\left\{ \begin{array}{l}S_{ref,m}(t)=b_{m}s_{m} (t)+n_{ref,m}(t),\\ S_{surv,m}(t)=n_{surv,m}(t),\\ S_{ref,m}(t)=b_{m}s_{m}(t)+n_{ref,m}(t),\\ S_{surv,m}(t)=\alpha_{m}s_{m}(t-\tau_{m})e^{j2\pi f_{d,m}t}+n_{surv,m}(t).\end{array} \right. \tag{4}\]  
Next, we further discuss the target's signatures, including \(\alpha_{m}\), \(\tau_{m}\), and \(f_{d,m}\). Due to the same physical path and the consistent bandwidth at each selected frequency channel, the target delay at each frequency shows consistency, that is to say, \(\tau_{1}=\ldots=\tau_{M}\). However, the complex reflectivity \(\alpha_{m}=a_{m}\exp(j\phi_{m})\) could be different because of the fluctuated radar cross-section (RCS) at different frequency channels, especially for the target's phase \(\phi_{m}\), resulting in substantial residual phase error. In addition, for the \(m\)-th channel, the target's Doppler frequency is expressed as [37],  
\[f_{d,m}=\frac{2V_{b}cos(\beta/2)}{\lambda_{m}}=\frac{gf_{m}V_{b}}{c}, \tag{5}\]  
Figure 1: The geometric configuration of a DVB-S-based MF-PBR.