## 3 Signal Model
In the presence of ICM-CP and ICM-MP, the MP clutter space-time snapshot is given by:  
\[\mathbf{x}_{pc;l}=\varepsilon_{p}{}^{\prime}\sum_{i=1}^{N_{\rm c}}\xi_{l+l_{p};i} \mathbf{v}_{\rm\_}{s}(\theta_{i})\otimes\big{(}\mathbf{v}_{\rm\_}{t}\big{(}\omega_{i}- \omega_{p}\big{)}\odot\mathbf{\gamma}_{\rm\_}1\odot\mathbf{\gamma}_{\rm\_}2\big{)}= \varepsilon_{p}{}^{\prime}\mathbf{\Gamma}\sum_{i=1}^{N_{\rm c}}\xi_{l+l_{p};i}\mathbf{ T}\mathbf{v}\big{(}\omega_{p;i},\theta_{i}\big{)}, \tag{4}\]  
where \(\varepsilon_{p}{}^{\prime}=\varepsilon_{p}^{*}/\varepsilon_{\rm\_}{d}^{*}\) represents the relative relative to the DP signal) amplitude and \(\varepsilon_{p}\) is the complex amplitude of the \(p\)-th MP signal [8]; \(\omega_{p;i}=\omega_{i}-\omega_{p}\) is the relative (relative to the \(p\)-th MP signal) normalized Doppler frequency, where \(\omega_{p}=T_{r}\omega_{p}{}^{\prime}\) and \(\omega_{p}{}^{\prime}\) are  
Figure 1: Airborne passive radar geometry. the normalized Doppler frequency and Doppler frequency corresponding to the \(p\)-th MP signal, respectively; \(\gamma_{2}\) is the \(M\times 1\) temporal fluctuation vector corresponding to ICM-MP, which means the small pulse-to-pulse fluctuations exist in MP signals; the vector \(\gamma_{2}\) is also called the ICM-MP fluctuation vector in what follows; the ICM-MP fluctuation matrix \(\mathbf{\Gamma}=\mathbf{I}_{N}\otimes diag(\gamma_{2})\). By defining the modified steering vector \(f\!\left(\omega_{\mathrm{d}}-\omega_{p}\right)=\mathbf{I}_{N\times 1}\otimes \boldsymbol{v}_{t}\!\left(\omega_{\mathrm{d}}-\omega_{p}\right)\), we obtain \(\boldsymbol{v}\!\left(\omega_{p,i},\theta_{i}\right)=\boldsymbol{v}\!\left( \omega_{\mathrm{d,i}},\theta_{i}\right)\odot f\!\left(\omega_{\mathrm{d}}- \omega_{p}\right)\). Thus, Equation (4) is reformulated as [8]:  
\[\boldsymbol{x}_{pc,l} =\varepsilon_{p}/\mathbf{\Gamma}\boldsymbol{x}_{\mathrm{d},L+(l_{ p}-l_{\mathrm{d}})}\odot f\!\left(\omega_{p}-\omega_{\mathrm{d}}\right) \tag{5}\] \[=\varepsilon_{p}/\mathbf{\Gamma}\boldsymbol{x}_{l+(l_{p}-l_{ \mathrm{d}})}\odot f\!\left(\omega_{p}-\omega_{\mathrm{d}}\right)- \varepsilon_{p}/\mathbf{\Gamma}\sum_{p=1}^{N_{T}}\varepsilon_{p}/\mathbf{ \Gamma}\boldsymbol{x}_{l+(l_{p}+l_{p}-2l_{\mathrm{d}})}\odot f\!\left( \overline{\omega}_{p}+\omega_{p}-2\omega_{\mathrm{d}}\right)\.\]  
where the first term and second term of the second equation in Equation (5) are called the true MP clutter and the derived ghost MP clutter, respectively [8]. Thus, the MP clutter snapshot correlates with three parameters, which are the DP clutter snapshot, the ICM-MP fluctuation matrix, and the modified steering vector. Then, the received snapshot at the \(l\)-th range cell, consisting of the DP clutter, the MP clutter, and the receiver thermal noise, is expressed as  
\[\boldsymbol{x}_{l}=\boldsymbol{x}_{\mathrm{d},L}+\sum_{p=1}^{N_{T}}\boldsymbol {x}_{pc,l}+\boldsymbol{x}_{n}, \tag{6}\]  
where \(\boldsymbol{x}_{n}\) is the noise vector, which is assumed to be spatially and temporally uncorrelated. The complex amplitudes of clutter patches \(\xi_{l,i}\), \(i=1,2,\ldots,N_{c}\), are assumed to be independent and identically distributed random variables. Hence, the corresponding covariance matrix \(\boldsymbol{R}_{x}\) is given by:  
\[\boldsymbol{R}_{x}=E\!\left[\boldsymbol{x}_{l}\boldsymbol{x}_{l}^{\mathrm{H} }\right]=\boldsymbol{R}_{\mathrm{d}}+\sum_{p=1}^{N_{T}}\boldsymbol{R}_{p}+ \boldsymbol{R}_{n}, \tag{7}\]  
where the DP clutter covariance matrix \(\boldsymbol{R}_{\mathrm{d}}=E\!\left[\boldsymbol{x}_{\mathrm{d},L}\!\boldsymbol {x}_{\mathrm{d},L}^{\mathrm{H}}\right]\); the \(p\)-th MP clutter covariance matrix \(\boldsymbol{R}_{p}=E\!\left[\boldsymbol{x}_{pc,l}\boldsymbol{x}_{pc,l}^{ \mathrm{H}}\right]\); \(\boldsymbol{R}_{n}=E\!\left[\boldsymbol{x}_{n}\boldsymbol{x}_{n}^{\mathrm{H}}\right]\) is the noise covariance matrix. From the above analyses, it is evident that the presence of MP clutter severely affects the estimated covariance matrix.