## 3 Signal Model  
The system being considered is an airborne passive radar. We assume that the transmitter is a ground-based non-cooperative emitter and that the airborne platform moves in a straight line with constant altitude \(H_{R}\) and velocity \(v_{R}\). The side-looking airborne passive radar geometry is shown in Figure 1. The measurement signals are received by a uniform linear array consisting of \(N\) elements. The reference antenna is directed toward the non-cooperative transmitter and receives the reference signal. The impure reference signal involves the DP signal, whose time bin is \(l_{d}\), and the \(N_{T}\) MP signals, whose time bins are \(l_{p}\) (\(p=1,\ldots,N_{T}\)), respectively. Let the reference signal and measurement signals be segmented uniformly. Then, signals are equivalent to \(M\) pulses with a constant pulse repetition interval \(T_{r}\) in a coherent integration time. \(L\) is the number of fast-time samples in \(T_{r}\). After using the segmented signals for matched filtering, the received data set comprises _LMN_ samples. The radar data corresponding to a fast-time sample can be arranged as a \(MN\times 1\) space-time snapshot. Considering an impure reference signal and ignoring the influence of range ambiguities, the clutter snapshot at the \(l\)-th range cell is composed of two parts. The first part is the DP clutter snapshot, that is, the matched result of the scatterer echoes whose time bin is \((l+l_{\rm d})\) with the DP signal. The DP clutter space-time snapshot, including ICM-CP, is given by:  
\[x_{\rm d;\_}l=\sum_{i=1}^{N_{\rm c}}\xi_{l+l_{\rm d};i}x_{\rm s}(\theta_{i}) \otimes(\mathbf{v}_{\rm\_}l(\omega_{i}-\omega_{\rm d})\odot\mathbf{\gamma}_{\rm\_}1)= \sum_{i=1}^{N_{\rm c}}\xi_{l+l_{\rm d};i}\mathbf{T}\mathbf{v}(\omega_{\rm\_}{\rm d;\_}i ), \tag{1}\]  
where \(N_{\rm c}\) is the number of statistically independent clutter patches; \(\xi_{l+l_{\rm d};i}=\varepsilon_{l+l_{\rm d};i}\varepsilon_{\rm d}^{*}\), where \(\varepsilon_{\rm d}\) and \(\varepsilon_{l+l_{\rm d};i}\) are the complex amplitudes of the DP signal and the \(i\)-th clutter patch at the \((l+l_{\rm d})\)-th time bin, respectively; \(\theta_{i}=d\cos\varphi_{i}/\lambda\) and \(\omega_{i}=T_{r}v_{\rm\_}R\cos\varphi_{i}/\lambda\) are spatial frequency and normalized Doppler frequency of the \(i\)-th clutter patch, respectively, where \(\lambda\) is the wavelength, \(d\) is the inter-channel spacing, and \(\varphi_{i}\) is the spatial cone angle of the \(i\)-th clutter patch; \(\omega_{\rm\_d;\_}i=\omega_{i}-\omega_{\rm\_}{\rm d}\) indicates the relative relative to the DP signal normalized Doppler frequency, where \(\omega_{\rm\_d}=T_{r}\omega_{\rm\_}{\rm\_}{d}^{\prime}\) and \(\omega_{\rm\_}{\rm\_}{d}^{\prime}\) are the normalized Doppler frequency and Doppler frequency of the DP signal, respectively; \(\mathbf{\gamma}_{\rm\_}1\) is the \(M\times 1\) temporal fluctuation vector corresponding to ICM-CP, which leads to the small pulse-to-pulse fluctuations in the scatterer echoes; the ICM-CP fluctuation matrix \(\mathbf{T}=\mathbf{I}_{N}\otimes\textit{diag}(\mathbf{\gamma}_{\rm\_}1)\); \(\mathbf{v}(\omega,\theta)=\mathbf{v}_{\rm\_}{s}(\theta)\otimes\mathbf{v}_{\rm\_}{t}(\omega)\) is the space-time steering vector with  
\[\mathbf{v}_{\rm\_}{s}(\theta)=[1,\exp(j2\pi\theta),\cdots,\exp(j2\pi(N-1)\theta) ]^{\rm T}, \tag{2}\]  
\[\mathbf{v}_{\rm\_}{t}(\omega)=[1,\exp(j2\pi\omega),\cdots,\exp(j2\pi(M-1)\omega) ]^{\rm T}, \tag{3}\]  
denoting the spatial and temporal steering vectors, respectively. The second part is the matched result of the scatterer echoes, whose time bin is \((l+l_{p})\) (\(p=1,\ldots,N_{\rm T}\)) with the MP signals, that is, the MP clutter snapshot.