# A Target Detection Method of Distributed Passive Radar without Direct-Path Signal
## 3 GLRT Detection Algorithm  
### Detection Algorithm  
For the receivers of the passive radar, the waveform emitted by the non-cooperative source is generally unknown. Therefore, traditional matched filtering cannot be used to achieve object detection, and multiple receivers can be used to collect target echo signals from different directions to achieve object detection. Figure 2: Passive radar geometric model. In passive radar systems, many radar systems require an additional independent channel as a reference channel, which is often used to collect direct-path signals from the irradiation source as a reference for target detection. Direct-path signals can be used to cross-correlate with echoes to achieve target detection, so obtaining a relatively pure direct-path signal is the key to achieving target detection. However, since the passive radar may not receive direct-path signals, the above method does not apply, so this paper adopts a target detection method based on without-direct-path signals. In the radar system based on the passive radiation source, the passive radiation source will send an electromagnetic pulse if there is a target, the electromagnetic wave transmitted to the target will be transmitted to the receiver, and the target echo signal received by the radar system will contain various parameter information of the target. If the target is not present, the received signal contains only noise. Obviously, the presence or absence of the target echo waves constitutes two assumptions. According to two assumptions, a decision is made that the target signal is present or that only noise is present. The assumption \(H_{1}\) indicates that the signal received by the receiving base station contains the target echo signals, and the assumption \(H_{0}\) indicates that the signal received by the receiving base station does not contain the target echo, and the target detection problem becomes a binary hypothesis problem, as shown in Equation (5). \[H_{1}:s_{j} =\gamma_{j}\Big{(}D_{j}\otimes\mathbf{a}_{j}\Big{)}\mathbf{u}+ \mathbf{n}_{j} \tag{5}\] \[H_{0}:s_{j} =\mathbf{n}_{j}\]  
Let \(s=[s_{1}^{T},\cdots,s_{N_{t}}^{T}]\in\mathbb{C}^{\mathrm{MN}_{t}L\times 1}\),\(\gamma=[\gamma_{1}\cdots\gamma_{N_{t}}]^{T}\in\mathbb{C}^{N_{t}\times 1}\). Due to the independence of receiver noise across the transmitter channels, for different receivers, it can be assumed that the received echo signals are independent of each other and that the echo signal is independent of the noise. The probability density function (PDF) under \(H_{1}\) conditions can be written as \(p_{1}(s|\gamma,\mathbf{u})\),\(p_{1}(s|\gamma,\mathbf{u})\) as shown in Equation (6). \[p_{1}(s|\gamma,\mathbf{u})=\exp\Biggl{\{}-\frac{1}{\sigma^{2}}\sum_{j=1}^{N_{t }}\ \left\|s_{j}-\gamma_{j}\Big{(}D_{j}\otimes\mathbf{a}_{j}\Big{)}\mathbf{u} \right\|^{2}\Biggr{\}} \tag{6}\]  
\(\left\|\cdot\right\|\) represents the Euclidean norm. The probability density function (PDF) under \(H_{0}\) condition can be written as \(p_{0}(s)\),\(p_{0}(s)\) as shown in Equation (7). \[p_{0}(s)=c_{n}\exp\Biggl{\{}-\frac{1}{\sigma^{2}}\left\|s\right\|^{2}\Biggr{\}} \tag{7}\]  
where \(c_{n}=\bigl{(}\pi\sigma^{2}\bigr{)}^{-N_{t}L}\) is a normalization constant. Because the transmitted signal and channel coefficients are non-deterministic and unknown, this optimal detector cannot be obtained directly. However, the actual detection algorithm can be designed according to the GLRT criterion, that is, to replace the unknown parameter with the maximum likelihood estimation (MLE). Let \(l_{1}(\gamma,\mathbf{u}|s)=\log p_{1}(s|\gamma,\mathbf{u})\) represent the log-likelihood function under \(H_{1}\).