## 2 \(N\)-Way GPR Target and Detection
### Detection  
Based on the \(N\)-way GPR targets identified, we designed the detection for each \(N\)-way GPR target. First, with the 2-way GPR target, we used the conventional VA to implement detection. The trellis, in this case, includes four states and two branches in each state, as shown in Figure 3. Next, we considered a 3-way GPR target. The trellis includes eight states and four branches in each state (Figure 4). However, we can use the MVA with feedback in [21] to reduce the complexity of detection for a 3-way GPR target. Simultaneously, the detection performance of [21] was better than that of the trellis, as shown in Figure 4.  
Figure 3: Trellis for the detection of the 2-way GPR target.  
Figure 2: Training model for estimating the GPR target and equalizer parameters.  
Next, we considered a 4-way GPR target. The trellis includes 16 states and eight branches in each state. Because this trellis has high complexity, [PERSON] and [PERSON] [21] also proposed the detection of a 4-way GPR target.  
Finally, the detection trellis for the 8-way GPR target exhibited the highest complexity. The trellis includes 64 states and eight branches in each state. [PERSON] and [PERSON] [22,23] designed serial detection (Figure 5) for when the GPR target is closer to the 8-way GPR. This is an approximation of the true 8-way GPR and can be decomposed as follows:  
\[\mathbf{G}=\begin{bmatrix}rp&p&rp\\ r&1&r\\ rp&p&rp\end{bmatrix}=\begin{bmatrix}0&p&0\\ 0&1&0\\ 0&p&0\end{bmatrix}\begin{bmatrix}0&0&0\\ r&1&r\\ 0&0&0\end{bmatrix}. \tag{3}\]  
The trellis of the horizontal detector includes 36 states and six branches in each state, and the trellis of the vertical detector has four states and two input/output branches in each state.  
The true 8-way GPR target has the following form:  
\[\mathbf{G}=\begin{bmatrix}t&p&t\\ r&1&r\\ t&p&t\end{bmatrix}, \tag{4}\]  
Figure 4: Trellis for the detection of the 3-way GPR target.  
Figure 5: Model for serial detection [22,23].  
where the parameter \(t\) must be independent of the parameters \(p\) and \(r\). This fact was not addressed in [22; 23] and is resolved in Section 3.  
We also calculated the MMSE of each GPR target. The results are listed in Table 1. Here, we can see that the performance improved as the value of \(N\) increased (the best results are bold). Except for the 3-way GPR target, because the channel is affected by asymmetric interference, when applying the 3-way GPR target, these interferences will be deflected in one direction, and the equalizer cannot estimate the signal to achieve better MMSE compared to that of the 2-way GPR target. The \(\widetilde{8}\)-way GPR target (Equation (3)) and 8-way GPR target (Equation (4)) appear to achieve similar performance. However, the 8-way GPR target was the most accurate. Therefore, although designing the detection for the 8-way GPR target is very difficult, building a suitable detection method for the 8-way GPR target is still needed.  
Next, we calculated the complexity of detection for each GPR target. Here, we only considered the raw VA with a trellis, as shown in Figures 3 and 4, ignoring the modified models for the VA. Table 2 lists the number of operations per bit of detection for each GPR target.  
From Table 2, we see that the detection for the 8-way GPR is highly complex. Therefore, researchers avoided using the raw VA model in this case. Instead, they always attempted to modify the VA to match the 8-way GPR target, as in [22; 23]. In the next section, we propose models that improve the performance and reduce the complexity of an 8-way GPR target.