## 3 The Proposed Algorithm
### NC-Euler-WESPRIT-DPD  
However, the joint spectral function in Equation (52) ignores the influence of propagation loss matrix \(\mathbf{\Lambda}_{g}\) and treats the data of all observation stations equally. Therefore, the joint spectral function is easily disturbed by the observation station with poor performance. For the sake of balancing the projection error and obtain better performance, we should assign a weight to compensate for the error. As we all know, higher SNR will lead to smaller error. Then, according to the principle of power allocation [30], we can reasonably allocate a weight \(w_{g}\) according to the SNR of each observation station.  
\[\mathbf{f}_{NC-Euler-WESPRIT}(\mathbf{p})=1/\sum_{g=1}^{G}w_{g}\bigg{(}\min_{q}\ |-2 \text{arccot}\big{(}\
u_{g,q}\big{)}-\frac{2\pi d}{\lambda}\frac{u_{g}(1)-\mathbf{p} (1)}{||\mathbf{u}_{g}-\mathbf{p}||}|\bigg{)},\ q\in[1,Q] \tag{53}\]  
Then, we rewrite Equation (23) by substituting Equation (34) into Equation (23).  
\[R_{g}=\frac{1}{T}\sum_{l=1}^{T}\left[\sum_{q=1}^{Q}\alpha_{g,q}^{2}\text{P}_{q }\mathbf{h}_{g}\mathbf{h}_{g}^{H}+\sigma_{n,g}^{2}\mathbf{I}_{2M\times 2M}\right] \tag{54}\]  
Here, it is necessary to assume that noise power \(\sigma_{n,g}^{2}\) is invariant in the process of observation [28]. According to Equation (54), SNR of different observation stations is proportional to \(\alpha_{g,q}^{2}\text{P}_{q}=\text{P}_{g,q}\), which is unknown. Nevertheless, we note that Equation (54) can be rewritten as  
\[\mathbf{R}_{g}=\mathbf{R}_{s}+\mathbf{R}_{n}=\mathbf{H}_{g}\text{diag}\big{\{}\text{P}_{g,1}, \cdots,\text{P}_{g,Q}\big{\}}\mathbf{H}_{g}^{H}+\sigma_{n,g}^{2}\mathbf{I}_{2M\times 2M} \tag{55}\]  
The eigenvalues of \(\mathbf{R}_{g}\) can be stated as  
\[\lambda_{g,m}=\left\{\begin{array}{ll}\sigma_{s,m}^{2}+\sigma_{n,g}^{2},\ 1\leq m\leq Q\\ \sigma_{n,g}^{2}\qquad\quad,\ Q+1\leq m\leq 2M\end{array}\right., \tag{56}\]  
where \(\sigma_{s,m}^{2}\) are eigenvalues of \(\mathbf{R}_{s}\) and can denote the estimated value of signal power. Then, we estimate noise power at the \(g\)th observation location:  
\[\hat{\sigma}_{n,g}^{2}=\frac{1}{2M-Q}\sum_{m=Q+1}^{2M}\lambda_{g,m} \tag{57}\]  
The corresponding estimated value of the received signal power is  
\[\hat{\text{P}}_{g}=\sum_{m=1}^{Q}\Big{(}\lambda_{g,m}-\hat{\sigma}_{n,g}^{2} \Big{)} \tag{58}\]  
Finally, the joint spectral function of NC-Euler-WESPRIT is constructed as  
\[f_{NC-Euler-WESPRIT}(\mathbf{p})=\frac{1}{\sum_{g=1}^{G}\frac{\hat{\text{P}}_{g}} {\hat{\sigma}_{n,g}^{2}}\bigg{(}\min_{q}\ |-2\text{arccot}\big{(}\
u_{g,q}\big{)}-\frac{2\pi d}{ \lambda}\frac{u_{g}(1)-\mathbf{p}(1)}{||\mathbf{u}_{g}-\mathbf{p}||}|\bigg{)}},\ q\in[1,Q] \tag{59}\]  
Acquire \(Q\) maximum points through spectral peak search, which corresponds to the target positions. Moreover, it can be seen that the spectral function in Equation (59) does not include complex number multiplication, so compared with the original subspace data fusion (SDF) algorithm [27], the computational complexity of NC-Euler-WESPRIT-DPD algorithm is greatly reduced. The main steps of NC-Euler-WESPRIT-DPD algorithm are summarized as Algorithm 1.  
```
0: all the data collected from \(G\) observation stations
0: target positions
1: Obtain the extended received signal \(\mathbf{z}_{g}(k)\) according to Equations (14) and (34).
2: Calculate the covariance matrix \(\mathbf{R}_{g}\) according to Equation (23).
3: According to Equation (24), perform EVD on \(\mathbf{R}_{g}\) in step 2.
4: Estimate the noise power \(\delta_{n,g}^{2}\) and signal power \(\tilde{\mathbf{P}}_{g}\) according to Equations (57) and (58).
5: Perform EVD on Equation (51) to obtain diagonal matrix \(\mathbf{\Xi}_{g}\).
6: Construct the weighted joint spectral function \(\mathbf{f}_{\text{NC-Euler-WESPRIT}}(\mathbf{p})\) according to Equation (59).
7: The target positions can be acquired by spectral peak search.
```  
**Algorithm 1** NC-Euler-WESPRIT-DPD