## Appendix A CRLBs Deduction  
### CRLB Deduction in the Known Target Transmit Power Case  
In the known target transmit power case, according to the observation vectors \(\mathbf{P}=[P_{1},P_{2},\cdots,P_{N}]^{T}(\mathbf{P}\in\mathbb{R}^{N})\) in Equation (1), the condition PDF in the given \(\mathbf{x}\) is  
\[f(\mathbf{P}|\mathbf{x})=\frac{1}{(2\pi)^{\frac{1}{2}}}\frac{1}{(|\mathbf{Q}|)^{\frac{1}{2 }}}\exp\left\{-\frac{1}{2}\left(\mathbf{P}-\hat{\mathbf{P}}(\mathbf{x})\right)^{T}\mathbf{Q}^ {-1}\left(\mathbf{P}-\hat{\mathbf{P}}(\mathbf{x})\right)\right\} \tag{1}\]  
where \(\hat{\mathbf{P}}(\mathbf{x})=[\hat{P}_{1}(\mathbf{x}),\cdots,\hat{P}_{N}(\mathbf{x})]^{T}\), and \(\hat{P}_{l}(\mathbf{x})=P_{0}-10\gamma\log_{10}\frac{\|\mathbf{x}-\mathbf{s}_{l}\|}{d_{0}} -\alpha_{f}(\|\mathbf{x}-\mathbf{s}_{l}\|-d_{0}),i=1,\cdots,N\).  
Thus, the log-likelihood function w.r.t the variable \(\mathbf{x}\) has the following form  
\[\ln f(\mathbf{P}|\mathbf{x})=-\frac{N}{2}\ln(2\pi)-\frac{1}{2}\ln(|\mathbf{Q}|)-\frac{1}{ 2}\left(\mathbf{P}-\hat{\mathbf{P}}(\mathbf{x})\right)^{T}\mathbf{Q}^{-1}\left(\mathbf{P}-\hat{ \mathbf{P}}(\mathbf{x})\right). \tag{2}\]For any unbiased estimator, i.e., \(E(\widehat{\mathbf{x}})=\mathbf{x}\), the covariance matrix \(cov(\widehat{\mathbf{x}})\) is lower bound by \(cov(\widehat{\mathbf{x}})\geq J^{-1}\), where \(\mathbf{J}\) is the Fisher information matrix (FIM). The CRLB is the inverse of FIM, \(J^{-1}\), which can be derived from Equation (A2) as follows:  
\[\frac{\partial\ln f(\mathbf{P}|\mathbf{x})}{\partial\mathbf{x}}=\left(\mathbf{P}-\hat{\mathbf{P}}( \mathbf{x})\right)^{T}\mathbf{Q}^{-1}\frac{\partial\hat{\mathbf{P}}(\mathbf{x})}{\partial\mathbf{x}}.\] (A3)  
Therefore,  
\[\frac{\partial^{2}\ln f(\mathbf{P}|\mathbf{x})}{\partial\mathbf{x}\partial\mathbf{x}^{T}}=- \left[\frac{\partial\hat{\mathbf{P}}(\mathbf{x})}{\partial\mathbf{x}}\right]^{T}\mathbf{Q}^{ -1}\frac{\partial\hat{\mathbf{P}}(\mathbf{x})}{\partial\mathbf{x}}+\left(\mathbf{P}-\hat{\mathbf{ P}}(\mathbf{x})\right)^{T}\mathbf{Q}^{-1}\frac{\partial\hat{\mathbf{P}}^{2}(\mathbf{x})}{ \partial\mathbf{x}\partial\mathbf{x}^{T}}.\] (A4)  
Based on the assumption that \(n_{i}\) is a zero mean Gaussian random variable with the covariance matrix \(\mathbf{Q}\), and taking the negative expectation on both sides of Equation (A4), we have  
\[-E\left(\frac{\partial^{2}\ln f(\mathbf{P}|\mathbf{x})}{\partial\mathbf{x} \partial\mathbf{x}^{T}}\right)=\left[\frac{\partial\hat{\mathbf{P}}(\mathbf{x})}{\partial \mathbf{x}}\right]^{T}\mathbf{Q}^{-1}\frac{\partial\hat{\mathbf{P}}(\mathbf{x})}{\partial\mathbf{x }}=\mathbf{G}^{T}\mathbf{Q}^{-1}\mathbf{G}\] (A5)  
where \(\mathbf{G}=\frac{\partial\hat{\mathbf{P}}(\mathbf{x})}{\partial\mathbf{x}}=[g_{1},g_{2},\cdots,g_{N}]^{T}\) with  
\[g_{i}=\frac{\partial\hat{P}_{i}(\mathbf{x})}{\partial\mathbf{x}}=-\frac{1 0\gamma}{\ln 10}\frac{(\mathbf{x}-\mathbf{s}_{i})^{T}}{\|\mathbf{x}-\mathbf{s}_{i}\|^{2}}-\alpha_{ f}\frac{(\mathbf{x}-\mathbf{s}_{i})^{T}}{\|\mathbf{x}-\mathbf{s}_{i}\|}.\] (A6)  
Finally, we derive the closed-form expression of the CRLB in the known target transmit power case as  
\[\text{CRLB-K}=\text{trace}(\mathbf{B}_{1})\] (A7)  
where \(\mathbf{B}_{1}=(\mathbf{G}^{T}\mathbf{Q}^{-1}\mathbf{G})^{-1}\).