# A Maximum Likelihood Estimation Method for Underwater Radiated Noise Power
## 5 The Cramer-Rao Lower Bound Derivation  
To analyze the estimation performance of the proposed method, we theoretically verified whether its estimation variance achieves the CRLB. First, the theoretical value of CRLB is derived. For Equation (24), taking the second partial derivative with respect to the parameter \(\sigma_{s}^{2}\) yields  
\[\frac{\partial^{2}\ln(\mathrm{p}(\mathbf{X}))}{\partial\left(\sigma_{s}^{2} \right)^{2}}=-K\left(\frac{2\mathrm{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{ G}\mathrm{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{R}_{x}^{\prime}\mathbf{R}_{n}^{-1} \mathbf{G}}{\left(\sigma_{s}^{2}\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1} \mathbf{G}+1\right)^{3}}-\frac{\left(\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1 }\mathbf{G}\right)^{2}}{\left(\sigma_{s}^{2}\mathbf{G}^{\mathrm{H}}\mathbf{R}_ {n}^{-1}\mathbf{G}+1\right)^{2}}\right). \tag{26}\]  
According to Reference [36], the Fisher information matrix \(I\left(\sigma_{s}^{2}\right)\) can be obtained by taking the negative expectation of Equation (26), denoted as  
\[ \begin{array}{ll}\mathbf{I}\left(\sigma_{s}^{2}\right)&=-\mathrm{E}\left( \frac{\partial^{2}\ln(\mathrm{p}(\mathbf{X}))}{\partial\left(\sigma_{s}^{2} \right)^{2}}\right)\\ &=K\left(\frac{2\mathrm{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}\mathrm{G }^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}+1}{\left(\sigma_{s}^{2}\mathbf{G} ^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}+1\right)^{3}}-\frac{\left( \mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}\right)^{2}}{\left( \sigma_{s}^{2}\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}+1\right)^ {2}}\right)\\ &=K\left(\frac{2\mathrm{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}\mathrm{G }^{\mathrm{H}}\mathbf{R}_{n}^{-1}\left(\sigma_{s}^{2}\mathbf{G}\mathrm{G}^{ \mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}+1\right)^{3}}-\frac{\left(\mathbf{G} ^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}\right)^{2}}{\left(\sigma_{s}^{2} \mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}+1\right)^{2}}\right)\\ &=\frac{K\left(\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}\right)^{2} }{\left(\sigma_{s}^{2}\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}+1 \right)^{2}},\end{array}  \tag{27}\]  
where \(\mathrm{E}(\mathbf{R}_{x}^{\prime})=\mathbf{R}_{x}=\sigma_{s}^{2}\mathbf{G} \mathbf{G}^{\mathrm{H}}+\mathbf{R}_{n}\). Subsequently, the CRLB can be calculated and represented as  
\[\mathrm{CRLB}=\left(I\left(\sigma_{s}^{2}\right)\right)^{-1}=\frac{1}{K}\left( \sigma_{s}^{2}+\frac{1}{\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}} \right)^{2}. \tag{28}\]  
Subsequently, we prove that the proposed power estimation method achieves the CRLB. For Equation (24), it can be transformed into  
\[ \begin{array}{ll}\frac{\partial\ln(\mathrm{p}(\mathbf{X}))}{\partial\sigma_ {s}^{2}}&=\frac{K\left(\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G} \right)^{2}}{\left(\sigma_{s}^{2}\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1} \mathbf{G}+1\right)^{2}}\left(\frac{G^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{ R}_{n}^{\prime}\mathbf{R}_{n}^{-1}\mathbf{G}}{\left(\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1} \mathbf{G}\right)^{2}}-\frac{1}{\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1} \mathbf{G}}-\sigma_{s}^{2}\right)\\ &=I\left(\sigma_{s}^{2}\right)\left(\sigma_{s\ MLE}^{2}-\sigma_{s}^{2}\right).\end{array}  \tag{29}\]  
According to CRLB theorem [36], the variance of estimation meets the CRLB. In addition, the expectation of Equation (25) is given by  
\[ \begin{array}{ll}\mathrm{E}\left(\sigma_{s\ MLE}^{2}\right)&=\frac{G^{ \mathrm{H}}\mathbf{R}_{n}^{-1}\mathrm{E}\left(\mathbf{R}_{n}^{\prime}\mathbf{R }_{n}^{-1}\mathbf{G}-\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G} \right.}{\left(\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}\right)^{2 }}\\ &=\frac{G^{\mathrm{H}}\mathbf{R}_{n}^{-1}\left(\sigma_{s}^{2}\mathbf{G}\mathbf{G} ^{\mathrm{H}}+\mathbf{R}_{n}\right)\mathbf{R}_{n}^{-1}\mathbf{G}-\mathbf{G}^{ \mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}}{\left(\mathbf{G}^{\mathrm{H}} \mathbf{R}_{n}^{-1}\mathbf{G}\right)^{2}}\\ &=\sigma_{s}^{2}.\end{array}