# A Maximum Likelihood Estimation Method for Underwater Radiated Noise Power
## 4 Maximum Likelihood Estimation Method
\begin{array}{ll}\sum\limits_{k}\Big{(}\mathbf{X}_{k}^{\mathrm{H}}\mathbf{R}_{n}^{- 1}\mathbf{G}\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{X}_{k}\Big{)}&=\sum\limits_{k} \Big{(}\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{X}_{k}\mathbf{X}_{k}^{\mathrm{H}}\bm {R}_{n}^{-1}\mathbf{G}\Big{)}\\ &=G^{\mathrm{H}}\mathbf{R}_{n}^{-1}\sum\limits_{k}\big{(}\mathbf{X}_{k}\mathbf{X}_{k}^{ \mathrm{H}}\big{)}\mathbf{R}_{n}^{-1}\mathbf{G}\\ &=KG^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{R}_{x}^{\prime}\mathbf{R}_{n}^{-1}\mathbf{G},\end{array}  \tag{23}\]  
where \(\mathbf{R}_{x}^{\prime}=\frac{1}{k}\sum\limits_{k}\big{(}\mathbf{X}_{k}\mathbf{X}_{k}^{ \mathrm{H}}\big{)}\) denotes the sampling covariance matrix of the received data. To find the parameter \(\sigma_{s}^{2}\) that maximizes Equation (22), by substituting Equation (23) into Equation (22) and taking the partial derivative of the resulting expression with respect to the parameter \(\sigma_{s}^{2}\)[36], one obtains  
\[ \begin{array}{ll}\frac{\partial\ln(\mathbf{p}(\mathbf{X}))}{\partial\sigma_{s}^{2}} =&-K\frac{G^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}}{\sigma_{s}^{2}\mathbf{G}^{ \mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}+1}+K\frac{G^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{R}_ {n}^{\prime}\mathbf{R}_{n}^{-1}\mathbf{G}}{\big{(}\sigma_{s}^{2}\mathbf{G}^{\mathrm{H}}\mathbf{ R}_{n}^{-1}\mathbf{G}+1\big{)}^{2}}\\ &=\frac{K\big{(}G^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}\big{)}^{2}}{\big{(}\sigma_ {s}^{2}\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}+1\big{)}^{2}}\bigg{(}\frac{G^{ \mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{R}_{n}^{\prime}\mathbf{R}_{n}^{-1}\mathbf{G}}{\big{(}G^{ \mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}\big{)}^{2}}-\frac{1}{G^{\mathrm{H}}\mathbf{R}_{n} ^{-1}\mathbf{G}}-\sigma_{s}^{2}\bigg{)}.\end{array}  \tag{24}\]  
By setting Equation (24) as equal to 0, the estimation of \(\sigma_{s}^{2}\) can be obtained, expressed as  
\[\sigma_{s}^{2}\,{}_{MLE}=\frac{\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{R}_{x}^{ \prime}\mathbf{R}_{n}^{-1}\mathbf{G}}{\Big{(}\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G} \Big{)}^{2}}-\frac{1}{\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}}=\frac{\mathbf{G}^{ \mathrm{H}}\mathbf{R}_{n}^{-1}(\mathbf{R}_{x}^{\prime}-\mathbf{R}_{n})\mathbf{R}_{n}^{-1}\mathbf{G }}{\Big{(}\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}\Big{)}^{2}}. \tag{25}\]  
For Equation (25), it can be interpreted as the outputs of Minimum Variance Distortionless Response (MVDR) beamforming of the covariance matrix difference between the received data and the ambient noise, thereby eliminating the influence of ambient noise on the estimation. Additionally, the ambient noise covariance matrix \(\mathbf{R}_{n}\) for radiated noise power estimation can be obtained by collecting ambient noise before radiated noise measurement. The channel transfer function \(\mathbf{G}\) can be calculated from marine environmental parameters using acoustic field simulation toolbox, such as the KrakenC program based on normal mode [37,38] in the Acoustics Toolbox [39]. The toolbox we are using was updated on 4 November 2020. Combining these with the sampling covariance matrix of the received data \(\mathbf{R}_{x}^{\prime}\), the radiated noise power can be accurately determined.