# A Maximum Likelihood Estimation Method for Underwater Radiated Noise Power
## 4 Maximum Likelihood Estimation Method  
Differing from beamforming-based approaches, this section introduces a radiated noise power estimation method based on probability density maximization. Since the array elements are mutually independent, the signal received by the array follows a joint normal distribution. In the complex domain, the Probability Density Function (PDF) is expressed as [36]  
\[\text{p}(\mathbf{X}_{k})=(\pi)^{-M}|\mathbf{R}_{x}|^{-1}\exp\Bigl{(}-\left(\mathbf{X}_{k}^ {\text{H}}\mathbf{R}_{x}^{-1}\mathbf{X}_{k}\right)\Bigr{)}. \tag{16}\]  
where \(|*|\) denotes the determinant of a matrix. For the received \(K\) independent and identically distributed frequency snapshots, the corresponding joint PDF is  
\[ \begin{array}{ll}\text{p}(\mathbf{X})&=\prod\limits_{k}\text{p}(\mathbf{X}_{k})\\ &=\prod\limits_{k}(\pi)^{-M}|\mathbf{R}_{x}|^{-1}\exp\bigl{(}-\left(\mathbf{X}_{k}^{ \text{H}}\mathbf{R}_{x}^{-1}\mathbf{X}_{k}\right)\bigr{)}\\ &=(\pi)^{-MK}|\mathbf{R}_{x}|^{-K}\exp\biggl{(}-\sum\limits_{k}\left(\mathbf{X}_{k}^ {\text{H}}\mathbf{R}_{x}^{-1}\mathbf{X}_{k}\right)\biggr{)}.\end{array}  \tag{17}\]  
For the matrix \(\mathbf{R}_{x}\), it can be transformed into  
\[\mathbf{R}_{x}=\sigma_{s}^{2}\mathbf{G}\mathbf{G}^{\text{H}}+\mathbf{R}_{n}=\mathbf{R}_{n}\Bigl{(} \sigma_{s}^{2}\mathbf{R}_{n}^{-1}\mathbf{G}\mathbf{G}^{\text{H}}+\mathbf{I}\Bigr{)}, \tag{18}\]  
where \(\mathbf{I}\) is the identity matrix. For the rank-1 matrix \(\sigma_{s}^{2}\mathbf{R}_{n}^{-1}\mathbf{G}\mathbf{G}^{\text{H}}\) in Equation (18), it has only one non-zero eigenvalue \(\lambda\). Additionally, since the sum of all eigenvalues of a matrix equals its trace, it follows that  
\[\lambda=\text{tr}\Bigl{(}\sigma_{s}^{2}\mathbf{R}_{n}^{-1}\mathbf{G}\mathbf{G}^{\text{H}} \Bigr{)}=\text{tr}\Bigl{(}\sigma_{s}^{2}\mathbf{G}^{\text{H}}\mathbf{R}_{n}^{-1}\mathbf{G }\Bigr{)}=\sigma_{s}^{2}\mathbf{G}^{\text{H}}\mathbf{R}_{n}^{-1}\mathbf{G}, \tag{19}\]  
where \(\text{tr}(*)\) denotes the trace of a matrix. Moreover, as the determinant of a matrix is equal to the product of all its eigenvalues, then  
\[|\mathbf{R}_{x}|=|\mathbf{R}_{n}|\Bigl{|}\sigma_{s}^{2}\mathbf{R}_{n}^{-1}\mathbf{G}\mathbf{G}^{ \text{H}}+\mathbf{I}\Bigr{|}=|\mathbf{R}_{n}|\Bigl{(}\sigma_{s}^{2}\mathbf{G}^{\text{H}} \mathbf{R}_{n}^{-1}\mathbf{G}+1\Bigr{)}. \tag{20}\]  
According to the matrix inversion lemma, the inverse of the matrix \(\mathbf{R}_{x}\) can be expressed as \[ \begin{array}{ll}\mathbf{R}_{x}^{-1}&=\Big{(}\mathbf{R}_{n}+\mathbf{G}\sigma_{s}^{2}\mathbf{G}^{ \mathrm{H}}\Big{)}^{-1}\\ &=\mathbf{R}_{n}^{-1}-\mathbf{R}_{n}^{-1}\mathbf{G}\Big{(}\sigma_{s}^{-2}+\mathbf{G}^{\mathrm{H} }\mathbf{R}_{n}^{-1}\mathbf{G}\Big{)}^{-1}\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\\ &=\mathbf{R}_{n}^{-1}-\frac{\sigma_{s}^{2}}{\sigma_{s}^{2}\mathbf{G}^{\mathrm{H}}\mathbf{R }_{n}^{-1}\mathbf{G}+1}\mathbf{R}_{n}^{-1}\mathbf{G}\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}. \end{array}  \tag{21}\]  
Substituting Equations (20) and (21) into Equation (17), and additionally, to facilitate computations, Equation (17) is represented in logarithmic form as  
\[ \begin{array}{ll}\ln(\mathbf{p}(\mathbf{X}))&=-MK\ln(\pi)-K\ln(|\mathbf{R}_{x}|)-\sum \limits_{k}\big{(}\mathbf{X}_{k}^{\mathrm{H}}\mathbf{R}_{x}^{-1}\mathbf{X}_{k}\big{)}\\ &=-MK\ln(\pi)-K\ln(|\mathbf{R}_{n}|)-K\ln\Big{(}\sigma_{s}^{2}\mathbf{G}^{\mathrm{H}} \mathbf{R}_{n}^{-1}\mathbf{G}+1\Big{)}\\ &-\sum\limits_{k}\big{(}\mathbf{X}_{k}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{X}_{k}\big{)} +\frac{\sigma_{s}^{2}}{\sigma_{s}^{2}\mathbf{G}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G }+1}\sum\limits_{k}\Big{(}\mathbf{X}_{k}^{\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{G}\mathbf{G}^ {\mathrm{H}}\mathbf{R}_{n}^{-1}\mathbf{X}_{k}\Big{)}.\end{array}  \tag{22}\]  
In Equation (22),  
\[