# Array Gain of a Linear Array in an Ocean Waveguide
## 3 Array Gain in the Ocean Waveguide
We define the spatial correlation coefficient of the acoustic channels \(i\) and \(k\) as:  
\[\rho_{ik}=\frac{\text{Re}\left[\int_{f_{L}}^{f_{H}}H_{i}(f)H_{k}(f)^{*}e^{j 2\pi f\tau}df\right]}{\sqrt{\int_{f_{L}}^{f_{H}}|H_{i}(f)|^{2}df}\sqrt{\int_{f _{L}}^{f_{H}}|H_{k}(f)|^{2}df}}, \tag{12}\]  
where \(2\pi f\tau\) is the phase-shift that can maximize \(\rho_{ik}\). The correlation coefficient matrix is then constructed as:  
\[\rho=\left[ \begin{array}{cccc}\rho_{11}&\rho_{12}&\cdots&\rho_{1M}\\ \rho_{21}&\rho_{22}&\cdots&\rho_{2M}\\ \vdots&\vdots&\ddots&\vdots\\ \rho_{M1}&\rho_{M2}&\cdots&\rho_{MM}\end{array} \right]. \tag{13}\]  
We assume that the phase difference between the weights at the \(i\)th and the \(k\)th elements equal to \(2\pi f\tau\) (maximize \(\rho_{ik}\)). When the uniform amplitude weighting (i.e., \(|w_{i}|=1/M\)) is applied, Equation (11) can be compacted by substituting Equations (12) and (13) into Equation (11), and after converting the integral to the summation, we obtain:  
\[S_{\text{array}}=\frac{B|S(f)|^{2}\mathbf{h}^{\mathsf{T}}\rho\mathbf{h}}{M^{2 }T_{s}}. \tag{14}\]  
The weighting amplitude has a slight influence on \(AG\) for a linear beamformer [21]. Hence, we are only concerned with the uniform amplitude weighting in the subsequent research. According to Equations (10) and (14), the array signal gain can be obtained as:  
\[asg=\frac{S_{\text{array}}}{S_{\text{avg}}}=\frac{\mathbf{h}^{\mathsf{T}} \rho\mathbf{h}}{M\|\mathbf{h}\|_{2}^{2}}. \tag{15}\]  
For an equal inter-element spacing \(d\) in an isotropic ambient noise environment, \(ang\) has been derived as [18]:  
\[ang=\sum_{i=1}^{M}{w_{i}}^{2}  
where the subscript \"\(W\)\" denotes the formula for \(AG\) in a waveguide. For \(d=\lambda/2\), Equation (17) can be simplified to:  
\[AG_{\text{W}}=\frac{\mathbf{h}^{\text{T}}\rho\mathbf{h}}{\|\mathbf{h}\|_{2}^{2}}, \tag{18}\]  
where \(AG\) is determined by the propagation loss and the acoustic channel coherence. Then, we discuss the \(AG\) given by Equation (18) in three special cases. Case 1: If the sound wave propagates in a free space and arrives as a plane wave, the \(ATL\)s at two arbitrary elements will be equal, or \(\sqrt{p_{i}}=\sqrt{p_{k}}=\sqrt{p},\mathbf{h}=\sqrt{p}[1,1\cdots,1]\), and the corresponding acoustic channels are fully coherent (\(\rho_{ik}=1\)). In this case, Equation (18) yields the ideal value \(10\log_{10}M\). Case 2: If the acoustic channel coherence decreases with the element separation, and the \(ATL\)s at two arbitrary elements are equal, \(AG\) will deviate from the ideal value according to Equation (18). Assuming \(\mathbf{h}=\sqrt{p}[1,1,\cdots,1]\), Equation (18) can be simplified as:  
\[AG_{\text{W}}=(\sqrt{p}[1,1,\cdots,1])\rho\Big{(}\sqrt{p}[1,1,\cdots,1]^{T} \Big{)}/p\|[1,1,\cdots,1]\|_{2}^{2}=\frac{1}{M}\sum_{i=1}^{M}\sum_{k=1}^{M} \rho_{ik}. \tag{19}\]  
In this case, \(AG_{\text{W}}\) is positively correlated with the correlation coefficients. Equation (19) has the same form as Equation (2) given by [PERSON] (\(AG_{\text{U}}\)), when the ambient noise is uncorrelated. The only difference is that the \(\rho_{ik}\) in \(AG_{\text{W}}\) is the acoustic channel spatial correlation coefficient that represents the spatial fluctuation of acoustic channels over the different elements. Generally, the acoustic channel correlation coefficient can be a measure of the corresponding signal coherence. When the acoustic channel coherence decreases exponentially or linearly, the expressions for \(AG\) given in Refs. [6; 7], respectively, can be obtained. It is observed that the work by [PERSON], [PERSON] and [PERSON] were carried out based on the assumption that \(ATL\)s are equal, which is true for the special case of Equation (18).