# Vertical Correlation and Array Gain Analysis for Vertical Line Array in Deep Water
## 2 Basic Theory
### Signal Gain and Noise Gain  
As mentioned above, array gain is defined as the improvement of the signal-to-noise ratio (SNR) between the hydrophone array and the single element, which can be written as  
\[AG=10\log\frac{(S/N)_{array}}{(S/N)_{element}}=10\log\frac{\left|\sum\limits_{g =1}^{M}s_{g}{{}^{\prime}(f)}\right|^{2}/\left|\sum\limits_{g=1}^{M}n_{g}{{}^ {\prime}(f)}\right|^{2}}{{\sigma_{s}}^{2}/{\sigma_{n}}^{2}} \tag{11}\]  
where \(s_{g}{{}^{\prime}(f)}\) and \(n_{g}{{}^{\prime}(f)}\) are spectrums of signal and noise of \(g\)th hydrophone after time-delay compensation, respectively, \(\sigma_{s}{{}^{2}}\) and \(\sigma_{n}{{}^{2}}\) are signal power and noise power of the single element. After expanding the square term of absolute value in Equation (11), the expression can be deduced as  
\[AG=10\log\frac{\left({\sigma_{s}}^{2}/{\sigma_{n}}^{2}\right)\cdot\sum\limits_ {u=1}^{M}\sum\limits_{v=1}^{M}(\rho_{s})_{uv}/\sum\limits_{u=1}^{M}\sum\limits _{v=1}^{M}(\rho_{n})_{uv}}{{\sigma_{s}}^{2}/{\sigma_{n}}^{2}} \tag{12}\]Taking the further simplification of Equation (12), the AG is only decided by the correlation coefficients of signal and noise, which can be calculated by  
\[AG=10\log\left[\begin{array}{c}\sum\limits_{u=1}^{M}\sum\limits_{v=1}^{M}(\rho_ {s})_{uv}\\ \sum\limits_{u=1}^{M}\sum\limits_{v=1}^{M}(\rho_{n})_{uv}\end{array}\right] \tag{13}\]  
The expression of AG in Equation (13) can be divided into two parts, the numerator and the denominator, into which a common factor is introduced. The new numerator and denominator are redefined as the signal gain and noise gain, which takes the following form  
\[SG=10\log\left[\begin{array}{c}\sum\limits_{u=1}^{M}\sum\limits_{v=1}^{M}( \rho_{s})_{uv}\\ \sum\limits_{u=1}^{M}\sum\limits_{v=1}^{M}(\rho_{is-n})_{uv}\end{array}\right] \tag{14}\]  
\[NG=10\log\left[\begin{array}{c}\sum\limits_{u=1}^{M}\sum\limits_{v=1}^{M}( \rho_{n})_{uv}\\ \sum\limits_{u=1}^{M}\sum\limits_{v=1}^{M}(\rho_{is-n})_{uv}\end{array}\right] \tag{15}\]  
where \(\rho_{is-n}\) denotes the correlation coefficient of isotropic noise, whose formula is shown in Equation (9). The array gain is redefined as the difference between the signal gain and noise gain, and thus it can be written in the form of dB as  
\[AG=SG-NG \tag{16}\]  
When the noise is isotropic, and the element interval is equal to half wavelength, the cross-correlation terms of the noise correlation matrix degenerate to zeroes. When the sound source radiating a single plane wave is placed in the far-field, the received signals of two arbitrary hydrophones are entirely correlated. Therefore, the numerator of Equations (14) and (15) can be reduced to  
\[\sum\limits_{u=1}^{M}\sum\limits_{v=1}^{M}(\rho_{s})_{uv}=M^{2} \tag{17}\]  
\[\sum\limits_{u=1}^{M}\sum\limits_{v=1}^{M}(\rho_{n})_{uv}=M \tag{18}\]  
Further, the signal gain and noise gain take the simplified form as  
\[SG=10\log\frac{M^{2}}{M}=10\log M \tag{19}\]  
\[NG=10\log\frac{M}{M}=0 \tag{20}\]  
Under these assumptions, AG of the VLA can reach its ideal value of \(10\log M\) in the isotropic noise field through conventional beamforming technology. It should be noted that NG is probably negative because of weak correlation, and thus AG can exceed the ideal value. This situation significantly improves array performance.