# Array Gain of a Linear Array in an Ocean Waveguide
## 3 Array Gain in the Ocean Waveguide  
In a real ocean environment, the elements of the array will have different outputs. We take the average of the _SNRs_ at all the elements as the \"single hydrophone\" reference required in the definition in Equation (1). Moreover, the ocean waveguide has different effects on the signal field and the noise field, and the influence of the array processing on signal and noise should be considered separately. The ratio of the signal powers of the array output (\(S_{\text{array}}\)) and the average powers (\(S_{\text{avg}}\)) is defined as the array signal gain, denoted by _asg_. Similarly, the ratio of the noise powers of the array output (\(N_{\text{array}}\)) and the average powers (\(N_{\text{avg}}\)) is defined as the array noise gain, denoted by _ang_. Then, _AG_ can be rewritten as:  
\[AG_{\text{W}}=\frac{SNR_{\text{array}}}{SNR_{\text{avg}}}=\frac{(S/N)_{\text{ array}}}{(S/N)_{\text{avg}}}=\frac{S_{\text{array}}}{S_{\text{avg}}}/\frac{N_{ \text{array}}}{N_{\text{avg}}}=\frac{asg}{ang}, \tag{3}\]  
where \(SNR_{\text{avg}}\) is the average \(SNR\) at all the elements. For a short duration pulse, the acoustic channel can be regarded as linear time-invariant and characterized by the transfer function in the frequency domain. We assume that the array receives a narrow-band signal with a frequency band \([f_{\text{L}},f_{\text{H}}]\). The signal in the frequency domain at the \(i\)th element can be described as:  
\[X_{i}(f)=S(f)H_{i}(f), \tag{4}\]  
where \(S(f)\) is the source spectrum, and \(H_{\text{f}}(f)\) is the acoustic channel transfer function (or the Green's function) between the source and the \(i\)th element, at frequency \(f\). According to [PERSON]'s theorem, the signal energy on the \(i\)th element is given by:  
\[E_{i}=\int_{-\infty}^{\infty}\lvert X_{i}(f)\rvert^{2}df=\int_{-\infty}^{ \infty}\lvert S(f)H_{i}(f)\rvert^{2}df. \tag{5}\]  
Dividing the signal band into a number of subbands, and assuming the transfer function to be constant in each subband, the integral in Equation (5) can be converted into a summation. Further, the power of the signal on the \(i\)th element can be rewritten as:  
\[P_{i}=\frac{1}{T_{s}}\int_{f_{\text{L}}}^{f_{\text{H}}}\lvert S(f)H_{i}(f) \rvert^{2}df=\frac{\lvert S(f)\rvert^{2}}{T_{s}}\sum_{b=1}^{B}\lvert H_{i}(f_ {b})\rvert^{2}, \tag{6}\]  
where \(T_{s}\) is the duration of the signal, and \(B\) is the number of frequency bins in \([f_{\text{L}},f_{\text{H}}]\). \(\lvert H_{i}(f)\rvert\) is the amplitude of the transfer function that takes the pressure at 1 m away from the source as the reference, which also denotes the pressure at the \(i\)th element. Denoting \(p_{i}\) as the mean square of the pressure at the \(i\)th element:  
\[p_{i}=\frac{1}{B}\sum_{b=1}^{B}\lvert H_{i}(f_{b})\rvert^{2}, \tag{7}\]  
the average transmission loss (ATL) (averaged over the frequencies) is:  
\[\text{ATL}_{i}=-10\log_{10}p_{i}, \tag{8}\]  
and the average signal power \(S_{\text{avg}}\) of all elements is:  
\[S_{\text{avg}}=\frac{1}{M}\sum_{i=1}^{M}P_{i}=\frac{B\lvert S(f)\rvert^{2}}{T _{s}M}\sum_{i=1}^{M}p_{i}. \tag{9}\]Denoting \(\mathbf{h}\triangleq\left[\sqrt{p_{1}},\sqrt{p_{2}},\cdots,\sqrt{p_{M}}\right]^{ \mathsf{T}}\), with the superscript \"T\" representing the transpose operation, Equation (9) can be written in a compact form as:  
\[S_{\text{avg}}=\frac{B|S(f)|^{2}}{T_{s}M}\|\mathbf{h}\|_{2}^{2}, \tag{10}\]  
where \"\(\left\|\cdot\right\|_{2}\)\" denotes the 2-norm of a vector. The signal power at the array output after weighting is:  
\[S_{\text{array}}=\frac{1}{T_{s}}\int_{f_{L}}^{f_{H}}\left|\sum_{i=1}^{M}w_{i}X_ {i}(f)\right|^{2}df=\frac{|S(f)|^{2}}{T_{s}}\sum_{i=1}^{M}\sum_{k=1}^{M}\int_{f _{L}}^{f_{H}}w_{i}H_{i}(f)H_{k}(f)^{*}{w_{k}}^{*}df, \tag{11}\]  
where \(w_{i}\) is the weighting coefficient at the \(i\)th element using to compensate for the phase difference between acoustic channels, and the superscript \"\({}^{*}\)\" denotes the complex conjugate operation.