# Array Gain of a Linear Array in an Ocean Waveguide
## 1 Introduction  
Sonar, no matter if passive or active, often uses array to improve the performance of communication, detection or localization depending on the end use. The improvement can be measured by array gain (\(AG\)). Array gain is defined as an improvement in the signal-to-noise ratio (\(SNR\)) obtained for an array output compared to that for a single element [1]. It is one of the most important measures of the sonar system performance which is subjected to the inhomogeneous waveguide. For the \(AG\) of a linear array, critical questions are usually posed as: How much improvement will be obtained with an array in a specific ocean waveguide? What is the length of the designed array that can ensure that the array still provides the added gain? Before addressing these questions, one should find out the mechanism of the gain affected by the ocean waveguide. Under ideal assumptions, i.e., the noise is uncorrelated, and the signal is a perfect plane wave (far field), \(AG\) can reach its ideal value \(10\log_{10}M\), where \(M\) is the number of elements in the array [2]. However, both assumptions are violated in practical sonar applications, where a waveguide must be considered. The real ocean waveguide manifests as a complex acoustic channel with spatial and temporal fluctuations that are caused mainly by the rough sea surface, the sound speed profile (SSP) and the seabed topography [3]. These factors cause the wave-front to undergo distortion and the signals to show amplitude/phase fluctuations varying across different elements [4]. This outcome ultimately leads to the practical \(AG\) lower than the ideal value [5], especially for a long-towed array [6, 7]. Nevertheless, both the decline in \(AG\) and its underlying mechanism have not been explicitly explored, due to the lack ofresearch efforts to the study of the acoustic channel characteristics. A precise formula is needed to describe the effect of the acoustic channels on \(AG\). Previous research has derived the expression for \(AG\) when the signal phase fluctuations are governed by a Gaussian joint-probability density function [8; 9; 10]. In the ocean waveguide, the fluctuations of the acoustic channel transfer functions that lead to the degradation of the coherence [11; 12] determine the signal phase fluctuations. [PERSON] calculated \(AG\) using the correlation coefficients of the signal and noise between all pairs of elements. Based on the Urick formula, [PERSON] [13] and [PERSON] [14] have derived the expression for \(AG\) in the uncorrelated noise, under the assumptions that the signal coherence decreases exponentially or linearly with the element separation, respectively. However, the signal coherence is determined by the waveguide that has uncertain change and the coherence lengths are different in shallow water or deep sea [15; 16; 17]. Furthermore, the formulas cannot predict the effect of acoustic channels on AG. In practice, different mechanisms give rise to the influence of the waveguide on the signal and the noise. In this case, \(AG\) can be divided into the array signal gain and noise gain, which has been used to study the \(AG\) of a passive vertical array [18]. In shallow water, \(AG\) can be expressed in terms of discrete normal modes [19], providing insight into the problem of the \(AG\) affected by the range-independent waveguide. However, the modes are coupled in a range-dependent waveguide, and the analyses based on the normal mode have not yet been developed for this case. A general formula of \(AG\) for a linear array in the ocean waveguide is derived in this paper based on the acoustic channel spatial coherence and the propagation losses. The physical problems involved in underwater acoustic signal processing that affect \(AG\) are investigated from the acoustic channel. The paper has the following organization. The traditional method presented by [PERSON] [1] is outlined in Section 2. Then, the analytical expression of \(AG\) in an isotropic noise field is derived in Section 3.