In the underwater sound source localisation systems, the sound wave signal is accompanied by time delay and Doppler shift, where the cross-ambiguity function (CAF) is commonly applied as a method of estimating them. This study proposes a novel method for calculating the exact value of the CAF,based on the theory of curved surface interpolation. Firstly, two optimisation problems with equality and inequality constraints are developed, where the penalty function method is employed to turn the equality constraint problem into an unconstrained problem. Secondly, the time delay and Doppler shift of the CAF solution are processed to obtain the measurement values required for accurate localisation. Furthermore, the authors propose an improved time difference of arrival and frequency difference of arrival (TDOA/FDOA)-based joint localisation algorithm, which changes the structure of traditional localisation equations and eliminates the localisation imprecision caused by neglecting the square term of the noise. The performance of the proposed algorithm is verified by extensive simulations and comparisons with several established methods. Remarkably, the proposed localisation method is confirmed to be noticeably superior and effective for the considered application.

中文翻译：

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基于互歧函数的水下TDOA / FDOA联合定位方法

在水下声源定位系统中，声波信号会伴有时间延迟和多普勒频移，其中交叉歧义函数（CAF）通常用作估计它们的方法。本研究基于曲面插值理论提出了一种计算CAF精确值的新方法。首先，提出了两个具有等式和不等式约束的优化问题，其中采用罚函数法将等式约束问题转化为无约束问题。其次，处理CAF解决方案的时间延迟和多普勒频移以获得精确定位所需的测量值。此外，作者提出了一种改进的基于到达时间差和到达频率差（TDOA / FDOA）的联合定位算法，从而改变了传统定位方程的结构，并消除了由于忽略噪声平方项而导致的定位不精确性。通过广泛的仿真和与几种已建立方法的比较，验证了所提出算法的性能。显着地，所提出的定位方法被证实对于所考虑的应用是明显优越和有效的。