In this study, a novel closed expression of direction of arrival (DOA) information (DI) in a single-source scenario is proposed, which is more convenient than the theoretical expression to evaluate the performance of DOA estimation. Firstly, the authors give a uniform linear array system model, where the DI is expressed as the mutual information between the source and the received signal with complex additive white Gaussian noise. Then the posterior probability density function of DOA is deduced. Moreover, by dividing the integral interval into a signal part and a noise part, a novel closed approximate expression of DI is derived. It contains the posteriori entropy in low and high signal-to-noise ratio (SNR) cases and the normalised probability. The normalised probability is obtained through a normalisation procedure on the basis of the correlation functions. As special cases, the closed expression of DI in low and high SNR cases is given and some interesting relationships between physical quantities are revealed. Finally, numerical results are presented to verify the effectiveness of the proposed approximate closed expression.

中文翻译：

---

传感器阵列中源信号的DOA信息的闭合表达式

在这项研究中，提出了一种新颖的在单源情况下到达方向（DOA）信息（DI）的封闭表达式，它比理论表达式更方便地评估DOA估计的性能。首先，作者给出了一个统一的线性阵列系统模型，其中DI表示为源和接收信号之间的互信息，并带有复杂的加性高斯白噪声。然后推导了DOA的后验概率密度函数。此外，通过将积分间隔划分为信号部分和噪声部分，可以得出新颖的DI闭合近似表达式。它包含在高信噪比（SNR）情况下的后验熵和归一化概率。归一化概率是基于相关函数通过归一化过程获得的。作为特殊情况，给出了在低信噪比和高信噪比情况下DI的闭合表达式，并揭示了物理量之间的一些有趣关系。最后，给出数值结果以验证所提出的近似闭合表达式的有效性。