Joint source and sensor localization is a challenging problem, nevertheless it has numerous potential applications. This paper takes time of flight measurements and develops an approximate and yet reasonably accurate algebraic solution to the problem by converting it to the identification of an upper triangular linear transformation matrix in the localization space. The proposed solution is in closed-form and not iterative, and it reduces the solution evaluation to the positive root of a polynomial of degree 3 or less. It is among the first in solving the problem when the source and/or sensor positions can be embedded in a lower dimensional space than the localization space, leading to a complete and robust solution. The solution derivation establishes naturally the conditions for the solvability of the localization problem. The resulting solution is sufficiently close to the optimum estimate and gives better results than those from the literature. Iterative refinement of the proposed solution on the source and sensor positions provides the Cramér-Rao Lower Bound accuracy under Gaussian noise with sufficient number of sources and sensors. Performance evaluation using the data from two experiments under real environment confirms the reliability and promising behavior of the proposed solutions.

中文翻译：

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使用飞行时间测量的联合源和传感器定位的代数完整解决方案

联合源和传感器定位是一个具有挑战性的问题，但它有许多潜在的应用。本文采用飞行时间测量，并通过将其转换为定位空间中上三角线性变换矩阵的识别，为该问题开发了一个近似但合理准确的代数解决方案。建议的解决方案是封闭形式的，而不是迭代的，它将解决方案评估减少到 3 次或更小多项式的正根。当源和/或传感器位置可以嵌入到比定位空间低的维空间中时，它是最先解决问题的方法之一，从而得到完整而稳健的解决方案。解的推导自然地为定位问题的可解性建立了条件。由此产生的解决方案非常接近最佳估计，并提供比文献中的结果更好的结果。在源和传感器位置上对所提议的解决方案进行迭代改进，在高斯噪声下提供了具有足够数量的源和传感器的 Cramér-Rao 下界精度。使用来自真实环境下的两个实验数据的性能评估证实了所提出的解决方案的可靠性和有希望的行为。