In this paper, we mainly establish the uncertainty principle (UP) for a function and its quaternion Fractional Fourier transform (QFrFT), as well as the UP for two QFrFTs. Using the polar representation of quaternion-valued signals, we give the UP for QFrFT in both the spatial and directional domains, providing a more precise condition for equality, example is given to verify the results. Furthermore, we extend the time-frequency UP to a frequency-frequency setting.

翻译：本文主要建立了函数及其四元数分数阶傅里叶变换（QFrFT）的不确定性原理（UP），以及两个QFrFT之间的不确定性原理。利用四元数值信号的极坐标表示，我们给出了空间域和方向域上QFrFT的不确定性原理，提供了更精确的等式成立条件，并通过实例验证了相关结果。此外，我们将时频不确定性原理推广至频-频场景。