Convolution Theorem Associated with the QWFRFT

The quaternion windowed fractional Fourier transform (QWFRFT) is a generalized form of the quaternion fractional Fourier transform (QFRFT), it plays a crucial role in signal processing for the analysis of multidimensional signals. In this paper, we first give the definition of the two-sided QWFRFT and some fundamental properties. Then, the quaternion convolution is proposed, and the relation between the quaternion convolution and the classical convolution is also given. Based on the quaternion convolution of the QWFRFT, relevant convolution theorems for the QWFRFT are studied. Moreover, the fast algorithm for QWFRFT is discussed. Finally, the complexity of QWFRFT and the quaternion windowed fractional convolution are given.