Convolution And Its Properties For Quaternion Fractional Fourier Transform

Quaternion fractional Fourier transform is a new transform combining quaternion and fractional Fourier transform,which has been highly valued by many researchers in recent years.However,quaternion fractional Fourier transform is not a local transform but a global transform.Therefore,in order to overcome this defect,the windowed quaternion fractional Fourier transform is proposed based on the windowed quaternion fractional Fourier transform are discussed.The main work of this paper includes the following aspects:1.Firstly,the definition of quaternion and fractional Fourier transform is introduced;Then the definition of quaternion Fourier transform and some related properties are given;Finally,the definition of quaternion fractional Fourier transform is also proposed,and some corresponding properties,such as reversibility,linearity,time shift and modulation are discussed.2.Based on the relationship between the quaternion fractional Fourier transform and windowed fractional Fourier transform,three types of quaternion windowed fractional Fourier transform(QWFRFT)are defined.Namely,the two-sided windowed quaternion fractional Fourier transform(two-sided QWFRFT),the left-sided windowed quaternion fractional Fourier transform(left-sided QWFRFT)and the right-sided windowed quaternion fractional Fourier transform(right-sided QWFRFT).This paper focuses on the properties of two-sided quaternion windowed fractional Fourier transform,a new convolution operation is proposed,the convolution theorem is derived,and the algorithm complexity analysis of quaternion windowed fractional Fourier transform is also given.3.Based on quaternion fractional Fourier transform,three kinds of quaternion offset fractional Fourier transform(QOFRFT)are defined,including two-sided quaternion offset fractional Fourier transform(two-sided QOFRFT),left-sided quaternion offset fractional Fourier transform(left-sided QOFRFT)and right-sided offset fractional Fourier transform(right-sided QOFRFT).The convolution and correlation operations of three kinds of QOFRFT are defined,respectively,and the corresponding convolution and correlation theorems are also derived.