Biquaternion Fourier transform and its applications

Abstract

In this paper, based on the biquaternion algebra, we proposed three kinds of biquaternion Fourier transforms (BiQFTs). These transforms are the extension of the
complex Fourier transform. Then, the relationships between the three kinds of transforms are obtained, and it is shown that the transform can be computed by four
complex Fourier transforms. Next, the inversion transforms and Plancherel theorems of the BiQFTs are proved. Moreover, the convolution theorems of the BiQFTs are studied by new convolution operators of the biquaternion. Finally, according to the convolution operator and convolution theorem associated with the right-side BiQFT, the biquaternion linear time-invariant systems are analyzed, and the biquaternion linear time-invariant systems for the right-side BiQFT is verified by the actual signal.