Fractional quaternion Fourier transform, convolution and correlation

Authors:
Xu Guanlei;Wang Xiaotong;Xu Xiaogang
Affiliations:
Department of Navigation, Dalian Naval Academy, Dalian 116018, China and Institute of Photoelectric Technology, Dalian Naval Academy, Dalian 116018, China;Department of Navigation, Dalian Naval Academy, Dalian 116018, China and Institute of Photoelectric Technology, Dalian Naval Academy, Dalian 116018, China;Department of Automatization, Dalian Naval Academy, Dalian 116018, China and Institute of Photoelectric Technology, Dalian Naval Academy, Dalian 116018, China
Venue:
Signal Processing
Year:
2008

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Abstract

The concept of fractional quaternion Fourier transform (FRQFT) is defined in this paper, and the reversibility property, linear property, odd-even invariant property, additivity property and other properties are presented. Meanwhile, the fractional quaternion convolution (FRQCV), fractional quaternion correlation (FRQCR) and product theorem are deduced, and their physical interpretations are given as classical convolution, correlation and product theorem. Moreover, the fast algorithms of FRQFT (FFRQFT) are yielded as well. In addition, we have discovered the relationship between the convolution and correlation in the FRQFT domain, so that the convolution and correlation can be implemented via product theorem in the Fourier transform domain using fast Fourier transform (FFT). Our paper proved that the computation complexities of FRQFT, FRQCV and FRQCR are similar to FFT.