The Properties and Fast Algorithm of Quaternion Linear Canonical Transform

The quaternion linear canonical transform (QLCT) is defined in this paper, with proofs given for its reversibility property, its linear property, its odd-even invariant property and additivity property. Meanwhile, the quaternion convolution (QCV), quaternion correlation (QCR) and product theorem of LCT are deduced. Their physical interpretation is given as classical convolution, correlation and product theorem. Moreover, the fast algorithm of QLCT (FQLCT) is obtained, whose calculation complexity for different signals is similar to FFT. In addition, the paper presents the relationship between the convolution and correlation in LCT domains, and the convolution and correlation can be calculated via product theorem in Fourier transform domain using FFT.