Fractional, canonical, and simplified fractional cosine transforms

Fourier transform can be generalized into the fractional Fourier transform (FRFT), linear canonical transform (LCT), and simplified fractional Fourier transform (SFRFT). They extend the utilities of original Fourier transform, and can solve many problems that can’t be solved well by original Fourier transform. In this paper, we will generalize the cosine transform. We will derive fractional cosine transform (FRCT), canonical cosine transform (CCT), and simplified fractional cosine transform (SFRCT). We will show they are very similar to the FRFT, LCT, and SFRFT, but they are much more efficient to deal with the even, real even functions. For digital implementation, FRCT and CCT can save 112 of the real number multiplications, and SFRCT can save 314. We also discuss their applications, such as optical system analysis and space-variant pattem recognition.