Fractional cosine and sine transforms

The fractional Fourier transform, which is a generalization of the ordinary Fourier transform (FT), was introduced 70 years ago, but only in the last decade has it been actively applied in signal processing, optics and quantum mechanics. The fractional FT gives a more complete representation of the signal in phase space and enlarges the number of applications of the ordinary FT [1]. In addition to the FT, the cosine and sine transforms (CT, ST), which are based on half-range expansions of a function over cosine and sine basis functions, respectively, are also important tools in signal processing. Despite of some lack of elegance in their properties compared to the FT, the CT and ST have their own areas of applications. The idea of fractionalization of the CT and ST was proposed in [2], where the real and imaginary parts of the fractional FT kernel were chosen as the kernels for a fractional CT and a fractional ST, respectively. Nevertheless, the authors note that their fractional transforms are not index additive and, in our point of view, cannot be considered as a fractional version of the CT and ST. In this paper we introduce fractional cosine and sine transforms that are additive on the index and preserve the similar relationships with the fractional FT as the ordinary CT and ST have with the FT. We derive the main properties of the fractional CT and ST and show, as examples, the fractional CT of some selected signals. Note that, although there are different ways for the fractionalization of cyclic transforms [3] like the FT, the CT, and the ST, in this paper we consider the fractional CT and ST in relation to the fractional FT, which is more useful for signal analysis because the fractional FT corresponds to a rotation of the Wigner distribution and the ambiguity function in phase space. 1. L.B. Almeida, "The fractional Fourier transform and time-frequency representations," IEEE Trans. Signal Process., vol. 42, pp.3084-3091, 1994. 2. A.W. Lohmann, D. Mendlovic, Z. Zalevsky, and R. G. Dorch, "Some important fractional transformations for signal processing," Opt. Commun., vol. 125, pp.18-20, 1996. 3. T. Alieva and M.L. Calvo, "Fractionalization of cyclic transforms," J. Opt. Soc. Am., vol. 17, pp. 2330-2338, 2000