Discrete generalized Fresnel functions and transforms in an arbitrary discrete basis

The idea of generalized Fresnel functions, which traces back to expressing a discrete transform as a linear convolution, is developed in this paper. The generalized discrete Fresnel functions and the generalized discrete Fresnel transforms for an arbitrary basis are considered. This problem is studied using a general algebraic approach to signal processing in an arbitrary basis. The generalized Fresnel functions for the discrete Fourier transform (DFT) are found, and it is shown that DFT of even order has two generalized Fresnel functions, while DFT of odd order has a single generalized Fresnel function. The generalized Fresnel functions for the conjunctive and Walsh transforms and the generalized Fresnel transforms induced by these functions are considered. It is shown that the generalized Fresnel transforms induced by the Walsh basis and the corresponding generalized Fresnel functions are unitary and that the generalized Fresnel transforms induced by the conjunctive basis and the corresponding generalized Fresnel functions consist of powers of the golden ratio. It is also shown that the Fresnel transforms induced by the generalized Fresnel functions for the Walsh and conjunctive transforms have fast algorithms.