  Discrete trigonometric transformations, such as the discrete Fourier and
cosine/sine transforms, are important in a variety of applications due to their
useful properties. For example, one well-known property is the convolution
theorem for Fourier transform. In this letter, we derive a discrete Fresnel
transform (DFnT) from the infinitely periodic optical gratings, as a linear
trigonometric transform. Compared to the previous formulations of DFnT, the
DFnT in this letter has no degeneracy, which hinders its mathematic
applications, due to destructive interferences. The circular convolution
property of the DFnT is studied for the first time. It is proved that the DFnT
of a circular convolution of two sequences equals either one circularly
convolving with the DFnT of the other. As circular convolution is a fundamental
process in discrete systems, the DFnT not only gives the coefficients of the
Talbot image, but can also be useful for optical and digital signal processing
and numerical evaluation of the Fresnel transform.
