Time scale discrete Fourier transforms

The discrete and continuous Fourier transforms are applicable to discrete and continuous time signals respectively. Time scales allows generalization to to any closed set of points on the real line. Discrete and continuous time are special cases. Using the Hilger exponential from time scale calculus, the discrete Fourier transform (DFT) is extended to signals on a set of points with arbitrary spacing. A time scale D_N consisting of N points in time is shown to impose a time scale (more appropriately dubbed a frequency scale), U_N, in the Fourier domain The time scale DFT´s (TS-DFT´s) are shown to share familiar properties of the DFT, including the derivative theorem and the power theorem. Shifting on a time scale is accomplished through a boxminus and boxplus operators. The shifting allows formulation of time scale convolution and correlation which, as is the case with the DFT, correspond to multiplication in the frequency domain.