Optimal Discrete Gaussian Function: The Closed-Form Functions Satisfying Tao’s and Donoho’s Uncertainty Principle With Nyquist Bandwidth

In this paper, closed-form discrete Gaussian functions are proposed. The first property of these functions is that their discrete Fourier transforms are still discrete Gaussian functions with different index parameter. This index parameter, which is an analogy to the variance in continuous Gaussian functions, controls the width of the function shape. Second, the discrete Gaussian functions are positive and bell-shaped. More important, they also have finite support and consecutive zeros. Thus, they satisfy Tao's and Donoho's uncertainty principle of discrete signal. The construction of these discrete Gaussian functions is inspired by Kong's zeroth-order discrete Hermite Gaussian functions. Three examples are discussed.