Dilated and Shifted Discrete Hermite Functions With Minimum Time-Bandwidth Product

In this letter, a novel method is proposed to generate discrete approximations of continuous Hermite-Gaussian functions with noninteger shift and dilation. Moreover, the approximations preserve orthogonality. In addition, the discrete Hermite-Gaussian functions reach the minimum time-bandwidth product under discrete version of Gabor uncertainty principle. Two applications are demonstrated. In fractional delay application, the proposed method is compared with the windowing method and proved to have smaller delay error. Another one is the signal expansion based on the proposed complete orthogonal basis.