Applications of the discrete-time Fourier transform to data analysis

We define a discrete analogue of the characteristic function for discrete random variable and develop numerical procedures for computing the discrete characteristic function for several well-known discrete random variables. We rigorously define what is meant by the Fourier transform of a probability mass function and also show how to reverse the process to recover the probability mass function of a discrete random variable, given a procedure for computing the characteristic function. Unlike previous work on the subject, our approach is novel in that we are not computing closed-form solutions of a continuous random variable but are focused on applying the methods to real-world data sets.