Further results on the Beaulieu series

A frequent problem in digital communications is the computation of the probability density function (pdf) and cumulative distribution function (cdf), given the characteristic function (chf) of a random variable (RV), This problem arises in signal detection, equalizer performance, equal-gain diversity combining, intersymbol interference, and elsewhere. Often, it is impossible to analytically invert the chf to get the pdf and cdf in closed form. Beaulieu has derived an infinite series for the cdf of a sum of RVs that has been widely used. In this letter, we rederive his series using the Gil-Pelaez inversion formula and the Poisson sum formula. This derivation has several advantages including both the bridging of the well-known sampling theorem with Beaulieu's series and yielding a simple expression for calculating the truncation error term. It is also shown that the pdf and cdf can be computed directly using a discrete Fourier transform.