Distribution of the Summation of A-Sub N/N; A-Sub N Randomly Equal to Plus Or Minus 1

The distribution of the random variable X = n--1 J2 an/n, 00 (1) where ah a2, · · · are independent random variables equal to +1 or -- 1 with probability is of some interest in the study of intersymbol interference in a digital communication system. For example, the sum of two independent expressions of the form (1) occurs when the pulse train a,, sin (t -- nir)/(t -- nir), an randomly equal to ± 1 , is sampled at regularly spaced instants which are slightly out of step with the zeros of sin t. The theory of random variables of type (1) (in particular with a,,/3" in place of a,,/n) has been studied by a number of investigators. A survey of the field has been made recently by Hill and Blanco.1 Here we evaluate the distribution of x numerically and give expressions for its behavior when x is large. Questions of continuity and convergence are put aside. Since the distribution is even about x = 0, only values for x ^ 0 need be considered. From the characteristic function f(u) = avg [exp (ixu)J = II [cos (u/n)J n-l 0 0