Some considerations of error bounds in digital systems

Simple upper and lower bounds on the distribution function of the sum of two random variables are presented in terms of the marginal distribution functions of the variables. These bounds are then used to obtain upper and lower bounds to the error probability of a coherent digital system in the presence of intersymbol interference and additive gaussian noise. The bounds are expressed in terms of the error probability obtained with a finite pulse train, and the bounds to the marginal distribution function of the residual pulse train. Since the difference between the upper and lower bounds can be shown to be a monotonically decreasing function of the number of pulses in the finite pulse train, the bounds can be used to compute the error probability of the system with arbitrarily small error. Also when the system performance is evaluated by simulation techniques, the methods presented in our paper can be utilized to estimate the error caused by using a finite pulse train approximation.