The Time-Domain Synthesis of Distributions

A new dimension to the time-domain synthesis problem is proposed. In particular, the objective is to find a realizable signal that approximates a given distribution (ie., generalized function). A solution is then presented, several convergence criteria are discussed, and some examples are given. A feature of the present technique is that, if the Laplace transform of the given distribution is known, the realizable approximating signal can be written down without any computation; it only requires values of the Laplace transform at various points in its region of convergence. The method also yields a convenient technique for "visualizing" those distributions that cannot be plotted. Since ordinary locally integrable functions are special cases of distributions, the technique is significant for the customary time-domain synthesis problem as well.