Optimum Filters for Independent Measurements of Two Related Perturbed Messages

A general technique is developed for designing an optimum filter to smooth a perturbed message (signal plus noise) by employing separate and independent information about a second perturbed message whose signal portion is related to the first. For example, the second signal might be the first derivative of the original signal, or its second derivative, or its delay value, etc. The two noises are required to be statistically distinguishable. The method involves four steps: 1) feeding both sources of information into the same device, 2) utilizing known relations between the signal portions so as to remove the signal portions and leave noise terms only, 3) designing a subsidiary system to pass one of the two noise terms in an optimal way, and 4) recombining with one of the original messages so as to furnish an optimum smoothed estimate of the desired signal. A second question considered in the paper shows how one might smooth simultaneously both of the perturbed messages with a single compact system. The main ideas are illustrated in detail in a practical physical problem involving independent measurements of a signal and its derivative, such as occurs in guidance and control problems using radar equipment. Optimum physically realizable system designs are derived for handling this example.