Signal reconstruction by path integral methods

As a signal travels through a waveguide or through the atmosphere, natural conditions such as inhomogeneity and dissipation can cause alterations in the signal quality. While generally we determine the received signal given the initial waveform along with its distribution in space and the conditions between the source and received points, here we attempt to solve the inverse problem. That is, given the received signal at a point in space, reconstruct the source signal. Naturally this can be accomplished by a number of methods given the field distribution over a cross section of a region in space, but here we only wish to consider the possibility of being provided with the material conditions of the space between the source and receiver and the actual signal received at a point in space. A scheme for restoring such distorted waves has been presented by Foong (1959), based on a generalized Kac method (1959). Our approach is similar in that it relies upon a Feyman path integral formulation, but in a form quite different from that of Kac.