Maximum Entropy Signal Transmission

Abstract

There are two means at our disposal to understand the behavior of physical systems: observation and experimentation. Observation becomes increasingly difficult as an object becomes more remote or obscure. Experimentation is impossible for objects that cannot be manipulated or directly contacted. In such cases it is necessary to use numerical simulation, drawing upon perceived virtual systems expressed through models. Two main approaches to deal with remote or obscure objects come under the headings of the “inverse source problem” and the “inverse medium problem.” In the typical inverse source problem, the source of energy is remote, the medium transmits the source signal to an accessible receiver, and information about the source is required. An example of an inverse source problem is classical earthquake seismology where received seismic data are used to determine locations of remote earthquakes. Another example is passive sonar where engine noise from a hidden submarine is used to locate its position. In the typical inverse medium problem the source of energy (usually man-made) is local, the signal penetrates an inaccessible medium that reflects energy back to accessible points, and information about the internal structure of the medium is required. Examples are reflection seismology, radar, and active sonar. The usual approach to either type of inverse problem is first to devise a theoretical model that admits a solution from the available data. Implementation then involves using the theoretical model to find the required solution, often through an iterative improvement method. One of the most basic models is a system with parallel plane layers (the so-called layer-cake model of geophysics). An important characteristic of the layer-cake model is that it yields a transmitted signal that has maximum entropy. Because of this property, a signal from a distant source can be deconvolved to remove the unwanted reverberations that occurred during transmission. Thus it is possible to obtain a good representation of the unknown source signal, and so the inverse source problem for the layer-cake model has an effective computer solution. The layer-cake model also has the characteristic that it yields a reflected signal with both feedforward and feedback components, where the feedback component has maximum entropy. In practical terms, this maximum entropy property means that the received reflected signal preserves the information about the structure of the medium. As a result, the received reflected signal can be deconvolved to give a picture of the internal structure of the medium, and so the inverse medium problem for the layer-cake model also has a computer solution.