Problems in Network Theory

After a brief philosophical consideration of the status and function of the theoretician in modern technology, the central objectives of the paper are stated; to investigate some of the restrictions on the design and synthesis of linear physical systems which are inherent in the mathematical constructs and methods by which systems are studied. The principal tools are the Laplace transformation and the theory of functions of a complex variable. It is shown that many of the commonly encountered generalizations to linear distributed-parameter systems of familiar lumped-parameter-system ideas are valid. These generalizations are perhaps intuitively obvious, but the details are gone through here once and for all. The implications in regard to these matters of the principle of analytic continuation are considered. Tests are derived to enable one to decide whether or not there is any chance of realizing a prescribed transfer characteristic T(jw), w1 less than w less than w2 (analytically expressed data), and what would be the consequences in terms of T(s) outside this range. The paradox of the idealized low-pass filter is examined in this light. The questions are shown to be unanswerable in the case of graphically expressed data. It is shown that the results of the study are in agreement with the allied work of others, and a problem of filter realization posed by Wallman is solved. The results of the investigations are summarized and reviewed in terms of what theoreticians can accomplish in general.