The role of losslessness in the network representation of stationary stochastic processes via Cauer-Darlington type synthesis

A brief review is given on the work of Wilhelm Cauer, who recognized the importance of the basic notions positive real and lossless in network synthesis, and laid the foundation of the theory of synthesis, a paradigm in technology. The post-Cauer history of progress in theoretical research on the main theme proposed by Cauer is indicated as well. Then, the synthesis of modeling and whitening filters for signal processing of stationary stochastic processes is discussed on the basis of the Cauer-Darlington synthesis of classical networks, in which losslessness plays an important role. As one of the main results, a basic theorem is presented to resolve a difficulty in the network-theoretical approach caused by the transcendency of the power spectral density of the stochastic processes. The convergence of synthesis procedures for whitening filters as well as for modeling filters is demonstrated under the Payley-Wiener criterion.