Dedicated to Yuri Anatolyevich Kropotov: Problems of Analysis and Modeling of Systems with Lagging Inverse Communication
Statement of the problem: systems with distributed parameters often require, in their analysis and modeling, taking into account delays in signal transmission circuits. The need to take into account the lag is inherent in the problems of constructing models of systems for adaptive filtering and compensation of acoustic echo and reverberation in terms of increasing the stability and intelligibility of communication. The purpose of the work is to develop methods of analysis and formation of models of information exchange systems with delayed feedback, functioning in a difficult interference environment. Used methods: the models used in the work are represented by equations of two types - differential equations that establish a direct mapping of the set of inputs to the set of outputs of the model, and differential equations in Cauchy normal form, more precisely, equations in state variables. The scientific novelty the equivalence of the equations of the two above types has been established. The models and their corresponding equations also differ in the nature of the delay in the feedback loop. Some models are represented by equations with discrete delay, while others - with distributed one. The former simulate the echo phenomenon, and the latter simulate reverberation, that is, the multiple nature of reflections. Results: the paper obtained equations for models of systems with delayed feedback. Their applicability to the problems of analyzing systems with acoustic feedback is shown. Potentially high sensitivity of any models to variations in the propagation paths of reflected sound signals, which are responsible for the occurrence of echo and reverberation phenomena caused by the multiplicity of reflections, is noted. Practical value: this paper gives an example of the distribution function of delays in magnitude. The questions of application, numerical analysis and generalization of the presented models are discussed.