Development of a Method for Researching the Occurrence of Additional Distortions in the Presence of Errors in the Digital Channel and Transformation of Different Metric Spaces

Signals from a message source can be of various types. At the same time, for their description, appropriate models are used, which make it possible to represent the generated signals in various metric spaces that are adequate to the type of signals under consideration. So, for example, the Euclidean space is used to describe analog signals, while Hamming space is used to describe a discrete digital signal. It is obvious that the mapping of signals first in the Euclidean space and then in the Hamming space corresponds to a nonlinear transformation, part of which is the need to match the powers of the corresponding sets under the condition of a one-to-one (one-to-one) mapping and with the introduced quality criterion. The article proposes a method for analyzing the bijection of the original set and the set of signal values at the reception, corresponding to the transformation of the elements of the original signal into digital elements of the Hamming space. This method makes it possible to carry out quantitative calculations in order to select the best one-digit match (bijection) method that provides the minimum amount of additional distortions arising in the elements of the original set due to errors in the elements of the set of signal values at the reception.