Cohen’s Class Time-Frequency Distributions for Measurement Signals as a Means of Monitoring Technological Processes

The article presents and describes Cohen’s class time-frequency distributions which are expedient to use as a mathematical tool that allows you to create a convenient—in terms of information content and semantic clarity—visual-graphical representation of the operating modes of various technological processes including ferrous metallurgy processes. It was noted that a controlling process is usually implemented without simultaneous visual monitoring of each scalar (one-dimensional) coordinate that is under control, but the presence of such monitoring is an important condition for computer-aided controlling the dynamics of non-stationary technological processes. To eliminate this drawback, it was proposed to perform synchronous monitoring using the multidimensional Cohen’s class time-frequency distributions, when each measurement scalar signal is specifically represented through one of these distributions, for example, the Wigner–Ville distribution. An expression is given for the generalized distribution of Cohen’s class with a distribution kernel and an ambiguity function available. The latter allows you to receive distributions of various types from the maternal function. The most typical representatives of time-frequency distributions forming this class, with their kernels available, are given. The possibility of the appearance of interference elements on a signal distribution map, which ones make it difficult to identify controlled modes, is proved. The case of the formation of virtual elements within the Wigner–Ville distribution, which represents a two-component one-dimensional signal, is considered. The conditions are explained for the emergence of parasitic elements on the distribution map, obtained, for example, during realizing the process of multi-component feeding the bulk blast furnace charge materials in the production of sintering mixture. An analytical expression is obtained for the Wigner distribution, which displays a multi-component scalar signal and contains the information (useful) and virtual (parasitic) parts of the time-frequency distribution. A link is made known between the number of bulk material feeders available in the feeding devices unit and the number of parasitic (virtual) elements in the Wigner distribution. Using the feeding process as an example, the effect of the noise components propagation in the Wigner distribution is demonstrated. An example is given to illustrate the penetration of noise into the Wigner distribution and the appearance of a virtual concentration in it when displaying a signal waveform with a noisy pause and two sections with different frequencies. An expression for the Wigner distribution in the form of a comb function is obtained. The conclusion was made about the parameters of the distribution periodicity and the required sampling frequency of measurement signals.