A New Effectivemethod to Determine the Discrete Finite Real Signalenvelopes Based on The Parametric Discrete Fourier Transform of the Second Type

Abstract

The article shows that expanding the applicationscope of digital signal processing (DSP) systems, increasing the scale of tasks and problems solved by such systems, led to the need to develop a theory,improve DSP methods and algorithms, including those based on discrete finite Fourier and Hilbert transforms (DFT) and (DPG). DFT and DPG, due to their properties, the emergence of the fast Fourier transform (FFT) algorithm (Cooley J.W., Tukey J.W., 1965), have found the widest application in DSP systems. It is shown that DFTs, along with their advantages, also have fundamental disadvantages, thatreveal in the time and frequency domains in a number of negative effects, and the calculations of DFTs are accompanied by a number of difficulties. The paper briefly examines the fundamentals of the DSP theory in parametric Fourier bases. Parametric discrete Fourier transforms (DFT-P) are the two generalizations of the classical DFT. At the same time, introducing a parameter into the DFT-P allows you to “control” the properties of the unitary transformation within the frequency or time domain. The article considers two types of mathematically timeequivalent descriptions of discrete finite real (DFR) signals: in the form of a spectrum (the sum of discrete harmonic components) and in the form that uses the instantaneous parameters of the DFD signal: instantaneous amplitude, instantaneous phase and envelope. From the information descriptionviewpoint, instantaneous parameters provide a more complete representation and informationidentification about the properties and states of the objects, phenomena, processes and systems under study. DFT and DPG transformations play an important role in describing DPD signals. The article, for example, shows that the DPG is the only linear operator that allows you to determine the instantaneous parameters of the DPD signalunambiguously, subject to the fulfillment of completely understandable requirements. In this work, a new effective method for determining envelopes based on parametric Fourier transforms of the second type has been developed. The theoretical results obtained in the article are confirmed by mathematical modeling.