The significance and unity of the time and frequency based approach
The possible transformations within a group open up the way for representations by different aspects. Of these the most important is the transmission of the polynomial time representations to the frequency world of the Fourier and Laplace transformations and to the domains of exponential and hyperbolic function variables. They enable us to approach the process behaviour from different aspects and to apply the approach related representation when analysing and designing the process.
The time approach and the frequency approach have different origins both historically and in terms of the physical root of the problem. The original goal was to create a mathematical description of natural processes viewed in time and space.
It took a long way to arrive at the representation of processes first with ordinary and then with partial differential equations, from the end of the 16th century to Galilei and Newton, and later from the study of the time and space aspects of gravity and other mass effects to the chain of ideas leading to the concepts of energy and potential. Their equilibriums and content-preserving properties became the mathematical representations of the physical concepts, the calculation framework of the entire real world of systems, including the Hamiltonian equations in mechanics, Maxwell’s equations in electrodynamics and partly the basic relations of quantum theory.
The observation of thermal conduction inspired Joseph Fourier’s theory, the expansion of phenomena into frequency functions and all the things that contributed to the development of the interrelations and inter-transformations of the two kinds of representation which however related to the very same process; and these are the relations which we use today in our calculations. This frequency representation is used to examine all wave phenomena, and in today’s physical reality the dual nature of substance, light and energy is united in this dual representation. Holographic images are illustrative of this. With digital tools an image is delivered in series in time, however, in holography—as the name suggests—the full image is transmitted as its approach within the given frequency domain.
A nice example of group theory is the bijective transformation of the mathematics of the two approaches: the correspondence of the Taylor series of time functions in the differential equations and the Fourier series. In this application, this is the representation tool set which we use to handle the problems of system science and control in practice.
The significance and unity of the time and frequency based approach
The possible transformations within a group open up the way for representations by different aspects. Of these the most important is the transmission of the polynomial time representations to the frequency world of the Fourier and Laplace transformations and to the domains of exponential and hyperbolic function variables. They enable us to approach the process behaviour from different aspects and to apply the approach related representation when analysing and designing the process.
The time approach and the frequency approach have different origins both historically and in terms of the physical root of the problem. The original goal was to create a mathematical description of natural processes viewed in time and space.
It took a long way to arrive at the representation of processes first with ordinary and then with partial differential equations, from the end of the 16th century to Galilei and Newton, and later from the study of the time and space aspects of gravity and other mass effects to the chain of ideas leading to the concepts of energy and potential. Their equilibriums and content-preserving properties became the mathematical representations of the physical concepts, the calculation framework of the entire real world of systems, including the Hamiltonian equations in mechanics, Maxwell’s equations in electrodynamics and partly the basic relations of quantum theory.
The observation of thermal conduction inspired Joseph Fourier’s theory, the expansion of phenomena into frequency functions and all the things that contributed to the development of the interrelations and inter-transformations of the two kinds of representation which however related to the very same process; and these are the relations which we use today in our calculations. This frequency representation is used to examine all wave phenomena, and in today’s physical reality the dual nature of substance, light and energy is united in this dual representation. Holographic images are illustrative of this. With digital tools an image is delivered in series in time, however, in holography—as the name suggests—the full image is transmitted as its approach within the given frequency domain.
A nice example of group theory is the bijective transformation of the mathematics of the two approaches: the correspondence of the Taylor series of time functions in the differential equations and the Fourier series. In this application, this is the representation tool set which we use to handle the problems of system science and control in practice.