Research paperGeneralization of Kramers-Krönig relations for evaluation of causality in power-law media
Introduction
For a linear time-invariant system, causality means that the response of the system can depend on past, but not on future values of a signal exciting the system [1], [2], [3], [4]. In other words, no output can occur before the input. Hence, causality restricts the characteristics of the system in both time and frequency domains. The Kramers-Krönig (K–K) relations are a fundamental tool for causality analysis of systems in electromagnetics [5], [6], [7], [8], [9], acoustics [10], [11], [12], [13], [14], solid mechanics [15], [16] and control theory [17].
In electrical sciences, the K–K relations connect real and imaginary parts of the frequency-domain response, i.e. the system transfer function [18]. The K–K relations also hold between the logarithm of modulus and the argument of the frequency-domain response, e.g. between the attenuation and the phase shift of a solution to a wave-propagation problem. These relations also work in circuit theory, connecting resistance and reactance in the frequency domain for lumped elements. Therefore, one can conclude that the K–K relations are very general and important in electrical sciences.
For square-integrable functions of the frequency, the satisfaction of classical K–K relations implies causality in the time domain [1], [4], [15]. Things become complicated when the K–K relations are checked between the logarithm of modulus and the argument. Then a considered function is not square integrable, and it is possible to employ classical K–K relations with subtractions, but their satisfaction does not imply causality of the originally considered function [11], [15]. That is, the dispersion relation can be obtained for the system, which is associated with the assumption that the subtracted logarithm of the response is causal, but not the considered response function. Therefore, we decided to generalize the K–K relations towards not square-integrable functions which, under additional assumptions, also allow for causality evaluation. That is, the K–K relations with one and two subtractions are proposed. Their validity and satisfaction of additional assumptions imply causality of a considered function.
The derived theory is then applied to electromagnetic media characterized by power-law frequency dispersion. Such media are described by fractional-order (FO) models (FOMs) [19], which stem from generalization of Maxwell’s equations with the use of FO derivatives [19], [20], [21], [22], [23], [24]. Hence, the inclusion of FO derivatives in mathematical models allows for representation of evolution of complicated electromagnetic systems with memory and dissipation [25]. In [19], propagation of a non-monochromatic plane wave is simulated in such media. It is demonstrated that a small perturbation of time-derivative orders in Maxwell’s equations makes it possible to observe pulses at an observation point faster than in the reference case of linear medium described by an integer-order (IO) model (IOM). Such FO generalization of Maxwell’s equations implies that electromagnetic field satisfies the diffusion-wave equation [26], [27], [28], [29], which interpolates between diffusion and wave equations. The diffusion equation describes a process for which a disturbance spreads infinitely fast, whilst the wave equation describes a disturbance whose propagation velocity is a constant. These properties are mixed in the time-fractional diffusion-wave equation for which propagation velocity of a disturbance is infinite, but its fundamental solution possesses a maximum which disperses with a finite velocity. It implies fundamental questions related to causality of solutions to the time-fractional diffusion-wave equation, as well as the media described by FOMs.
To the Authors’ best knowledge, although several literature sources generally claim that satisfaction of the K–K relations between the phase and the attenuation implies causality, there is no rigorous mathematical proof for the case of formulas with subtractions [11], [14]. However, the classical subtraction procedure generates functions which may be not square integrable, or even not locally integrable. On the other hand, the K–K relations require a formal assumption that the considered functions are square integrable on the entire real line, which is not valid for the media described by FOMs. It motivates us to put in order the theory of causality and rigorously prove that, for power-law frequency dispersion, one can generalize the K–K relations with subtractions in such a way that their satisfaction for the phase and attenuation functions also implies causality. Hence, we can rigorously analyse causality of the media described by FOM using the derived theory, as well as parameter ranges for which such models are causal. To the Authors’ best knowledge, these are novel results, and they can be found useful in all these branches of physics and engineering, which require procedures for causality evaluation.
The paper starts with a short introduction to the notation and basic mathematical tools used throughout the paper. In Section 3, classical K–K relations (i.e. valid for square-integrable functions), implying causality of a real time-domain function, are analysed. These are further extended towards a more general case of distributions in a series of abstract, theoretical results. Theorem 5 presents the K–K relations with subtractions, in a distributional sense, which can be applied to functions with the growth restricted by the function for . However, the differentiability assumption is not needed in this theorem, which is opposite to [4, Section 1.7]. That is, we can employ the subtraction method to power-law dispersion characteristics in a mathematically rigorous way. To the Authors’ best knowledge, this is a novel contribution of the paper. On the other hand, in Theorem 6, the K–K relations with subtractions are stated in a classical (not distributional) sense, without an assumption of differentiability. This theorem can be applied to prove causality of the response function by means of relations between real and imaginary parts of its complex logarithm. It leads to Theorem 8, where sufficient conditions for causality of the response function are formulated based on the K–K relations between the attenuation and the phase. Then, in the next section, the case of signal propagation in electromagnetic media described by FOMs is discussed. It results in formal problems connected with application of the classical causality theory valid for square-integrable functions. Theorem 10 states that such media are causal for a fractional parameter only when . These problems are discussed and solved with the use of mathematical tools introduced in Section 3. Long proofs of theorems are collected in Appendices A-C.
Section snippets
Basic notations
A standard engineering notation for an imaginary unit which is widely used in electrical sciences, is employed throughout the paper. Real and imaginary parts of the complex number are denoted as and respectively. The conjugate to the complex number is denoted as . The modulus of the complex number is given by . The signum function defined on the real axis is given by for whereas . The right half-plane of the complex plane is denoted as
Causality
The function or the distribution is called causal if its support . The Fourier transform is called a causal transform if . The other way of looking into causality is to establish that is causal if it may be represented as where is the Heaviside step function and for [45]. In practical terms, we should also assume that is the function whose Laplace transform has a non-degenerate region of convergence.
Signal propagation in electromagnetic media described by FOM
In [19], propagation of the plane wave is analysed in isotropic and homogeneous media described by FOM. It is assumed that there is no current or charge sources in the considered space. Hence, one can formulate Maxwell’s equations based on and fields only asLet us assume (as in [19]) that there is no power dissipation due to Joule’s heating. That is, the current density is related to the electric field intensity by the classical Ohm law ()
Conclusion
In this paper, the K–K relations are generalized towards not square-integrable functions, also allowing for causality evaluation. We propose generalization of the K–K relations with subtractions, whose validity for the logarithm of the system response and satisfaction of additional assumptions imply causality of an originally considered function. The derived theory is applied to electromagnetic media characterized by the power-law frequency dispersion, which are described by FOMs. In this case,
CRediT authorship contribution statement
Jacek Gulgowski: Conceptualization, Funding acquisition, Formal analysis, Writing - original draft, Writing - review & editing, Visualization. Tomasz P. Stefański: Conceptualization, Funding acquisition, Formal analysis, Writing - original draft, Writing - review & editing, Visualization.
Declaration of Competing Interest
We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.
References (56)
Fractional vector calculus and fractional Maxwell’s equations
Ann Phys
(2008)- et al.
Propagation speed of the maximum of the fundamental solution to the fractional diffusion-wave equation
Computers and Mathematics with Applications
(2013) Fractional wave equation and damped waves
J Math Phys
(2013)- et al.
Cauchy and signaling problems for the time-fractional diffusion-wave equation
J Vibr Acoust
(2014) Hilbert transforms
(2009)Causality and the dispersion relation: logical foundations
Phys Rev
(1956)Validity conditions for the Kramers-Kronig relations
Am J Phys
(1964)- et al.
An introduction to dispersion relations
Am J Phys
(1964) Causality and dispersion relations
(1972)Classical electrodynamics
(1998)
Waves and fields in inhomogenous media
Primitive causality and optically active molecules
Journal of Physics B: Atomic and Molecular Physics
Causality and Kramers-Kronig relations for waveguides
Optics Express
Alternative approach to the derivation of dispersion relations for optical constants
J Phys A
Kramers-Kronig relationship between ultrasonic attenuation and phase velocity
J Acoust Soc Am
On the applicability of Kramers-Kronig relations for ultrasonic attenuation obeying a frequency power law
J Acoust Soc Am
On a time-domain representation of the Kramers-Krönig dispersion relations
J Acoust Soc Am
Differential forms of the Kramers-Kronig dispersion relations
IEEE Trans Ultrason Ferroelectr Freq Control
Causality-imposed (Kramers-Kronig) relationships between attenuation and dispersion
IEEE Trans Ultrason Ferroelectr Freq Control
Dispersion relations for linear wave propagation in homogeneous and inhomogeneous media
Journal of Mathematical Physics
Generalization of Kramers-Kronig transforms and some approximations of relations between viscoelastic quantities
Rheol Acta
Kramers-Kronig, Bode, and the meaning of zero
Am J Phys
Fundamentals of circuit theory (in Polish)
Signal propagation in electromagnetic media described by fractional-order models
Commun Nonlinear Sci NumerSimul
An approach to introducing fractional integro-differentiation in classical electrodynamics
Moscow Univ Phys Bull
Fractional integro-differential equations for electromagnetic waves in dielectric media
Theor Math Phys
A note on fractional electrodynamics
Commun Nonlinear Sci NumerSimul
From a generalised Helmholtz decomposition theorem to fractional Maxwell equations
Commun Nonlinear Sci NumerSimul
Cited by (8)
Electromagnetic field in a conducting medium modeled by the fractional Ohm's law
2022, Communications in Nonlinear Science and Numerical SimulationCitation Excerpt :The multi-component problem of moving conductor is treated in [14]. Constitutive models of the dielectric media are generalized in [15] by generalization of the Kramers–Krönig relations, while in [16] the Debye media is considered. Further, the two-sided fractional derivative is used in [17] to model the medium, while in [18] the Atangana–Baleanu fractional derivative is used to model wave propagation in dielectric media.
Investigation of Optical Properties and Activity of Wheat Stripe Rust Urediospores
2023, Agriculture (Switzerland)Discrete-Time Fractional Difference Calculus: Origins, Evolutions, and New Formalisms
2023, Fractal and FractionalQuantitative, in situ visualization of intracellular insulin vesicles in pancreatic beta cells
2022, Proceedings of the National Academy of Sciences of the United States of America