Research paper
Generalization of Kramers-Krönig relations for evaluation of causality in power-law media

https://doi.org/10.1016/j.cnsns.2020.105664Get rights and content

Highlights

  • The Kramers-Krönig relations are generalized towards functions which are not square integrable.

  • We propose generalization of the Kramers-Krönig relations with subtractions, whose validity for the logarithm of the system response and the satisfaction of additional assumptions imply causality of the originally considered function.

  • The derived theory is then applied to electromagnetic media characterized by power-law frequency dispersion, i.e. the media which are described by fractional-order models.

Abstract

Classical Kramers-Krönig (K–K) relations connect real and imaginary parts of the frequency-domain response of a system. The K–K relations also hold between the logarithm of modulus and the argument of the response, e.g. between the attenuation and the phase shift of a solution to a wave-propagation problem. For square-integrable functions of frequency, the satisfaction of classical K–K relations implies causality in the time domain. On the other hand, when the K–K relations are checked for the logarithm of the system response, the function is not a square integrable one. Then one can employ classical K–K relations with subtractions, but their satisfaction for the logarithm of the system response does not imply causality of the original function. In this paper, the K–K relations are generalized towards functions which are not square-integrable, also allowing for causality evaluation when the logarithm of the system response is considered. That is, we propose generalization of the K–K relations with subtractions, whose validity for the logarithm of the system response and the satisfaction of additional assumptions imply causality of the originally considered function. The derived theory is then applied to electromagnetic media characterized by power-law frequency dispersion, i.e. the media which are described by fractional-order models (FOMs). In this case, the subtraction procedure generates functions which may be not square integrable, or even not locally integrable. However, we can rigorously analyse causality of the media described by FOM using the derived theory, as well as the parameter ranges for which such models are causal.

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 ων, for ν>0. 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 ν>0 only when ν(0,1). 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 j=1, which is widely used in electrical sciences, is employed throughout the paper. Real and imaginary parts of the complex number s are denoted as Re(s) and Im(s), respectively. The conjugate to the complex number s is denoted as s¯. The modulus of the complex number s is given by |s|=ss¯. The signum function defined on the real axis is given by sgn(x)=x/|x| for x0, whereas sgn(0)=0. The right half-plane of the complex plane is denoted as

Causality

The function f:RR or the distribution fD is called causal if its support supp(f)[0,+). The Fourier transform F=F(f) is called a causal transform if supp(F1(F))[0,+). The other way of looking into causality is to establish that f(t) is causal if it may be represented as f(t)=u(t)g(t), where u(t) is the Heaviside step function and g(t)=f(t) for t>0 [45]. In practical terms, we should also assume that g(t) 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 E and H fields only as·E=0×E=μγDtγH·H=0×H=ϵβDtβE+σαDt1αE.Let 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 (α=1)J=σ1E

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)

  • W.C. Chew

    Waves and fields in inhomogenous media

    (1995)
  • W.P. Healy

    Primitive causality and optically active molecules

    Journal of Physics B: Atomic and Molecular Physics

    (1976)
  • M.W. Haakestad et al.

    Causality and Kramers-Kronig relations for waveguides

    Optics Express

    (2005)
  • F.W. King

    Alternative approach to the derivation of dispersion relations for optical constants

    J Phys A

    (2006)
  • M. O’Donnell et al.

    Kramers-Kronig relationship between ultrasonic attenuation and phase velocity

    J Acoust Soc Am

    (1981)
  • K.R. Waters et al.

    On the applicability of Kramers-Kronig relations for ultrasonic attenuation obeying a frequency power law

    J Acoust Soc Am

    (2000)
  • K.R. Waters et al.

    On a time-domain representation of the Kramers-Krönig dispersion relations

    J Acoust Soc Am

    (2000)
  • K.R. Waters et al.

    Differential forms of the Kramers-Kronig dispersion relations

    IEEE Trans Ultrason Ferroelectr Freq Control

    (2003)
  • K.R. Waters et al.

    Causality-imposed (Kramers-Kronig) relationships between attenuation and dispersion

    IEEE Trans Ultrason Ferroelectr Freq Control

    (2005)
  • R. Weaver et al.

    Dispersion relations for linear wave propagation in homogeneous and inhomogeneous media

    Journal of Mathematical Physics

    (1981)
  • H. Booij et al.

    Generalization of Kramers-Kronig transforms and some approximations of relations between viscoelastic quantities

    Rheol Acta

    (1982)
  • J. Bechhoefer

    Kramers-Kronig, Bode, and the meaning of zero

    Am J Phys

    (2011)
  • J. Osiowski et al.

    Fundamentals of circuit theory (in Polish)

    (1995)
  • T.P. Stefański et al.

    Signal propagation in electromagnetic media described by fractional-order models

    Commun Nonlinear Sci NumerSimul

    (2020)
  • A.N. Bogolyubov et al.

    An approach to introducing fractional integro-differentiation in classical electrodynamics

    Moscow Univ Phys Bull

    (2009)
  • V.E. Tarasov

    Fractional integro-differential equations for electromagnetic waves in dielectric media

    Theor Math Phys

    (2009)
  • H. Nasrolahpour

    A note on fractional electrodynamics

    Commun Nonlinear Sci NumerSimul

    (2013)
  • M.D. Ortigueira et al.

    From a generalised Helmholtz decomposition theorem to fractional Maxwell equations

    Commun Nonlinear Sci NumerSimul

    (2015)
  • Cited by (8)

    View all citing articles on Scopus
    View full text