Kramers-Kronig relations Hilbert transform

This method is based on the Kramers-Kronig relation (1), saying that both s and e" carry the same information about relaxation processes and are related by a Hilbert transformation ... 

Luckily, the real and imaginary parts of the complex dielectric permittivity are not independent of each other and are connected by means of the Kramers-Kronig relations [11]. This is one of the most commonly encountered cases of dispersion relations in linear physical systems. The mathematical technique entering into the Kramers-Kronig relations is the Hilbert transform. Since dc-conductivity enters only the imaginary component of the complex dielectric permittivity the static conductivity can be calculated directly from the data by means of the Hilbert transform. 

Computationally important properties of the Fourier analysis were discussed by Ernst at quite an early stage. For example, when combined with Hilbert transforms, the Kramers-Kronig relation between the real and imaginary components of complex data makes it possible to apply post-acquisition phase corrections. Ernst made some proposals for an automatic phase correction algorithm, but this kind of method was only routinely used after the developments made by Levitt and Freeman. - Note that compared with phase problems, the use of power... 

Mathematically, integral Kramers-Kronig relations have very general character. They represent the Hilbert transform of any complex function s(co) = s (co) + s"(co) satisfying the condition s (co) = s(—co)(here the star means complex conjugate). In our particular example, this condition is applied to function n(co) related to dielectric permittivity s(co). The latter is Fourier transform of the time dependent dielectric function s(f), which takes into account a time lag (and never advance) in the response of a substance to the external, e.g. optical, electric field. Therefore the Kramers-Kronig relations follow directly from the causality principle. 

If the transfer function H is in accordance with the causality rule, the components R and X are no longer independent of each other. Causality in the meaning of system theory forces couplings between the real and imaginary part, which are known as Kramers-Kronig relations (KKT) or Hilbert relations (HT), for details see Section 3.1.2.9 (The use of Kramers-Kronig Transforms). 

The Kronig-Kramers relationships are a very general set of integral transforms that find wide application in phjreical problems. They are intimately related to Hilbert transforms which, subject to certain integrability and analyticity conditions, allow the real and imaginary parts of a complex function f(z) = u iv to >t expressed as a pair of transform mates. This property follows from the fact that u and v are not completely independent when / z) is analytic in the whole upper half of the complex plane. 

Impedance is by definition a complex quantity and is only real when 0=0 and thus Z(m) = Z(a>), that is, for purely resistive behavior. In this case the impedance is completely frequency-independent. When Z is found to be a variable function of frequency, the Kronig-Kramers (Hilbert integral transform) relations (Macdonald and Brachman [1956]), which holistically connect real and imaginary parts with each other, ensure that Z" (and 9) cannot be zero over all frequencies but must vary with frequency as well. Thus it is only when Z(linear resistance, that Z(m) is purely real. 

---