A fast Fourier transform implementation of the Kramers-Kronig relations:Application to anomalous and left handed propagationJérôme Lucas, Emmanuel Géron, Thierry Ditchi, and Stéphane Holé Citation: AIP Advances 2, 032144 (2012); doi: 10.1063/1.4747813 View online: http://dx.doi.org/10.1063/1.4747813 View Table of Contents: http://aipadvances.aip.org/resource/1/AAIDBI/v2/i3 Published by the AIP Publishing LLC. Additional information on AIP AdvancesJournal Homepage: http://aipadvances.aip.org Journal Information: http://aipadvances.aip.org/about/journal Top downloads: http://aipadvances.aip.org/features/most_downloaded Information for Authors: http://aipadvances.aip.org/authors

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AIP ADVANCES 2, 032144 (2012)

A fast Fourier transform implementation of theKramers-Kronig relations: Application to anomalous andleft handed propagation

Jerome Lucas,a Emmanuel Geron, Thierry Ditchi, and Stephane HoleLaboratoire de Physique et d’Etude des Materiaux, UPMC Univ Paris 06, ESPCI-ParisTech,CNRS UMR8213, France

(Received 22 May 2012; accepted 10 August 2012; published online 17 August 2012)

In this this paper we quickly derive the Kramers-Kronig relations from simple causal-ity considerations and propose a simple way to implement them using the Fast FourierTransform. This work focuses on how to make these relations a usable tool even whentheir conditions of validity are not strictly respected. In this respect we emphasizeon their application to the constant low level loss approximation at microwave fre-quencies. The method presented is demonstrated on various typical cases of fancypropagation: high velocity, negative phase velocity and evanescent waves. Copyright2012 Author(s). This article is distributed under a Creative Commons Attribution 3.0Unported License. [http://dx.doi.org/10.1063/1.4747813]

I. INTRODUCTION

Exotic propagation phenomena called anomalous dispersion at optic frequencies have beenreported long ago. Those phenomena often imply superluminal group velocity such as in photoniccrystals. Though Brillouin and Sommerfield1 early stated the conditions required for the groupvelocity to be the velocity of the energy, there are still numerous papers presenting and discussingsuperluminal group velocities.2–4 More recently left handed material lines or structures introduced byVelesago5 and Pendry6 came into focus. They are typically media where some anomalous dispersionoccurs: there are bands where the phase velocity is negative and forbidden bands where the wavesare evanescent. The thread binding those phenomena is the relation which links the phase shift to theamplitude of the signal. When the phase shift is due to propagation one has to consider the dispersionrelation along with the amplitude. At lower frequencies when no propagation is involved it is referredto as the frequency response. The Kramers Kronigs relations7–9 are an interesting theoretical toolto meet those considerations. They translate to the frequency domain the simple temporal paradigmthat the effect cannot occur before the cause. By doing so they relate the phase shift to the amplitude.

The Kramers-Kronig relations are commonly used in optics,10 acoustic11 or at terahertz12

frequencies to verify the consistency of performed measurements or to extrapolate for instancereflection coefficient measurements at frequencies where they are difficult to perform. Neverthelessthe methods employed are not directly presented or restricted to a given kind of application. Forinstance, Kendall10 presents a method without explicitly giving the algorithm used for the calculationand applies it to relate the attenuation coefficient to the phase velocity of acoustical waves. In thesame way, left handed propagation and evanescent waves are not considered.

On the basis of previous work,13 we present how to apply the Kramers-Kronig relations evenwhen they cannot be straightforwardly applied because of limited bandwidth or because of Fouriertransform convergence problem for instance. Its applications to S parameter measurements or to dataextrapolation are discussed. The method is then practically applied to some interesting typical cases

aElectronic address: jerome.lucas@espci.fr

2158-3226/2012/2(3)/032144/11 C© Author(s) 20122, 032144-1

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032144-2 Lucas et al. AIP Advances 2, 032144 (2012)

such as photonic crystals and right and left handed media, where the waves can be superluminal,evanescent or left handed.

II. THEORETICAL BACKGROUND

A. Kramers-Kronig relations

Whatever the introductory phenomenon presented (photonic crystal, left handed material andso on) they occur in real thus causal media. Consequently the media response r(t) to any excitationbeginning at time t = 0 verifies:

r (t) = u(t) r (t) (1)

where u(t) denotes the unity step function.By taking the Fourier transform of this relation, and considering Cauchy principal value, one

obtains:

R(ω) = −2 j

ω∗ R(ω) = j HT

(R(ω)

)(2)

In this Equation, j = √−1 , ω is the circular frequency, R(ω) is the Fourier transform of r(t),HT the Hilbert transform and * denotes the convolution product. Note that the minus sign in thisrelation may change accordingly to the definition used for the Fourier Transform. In this paper weuse the sign that is relevant with the FFT algorithm. By presenting separately the imaginary part Ri

and the real part Rr of R(ω), one has:

Ri (ω) = HT (Rr (ω))

Rr (ω) = −HT (Ri (ω)) (3)

These relations are known as the Kramers-Kronig (K-K) relations. In a linear and time invariantmedia, Equation (1) also applies to the impulsional response h(t), whose Fourier transform H(ω) isthe frequency response. Consequently a frequency response H(ω) respects the pair of relations (3).

In a very general way, a relation such as (4), valid over all frequencies between two physicalquantities as the electrical field �E(ω) and the displacement field �D(ω) where ε(ω) is the complexdielectric permittivity, describes a linear system.

�D(ω) = ε(ω) �E(ω) (4)

Indeed with the inverse Fourier transform, it comes �D(t) = ε(t) ∗ �E(t). Permittivity ε(t) can beinterpreted as an impulsional response and consequently ε(ω) verifies the K-K relations (2) and (3).

B. Practical computation of the Hilbert Transform

Let us consider a function f (x) of variable x whose Fourier transform is f (k) where k is thecorresponding variable in the Fourier space. Since the Hilbert transform of f (x) is the convolutionproduct with −2/x, it can be calculated as follows:

H( f (x)) = − 2

x∗ f (x)

= F−1

(F

(− 2

x∗ f (x)

))

= F−1

(j sgn(k) f (k)

)(5)

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032144-3 Lucas et al. AIP Advances 2, 032144 (2012)

where F denotes the Fourier transform, F−1 the inverse Fourier transform and where the sgn(k)function is +1 when k ≥ 0 and -1 when k < 0.

The numerical Hilbert transform HT corresponding to Equation (5) can be computed for a givensampled signal S(xi) using the FFT algorithm:

HT(S) = I F FT(

j × SG N (ki ) × F FT (S))

(6)

In this Equation SGN(ki) emulates the signum function. It is 1 for the samples correspondingto positive values of ki in the Fourier space and it is -1 for the others.

It is also possible with some precautions to use the direct convolution algorithm to compute theHilbert transform. It does not yield better results than with the FFT algorithm and it is much slower.

1. Making things work

The K-K relation of Equation (2) implies an integration from −∞ to ∞. It is indeed not possiblenumerically and, worse, no measurement can be performed over such a wide band. The FFT algorithmcircumvent this problem by calculating the numerical Fourier transform of the periodized sampledsignal S. This technique leads to “spectral leakage” when the original signal cannot be periodizedwith matching junctions.14 In our case we compute the Fourier transform R(ω) of some real signalr(t) as defined in section II A. Consequently, R(ω) is Hermitian which can be written as in Equation(7), where R* is the conjugate of R.

R(−ω) = R∗(ω) (7)

This condition is equivalent to the even symmetry of the real part of H(ω) and to the odd symmetryof its imaginary part. Consequently the response R(ω) must be symmetrized respecting Equation (7)before applying the algorithm (6) to take Equation (2) into account where Cauchy’s principal valueimplies symmetry over the integration variable ω.

As it will be described later in section III, the “spectral leakage” is dealt with by taming up theresponse R(ω) when required.

C. Computing a phase shift from an amplitude measurement and vice-versa

This is the main practical application of the K-K relations. Indeed R(ω) = G(ω)ejθ(ω) is oftendetermined by measuring its amplitude G(ω) and its argument θ (ω). The K-K relation relatesG(ω) cos (θ (ω)) to G(ω) sin (θ (ω)). But it is interesting from an experimental point of view to obtainG(ω) from the measurement of θ (ω) and vice-versa. It is all the more interesting when one of themeasurements is difficult or impossible to perform such as θ (ω) when using a spectrum analyzer.

In order to separate G(ω) and θ (ω), it is usual to consider the logarithm of R(ω): ln (R(ω))= ln (G) + jθ (ω). It would be interesting if the following pair of relations could be applied:{

ln(G(ω)

) = −HT(θ (ω)

)θ (ω) = HT

(ln

(G(ω)

) (8)

This is unfortunately not the general case. As an example, let us consider a system consistingin a piece of ideal coaxial cable of length L. For the TEM mode, the group and phase velocities areequal and constant. Therefore the frequency response is: H (ω) = exp(− j ω

VϕL). Function H(ω) is

the frequency response of a causal system since the transit time in such a system is LVg

≥ 0.

Considering ln(H (ω)

), one obtains:

{ln

(G(ω)

) = 0θ (ω) = − ω

VϕL (9)

ThereforeH(ln(G(ω)

)) = 0 which is obviously different from θ (ω). Consequently the Relations

(8) are not respected. In this case H(ω) corresponds to a causal system and the K-K relations applywhen they do not apply to ln

(H (ω)

).

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032144-4 Lucas et al. AIP Advances 2, 032144 (2012)

One can find in the literature methods using Cauchy’s integral theorem to demonstrate the K-Kpairs of relations between ln

(G(ω)

)and θ (ω) of Equation (8) by adding some convergence condition

to H(ω). For instance Wooten8 assuming that:

|H (z)| < |b × z−s | (10)

where b and s are real strictly positive constants, and z the analytical extension to the complex planeof ω with R(z) = ω and �(z) ≤ 0, proves that:

θ (ω) = −2ω

πP

∫ +∞

0

ln(G(u)

)ω2 − u2

du (11)

In Equation (11), P stands for Cauchy’s principal value.Considering that G(ω) is an even function, it can be rewritten as: θ (ω) = HT

(ln(G(ω))

).

Calculation details can be found in appendix. Finally by taking the Hilbert transform of this relationone obtains the other Equation of the pair of relations (8).

The condition presented in Equation (10) is often invoked to justify the application of the K-Krelations to the logarithm of the frequency response because it is a realistic physical condition for theresponse of a physical system. Nevertheless it seems to be a sufficient condition and not a necessaryone. Let us consider a simple first order high pass filter (HPF) with a cutoff frequency of ωc:

HH P F (ω) = 1

1 − j ωcω

(12)

Nevertheless it corresponds to what is measured when the frequency remains sufficiently low. Thissystem is causal because it respects the causality condition of Equation (1), but it does not respectthe condition of Equation (10). However one obtains with a very good accuracy ln (GHPF(ω)) fromθHPF(ω) and vice-versa by simply applying Relations (8) using the algorithm based on the Fouriertransform.That example shows that the condition (10) is a sufficient but not necessary one.

III. APPLICATIONS

In this section we demonstrate how to successfully apply K-K relations to various exampleschosen to cover most cases that can be encountered. Theses examples show how to deal withalgorithm discrepancy. They emphasize on the case of simple propagation which in any case has tobe dealt particularly. Finally most kind of wave propagation such as in photonic crystal like lines,evanescent waves, left handed waves are illustrated through application to measurements.

A. Second order low pass Filter:

Let us consider as an example a plain second order low pass filter of central frequency ω0 andquality factor Q, whose frequency response is:

HL P F2(ω) = 1(1 + j ω

Qω0− ω2

ω0

) (13)

It is a causal system, and it respects the condition (10). However applying straightforwardly theK-K relation pair (8), does not yield the expected result as presented in Figure 1. In this figure onecan compare ln

(GL P F2(ω)

)(1(a) dashed black) to −H

(θL P F2(ω)

)(1(a) solid red) and θLPF2(ω)

(1(b) dashed black) to H(ln

(GL P F2(ω)

))(1(b) solid red) and be convinced of the difference. As

condition (10) is respected, the discrepancy can only be a consequence of the algorithm used, whichis to say in our case some “frequency leakage”.

In order to solve this problem one can use a gain compensation technique.Let H1(ω) be the frequency response of a given physical quantity. According to the case, whether

the real part of ln (G1(ω)) = e

(ln

(H1(ω)

))or the imaginary part of θ1(ω) = �m

(ln

(H1(ω)

))are

known or can be measured between 0 rad/s and ωmax. Let H2(ω) be a frequency response such thatthe pair of Equations (8) applies.

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032144-5 Lucas et al. AIP Advances 2, 032144 (2012)

K-K Direct

K-K Direct

Theoretical

Theoretical

K-K Comp

K-K Comp

-5

-5-5

-4

-4

-3

-3

-2

-2

-1-1

-1

0

0

0

0

5

5

5

4

4

3

3

2

2

1

1

1

ln(G

(ω))

θ (ω

)×π

rad

0.5

-0.5

Frequency kHz

Frequency kHz

a)

b)

FIG. 1. Application of the compensation gain method to HLPF2(ω).

By calculating the product H(ω) = H1(ω) × H2(ω), one has θ (ω) = θ1(ω) + θ2(ω) and ln(G(ω)

)= ln

(G1(ω)

) + (G2(ω)

). To compute θ1(ω) from ln

(G1(ω)

)for instance, we choose H2(ω) such

that: ln (G1(ω)) + ln (G2(ω)) is asymptotic to zero when ω → ωmax. With this condition one reducesthe “frequency leakage” problem when computing the FFT. It is simply sampled at 1/ωmax rate.Consequently one can expect to compute the Hilbert transform with a minimal error. The phase

θ1(ω) can then be retrieved from the computation of θ (ω) = HT

(ln

(G1(ω)

) + ln(G2(ω)

))by

simply computing: θ1(ω) = θ (ω) − θ2(ω).This has been applied successfully using H 2

H P F (ω) as a compensation gain. The result ispresented in Figure 1: “K-K Comp” plots using blue o markers. One can see it behaves as expected.Let us emphasize again that it is possible to use H 2

H P F (ω) as a compensating frequency responsebecause HHPF(ω) respects Equation (8). It is not possible to use any built up function of a complexvariable for whose Hilbert transform may yield anything.

Of course there are many ways to respect the condition: “ln(G1(ω)

) = ln(G2(ω)

)is asymptotic

to zero when ω → ωmax” even when simply using HHPF(ω)2 as a compensation gain. Indeed the cutofffrequency of HHPF(ω) has to be chosen at best. Noticing that when there is “frequency leakage”,some imaginary part appears when computing the Hilbert transform, the compensation optimizationcan be performed by minimizing �m (HT (ln (G(ω)))) as the Hilbert transform of a real functionmust be real. This is what has been performed in the example presented in Figure 1.

B. Plain propagation in lossy FR4 media

In the case of plain propagation, such as the propagation of TEM waves in the piece of cablepreviously presented, the condition (10) does not apply. We have verified in section II C that the pairof relations (8) does not apply nether. Nevertheless, as propagation cannot be neglected when thefrequency gets sufficiently high, it is necessary to handle the case.

1. A quiet general case: the Debye model

It is interesting at this point to shift from an ideal coaxial wire model to a more realistic modelwhere the wire presents dielectric losses. Those losses result from the response time of the dielectricused in the wire and of its conductivity σ .

With the Maxwell’s equations the propagation constant γ for a TEM wave and the associatedfrequency response H(ω) can be obtained:15

γ 2 = −ω2εμ0 + j(μ0σω) (14)

H (ω) = exp(−γ L)

where μ0 is the permeability of vacuum.

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032144-6 Lucas et al. AIP Advances 2, 032144 (2012)

Ghz

0

0

0 00

5

5

10

10

15

15

20

20

25

25

30

30

35

35

40

40

45

45

50

50

200

400

600

800

-10

-20

-30

0.5

1

1.5

log(

G( ω

))/m

−θ(

ω)×

πra

d/m

Δθ(ω

)×π

rad/

m

Debye model

Debye model

Difference = Δθ

tan(δ ) = 0.02

tan(δ ) = 0.02

a)

b)

FIG. 2. Plane wave propagation in FR4 substrate considering the Debye model compared to the constant loss angle approx-imation. θ is the phase difference between models

With for instance the Debye model,16 along with the numerical values from,17 one obtains(dashed black) the dispersion plot of Figure 2(a) and the attenuation plot of Figure 2(b). Let usintroduce at this point the classical approximation where the losses are sufficiently small so thatthe transfer function can be approximated as follows: H (ω) = G(ω) × e− j ω

VϕL = e− 1

2ωc

√εr tan(δ)L

× e− j ωVϕ

L . In this Equation c, is the speed of light in vacuum, εr the relative dielectric constant, Lthe length of the wire and δ the loss angle considered as constant. This yields:

1

�= − ω

2c

√εr tan(δ)L

ln(H (ω)

) = − jL

V�

ω − ω1

�(15)

Let us note that this approximation is not physical. Indeed, considering the pair of Equations(8), the phase velocity cannot remain constant when the amplitude varies. The phase shift and theamplitude corresponding to this approximation has also been plotted in Figure (2) (green solid).One can verify that the approximation is very good insofar θ (ω) is concerned (both curves seemssuperimposed), and correct for log (G(ω)) up to around 15 GHz.

At this point, it is interesting to rewrite Equation (14) splitting the imaginary part εi(ω) andreal part εr(ω) of ε(ω): γ 2 = −ω2εrμ0 + j(μ0σω − ω2εiμ0). Without losses only the real part ofthis Equation remains which corresponds to the phase variation L

V�ω of the small and constant δ

approximation: 1V�

= √limω→∞ εr (ω)μ0. Finally by considering the difference between the constant

loss angle model and the Debye model one obtain the blue θ curve plotted in Figure 2. We haveverified that when applying the K-K relations to θ , the attenuation corresponding to the Debyemodel (Figure 2(b) Black dashed) is obtained. This is in fact very general and is demonstrated in thenext section.

2. Making K-K relations work with the small and constant loss angle approximation.

Many times the measurement bandwidth is restricted enough so that the constant loss angleapproximation applies, and one obtains the aspect of the plots presented in Figure (3) for thedispersion and the attenuation. In this figure dispersion curves with and without some so called“anomalous dispersion” are presented.

With that approximation, it is realistic to consider media where the two following limits exist:⎧⎨⎩

limω→±∞ − θ(ω)

ω=

limω→±∞

ln(

G(ω))

|ω| = − 1�

(16)

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032144-7 Lucas et al. AIP Advances 2, 032144 (2012)

ωln G ω)

slope :− 1Ω

−θ (ω)slope : Θ

Arb

itra

ryun

its

Standard propagationAnomalous dispertion

FIG. 3. Dispersion and attenuation with the low and constant loss angle approximation.

In plain words, it means that both G(ω) and θ (ω) are asymptotic to straight lines whose slopes arerespectively − 1

�and . Using those two limits, one obtains deviations to standard propagation that

are limited in frequency: ⎧⎨⎩

limω→±∞( − θ (ω) − ω

) = 0

limω→±∞(

ln(G(ω)

) + |ω|�

)= 0

(17)

Using the pair of conditions (17), via the alternate classical demonstration of the K-K relations8, 18

which consists in integrating in the complex plane using Cauchy integral theorem, one can obtainthe following new relations:

ln(G(ω) + |ω|

�

) = −HT( − θ (ω) − ω

)(18)

−θ (ω) − ω = HT

(ln

(G(ω)

) + |ω|�

)(19)

The absolute value in Equations (16) and (19) turns ln(G(ω)

)into an even function when

symmetrizing over negative frequencies.In order to use these two Equations one must have some knowledge of the system studied.

The limits or 1�

must be known. Note that = limω→+∞(− θ(ω)ω

), is sometime called the frontvelocity.19

In what follows these results are applied to various cases chosen to cover some of the mosttypical situations that can be encountered when working at microwave frequencies.

C. Photonic Crystal at microwave frequencies

A photonic crystal as the one presented in the Figure 4 has been realized. This structure is madeof microstrip lines whose characteristic impedance is alternately Z0 = 50 � and Z1 = 100 �. Thesubstrate used is standard FR4 substrate 0.8 mm thick. The length of each segment has been chosento be equal to a quarter of the wave length for each characteristic impedance at 2 GHz. This way at2 GHz, the wave reflected at the end of each Z0 line segment interferes destructively with the waveentering the segment thus creating the photonic effect. This is often referred to as the Bragg condition.It can be written for this line ω = ω0 + 2nω0. Where ω0 is the frequency for which the length of eachsegment is λ

4 (2 GHz) and n is a positive integer. At these frequencies, the amplitude of the waves isgreatly diminished which leads to frequency bands where the transmission coefficient is very low.Those structures have known some renown recently because the group velocity is superluminal inthese bands.

In order to demonstrate the application and efficiency of the method we have performed themeasurement of ln (G) = ln (|S21|) using spectrum analyzer (SPA) and a generator. The samemeasurement has also been performed using a vector analyzer (VNA) along with the measurement

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032144-8 Lucas et al. AIP Advances 2, 032144 (2012)

Z1Z1Z1 Z0Z0Z0

Z0

Z0

Gen

erat

or

Elementary cell

FIG. 4. Photonic crystal built up from microstrip lines.

VNA

SPA

GHz

S 21

dBm

2 4 6 8 10 12 14 16 18 20

0

0

-5

-10

-15

-20

-25

-30

-35

-40

FIG. 5. |S21| measured using a Vectorial Network Analyzer (VNA) and a Spectrum Analyzer (SPA).

of θ . The results are presented Figure (5). The results match up to 12 GHz. Beyond this frequencya difference is visible. This difference can be accounted for the calibration procedure (simple twoports SOLT) used for the VNA which is certainly not the most adapted for such a large bandwidth.The measurement from the SPA were compensated using a THRU measurement from the generator.

One can verify in Figure 5 that log (|S21|) behaves asymptotically. By mean of asymptoticanalysis a “rectified” gain as defined by the left member of Equation (18) can be obtained: G(ω) + |ω|

�.

This gain has been filled in with zeros for frequencies below the first effectively measured frequencyby the analyzer in order to extrapolate the signal down to the null frequency. It has also beensymmetrized over the negative frequencies to take into account the considerations presented insection II C. Finally the argument of S21 has been calculated using Equation (19). Figure 6 presentsboth the calculated and the measured argument of S21. Of course to compare both results one needsto know the asymptotic behavior of θ as well. To obtain a figure with an interesting magnificationfactor the arguments are presented without the asymptotic behavior.

The results are identical as long as the amplitudes measured by the VNA and the SPA areidentical which validates the method.

D. Left/right handed hybrid media presenting evanescent waves

A left handed media is a media where the phase velocity Vϕ = ωk is negative over a frequency

band.5 A simple way to build up a left handed line is presented in Figure 7 between Port 2 and Port 3.The lumped capacitors and inductors yield the left handed properties of the line at low frequencies.The transmission lines segments used to connect the lumped components turn the line into a standardright handed line by adding their distributed capacitance C and inductance L when the frequencygets higher. At even higher frequency, when the physical length of the line gets small compared thewavelength, propagation has to be considered. A thorough study of that kind of line can be foundin.20 This kind of line can be used to realize couplers21 with special properties by taking advantageof the left handed behavior. These couplers are made as presented in Figure 7 of a left handed linescoupled to a standard microstrip line. In this figure for instance the coupled ports are ports 1 and 2whereas port 4 is the direct port.

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032144-9 Lucas et al. AIP Advances 2, 032144 (2012)

GHz

6 8 10 12 14 16 18 20

0

0

-1

-2

-3

-4

1

2

2

3

4

Rad

ians

− Arg(S21)+Θ Meas

− Arg(S21)+Θ Calc

FIG. 6. Phase measured using a VNA compared to the phase calculated from a spectrum analyzer measurement using K-K.

LC LC LC LC LC

ZCZCZCZCZC

Z0Z0Z0Z0Z0

Cell n Cell n+1

Port 1

Port 3Port 2

Port 4

FIG. 7. Hybrid left handed line coupled to a standard microstrip line.

At low frequencies, when the effect of line segments and coupling are negligible, the capacitorsand inductors induce a phase shift that can be calculated using the Kirchhoff’s Laws:

cos(θ ) = 1 − 1

LCω2(20)

where θ is the phase shift per cell. At frequencies below 12π

√2LC

, the waves in the structure are beevanescent and no phase shift occurs. Consequently the phase shift remains fixed at −π . Furthermorebeing in a forbidden band implies a very high attenuation at consequently a very low signal to noiseration. In Figure 8(a), the measurement of |S32| has been plotted. One can see indeed that belowabout 1 Ghz only noise is measured.

When the frequency gets higher, both the distributed capacitance and inductance of the lineand the coupling effect, introduce an attenuated frequency band (centered about 1.8 Ghz) which hasconsequences on the dispersion. At even higher frequencies the propagation occurring in the linesegments becomes preponderant, and |S32| remains constant. This line is interesting regarding theapplication of K-K relations. Indeed, the waves are successively evanescent, left-handed and righthanded. Further more the measurement starts with noise which is an additional interesting issue.

In Figure 8(b) the solid blue curve represents the measured argument of S32. As the signal islost in noise at low frequency, the argument of S32 is not significant below 1 GHz. When the signalis great enough to be measured above 1 Ghz as the initial phase value has been lost, the argument isbiased and only the phase variations are correct.

In order to be able to apply K-K relations with success and to obtain the black dashed plot ofFigure 8(b) it is first necessary to deal with the data below 1 GHz which are not significant. We usehere the theoretical value of |S32| obtained from Equation (20) which has been plotted (black dashed )in Figure 8(a) to extrapolate the data below 1 Ghz. The black curve of Figure 8(b) is then obtained bysuccessively computing the Hilbert transform and adding the phase due to the propagation knowingthe phase velocity in the line segments. The result obtained is very good and has allowed us to

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032144-10 Lucas et al. AIP Advances 2, 032144 (2012)

0

0

0

0

-2

-2

-4-6-8

-101

1

1

-1

2

2

3

3

4

4

5

5

GHz

GHz

K-K

Extrapolation dataRaw Mes

Raw Mes

|S32

|-a

rg(S

32)×

π

Forbiden Left handed Right handed

a)

b)

FIG. 8. Application of the K-K relations to measurements obtained on an hybrid left/right handed line (per cell data).

verify the left handed behavior of this line between 1 and 3 Ghz when the argument is negative andincreasing with the frequency.

IV. CONCLUSION

Though Kramers-Kronig relations are an interesting tool, it simply relates the real part to theimaginary part of the response of a physical system. Practically, it is often required to be able toextrapolate the phase shift from the attenuation and vice-versa. In a general way, it is not possibleto apply the Kramers-Kronig relations to perform such calculation. Nevertheless, by means of someassumptions on the modulus of the frequency response, it is possible to apply the Kramers-Kronigrelation on many physical systems. Furthermore we have shown that the case of plain propagationhas to be dealt with in a particular way.

In this work, we have firstly recalled how the Kramers-Kronig relations can be obtained fromcausality considerations. We have proposed and thoroughly tested a simple and practical method tocompute the Hilbert transform that subtend the Kramers-Kronig relations. This method takes intoaccount algorithm discrepancy and the corresponding required manipulation of the data.

We have applied successfully the method to various special cases chosen because of theirrepresentatives behaviors. We have verified that method can be applied whether there is propagationor not. In the case of propagation the waves can be evanescent, right or left handed, or be the steadystate resulting from interferences as in the case of photonic crystals.

The precision reached for data extrapolation is very good. It greatly depends nevertheless onthe knowledge of the asymptotical behavior of the dispersion curve or of the attenuation curve ofthe media at high frequencies.

APPENDIX

1. Converting from K-K integral form over positive frequencies to its Hilbert transformform

θ (ω) = −2ω

π

∫ +∞

0

ln(G(u)

)ω2 − u2

du

= − 1

π

∫ +∞

0

[(ω − u) ln

(G(u)

)ω2 − u2

+ (ω + u) ln(G(u)

)ω2 − u2

]du

= − 1

π

[∫ +∞

0

ln(G(u)

)(ω + u)

du +∫ +∞

0

ln(G(u)

)(ω − u)

du

]

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032144-11 Lucas et al. AIP Advances 2, 032144 (2012)

Changing the integration variable from u to −u in the left integral with G(−u) = G(u):

θ (ω) = − 1

π

[∫ 0

−∞

ln(G(u)

)(ω − u)

du +∫ +∞

0

ln(G(u)

)(ω − u)

du

]

θ (ω) = 1

π

∫ ∞

−∞

ln(G(u)

)(u − ω)

du

θ (ω) = H(ln(G(ω)

))

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