Properites of the transfer function in multi-dimensional ...€¦ · Self Compl. Gustafsson2006 Maloney2011 SCADA, Kolitsidas 14,16 JJH. Wang2016 BAVA, Elsallal2011 Stasiowski2008

Properites of the transfer function in multi-dimensionalsystems

Lars Jonsson1

1School of Electrical EngineeringKTH Royal Institute of Technology, Sweden

Mittag Leffler workshop, Stockholm, 2017-05-10

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 1 / 1

Table of contents


Motivation: Arrays are passive, (unit-cell approach)

Ground plane

Matchingnetwork

Antennaelement

TE or TM-mode

unit cell

Array feed

z

Γ ΓTE,TM

d

Ei

Er

θ

Sum-rule (Bode-Fano type result) for reflection coefficient ΓTE.

The lowest Floquet mode in array system is scattering passive, hence:

I(θ) :=

∫ ∞0

ln(|ΓTE(λ, θ)|−1) dλ ≤ 2π2µsd cos θ

where d is thickness, µs, maximum relative static permeability, λwavelength at frequency f .

[Refs: Rozanov 2000, Sjoberg, 2011]Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 3 / 1

Methods to estimate bandwidth(s) and scan-range

Brick-wall estimate

Given M wavelength (or frequency) bands Bm := [λ−,m, λ+,m],

Define |Γm| := maxλ∈Bm,θ∈[θ0,θ1] |Γ(λ, θ)|.Clearly ln(|Γ(λ, θ)|−1) ≥ ln(|Γm|−1)

Hence:

0 ≤ ηTEM :=

∑Mm=1 ln(|Γm|−1)(λm,+ − λm,−)

2π2µsd cos θ1≤ 1 (1)

Here ηTEM is the Array Figure of Merit for a M -band antenna.


Array figure of merit (J, Kolitsidas, Hussain 2013)

0.5 1 1.5 2 2.5 3 3.5

0.2

0.4

0.6

0.8

1

Schaubert’07Vivaldi, Schuneman2001

Kindt2010

Microstrip Huss2005

PUMA, Holland2012Patch, Infante2010

Dipole, Doane2013

Dipole, Jones2007Self Compl. Gustafsson2006

Maloney2011

SCADA, Kolitsidas 14,16

JJH. Wang2016

BAVA, Elsallal2011

Stasiowski2008

d/λhf

ηTE

ηTE1 =(λ+ − λ−) ln |Γmax|−1

2π2µsd cos θ1≤ 1


Strongly Coupled Array Antenna (SCADA)

Picture removed. Patent process in progress.My PhD student Kolitsidas has developed SCADA, which hasfmax/fmin = 7, with reflection coefficient Γ < 0.35 (∼ −9.1 dB), andwith large scan-range. ηTE1 ∼ 0.84.— Potential candidate to next generation base stations.


Questions: Extension possibilities?

Observation

Sum-rules yield interesting results for antennas and other electrical devices.

Questions:

Are there multi-dimensional extensions? I.e. can we have additionalcoordinates beyond time like θ or kx and obtain a higher ordersum-rule, giving bounds in two or more variables (ω, kx)?.

Are there Hilbert-transform pairs in multi-dimension?

Are there (Herglotz) representations in higher dimensions? (Yes)

What applications use multi-dimensional Herglotz-functions, and howdo they connect to e.g. antenna theory?


Work in progress

This work aims towards using the connection between analyticproperties in multidimensional linear systems to electromagneticproblems.

It is based on results from:

– Vladimirov, Methods of the theory of Generalized Functions, 2002– Reed & Simon Methods of Modern Mathematical Physics Part II,

Fourier Analysis, Self-Adjointness 2003,– King, Hilbert Transforms, 2009;– Zemanian 1965, Bernland, Luger, Gustafsson 2011.– Agler, et.al. 2012,– Luger, Nedic 2016


Linear, passive system: 1 dimension

f(t)w∗

v(t) = (w ∗ f)(t)

Requirements

Linear: f1 7→ v1, f 7→ v2 ⇒(αf1 + βf2) 7→ αv1 + βv2.

Time translational invariant.

Continuity: If f → 0 in E ′then v → 0 in D′.Real-valued: f ∈ R⇒ v ∈ R.

Passivity: Let f ∈ D, then∫ t−∞ v(τ)f(τ) dτ > 0 for all t

W (s) = L[w](s) =

∫Rw(s)e−st dt

Properties (Zemanian 1965)

W (s) is analytic forRe(s) > 0

W (s) ∈ R for 0 < s ∈ RReW (s) ≥ 0, Re(s) > 0

h(z) = jW (−jz) is a Herglotzfunction i.e., h : C+ 7→ C+ ∪ Rand it is holomorphic.


Properties of Herglotz functions (1D)

Nevanlinna representation:

h(z) = a+ bz +1

π

∫R

1

x− z− x

1 + x2dµ(x)

Let N0, N∞ ≥ 0. If h has the asymptotic behavior (+conditions):

h(ω) =

N0∑n=−1

a2n−1ω2n−1 + o(ω2N0−1), ω→0

h(ω) =

−N∞+1∑n=1

b2n−1ω2n−1 + o(ω1−2N∞), ω→∞

then we have:

Sum-rule [see e.g. Bernland etal 2011]

limδ→0+

limy→0+

2

π

∫ δ−1

δ

Imh(ω + iy)

ω2ndω = a2n−1 − b2n−1


Example: Extinction cross section

Optical theorem, and forward scattering

The extinction cross section σe(ω, k) with ω = ck, is the imaginary part ofa Laplace transform of a linear passive operator. We have

0 ≤ σe(ω, k) =4π

kIm e∗ · S(

ω

c, k, k) · e = Imhk(ω).

Here hk is a Herglotz function. We have

hk(ω)→ γ(k)ω

c, as ω → 0, and hk(ω)→ 2iA(k) as ω →∞.

γ = e∗ · γe · e+ k× e∗ · γm · k× e, A is the projected area in direction k.

k

Ei = eE0eikk·r

S(k, k, ks) ks

Es

ks = k


Bound on D/Q, Gustafsson etal 2007

Extinction cross section and a D/Q-bound

The extinction cross section σe(ω, k) is a the imaginary part of aHergloz-function hk(ω) we have∫ ∞

0

σe(k)

k2dk =

π

2γ ⇒ D

Q≤ ηk3

0

2πγ, η ∈ [0, 1].

for n = 1, since a1 = γ, b−1 = 0.

0.1 1 10 100 10000.01

0.1

1

` /`

Chu bound,D/Q/(k a)3

0

´=1

k a¿10

´=1/2

`a

2

1

`

1 2

physical bounds

e

Ref:Gustafsson etal ’07, ’09

Sum-rule applications:Array figure of meritHigh-impedance surfacesScattering with circularpolarization


Property: Dispersion relations – Hilbert pair

Let D = ε(ω)ε0E, where E is the electric field and D is the displacementvector. The ε(ω) is a system response and it satisfy the following relation:

Example: Dielectric constant – Titchmarsh thm, L2

Appropriate assumptions on ε(ω) (bounded, continuous, asymptotic etc.)we have the dispersion relation [Landau etal; King; Bernland]

Re ε(ω) = ε∞ + limδ→0

1

π

∫|ξ−ω|>δ

Im(ε(ξ))

ξ − ωdξ, ω ∈ R

Underlying structure

1D PassivityHerglotz,

representationSum-rule,

Hilbert-pair


Linear passive system, (n-dim)

Input: u(x) = (u1(x), . . . , uN (x)). Output: f(x) = (f1, . . . , fN ).

Linearity. ua 7→ fa, ub 7→ fb then αua + βub 7→ αfa + βfb.Reality: u ∈ RN then f ∈ RN .Continuity: If uj → 0 ∀j ∈ [1, N ] in E ′ then fk → 0 in D′ for all k.Translational invariance: Let u(x) 7→ f(x) then ∀h ∈ Rnu(x+ h) 7→ f(x+ h)Admittance Passive w.r.t the cone Γ: Re

∫−Γ(Z ∗ φ) · φ dx ≥ 0

There exists a unique N ×N matrix Z(x), with Zjk ∈ D′(Rn) such thatf = Z ∗ u.

Examples

linear n-port circuit theory with RLC-components, with zero initialconditions.

Passive Cauchy systems:∑

j Zj∂j + Z0 with constant matrices Zj ,real and symmetric with

∑j qjZj ≥ 0,∀q ∈ intC∗ and ReZ0 ≥ 0.

(Maxwell, Linear acoustics) [Vladimirov 20.6 Thm 1]


Laplace transform gives n-dim Herglotz function

Theorem 1: see Vladimirov 20.2.7

The Laplace transform Z(z) = L[Z](z) of a passive linear system matrixZ is holomorphic for z ∈ TC where TC = Rn + iC, C = int Γ∗,

furthermore ReL(Z) ≥ 0⇒ (L(Z)a+ L(Z)Ta) · a ≥ 0 in TC , e.g.,

jZ(−jz) is a n-dim Herglotz function

z = x+ iy, x ∈ R, y ∈ R+

Im z

Re z

T 1 = C+ = R + iR+ TC = C+2, C = R2+

x1

z = x+ iy, x ∈ R2, y ∈ R2+

x2

y2

y1


Cauchy Kernel

Cauchy(-Szego) Kernels KC [Vladimirov 10.2]

The Cauchy kernel for a connected open cone in Rn with vertex 0 is:

KC(z) =

∫C∗

eiz·ξ dξ = F [θC∗e−y·ξ], z = x+ iy

Here θC∗ is the characteristic-function of C∗, the conjugate cone.

KRn+

(z) = in

z1···zn ⇒ K1(x) = ix+i0 = πδ(x) + iP 1

x .

KV +(z) = 2nπ(n−1)/2Γ(n+12 )(−z2)−

n+12 , z ∈ T V +

,z2 = z2

0 − z21 − · · · − z2

n.

KPn(Z) = πn(n−1)/2jn2 1! . . . (n− 1)!

(detZ)n, Z ∈ TPn ,

Properties: K−C(x) = (−1)nKC(x), x ∈ C ∪ (−C);ImKC(x) = 1

2iF(θC∗ − θ−C∗). KC holomorphic in TC


Property: Generalized Titchmarsh’s theorem

Theorem 2: (V10.6) Generalized Titchmarsh’a relation, n-dim

Let f+ = Fg ∈ Hs, i.e., g ∈ L2s the following things are equivalent:

supp g = suppF−1(f+) ⊂ C∗. [g is causal]

(‘Hilbert’-transform pair)

Re f+(x) =−2

(2π)n

∫Rn

(Im f+)(x′)(ImKC)+(x− x′) dx′,

Im f+(x) =2

(2π)n

∫Rn

(Re f+)(x′)(ImKC)+(x− x′) dx′,

f+ is a boundary value of some f ∈ H(s)(TC). (Holomorphic in TC)

Note: Re f+ and Im f+ form a ‘Hilbert’-transform pair.The above are derived from the Cauchy-Bochner-transform

f(z) =1

(2π)n

∫Rn

KC(z − x′)f(x′) dx′ =1

(2π)n(f(x′),K(z − x′)),

where z ∈ TC ∪ T−C :Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 17 / 1

Example, 2 dimension

2-dim case: Herglotz + L2 in R2+

We have KR2+

(z) = −1z1z2

. Note that in distributional sense

limy→0KR2

+(x+ iy) = −(P

1

x1− iπδ(x1))(P

1

x2− iπδ(x2))

Thus ImKR2+

(x) = πP 1x1δ(x2) + πP 1

x2δ(x1). The Theorem 2

‘Hilbert-transform’ pair becomes:

Re f+(x1, x2) =1

2πP

∫R

Im f+(x′, x2)

x1 − x′+

Im f+(x1, x′)

x2 − x′dx′

Im f Re f −12π2 (ImKC) ∗ Im f

x→ x+ iε

ε = 10−3

f = Fχ>0e−a·r


Poisson Kernel and Schwarz kernel [Vladimirov 11, 12]

Poisson Kernel

PC(x, y) = KC(x+iy)πnKC(iy) , (x, y) ∈ TC

PRn+

(x, y) = y1···ynπn|z1|2···|zn|2

PV +(x, y) =2nΓ(n+1

2)

πn+32

(y2)n+12

|(x+iy)2|n+1

Schwarz kernel

SC(z, z0) = 2KC(z)KC(−z0)

(2π)nKC(z−z0)− PC(x0, y0)

SRn+

=2in

(2π)n

(1

z1− 1

z01

)· · ·

(1

zn− 1

z0n

)− PRn

+(x0, y0)

SV + is also known explicitly.


A representation theorem [Vladimirov 17.6]

Properties of Herglotz functions

Let 0 ≤ Im f with f ∈ H+(TC), with cone Rn+ and µ is a non-negativetempered measure. Then

f(z) = i

∫Rn

SRn+

(z − x′; z0 − x′) dµ(x′) + (a, z) + b(z0), z, z0 ∈ TC ,

where µ = Im f+, b(z0) = Re(f(z0))− (a, x0), aj = limyj→0Im f(iy)yj

,

j = 1 . . . , n, y ∈ Rn

Note 1) H+ are Herglotz-functions condition on the tubular cone TC .Note 2) For n > 1: Agler et.al. 2012 have operator representationtheorems. Integral representations: Vladimirov 2002, Luger + Nedic 2016ArXiv 2016 ⇒ (a, z)→

∑j ajzj . Condition on measure.

Note 3: Generalizations to other regular cones are known (Vladimirov).


Herglotz-Nevanlinna representation, 2-dim

Let z = (z1, z2) ∈ C+2+, t = (t1, t2) ∈ R2 and

S2(z, t) :=−i

2

(1

t1 − z1− 1

t1 + i

)(1

t2 − z2− 1

t2 + i

)+

1

(1 + t21)(1 + t22)

Theorem 3 [Luger, Nedic 2016]

A function q : C+2 7→ C is a Herglotz function iff

q(z) = a+ b · z +1

π2

∫R2

S2(z, t) dµ(t)

where a ∈ R, bj ≥ 0 and µ is a postive Borel measure on R2 such that∫R2

1

(1 + t21)(1 + t22)dµ(t) <∞

and ∫R2

Re

[(1

t1 + z1− 1

t1 + i

)(1

t2 − z2− 1

t2 − i

)]dµ(t) = 0

for all z ∈ C+2.Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 21 / 1

Two classes: Dependent and independent variables

Observation: Real and imaginary part of the kernel LZ for a passivesystem are connected with through the Cauchy-kernel KC , which dependson domain, (cone) Γ of the variables x ∈ Γ.

Case 1: Light cone Γ = V +n

Dispersion-relations for solutions to Cauchy-problem in homogeneousspace, (t, x) ∈ V +

n . [Vladimirov 2002].

Spatial dispersion properties V +4 .

Case 2: Cone Γ = RN+ – independent variables

Examples:

Nonlinear susceptibility, variables ωk ∈ Rn+.

Certain elements of nonlinear circuit theory

Homogenization of ωεj(ω)


Examples of applications, Hilbert transform pairs

Spatial dispersion, periodic structure (case 1)

Let ε(ω,k) be analytic in (ω,k) ∈ T V +, and with boundary value

ε+(ω,k) in Hs for (ω,k) ∈ R4 then

Re ε+(ω,k) =−2

(2π)n(ImKV +) ∗ Im ε+ =

Γ(2)

π3

∫R

∫R3

(ImKV +)(ω − ω′,k − k′) Im ε+(ω′,k′) dω′ dVk′


Case 2: n-dimensional Hilbert transform on cone Rn+

(Hnf)(x) =1

πnP

∫Rn

f(s)Πnk=1

1

xk − skds

Furthermore we have that (H2nf)(x) = (−1)nf(x)

Examples: King 2009

Hn[sin(a · s)](x) =

(−1)(n−1)/2 cos(a · x)Πk sgn ak n odd

(−1)n/2 sin(a · x)Πk sgn ak n even

Hn[cos(a · s)](x) =

(−1)(n−1)/2 sin(a · x)Πk sgn ak n odd

(−1)n/2 cos(a · x)Πk sgn ak n even

Hn[eja·s](x) = (−1)neja·xΠk sgn ak

Hn[e−as2](x) = (−j)ne−ax

2Πk erf(jxk

√a)

Hn is a special case of a Calderon-Zygmund singular operator.


Application, con’t

Nonlinear electric susceptibilities:

P (t) =∑n

P (n)(t),

where

P(n)k (ω) = ε0

∫R

dω1E`1(ω1) · · ·∫R

dωnE`n(ωn)·

χ(n)k`1`2···`n(ω1, . . . , ωn)δ(ω − ω1 − ω2 − · · ·ωn)

Nonlinear electric susceptibilities [Peiponen 1988, King 2009]

Ref: Peiponen 1988 (see also King: Hilbert transforms Chapt 22.9)

χ(n)(ω1, . . . , ωn) = jnHn[χ(n)]

n-odd:

Reχ(n)(ω1, . . . , ωn) =jn+1

πnP

∫R· · ·P

∫R

Imχ(n)(ω′1, . . . , ω′n) dω′1 · · · dω′n

(ω1 − ω′1) · · · (ωn − ω′n)Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 25 / 1

Sum-rules, case 2, King (Chapt 22.11) 2009

Using that χ(n)(ω1, . . . , ωk, . . . , ωn) = O(ω−1−δk ) as ω →∞, the result:

Sum-rule nonlinear susceptibility [Peiponen 1988]:∫R· · ·∫R

(ω1 · · ·ωn)s−1[χ(n)(ω1, . . . , ωn)]t dω1 · · · dωn = 0

where s = 1, 2, . . ., t = 1, 2, . . ., s ≤ t.

Note: The claimed dispersion-relations and sum-rule differ from thepassive system approach outlined above if n even. They are discussed inKing.


Examples of Applications con’t

Homogenization, Milton 2002, Orum etal 2011

Find an efficient media paramter: λ∗ representingfrom a microscopicλ(r) = λ1χ1(r) + λ2χ2(r) + λ3χ3(r). Note thatλ∗ is Herglotz in λjj . The representation:

λ∗ = 1−∫Tn

K(λjj , θkk) dµ(θ1, θ2, θ3)

separate geometry and amplitude

DtN-map, Cassier etal 2016

Let f on ∂Ω be a tangential electric field forMaxwells eqn’s on Ω consisting ofz = ωεj , ωµjj-materials. The (generalized)Dirichlet-to-Neumann-map Λz generates an-dimensional Herglotz-functionhf (z) = 〈f ,Λzf〉. ⇒ Representation thm’s.

Two parameter space:Orum, Cherkaev,Golden 2011 – inverseproblem for sea icegeometry recovery.

∂Ω


Conclusions

1D Properties of passive system ⇒ n-dimensional passive problems.[Herglotz-functions]

n-dim Herglotz functions have representation theorems. [SchwarzKernel]

Herglotz + supy∈C ‖f(x+ iy)‖s <∞ yields a GeneralizedTitchmarsh theorem [Cauchy-Kernel, and Cauchy-Bochner transform]

Representation theorems [Vladimirov ’02, Luger etal ’16].

Applications: Homogenization, DtN map, Dispersion relations.

Potential applications in spatial dispersion, nonlinear susceptibilities,multi-dimensional phase reconstruction.


Documents

Properites of the transfer function in multi-dimensional ...€¦ · Self Compl. Gustafsson2006 Maloney2011 SCADA, Kolitsidas 14,16 JJH. Wang2016 BAVA, Elsallal2011 Stasiowski2008