Properites of the transfer function in multi-dimensional ...€¦ · Self Compl. Gustafsson2006...

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

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 2 / 1

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.

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 4 / 1

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

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 5 / 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.

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 6 / 1

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?

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 7 / 1

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

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 8 / 1

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.

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 9 / 1

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

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 10 / 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

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 11 / 1

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

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 12 / 1

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

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 13 / 1

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]

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 14 / 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

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 15 / 1

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

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 16 / 1

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

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 18 / 1

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.

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 19 / 1

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).

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 20 / 1

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)

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 22 / 1

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′

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 23 / 1

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.

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 24 / 1

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.

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 26 / 1

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

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 27 / 1

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.

Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 28 / 1