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