Properites of the transfer function in multi-dimensional systems Lars Jonsson 1 1 School of Electrical Engineering KTH Royal Institute of Technology, Sweden Mittag Leffler workshop, Stockholm, 2017-05-10 Jonsson (KTH) Properties of the transfer function Mittag Leffler workshop 1/1
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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],
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,
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
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
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]