ARSAdvances in Radio ScienceARSAdv. Radio Sci.1684-9973Copernicus PublicationsGöttingen, Germany10.5194/ars-15-21-2017Differential form representation of stochastic electromagnetic fieldsHaiderMichaelmichael.haider@tum.dehttps://orcid.org/0000-0002-5164-432XRusserJohannes A.Institute for Nanoelectronics, Technical University of Munich,
Arcisstraße 21, 80333 München, GermanyMichael Haider (michael.haider@tum.de)21September201715212823December201625February201727February2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://ars.copernicus.org/articles/15/21/2017/ars-15-21-2017.htmlThe full text article is available as a PDF file from https://ars.copernicus.org/articles/15/21/2017/ars-15-21-2017.pdf
In this work, we revisit the theory of stochastic electromagnetic fields
using exterior differential forms. We present a short overview as well as a
brief introduction to the application of differential forms in
electromagnetic theory. Within the framework of exterior calculus we derive
equations for the second order moments, describing stochastic electromagnetic
fields. Since the resulting objects are continuous quantities in space, a
discretization scheme based on the Method of Moments (MoM) is introduced for
numerical treatment. The MoM is applied in such a way, that the notation of
exterior calculus is maintained while we still arrive at the same set of
algebraic equations as obtained for the case of formulating the theory using
the traditional notation of vector calculus. We conclude with an analytic
calculation of the radiated electric field of two Hertzian dipole, excited by
uncorrelated random currents.
Introduction
The most widely used concept for the formulation of Maxwell's
equations is the vector field approach. Even though vector calculus became a
quasi-standard for engineering applications, other formulations like tensor
calculus, quaternions and differential forms could provide deeper insight
into the underlying physics . The theoretic
treatment for stochastic electromagnetic fields, originating from noise
sources with Gaussian probability distribution, has been given by
, using the traditional notation of vector
calculus.
Characterization and modeling of stationary stochastic electromagnetic fields
using field correlations has been expanded to the transmission line matrix
method in , to noisy cyclostationary fields
, and it has been used for
source localization
and imaging of radiated electromagnetic interference (EMI) sources
. Free space electromagnetic field propagation
considering field-field correlations has been addressed in
. Near field measurements for radiated EMI is
discussed in , while analysis of
measurement data by principal component analysis, critical for large data
sets acquired in characterizing fields by correlation information, is
discussed in .
As in the case of deterministic fields, it can be expected that differential
forms may lead to a better understanding of stochastic electromagnetic
fields. Within this work, we take advantage of differential form
representation for the modeling of noisy processes. Noise is an inevitable
perturbation in wireless communication scenarios. While noise is a stochastic
process also interfering signals originating from deterministic processes may
have to be treated as quasi-stochastic signals due to lack of knowledge or
the prohibitive complexity to model the deterministic process. Accurate noise
modeling is crucial with respect to electromagnetic compatibility (EMC),
electromagnetic interference (EMI), and signal integrity (SI) considerations
for the design of electronic components and systems. A careful modeling of
noisy processes and stochastic electromagnetic fields shows also potential to
improve wireless- and on-chip communication .
Differential forms
The calculus of exterior differential forms was introduced by Élie
, based on previous work from Hermann Grassmann,
who himself introduced the exterior algebra in
. The exterior algebra is based upon
the exterior product,
a∧b=-b∧a,
which is defined in more detail in Russer
(, p. 631 ff) as well as in
. We begin
with introducing the fundamental elements of the mathematical framework of
exterior calculus. Let A:U⊆R3→R3 be a
vector field. We consider the line integral of A along a piece-wise
smooth curve C⊂U, given by
∫CAr⋅dr.
The dot-product in Eq. () assigns each
element of A a differential for the respective direction in
R3. By including these differentials into the vector field
itself, a new quantity is introduced. With the choice of a certain basis in
R3, this quantity can be expressed by its coefficients,
A=Axdx+Aydy+Azdz.
Using this expression, the integral from
Eq. () can be rewritten as
∫CA.
The quantity A is called differential form. It is of
degree one, i.e. it can be integrated over a curve C, which is a function
possessing one degree of freedom. Differential forms of degree two and three
are introduced in the same way, where a two-form is a quantity which can be
integrated over area, while a three-form is integrated over volume. For the
definition of two- and three-forms we make use of the exterior product, shown
in Eq. (). The exterior product accounts for
the orientation of a surface in a natural way, when we integrate a two-form
over an area. Two-forms, often referred to as bivectors
, are themselves dual to a corresponding
one-form. This duality is reflected by means of the Hodge star operator
⋆. One can say that the star operator takes a differential form and
converts it to a new differential form consisting of the differentials
missing in the old form . The action of the
Hodge dual for one-forms and two-forms is given by
⋆dx=dy∧dz⋆dy∧dz=dx,⋆dy=dz∧dx⋆dz∧dx=dy,⋆dz=dx∧dy⋆dx∧dy=dz,
and for the three- and zero-form by
⋆dx∧dy∧dz=1.
A one-form is naturally dual to a vector in R3. As long as
Cartesian coordinates are used, the components of the dual vector are just
given by the coefficients of the respective differential form. This duality
relation changes, for different coordinate systems. We see that
Eq. () keeps its form, irrespective
of the considered coordinate system. Only the components of the dual vectors
must be evaluated for each coordinate system separately, if a representation
by components is needed. When considering electromagnetic fields, the
notation becomes completely independent of the choice of a specific
coordinate system, an advantage of differential forms over vectors.
Furthermore, presented a very intuitive way
for visualizing one-, two- and three-forms.
Maxwell's equations are often given in terms of differential equations,
incorporating “curl” and “div” operators in traditional vector calculus
notation. This representation can be generalized to differential forms, by
introducing the exterior differential operator “d”. The exterior
differential operator acting on an n-form A yields a new
n+1-form, representing the spatial variations of
A. In a three-dimensional Cartesian coordinate system, the
exterior derivative operator has the representation,
d=∂∂xdx+∂∂ydy+∂∂zdz∧,
where the differentials of the operator are connected with those of the
differential form by the exterior product, while the partial derivatives act
on the respective coefficients.
With the framework presented so far, we can postulate Maxwell's equations in
a very general and coordinate independent way, starting with Gauss' law,
which relates an electric displacement field D to a charge
density Q. The charge density Q is represented by a
form of degree three, since it is a quantity which can be integrated over
volume. We call Q closed, since dQ=0. This
fact can be easily verified by using Eqs. ()
and (). Poincaré's lemma
(, p. 137 ff), then ensures the existence of
a two-form D, such that dD=Q. By
this, Gauss' law is a direct consequence of how we describe fields by
differential forms. Altogether, time harmonic Maxwell's equations expressed
using differential forms are given by
dE=-jωB,dH=J+jωD,dD=Q,dB=0.
Compared to vector calculus notation, we do not have the “div” and “curl”
operators within Maxwell's equations. Hence, we unified their form by
introducing exterior calculus to electromagnetics.
Stochastic electromagnetic fields
As pointed out by , noise has to be modeled as a
stochastic electromagnetic field. Under the assumption that the considered
fields can be assumed as Gaussian stationary random processes, it suffices to
focus on the first and second order moments . The
first order moment, i.e. the mean, can be set to zero without loss of
generality. The second order moments are described by auto- and
cross-correlation spectra. Consider a random current density J
representing the source of a stochastic electromagnetic field. We obtain the
electric field E, excited from J by the integral
E=∫V′G∧J′,
with a suitable Green's double one-form G
(, p. 129). The Green's double one-form
relates the source current J′ in primed coordinates, i.e. the
domain where J′ is non-vanishing, to the electric field
E in the observation domain .
The integral is extended over the complete volume of the source domain in
primed coordinates. In order to represent the second order moments of the
electric field E and source current J, we define
auto- and cross-correlation functions of random signals {si}i=1N
in time-domain by
cijτ=∫-∞∞sitsjt-τdt.
We call ciiτ auto-correlation function for i=j and
cross-correlation function for i≠j. It should be pointed out here, that
a stationary random process is not square integrable over time in general.
Therefore, one needs to introduce time-windowed quantities in order to
properly define frequency domain variables by Fourier transform. We treat
each spatial component of ET and JT as the Fourier
transform of a time-windowed stationary random processes with Gaussian
statistics. For a compact notation, subscripts 1 and 2 denote the
dependency on different spatial coordinates, r1 and r2,
while a subscript T indicates time-windowing. The correlation double
one-form for the electric field, and the double two-form for the current
density are given by
ΓEr1,r2,ω=limT→∞12TE1T⊗E2T*,ΓJr1,r2,ω=limT→∞12TJ1T⊗J2T*,
where the brackets ⋅ denote the forming of
an ensemble average, the star superscript ⋅* means complex
conjugate, and ⊗ is an implied tensor product. By inserting
Eq. () into Eq. (), we obtain
the correlation double one-form of the electric field from the correlation
double two-form of the source currents,
ΓEr1,r2,ω=∫V1′∫V2′G1∧ΓJr1′,r2′,ω∧G2*.
Similar results have been obtained in
,
however, with traditional vector calculus notation.
Method of moments
The correlation double one-forms and two-forms for the electric field and the
source currents, respectively are continuous in space. In order to enable a
numerical treatment of problems related to stochastic electromagnetic fields,
a discretization scheme has to be introduced. We use the method of moments
(MoM) to transform field problems to network problems following
. The
MoM is based on expanding an unknown function f into a series of known
basis functions {un}n=1N with unknown coefficients. The
dimension of the problem needs to be truncated after a finite
N∈N in order to facilitate numerical evaluation on a computer.
The goal is to establish a linear equation, relating the unknown series
coefficients to known source coefficients. So let f be an unknown function
which is mapped to a known function g by a linear operator L,
Lf=g.
By developing f into a series of basis functions {un}n=1N, we
obtain
∑n=1NanLun=g.
Note that so far, we did not make any approximations, if {un}n=1N
forms a complete basis. Also the known function g can be expanded by a sum
over a set of so called weighting functions {wm}m=1N,
g=∑m=1Nbmwm,
where the coefficients bm are obtained by the inner product
bm=wm,g.
After the second series expansion, the final problem reads as
∑n=1Nanwm,Lun=wm,g,
for each index m≤N. By summarizing the coefficients {an}n=1N and {wm,g}m=1N into
vectors, we arrive at our desired system of linear equations, relating
unknown coefficients to source coefficients,
w1,Lu1⋯w1,LuNw2,Lu1⋯w2,LuN⋮⋱⋮wN,Lu1⋯wN,LuNa1a2⋮aN=w1,gw2,g⋮wN,g.
After addressing some fundamentals of the differential form representation of
the electromagnetic field we proceed to apply the method of moments as
outlined above. First of all, we need to define a proper inner product for
imposing Hilbert space structure in our solution domain. Let our solution
domain be U⊆R3 and let ω and
ν be differential forms of the same degree D∈{1,2} on
U. We define an inner product for one-forms and two-forms on U by
ω,ν=∫Uω*∧⋆ν.
This definition satisfies the requirements for an inner product
, as we will verify in the following. Let a,b∈C be constants, and let ω, μ,
ν be differential forms of degree D∈{1,2}. Then the
following holds,
ω+μ,ν=∫Uω+μ*∧⋆ν=∫Uω*∧⋆ν+∫Uμ*∧⋆ν,ω,aν=∫Uaω*∧⋆ν=a∫Uω*∧⋆ν=aω,ν,ω,ν=∫Uω*∧⋆ν=∫Uν∧⋆ω*=ν,ω*.
The last requirement for a valid inner product is that it has to be a
positive semi-definite functional. Since ω is either a
one-form or a two-form, ⋆ω has exactly the opposite
degree. For showing positive semi-definiteness, we choose ω
to be a one-form, without loss of generality. The Hodge dual
⋆ω is therefore given as a two-form, with the same
coefficients as the one-form ω. We proceed by expressing
ω by a sum over its components. Hence, we have
ω,ω=∫Uω*∧⋆ω=∫U∑nωndxn∧12∑i,jωijdxi∧dxj,
where ωij is an anti-symmetric tensor, i.e.
ωij=-ωji. The coefficients ωij are related to the
coefficients ωn, by
ω1=ω23=-ω32,ω2=ω31=-ω13,ω3=ω12=-ω21.
as can be seen from Eqs. ()–(). For
n=i, n=j, or i=j we do not get any contribution, since dxi∧dxi=0. There are only contribution, if n≠i≠j
holds. Thus, we get
ω,ω=∫Uω1*ω23dx1∧dx2∧dx3+ω2*ω13dx2∧dx1∧dx3+ω3*ω12dx3∧dx1∧dx2,=∫Uω12+ω22+ω32dx1∧dx2∧dx3,
where we used the Hodge duality relations given in
Eqs. ()–(). Since each contribution is
non-negative, we can conclude that
0≤ω,ω,
where ω,ω=0 only
holds, if ω=0. We have defined the zero-element to be a form
with all coefficients equal to 0.
□
In the next step, we expand the electric field E and the current
density J into basis elements. Since E and
J are differential forms, the basis elements also need to be
one-forms and two-forms, respectively. The series expansions are given as
E=∑n=1NVnωUn,J=∑n=1NInωVn,
where the basis one-forms Un for the electric field and the
basis two-forms Vn for the current density are of the same
family, i.e. Un=⋆Vn. This choice can be
interpreted as the differential forms version of Galerkin's method
. The differential form bases are chosen in such a
way, that the orthogonality relation
Um,Un=∫VUm*∧⋆Un=δmn,
holds. In practice, this is quite uncommon since numerical solvers utilizing
the method of moments often use triangular or sinusoidal basis functions with
an extent over several mesh cells. Nevertheless, we use this restriction in
order to make the following equations more readable. The coefficients Vn
and In can be considered as generalized voltages and generalized currents
. Inserting the series
expansions ()
and () into the electric field
integral Eq. () and applying the inner
product () yields
Vmω=∑n=1NInω∫V∫V′Vm*∧G∧Vn′.
We can identify the MoM matrix elements by
Zmnω=∫V∫V′Vm*∧G∧Vn′,
and simplify the Eq. () to
Vmω=∑n=1NInωZmnω.
The same result has been obtained in , but with
traditional vector calculus notation. Since differential forms are naturally
dual to vectors, and a double one-form can be expressed by a dyadic function
in a chosen coordinate system, it can be argued that both results are indeed
equal.
We can now calculate correlation matrices for the generalized voltages
Vnω and the generalized currents Inω,
CV,mnω=limT→∞12TVmTωVnT*ω,CI,mnω=limT→∞12TImTωInT*ω.
The matrix elements can be obtained directly from the correlation double
one-forms and two-forms, by
CV,mnω=∫V1∫V2Vm*∧ΓEr1,r2,ω∧Vn,CI,mnω=∫V1′∫V2′Um′*∧ΓJr1′,r2′,ω∧Un′.
The correlation matrix of generalized voltages CVω, given by its coefficients CV,mnω can be
related to the matrix of generalized currents CIω, given by CI,mnω using
Eq. (). Hence, we obtain
CVω=ZωCIωZ†ω,
By this we have established a direct connection between the correlation
matrix of the sources and the correlations of the observed field.
Analytic calculations
In order to show, that exterior calculus could improve our understanding of
electromagnetic fields in general, and stochastic electromagnetic fields in
particular, we perform analytic calculations on the stochastic emission of
two Hertzian dipoles.
The Green's double one-form G, we introduced in
Eq. (), can be expanded as
G=Gxx′dx⊗dx′+Gxy′dx⊗dy′+Gxz′dx⊗dz′+Gyx′dy⊗dx′+Gyy′dy⊗dy′+Gyz′dy⊗dz′+Gzx′dz⊗dx′+Gzy′dz⊗dy′+Gzz′dz⊗dz′,
when choosing a Cartesian coordinate system. For the sake of a more
streamlined notation, we write
r-r′=x-x′2+y-y′2+z-z′2,
instead of writing down each spatial component separately. Let us also
introduce the functions g1r,r′ and
g2r,r′ as part of the Green's double one-form as
g1r,r′=-3jr-r′5-3kr-r′4+jk2r-r′3,g2r,r′=-jk2r-r′+kr-r′2+jr-r′3.
With Eq. (), ()
and (), the coefficients of the Green's double
one-form from Eq. () can be expressed
as
Gxx′=Z0e-jkr-r′4πkg1⋅x-x′2+g2,Gxy′=Gyx′=Z0e-jkr-r′4πkg1⋅x-x′y-y′,Gxz′=Gzx′=Z0e-jkr-r′4πkg1⋅x-x′z-z′,Gyy′=Z0e-jkr-r′4πkg1⋅y-y′2+g2,Gyz′=Gzy′=Z0e-jkr-r′4πkg1⋅y-y′z-z′,Gzz′=Z0e-jkr-r′4πkg1⋅z-z′2+g2.
We consider two Hertzian dipoles oriented in x direction, excited with
random currents, I1ω and I2ω.
The model of two Hertzian dipoles is simple enough, to allow for an analytic
treatment.
A sketch of the setup we use for our analytic considerations is given by
Fig. .
Setup for analytic calculations.
The distance between the dipoles is 2d in y direction. Thus, we define
the source-current density as
J=I1ωlδxδy-dδzdy∧dz+I2ωlδxδy+dδzdy∧dz.
According to Eq. (), we can obtain the electric
field, excited from this current density by
E=∫V′G∧J′=∫V′Gxx′dx+Gyx′dy+Gzx′dz××I1ωlδx′δy′-dδz′dx′∧dy′∧dz′+∫V′Gxx′dx+Gyx′dy+Gzx′dz××I2ωlδx′δy′+dδz′dx′∧dy′∧dz′.
The Dirac delta functions within J act as a spatial shift, such
that we need to evaluate the coefficients for the Green's double one-form
only for specific source points,
r1′=0,d,0T,r2′=0,-d,0T.
We introduce the following functions
g1±=-3jx2+y±d2+z25-3kx2+y±d2+z24+jk2x2+y±d2+z23,g2±=-jk2x2+y±d2+z2+kx2+y±d2+z22+jx2+y±d2+z23,
where gi- is given by gir,r1′, and gi+ is given by gir,r2′.
Using the shorthand functions gi- and
gi+, we obtain the solution for the electric field
one-form in the whole observation domain as
E=I1ωlZ0e-jkx2+y-d2+z24πk××g1-x2dx+xy-ddy+xzdz+g2-dx+I2ωlZ0e-jkx2+y+d2+z24πk××g1+x2dx+xy+ddy+xzdz+g2+dx.
The current density two-forms at locations r1′ and r2′ are
simply given by
Jr1′=I1ωdy∧dz,Jr2′=I2ωdy∧dz.
With this, we are able to calculate the correlation matrix elements for the
source currents,
ΓJr1′,r1′,ω=Jr1′⊗J*r1′=I1ωI1*ωdy1′∧dz1′⊗dy1′∧dz1′,ΓJr1′,r2′,ω=Jr1′⊗J*r2′=I1ωI2*ωdy1′∧dz1′⊗dy2′∧dz2′,ΓJr2′,r1′,ω=Jr2′⊗J*r1′=I2ωI1*ωdy2′∧dz2′⊗dy1′∧dz1′,ΓJr2′,r2′,ω=Jr2′⊗J*r2′=I2ωI2*ωdy2′∧dz2′⊗dy2′∧dz2′.
The currents I1ω and I2ω are
chosen to be uncorrelated, i.e. I1ωI1*ω=I2ωI2*ω=1 and
I1ωI2*ω=I2ωI1*ω=0. The correlations double one-forms of the
excited fields are continuous in space, just like the correlation two-forms
of the source currents. But in contrast to the sources, defined by Dirac
delta functions, the fields do not vanish everywhere, except for the
observation points. Hence, we want to observe the correlations of the
electric field only at two distinct locations at some distance h from the
Hertzian dipoles in z direction, in order to obtain finite dimensional
correlation matrices, which we can compare to the source correlations.
The locations of our observation points are given by the coordinates,
r1=0,d,hT,r2=0,-d,hT.
Figure also shows how source and observation
plane are aligned.
The electric fields at locations r1 and r2 are given by
Er1=I1ωlZ0e-jkh4πkg2-r1dx+I2ωlZ0e-jk4d2+h24πkg2+r1dx,
and
Er2=I1ωlZ0e-jk4d2+h24πkg2-r2dx+I2ωlZ0e-jkh4πkg2+r2dx.
From this, we can evaluate the matrix elements of the correlations of the
electric field at locations r1 and r2. The auto- and
cross-correlation spectra are given by
ΓEr1,r1,ω=Er1⊗E*r1=l2Z0216π2k2g2-r12dx1⊗dx1+l2Z0216π2k2g2+r12dx1⊗dx1,ΓEr1,r2,ω=Er1⊗E*r2=l2Z02e-jkh-4d2+h216π2k2××g2-r1g2*-r2dx1⊗dx2+l2Z02e-jk4d2+h2-h16π2k2××g2+r1g2*+r2dx1⊗dx2,ΓEr2,r1,ω=Er2⊗E*r1=l2Z02e-jk4d2+h2-h16π2k2××g2-r2g2*-r1dx2⊗dx1+l2Z02e-jkh-4d2+h216π2k2××g2+r2g2*+r1dx2⊗dx1,ΓEr2,r2,ω=Er2⊗E*r2=l2Z0216π2k2g2-r22dx2⊗dx2+l2Z0216π2k2g2+r22dx2⊗dx2.
This shows that the excited fields at a distance are no longer uncorrelated,
since the radiation from the source at r1′ also affects the field
observed in r2. As expected, the magnitude of the cross-correlation
spectra increases compared to the auto-correlation spectra with increasing
observation distance. The analytic expression presented here describes how
our ability to distinguish between the two dipole sources recedes as the
observation distance h increases while d remains constant. The
correlation matrix of the field samples exhibits block Toeplitz character in
the far-field .
Conclusions
We have revisited the the theoretic description of stochastic
electromagnetic fields and introduced exterior calculus for their
description. The method of moments was applied within the framework of
differential forms in order to obtain equations which can be treated
numerically. These equations are equivalent to those obtained from
traditional vector calculus. The main advantage, however, of the formulation
presented over the traditional formulation, given in terms of the vector
calculus notation, is the formulation's independence of any choice of a
particular coordinate system. Other benefits, like the graphical
representation of stochastic field forms, may be worth to be further
explored. We derived the correlation matrix of stochastic generalized
voltages from the correlation matrix of generalized currents. The involved
linear operators, i.e. the impedance matrices
Zω are the same as calculated for
deterministic problems. The impedance matrices, in turn only depend on the
basis functions and the Green's dyadic for the considered field problem.
Thus, the transformation matrix computed for deterministic field problems can
also be applied to noisy field problems . This
facilitates the use of numerical tools, developed for solving deterministic
field problems, to treat, in combination with the method presented, problems
related to stochastic electromagnetic fields. Finally, we have given an
analytic example for calculating field-field-correlations at distinct
observation points in free space, using the framework of exterior calculus on
stochastic electromagnetic fields. While the example of two dipole sources
presented here is simple, it can be easily extended to the case of a
two-dimensional source (dipole-) array, representing for example the currents
on a printed circuit board (PCB). This allows the investigation of noisy
electromagnetic field propagation in complex environments or the
investigation of inverse problems such as stochastic source localization
based on near-field measurements.
The authors declare that they have no conflict of interest.
Acknowledgements
This work was supported by the European Union's Horizon 2020 research and
innovation programme under grant no. 664828 (NEMF21).
Edited by: R. Schuhmann
Reviewed by: two anonymous referees
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