Frame Based Beam Launching for 3D Field Simulations in ... · ib 11 ib 12 ib 12 z ib 22 1 1(z ) = 1(0) + z I d Christine LETROU Frame Based Beam Launching for 3D Field Simulations

Frame Based Beam Launching for 3D FieldSimulations in Urban Environments

Christine LETROU

Lab. SAMOVAR (UMR CNRS 5157) - TELECOM SudParis - FRANCE

ACC Antennas Mini-Symposium - Tel-Aviv University -November 21, 2013

Christine LETROU Frame Based Beam Launching for 3D Field Simulations in Urban Environments

IntroductionFrame-based Gaussian beam shooting (FB-GBS)

FB-GBS applied to indoor propagation simulation in the millimetric rangeFB-GBS for propagation simulations in urban environments

Conclusion

Gaussian beams as an alternative to rays ?

Good localization proper-ties selection of beamsand partial account fordiffraction

0 10 20 30 40 50 60−10

−5

0

5

10

15

20

25

30

z/λ

x/λ

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Each beam tracked like aray (axis path). Paraxialoperators computed only

along the beam axes.

“Big rays” but withoutdiscontinuity effects(smooth decrease).

Paraxial propagatorsNo far field approximation, no caustics.




Conclusion

Source field representation

The decomposition into paraxial GBs can be performed rigorouslythrough frame based algorithms.

Discrete sum of GB fieldsGaussian beam launching - or

shooting (GBS)Christine LETROU Frame Based Beam Launching for 3D Field Simulations in Urban Environments



Conclusion

Outline

1 Frame-based Gaussian beam shooting (FB-GBS)Frame basicsParaxial beams launched from frame windows

2 FB-GBS applied to indoor propagation simulation in themillimetric range

Description of the problemAmplitude-delay profilesFrame to frame channel transfer matrix

3 FB-GBS for propagation simulations in urban environmentsGBS from non directive sources: spectral partitioning solutionDiffraction: playing with frame “resolution”

4 Conclusion




Conclusion

Frame basicsParaxial beams launched from frame windows

Outline





4 Conclusion




Conclusion


E0(x, y): a planar source distribution of electric field in plane xOy, radiatinginto the half space z > 0.

Eα0 (x, y), α = x, y: one of the two components of E0.

We want to express the field radiated E0(x, y) as a GB summation:

E(r) =∑

α∈x,y

∑j∈J

ajBαj (r)

with Bαj a Gaussian beam field radiated by an α-polarized source.

Hence, we wish to write Eα0 (x, y), α = x, y, as:

Eα0 (x, y) =∑j∈J

ajΨj(kx, ky)

where the set Ψj, j ∈ J is a set of Gaussian functions in L2(R2).




Conclusion


The set Ψj, j ∈ J must be a set of Gaussian functions with the followingproperties:

be a complete set of functions, providing stable analysis and synthesisof any function in L2(R2),

span the whole phase-space domain the Gaussian functions Ψj aretranslated in the spatial and spectral domains.

=⇒ Ψj, j ∈ J is a Gabor frame of Gaussian functions.




Conclusion


Gabor frames in L2(R)

Sets of translated and modulated functions:

ψmn(x) = ψ(x− mx)einkxx , (m, n) ∈ Z2

with x and kx : spatial and spectral translation steps.

Such a set is a Gabor frame⇔ x kx ≤ 2π

The set ψmn is a frame of Gaussian functions⇔ x kx ≤ 2πOversampling factor: ν = x k/2π < 1




Conclusion


Phase-space coverage by a frame of Gaussian functions

SpatialLx

νx

x = νxLx

SpectralΩx

νk = ν/νx

k = νkΩx




Conclusion


Gaussian “mother” window ψ

ψ(x) =

√√2

Le−π

x2

L2 =

(kπb

)1/4

e−kx2/2b =1√σ√π

e−x2

2σ2

L: Gaussian window “width” (half-width at 1/e)b: collimation distance; b = L2/λ σ: variance

Gabor frames of Gaussian functions in L2(R2)

Defined as product frames:

Ψµ(x, y) = ψx|mn(x)ψy|pq(y) with µ = (m, n, p, q) ∈ Z4

ψx|mn, ψy|mn: Gabor frames of Gaussian functions in L2(R).




Conclusion


Frame analysis/synthesis

f =∑

(m,n)∈Z2

amnψmn ⇔ f =∑

(n,m)∈Z2

amneimnkx xψnm

Frame coefficients calculationProjection of f on the dual frame functions ψmn:

amn = 〈f , ψmn〉 =

∫ ∞−∞

f (x)ψ∗mn(x)dx =

∫ ∞−∞

f (x)ψ∗(x− mx)e−inkxxdx

... or projection of f on the dual frame functions ˆψnm:

amn = e−imnxkx 〈f , ˆψmn〉 =

e−imnxkx

2π

∫ ∞−∞

f (kx)ˆψ∗(kx − nkx)eimxkx dkx




Conclusion


For high enough oversampling (ν ≤ 0.33): ψ ≈ ν

‖ψ‖2ψ.

−4 −3 −2 −1 0 1 2 3 40

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

x/Lx

FONCTION DUALE

ν = 0.25

−4 −3 −2 −1 0 1 2 3 4−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x/Lx

FONCTION DUALE

ν = 0.5

−4 −3 −2 −1 0 1 2 3 4−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x/Lx

FONCTION DUALE

ν = 0.95

Dual functions for different ν (x =√νL)




Conclusion


Eα0 (x, y) =∑µ

aαµΨµ(x, y) , α ∈ x, y , µ = (m, n, p, q) ∈ Z4

Eα0 (kx, ky) =∑µ∈Z4

aαµΨµ(kx, ky)

E(r) =∑

α∈x,y

∑µ∈Z4

aαµei(mnxkx+pqyky)Bαµ(r)

Bαµ(r) =

√2LxLy

4π2

∫∫ ∞−∞

fα(kx, ky)eiφ(kx,ky)dkxdky

φ(kx, ky) = kx(x−mx)+ky(y−py)+kzz +iπΩ2

x(kx−nkx)

2 +iπΩ2

y(ky−qky)

2

f x(kx, ky) = x− kxkz

z and f y(kx, ky) = y− ky

kzz




Conclusion


Paraxial approximation (for spectrally narrow windows)

Beams radiated by frame windows withq = 0. Cut in the xOz plane.

Beam coordinate system:

(Omp, xµ, yµ, zµ)

with:zµ along the “beam axis”:

zµ · x = nkx/k , zµ · y = qky/k

ξµ = (xµ, yµ)

the transverse position vectorof the observation point




Conclusion


Paraxial beam expression

Bαµ(r) ∼ B0 fα(nkx, qky)

(det Γ(zµ)

det Γ(0)

)1/2

eikzµeik2 ξ

TµΓ(zµ)ξµ

B0 =(

2LxLy

)1/2

and Γ is the “complex curvature matrix” of Bαµ.

Complex curvature matrix of Bαµ

Γ(zµ) =

[zµ − ib11 ib12

ib12 zµ − ib22

]−1

Γ−1(zµ) = Γ−1(0) + zµId




Conclusion


Outline





4 Conclusion




Conclusion


Description of the problem

MVNA

Laboratory Tables

Windows

Metallic Door

Wood Closet

Tx .

.

.

.

Rx3

3.5m

6.5

m

wood

metallic structure

plasterboard

O x

z

Rx1

Rx2

Rx4

Emitting and receivingantennas:open waveguides

Frequency: 60GHz

Frame: ν = 0.25, L = 6λ

625 frame windows:M=P=0, N=Q=12

Number of interactions:7 reflexions




Conclusion


* Tx

* Rx

z

xO

3.5m

6.5m

Beam launchingsimilar to ray launching:

paths along beam axes

image principle (imagebeams)

reflexion/transmissionoperators along beam axes

signal coupled to receiver viareceived antenna

time-delays easily derivedfrom path lengths andobservation point coordinates




Conclusion


x

z

zi

beam axis

equiphase

surfaces

oM

oP

t =Re (zµ + 1

2~ξ Tµ Q(zµ)~ξµ)

c




Conclusion


Amplitude-delay profiles: LOS

RX2

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

−5

retard [ns]

am

plit

ud

e

mesures

simulations

Measurements:

Simulations:

τm

=30.8780

τrms

=19.8964

τm

=28.5086

τrms

=11.2615




Conclusion


Amplitude-delay profiles: LOS

RX3

0 20 40 60 80 100 1200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−6

retard [ns]

am

plit

ud

e

mesures

simulations

Measurements:

Simulations:

τm

=41.8237

τrms

=16.7361

τm

=34.2219

τrms

=7.7589




Conclusion


Frame to frame channel transfer matrix HC

100

200

300

400

500

10

20

30

40

0.2

0.4

0.6

0.8

1

Tx frame index (23*23)Rx frame index (7*7)




Conclusion


Azimuth DoD-DoA channel transfer matrix

Directions of Departure (φ in deg.)

Dir

ecti

on

s o

f A

rriv

al (φ

in

deg

.)

−60 −40 −20 0 20 40 60

−100

−80

−60

−40

−20

0

20

40

60

80

100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1




Conclusion

GBS from non directive sources: spectral partitioning solutionDiffraction: playing with frame “resolution”

Outline





4 Conclusion




Conclusion


Limitation of the region of validity of FB-GBS from oneplane

Half-wavelength dipole far field modulus at r = 50λ.

Reference

Synthetized by FB-GBS from oneplane




Conclusion


Proposed approach

Define 6 overlappingPWS in the 6 planesPj (deduced from farfields in 6half-spaces)

Multiply these PWSby partitioningfunctions χj

Shoot beams from allplanes, and sum allthe radiated fields.




Conclusion


Partitioning functions

Partitioning function χ5 Partitioning function χ1




Conclusion


Partitioned PWS in plane P1: x1-component




Conclusion


Partitioned PWS in plane P1: y1-component




Conclusion


Partitioned PWS in plane P1: z1-component




Conclusion


Radiated field synthesized by GBS from the 6 Pj planes

Synthesized field normNormalized absolute error ofsynthetized complex vector




Conclusion


A challenge for GBS !

Proposed scheme:

re-shooting of paraxial GBsfrom the surface of the obstacle,using frame discretizationand narrow/wide Gaussian windowspatial/spectral localization.




Conclusion





Conclusion


Spectrum of the incident beam field in the plane P′:

ψP′

µ (kx′) = ψµ(kx)kz

k′zeik·−−→OO′

Projection integral of this spectrum on the approximated dualfunctions of the Fourier transforms of the spatially narrow windows:

a′µ′ =

(1

2π

)2 ∫∫ ∞−∞

ψP′

µ (kx′)ν ′xν′y

‖ψ′‖2ψ′×µ′(kx′) d2kx′

Paraxial approximation ... leads to closed form expression for theframe coefficients a′µ′ for the incident field

Transformation at the interface: applied to narrow windowscoefficients (truncation), or later on wide windows coefficients andfields.




Conclusion


Frame change

u(x′) =∑µ∈Z4

A′µ′ψ′µ′(x′) =

∑µ∈Z4

Aµψµ(x′)

Aµ =

∫ ∞−∞

u(x′) ϕ×µ(x′) d2x′

=∑µ′

A′µ′

∫ ∞−∞

ψ′µ′(x′) ϕ×µ(x′) d2x′

=∑µ′

Cµ′

µ A′µ′ with Cµ′

µ =

∫ ∞−∞

ψ′µ′(x′) ϕ×µ(x′) d2x′

Approximate ϕµ closed form expression of Cµ′

µ

A = CA′

where the matrix C can be precomputed in closed form.Christine LETROU Frame Based Beam Launching for 3D Field Simulations in Urban Environments



Conclusion


Scenario

Test caseTilt in the horizontal plane:L = 10λ, ν = 0.09, n=25, q=0 θµ = 48.6 deg tilt

Obstacle description

Perfectly conductingIn the vertical lateral planedefined by x0 = 50λCenter: y0 = 0, z0 = 41.5λSquare with side length = 3λ

Narrow window frameν′ = 0.09 and L′ = 0.075λ

Observation region

In the vertical plane x0 = 30λCenter: y0 = 0, z0 = 60λSquare with side length = 80λ




Conclusion


Truncated field on the obstacle surface

Ey incident field component magnitude onthe obstacle surface.

Comparison of frame summationsand PWS integral referencealong the horizontal axis.




Conclusion


Coefficients de re-decomposition

Fenetres etroites

A′µ′(m′, p′)pour n′ = q′ = 0

Fenetres larges

Aµ(n, q)pour m = p = 0sans “selection”

spectrale

Fenetres larges

Aµ(n, q)pour m = p = 0avec “selection”

spectrale




Conclusion


Reflected-diffracted field in the observation region

Ey component magnitude obtained by GBS after re-expansion

Comparison with reference PWS integralalong the horizontal axis in the

observation region




Conclusion


Reflected-diffracted field in the observation region

Normalized error of GB summation (in dB).Reference: PWS integral.




Conclusion

Outline





4 Conclusion




Conclusion

Gaussian beams:well-suited for intensive multipath propagation, including non farfield interactions.

Use of frame decomposition:- guarantees the “completeness” and stability of initial and successivediscretisations,- offers flexibility to cope with non uniform surfaces (rough or withmoderately small details).

Numerical efficiency to be evaluated.


Documents

Frame Based Beam Launching for 3D Field Simulations in ... · ib 11 ib 12 ib 12 z ib 22 1 1(z ) = 1(0) + z I d Christine LETROU Frame Based Beam Launching for 3D Field Simulations