Channel modeling based on 3D time-varying fields of information

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FINNISH WIRELESS COMMUNICATIONS WORKSHOP ’01

Spatial Channel Modeling Based onWave-Field Representation

Pavel Loskot, Matti Latva-aho

Centre for Wireless CommunicationsUniversity of Oulu, Finland

loskot,[email protected]

24th October 2001

– FWCW’01 –

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Outline

• How to apply electromagnetic (EM) theory to channel modeling incommunication signal processing ? i.e. signal −→ wave

• Overview of existing spatial channel models (literature)

• A new approach to spatial channel modeling is suggested

• A necessary EM theory background is discussed

• The method illustrated on linear stochastic and geometrical channel models

– FWCW’01 – c©Pavel Loskot 2001/10/24 2(14)

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Spatial Channel Models

1. [Hedddergott,Bernhard,Fleury, PIMRC’97]

• time-invariant channel impulse response (CIR) for Rx antenna at location xwith response aRx(Ω); models delay, direction and polarization

E(x, τ,Ω) =M(x)∑m=1

Em(x, τ,Ω)

h(x, τ) =∫aRx(Ω)E(x, τ,Ω)dΩ

2. [Blanz,Jung, TrCom’98]

• time-variant CIR for Tx antenna response aTx(τ,Ω) convolved withdirectional CIR distribution ϑ(τ, t,Ω)

h(τ, t) =∫ϑ(τ, t,Ω)⊗ aTx(τ,Ω)dΩ


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Spatial Channel Models (cont.)

3. [Fleury, TrIT’00]• relates input signal s(t) and received signal r(x, t) at location x

r(x, t) =∫ ∫ ∫

expj2πλ−10 Ωx expj2πνt s(t− τ)h(Ω, τ, ν)dΩdτdν

h(Ω, τ, ν) =L∑l=1

αlδ(Ω− Ωl)δ(τ − τl)δ(ν − νl)

4. [Zwick,Fischer,Didascalou,Wiebeck, JSAC’00]• time-variant CIR through spatial impulse response Φ(t, τ,ΩTx ,ΩRx) and Rx,

Tx antenna responses aRx(t,ΩRx), aTx(t,ΩTx), respectively

h(t, τ) =∫ ∫

aRx(t,ΩRx)Φ(t, τ,ΩTx ,ΩRx)aTx(t,ΩTx)dΩTxdΩRx

Φ(t, τ,ΩTx ,ΩRx) =L(t)∑l=1

αl(t, τ − τl(t))δ(ΩTx − ΩTx ,l)δ(ΩRx − ΩRx ,l)


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Representation of Wireless Transmission

• let us assume a global 3D-space, time and frequency coordinates

• x(t, s) could be an electromagnetic wave

signal,wave system (channel)x(t)↔ X(f) h(t, τ)

x(t, s) h(t, τ ; s, σ)

• when does h(t, τ) or h(t, τ ; s, σ) form a channel impulse response ?


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System Model


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Antenna Representation

Electromagnetic wave

• is a function of time t ∈ R and space s ∈ R3, i.e., it is time-varying field

• fields are invariant w.r.t. coordinate system

• fields are scalar |E|, |H| or vector E,H where given E (H) we know H (E)

Transmit Antenna

• radiating (source) field

x(t, s− sTx) = ATx [x(t)] = x(t)xc(t, s− sTx)

where xc(t, s) is carrier field (hence, amplitude modulator)

Receive Antenna

• observable field

y(t) = ARx [y(t, s)] =∫R3y(t, s)aRx(s− sRx)ds

where aRx(s) is time-invariant infinite bandwidth antenna response


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Wave Propagation• obstacles, atmosphere (rain, fog, smoke), noise and interference (cosmic,

atmospheric, industrial) −→ EM energy absorbed or scattered

• indoor, outdoor and deep-space different propagation conditions, hencechannel models with different accuracy (= prediction)

Near-field• reactive and radiating field with very complex structure

Far-field• ≈ spherical wave


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Maxwell Theory• every medium: permitivity εr, permeability µr conductivity σ

E.g. raindrops, trees, walls (dielectric material), cars (conductive material)

Homogeneous medium

• propagation along straight lines (at least locally)

ε(s)→ ε µ(s)→ µ

Dispersive medium

ε = ε(ω) µ = µ(ω)

Isotropic

• energy flow along the direction of propagation (ε, µ direction independent)

Linear

• ε, µ and σ are independent of applied field E,H

• Maxwell equations are linear and superpozition applies

Etotal = Eincident +Escattered


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Plane Waves• monochromatic, time-harmonic plane wave with wave-vector k

E(s) = e−jks(E0

|k|0 +E1

|k|1 +E2

|k|2 + . . .

)≈ E0e

−jks

• good approximation for far-field and sufficiently short wavelengths

Complex Envelope (Phasor Representation)

E(t, s) = <[E(s)ejωct

]= E0 cos(ωct+ ks + φ)

where ωc is carrier frequency, wave-vector k = 2πΩλ , |k| = 2π/λ and ||Ω|| = 1

is direction of propagation

• for Doppler frequency ωd = 2πfd

E(s) −→ E(s)ejωdt = E0ej(ωdt−ks) 4= x(t, s)


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Spatial Channel Model

• radio channel = mapping from radiating field to observable field

y(t, s) = H [x(t, s)]

spatial channel model

temporal channel model

• linearity

y(t, s) = H[∑i

xi(t, s− sTxi )] =

∑i

H[xi(t, s− sTxi )]


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Linear Stochastic Model

• let the linearity assumption holds and x(t, s) =∑i

xi(t)xci(t, s− sTxi )

H[x(t, s)] = H[∫R3

∫Rx(τ, σ)δ(t− τ, s− σ)dτdσ]

• spatio-temporal channel impulse response

h(t, τ ; s, σ) = H[δ(t− τ, s− σ)]

• temporal channel impulse response

g(t, τ ; sTxi , s

Rxj ) =

∫R3

∫R3xci(τ, σ − sTx

i )h(t, τ ; s, σ)aRxj (s− sRx

j )dσds

• finally

yj(t) =∑i

∫Rxi(τ)g(t, τ ; sTx

i , sRxj )dτ


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Linear Geometrical Model

Geometrical Optics

• high frequency approximation, diffraction neglected

• asymptotic solution of field integrals (Maxwell equations)

• approximation of the field by rays (= locally plane waves)

• ray tracing→ asymptotically accure time-invariant channel impulse response

• assume time-harmonic plane-wave sum approximation of radiating field

xi(t, s− sTxi ) = xi(t)

L∑l=1

Alejωlte

−j2πΩlλ0

(s−sTxi )

• assume separate channels with attn. αl, delay τl, shift Ωl → Ω′l, ωl → ω′l

yj(t) =∑i

L∑l=1

Alαlxi(t− τi)ej2πw′ltej2π

Ωlλ0

sTxi e−j2πΩ′l

λ′lsRxj


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Conclusions

• spatial channel modeling based on wave-fields was presented as an attemptto bring EM theory into communication signal processing

• it was demonstrated for the case of linear stochastic model and lineargeometrical model where plane-wave propagation is assumed

• necessary but not sufficient conditions of radio channel linearity are

– far-field, isotropic and homogeneous medium, no diffraction– but further investigation is still required

• receiving antennas provide us with (some) knowledge on signal distribution

Y (t,Ω/λ) = F (Ω/λ)s [y(t, s)]

where F [.] is 3D-Fourier transform


Channel modeling based on 3D time-varying fields of information

Engineering

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