In wireless communications, information is encoded in time-varying 3D patterns which are sampled by antennas. Very fundamental problem.

1. C
W
C
FINNISH WIRELESS COMMUNICATIONS WORKSHOP ’01
Spatial Channel Modeling Based on
Wave-Field Representation
Pavel Loskot, Matti Latva-aho
Centre for Wireless Communications
University of Oulu, Finland
{loskot,matla}@ee.oulu.fi
24th
October 2001
– FWCW’01 –

2. C
W
C
Outline
• How to apply electromagnetic (EM) theory to channel modeling in
communication 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)

3. C
W
C
Spatial Channel Models
1. [Hedddergott,Bernhard,Fleury, PIMRC’97]
• time-invariant channel impulse response (CIR) for Rx antenna at location x
with 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 with
directional CIR distribution ϑ(τ, t, Ω)
h(τ, t) = ϑ(τ, t, Ω) ⊗ aTx
(τ, Ω)dΩ
– FWCW’01 – c Pavel Loskot 2001/10/24 3(14)

4. C
W
C
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−1
0 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 )dΩTx dΩRx
Φ(t, τ, ΩTx , ΩRx ) =
L(t)
l=1
αl(t, τ − τl(t))δ(ΩTx − ΩTx,l)δ(ΩRx − ΩRx,l)
– FWCW’01 – c Pavel Loskot 2001/10/24 4(14)

5. C
W
C
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 ?
– FWCW’01 – c Pavel Loskot 2001/10/24 5(14)

6. C
W
C
System Model
– FWCW’01 – c Pavel Loskot 2001/10/24 6(14)

7. C
W
C
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)] =
R3
˜y(t, s)aRx
(s − sRx
)ds
where aRx
(s) is time-invariant infinite bandwidth antenna response
– FWCW’01 – c Pavel Loskot 2001/10/24 7(14)

8. C
W
C
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, hence
channel models with different accuracy (= prediction)
Near-field
• reactive and radiating field with very complex structure
Far-field
• ≈ spherical wave
– FWCW’01 – c Pavel Loskot 2001/10/24 8(14)

9. C
W
C
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
– FWCW’01 – c Pavel Loskot 2001/10/24 9(14)

10. C
W
C
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 = 2πΩ
λ , |k| = 2π/λ and ||Ω|| = 1
is direction of propagation
• for Doppler frequency ωd = 2πfd
E(s) −→ E(s)ejωdt
= E0ej(ωdt−ks)
= ˜x(t, s)
– FWCW’01 – c Pavel Loskot 2001/10/24 10(14)

11. C
W
C
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 − sTx
i )] =
i
H[˜xi(t, s − sTx
i )]
– FWCW’01 – c Pavel Loskot 2001/10/24 11(14)

12. C
W
C
Linear Stochastic Model
• let the linearity assumption holds and ˜x(t, s) =
i
xi(t)˜xci(t, s − sTx
i )
H[˜x(t, s)] = H[
R3 R
˜x(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; sTx
i , sRx
j ) =
R3 R3
˜xci(τ, σ − sTx
i )h(t, τ; s, σ)aRx
j (s − sRx
j )dσds
• finally
yj(t) =
i R
xi(τ)g(t, τ; sTx
i , sRx
j )dτ
– FWCW’01 – c Pavel Loskot 2001/10/24 12(14)

13. C
W
C
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 − sTx
i ) = xi(t)
L
l=1
Alejωlt
e
−j2π
Ωl
λ0
(s−sTx
i )
• 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 lt
e
j2π
Ωl
λ0
sTx
i
e
−j2π
Ω l
λ l
sRx
j
– FWCW’01 – c Pavel Loskot 2001/10/24 13(14)

14. C
W
C
Conclusions
• spatial channel modeling based on wave-fields was presented as an attempt
to bring EM theory into communication signal processing
• it was demonstrated for the case of linear stochastic model and linear
geometrical 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
– FWCW’01 – c Pavel Loskot 2001/10/24 14(14)