Modelling Polarized Wireless Channels Using Propagation Graphs

Modelling Polarized Wireless Channels Using
Propagation Graphs
August 28, 2018
Ramoni Adeogun and Troels Pedersen
Email: [ra,troels]@es.aau.dk
Department of Electronic Systems
Aalborg University
Denmark

Ramoni Adeogun and Troels Pedersen Email: [ra,troels]@es.aau.dk | Modelling Polarized Wireless Channels Using Propagation Graphs
1Graph Theory Based Channel Modelling:
Why Bother?
Motivation
Develop a model which allows us to simulate multiple-bounce
scattering up to an arbitrary number of interactions in a
very efﬁcient manner. The model should include both specular
components and the diffuse components, including reverbera-
tion effects for each of them and transition from specular to
diffuse within a single framework.

2Modelling Propagation Channels Using
Graphsa
aT. Pedersen, G. Steinbock, and B. H. Fleury. “Modeling of Reverberant Radio
Channels Using Propagation Graphs”. In: IEEE Trans. Antennas Propag. (2012).
We consider a simple directed graph G = (V, E)
Vertex set V: The transmitters, scatterers and receivers are
represented by vertices in the set: V = Vt ∪ Vs ∪ Vr
Edge set E: Wave propagation between vertices is modeled by
edges in E. (v, v ) ∈ E, iff wave propagation from v ∈ V to v ∈ V
is possible.

3Modelling Propagation Channels Using
Graphs(2)
We deﬁne two rules for propagation:
• The sum of signals impinging via the incoming edges of a scatterer
are re-emitted via the outgoing edges.
• An edge e = (v, v ) ∈ E transfers the signal from v to v according
to its transfer function Ae(f) where f denotes frequency. We set
Ae(f) = 0 for e /∈ E.
The transfer matrix of the channel is given as
H(f) = D(f) +
1 bounce
R(f)T(f) +
2 bounce
R(f)B(f)T(f)
+
3 bounce
R(f)B2
(f)T(f) + . . .
= D(f) + R(f)[I − B(f)]−1
T(f); ρ(B) < 1

4Propagation Graph Model: State of the
Art
Propagation
Graph Model
(PGM)
SISO [2006] MIMO [2007]
Modelling
[2012]
Analysis
POLARIZED
CHANNELS
Indoor Dis-
tributed [2013]
HSR/Tunnel
[2012-2016]
Outdoor-to-
Indoor [2014]
mmWave
[2015/17]
Hybrid
BAN/BSE
[2013]
Rank and Ca-
pacity [2013]
Correlation
and Angu-
lar [2014]
METIS+PGM
[2016]
Ray Trac-
ing+PGM
[2016]

5Polarized Channel Modelling
Ideally, cross-polar transmissions should be equal to zero.
Depolarization mechanisms: results in cross-talks between
orthogonal polarizations.
1. Antenna cross-polar isolation.
2. Polarization mismatch due to array tilting.
3. Electromagnetic channel depolarization resulting from interactions
of electromagnetic waves with the scatterers.
Developing propagation graph based model for polarized
channels requires incorporating these effects into the classical
graph model
1. Cross-polar isolation and array tilting can be accounted for via
appropriate operations at the transmitter and receiver.
2. Scatterer based depolarization results from the environment.

6From Unipolarized to Multipolarized Prop-
agation Modelling
Assumption: Delay dispersion is only due to propagation
between scatterers.
Each edge is represented in the unpolarized graph by the scalar
transfer function
Ae(f) = ge(f) exp[j(Ψe − 2πfτe)]
In the graph description of a multi-polarized propagation channel:
1. Signal propagating on the edges is an Np dimensional vector;
2. Depolarization effect at the nth scattering vertex is modelled as a
Np × Np matrix, Pn.
Each edge in the polarized graph is described via a
vector/matrix transfer function as
Ae(f) = Ge(f) exp[j(Ψe − 2πfτe1)]

7From Unipolarized to Multipolarized Prop-
agation Modeling(2)
The polarimetric edge gain for are modelled as
Ge(f) =



Fe
(4πfτe)
; e ∈ Ed
Fe√
4πτ2
e fµ(Et)S(Et)
; e ∈ Et
gFe
odi(e)
; e ∈ Es
Fe√
4πτ2
e fµ(Er)S(Er)
; e ∈ Er
odi(e) denotes the number of edges from the initial vertex of edge, e to
other scatterers and
µ(Ea) =
1
|Ea| e⊂Ea
τe, S(Ea) =
e⊂Ea
τ−2
e , Ea ⊂ E
Edge depolarization matrix, Fe deﬁned as
Fe =



XT
t Xr; e ∈ Ed
Xt; e ∈ Et
Pn; e ∈ Es
PnXT
r ; e ∈ Er

8Stochastic Generation of Polarized Prop-
agation Graph
The procedure for generating a realization of the graph is as
follows:
1. Specify the coordinates of the transmitter(s) and receiver(s).
2. Draw the positions of N scatterers uniformly on the room volume or
walls.
3. Generate edges according to the edge occurrence probability
Pr[e ∈ E] =



Pdir, e ∈ Ed
Pvis, e ∈ (Et, Es, Er)
0, otherwise
.
4. Compute polarimetric gains and edge transfer functions
5. Compute H(f); f = fmin, fmin + ∆f, · · · , fmax
6. Compute channel impulse response via inverse discrete Fourier
transform.

9Application to Dual Polarized Channels
Assumption: Depolarization effect independent of propagation
direction and equal for all scattering edges.
Scattering matrix modelled as
Pn =
1
√
γn
1
√
γn
√
γn 1
; 0 ≤ γn < 1
We consider a half-wave dipole antenna oriented in the y-z plane
at a tilt angle, β, to the z-axis.
Collocated antennas tilted at β = 0o
, 90o
are considered.
Simulation Parameters:
Parameters Ns Pvis g Freq. B
Value 10 0.9 0.6 2.5 GHz 1 GHz

10Simulation and Results
Power[dB]
Delay [ns]
0 50 100 150
-110
-100
-90
-80
-70
-60
Co-pol Cross-pol
0 50 100 150
-110
-100
-90
-80
-70
-60
0 50 100 150
-110
-100
-90
-80
-70
-60
0 50 100 150
-110
-100
-90
-80
-70
-60
0 50 100 150
-110
-100
-90
-80
-70
-60
0 50 100 150
-110
-100
-90
-80
-70
-60
0 50 100 150
-110
-100
-90
-80
-70
-60
0 50 100 150
-110
-100
-90
-80
-70
-60
NB
=2 NB
=3 NB
=4
NB
=1
NB
=5 NB
=6 NB
=10 NB
=

11Simulation and Results (2)
0 20 40
Delay [ns]
-80
-60
-40
Power[dB]
= 0.1
0 10 20 30 40 50
Delay [ns]
0
5
10
15
20
XPR[dB]
= 0.1
= 0.4
0 20 40
Delay [ns]
-80
-60
-40
Power[dB]
= 0.4
Co-pol
Cross-pol
Depolarization Time
Different decay rate for early arrivals
Increased mixing parameter = decrease depolarization time

12Simulation and Results (3)
Averaged PDP and XPR: Effect of room sizes
0 50 100
Delay[ns]
-100
-80
-60
-40
Power[dB]
3 4 3
0 10 20 30 40 50 60 70
Delay[ns]
0
5
10
15
20
XPR[dB]
3 4 3
3 6 3
4 10 3
0 50 100
Delay[ns]
-100
-80
-60
-40
Power[dB]
3 6 3
0 50 100
Delay[ns]
-100
-80
-60
-40
Power[dB]
4 10 3
Co-pol
Cross-pol
Larger rooms = weaker early arrivals and longer depolarization
region.

13Conclusion
A framework for modelling polarized channels via propagation
graphs is presented.
Reverberation due to multiple scattering, polarimetric antenna
responses and scattering based depolarization are captured in
the model.
• Recursive structure of PG results in exponentially decaying PDP
for both co- and cross-channels
• Co- and cross-polar channel has different decays rates in early
part of PDP
• Cross-polarization ratio decreases with delay before becoming
nearly constant.

Modelling Polarized Wireless Channels Using Propagation Graphs

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