This document presents a reduced complexity method for computing the transfer matrix of wireless channels in complex indoor environments using propagation graphs. It describes how propagation environments are modeled as directed graphs and introduces a matrix signal flow graph approach that significantly lowers computational complexity. Simulation results indicate that this new method yields similar predictions to the original model with reduced computational demands.

1. Reduced Complexity Transfer Function
Computation for Complex Indoor Channels
Using Propagation Graphs [TD(18)08038]
Oct 1, 2018
Ramoni Adeogun, Troels Pedersen and Ayush Bharti
Email: [ra,troels,ayb]@es.aau.dk
Department of Electronic Systems
Aalborg University
Denmark

2. | Reduced Complexity Transfer Function Computation for Complex Indoor Channels Using Propagation Graphs [TD(18)08038]
1Motivation
Propagation graph (PG) provides a simple and flexible model
structure for MIMO and multilink channels.
PG model account for multiple scattering and provide closed
form expression for the transfer matrix.
Due to a matrix inversion involved in the model, the
complexity increases with increased number of scatterers.
For large number of scatterers, this work presents a reduced
complexity method for computing the transfer matrix of wireless
channels in complex indoor environments using propagation
graphs.
The transfer matrix can be computed using Mason’s theorem
which leads to a much-reduced computational complexity.

3. | Reduced Complexity Transfer Function Computation for Complex Indoor Channels Using Propagation Graphs [TD(18)08038]
2Modelling Propagation Channels Using
Graphs
Propagation environment is modelled via a 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 modelled by
edges in E. (v, v ) ∈ E, iff wave propagation from v ∈ V to v ∈ V
is possible.
Each edge is described by a transfer function Ae(f), where f
denotes the frequency. Ae(f) = 0 for e /∈ E.

4. | Reduced Complexity Transfer Function Computation for Complex Indoor Channels Using Propagation Graphs [TD(18)08038]
3Modelling Propagation Channels Using
Graphs II
Edge transfer functions are collected into:
D(f) : transmitters → receivers T(f) : transmitters → scatterers
R(f) : scatterers → receivers B(f) : scatterers → scatterers.
The transfer function of the PG is given as
H(f) = D(f) + R(f)[I − B(f)]−1
T(f); ρ(B(f)) < 1
Computation is dominated by the matrix inversion with
complexity O(|Vs|3
) which is prohibitive for complex
environments where |Vs| is large.
For an indoor scenario with N rooms and Ns scatterers per room,
this results in a complexity of O((NNs)3
).

5. | Reduced Complexity Transfer Function Computation for Complex Indoor Channels Using Propagation Graphs [TD(18)08038]
4Matrix Signal Flow Graph for Multi-room
Indoor Environments
Multi-room indoor environments can be represented in a matrix
signal flow graph (MSFG) with the rooms as nodes and
propagation between rooms as branches.
We limit our consideration to simple building structure with N
adjacent rooms:
From the floor plan and/or PG, a MSFG is constructed with one
node per room (for transmitter(s) in room 1 and receiver(s) in
room 4):

6. | Reduced Complexity Transfer Function Computation for Complex Indoor Channels Using Propagation Graphs [TD(18)08038]
5Reduced Complexity Transfer Matrix
Computation
Denoting the number of paths in the MSFG from the nth to mth
room as K, the transfer matrix for the channel using the matrix
equivalent of Mason’s rule:
Hnm(f) = Dnm(f) + Rm(f)Σ(f)−1
K
k
(Pnmk (f)Σnmk (f))Tn(f)
Pnmk (f): transfer matrix of kth path computed as the product of all
transfer matrices on the k path from n to m
Σ(f): determinant of the MSFG defined as
Σ = I − Lz
all loops
+ Li Lj
non-touching
− Li Lj Lc
non-touching
+ . . . .
where Lz is the transfer matrix of the zth
loop.
Σnmk (f): co-factor of Pnmk (f) and is equal to Σ(f) with all loops
touching Pnmk removed.
For the simple building structure, the computation is dominated
by Σ(f)−1
with complexity O(N3
s ).

7. | Reduced Complexity Transfer Function Computation for Complex Indoor Channels Using Propagation Graphs [TD(18)08038]
6Simulation Example: Two Adjacent
Rooms
Three computation methods are compared:
PG. Original Propagation Graph Model
VG1. Mason’s Rule: Σ(f) = I − (B11 + B22 + B21B12) + B22B11
VG2. Approximate Mason’s Rule: Σ(f) = I − (B11 + B22 + B21B12)
Signal frequency range: [58, 62] GHz.
Edge transfer functions are defined as in1
with an extra
attenuation factor , η for inter-room propagation.
1T. Pedersen, G. Steinbock, and B. H. Fleury. “Modeling of Reverberant Radio
Channels Using Propagation Graphs”. In: IEEE Trans. Antennas Propag. (2012).

8. | Reduced Complexity Transfer Function Computation for Complex Indoor Channels Using Propagation Graphs [TD(18)08038]
7Results
Averaged power delay profile and computer time:
0 20 40 60 80 100 120
Delay [ns]
-160
-150
-140
-130
-120
-110
-100
PDP[dB]
PG: =0.4
VG1: =0.4
VG2: =0.4
PG: =0.8
VG1: =0.8
VG2: =0.8
5 10 15 20 25 30 35 40
Number of Scatterers Per Room
0
0.05
0.1
0.15
0.2
0.25
ComputationTime[s]
PG
VG1
VG2
Ns = 10, Pdir = 1, Pvis = 0.92 and g = 0.8.
As expected, VG1 and PG yield same transfer function and PDP.
VG1 requires much lower computation time than PG.
Eliminating the last term in Σ (in VG2) impacts the PDP and only
marginally reduces the complexity.

9. | Reduced Complexity Transfer Function Computation for Complex Indoor Channels Using Propagation Graphs [TD(18)08038]
8Conclusion and Ongoing Work
A low complexity implementation of the propagation graph model
is proposed via a representation of the complex environment as
matrix signal flow graph.
Simulation results show that the MSFG equivalent yield same
prediction of the channel as the original model with much lower
computational complexity.
Our ongoing work is generalizing the computation method to
more realistic building structures and exploring other methods.

10. Thanks for your attention!