Propagation graphs are a mathematical modelling method for radio propagation channels. A propagation graph is a signal flow graph in which vertices represent transmitters, receivers or scatterers. Edges in the graph model propagation conditions between vertices. Propagation graph models were initially developed by Troels Pedersen, et al. for multipath propagation in scenarios with multiple scattering, such as indoor radio propagation. It has later been applied in many other scenarios.
Mathematical definition A propagation graph is a simple directed graph G = ( V , E ) {\displaystyle {\mathcal {G}}=({\mathcal {V}},{\mathcal {E}})} with vertex set V {\displaystyle {\mathcal {V}}} and edge set E {\displaystyle {\mathcal {E}}} . The vertices models objects in the propagation scenario. The vertex set V {\displaystyle {\mathcal {V}}} is split into three disjoint sets as
V = V t ∪ V r ∪ V s {\displaystyle {\mathcal {V}}={\mathcal {V}}_{t}\cup {\mathcal {V}}_{r}\cup {\mathcal {V}}_{s}} where V t {\displaystyle {\mathcal {V}}_{t}} is the set of transmitters,
V r {\displaystyle {\mathcal {V}}_{r}} is the set of receivers and
V s {\displaystyle {\mathcal {V}}_{s}} is the set of objects named "scatterers". The edge set E {\displaystyle {\mathcal {E}}} models the propagation models propagation conditions between vertices. Since G {\displaystyle {\mathcal {G}}} is assumed simple, E ⊂ V 2 {\displaystyle {\mathcal {E}}\subset {\mathcal {V}}^{2}} and an edge may be identified by a pair of vertices as e = ( v , v ′ ) {\displaystyle e=(v,v')}
An edge e = ( v , v ′ ) {\displaystyle e=(v,v')} is included in E {\displaystyle {\mathcal {E}}} if a signal emitted by vertex v {\displaystyle v} can propagate to v ′ {\displaystyle v'} . In a propagation graph, transmitters cannot have incoming edges and receivers cannot have outgoing edges. Two propagation rules are assumed
A vertex sums the signals impinging via its ingoing edges and remits a scaled version it via the outgoing edges. Each edge e = ( v , v ′ ) {\displaystyle e=(v,v')} transfers the signal from v {\displaystyle v} to v ′ {\displaystyle v'} scaled by a transfer function. The definition of the vertex gain scaling and the edge transfer functions can be adapted to accommodate particular scenarios and should be defined in order to use the model in simulations. A variety of such definitions have been considered for different propagation graph models in the published literature.
The edge transfer functions (in the Fourier domain) can be grouped into transfer matrices as
D ( f ) {\displaystyle \mathbf {D} (f)} the direct propagation from transmitters to receivers
T ( f ) {\displaystyle \mathbf {T} (f)} transmitters to scatterers
R ( f ) {\displaystyle \mathbf {R} (f)} scatterers to receivers
B ( f ) {\displaystyle \mathbf {B} (f)} scatterers to scatterers, where f {\displaystyle f} is the frequency variable. Denoting the Fourier transform of the transmitted signal by X ( f ) {\displaystyle \mathbf {X} (f)} , the received signal reads in the frequency domain
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