Propagation graph
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.
The vertices models objects in the propagation scenario.
is split into three disjoint sets as
is the set of objects named "scatterers".
and an edge may be identified by a pair of vertices as
In a propagation graph, transmitters cannot have incoming edges and receivers cannot have outgoing edges.
Two propagation rules are assumed 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 where
Denoting the Fourier transform of the transmitted signal by
, the received signal reads in the frequency domain
of a propagation graph forms an infinite series[3]
The transfer function is a Neumann series of operators.
Alternatively, it can be viewed pointwise in frequency as a geometric series of matrices.
This observation yields a closed form expression for the transfer function as
is the spectral radius of the matrix given as argument.
The transfer function account for propagation paths irrespective of the number of 'bounces'.
are obtained by inverse Fourier transform of
Closed form expressions are available for partial sums, i.e. by considering only some of the terms in the transfer function.
The partial transfer function for signal components propagation via at least
denotes the number of interactions or the bouncing order.
Special cases: One application of partial transfer functions is in hybrid models, where propagation graphs are employed to model part of the response (usually the higher-order interactions).
The propagation graph methodology have been applied in various settings to create radio channel models.
Such models have been derived for scenarios including To calibrate a propagation graph model, its parameters should be set to reasonable values.
Certain parameters can be derived from simplified geometry of the room.
In particular, reverberation time can be computed via room electromagnetics.
Alternatively, the parameters can ben set according to measurement data using inference techniques such as method of moments (statistics),[5] approximate Bayesian computation.,[16] or deep neural networks[17] The method of propagation graph modeling is related to other methods.
