Propagation graph
Propagation graphs are a
Mathematical definition
A propagation graph is a
The vertices models objects in the propagation scenario. The vertex set is split into three disjoint sets as where is the set of transmitters, is the set of receivers and is the set of objects named "scatterers".
The edge set models the propagation models propagation conditions between vertices. Since is assumed simple, and an edge may be identified by a pair of vertices as An edge is included in if a signal emitted by vertex can propagate to . 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 transfers the signal from to 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
- the direct propagation from transmitters to receivers
- transmitters to scatterers
- scatterers to receivers
- scatterers to scatterers,
where is the frequency variable.
Denoting the Fourier transform of the transmitted signal by , the received signal reads in the frequency domain
Transfer function
The transfer function of a propagation graph forms an infinite series[3]
The series is similar to the
The impulse responses are obtained by inverse Fourier transform of
Partial transfer function
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 and at most interactions is defined as
The partial transfer function is then[3]
- : Full transfer function.
- : Inderect term only.
- : Only terms with or fewer bounces are kept (-bounce truncation).
- : Error term due to an -bounce truncation.
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 partial impulse responses are obtained from by the inverse Fourier transform.
Propagation graph models
The propagation graph methodology have been applied in various settings to create radio channel models. Such a model is referred to as a propagation graph model. Such models have been derived for scenarios including
- Unipolarized inroom channels. The initial propagation graph models [1][2][3] were derived for unipolarized inroom channels.
- In [5] a polarimetric propagation graph model is developed for the inroom propagation scenario.
- The propagation graph framework has been extended in [6] to time-variant scenarios (such as the vehicle-to-vehicle). For terrestrial communications, where relative velocity of objects are limited, the channel may be assumed quasi-static and the static model may be applied at each time step.
- In a number of works including [7][8][9][10] propagation graphs have been integrated into ray-tracing models to enable simulation of reverberation phenomena. Such models are referred to as hybrid models.
- Complex environments including outdoor-to-indoor cases.[11] can be studied by taking advantage of the special structure of propagation graphs for these scenarios. Computation methods for obtaining responses for very complex environments have been developed in [12]
- The graph model methodology has been used to make spatially consistent MIMO channel models.[13]
- Several propagation graph models have been published for high-speed train communications.[14][15]
Calibration of propagation graph models
To calibrate a propagation graph model, its parameters should be set to reasonable values. Different approaches can be taken. 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
Related radio channel model types
The method of propagation graph modeling is related to other methods. Noticeably,
- Multiple scattering theory
- Radiosity
- Ray tracing
- Geometry-based stochastic channel models (GBSCM)
References
- ^ a b Pedersen, Troels; Fleury, Bernard (2006). "A Realistic Radio Channel Model Based in Stochastic Propagation Graphs" (PDF). Proceedings 5th MATHMOD Vienna: 324–331.
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