Multipath Fading Channels

This module provides several implementations for statistical time-variant channel models. A single multipath propagation path is modeled as a Rician fading distribution by a sum-of-sinusoids approach

\[h_{\ell}(t) = \sqrt{\frac{K_{\ell}}{1 + K_{\ell}}} \mathrm{e}^{\mathrm{j} t \omega_{\ell} \cos(\theta_{\ell,0}) + \mathrm{j} \phi_{\ell,0} } + \sqrt{\frac{1}{N(1 + K_{\ell})}} \sum_{n=1}^{N} \mathrm{e}^{\mathrm{j} t \omega_{\ell} \cos\left( \frac{2\pi n + \theta_{\ell,n}}{N} \right) + \mathrm{j} \phi_{\ell,n}}\]

as proposed by Xiao et al.[1]. Each propagation path is composed of a specular component and a diffuse component, with rice factor \(K_{\ell}\) balancing the power distribution between the both components. The more sinusoids \(N\) are summed to model the fading, the more accurate the model is, however, Xiao et al.[1] indicate that \(N = 8\) is a good starting point for balancing accuracy with computational complexity. The overall path has a doppler shift \(\omega_{\ell}\) that is balanced by the angles \(\theta_{\ell,0}\) and \(\theta_{\ell,n}\) for line of sight and diffuse components, respectively, with a random phase \(\phi_{\ell,0}\) and \(\phi_{\ell,n}\). Note that for \(K_{\ell} = 0\), i.e. a non line of sight fading consisting of diffuse components only, this approximates a Rayleigh distribution. This model can is extended by \(L\) spatial delay taps, meaning there are multiple spatial propagation paths resulting in a delay spread of the transmitted signal at the receiver, so that the overall channel is the sum of all paths

\[\mathbf{H}(t,\tau) = \mathbf{A}^{(0)} \sum_{\ell=1}^{L} g_{\ell} h_{\ell}(t) \delta(\tau - \tau_{\ell}) \mathbf{A}^{(1)} \ \text{,}\]

with \(g_{\ell}\) being the gain factor of the \(\ell\)-th path, \(\tau_{\ell}\) the delay of the \(\ell\)-th path, and \(\mathbf{A}^{(0)}\) and \(\mathbf{A}^{(1)}\) being the antenna correlation matrices of the transmitter and receiver, respectively. For MIMO configurations with Devices featuring multiple transmit- or receive-antennas, custom antenna correlations \(A^{(0)}\) and \(A^{(1)}\) can be specified for both linked devices, respectively, as proposed by Yu et al.[2]. However, these antenna correlations do not model antenna array responses from spatial wave impingements. Instead, they are purely statistical approximations.

classDiagram class MultipathFadingChannel { +realize() +propagate() } class MultipathFadingRealization { +propagate() } class PathRealization { +propagate() } class MultipathFading5GTDL { +realize() +propagate() } class MultipathFadingCost259 { +realize() +propagate() } class MultipathFadingExponential { +realize() +propagate() } MultipathFadingChannel --o MultipathFadingRealization : realize() PathRealization --* MultipathFadingRealization MultipathFading5GTDL --|> MultipathFadingChannel MultipathFadingCost259 --|> MultipathFadingChannel MultipathFadingExponential --|> MultipathFadingChannel click MultipathFadingChannel href "channel.multipath_fading_channel.MultipathFadingChannel.html" click MultipathFadingRealization href "channel.multipath_fading_channel.MultipathFadingRealization.html" click PathRealization href "channel.multipath_fading_channel.PathRealization.html" click MultipathFading5GTDL href "channel.multipath_fading_templates.MultipathFading5GTDL.html" click MultipathFadingCost259 href "channel.multipath_fading_templates.MultipathFadingCost259.html" click MultipathFadingExponential href "channel.multipath_fading_templates.MultipathFadingExponential.html"

The base equations are implemented in the Multipath Fading Channel and its respective Multipath Fading Realization. Users may directly use the Multipath Fading Channel with their own parameter sets. More conveniently, several standard parameterizations such as 5G TDL Channel Model, Cost 259 Channel Model and Exponential Channel Model are provided.

[1] (1,2)

Chengshan Xiao, Yahong Rosa Zheng, and Norman C. Beaulieu. Novel sum-of-sinusoids simulation models for rayleigh and rician fading channels. IEEE Transactions on Wireless Communications, 5(12):3667–3679, 2006. doi:10.1109/TWC.2006.256990.

[2]

Kai Yu, M. Bengtsson, B. Ottersten, D. McNamara, P. Karlsson, and M. Beach. A wideband statistical model for nlos indoor mimo channels. In Vehicular Technology Conference. IEEE 55th Vehicular Technology Conference. VTC Spring 2002 (Cat. No.02CH37367), volume 1, 370–374 vol.1. 2002. doi:10.1109/VTC.2002.1002729.