Three-dimensional channel modelling using spherical statistics for smart antennas K. Mammasis, E. Pfann, R.W. Stewart and G. Freeland The distribution of angle of arrival of multipaths in a novel threedimensional approach is modelled. The approach followed takes into consideration a cluster of scatterers local to the mobile and models it using mixtures of Von Mises Fisher (VMF) distributions. Real MIMO experimental data was collected in a drive test campaign in Ilmenau city of Germany and was used to validate the proposed model. Results show a good match between the proposed model and the two-dimensional experimental histogram.
Introduction: In a typical mobile communication system multipath components arrive with different angles of arrival (AoA), amplitudes and delays. Some of the multipath components experience similar fading effects owing to the nature of the physical structures (scatterers) encountered on their way to the receiver. The channel clusters the multipath components according to the nature of those scatterers, forming many and different propagation paths. Some clusters of scatterers can impose greater fading effects on the transmitted signal depending on characteristics such as the density of a cluster of scatterers. It is evident that an accurate estimation of the distribution of scatterers in space and hence the AoA distribution of multipaths will result in more realistic propagation channel models. This spatial channel information is very important in numerous applications. Evaluating the performance of sophisticated adaptive signal processing algorithms, widely implemented in smart antenna beamforming, is one such application. Various researchers have considered the distribution of scatterers around the mobile [1– 3]. Clarke’s widely used scattering model is a two-dimensional (2D) model because of the assumption that the waves arrive only from the azimuthal direction. In particular, the model assumes isotropic scattering which means uniform distribution of the AoA of multipaths arriving at the MS [1]. However, it is now commonly accepted that the scattered electromagnetic waves do not strictly travel in azimuth but in elevation too. In [2] the distribution of scatterers is modelled with a Gaussian probability density function (PDF). In [3] the AoA distribution of multipath components is modelled using the Laplacian distribution. Both the Gaussian and Laplacian distributions lack the physical meaning of the underlying angles. Secondly, these two approaches can only be applied in 2D channel modelling. In this Letter we propose a new 3D model for the AoA distribution of multipaths. Since the AoA information is concerned with angular variables this distribution is more suitable for this type of application. The proposed model originates from the field of directional statistics and uses the VMF as its underlying PDF. In particular, the proposed model uses mixtures of VMF distributions (multimodality).
k is known as the concentration parameter k 0; and Ip/2 21 (k) is the modified Bessel function of the first kind and order p/2 21. As a result of the above definition the overall density of the mixture model comprising M VMF distributions can be described as f ðVjVÞ ¼ f Vj k1 ; m1 ; k2 ; m2 ; . . . ; kM ; mM M M P P ¼ ri fi Vj ki ; mi ¼ ri fi ðVjvi Þ ð5Þ i¼1
i¼1
with parameter vi ¼ (ki , mi). The parameter vector V ¼fr1 , . . . , rM; v1 , . . .,vMg, where ri defines the prior probability that vector V was generated by the ith component.
Table 1: Parameter estimates mean direction and concentration using EM Cluster ID x-co-ordinate y-co-ordinate z-co-ordinate k Priors 1 0.9845 0.2281 0.056907 277 0.0269 2 0.5965 20.7129 0.06712 183 0.0734 3 4 5 6 7
0.16251 0.2266 0.2515 0.5515 0.022
0.9411 0.9169 0.9845 20.8191 0.9714
0.1723 0.2869 0.07415 0.0419 0.1997
258 94 32 129 226
0.1889 0.1381 0.0235 0.1252 0.3312
8
20.8014
20.6811
0.2136
277 0.0928
Fig. 1 2D histogram of multipath data
3D non-isotropic scattering model using mixtures of VMF distributions: The VMF is a model for directions distributed unimodally with rotational symmetry, and serves more generally as an allpurpose probability model for directions in space [4, 5]. Any 3D unit vector V can be written as V ¼ ½sin u cos w sin u sin w cos uT
ð1Þ
V is determined by its spherical co-ordinates (u,f ) with 08 u 1808, 08 w , 3608. In the following, each vector represents the physical location of a scatterer in three dimensions. As a result, a multipath can arrive from any azimuth co-elevation pair of angles defined by that scatterer’s position within the propagation path. The unimodal VMF distribution used to model the direction of scatterers in a single cluster is given by fp ðV;m; kÞ ¼ Cp ðkÞ expðkmT VÞ
ð2Þ
where C p ðk Þ ¼
kp=21 p=2 2p Ip=21 ðkÞ
m is the mean direction unit vectorkmk ¼ 1parameter; T m ¼ sin u cos w sin u sin w cos u
Fig. 2 3D view of mixtures of VMFs with different k
ð3Þ
ð4Þ
MIMO experiment setup: MIMO measurement data from a drive test campaign in Ilmenau city was obtained from [6]. The MIMO system was equipped with a 16-element uniform circular array at the mobile station (MS) and a 12-element uniform linear array at the base station
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(BS). The measurement data contained the angle of arrival information in azimuth and elevation at the receiver as these were estimated by the RiMAX algorithm [7]. Algorithms for estimating parameters of mixture components: The EM algorithm was used to estimate the number of clusters, cluster the multipath and estimate the parameters of each component in the mixture of distributions. The optimum number of clusters was chosen to be eight according to the maximisation of the expected log-likelihood of the observed data. Therefore the EM was initialised with eight clusters. The EM algorithm for VMF mixtures was implemented as in [8]. The estimated parameters of each component are tabulated in Table 1. From our simulation we conclude that the VMF mixture model can provide accurate estimation of the clusters that exist in the channel and the directions arriving at the mobile by comparing the 2D histogram of the experimental data (Fig. 1) and the estimated VMF components (Fig. 2). Note that the co-latitude range is restricted from 608 to 1208 since the number of multipaths found outside this range was insignificant and hence considered to be outliers. Discussion: The next stage of work is to investigate more formal goodness of fit tests of the proposed VMF mixture model [9]. A simpler clustering algorithm (Spherical K-Means) is also currently under investigation [8]. This algorithm uses a hard assignment procedure as opposed to the soft assignment used by the EM algorithm. The main difference between the two algorithms is the method of assigning labels to multipaths. The EM assigns soft or probabilistic labels compared to the Spherical K-Means where a multipath can only belong to one cluster. It is worth mentioning that the Spherical K-Means can be viewed as a special version of the EM algorithm. Our preliminary findings using the Spherical K-Means algorithm are shown in Fig. 3. The concentration parameter k was kept constant for all clusters in the mixture. The goodness of fit test was performed using the co-latitude and azimuth plots [4, 5].
Conclusions: In this Letter we have demonstrated and explained theoretical aspects of the VMF distribution and how it can be applied in a new way to model the distribution of scatterers and thus the distribution of the AoA in a mobile propagation environment. In an attempt to show that the VMF model provides a good fit in such scenarios we obtained data from a MIMO drive test campaign carried out in Ilmenau city. The multipaths obtained were clusterised using the EM algorithm. By comparison of the 2D experimental histogram and the estimated mixture VMF components we conclude that the proposed model is suitable for this application. Hence synthetic data can be generated using mixtures of VMF distribution to describe the AoA PDF and hence the distribution of scatterers in space. # The Institution of Engineering and Technology 2008 14 September 2007 Electronics Letters online no: 20082462 doi: 10.1049/el:20082462 K. Mammasis, E. Pfann and R.W. Stewart (Electrical and Electronic Engineering, University of Strathclyde, Glasgow, United Kingdom) E-mail:
[email protected] G. Freeland (Steepest Ascent Ltd, Ladywell, Glasgow, United Kingdom) References 1 Lotter, M., and Van Rooyen, P.: ‘Modeling spatial aspects of cellular CDMA/SDMA Systems’, IEEE Commun. Lett., 1999, 3, (5), pp. 128– 131 2 3GPP-3GPP2 SCM-121: ‘Spatial channel model text description’, 2003 3 Clarke, R.H.: ‘A statistical theory of mobile-radio reception’, Bell Syst. Tech. J., 1968, 47, (6), pp. 957– 1000 4 Fisher, N.I., Lewis, T., and and Embleton, B.J.J.: ‘Statistical analysis of spherical data’ (Cambridge University Press, 1987) 5 Mardia, K.V., and and Jupp, P.E.: ‘Directional statistics’ (John Wiley & Sons, 2000) 6 http://www.channelsounder.de/, accessed February 2007 7 Richter, A., Landmann, M., and Thoma¨, R.: ‘RIMAX – A flexible algorithm for channel parameter estimation from channel sounding measurements’, (COST 273 TD(04)045) 8 Banerjee, A., Dhillon, I.S., Ghosh, J., and Sra, S.: ‘Clustering on the unit hypersphere using von Mises-Fisher distributions’, J. Mach. Learn. Res., 2005, 6, pp. 1345– 1382 9 Fisher, NI., and Best, DJ.: ‘Goodness-of-fit tests for Fisher’s distribution on the sphere’, Aust. J. Stat., 1984, 26, pp. 142– 150
Fig. 3 Spherical K-Means clusters
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