Integral Equation Formulation for Scatter Density Problem

Integral equation formulation for the problem of finding a circularly symmetric scatter density (SD) around the mobile is deduced, and the SD is computed assuming that the distribution of angle-of-arrival (AoA) in base station is known. The corresponding integral equation is solved by using the spline collocation method. Thus, instead of fitting a priori selected SD to estimated AoA distribution, we solve the SD based on AoA characteristics. Introduction: Recently, geometrical-based single bounce channel models have been proposed by different authors [1],[2],[3], where the distribution of the scatterers defines the model through the selected simple geometry. In [1], it was assumed that scatterers are uniformly spread over a disc with radius R and vanishes outside of this region. Similar analysis was carried out in [2] where the scatterers are uniformly distributed inside the ellipse with foci at Base Station (BS) and Mobile Station (MS). In [3], it was shown that angle-of-arrival (AoA) in BS, computed assuming Gaussian scatter density (SD), provides a better fit to measurements of [4] when compared to model applying uniform distribution of scatterers. In [1],[2,[3], the form of SD is first selected, after that the distribution of AoA is computed, and finally the result is compared to existing measurements. However, in order to maximise the fit between the AoA provided by the model and measurements the direction in the selection process concerning to SD should be reversed. We propose a novel approach where AoA in BS is first selected based on measurements and SD is then computed from the integral equation that defines the relation between SD and AoA seen by BS. The resulting SD can be used when building up a channel simulator modelling a certain environment. System Model and Notations: The notation used in the sequel is introduced in Fig. 1, where r refers to the distance between MS and scatterer, d is the distance between scatterer and BS, D is the distance between MS and BS, and α and β define the departing (arriving) and arriving (departing) angles of the signal path (dashed lines) respectively. October 6, 2004 DRAFT