International Journal of Antennas and Propagation

Volume 2015 (2015), Article ID 936406, 14 pages

http://dx.doi.org/10.1155/2015/936406

## Estimation of the Reception Angle Distribution Based on the Power Delay Spectrum or Profile

Institute of Telecommunications, Faculty of Electronics, Military University of Technology, Gen. Sylwestra Kaliskiego Street No. 2, 00-908 Warsaw, Poland

Received 25 September 2015; Accepted 25 November 2015

Academic Editor: Ana Alejos

Copyright © 2015 Cezary Ziółkowski and Jan M. Kelner. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The paper presents an estimation of the reception angle distribution based on temporal characteristics such as the power delay spectrum (PDS) or power delay profile (PDP). Here, we focus on such wireless environment, where the propagation phenomenon predominates in azimuth plane. As a basis to determine probability density function (PDF) of the angle of arrival (AOA), a geometrical channel model (GCM) in form of the multielliptical model for delayed scattering components and the von Mises’ PDF for local scattering components are used. Therefore, this estimator is called the distribution based on multielliptical model (DBMM). The parameters of GCM are defined on the basis of the PDS or PDP and the relative position of the transmitter and the receiver. In contrast to the previously known statistical models, DBMM ensures the estimation PDF of AOA by using the temporal characteristics of the channel for differing propagation conditions. Based on the results of measurements taken from the literature, DBMM verification, assessment of accuracy, and comparison with other models are shown. The results of comparison show that DBMM is the only model that provides the smallest least-squares error for different environments.

#### 1. Introduction

In wireless communications, an angle distribution of the radio wave multipath components significantly affects the statistical properties of the received signals. Numerous measurements show that the environmental properties have a significant effect on the angle of arrival (AOA) of the receiving radio wave. Therefore, to perform statistical modeling of the impact of the propagation environment on deformations in time-spectrum of transmitted signals, knowledge of the probabilistic characteristics of angle is required. The search for new ways of transmitting information in wireless systems is primarily based on the results of simulation studies. Therefore, these characteristics are particularly important in these studies because their use provides the mapping of the effect of the object movement direction on the received signal.

One of the main ways of reception angle modeling is approximation of the measured data using standard probability density functions (PDFs). In this case, we talk about empirical models of angle distributions. In practice, Laplacian, Gaussian functions and Aulin’s, Parson’s models are the most commonly used distributions for azimuth [1] and elevation [2, 3] angle, respectively. Unfortunately, there is no relationship that explicitly associates the parameters of these distributions with different types of propagation environments and the distance between the transmitter (Tx) and the receiver (Rx). Therefore, the use of geometric channel models (GCMs) gives the possibility of taking into account the impact of changes in the position of the objects (Tx, Rx) on the spatial properties of the received signals. These models reproduce the geometry of the spatial relationships among Tx, Rx, and the location scattering areas. The shape of the scattering areas and spatial density of the scatterers are the criteria that differentiate the individual models. In literature, models of PDF of AOA are based primarily on such scatterer regions as circle [4–6], ellipse [5–7], elliptical disc [8], hollow-disc [9, 10], semispheroid [4, 11], clipping semispheroid [12], ellipsoid [13], and bounded ellipsoid [14] for two-dimensional (2D) and three-dimensional (3D) models, respectively. To the description of the spatial density of scatterers, such distributions are used as 2D uniform [4, 5, 9, 15, 16], 3D uniform [4, 12], 2D Gaussian [17, 18], 3D Gaussian [12], hyperbolic [19], Rayleigh and exponential [20], parabolic [6, 14], “inverted parabolic” [21], and conical [22]. GCMs provide the basis for theoretical analysis of PDFs of AOA. Each of these models is described by one, two, or more parameters. However, it is difficult to identify the physical premises that bind the values of these parameters with the propagation properties of environment. Therefore, there is no clear relationship between the PDF model parameters and the propagation conditions. For this reason, the verification of these PDFs in the literature is reduced to a measurement data approximation only. In this paper, the method of determining the structure of GCM is significantly different from existing solutions. In our proposition, the parameters of the temporal characteristics of channel are the basis for GCM. As a result, statistical model of AOA is PDF estimator closely related to the type of propagation environment.

In most land mobile access systems, the half power beamwidth (HPBW) of antenna pattern is (omnidirectional antenna), or at least several tens of degrees in the azimuth plane. In elevation plane, HPBW has a few or at most a dozen degrees only. In this case, the phenomena modeling in the azimuth determines the mapping quality of the real signal. Therefore, these modeling conditions are decisive in simulation studies of wireless mobile channel. Unfortunately, none of the existing models does ensure the accuracy of the analytical description of the actual statistical properties of the AOA for different environmental conditions. Confirmed with the correctness of these observations are the results of a comparative analysis of the PDF of AOA for azimuth models that are presented in [23]. This paper presents an evaluation of the accuracy of the mapping format of azimuth AOA by various 2D GCMs for different propagation scenarios. The results for selected measuring scenarios are reference data, whereas the accuracy measure of the models is the least-square error (LSE). The summary of [23] is that “…no one geometric model is best by all criteria and for all environments….” As an exception to this conclusion, we present an analytical model, estimator of the azimuth AOA distribution for multipath homogeneous environments. This PDF estimator is based on temporal characteristics of the channel such as the power delay spectrum (PDS) or power delay profile (PDP). For a given type of environment, the parameters of these characteristics for different positions of Tx/Rx are the estimation parameters.

The developed PDF estimator is based on the Parson-Bajwa multielliptical GCM [19]. The geometry of this model results from the PDS or PDP. Literature analysis of the measurement results justifies such a methodology for determining PDF estimator. In [1], the correlation coefficient between delay and azimuth angle parameters is computed on the basis of the measurement data. Obtained linear regression line shows that these parameters are highly correlated [1, Figure ]. Therefore, the developed model ensures a clear reproduction of the angle statistical properties with respect to temporal characteristic that is associated with the type of environment. In simulation studies, this fact makes it possible to adopt the model to the specific environment that is described by temporal characteristics of radio channel.

The purpose of this study is to show that the developed PDF model of azimuth AOA for different environments provides the minimum value of the estimation error in comparison to the models previously presented in the literature. Our work concerns the uniform environments, which means that the comparative analysis is focused on the propagation scenarios with unimodal PDF of azimuth angle.

The paper is organized as follows. Section 2 presents the analysis principles of reception angle distribution for the developed model. Section 3 describes in detail AOA analysis for delayed components as well as methodology of von Mises’ distribution application for local scattering components. Methodology of adaptation of model parameters is presented in Section 4. This adaptation is based on the channel characteristics and the parameters of measurement scenario. Based on the results of measurements taken from the literature, model verification, assessment of accuracy, and comparison with other PDFs of AOA models are shown in Section 5. Conclusion, which highlights the practical use of the model to predict the angle distribution of received signal at different environments, is presented in Section 6.

#### 2. Distribution of Reception Angle: Principles of Analysis

As shown by numerous measurement data, for example, [24–26], due to multipath propagation of environment, a few to several replicas of the transmitted signal are received with various powers and time delays. This means that the received signal consists of the components. These components make up the time-clusters with a specific time delay, ( is number of clusters), relative to the direct path.

The directivity, polarization, and gain from antenna patterns that are practical applications in wireless mobile communications are the reason for the dominance of the signal components, which are concentrated in the azimuth plane. It follows that the statistical properties of the received angle can be limited to the angle in the azimuth plane. These comments indicate that the angular distribution of power can be represented as power azimuth spectrum (PAS), , in the form where is PAS of the th time-cluster and is PAS for local scatterings, that is, for cluster with negligibly small delay relative to the direct path.

However, the angle distribution and intensity of the direct path signal change with the change of Tx/Rx position. In this case, PDF of AOA, , provides a basis for evaluation of the angular power dissipation and the Rice -factor describes the power distribution on direct path component and the components of local scatterings. The statistical description of PAS can be presented as [27]where is the average power of the received signal, is the power of all components delay by (the th cluster), is PDF of AOA of multipath components delay by , is the power of all components delay by zero, is PDF of AOA of multipath components from local scatterings (delay by zero), and is the delta distribution.

Considering that , we havewhere is the normalized power of the th time-cluster.

Equation (3) shows that the statistical properties of the signal reception angle are determined by the components arriving with a delay to Rx and components from local scatterings. For delayed components, an analysis of reception angle is based on multielliptical channel model, whereas, for local scattering components, von Mises’ PDF is used. Thus, the analytical form of PDF of AOA is called the distribution based on multielliptical model (DBMM). The basis for the DBMM parameters can be characteristics such as power delay profile (PDP) or power delay spectrum (PDS) that determine values and the size of each ellipse. An additional input parameter is the height of the receiving antenna relative to the height of surrounding objects. This parameter is the basis for the assessment of local scattering intensity and determines the value of von Mises’ PDF parameter. An analysis of the statistical properties of reception angle refers to homogeneous propagation environments. This means that such environments are dealt with for which the probability of a scatterer occurrence is the same for each direction when viewed from Tx. In practice, this means that the analysis covers those propagation scenarios that are characterized by unimodal PDF of the reception angle. In addition to the above considerations, to simplify the analyzed issues, the general assumptions that constitute the basis for most of GCMs are used [12, 16, 17]:(1)The radiation characteristics of the transmitting and receiving antennas are omnidirectional.(2)Each propagation path from Tx to Rx consists of scatters on exactly one scattering element.(3)For individual ellipses, each scatterer is a reradiating omnidirectional element with the same probability properties of scattering coefficient and uniform phase distribution.

#### 3. Statistical Model of Angle of Arrival

The purpose of this work is to determine the analytical form of PDF of AOA of radio waves for a homogeneous multipath environment. The presented analysis uses the geometric mapping and statistical description of propagation phenomena. Propagation of delayed components is mapped by multielliptical GCM, while PDF of AOA of local scattering components is described by von Mises’ PDF. In [27], a similar method of modeling the effects of propagation phenomena is used in analysis of the power azimuth spectrum (PAS).

The empirical studies demonstrate that, in time domain, the signal structure at the output of channels is shaped by a few or several clusters of the multipath components. These time-clusters are differentiated according to the values of power and time delay, after which the components arrive to Rx. This means that, in the azimuth plane, the location of scatterers defines an ellipse for components that make up a single time-cluster. This fact is the basis for the use of the Parson and Bajwa model in the statistical analysis of AOA. In this case, GCM creates a set of ellipses that represent time-clusters with various . The foci are common to all the ellipses. Their location determines the position of Tx and Rx, which are located at a distance . Signal components, for which the arrival delay is negligibly small and is expressed only in the phase difference relative to the direct component, form a particular time-cluster. These components are the result of local scattering that occurs in proximity to Tx/Rx antenna. The above model reproduces statistical location of the scatterers and thus provides the basis for determining PDF of AOA. The geometric structure of the considered problem and the adopted notations on the example of a chosen ellipse are shown in Figure 1. In addition, the local scattering area is highlighted in this figure.