Wireless Communications and Mobile Computing

Volume 2018, Article ID 4704218, 13 pages

https://doi.org/10.1155/2018/4704218

## Improved Model for Estimation of Spatial Averaging Path Length

^{1}Department of Telecommunications, Faculty of Electrical Engineering, University of Sarajevo, Sarajevo, Bosnia and Herzegovina^{2}Department of Electrical Engineering and Computing, University of Dubrovnik, Dubrovnik, Croatia

Correspondence should be addressed to Pamela Njemčević; ab.asnu.fte@civecmejn.alemap

Received 16 March 2018; Revised 23 May 2018; Accepted 31 May 2018; Published 12 July 2018

Academic Editor: Enrico M. Vitucci

Copyright © 2018 Pamela Njemčević et al. 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

In mobile communication systems, the transmitted RF signal is subject to mutually independent deterministic path loss and stochastic multipath and shadow fading. As at each spatial location mostly the composite signal samples are measured, their components are distinguished by averaging out the multipath-caused signal level variations, while preserving just the ones due to shadowing. The prerequisite for this is the appropriateness of the local area averaging path length that enables obtaining the local mean (composed of mean path loss and shadow fading) and the multipath fading as difference between the composite signal sample and the local mean. However, the so far reported analytical approaches to estimation of the averaging path length are based on considering either the multipath or just the shadow fading, with applicability limited to only specific topologies and frequencies. Therefore, in this paper, the most widely used Lee analytical method is generalized and improved by considering multipath and shadowing concurrently, so providing the general closed-form elementary-function based estimation of the optimal averaging path length as a function of common multipath and shadow fading parameters characterizing particular propagation environment. The model enables recommendations for the optimal averaging length for all propagation conditions facing the mobile receiver.

#### 1. Introduction

In a mobile communication system, the received RF signal, attenuated by deterministic path loss, varies in time and space stochastically [1]. Temporal variations are caused by moving objects near the stationary transmitter and receiver, while overall spatial variations are due to (mutually independent) shadowing and multipath propagation effects [2]. The shadowing-caused variations correspond to slow fading of the signal local mean that is noticeable only over large-scale receiver location changes (of the order of tens of wavelengths), implying constant local mean within these so-called local areas [3]. On the contrary, multipath fading occurs on a small-scale change of distance between transmitter and receiver (of the order of less than a wavelength), when constructive and destructive interference of multiple propagation paths (each exhibiting various amplitude, delay, and phase) cause fast signal variations even within local areas, so that the composite spatially variable received signal is finally affected by both fading types. Accordingly, with the real-life RF measurements, only the composite signal samples can be obtained. However, if the collected samples are to be used for particular channel modelling, it is necessary to distinguish slow and fast signal variations and process them separately in order to estimate their statistical distribution and relevant metrics, extensively used in wireless digital communication systems. So, e.g., while the multipath fading statistics is important for the transceiver design, as well as for the performance estimation (e.g., bit error rate (BER), outage probability), the shadow fading statistical description determines the automatic gain control parameters, as well as enabling optimization of the macroscopic propagation models for improved coverage on the area of interest and for evaluation of cell coverage area within already installed wireless systems.

Accordingly, it is necessary to correctly average the composite signal within each local area, i.e., to choose the optimal local averaging path length value, along which the multipath-caused received signal level variations would be sufficiently integrated out, while preserving intact the ones caused exclusively by shadowing. This enables obtaining the shadow fading samples (proportional to the local mean) and so the multipath fading as a difference between the composite signal sample and the appropriate local mean at each spatial location.

With this regard, after some empirical methods with questionable accuracy, the first generally applicable analytical procedure for calculation of the appropriate averaging length [4, 5] evolved to the well-known Lee criteria, suggesting averaging the composite signal samples along the length between 20*λ* and 40*λ*. Although such recommendations have served for many years as a reference for spatial local averaging in outdoor environment with the Ultrahigh Frequency (UHF) band, these recommendations are not compliant with empirically obtained values for indoor environment at 900 MHz and 2 GHz (10*λ*) [6], or for urban cells with the lower Very High Frequency (VHF) band (6*λ* – 8*λ*) [7].

The lack of generality of Lee results is partially overcome in [1, 8–10], but the proposed models still cannot provide meaningful recommendations for urban and indoor environments with UHF band, as well as for mobile systems operating outside that band [11]. Namely, the models from [8–10] are focused on multipath averaging but neglect that the variations caused by shadowing should not be averaged along the (optimal) length, while the model in [1] is focused only on preserving the shadow fading constancy over the averaging length. Although these models can be used for estimation of the averaging length for limited combinations of multipath and shadow fading statistical parameters characterizing specific propagation conditions, it is shown [11] that still no recommendation can be derived for many other remaining parameter combinations. That is why the so far proposed analytical solutions lack generality with respect to propagation environment and frequency band [11].

Accordingly, in this paper, we extend the Lee method for estimating the appropriate averaging length by taking into account the variations caused by shadowing and multipath concurrently and developed the unified model which is applicable for all relevant propagation conditions facing the mobile receiver.

Since the herein proposed model is based on the Lee procedure, the latter is first introduced in detail in Section 2, and then the overview of other reported analytical models for estimating the averaging length is given. These models’ drawbacks motivated derivation of the analytical expression as a function of common multipath and shadow fading parameters, to enable accurate local mean estimation and differentiation between shadow and multipath fading, without propagation and system restrictions, as it is presented in Section 3. With this regard, the optimal averaging path length values, obtained by applying the corresponding channel parameters’ empirical values for particular propagation environments onto the analytical model, are presented in Section 4. Comparisons of recommendations for the averaging length values derived using the herein proposed model with widely used Lee's recommendations, as well as with the existing empirically based ones, are presented in Section 4.1, while conclusions are summarized in Section 5.

#### 2. Models for Estimation of Spatial Averaging Path Length

##### 2.1. Lee Model

Having passed the narrowband channel, the composite spatially variable received RF signal can be expressed as [4]where and denote its true mean value and fast variations of the signal at spatial location within a local area, respectively. Since only the composite signal values can be obtained by measurements, in order to perform accurate statistical modeling of slow and fast variations separately, it is a priori necessary to estimate the local means. Accordingly, in the Lee procedure, it is assumed that the estimated local mean can be obtained by integrating along a certain path length [4]and that it approaches when becomes close to the optimal value .

Since the variance of the estimated mean around can be expressed as a function of the autocorrelation function of the received signal and its true mean value as well as of the length* 2L* [5]: it is used in the Lee procedure as the local mean estimation error metric in terms of , based on the condition that 68% of values on (where is assumed to be Gaussian random variable with the mean [5] and variance ) should be within the interval of two standard deviations around the true mean [5]. Accordingly, the optimal averaging length is estimated by minimizing the local mean estimation error defined in logarithmic scale as [4, 5]: With Lee method, the proposed metric (5) was used for calculation of under assumptions of constant independent of and Rayleigh-distributed with its mean . Consequently, is extracted in front of the integral in (2), implying to be also Rayleighian (with its easy-to-calculate autocorrelation function and mean [5], which are substituted in (4)-(5) for calculation of ). This provided expressed as monotonically decreasing function of (see Figure 1), which is used to derive recommendations for the appropriate averaging length value () by applying the analytically groundless assumption that the overall multipath variations can be neglected for the local mean estimation error up to 1 dB (i.e., ).