# Improved Model for Estimation of Spatial Averaging Path Length

Authors: Pamela Njemčević; Adriana Lipovac; Vlatko Lipovac
Journal: Wireless Communications and Mobile Computing
 (2018)

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## 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.


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## Body

## 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 20λ and 40λ. 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 (10λ) [6], or for urban cells with the lower Very High Frequency (VHF) band (6λ – 8λ) [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 Section2, 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 signalr(x) can be expressed as [4](1)rx=μxr0x.where μ(x) and r0(x) denote its true mean value μx=r(x) and fast variations of the signal at spatial location x 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 μ^x can be obtained by integrating r(x) along a certain path length 2L [4](2)μ^x=12L∫x-Lx+Lrydy=12L∫x-Lx+Lr0yμydyand that it approaches μx when 2L becomes close to the optimal value 2La.Since the varianceσμ^2 of the estimated mean μ^x around μ(x)(3)σμ^2=14L2∫x-Lx+L∫x-Lx+Lry1ry2–ry1ry2dy1dy2can be expressed as a function of the autocorrelation function Rrx of the received signal and its true mean value r(x) as well as of the length2L [5]: (4)σμ^2=1L∫02L1-x2LRrx-rx2dxit is used in the Lee procedure as the local mean estimation error metric in terms of 2L, based on the condition that 68% of μ^x values on 2L (where σμ^2 is assumed to be Gaussian random variable with the mean μ^x=μ(x) [5] and variance σμ^2) should be within the interval of two standard deviations (μ(x)-σμ^,μ(x)+σμ^) around the true mean μ(x) [5]. Accordingly, the optimal averaging length 2La is estimated by minimizing the local mean estimation error defined in logarithmic scale as σμ^spread2L [4, 5]: (5)σμ^spread2L=20log⁡1+σμ^/rx1-σμ^/rx.With Lee method, the proposed metric (5) was used for calculation of 2La under assumptions of constant μ(x) independent of 2L and Rayleigh-distributed r0(x) with its mean μ(x). Consequently, μ(x) is extracted in front of the integral in (2), implying r(x) to be also Rayleighian (with its easy-to-calculate autocorrelation function Rrx and mean r(x) [5], which are substituted in (4)-(5) for calculation of σμ^spread2L). This provided σμ^spread(2L) expressed as monotonically decreasing function of 2L (see Figure 1), which is used to derive recommendations for the appropriate averaging length value (2La=40λ) 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., σμ^spread2La=1dB).Figure 1
σ μ ^ s p r e a d 2 L functions, obtained analytically by assuming Rayleigh multipath fading distribution and constant local mean for any2L [4].


### 2.2. Other Models

However, in reality, the Rayleigh distribution presumption for multipath fading is mostly worst case, i.e., describing the most severe fast signal variations, particularly if line-of-sight between the transmitter and the receiver does not exist, implying applicability of previously developed recommendation only to some very specific propagation conditions.Furthermore, considering channels with the multipath fading intensity less severe than in Rayleigh channels, by assuming the Rician [8, 9] or Nakagami-m multipath fading distribution [10] instead of the Rayleigh one, while still following the Lee approach (by assuming the local mean constancy on any 2L and maximal acceptable estimation error of 1 dB), this resulted with monotonically decreasing σμ^spread2L curves with recommendations for 2La dependent on the multipath fading severity (expressed byK orm parameter for Rician or Nakagamian distribution, respectively) of the observed propagation environment.However, this still neglected that the assumption of constantμ(x) [4, 5, 8–10] is not applicable to just any 2L [1], but only if it does not exceed certain upper bound (from now on (6)∫02Lx·J022πxλdx=2L2J022π·2Lλ+J122π·2Lλdenoted as2Lsh), after which, in reality, the local mean μ(x) becomes function of 2L and the local mean estimation error (i.e., σμ^spread2L) starts to increase again. Accordingly, before determination of 2La, it is first necessary to calculate 2Lsh for the target environment with its shadow fading severity reflected by the lognormal standard deviation (in logarithmic scale) σdB and decorrelation distanceD. This task was accomplished in the previous work [1], where 2Lsh was identified as the maximal averaging length for which the local mean correlation coefficient ρ(2L): (7)ρx=Rμx-μx2σμx2(with μx, σμ(x)2, and Rμ(x) representing the mean, the variance, and the autocorrelation function of the true mean μ(x), respectively) is larger than or equal to 0.8 (in linear domain), indicating that, for any 2L≤2Lsh, the local mean can be considered highly autocorrelated or approximately constant [1].Accordingly, the so far available analytical expressions applies for estimation of the averaging length lower bound2Lmp that enables good enough averaging out of multipath fading [4, 5, 8–10], as well as for estimation of the averaging length upper bound (2Lsh) [1] that enables μ(x) to remain constant on 2L. If, for the adopted shadow and multipath fading parameters describing particular propagation conditions, 2Lmp≤2Lsh (as it is the case form =2, σdB=7dB, andD= 200 m atf = 1 GHz, which characterize suburban outdoor environment with UHF band; see Section 3), the optimal averaging length 2La can be considered to be in between 2Lmp and 2Lsh (since 2Lmp≈30λ [10] and 2Lsh≈100λ [1]).However, if2Lmp>2Lsh is obtained (which is the case for the following combination of parameters:m= 1, σdB=7dB, andD= 40 m atf = 1 GHz, characterizing outdoor urban cells with UHF, where 2Lmp=45λ [10] and 2Lsh=20λ [1]), simultaneously fulfilling both criteria: σμ^spread2La≤1dB (obtained by assuming just multipath variations) and ρ(2L)≥0.8 (obtained by assuming just shadowing variations) is not possible and so is to proceed with estimation of 2La that would have been appropriate for these propagation environments [11]. Moreover, even in the case when 2Lmp≤2Lsh, still both bounds arise from the analytically groundless, i.e., subjective, criteria that σμ^spread(2Lmp)≈1dB [4]) and ρ2Lsh≈0.8 [1], with no analytical proof that multipath and shadow fading are really constant enough at these bounds.In addition, though so far no analytical hint exists for estimating2La regardless of propagation specifics and beyond subjectively chosen criteria, still some recent empirical 2La estimations were proved to be accurate, unambiguous, and objective by processing measured composite signal samples, i.e., by treating shadow and multipath fading on 2L not separately but concurrently [12]. This approach resulted with the U-shaped curves of the σμ^spread(2L) function (reflecting both multipath and shadowing residual variations along 2L) [7, 12] instead of the monotonically decreasing curves for residual multipath-only variations, thus identifying 2La as the curve minimum, rather than using the subjective 1dB Lee criterion.Still, this empirical approach requires acquisition of many composite signal samples, which is time-consuming and implies lower significance of empirical methods for2La estimation and motivates development of a general analytical algorithm. However, the idea of concurrent treatment of shadow and multipath fading is definitely what must be adopted from the referenced empirical methods, since thus obtained U-shaped spread curves exhibit 2La as a minimum for all possible combinations of m, σdB, and D, thus avoiding potentiality intuitive selection of 2La in the propagation conditions with 2Lmp>2Lsh (2La should be chosen so as to comply with 2Lmp≤2La≤2Lsh).


## 2.1. Lee Model

Having passed the narrowband channel, the composite spatially variable received RF signalr(x) can be expressed as [4](1)rx=μxr0x.where μ(x) and r0(x) denote its true mean value μx=r(x) and fast variations of the signal at spatial location x 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 μ^x can be obtained by integrating r(x) along a certain path length 2L [4](2)μ^x=12L∫x-Lx+Lrydy=12L∫x-Lx+Lr0yμydyand that it approaches μx when 2L becomes close to the optimal value 2La.Since the varianceσμ^2 of the estimated mean μ^x around μ(x)(3)σμ^2=14L2∫x-Lx+L∫x-Lx+Lry1ry2–ry1ry2dy1dy2can be expressed as a function of the autocorrelation function Rrx of the received signal and its true mean value r(x) as well as of the length2L [5]: (4)σμ^2=1L∫02L1-x2LRrx-rx2dxit is used in the Lee procedure as the local mean estimation error metric in terms of 2L, based on the condition that 68% of μ^x values on 2L (where σμ^2 is assumed to be Gaussian random variable with the mean μ^x=μ(x) [5] and variance σμ^2) should be within the interval of two standard deviations (μ(x)-σμ^,μ(x)+σμ^) around the true mean μ(x) [5]. Accordingly, the optimal averaging length 2La is estimated by minimizing the local mean estimation error defined in logarithmic scale as σμ^spread2L [4, 5]: (5)σμ^spread2L=20log⁡1+σμ^/rx1-σμ^/rx.With Lee method, the proposed metric (5) was used for calculation of 2La under assumptions of constant μ(x) independent of 2L and Rayleigh-distributed r0(x) with its mean μ(x). Consequently, μ(x) is extracted in front of the integral in (2), implying r(x) to be also Rayleighian (with its easy-to-calculate autocorrelation function Rrx and mean r(x) [5], which are substituted in (4)-(5) for calculation of σμ^spread2L). This provided σμ^spread(2L) expressed as monotonically decreasing function of 2L (see Figure 1), which is used to derive recommendations for the appropriate averaging length value (2La=40λ) 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., σμ^spread2La=1dB).Figure 1
σ μ ^ s p r e a d 2 L functions, obtained analytically by assuming Rayleigh multipath fading distribution and constant local mean for any2L [4].


## 2.2. Other Models

However, in reality, the Rayleigh distribution presumption for multipath fading is mostly worst case, i.e., describing the most severe fast signal variations, particularly if line-of-sight between the transmitter and the receiver does not exist, implying applicability of previously developed recommendation only to some very specific propagation conditions.Furthermore, considering channels with the multipath fading intensity less severe than in Rayleigh channels, by assuming the Rician [8, 9] or Nakagami-m multipath fading distribution [10] instead of the Rayleigh one, while still following the Lee approach (by assuming the local mean constancy on any 2L and maximal acceptable estimation error of 1 dB), this resulted with monotonically decreasing σμ^spread2L curves with recommendations for 2La dependent on the multipath fading severity (expressed byK orm parameter for Rician or Nakagamian distribution, respectively) of the observed propagation environment.However, this still neglected that the assumption of constantμ(x) [4, 5, 8–10] is not applicable to just any 2L [1], but only if it does not exceed certain upper bound (from now on (6)∫02Lx·J022πxλdx=2L2J022π·2Lλ+J122π·2Lλdenoted as2Lsh), after which, in reality, the local mean μ(x) becomes function of 2L and the local mean estimation error (i.e., σμ^spread2L) starts to increase again. Accordingly, before determination of 2La, it is first necessary to calculate 2Lsh for the target environment with its shadow fading severity reflected by the lognormal standard deviation (in logarithmic scale) σdB and decorrelation distanceD. This task was accomplished in the previous work [1], where 2Lsh was identified as the maximal averaging length for which the local mean correlation coefficient ρ(2L): (7)ρx=Rμx-μx2σμx2(with μx, σμ(x)2, and Rμ(x) representing the mean, the variance, and the autocorrelation function of the true mean μ(x), respectively) is larger than or equal to 0.8 (in linear domain), indicating that, for any 2L≤2Lsh, the local mean can be considered highly autocorrelated or approximately constant [1].Accordingly, the so far available analytical expressions applies for estimation of the averaging length lower bound2Lmp that enables good enough averaging out of multipath fading [4, 5, 8–10], as well as for estimation of the averaging length upper bound (2Lsh) [1] that enables μ(x) to remain constant on 2L. If, for the adopted shadow and multipath fading parameters describing particular propagation conditions, 2Lmp≤2Lsh (as it is the case form =2, σdB=7dB, andD= 200 m atf = 1 GHz, which characterize suburban outdoor environment with UHF band; see Section 3), the optimal averaging length 2La can be considered to be in between 2Lmp and 2Lsh (since 2Lmp≈30λ [10] and 2Lsh≈100λ [1]).However, if2Lmp>2Lsh is obtained (which is the case for the following combination of parameters:m= 1, σdB=7dB, andD= 40 m atf = 1 GHz, characterizing outdoor urban cells with UHF, where 2Lmp=45λ [10] and 2Lsh=20λ [1]), simultaneously fulfilling both criteria: σμ^spread2La≤1dB (obtained by assuming just multipath variations) and ρ(2L)≥0.8 (obtained by assuming just shadowing variations) is not possible and so is to proceed with estimation of 2La that would have been appropriate for these propagation environments [11]. Moreover, even in the case when 2Lmp≤2Lsh, still both bounds arise from the analytically groundless, i.e., subjective, criteria that σμ^spread(2Lmp)≈1dB [4]) and ρ2Lsh≈0.8 [1], with no analytical proof that multipath and shadow fading are really constant enough at these bounds.In addition, though so far no analytical hint exists for estimating2La regardless of propagation specifics and beyond subjectively chosen criteria, still some recent empirical 2La estimations were proved to be accurate, unambiguous, and objective by processing measured composite signal samples, i.e., by treating shadow and multipath fading on 2L not separately but concurrently [12]. This approach resulted with the U-shaped curves of the σμ^spread(2L) function (reflecting both multipath and shadowing residual variations along 2L) [7, 12] instead of the monotonically decreasing curves for residual multipath-only variations, thus identifying 2La as the curve minimum, rather than using the subjective 1dB Lee criterion.Still, this empirical approach requires acquisition of many composite signal samples, which is time-consuming and implies lower significance of empirical methods for2La estimation and motivates development of a general analytical algorithm. However, the idea of concurrent treatment of shadow and multipath fading is definitely what must be adopted from the referenced empirical methods, since thus obtained U-shaped spread curves exhibit 2La as a minimum for all possible combinations of m, σdB, and D, thus avoiding potentiality intuitive selection of 2La in the propagation conditions with 2Lmp>2Lsh (2La should be chosen so as to comply with 2Lmp≤2La≤2Lsh).


## 3. Extension of the Lee Method for Estimating Optimal Averaging Path Length

In contrast to the Lee analytical approach, based on (2)–(5), in which the local mean μ(x) was assumed constant regardless of 2L value, thus effectively enabling processing of the composite signal r(x) distribution as if only multipath fading was present and results with monotonically decreasing σμ^spread(2L) function, no such limiting assumption is made here. Instead, both the residual slow and fast variations of the entire composite signal samples on 2L are taken into account for estimating σμ^spread2L.Thus, in order to calculateσμ^spread(2L) by (5), we first assume the distribution of r(x), which we consider to be the most convenient for describing propagation in mobile systems and then derive the appropriate first- and second-order statistics, i.e., its mean value rx and autocorrelation function Rrx. Thereby, similarly as Lee [4, 5] and his successors [8, 9], we also focus our analysis on narrowband channels, as, e.g., in 2G, channel bandwidth is 200 kHz in the 1 GHz band, while in 4G, the OFDM subchannels are 15 kHz wide etc.So, following the above scenario, in this paper, we consider the Generalized K distribution (mixture of Nakagami-m and Gamma distributions describing multipath and shadow fading statistics, respectively) as most convenient for the target systems [13]: (8)frx=4ΓmΓkmΩk+m/2rxm+k-1Kk-m2mΩrxm≥12,k>0,Ω>0where Γ(·) denotes the Gamma function and Kk-m(·) the modified Bessel function of the second kind and order (k-m). The Nakagami parameter m inversely reflects severity of multipath fading [14], while scale parameter Ω is the mean of local average power envelope [13]. The shape parameter k inversely reflects Gamma shadow fading severity [13] and is related to the lognormal shadow fading standard deviation σdB as follows [13, 14]: (9)k=1e0.115σdB2-1where σdB is expressed in dB units, reflecting the common practice with regard to normal probability density function (pdf) coming out of measurement campaigns.Although any composite signal distribution could be adopted into (3)–(5), the Nakagami-Gamma distribution of rx is chosen here since the Nakagami-m distribution is flexible enough to describe multipath fading conditions of variable severity, in a mathematically convenient way [15] and consistently with empirical data [14, 16–18], while the Gamma distribution is much superior to other ones (including the usually used lognormal distribution) for shadow fading analytical modeling [15], still remaining consistent with empirical data regarding terrestrial and satellite channels in variety of frequency bands [19].Having adopted these assumptions,rx is calculated using the integral transformation form, as follows [20]:(10)rx=ΩmΓk+1/2Γm+1/2ΓmΓkwhile Rrx is derived based on the bivariate Generalized K pdf [21] and the calculated product of its moments of the order n1+n2 [21, eq.6] (since the autocorrelation function of the Generalized K-distributed process can be calculated as the moment of the product of joint Generalized K variables of the first order, i.e., when n1=n2=1), as(11)Rrx=rx2F1-12,-12;m;ρNx2F1-12,-12;k;1-ρGx2where the first two parameters -0.5 within both hypergeometric functions are obtained by substitution of n1=n2=1 values in n1+n2 product moment expression [21, eq. 6]. ρNx and ρG(x) denote the power correlation coefficient between two squared Nakagami envelope samples and the correlation coefficient between two samples of Gamma local mean power [21], respectively. Here 2F1(.) represents the Gaussian hypergeometric functions whose utmost right parameter is adopted as monotonically decreasing function of distance x when it reflects multipath fading, and as monotonically increasing function ofx when it reflects variations caused by shadowing, since local mean estimation error due to residual multipath variations along the chosen averaging length decreases with distance, while the error caused by inappropriately averaged shadow fading increases when averaging length exceeds the local area.Furthermore when the squared Nakagami-m envelope is expressed as a sum of sinusoidal, independent, and identically distributed squared Rayleigh envelopes [22], its correlation coefficient for narrowband channels can be calculated as [10](12)ρNx=J022πxλwhere J0(.) is the first-kind Bessel function of order zero and the simplest isotropic multipath fading scenario with uniformly distributed angles of arrivals (AoA) is assumed (although more general expression, based on nonisotropic fading scenario with nonuniform distribution of AoA, can be used instead) [10].On the other hand, the Gamma local mean correlation coefficientρG(x) is approximated by lognormal local mean correlation coefficient ρL(x) [1] (obtained following the Gudmundson shadowing correlation models [23]), due to excellent match between the lognormal and Gamma pdfs [13, 14]: (13)ρGx≈e0.115σdB2e-x/D-1e0.115σdB2-1where D denotes the shadowing decorrelation distance (expressed in meters) of the propagation environment, at which the signal autocorrelation equals 1/e of its maximum value [23].Accordingly, by substituting (9), (12), and (13) in (11), (9), (10), and (11) in (4) and finally (4) and (10) in (5), the relative spread σμ^spread(2L) of the estimated mean μ^x around the true mean μx of the Generalized-K-distributed composite signal for a particular propagation environment of interest, is finally expressed as a function of 2L:(14)σμ^spread2L;m,σdB,D=20log⁡1+1/L∫02L1-x/2LF1-1/2,-1/2;m;J022πx/λ2F1-1/2,-1/2;1/e0.115σdB2-1;1-e0.115σdB2e-x/D-1/e0.115σdB2-12-1dx1-1/L∫02L1-x/2LF1-1/2,-1/2;m;J022πx/λ2F1-1/2,-1/2;1/e0.115σdB2-1;1-e0.115σdB2e-x/D-1/e0.115σdB2-12-1dxIn Figure 2, typical σμ^spread2L plots according to (14) are presented for 4 ad hoc chosen exemplar combinations of m, σdB, and D values. The curves are obviously U-shaped, just as their empirically obtained counterparts [7, 12], which has already been mentioned in Section 2 to enable straightforward identification of the optimal averaging path length 2L=2La as the minimum.Figure 2
σ μ ^ s p r e a d 2 L functions for different sets of input parameters m, σdB, and D.Comparing the observations from Figure2 with the results obtained by using other analytical models, it can be seen that, by reducing the multipath fading severity m, 2La shifts left taking smaller values, while σμ^spread2L drops significantly, which is consistent with analytical 2Lmp estimations [8–10]. Moreover, by increasing σdB or decreasing D, σμ^spread2L increases, while 2La shifts left toward smaller values, which is consistent with analytical 2Lsh estimation [1].However, the main task of the proposed approach is to provide recommendations for the optimal averaging length2La enabling minimal values of σμ^spread(2L) function for specific environments described by parameters m, σdB, and D. With this regard, differentiating very complex expression (14) with respect to 2L is anything but simple and can only be accomplished through a number of successive approximations, based on prior assessment and comparison of particular m, σdB, and D values.Accordingly, for the environments withm≥1 (describing Rayleigh and less severe multipath fading channels) and 0<2L≪D, the elementary-function-based implicit expression for the optimal averaging path length 2L=2La is derived in the Appendix as (A.37) and accordingly rewritten here as follows:(15)0.115σdB2e0.115σdB23D/λ2Laλ2-1mπ2ln⁡2π2Laλ+0.67=0which is wavelength-independent (ifD and2L are expressed as multiples of λ) and obviously much more computationally efficient than finding the minimum from an integral equation such as (14).Thus, by means of (15), 2La can be evaluated for any combination of the relevant parameters m, σdB and D, which was not possible with other analytical models.


## 4. Recommendations for Optimal Averaging Path Length in Various Propagation Conditions

The derived expression (15) is to be used now to derive recommendations for optimal averaging path length in various propagation environments, represented in (15) by the corresponding values of multipath and shadowing parameters obtained empirically. Thereby, in order to avoid any subjectivity and partiality which might arise from our own measurements, the values of m,σdB,D are collected from the relevant literature. However, empirical determination of any particular values of parameters m,σdB,D, presumes prior separation of shadow and multipath fading, for what assuming a certain value of the averaging length is a prerequisite. Accordingly, in order to avoid adopting potentially inaccurate specific values of m,σdB,D in relevant papers, we determine boundaries of each parameter in various propagation environments and frequency bans, instead.So, considering the shadow and multipath fading facing the mobile receiver in urban cells operating in 0.9 – 2.4 GHz band (e.g., [23–26]), the multipath fading was found to be very severe – with quite pronounced envelope variations usually described by Rayleigh distribution (or Nakagami with m=1). The shadowing standard deviations from the mean path losses are mostly much bigger than in other environments and typically take values between 7 and 9 dB, while the decorrelation distance can be considered to be between 30 and 50 m.The multipath fading in suburban cells with UHF band is less severe than in urban ones, and can be modelled by Nakagami distribution withm between 2 and 4. Similarly, the suburban shadowing standard deviations are mostly smaller than in urban environments, and can be described with σdB between 5 and 7 dB, while the decorrelation distance due to topological characteristics, can be taken to be between 200 and 300 m (e.g., [23, 27, 28]).In rural cells, the shadow fading shows similar characteristics as in suburban cells, with slightly bigger decorrelation distanceD > 300 m, while the multipath fading usually exhibits smaller dips and slower changes, and can be described by Nakagami-m distribution, with m between 4 and 8 (e.g., [27, 29]).Finally, by analysing the shadow and multipath fading in indoor cells, it can be seen that the multipath has similar characteristics as in suburban cells (m between 2 and 4), while the shadowing standard deviation is the smallest among all of the environments, and the decorrelation distance due to topology is usually between 3 and 5 m (e.g., [30, 31]).The adopted boundaries of the multipath and shadow fading parameters, for various environments with UHF band, are listed in Table1.Table 1
Adopted boundaries of multipath and shadowing parameters for various environments with UHF band.
Cell type m σ d B (dB) D (m) Urban 1 7 – 9 30 – 50  Suburban 2 – 4 5 – 7 200 – 300  Rural 4 – 8 5 – 7 > 300  Indoor 2 – 4 < 5 3 – 5Moreover, as beside the UHF band, the VHF band and the lower Super High Frequency (SHF) band are also used for mobile communications (where similar1st and 2nd order signal statistics applies as in UHF), boundaries of multipath and shadow fading parameters have to be adopted also for frequencies slightly beyond the UHF band.From a number of papers such as [32], it has come out that the decorrelation distance is practically independent on frequency, while the shadowing standard deviation changes for about 1 to 2 dB between different frequency bands (e.g., VHF and UHF). Accordingly, in urban cells with mobile systems operating below the UHF band, the shadowing standard deviation should be adopted between 5 and 7 dB, and between 8 and 10 dB for the lower SHF band [33, 34], while the multipath shows similar behaviour regardless the operating frequency band (Table 2).Table 2
Adopted boundaries of multipath and shadowing parameters for various frequency bands in urban cells.
Frequency band (urban cells) m σ d B (dB) D (m) Below the UHF band 1 5 – 7 30 – 50  UHF 1 7 – 9 30 – 50  Lower SHF 1 8 – 10 30 – 50Accordingly, substitution of the adopted values for the multipath and shadow fading boundaries from Tables1 and 2 into (15) provides the recommendations for the optimal averaging length values listed in Table 3.Table 3
Recommendations for optimal averaging length in various propagation environments and frequency bands.
Frequency band Cell type 2L a UHF Urban 10λ – 20λ  UHF Suburban 20λ – 50λ  UHF Rural 20λ – 50λ  UHF Indoor 5λ – 10λ  Below the UHF band Urban <10λor ~5λ  Lower SHF Urban >50λor ~80λSince the boundaries rather than exact values of recommended2La values are given in Table 3, it is important to investigate the impact of the adopted 2La values on the local mean estimation error. For this purpose, the proposed values of shadow and multipath fading parameters are substituted into (14), which is graphed in Figure 3 for UHF band.Figure 3
σ μ ^ s p r e a d 2 L function for various input parameters representing typical topologies with UHF band.From Figure3 it can be seen that, in urban UHF cells, the local mean estimation error remains within 5% of its minimal value if 2La is chosen anywhere within the proposed boundaries from Table 3. However, if local averaging is performed according to the Lee recommendation of 40λ, σμ^spread2L becomes 30% worse than it could be obtained by adopting the optimal averaging length, thus indicating the importance of appropriate selection of the spatial averaging length in these environments.Similarly, with indoor cells, averaging along the path length within 5λ – 15λ results with about 5% deviation of the estimation errors with respect to their minimal values, while, for larger lengths, the errors become unacceptable (e.g., equal to 50% for 2L=40λ).However, in suburban and rural UHF cells, due to small estimation errors obtained within large2L range, local averaging can be performed along any path length between 20λ and 70λ with acceptable sample mean error.Analogously, in urban cells with mobile systems operating below the UHF band, local averaging should be performed along any path length between 1λ and 10λ, due to small relative difference between the so obtained and the minimal local mean estimation error for a given set of parameters. If local averaging is performed along 40λ, the local means include not only the multipath, but the averaged shadowing-caused variations, too, leading to huge local mean estimation error (with σμ^spread equal to 50% with respect to its minimal value).Moreover, with high frequency mobile systems, local averaging should be done along the lengths longer than 50λ, since smaller lengths would lead to huge local mean estimation error due to insufficiently smoothed multipath fading along that length.


### 4.1. Comparison with the Existing Recommendations

Finally, the recommendations for adoption of2La coming out of the herein proposed model are compared with the widely used Lee recommendations and the existing empirical ones.From Table4, it can be seen that the recommendations based on (14) and (15): 10λ – 50λ for UHF outdoor, considerably overlap with Lee’s and other widely used recommendations for 2La: 20λ – 40λ, for UHF outdoor in such environments. However, the former are more precisely segmented regarding the environment type, thus significantly improving the accuracy of local mean estimation, expressed by the widely used Lee metric σμ^spread2L. This explains why Lee recommendations, though based on incomplete analytical model, provide acceptable recommendations for UHF band in outdoor cells.Table 4
Comparison of derived recommendations with existing analytical and empirical results.
Frequency band /cell type Modeled 2L a Other analytical or empirical 2L a UHF/urban 10λ – 20λ Lee analytical recommendations [4, 5]: 20λ – 40λ UHF/suburban 20λ – 50λ UHF/rural 20λ – 50λ  UHF/indoor 5λ – 10λ Empirical recommendations [6]: 8λ – 10λ  Below the UHF band/urban λ – 10λ Empirical recommendations [7]: 6λ – 8λ  Lower SHF/urban 50λ – 100λ -Moreover, the model, articulated by (14)/(15) analytically derives the recommendations for UHF indoor, as well as for the mobile systems operating slightly beyond the UHF band. Recommendations for 2La, as well as the assumptions about shadowing and multipath parameters boundaries (given in Tables 1 and 2), can be verified by the corresponding empirically backed recommendations [6, 7] for indoor cells at 0.9 GHz and 2 GHz (8λ-10λ) [6] and for urban cells operating below the UHF band (6λ-8λ) [7] (see Table 4 for comparison).Additionally, the idea behind the model treating composite multipath and shadow fading concurrently could be further extended to any frequency band and wireless system, taking into consideration the appropriate1st and 2nd order multipath and shadow fading statistics (instead of expressions (7) – (13)).It is worth pointing out that improved2La estimation accuracy enhances the local mean estimation accuracy. Thereby, more accurate local mean, used within handoff and adaptive modulation/coding algorithms, improves mobile system capacity, as well as improving coverage.


## 4.1. Comparison with the Existing Recommendations

Finally, the recommendations for adoption of2La coming out of the herein proposed model are compared with the widely used Lee recommendations and the existing empirical ones.From Table4, it can be seen that the recommendations based on (14) and (15): 10λ – 50λ for UHF outdoor, considerably overlap with Lee’s and other widely used recommendations for 2La: 20λ – 40λ, for UHF outdoor in such environments. However, the former are more precisely segmented regarding the environment type, thus significantly improving the accuracy of local mean estimation, expressed by the widely used Lee metric σμ^spread2L. This explains why Lee recommendations, though based on incomplete analytical model, provide acceptable recommendations for UHF band in outdoor cells.Table 4
Comparison of derived recommendations with existing analytical and empirical results.
Frequency band /cell type Modeled 2L a Other analytical or empirical 2L a UHF/urban 10λ – 20λ Lee analytical recommendations [4, 5]: 20λ – 40λ UHF/suburban 20λ – 50λ UHF/rural 20λ – 50λ  UHF/indoor 5λ – 10λ Empirical recommendations [6]: 8λ – 10λ  Below the UHF band/urban λ – 10λ Empirical recommendations [7]: 6λ – 8λ  Lower SHF/urban 50λ – 100λ -Moreover, the model, articulated by (14)/(15) analytically derives the recommendations for UHF indoor, as well as for the mobile systems operating slightly beyond the UHF band. Recommendations for 2La, as well as the assumptions about shadowing and multipath parameters boundaries (given in Tables 1 and 2), can be verified by the corresponding empirically backed recommendations [6, 7] for indoor cells at 0.9 GHz and 2 GHz (8λ-10λ) [6] and for urban cells operating below the UHF band (6λ-8λ) [7] (see Table 4 for comparison).Additionally, the idea behind the model treating composite multipath and shadow fading concurrently could be further extended to any frequency band and wireless system, taking into consideration the appropriate1st and 2nd order multipath and shadow fading statistics (instead of expressions (7) – (13)).It is worth pointing out that improved2La estimation accuracy enhances the local mean estimation accuracy. Thereby, more accurate local mean, used within handoff and adaptive modulation/coding algorithms, improves mobile system capacity, as well as improving coverage.


## 5. Conclusion

Real-life RF measurements provide composite signal samples, but often distinct multipath and shadow fading statistics and related metrics are needed in practice. The former is important for transceiver design, as well as for estimation of the performance,BER, outage probability etc., while the latter determines the automatic gain control parameters, as well as enableing optimization of the macroscopic propagation models. Thereby, for distinguishing of variations caused by shadowing and multipath, it is necessary to average the composite samples along the appropriate averaging length. With this regard, the so far proposed analytical solutions lack generality with respect to propagation environment and frequency band, providing recommendations only for specific combinations of multipath and shadow fading parameters. So, there is a need for a more general averaging length estimation approach, based on realistic and integral assumptions on both shadow and multipath fading and thus being applicable for all relevant propagation conditions.By using the derived expression for the optimal averaging length, the recommendations for the appropriate values of averaging length in various propagation environments and frequency bands are derived: the optimal averaging path length values, obtained by applying the according parameters’ empirical values boundaries for particular propagation environments onto the analytical model, pointed out that the optimal averaging length range in UHF band for rural and suburban cells is between20λ and 50λ; for urban cells, it is between 10λ and 20λ, while, indoors, the values between 5λ and 10λ are shown to be most appropriate; in urban cells with mobile systems operating slightly above the UHF band, the local averaging should be done along the length of about 80λ, while below the UHF band, it is less than 10λ.It is also shown that, compared to other state-of-the-art analytically backed recommendations, the herein proposed ones provide much smaller local mean estimation error in various propagation conditions and so they potentially improve performance of handoff and adaptive modulation/coding algorithms within real-life mobile systems.Besides, the herein proposed recommendations, regarding averaging length values, are verified to be consistent with other analytically backed ones (but just in specific propagation environments where their related models apply), as well as with the corresponding empirically evaluated values in various frequency bands and propagation environments.



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*Source: 4704218-2018-07-12.xml*