An adaptive q-Lognormal model towards the computation of average channel capacity in slow fading channels

The characterization of multipath fading and shadowing in wireless communication systems is essential towards the evaluation of various performance measures. It is well known that the statistical characterization of shadowing phenomena is captured by distributions viz., log-normal distribution, gamma distribution and other mixture distributions. However, it is observed that the log-normal distribution fails to characterize the outliers in the fading signal. The extreme fluctuations in the fading signal needs to be characterized efficiently for error free computation of the various performance metrics. In this context, this paper portrays an adaptive generalized Tsallis’ non-extensive q-Lognormal model towards the characterization of various fading channels. This model operates well with the synthesized fading signals and captures the wide range of tail fluctuations to adapt different fading scenarios. The significance and applicability of the proposed novel q-lognormal model in capturing the slow fading channels is validated using different statistical tests viz., chi-square test and symmetric JS measure. Furthermore, essential performance measures viz., the average channel capacity, closed form expression of cumulative distribution function (CDF) in terms of Gauss-Hypergeometric function 2F1a,b,c;z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_2{F_1}\left[ {\mathrm{{a, b, c; z}}} \right] $$\end{document}, moment generating function, higher order moments corresponding to q-Lognormal channel capacity and coefficient of variation is evaluated corresponding to the proposed q-lognormal model performing extensive Monte-Carlo simulation techniques up to O(107)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(10^7)$$\end{document}.


Introduction
In pragmatic wireless propagation environments, the channels gets impaired by shadowing phenomena.It arises in conditions when the coherence time of the transmission channel is large corresponding to the delay constraint of the channel [1,5].The shadowing effects will result in the signals bending i.e., the transmitted signals will not follow a straight propagation path.In this arrangement, the phase and amplitude variations exploited over the communication channel could be contemplated roughly constant throughout the period of its existence.The amplitude fluctuations as a result of shadowing is often modeled using the log-normal distribu-of the normalization constraint along with first and second moment constraints results in the q-Lognormal model [32] for the continuous range of parameter q, 1 < q < 3. The presented q-Lognormal model can characterize an entire scale of tail variations in the real valued range of non-extensive parameter q, and the model adapts to different fading scenarios.When q → 1, the proposed model mimics the conventional log-normal distribution.In Tsallis entropy framework, the value of the non-extensive parameter q between the proposed q-Lognormal distribution and synthesized fading data is estimated by optimizing the generic Jensen-Shannon (JS) divergence.In contrary to the commonly modeled lognormal distribution, as q > 1 the proposed model portrays the long-tailed characteristics that provides an excellent agreement over the outliers in the fading signals [32].Moreover, it is observed that the estimated non-extensive parameter q through symmetric JS measure leads to the accurate computation of performance metrics which results in significant improvement of the system performance gains viz., battery optimization [25], minimization of symbol error probabilty [27] and transmission delay.
The evaluation of channel capacity over q-lognormal fading is an important performance criterion for the estimation of the maximum transmission capacity that can be reached by a communication channel with an arbitrary low probability of error [21,30,31].In various fading situations, a wide range of theories concerning channel capability have been portrayed in literature.As defined in IEEE 802.15.3a wireless communication system , the lognormal probability density function (PDF) well characterizes small scale fading [33].In [34], the lower and upper range have been portrayed cor-responding to average channel capacity but in case of high decibel (dB) spread these appear to be in weak agreement.When the transmission time is sufficiently long, stretching over a large number of coherence periods and thus achieving long-term ergodicity in the fading process, the term ergodic channel capacity (Shannon capacity) is used [21].Moreover, this paper presents the theoretical model for the ergodic channel capacity over the proposed q-lognormal distribution as well as the computation of higher order statistics [28].However, it is interesting to observe that the theoretical derivation of CDF corresponding to the channel capacity model with respect to q-Lognormal model is obtained in terms of wellknown Gauss Hypergeometric function, 2 F 1 [a, b, c; z].
In this paper, the contribution of the presented q-Lognormal model in contrast to the well-known conventional log-normal model has been portrayed to characterize the shadowing environment in wireless communications.Moreover, to validate the proposed model and its adaptability in slow fading channels, different statistical tests viz., chi-square test and generalized symmetric JS measure have been tested.The remainder of the paper goes on as follows.In Sect.2, the probability density function (PDF) of the proposed q-Lognormal model is expressed in closed form.Section 3, deals with the evaluation of goodness of fit and the relative error of the proposed model in contrast to the conventional log-normal model.This section also presents a table of mathematical notations and functions used throughout the paper (Refer Table 1).In Sect.4, the generalized symmetric JS metric corresponding to theoretical and synthesized data is portrayed .The closed form solutions corresponding to the PDF and CDF of the average channel capacity in q-lognormal fading scenarios is derived in Sect. 5.The generalized q-Lognormal moment generating function (MGF) for respective derivation of moments have been computed in Sect.6.In Sect.7, K th order moments of log-normal model and the proposed q-Lognormal model for slow fading channels have been portrayed and validated using Monte-Carlo simulation schemes.In Sect.8, an important statistical measure i.e., Coefficient of Variation (CV) is evaluated corresponding to the proposed model.The final Sect. 9 includes the conclusion of the work.

q-Lognormal probability density function (PDF) and cumulative distribution function (CDF)
This section provides the formulation procedure of nonextensive q-Lognormal PDF.The desired PDF undergoes the difference of integral sign of the corresponding Lagrangian function, which is constructed following maximum entropy principle i.e., non-extensive q-Lognormal PDF is maximized Tsallis' entropy with normalized, first, and second moment constrains [10,11].
The expression for Tsallis' entropy incorporating the parameter 'q' [10,11] is given as The normalization constant, first and second moment of SNR are given as ( and where, ϕ es (γ ) = [ϕ q (γ )] q / [ϕ q (γ )] q dγ represents the escort probability distribution [13,16].The Lagrangian function is given as where Lagrangian multipliers are expressed as λ 1 , λ 2 and λ 3 .Employing Euler-Lagrange equation: The q-Lognormal PDF incorporating non-extensive Tsallis' parameter 'q' for long term fading channels with received SNR(γ ) is given from Eq. 6 together with Eq. 1 to Eq. 5 as [32] where α = q−1 3−q and the normalized constant ξ is defined as When q → 1, the above Eq.7 portrays the conventional log-normal model given as where σ and μ represents standard deviation and mean of ln(γ ).
In Fig. 1 it is perceived that when q > 1, the notion of long-tailed distribution is achieved which is important in characterizing the extreme fluctuations in the outliers.The CDF formulation for the proposed q-Lognormal distribution can be expressed in closed form as (for details see Appendix) where 2 F 1 is the Gauss Hypergeometric function and is defined as [6] 2 and Γ (.) denotes the gamma function.Figure 2 portrays the CDF plot of the proposed q-Lognormal distribution for q=1.2 and with varying range of the parameter σ .It is well-known that the lognormal distribution characterizes long term fading or shadowing effects [1] .However, the proposed model well characterizes the tail variations over the synthesized fading signals [32].From Fig. 3, it is noted that the log-normal model does not provide better agreement with the synthesized synthetic fading signals.To mimic realtime fading scenarios, the fading signals are produced with the help of MATLAB tool.It is observed that the proposed q-Lognormal distribution based on Tsallis' entropy widely captures the extreme fluctuations in the tail regions of the fading signals.In this setting, the presented model signified an excellent agreement to the outliers of the synthesized fading signals in the continuous range, 1 < q < 3. The presenetd q-Lognormal distribution well captured the synthetic fading signals with respect to q = 2.56 as shown in Fig. 4 [32].Moreover, the adaptability behavior of the proposed model characterizes a wide range of fading scenarios on varying the continuous range of non-extensive parameter q.For q → 1, the proposed model mimics the conventional log-normal model for capturing the slow fading channels.Similarly q-Gaussian distribution is obtained by the exponential transformation of q-Lognormal model [10] and characterizes the spectrum with non-Gaussian fading [14].Further, for q = 1.5, the model characterizes the Student-t fast fading channels [19] with degree of freedom=3 which is a wellknown model for characterizing the tail regions.The relation between Student-t distribution and q-Gaussian distribution has been established in [10].

Goodness of fit
The goodness of fit of q-Lognormal distribution in reference to synthesized fading signal is portrayed [10].The null hypothesis H 0 : the generated fading data is in agreement with the theoretical q-Lognormal distribution is validated by employing χ 2 -test.The value of degrees of freedom (d.f)= n − 1 − 1, where n=19 is the number of bins in the histogram [10] of the synthetic signal.Since, JS measure is used to estimate the parameter q, there is an unit decrement in the d.f.Furthermore, one d.f is also reduced by the normalization condition.The calculated χ 2 (obs)=1.39 is less than

Relative error
The presented q-Lognormal model in Eq. 7 is compared against the well known lognormal distribution in terms of relative error.The Fig. 5 illustrates that the proposed distribution has lower relative error in contrary to the conventional lognormal model towards defining the shadowing phenomena.

Generic symmetric JS estimation
The selection of an appropriate model is a challenge while characterizing various fading channels because significant prior information about the class of exact distributions is obscure [10,35].The proposed q-Lognormal distribution is equipped with the real valued Tsallis' parameter 'q', that can be evaluated from the observed histogram of the synthesized fading data.In this context, the generic Kullback-Leibler (KL) metric provides the relative distance amongst the proposed model and the referenced synthetic distribution viz., the synthesized fading data.As KL measure is asymmetric with respect to the synthetic fading data and the measure is not bounded, the calculated value of the non-extensive parameter 'q' may lead to erroneous characterization of the fading data.In this context, the estimated value of parameter 'q' is within the vicinity of the minimal value of symmetric JS measure.In Figs. 6 and 7 it is observed that the value of real-valued Tsallis' parameter 'q' is estimated and the measure is bounded i.e., the symmetric JS measure value is confined within the range 0 to 1.
The generic KL measure between the proposed model p ql (u, v) and referenced synthetic normalized histogram q(u, v) in Tsallis' context is expressed as Discretizing the above equation yields A generalized symmetric measure proposed by Jensen-Shannon is given as The different values of the parameter q corresponding to the synthesized fading data have been computed using the following designed algorithm.
The Fig. 6 highlights the estimated value of q in the vicinity of 2.5 and it is observed that the exact value of q for the generated synthetic signal is 2.56.The same synthetic data Fig. 6 Symmetric JS measure for q-Lognormal model and synthetic signal of O( 10 Algorithm 1: Algorithm for Tsallis' parameter estimation ('q log ') corresponding to proposed q-Lognormal model for synthesized fading data for q log = 1 to 3 do Evaluate : J S q log = τ1(q log )+τ2(qlog) 2 ; end Output: q log = min J S q log ; The bold values indicate the optimal estimation of q for minimal value of symmetric JS measure has been used in Fig. 4 along the with proposed q-Lognormal model and it is found to be in tight agreement with the generated data for q=2.56.For more clarity, Table 2 provides the tabulated value of JS divergence for various range of 'q' in the vicinity of 2.5.From Table 2, the optimal value for JS measure is obtained at 'q'=2.56 i.e., J S q Log =0.4451012.

Analytical model for channel capacity over q-Lognormal distribution
The channel capacity is stated as the probability of achieving a small error rate while maintaining the maximum transmission capacity of the channel [21].The instantaneous channel capacity is obtained as [15,22] Here γ represents the q-Lognormal random variable with PDF defined as where ξ denotes the normalization constant defined in Eq. 8, σ and μ represents the standard deviation and mean for the random variable log(γ ) which is expressed in decibels(dB) and ϑ = 10/ln (10).From Eqs. 15 and 16, the PDF of channel capacity is expressed in closed form as 123 Fig. 7 Cropping of Fig. 6 in the close neighborhood of q = 2.5  The mathematical expression representing the CDF of channel capacity over the proposed q-Lognormal model is obtained as Substituting ϑ ln (exp(c) − 1) = x and after some mathematical operations we obtain: Finally, the CDF of channel capacity corresponding to q-Lognormal model is obtained in closed form as where 2 F 1 is the Gauss Hypergeometric function and is defined in Eq. 11 [6].From Eqs. 17 and 21, PDF and CDF graphs are plotted correponding to q=1.2 and different range of σ d B for realistic dB spread as portrayed in Figs. 8 and 9.

Moment generating function (MGF) for channel capacity
In context of the assumption that the receiver has the channel state information, the instantaneous channel capacity is given as [23] C = ln(1 + γ ), (22) where γ represents the q-Lognormal random variable with PDF defined in Eq. 16.
The MGF expression w.r.t real z for the channel capacity over the proposed q-Lognormal environment is defined as Further simplifying, and performing certain mathematical computations, the expressions of MGF for channel capacity is obtained as (for details see Appendix) which can be further written as

K th order q-Lognormal moments corresponding to channel capacity in shadowing environment
It is worth noting that the computation of higher order moments is essential for analyzing important statistical behavior of channel capacity.In this context, this section derives the analytical expressions of a few higher moments of the instantaneous channel capacity.Laourine et.al [29] have computed the accurate expressions in terms of convergent infinite series portraying the higher order moments of channel capacity over lognormal fading scenarios.In context of maintaining completeness, we provide only the first and second order moments corresponding to channel capacity as derived in [29].The first order moment is mathematically expressed as From this Equation the analytical expression for first order moment is obtained due to Laourine et.al.[29] as The expression for second order moment is given as, From this Equation the analytical expression for second order moment is obtained due to Laourine et.al.[29] as: However, we need to compute the higher order moments of channel capacity corresponding to the proposed q-Lognormal model.Since the proposed model is not analytically tractable we perform rigorous Monte-Carlo simulation schemes averaging over O(10 7 ) simulation paths.Figures 10  and 11 represents the plot of the first and second order moments corresponding to the channel capacity i.e.E[C] and E[C 2 ] over the proposed model for q=1.2.The obtained results have been validated using the Symbolic Math Toolbox of MATLAB and is observed that the simulation results are in close agreement with the results obtained from symbolic integration.Moreover, for q → 1, the proposed model mimics the conventional log-normal distribution for characterizing the shadowing phenomena [3,32].In this context, we obtain the closed form expressions of the moments for channel capacity using Holtzman approximation [12].
The first moment corresponding to channel capacity over the proposed q-Lognormal model is expressed as The second moment corresponding to channel capacity over the proposed q-Lognormal model is expressed as When q → 1 Eq.30 reduces to the first order moment of channel capacity corresponding to log-normal distribution and is expressed as The closed form expression of the integral in Eq. 32 is obtained by employing the method defined by Holtzman [12,26,32].The PDF incorporating the standard normal distribution, with random variable B ∼ N(μ, σ 2 ) can be portrayed as three points at μ, μ + √ 3σ, μ − √ 3σ .The normal variate Φ(β) is obtained as Let ζ be a function with random variable B having variance σ 2 and mean μ.Performing Taylor's series expansion it is obtained By performing mathematical expectation, Holtzman [12] resulted in the approximation given as Employing Stirling's formula for evaluating expansion in differences and taking expectation, where h denotes the difference parameter and is appropriated as h = σ √ 3, hence Eq. 36 evaluates to Now, k th order moment of the channel capacity is defined as Substituting ln(γ ) = β, so, γ = exp(β), the Eq.38 is transformed as From Eqs. 37 and 39, an accurate analytical expression for k th moment corresponding to the channel capacity is obtained as where In Fig. 12, the plot of analytical expression for first moment of channel capacity corresponding to log-normal model Eq.41 has been presented.Moreover, the first order moment portraying q-Lognormal model for q → 1 i.e., mimicking the log-normal distribution has also been plotted by performing rigorous Monte-Carlo simulation schemes.It can be observed that both the results are in good agreement with each other which proves the validity and accuracy of the simulation schemes employed.

Computation of coefficient of variation (CV)
The statistical measure, coefficient of variation (CV), is an important metric for the channel capacity over the proposed q-Lognormal model.It represents the dispersion of the signal points over the wireless fading signals around the mean.It is expressed as the relative change in the square root of variance with respect to its mean, and is an important measure for estimating the fluctuation rate in channel capacity of the fading signals [9].It is obtained as, The plot of CV against average SNR over q-Lognormal channel capacity is presnted in Fig. 13, and it signifies that the coefficient of variation portrays higher values corresponding to lower arguments of received average SNR.The fluctuation rate in the channel capacity drops with an increase in the average SNR.It is noted that the simulation results and the symbolic integration results are in closed agreement with each other.The symbolic integration has been performed using Symbolic Math Toolbox in MATLAB.

Conclusion
The characterization of extreme fluctuations over shadowing phenomena due to random obstacles in wireless propagation channels is essential for the evaluation of various performance metrics.The proposed q-Lognormal model well characterized these fluctuation phenomena in various fading environments than the conventionally used shadowing models.The proposed model operates well with the synthesized fading data in the continuous range of real valued Tsallis' parameter q, 1 < q < 3. Furthermore, the adaptability of the presented q-Lognormal model in characterizing the shadowing phenomena have been validated using different statistical tests viz., chi-square test and symmetric JS measure.Furthermore, the theoretical model for average channel capacity in slow fading environments is obtained corresponding to the proposed q-lognormal model.It is profound that the q-Lognormal model adapts to varying fading scenarios in the real valued domain of Tsallis' parameter 'q' and also mimics various well-known shadowing models, which explicitly reinforced the computation of higher dimensional performance metrics viz., MGF, CDF with respect to channel capacity in terms of the well-known Gauss Hypergeometric function ( 2 F 1 (.)), evaluation of generalized k th moments, the relative fluctuations i.e., CV over the decibel spread.These measures have been computed and are found to be in closed agreement with the simulation results.

Fig. 1 7 Fig. 2
Fig. 1 Illustration of q-Lognormal model of SNR (γ ) with respect to different values of parameter q

Fig. 3 Fig. 4
Fig. 3 Illustration of log-normal model corresponding to synthesized signal

Fig. 5
Fig. 5 Comparison of Relative error of q-Lognormal model in contrast to conventional Lognormal distribution

Fig. 8
Fig. 8 PDF of q-Lognormal channel capacity over q=1.2, μ d B = 1 with different range of σ d B

Fig. 9
Fig. 9 CDF of q-Lognormal channel capacity over q=1.2, μ d B = 1 with different range of σ d B

Fig. 10
Fig. 10 Plot of first moment of Average channel capacity (E[C]) over q-Lognormal model (q=1.2) corresponding to different σ d B

Fig. 13
Fig.13 Plot of CV for q-Lognormal (q=1.2) channel capacity vs Average SNR corresponding to different range of σ d B

Table 1
List of used 2 F 1 [a, b, c; z]

Table 2
Tabulated value of JS divergence of the synthesized fading signal in the vicinity of q=2.5