Closed-Form Coverage Probability for Downlink Poisson Network with Double Shadowed Fading
Jingrui Chen Chaowei Yuan
cs.IT math.IT


./Figures/



Closed-Form Coverage Probability for Downlink Poisson Network With Double Shadowed Fading
    
Jingrui Chen and Chaowei Yuan
Jingrui Chen and Chaowei Yuan are with the School of Information and Communication Engineering, Beijing University of Posts and Telecommunications, Beijing, 100876, China, (e-mail: xiaojingyun604604@163.com; yuancw2000@bupt.edu.cn). 
Last updated: December 30, 2023.



    Received 25 September 2017 / Accepted 15 January 2018
================================================================================================================================================================================================================================================================================================================




Performances of cellular networks over κ-μ shadowed fading with long-term shadowing has been studied in the existing literature. However, the impact of κ-μ shadowed fading with instantaneous shadowing on performances of cellular networks is unknown. Therefore, this letter analyzes the downlink coverage probability of a Poisson network with double shadowed fading which is composed of a large-scale fading of lognormal distribution and κ-μ shadowed fading with integer fading parameters. The closest base station association rule without shadowing is considered. For analytical tractability, the double shadowed fading is approximated as a weighted sum of κ-μ shadowed distributions based on the Gaussian-Hermit quadrature. As a main theoretical result, a closed-form expression for the downlink coverage probability of a Poisson network under double shadowed fading for the desired signal and arbitrary fading for the interfering signals is successfully derived. Numerical simulations reveal that the double shadowed fading provides a pessimistic coverage on a Poisson network compared with the long-term shadowing which is incorporated into cell selection.



Stochastic geometry, coverage probability, double shadowed fading, Poisson network.





§ INTRODUCTION
 
Stochastic geometry as a useful tool has employed to the performance analysis of heterogeneous cellular networks (HetNets), where the irregular locations of users and base stations (BSs) are modeled by Point processes <cit.>. The single-tier cellular network is developed in <cit.> and then extended to a multi-tier HetNet <cit.>. To meet the needs of various fading scenarios in future communications, a versatile fading model, i.e. κ-μ shadowed fading is proposed in <cit.>. However, it is intractable to the performance analysis of HetNets over this fading model due to the existence of the confluent hypergeometric function (CHF). To overcome this problem, the moment matching method with Gamma distribution is utilized in <cit.> and the truncated series form of the CHF is used in <cit.>.

In order to accommodate the requirements of various fading scenarios in fifth-generation communications, a double shadowed fading model is first proposed in <cit.>, where the double shadowed fading is composed of a large-scale fading and κ-μ shadowed fading. Similar to <cit.>, the large-scale fading is incorporated into cell association, which can characterize the impact of long-term shadowing on the cell selection. As a result, the effect of κ-μ shadowed fading with long-term shadowing on performances of cellular networks is studied.

In this paper, we distinguish from the previous related works <cit.>-<cit.> by studying the impact of instantaneous shadowing on the coverage probability of a Poisson network. Specifically, we consider a double shadowed fading which is composed of a lognormal distribution and κ-μ shadowed fading with integer fading parameters <cit.>. The nearest BS selection without shadowing is also considered.

There are three contributions in this paper. Firstly, an accurate and tractable expression for the double shadowed fading is obtained according to the Gaussian-Hermit quadrature (GHQ) <cit.>. Secondly, a closed-form expression for the downlink coverage probability of a Poisson network is successfully derived, assuming that the desired signal experiences double shadowed fading and the interfering signals experience arbitrary fading. The resultant expression avoids complex integral evaluations or high order differential calculations. Meanwhile, it is generic due to the application of versatile model of double shadowed fading in the desired link and the assumption of an arbitrary fading distribution for all interfering links. Most importantly, the theoretical analysis of coverage probability with respect to the interfering fading is avoided under the independent assumptions of fading models. Thirdly, compared with the long-term shadowed environment, numerical simulations show that the double shadowed fading offers a smaller coverage to a Poisson network.





§ SYSTEM MODEL

Consider a downlink Poisson network where the locations of BSs are modeled by a homogeneous Poisson point process Φ _B with density λ. Assuming that users are uniformly distributed in the network. The received power of a user at origin (termed as a typical user) from a BS at x∈Φ _B is modeled as P( x) = PH_xx^ - α, where P is the transmit power, α is the path loss exponent, and H_x is the independent fading gain with unit power. Consider the nearest BS association policy, thus the location of serving BS is x* = max _x∈Φ _BPx^ - α. According to <cit.>, the probability density function (PDF) of serving distance is f_R( r ) = 2πλ rexp(  - πλr^2). Assuming that the channel power of the desired signal H_x* follows a double shadowed distribution, i.e. H_x* = H_χ·H_S, where the independent random variables H_χ follows a lognormal distribution with a standard deviation (SD) σ̃_S (σ̃_S = σ _Sln( 10)/
 .
 -10) and H_S follows a κ-μ shadowed distribution with integer fading parameters <cit.>. For notational simplicity, we denote the distributions H_χ and H_S as H_χ∼ln N( 0,σ̃_S) and H_S∼ S( κ ,μ ,m;h_S) respectively. According to <cit.>, the PDF and complementary cumulative distribution function (CCDF) of H_S are

    f_H_S( κ ,μ ,m;h_S) = ∑_i = 0^M C_ih_S^m_i - 1/Γ( m_i)Ω _i^m_iexp(  - h_S/Ω _i),


    F̅_H_S( κ ,μ ,m;h_S) = ∑_i = 0^M ∑_j = 0^m_i - 1C_i( h_S/
     .
     -Ω _i)^j/Γ( j + 1)exp( h_S/
     .
     -Ω _i),

where the coefficients M, m_i, Ω _i and C_i are depended on the fading parameters κ, μ and m. Let ω _1 = m/κμ  + m, ω _2 = κμ/κμ  + m and ω _3 = 1/μ( κ  + 1). According to <cit.>, if μ>m, then M=μ, m_i = μ  - m - i + 1, Ω _i = ω _3 and C_i = ( [ m + i - 2;     i - 1 ])(  - ω _1)^mω _2^ 1- m - i for i = 1,2, … ,μ  - m, and m_i = μ  - i + 1, Ω _i = ω _3/ω _1 and C_i = ( [          i - 2; i - μ  + m - 1 ])(  - ω _1)^i - μ  + m - 1ω _2^1 - i for i = μ  - m + 1, … ,μ, otherwise M = m - μ, m_i = m - i, Ω _i = ω _3/ω _1 and C_i = ( [ m - μ;     i ])ω _1^iω _2^m - μ  - i. Assuming that the channel powers of the interfering signals {H_x} are i.i.d., where x∈Φ _B\x*. In order to obtain a generic result, we consider H_x follows an arbitrary distribution. The orthogonal multiple access technology <cit.> is employed at each cell to eliminate intra-cell interference. Thus, by considering the interference-limited (noise is ignored) network, the signal-to-interference ratio (SIR) of a typical user can be written as SIR( r ) = PH_x*r^ - α/I, where the aggregate interference power I = ∑_x∈Φ _B\x*PH_xx^ - α.





§ DOUBLE SHADOWED FADING

In order to study the impact of κ-μ shadowed fading with instantaneous shadowing on performances of cellular networks, the exact PDF of the double shadowed distribution described in section II is given bellow.

Given two mutually independent random variables H_χ and H_S, where H_χ∼ln N( 0,σ̃_S) and H_S∼ S( κ ,μ ,m;h_S), the exact PDF of the double shadowed fading H_x* = H_χ·H_S is
 
    f_H_x*( h_x*) = ∑_i = 0^M ∫_0^∞h_S^m_i - 1/exp( h_S/Ω _i + ( lnh_x* - lnh_S)^2/2σ̃_S^2)dh_S/√(2π)σ̃_SΓ( m_i)C_i^ - 1Ω _i^m_ih_x*,

where the coefficients M, m_i, Ω _i and C_i are given by (<ref>).


Based on the density function of product of two random variables <cit.>, the PDF H_x* = H_χ·H_S is

    f_H_x*( h_x*) = ∫_0^∞f_H_S( h_S)f_H_χ( h_x*/h_S)1/| h_S|dh_S.

Substituting the distribution of H_χ∼ln N( 0,σ̃_S) and (<ref>) into (<ref>) yields (<ref>). This completes the proof.

It has been known that the κ-μ shadowed fading contains majority of the fading models proposed in the literature as special cases <cit.>, including Rayleigh, Rician, Nakagami-m, Nakagami-q, One-sided Gaussian, κ-μ, η-μ, etc. Therefore, Lemma <ref> encompasses various composite fading models and shows the instantaneous property of large scale fading of lognormal distribution. However, it is intractable to the performance analysis of cellular networks due to an integral existed in (<ref>). To conquer this problem, the GHQ rule <cit.> is employed and the derived expression is as follows.

The double shadowed distribution described in Lemma <ref> can be expressed as

    f_H_x*( h_x*) = ∑_l = 1^n_GHQa_lf_H_S( κ ,μ ,m;h_x*/
     .
     -b_l),

where a_l = w_l/∑_l_1 = 1^n_GHQw_l_1, b_l = exp( √(2)σ̃_St_l_1), w_l_1 and t_l_1 are the weights and abscissas of the l_1-th order Hermite polynomial respectively, and f_H_S( ·) is given by (<ref>).


The density function of product of two random variables <cit.> H_x* = H_χ·H_S can be rewritten as

    f_H_x*( h_x*) = ∫_0^∞f_H_χ( h_χ)f_H_S( h_x*/h_χ)1/| h_χ|dh_χ.

Substituting the distribution of H_χ∼ln N( 0,σ̃_S) and (<ref>) into (<ref>), we can obtain

    f_H_x*( h_x*) = ∑_i = 0^M ∫_0^∞exp(  - h_x*/Ω _ih_χ - ( lnh_χ)^2/2σ̃_S^2)/h_χ ^m_i + 1dh_χ/√(2π)σ̃_SΓ( m_i)C_i^ - 1Ω _i^m_ih_x*^1 - m_i.

Using the change of variable t = lnh_χ/√(2)σ̃_S in (<ref>), applying the GHQ rule and combing with the identity √(π)= ∑_l_1 = 1^n_GHQw_l_1 yields (<ref>). This completes the proof.

As illustrated in Fig. <ref>, the approximated expression provided in (<ref>) exactly matches the exact expression in Lemma <ref>. Most importantly, the f_H_x*( h_x*) in Lemma <ref> is expressed as a weighted sum of κ-μ shadowed PDFs f_H_S( ·), which simplifies the analytical expressions significantly in special cases. When the lognormal distribution is ignored, (<ref>) is reduced to (<ref>) as σ _S = 0dB. Another important expression is the CCDF of H_x*, which is directly utilized for the coverage analysis of a downlink Poisson network. Combing with (<ref>) and Lemma <ref>, we can obtain the CCDF of H_x* conveniently.

The CCDF of the double shadowed fading is

    F̅_H_x*( h_x*) = ∑_l = 1^n_GHQa_lF̅_H_S( κ ,μ ,m;h_x*/
     .
     -b_l),

where the coefficients a_l and b_l are given by Lemma <ref> and F̅_H_S( ·) is given by (<ref>).







§ COVERAGE PROBABILITY

The coverage probability in a downlink Poisson network is defined as the probability that the received power of a typical user is bigger than the SIR threshold θ <cit.>. Consider the nearest BS association policy, the closed-form expression for the downlink coverage probability in a Poisson network is obtained in the next Theorem.

When the desired signal experiences the double shadowed fading and the interfering signals experience arbitrary fading, the coverage probability of a typical user is

    p_c( θ) = ∑_i = 0^M ∑_j = 0^m_i - 1∑_l = 1^n_GHQ∑_( N_1, ⋯ ,N_j) ∈𝕋_ja_lC_iA_jΓ( B_j)/(  - 1)^jG_i^B_j,

where the coefficients M, m_i, Ω _i, C_i, a_l and b_l are given by Corollary <ref>, 𝕋_j = {. ( N_1, ⋯ ,N_j) ∈( ℕ∪{ 0 })^j|∑_q = 1^j N_qq  = j,∑_q = 1^j N_q  = q}, A_j = ∏_q = 1^j 1/Γ( N_q + 1)( 2(  - 1)^qE_h_q/αΓ( q + 1)( θ/Ω _ib_l)^2 /
 .
 -α)^N_q, B_j = 1 + ∑_q = 1^j N_q, G_i = 1 + ( θ/
 .
 -Ω _ib_l)^2 /
 .
 -αE_h_0, the values of the expectations with respect to the interfering fading distributions are E_h_0 = 𝔼_H_x[ ∫_( Ω _ib_l/
 .
 -θ)^2 /
 .
 -α^∞( 1 - exp(  - h_xt^ - α/
 .
 - 2))dt] and E_h_q = 𝔼_H_x[ h_x^2 /
 .
 -αγ( q - 2 /
 .
 -α,θh_x/Ω _ib_l)], and the functions Γ(·) and γ( a,x) are the Gamma and lower incomplete Gamma function <cit.> respectively.



Substituting the expression of SIR into the definition of the coverage probability, we have

    p_c( θ ,r) = ℙ( h_x* > θI_1r^α/P).

Since the H_x* follows the double shadowed fading, combined by (<ref>) in Corollary <ref>, we obtain

    p_c( θ ,r) = ∑_i = 0^M ∑_j = 0^m_i - 1∑_l = 1^n_GHQC_ia_l( θ Ir^α/PΩ _ib_l)^j/Γ( j + 1)exp( θ Ir^α/PΩ _ib_l).

Due to the fact that exp(  - α) = (  - α)^ - j. exp(  - α z)^( j )|_z = 1, where g^( j )( z ) is the j-th differential of z, (<ref>) can be written as

    p_c( θ ,r) = ∑_i = 0^M ∑_j = 0^m_i - 1∑_l = 1^n_GHQ. ∂ ^j/∂z^j𝔼[ exp(  - zsI/Ω _ib_l)]|_z = 1/a_l^ - 1C_i^ - 1(  - 1)^jΓ( j + 1),

where s = θr^α/
 .
 - P. Since H_x has to be identically distributed as well as for all interfering BSs, substituting I = ∑_x∈Φ _B\{x*}PH_xx^ - α into (<ref>) and using the result of the conditional generating function of PPP <cit.>, we get

    [ p_c( θ ,r) = ∑_i = 0^M ∑_j = 0^m_i - 1∑_l = 1^n_GHQC_ia_l(  - 1)^j/Γ( j + 1)∂ ^j/∂z^jexp(  - 2πλ.;                                               . ×. 𝔼_H_x[ ∫_r^∞( 1 - exp(  - zLh_x))vdv])|_z = 1, ]

where L = θr^α/Ω _ib_lv^α. According to the higher order derivatives <cit.>, we obtain

    [                  ∂ ^j/∂z^jexp(  - 2πλ𝔼_H_x[ ∫_r^∞( 1 - exp(  - zLh_x))vdv]);                       = j!exp(  - 2πλ𝔼_H_x[ ∫_r^∞( 1 - exp(  - zLh_x))vdv]); ×∑_( N_1, ⋯ ,N_j) ∈𝕋_j∏_q = 1^j 1/Γ( N_q + 1)( 1/Γ( q + 1).∂ ^q/∂z^q(  - 2.;                         . . ×πλ𝔼_H_x[ ∫_r^∞( 1 - exp(  - zLh_x))vdv]))^D_q, ]

where 𝕋_j is given in Theorem <ref>. Solving the derivative ∂ ^q/∂z^q(  - 2πλ𝔼_H_x[ ∫_r^∞( 1 - exp(  - zLh_x))vdv]), plugging in z=1 and L, and by using the change of variable t = ( θ/
 .
 -Ω _ib_l)^ - 1/
 .
 -αr^ - 1v, we have
 
    [ ∂ ^q/∂z^q(  - 2πλ𝔼_H_x[ ∫_r^∞( 1 - exp(  - zLh_x))vdv]);       = 2πλr^2(  - 1)^qE_h_xq/α( θ/Ω _ib_l)^2 /
     .
     -α, ]

where E_h_q = 𝔼_H_x[ h_x^2 /
 .
 -αγ( q - 2 /
 .
 -α,θh_x/Ω _ib_l)] . In addition, substituting z=1 and L into 𝔼_H_x[ ∫_r^∞( 1 - exp(  - zLh_x))vdv] and by using the change of variable t = θr^αh_x/Ω _ib_lv^α, we obtain
 
    [ exp(  - 2πλ𝔼_H_x[ ∫_r^∞( 1 - exp(  - zLh_x))vdv]);     = exp(  - πλr^2( θ/Ω _ib_l)^2 /
     .
     -αE_h_0), ]

where E_h_0 = 𝔼_H_x[ ∫_( Ω _ib_l/
 .
 -θ)^2 /
 .
 -α^∞( 1 - exp(  - h_xt^ - α/
 .
 - 2))dt]. Substituting (<ref>), (<ref>) and (<ref>) into (<ref>), we have
 
    [ p_c( θ ,r) = ∑_i = 0^M ∑_j = 0^m_i - 1∑_l = 1^n_GHQC_ia_l(  - 1)^jW_i,j( r^2)/exp( πλr^2( θ/
     .
     -Ω _ib_l)^2 /
     .
     -αE_h_0) ,;                             W_i,j( t ) = ∑_( N_1, ⋯ ,N_j) ∈𝕋_j∏_q = 1^j ( 2πλ tE_h_q/(  - 1)^qα q!( θ/Ω _ib_l)^2/α)^N_q/N_q!. ]
Note that if j=0, then W_i,j( t ) = 1.
 Substituting the PDF f_R( r ) = 2πλ rexp(  - πλr^2) and (<ref>) into the coverage probability p_c( θ) = 𝔼[ p_c( θ ,r)] and by some algebraic manipulations, we get
 
    p_c( θ) = ∑_i = 0^M ∑_j = 0^m_i - 1∑_l = 1^n_GHQC_ia_l/(  - 1)^j∫_0^∞W_i,j( r /
     .
     -2πλ)/exp( rG_i)dr,

 where G_i = 1 + ( θ/
 .
 -Ω _ib_l)^2 /
 .
 -αE_h_0. Because of the fact that W_i,j( t ) = ∑_( N_1, ⋯ ,N_j) ∈𝕋_jA_jt^∑_q = 1^j N_q, where A_j is given in Theorem <ref>, the integral term of (<ref>) can be rewritten as
 
    [                       ∫_0^∞exp(  - rG_i)W_i,j( r /
     .
     -2πλ)dr; = ∑_( N_1, ⋯ ,N_j) ∈𝕋_jA_j∫_0^∞exp(  - rG_i)r^∑_q = 1^j N_qdr. ]

 Finally, by evaluating the integral in (<ref>) based on <cit.> and with some algebraic manipulations in (<ref>) yields (<ref>). This completes the proof.
 
 Obviously, (<ref>) avoids complicated integrals or high order differential calculations.
 
 As the popular fading distributions such as small-scale fading models, line-of-sight shadowing models and traditional composite fading/shadowing models are special cases of double shadowed fading <cit.>, the closed-form expression of the coverage probability obtained is generic. When the instantaneous shadowing of lognormal distribution is ignored, i.e. σ_S=0dB, Theorem <ref> is reduced to the κ-μ shadowed fading scenario experienced by the desired signal. Most importantly, the theoretical analysis of coverage probability concerning the interfering fading is avoided and it is replaced by simple calculations E_h_0 and E_h_q.

 





§ NUMERICAL RESULTS

In this section, we validate the accuracy of derived coverage probability through Monte Carlo and make a comparison between the double shadowed fading and long-term shadowed scenario. For notational simplicity, we use DS and LT to stand for the double shadowed fading and comparative term of long-term fading respectively, which are experienced by the desired signal. The interfering signals experience Rayleigh/Lognormal fading and Rayleigh fading separately. The simulation parameters of the Poisson network are λ  = 10^ - 7, α=4 and P=1. For numeral calculations, we set n_GHQ = 32. In Fig. <ref>, the coverage probability is plotted for the DS model when M=μ (i.e. μ  > m). It can be observed that the analytically obtained results closely match the simulation results. Intuitively, the coverage probabilities increase as parameters of m and μ increase and decrease as the values of σ_S increase. The coverage probability of the LT model is also considered in Fig. <ref>. Comparing the coverage probabilities of the DS model and LT model, we can see that the instantaneous property of large scale fading makes a great difference in coverage. The DS model provides a pessimistic coverage. This is due to the fact that choosing the serving BS depends solely on the distance. Hence, the instantaneous property of large scale fading is presented. When M=m-μ (i.e. μ≤ m), the coverage probabilities of the DS model and LT model are shown in Fig. <ref> and exact matches can be seen for various fading parameters. Similarly, it reveals the same conclusions as the Fig. <ref>.







§ CONCLUSION

This paper analyzes the downlink coverage probability for a Poisson network when the desired signal experiences double shadowed fading and the interfering signals experience arbitrary fading. The nearest BS association policy without shadowing is considered. The exact and approximate expressions for the double shadowed fading composed of a lognormal distribution and κ-μ shadowed fading with integer fading parameters are provided. Based on the resulting expression, the generic and closed-form expression for the downlink coverage probability is successfully derived and verified through simulation. The obtained expression of the coverage probability indicates that the theoretical analysis of coverage probability about the interfering fading is avoided. In addition, compared to the long-term shadowed environment, numerical simulations show that the instantaneous shadowing of large scale fading is bad for the coverage of a Poisson network.



IEEEtran
14
Elsawy2013cstH. Elsawy, E. Hossain, and M. Haenggi, “Stochastic geometry for modeling, analysis, and design of multi-tier and cognitive cellular wireless networks: A survey,” IEEE Commun. Surveys & Tuts., vol. 13, no. 3, pp. 996–1019, 3rd Quart. 2013.
andrews2016onlineJ. G. Andrews, A. K. Gupta, and H. S. Dhillon, “A primer on cellular network analysis using stochastic geometry,” arXiv:1604.03183, Oct. 2016.
andrews2011tcJ. G. Andrews, F. Baccelli, and R. Ganti, “A tractable approach to coverage and rate in cellular networks,” IEEE Trans. Commun., vol. 59, no. 11, pp. 3122–3134, Nov. 2011.
dhillon2012jsacH. S. Dhillon, R. K. Ganti, F. Baccelli, and J. G. Andrews, “Modeling and analysis of K-tier downlink heterogeneous cellular networks,” IEEE J. Sel. Areas Commun., vol. 30, no. 3, pp. 550–560, Mar. 2012.
paris2014tvtJ. F. Paris, “Statistical characterization of κ-μ shadowed fading,” IEEE Trans. Veh. Technol., vol. 63, no. 2, pp. 518–526, Feb. 2014.
kumar2015wclS. Kumar, “Approximate out probability and capacity for κ-μ shadowed fading," IEEE Wireless Commun. Lett., vol. 4, no. 3, pp. 10–13, Jun. 2015.
parthasarathy2017wclS. Parthasarathy and R. K. Ganti, "Coverage analysis in downlink Poisson cellular network with κ-μ shadowed fading," IEEE Wireless Commun. Lett., vol. 6, no. 1, pp. 10–13, Feb. 2017.
chun2017twcY. J. Chun, S. L. Cotton, H. S. Dhillon, F. J. Lopez-Martinez, J. F. Paris, and S. K. Yoo, "A comprehensive analysis of 5G heterogeneous cellular systems operating over κ-μ shadowed fading channels," IEEE Trans. Wireless Commun., vol. 16, no. 11, pp. 6995–7010, Nov. 2017.
dhillon2014wclH. S. Dhillon and J. G. Andrews, "Downlink rate distribution in heterogeneous cellular networks under generalized cell selection," IEEE Wireless Commun. Lett., vol. 3, no. 1, pp. 42–45, Feb. 2014.
Saha2017twcC. Saha, M. Afshang, and H. S. Dhillon, “Enriched K-tier hetnet model to enable the analysis of user-centric small cell deployments,” IEEE Trans. Wireless Commun., vol. 16, no. 3, pp. 1953–1608, Mar. 2017.
martinez2017tvtF. J. Lopez-Martinez, J. F. Paris, and J. M. Romero-Jerez, “The κ-μ shadowed fading model with integer fading parameters,” IEEE Trans. Veh. Technol., vol. 66, no. 9, pp. 7653–7662, Sept. 2017.
abramowitz1972bookM. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, 9nd ed. New York, USA: Dover Publications, 1972.
papoulis1991bookA. Papoulis, Probability, Random Variables, and Stochastic Processes. New York, NY, USA: McGraw-Hill, 1991.
gradshteyn2007bookI. S. Gradshteyn and I. M. Ryzhik, Tables of integrals, series, products, 7nd ed. San Diego, CA, USA: Academic, 2007.

