Impact of Correlation between Interferers on Coverage Probability and rate in Cellular Systems
Suman Kumar Sheetal Kalyani
cs.IT math.IT





Impact of Correlation between Interferers on Coverage Probability and rate in Cellular Systems
    Suman Kumar  Sheetal Kalyani  

 Dept. of Electrical Engineering 

     IIT Madras,  Chennai 600036, India   
 
{ee10d040,skalyani}@ee.iitm.ac.in


    December 30, 2023
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When the user channel experiences Nakagami-m fading, the coverage probability expressions are theoretically compared for the following cases: (i). The N interferers are independent η-μ random variables (RVs). (ii). The  N interferers are correlated η-μ RVs. It is  analytically shown that  the coverage probability in the presence of correlated interferers is greater than or equal to the coverage probability in the presence of  independent interferers  when the  shape parameter of the channel between the user and its base station (BS)  is not greater than one. Further, rate is compared for the following cases: (i).  The user channel experiences η-μ RV and the N interferers are independent η-μ RVs. (ii). The  N interferers are correlated η-μ RVs. It is   analytically shown that the rate in the presence of correlated interferers is greater than  or equal to the rate in the presence of independent interferers.  Simulation results are provided  and these match with the obtained theoretical results. The utility of our results are also discussed. 


Majorization theory, Stochastic ordering, Gamma random variables, Correlation, Coverage probability, rate.



§ INTRODUCTION


Typically, in practical scenarios correlation exists among the interferers, as evidenced by experimental results reported in  <cit.>.    For example, in cellular networks when two base stations (BSs) from adjacent sectors act as interferers, the interferers are correlated and it is mandated that while performing system level simulation, this correlation be explicitly introduced in the system <cit.>. We refer the  reader to  <cit.>   for a structured synthesis of the existing literature on correlation among large scale fading.  Considering the impact of correlation in the large scale  shadowing component and the small scale  multipath component is also an essential step towards modeling the channel.  The decorrelation distance in multipath components is lower when compared to shadowing components since shadowing is related to terrain configuration and/or large obstacles between transmitter and receiver <cit.>. Having said this, there is a need to analyse the performance of cellular system in the presence of correlation among interferers. 


Coverage probability[It is a probability that a user can achieve a target Signal-to-Interference-plus-noise-Ratio (SINR) T, and outage probability is the complement of  coverage probability.] and rate are  important metric  for performance evaluation of cellular systems. Coverage probability in the presence of interferers has been derived in  <cit.> and references there in, and rate in the presence of interferers has been studied in <cit.>  and references therein for the case of η-μ fading. Moreover, the correlation among interferers is assumed in <cit.>. Fractional frequency reuse and soft frequency reuse have been compared in the presence of correlation among interferers <cit.>. The impact of correlation among interferers on symbol error rate performance has been analysed in <cit.>. However, to the best of our knowledge, no prior work in open literature has  analytically compared the coverage probability and rate for generalized fading when interferers are independent with the coverage probability and rate when interferers are correlated. In this paper, we compare coverage probability and rate using majorization theory and stochastic ordering theory, respectively. Majorization theory is an important theory for comparison of two vectors in terms of the dispersion of their components. It  has  been extensively used in various problems in information theory <cit.>. In particular, using the results of majorization theory, a new analysis on Tunstall algorithm is provided in <cit.> and the bounds of  the average length of the Huffman source code in the presence of limited knowledge of the source symbol probability distribution is provided in <cit.>. Our analysis of comparison of coverage probability shows further evidence of the relevance of majorization theory. Stochastic ordering theory is used extensively for comparison of  random variables (RVs). Recently, it has been used in comparing various metrics in wireless communication <cit.>. Our analysis of comparison of rate shows another evidence of the relevance of stochastic order theory in wireless communication.


In this work,  we compare the  coverage probability when user experience Nakagami-m fading and interferers experience η-μ fading. In other words, we compare the coverage probability when the interferers are independent   with the coverage probability when the interferers are positively correlated[If cov(X_i, X_j)≥ 0 then X_i and X_j  are positively correlated RVs, where cov(X_i, X_j) denotes the covariance between X_i and X_j <cit.>.] using majorization theory.   It is analytically shown that the coverage  probability in presence of correlated interferers is higher than the coverage probability when the interferers  are independent, when the user channel's shape parameter is lesser than or equal to one.   We also show that when the user channel's shape parameter is greater than one, one cannot say whether coverage 
probability is higher or lower for the correlated case when compared to the independent case, and in some cases coverage probability is higher while in other cases it is lower.

We then analytically compare the rate when the interferers are independent  with the rate when the interferers are correlated using  stochastic ordering theory.   It is shown that the rate in the presence of positively correlated interferers is higher than  the rate in the presence of independent interferers when both user channel and interferers experience η-μ fading. Our results show that correlation among interferers is beneficial for the desired user. We briefly discuss how the desired user can exploit this correlation among the interferers to improve its rate. Multi-user multiple input multiple output (MU-MIMO) system is also considered and it is shown that the impact of correlation is significant on MU-MIMO system. We have also carried out extensive simulations for both the independent interferers case and the correlated interferers case and some of these results are reported in the Simulation section.
In all the cases, the simulation results match with our theoretical results.  






§ SYSTEM MODEL

We consider a homogeneous macrocell network with hexagonal structure with radius R as shown in Fig. <ref>. The Signal-to-Interference-Ratio  (SIR)  of a user located at r meters from the BS is given by

    SIR=η(r)= Pgr^-α/∑_i∈ψPh_id_i^-α= gr^-α/∑_i∈ψh_id_i^-α=S/I

where ψ denotes the set of interfering BSs and  N=|ϕ| denotes the cardinality of the set ϕ. The transmit power of a BS is denoted by P.  A standard path loss model r^-α is considered, where α≥ 2 is the path loss exponent.  The distance between user to tagged BS (own BS) and the ith interfering BS is denoted by r and d_i, respectively. The user channel's power  and the channel power between i^th interfering BS and user are η-μ power RVs. The probability density function (pdf) f_g_η-μ(x) of the η-μ power RV g is given by <cit.>,

    f_g_η,μ(x)=2√(π)μ^μ+ 1/2h^μ x^μ- 1/2/Γ(μ)H^μ- 1/2e^-2μ h xI_μ- 1/2(2μ H x)

where μ is shape parameter. Parameters H and h  are given by

    H=η^-1-η/4,  and   h=2+η^-1+η/4.

where 0<η< ∞ is the power ratio of the in-phase and quadrature 	component of the fading signal in each multipath cluster. 
The parameters of pdf of h_i are η_i and μ_i corresponding to η-μ power RV. The gamma RV is a special case of η-μ  power RV with η=1 and μ=m/2 and the pdf f_g(x)  of the gamma RV g  is given by

    f_g(x)=m^m e^-mxx^m-1/Γ(m)

where m and 1/m are the shape parameter and scale  parameter, respectively, and Γ(.) denotes the gamma function. When one assumes that there is correlation among interferers then h_i and h_j are correlated ∀ i  and j.


The coverage probability expression when signal of interest (SoI) experience Nakagami-m fading and interferers experience independent η-μ fading with equal μ, i.e.,  μ_i=μ_c ∀ i is given by <cit.>

    C_p,η(r)=Γ(2Nμ_c+m)/Γ(2Nμ_c+1)1/Γ(m)∏_i=1^2N(1/Tr^αm λ_i +1)^μ_c×


    F_D^(2N)[1-m,μ_c,⋯,μ_c; 2Nμ_c+1;1/Tr^αλ_1m+1,⋯,1/Tr^αλ_2N m+1]


    where λ_2i-1=d_i^-α/μ_c(1+η_i^-1) and λ_2i=d_i^-α/μ_c(1+η_i).

Here F_D^(N)[a,b_1, ⋯, b_N;c;x_1,⋯, x_N] is the  Lauricella's function of the fourth kind <cit.>. 
The coverage probability expression when SoI experience Nakagami-m fading and interferers experience correlated η-μ fading with equal μ, i.e.,  μ_i=μ_c ∀ i is given by <cit.>

    Ĉ_p,η(r)=Γ(2Nμ_c+m)/Γ(2Nμ_c+1)1/Γ(m)∏_i=1^2N(1/Tr^αm λ̂_i +1)^μ_c×


    F_D^(2N)[1-m,μ_c,⋯,μ_c; 2Nμ_c+1;1/Tr^αλ̂_1m+1,⋯,1/Tr^αλ̂_2N m+1]

Here λ̂_is are the eigenvalues of  𝐀_η=𝐃_η𝐂_η. 𝐃_η is the diagonal matrix with entries λ_i given in (<ref>) and 𝐂_η is the s.p.d. 2N× 2N matrix as given by

    𝐂_η=[ [        1        0  √(ρ_13)      ...        0;        0        1        0      ... √(ρ_22N);        ⋯        ⋯        ⋯        ⋱        ⋯;        0 √(ρ_2N2)        ⋯        ⋯        1 ]],

here ρ_ij=0, when i+j=2n+1, where n is an integer. 

Now, we want to compare (<ref>) with   (<ref>). However both involve N- fold infinite series making any comparison fairly difficult.  In order to elegantly compare these two expressions we make use of Majorization theory.




§ PRELIMINARIES: MAJORIZATION AND  STOCHASTIC ORDERING THEORY

In this section, we recall the basic notions of majorization and  stochastic ordering theory, which are applicable to our context.  Note that the majorization theory is used to compare deterministic vectors, whereas stochastic order is applicable on RVs <cit.>. We refer the reader to  <cit.> and <cit.> as excellent references for the majorization theory and stochastic order theory, respectively.


 §.§ Majorization theory

definitionDefinition


Let A and B be m× n matrices with entries in C. The Hadamard product of A and B is defined by [A ∘ B]_ij=[A]_ij[B]_ij for all 1≤ i≤ m, 1≤ i≤ n.



Let vector a=[a_1,⋯ a_n] and vector b=[b_1,⋯ b_n] with  a_1≤, ⋯,≤  a_n and  b_1≤, ⋯,≤  b_n then vector b is majorized by vector a, denoted by a≻b, if and only if 

    ∑_i=1^kb_i≥∑_i=1^ka_i,  k=1,⋯, n-1,  and ∑_i=1^nb_i= ∑_i=1^na_i.


lemmaLemma 
theoremTheorem
We will briefly state few well known results from  majorization theory which we use in our analysis.


 Let A and B be positive semidefinite matrices of size n. Let λ_1,⋯, λ_n be the eigenvalues of A ∘ B and let λ̂_1, ⋯, λ̂_n be the eigenvalues of AB. Then

    ∏_i=k^nλ_i ≥∏_i=k^nλ̂_i, k=1,2,⋯,n



propositionProposition


If function ϕ is symmetric and convex, then ϕ is Schur-convex function. Consequently, x≻y implies ϕ(x)≥ϕ(y).


For the details of this proof please refer to <cit.>.



If ϕ_i is Schur-convex, i=1,⋯ k, and ϕ_i(x)≥ 0 for all i and x, then 

    ψ(x)=∏_i=1^kϕ_i(x)

is Schur-convex.


For the details of this proof please refer to <cit.>.



 §.§ Stochastic order theory

In this subsection, our focus will be on convex order. 

If X and Y are two RVs such that 

    E[ϕ(X)]≤ E[ϕ(Y)]

for all convex function ϕ: ℝ→ℝ,  provided the expectation exist, then X is said to be smaller than Y in the convex order, denoted by X ≤_cx Y.

Note that if (<ref>) holds then  Y is more variable than X <cit.>. We now briefly state the theorems in convex order theory  that is relevant to this work.


Let X_1, X_2,⋯, X_N be exchangeable RVs. Let a=(a_1,a_2,⋯,a_N) and b=(b_1,b_2,⋯,b_N) be two vectors of constants. If a≺b, then

    ∑_i=1^Na_iX_i≤_cx∑_i=1^Nb_iX_i.



The details of the proof is given in <cit.>.




If X≤_cx Y and f(.) is convex, then E[f(X)]≤ E[f(Y)].


The details of the proof is given in <cit.>.






§ COMPARISON OF COVERAGE PROBABILITY
 
In this section, we first  compare the coverage probability in the independent case and correlated case, and analytically quantify the impact of correlation when both  SoI and interferers experience Nakagami-m fading. Then we analyse the impact of correlation on coverage probability for the scenario where SoI experiences Nakagami-m fading and interferers experience η-μ fading. The coverage probability expression when SoI experiences Nakagami-m fading and interferers  experience independent Nakagami-m fading  is given by <cit.>

    C_p(r)=Γ(∑_i=1^Nm_i+m)/Γ(∑_i=1^Nm_i+1)1/Γ(m)∏_i=1^N(1/Tr^αmλ_i+1)^m_i×


    F_D^(N)[1-m,m_1,⋯,m_N; ∑_i=1^Nm_i+1;1/Tr^α mλ_1+1,⋯,1/Tr^αmλ_N+1]  where λ_i=d_i^-α/m_i

Here m_i is the shape parameter of the i^th interferer. Note that the coverage probability expression for the correlated case is derived in <cit.> when the interferers shape parameter are all equal, i.e., m_i=m_c ∀ i.   The coverage probability expression when  SoI experiences Nakagami-m fading and interferers  experience correlated Nakagami-m fading is given by <cit.>

    Ĉ_p(r)=Γ(Nm_c+m)/Γ(Nm_c+1)1/Γ(m)∏_i=1^N(1/Tr^αmλ̂_i +1)^m_c×


    F_D^(N)[1-m,m_c,⋯,m_c; Nm_c+1;1/Tr^αmλ̂_1+1,⋯,1/Tr^αmλ̂_N+1]

here λ̂_is are the eigenvalues of the matrix 𝐀=𝐃𝐂, where 𝐃 is the diagonal matrix with entries λ_i and 𝐂 is the symmetric positive definite (s.p.d.) N× N matrix defined by

    𝐂=[ [       1 √(ρ_12)     ... √(ρ_1N); √(ρ_21)       1     ... √(ρ_2N);       ⋯       ⋯       ⋱       ⋯; √(ρ_N1)       ⋯       ⋯       1 ]],

where ρ_ij denotes the correlation coefficient between h_i and h_j, and is given by,

    ρ_ij=ρ_ji=cov(h_i,h_j)/√(var(h_i)var(h_j)), 0≤ρ_ij≤ 1, i,j=1,2, ⋯, N.

cov(h_i,h_j)   denotes the covariance between  h_i and h_j. 𝐂 is the s.p.d. N× N matrix given in (<ref>).
 Since 𝐀=𝐃𝐂, it is given by

    𝐀=[ [        λ_1 λ_1√(ρ_12)          ⋯ λ_1√(ρ_1N); λ_2√(ρ_21)        λ_2          ⋯ λ_2√(ρ_2N);          ⋮          ⋮          ⋱          ⋮; λ_N√(ρ_N1)          ⋯          ⋯        λ_N ]].


Note that the coverage probability expression for the correlated case is derived when the interferers shape parameter are all equal and hence for a fair comparison we consider equal shape parameter for the independent case also, i.e., m_i=m_c ∀ i. Therefore, the coverage probability when interferers are independent and m_i=m_c ∀ i is given by

    C_p(r)=Γ(Nm_c+m)/Γ(Nm_c+1)1/Γ(m)∏_i=1^N(1/Tr^αm λ_i +1)^m_c×


    F_D^(N)[1-m,m_c,⋯,m_c; Nm_c+1;1/Tr^αλ_1m+1,⋯,1/Tr^αλ_N m+1]

Our goal is to compare the coverage probability in the presence of independent interferers, i.e., C_p(r) given in (<ref>) and the coverage probability in the presence of correlated interferers, i.e., Ĉ_p(r) given in (<ref>).
We first start with the special case when user channel's fading is Rayleigh fading (i.e, m=1) and interferers experience Nakagami-m fading. When m=1, the coverage probability given in (<ref>) reduces to

    C_p(r)= ∏_i=1^N(1/Tr^αλ_i +1)^m_c F_D^(N)[0,m_c,⋯,m_c; Nm_c+1;1/Tr^αλ_1+1,⋯,1/Tr^αλ_N+1]

A series expression for F_D^(N)( .) involving N-fold infinite sums is given by

    F_D^(N)[a,b_1,⋯, b_N;c;x_1,⋯, x_N]=∑_i_1⋯ i_N=0^∞(a)_i_1+⋯+i_N(b_1)_i_1⋯(b_N)_i_N/(c)_i_1+⋯+i_Nx_1^i_1/i_1!⋯x_N^i_N/i_N!,


    max{|x_1|,⋯|x_N|}<1,

where, (a)_n denotes the Pochhammer symbol which is defined as (a)_n=Γ(a+n)/Γ(a). With the help of series expression  and  using the fact   that (0)_0=1 and (0)_k=0 ∀ k≥ 1,  the coverage probability given in (<ref>) can be reduced to 

    C_p(r)= ∏_i=1^N(1/Tr^αλ_i +1)^m_c
 
Similarly, the coverage probability in correlated case when SoI experiences Rayleigh fading Ĉ_p(r) is given by

    Ĉ_p(r)= ∏_i=1^N(1/Tr^αλ̂_i +1)^m_c

We now state and prove the following theorem for the case where the SoI experiences Rayleigh fading  and interferers experience  Nakagami-m fading and then generalize it to the  case where user also experiences Nakagami-m fading. 
 


The coverage probability in correlated case is higher than that of the independent case, when user's  channel undergoes Rayleigh fading, i.e.,

    ∏_i=1^N(1/1+kλ̂_i)^m_c≥∏_i=1^N(1/1+kλ_i)^m_c

where λ̂_is  are the eigenvalues of  matrix 𝐀 and λ_is are the diagonal elements of  matrix 𝐀 given in (<ref>), and k=Tr^α is a non negative constant.


 Firstly, we show that  λ̂≻λ where λ̂=[λ̂_1, ⋯, λ̂_n] and λ=[λ_1, ⋯, λ_n]. Note that the Hadamard product of 𝐃 and 𝐂, i.e., 𝐃∘𝐂  is defined as[Definition of Hadamard product is given in Definition <ref>] 

    𝐃∘𝐂=[ [ λ_1   0   ⋯   0;   0 λ_2   ⋯   0;   ⋮   ⋮   ⋱   ⋮;   0   ⋯   ⋯ λ_N ]].

Observe that 𝐃∘𝐂 is a diagonal matrix, the eigenvalues of the 𝐃∘𝐂 are λ_is. Since the eigenvalues of  matrix 𝐀=𝐃𝐂 are λ̂_̂îs and 𝐃 and 𝐂 are the positive semi-definite matrices,  hence  from Theorem <ref> (given in Section II),

    ∏_i=k^Nλ_i ≥∏_i=k^Nλ̂_i, k=1,2,⋯,N

Note that λ̂_̂îs are the eigenvalues and λ_is are the diagonal elements of a symmetric matrix 𝐀=𝐃𝐂, Hence

    ∑_i=1^Nλ_i=∑_i=1^Nλ̂_̂î

 If conditions given in (<ref>) and (<ref>) are satisfied then λ̂≻λ <cit.> . Now if it can be shown that ∏_i=1^N(1/1+kλ_i)^m_c ( and ∏_i=1^N(1/1+kλ̂_i)^m_c) is a Schur-convex function then by a simple application of Proposition <ref> (given in Section II) it is evident that ∏_i=1^N(1/1+kλ̂_i)^m_c≥∏_i=1^N(1/1+kλ_i)^m_c. To prove that ∏_i=1^N(1/1+kx_i)^m_c is a Schur convex function we need to show that it is a symmetric and convex function <cit.>. 

It is apparent that the function ∏_i=1^N(1/1+kx_i)^m_c is a symmetric function due to the fact that any two of its arguments can be interchanged without changing the value of the function.
So we now  need to show  that the function  f(x_1,⋯,x_n)=∏_i=1^N(1/1+kx_i)^a is a convex function where x_i≥ 0, a > 0. Using the Theorem <ref>, it is apparent that if the function f(x)=(1/1+kx)^a is convex function then f(x_1,⋯,x_n)=∏_i=1^N(1/1+kx_i)^a would be convex function. Note that f(x)=(1/1+kx)^a is convex function when x ≥ 0 and a>0 due to the fact that double differentiation of f(x)=(1/1+kx)^a is always non negative, i.e., f”(x)=a (a+1) k^2 (1/k x+1)^a+2≥ 0. Thus, f(x_1,⋯,x_n)=∏_i=1^N(1/1+kx_i)^a is a convex function. 


Since ∏_i=1^N(1/1+kx_i)^m_c is a convex function and  a symmetric function therefore, it is a Schur-convex function. We have shown that  λ̂≻λ  and ∏_i=1^N(1/1+kx_i)^m_c is a Schur-convex function. Therefore, from Proposition 1,  
 ∏_i=1^N(1/1+kλ̂_i)^m_c≥∏_i=1^N(1/1+kλ_i)^m_c.

Thus, the coverage probability in the presence of correlation among the interferers  is greater than or equal  to the coverage probability in  the independent case,  when user channel undergoes Rayleigh fading and the interferers shape parameter m_i=m_c ∀ i. The same result was shown for the case of both Rayleigh user channel and Rayleigh interferers in <cit.> using Vieta formula. By exploiting the mathematical tool of Majorization we are able to provide a much simpler proof.
Next, we compare the coverage probability for general case, i.e., when   m is arbitrary.


The coverage probability in the presence of the correlated interferers is greater than   or equal to the coverage probability in presence of independent interferers, when user channel's shape parameter is less than or equal to 1, i.e., m ≤ 1. When  m>1, coverage probability in the presence of independent is not always lesser than the coverage probability in the presence of correlated interferers. 

Please see Appendix.

Summarizing, the coverage probability in the presence of correlated interferers is greater than or equal to the coverage probability in presence of independent interferers, when user channel's shape parameter is less than or equal to 1, i.e., m≤ 1. When m>1, one can not say whether coverage probability is better in correlated interferer case or independent interferer case. Note that when m≤ 1, usually the interferers m_i is also smaller than 1. However, the given proof   holds for both m_i>1 and m_i<1.




 §.§ η-μ fading 

In this subsection, we analyse the impact of correlation on coverage probability for the scenario where SoI experiences Nakagami-m fading and interferers experience η-μ fading.  Recently, the  η-μ fading distribution with two shape parameters η and μ has been proposed to model a general non-line-of-sight propagation scenario <cit.>.  It includes Nakagami-q (Hoyt), one sided Gaussian, Rayleigh and Nakagami-m as special cases. The coverage probability expression when SoI experience Nakagami-m fading and interferers experience independent η-μ fading with equal μ, i.e.,  μ_i=μ_c ∀ i is given in (<ref>). 
The coverage probability expression when SoI experience Nakagami-m fading and interferers experience correlated η-μ fading with equal μ, i.e.,  μ_i=μ_c ∀ i is given in (<ref>).  Note that the functional form of the coverage probability expressions given in (<ref>) and (<ref>)  are similar to the case when both SoI and interferers experience Nakagami-m fading.  Hence, the same analysis holds and the coverage probability in the presence of correlated interferers is higher than the coverage probability in the presence of independent interferers when  the user's  channel shape parameter is less than or equal to 1, and interfering channel experience η-μ fading. However, one can not conclude anything when the user's  channel shape parameter is higher than 1.


§ COMPARISON OF RATE

In this section, we analyse the impact of correlation on the rate for the scenario when both SoI and interferers experience η-μ fading with arbitrary parameters.
In other words, we compare the rate when interferers are independent with the rate when the interferers are correlated using  stochastic ordering theory. 


The rate of a user at a distance r when interferers are independent is R=E[ln(1+S/I)], where S is the desired user channel power. For correlated case, the average rate of a user at a distance r is R̂=E[ln(1+S/Î)].  Here I and Î are the sum of  independent and  correlated interferers, respectively.  Using iterated expectation one can rewrite the rate as    

    R=E_S[E_I[ln(1+S/I)|S=s]],  and R̂=E_S[E_Î[ln(1+S/Î)|S=s]].

Since the  expectation operator preserves inequalities, therefore if we can show that 

    E_Î[ln(1+S/Î) |S=s]≥ E_I[ln(1+S/I)|S=s],

then this implies R̂≥ R.

The sum of interference power in the independent case can be written as  

    I=∑_i=1^Nh_id_i^-α

where h_i is η-μ power RV.  It has been shown in <cit.> that the η-μ power RV can be represented as the sum of two gamma RVs with suitable parameters. In other words, if h_i is η-μ power RV then, 

    h_i=x_i+y_i  where  x_i∼𝒢(μ_c,1/2μ_c(1+η_i^-1))  and  y_i∼𝒢(μ_c,1/2μ_c(1+η_i))

Now, the sum of interference power when interference experience η-μ RV can be written as

    I=∑_i=1^Nh_id_i^-α=∑_i=1^2Nλ_iG_i  with  G_i∼𝒢(μ_c,1), λ_2i-1=d_i^-α/μ_c(1+η_i^-1) and λ_2i=d_i^-α/μ_c(1+η_i).

Similarly, for correlated case,

    Î=∑_i=1^Nĥ_i=∑_i=1^2Nλ̂_iG_i

  Recall that these ĥ_i  are correlated, and  λ̂_is are the eigenvalues of the matrix A_η=D_ηC_η. In other words one can obtain a correlated sum of gamma variates by multiplying independent and identical distributed (i.i.d.) gamma variates with weight λ̂_is. Now, we use Theorem <ref> (given in Section III) to show that R̂ is always greater than equal to R. Note that a sequence of RVs X_1,⋯ X_N is said to be exchangeable if for all N and π∈ S(N) it holds that X_1⋯ X_N 𝒟= X_π(1)⋯ X_π(N) where S(N) is the group of permutations of {1,⋯ N} and 𝒟= denotes equality in distribution <cit.>. Furthermore, if X_is are identically distributed,  they are exchangeable <cit.>. Hence G_is are exchangeable since they are identically distributed. It has already been shown that  λ̂≻λ in Section IV.  Hence by a direct application of  Theorem <ref> (given in Section III), one obtains, I ≤_cxÎ.





Note that ln(1+k/x) is a convex function when k≥ 0 and x ≥ 0 due to the fact that double differentiation of ln(1+k/x) is always non negative, i.e., ∂∂ln(k/x+1)/∂ x/∂ x=k (k+2 x)/x^2 (k+x)^2≥ 0. Note that S and I are non negative RVs, hence by a direct application of Theorem <ref> (given in Section III), one obtains 

    E_Î[ln(1+S/Î)|S=s]≥ E_I[ln(1+S/I)|S=s]

Since expectation preserve inequalities therefore, E_S[E_I[ln(1+S/I)]] ≤ E_S[E_Î[ln(1+S/Î)]]. In other words, positive correlation among the interferers increases the rate.

Summarizing, the rate in the presence of the positive correlated interferers is greater than or equal to the rate in the presence of independent interferers, when SoI and interferers both experience η-μ fading. Now we briefly discuss the utility of our results in the presence of log normal shadowing.


 §.§ Log Normal Shadowing

Although all the analysis so far  (comparison of the coverage probability and rate) considered only small scale fading and path loss, the analysis can be further extended to take into account shadowing effects. In general, the large scale fading, i.e, log normal shadowing   is modeled by zero-mean log-normal distribution which is given by,

    f_X(x)=1/x√(2π(σ_dB/8.686)^2)exp(-ln^2(x)/2(σ_dB/8.686)^2), x>0,

where σ_dB is the shadow standard deviation represented in dB. Typically the value of σ_dB varies from 3 dB to 10 dB <cit.>,<cit.>. It is shown in <cit.> that the pdf of the  composite fading channel (fading and shadowing) can be expressed using the generalized-K (Gamma-Gamma) model. Also in <cit.>, it has been shown that the generalized-K pdf can be well approximated by Gamma pdf 𝒢(β, γ) using the moment matching method, with β and γ are given by 

    β=1/(1/m+1)exp((σ_dB/8.686)^2)-1=m/(m+1)exp((σ_dB/8.686)^2)-m
 

    and γ=1+m/mexp(3(σ_dB/8.686)^2/2)-exp((σ_dB/8.686)^2/2)
 
Thus, SIR η_l of a user can now given by 

    η_l(r)=Pkr^-α/∑_i∈ϕPl_id_i^-α

where k∼𝒢(β, γ) and l_i∼𝒢(β_i, γ_i). Here 

    β_i=m_i/(m_i+1)exp((σ_dB/8.686)^2)-m_i and γ_i=(1+m_i)/m_iexp(3(σ_dB/8.686)^2/2)-exp((σ_dB/8.686)^2/2)

Further, the correlation coefficient between two identically distributed generalized-K RVs is derived in <cit.>, and it is in terms of correlation coefficient  of the RVs corresponding to the short term fading component (ρ_i,j) and the correlation coefficient of the RVs corresponding to the shadowing component(ρ^s_i,j). The resultant correlation coefficient (ρ^l_i,j) is then given by

    ρ^l_i,j=ρ_i,j/( exp(σ_dB/8.686)^2)-1)+ρ^s_i,j m_i+ρ_i,jρ^s_i,j/m_i+1/( exp(σ_dB/8.686)^2)-1)+1


Note that after approximation, the SIR expression in the presence of log normal shadowing is similar to the SIR expression given in (<ref>), where only small scale fading is present. Hence now  the coverage probability and  rate for the independent case and correlated case can be compared using the methods outlined in Section IV and Section V. In other words, it can be shown that  the coverage probability in the presence of correlated interferers is greater than or equal to the coverage probability in presence of independent interferers, when user's shape parameter is less than or equal to 1, i.e., β≤ 1, in the presence of shadow fading. Also, the rate in the presence of positive correlated interferers is always greater than or equal to the rate in the presence of independent interferers, in the presence of shadow fading.




 §.§ Physical interpretation of the impact of correlated Interferers

We know that the Nakagami-m distribution considers a signal composed of n number of clusters of multipath waves. Within each cluster, the phases of scattered waves are random and have similar delay times. The delay-time spreads of different clusters are relatively large. More importantly, the non-integer Nakagami parameter m is the real extension of integer n. One of the primary reason of parameter m being real extension of n is the non zero correlation among the clusters of multipath components<cit.>. In other words, if there exist  correlation among the  clusters, the shape parameter of Nakagami-m fading decreases. 
 
Now, in order to find the physical interpretation of the impact of correlated interferers, we consider a cellular system where the interferes are equidistant and also the shape parameters are identical for every interference. When all the interferers are independent, the shape parameter of the total interference to be mN where the shape parameter of each interfere is m and the total number of interference is N (since the sum of Gamma RV is Gamma RV). However, when there exists  correlation among the interferers, i.e., there exists correlation among the clusters of different interferers, the shape parameter of the total interference  decreases as observed in <cit.>. For example, if the interferers are completely correlated, the shape parameter of total interference is only m, whereas it was mN, when the interferers were independent. The smaller the shape parameter more faster is the power attenuation of interferers. Hence, the rate increases when there exists correlation among  interferers.















 
 In the next section, we will show simulation results and discuss how those match with the theoretical results.


§ NUMERICAL ANALYSIS AND APPLICATION

In this section, we study the  impact of correlation among interferers on the coverage probability and rate using simulations and numerical analysis. For simulations, we have considered the classic 19 cell system associated with a hexagonal structure as shown in Fig. <ref>. For a user we generate the channel fading power corresponding to its own channel as well as that corresponding to the 18 interferers  and then compute the SIR per user. For correlation scenario, we generate correlated channel fading power  corresponding to the 18 interferers and then compute SIR per user. Furthermore, based on the SIR, we find coverage probability and rate. Fig. <ref>  depicts the impact of correlation among the interferers on the coverage probability for different values of shape parameter. Note that only small scale fading is considered in Fig. <ref>. The  correlation among the interferers is defined by the correlation matrix in (<ref>)  with  ρ_pq=ρ^|p-q| where p,q=1,⋯ ,N <cit.>. From Fig. <ref>, it can be observed that for  m=0.5 and m=1, coverage probability in presence of correlation is higher than that of independent scenario (which match our analytical result).  For example, at m=0.5, coverage probability increases from 0.16 in the independent case to 0.20 in the correlated case and at m=1, coverage probability increases from 0.148 to 0.216 when user is at normalized distance 0.7 from the BS. Whereas m=3, one cannot say  that coverage probability in presence of correlation is higher or lower than that of independent scenario. In other words, the coverage probability of independent interferers is higher than the coverage probability of correlated interferers when user is close to the BS. However, the coverage probability of independent interferers is significantly lower than the coverage probability of correlated interferers when the user is far from the BS.

Fig. <ref> shows the  impact of correlation among the interferers on the coverage probability for different values of correlation coefficient. Here both small scale fading, i.e., η-μ fading and large scale fading, i.e., log normal shadowing are considered. The correlation among small scale fading is denoted by ρ_s and the correlation among the large scale fading is denoted by ρ_l. It can be seen that as correlation coefficient increases the coverage probability with increases for the correlated interferers case.




            
            


 §.§ How the User can Exploit Correlation among Interferers

We will now briefly discuss how the user in a cellular network can exploit knowledge of positive correlation among its interferers.
We  compare the coverage probability in the presence of correlated interferers for single input single output (SISO) network with the coverage probability in the presence of independent interferers for single input multiple output (SIMO) network to show that the impact of correlation is significant.  For the SIMO network, it is assumed that each user is equipped with 2 antennas and both antennas at the user are used for reception since downlink is considered. A linear minimum mean-square-error (LMMSE) receiver <cit.> is considered. In order to calculate rate with a LMMSE receiver, it is assumed that the closest interferer can be completely  cancelled  at the SIMO receiver. Fig. <ref> plots the SISO rate in the presence of independent and correlated interferers case and the rate in the presence of independent interferers for a SIMO network. It can be seen that for ρ_s=0.6, ρ_l=0.6, the SISO rate for the correlated case[The  correlation among the interferers is defined by the correlation matrix in (<ref>)  with  ρ_pq=ρ^|p-q| where p,q=1,⋯ ,N]  is slightly higher than the SIMO rate for independent case. However, for ρ_s=0.7,ρ_l=0.9, SISO rate with correlated interferers is significantly higher than the SIMO rate with independent interferers.   In other words, correlation among the interferers seems  to be as good as having one additional antenna at the receiver capable of cancelling the dominant interferer. Obviously, if one had correlated interferers in the SIMO system that would again lead to improved coverage probability and rate and may be compared to a SIMO system with higher number of antennas. In all three cases, it is apparent that if the correlation among the interferers is exploited, it leads to  performance results for a SISO system which are comparable to the   performance of a 1× 2 SIMO system with independent interferers.


We have now consider a MU-MIMO system and show that the impact of correlation on MU-MIMO is significant. It is assumed that each user and BS  are equipped with 2 receive antennas and 2 transmit antennas, respectively. It is also assume that both transmit antennas at the BS are utilized to transmit 2 independent data streams to its own 2 users. A LMMSE receiver is considered and assume that user can cancel the closest interferer. Hence, in this MU-MIMO system, the user will experience no intra-tier interference coming from the serving BS.  Fig. <ref> plots the MU-MIMO rate in the presence of independent and correlated interferers case. It can be seen that there is significant gain in rate for correlated case. For example, at normalized distance 0.6 from the BS, rate increases from 1.24 nats/Hz in the independent case to 2.55 nats/Hz in the correlated case when ρ_s=0.8 and ρ_l=0.95 and 1.985 nats/Hz in the correlated case when ρ_s=0.6 and ρ_l=0.6. The reason for significant gain in rate is due to the fact that interferers becomes double for 2× 2 MU-MIMO system.

            
 
 
We would also likely to briefly point out that the impact of correlation among the interferers is like that of introducing interference alignment in a system. Interference alignment actually aligns interference using appropriate precoding so as reduce the number of interferers one needs to cancel. Here the physical nature of the wireless channel and the presence of co-located interferers also “aligns” the interferer partially. This is the reason one can get a gain equivalent to 1× 2 system in a 1× 1 system with correlated interferers provided the user knows about the correlation. 

Summarizing, our work is able to analytically shows the impact of correlated interferers on coverage probability and rate. This can be used by the network and user to decide whether one wants to use the antennas at the receiver for diversity gain or interference cancellation depending on the information available about interferers correlation. Note that interferers from adjacent sector of a BS will definitely be correlated <cit.>. We believe that this correlation should be exploited, since the analysis shows that knowledge of correlation will lead to higher coverage probability and rate.  



§ CONCLUSIONS

In this work, the coverage probability  have been compared analytically for following two cases: (a)  User channel experiences Nakagami-m fading and interferers experience η-μ fading. (b) Interferers being correlated where the correlation is specified by a correlation matrix. We have  shown that the coverage probability in correlated interferer case is higher than that of the independent case, when the user channel's shape parameter is lesser than or equal to one, and the interferers have Nakagami-m fading with arbitrary parameters. Further, rate have been compared when both user channel and interferers experience η-μ fading. It has been shown that positive correlation among the interferers always increases the rate.  We have also taken into account the large scale fading component in our analysis. The impact of correlation seems even more pronounced in the presence of  shadow fading. Moreover, MU-MIMO system is considered and it has been shown that the impact of correlation among interferers is significant on MU-MIMO. Our results indicate that if the user is aware of the interferers correlation matrix then it can exploit it since the correlated interferers behave like partially aligned interferers. This means that if the user is aware of the correlation then one may be able to obtain a rate equivalent to a 1× 2 system in a 1× 1 system depending on the correlation matrix structure. Extensive simulations  were performed and these match with the theoretical results.




§ PROOF OF THEOREM <REF>

The coverage probability expressions for the scenario when interferers are independent and the scenario when interferers are correlated  are given as follows (given in (<ref>) and (<ref>), respectively). 

    C_p(r)=K F_D^(N)[1-m,m_c,⋯,m_c; Nm_c+1;1/Tr^αλ_1m+1,⋯,1/Tr^αλ_N m+1]


    Ĉ_p(r)= K̂ F_D^(N)[1-m,m_c,⋯,m_c; Nm_c+1;1/Tr^αmλ̂_1+1,⋯,1/Tr^αmλ̂_N+1]

where K=Γ(Nm_c+m)/Γ(Nm_c+1)1/Γ(m)∏_i=1^N(1/Tr^αm λ_i +1)^m_c and K̂=Γ(Nm_c+m)/Γ(Nm_c+1)1/Γ(m)∏_i=1^N(1/Tr^αmλ̂_i +1)^m_c.  From Theorem <ref> it is clear that K̂> K . Now, we need to compare the Lauricella's function of the fourth kind of (<ref>) and (<ref>). Here, for comparison we use the series expression for F_D(.).
We expand the  series expression for the Lauricella's function of the fourth kind in the  following form:

    F_D^(N)[a,b,⋯, b;c;x_1,⋯, x_N]=   1+K_1,1∑_i=1^Nx_i+K_2,1∑_i=1^Nx_i^2+K_2,2∑_1≤ i<j≤ Nx_ix_j+K_3,1∑_i=1^Nx_i^3
       +K_3,2∑_i,j=1, s.t.i≠ j^Nx_i^2x_j+K_3,3∑_1≤ i<j<k≤ Nx_ix_jx_k+⋯

 where
K_1,1=(a)_1(b)_1/(c)_1 1!, K_2,1=(a)_2(b)_2/(c)_2 2!, K_2,2=(a)_2(b)_1(b)_1/(c)_21!1!, K_3,1=(a)_3(b)_3/(c)_3 3!, K_3,2=(a)_3(b)_2(b)_1/(c)_3 2!1!, K_3,3=(a)_3(b)_1(b)_1(b)_1/(c)_31!1!1! and so on.

Hence the coverage probability for independent case given in (<ref>) can be written as

    C_p(r)=K[ 1+K_1,1∑_i=1^N(1/Tr^αm λ_i +1)+K_2,1∑_i=1^N(1/Tr^αm λ_i +1)^2+ K_3,1∑_i=1^N(1/Tr^αm λ_i +1)^3+


    K_2,2∑_1≤ i<j≤ N(1/Tr^αm λ_i +1)(1/Tr^αm λ_j +1)+
    K_3,2∑_i,j=1, s.t. i≠ j^N(1/Tr^αm λ_i +1)^2(1/Tr^αm λ_j +1)+


    K_3,3∑_1≤ i<j<k≤ N(1/Tr^αm λ_i +1)(1/Tr^αm λ_j +1)(1/Tr^αm λ_k +1)+⋯]

Similarly,  for the correlated  case  the coverage probability given in (<ref>) can be written as

    Ĉ_p(r)=K̂[ 1+K_1,1∑_i=1^N(1/Tr^αmλ̂_i +1)+K_2,1∑_i=1^N(1/Tr^αmλ̂_i +1)^2+ K_3,1∑_i=1^N(1/Tr^αmλ̂_i +1)^3+


    K_2,2∑_1≤ i<j≤ N(1/Tr^αmλ̂_i +1)(1/Tr^αmλ̂_j +1)+
    K_3,2∑_i,j=1, s.t. i≠ j^N(1/Tr^αmλ̂_i +1)^2(1/Tr^αmλ̂_j +1)+


    K_3,3∑_1≤ i<j<k≤ N(1/Tr^αmλ̂_i +1)(1/Tr^αmλ̂_j +1)(1/Tr^αmλ̂_k +1)+⋯]

Here 
K_1,1=(1-m)_1(m_c)_1/(N m_c+1)_1 1!, K_2,1=(1-m)_2(m_c)_2/(N m_c+1)_2 2!, K_2,2=(1-m)_2(m_c)_1(m_c)_1/(N m_c+1)_2 1!1!, K_3,1=(1-m)_3(m_c)_3/(N m_c+1)_3 3!, K_3,2=(1-m)_3(m_c)_2(m_c)_1/(N m_c+1)_3 2!1!, K_3,3=(1-m)_3(m_c)_1(m_c)_1(m_c)_1/(N m_c+1)_31!1!1! and so on. Note that here K_i,j are the same for both C_p(r) and Ĉ_p(r). Now, we want to show that each summation term in the series expression is a Schur-convex function.

Each summation term in the series expression is symmetrical due to the fact that any two of its argument can be interchanged without changing the value of the function. We have already shown that ∏_i=1^N(1/1+kx_i)^a is a convex function ∀ x_i≥ 0 and ∀ a>0. Now, the terms in the summation terms in (<ref>) and (<ref>) are of the form ∏_i=1^M(1/1+kx_i)^a_i where M≤ N.  Using Theorem <ref> (given in Section II), one can show that each  term of each summation term is a convex function. Using the fact that convexity is preserved under summation one can show that each summation term is a convex function. Thus, each summation term 
in 
series expression is a Schur-convex function.

Now we consider following two cases.

Case I when m< 1:
Since m< 1, so 1-m > 0 and hence all the constant K_i, j> 0 ∀ i, j.  Each summation term in series expression of coverage probability for correlated case is greater  than or equal to the corresponding summation term in the series expression of coverage probability for independent case. Thus, if user  channel's shape parameter m< 1 then coverage probability of  correlated case is  greater than or equal to  the coverage probability for independent case.

Case II when m> 1:
Since m > 1, then 1-m < 0 and hence K_i,j<0 ∀ i ∈ 2|ℤ|+1  and ∀ j  where set ℤ denote the integer number, due to  the fact that (a)_N<0  if  a<0  and  N ∈ 2|ℤ|+1.  Whereas, K_i,j>0 ∀ i ∈ 2|ℤ|  and ∀ j due of the fact that (a)_N>0  if  a<0  and  N∈ 2|ℤ|. Thus, if   m>1, we cannot state whether the coverage probability of one case is greater than or lower than the other case.

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