# An Efficient Approximation to the Correlated Nakagami-$m$ Sums and its
Application in Equal Gain Diversity Receivers

Nikola Zlatanov, Zoran Hadzi-Velkov, and George K. Karagiannidis Accepted for
IEEE TWirelessN. Zlatanov and Z. Hadzi-Velkov are with the Faculty of
Electrical Engineering and Information Technologies, Ss. Cyril and Methodius
University, Skopje, Email<EMAIL_ADDRESS><EMAIL_ADDRESS>K. Karagiannidis is with the Department of Electrical and Computer
Engineering, Aristotle University of Thessaloniki, Thessaloniki, Email:
<EMAIL_ADDRESS>
###### Abstract

There are several cases in wireless communications theory where the statistics
of the sum of independent or correlated Nakagami-$m$ random variables (RVs) is
necessary to be known. However, a closed-form solution to the distribution of
this sum does not exist when the number of constituent RVs exceeds two, even
for the special case of Rayleigh fading. In this paper, we present an
efficient closed-form approximation for the distribution of the sum of
arbitrary correlated Nakagami-$m$ envelopes with identical and integer fading
parameters. The distribution becomes exact for maximal correlation, while the
tightness of the proposed approximation is validated statistically by using
the Chi-square and the Kolmogorov-Smirnov goodness-of-fit tests. As an
application, the approximation is used to study the performance of equal-gain
combining (EGC) systems operating over arbitrary correlated Nakagami-$m$
fading channels, by utilizing the available analytical results for the error-
rate performance of an equivalent maximal-ratio combining (MRC) system.

###### Index Terms:

Nakagami-$m$ fading, arbitrary correlation, approximative statistics, equal
gain combining (EGC), maximal ratio combining (MRC)

##  I Introduction

The analytical determination of the the probability distribution functions
(PDF) and the cumulative distribution functions (CDF) of the sums of
independent and correlated signals’ envelopes is rather cumbersome, yielding
difficulties in the theoretical performance analysis of some wireless
communications systems [1]. A closed-form solution for the PDF and the CDF of
the sum of Rayleigh random variables (RVs) has not been presented for more
then 90 years, except when the number of RVs equals two. The famous Beaulieu
series for computing PDF of a sum of independent RVs were proposed in [2].
Later, a finite range multifold integral for PDF of the sum of independent and
identically distributed (i.i.d.) Nakagami-$m$ RVs was proposed in [3]. A
closed-form formula for the PDF of the sum of two i.i.d. Nakagami-$m$ RVs was
given in [4]-[6]. Exact infinite series representations for the sum of three
and four i.i.d. Nakagami-$m$ RVs was presented in [7], although their
usefulness is overshadowed by their computational complexity.

The most famous application, where these sums appear, deals with the
analytical performance evaluation of equal gain combining (EGC) systems
[8]-[13]. Only few papers address the performance of EGC receivers in
correlated fading with arbitrary-order diversity. In [14], EGC was studied by
approximating the moment generating function (MGF) of the output SNR, where
the moments are determined exactly only for exponentially correlated
Nakagami-$m$ channels in terms of multi-fold infinite series. A completely
novel approach for performance analysis of diversity combiners in equally
correlated fading channels was proposed in [15], where the equally correlated
Rayleigh fading channels are transformed into a set of conditionally
independent Rician RVs. Based on this technique, the authors in [16] derived
the moments of the EGC output signal-to-noise ratio (SNR) in equally
correlated Nakagami-$m$ channels in terms of the Appell hypergeometric
function, and then used them to evaluate the EGC performance metrics, such as
the outage probability and the error probability (using Gaussian quadrature
with weights and abscissas computed by solving sets of nonlinear equations).

All of the above works yield to results that are not expressed in closed form
due to the inherent intricacy of the exact sum statistics. This intricacy can
be circumvented by searching for suitable highly accurate approximations for
the PDF of a sum of arbitrary number of Nakagami-$m$ RVs. Various simple and
accurate approximations to the PDF of sum of independent Rayleigh, Rice and
Nakagami-$m$ RVs had been proposed in [17]-[21], which had been used for
analytical EGC performance evaluation. Based on the ideas given in [1], the
works [18]-[21] use various alternatives of the moment matching method to
arrive at the required approximation.

In this paper, we present a highly accurate closed-form approximation for the
PDF of the sum of non-identical arbitrarily correlated Nakagami-$m$ RVs with
identical (integer) fading parameters. By applying this approximation, we
evaluate the performance of EGC systems in terms of the known performance of
an equivalent maximal ratio combining (MRC) system [22], [24], thus avoiding
many complex numerical evaluations inherent for the methods presented in the
aforementioned previous works for the EGC performance analysis. Although
approximate, the offered closed-form expressions allow to gain insight into
system performance by considering, for example, operation in the low or high
SNR region.

##  II An accurate approximation to the sum of arbitrary correlated
Nakagami-$m$ envelopes

Let $Z$ be a sum of $L$ non-identical correlated Nakagami-$m$ envelopes,
$\\{Z_{k}\\}_{k=1}^{L}$, defined as

$Z=\sum_{k=1}^{L}Z_{k}\,.$ (1)

The envelopes $\\{Z_{k}\\}_{k=1}^{L}$ are distributed according to the
Nakagami-$m$ distribution, whose PDF is given by [1]

$f_{Z_{k}}(z)=\left(\frac{m_{z}}{\Omega_{k}}\right)^{m_{z}}\frac{2z^{2m_{z}-1}}{\Gamma(m_{z})}\exp\left(-\frac{m_{z}}{\Omega_{k}}z^{2}\right),\qquad
z\geq 0\,,$ (2)

with arbitrary average powers $E[Z_{k}^{2}]=\Omega_{k}$, $1\leq k\leq L$, and
the same (integer) fading parameter $m_{z}$. The power correlation coefficient
between any given pair of envelopes $(Z_{i},Z_{j})$ is defined as

$\rho_{ij}=\frac{{\rm{cov}}(Z_{i}^{2},Z_{j}^{2})}{\sqrt{{\rm{var}}(Z_{i}^{2})\
{\rm{var}}(Z_{j}^{2})}},\qquad i\neq j\,,$ (3)

where $E[\cdot]$, $\rm{cov}(\cdot,\cdot)$ and $\rm{var}(\cdot)$ denote
expectation, covariance and variance, respectively.

We propose the unknown PDF of $Z$ be approximated by the PDF of $R$ defined as

$R=\sqrt{\sum_{k=1}^{L}R_{k}^{2}}\,\,,$ (4)

where $R_{k}$, $1\leq k\leq L$, denote a set of $L$ correlated but identically
distributed Nakagami-$m$ envelopes with same average powers,
$E[R_{k}^{2}]=\Omega_{R}$, and same fading parameters, $m_{R}$. The power
correlation coefficients between any given pair $(R_{i},R_{j})$ is assumed
equal to that of the respective pair of the original envelopes
$(Z_{i},Z_{j})$, $\rho_{ij}$.

The statistics of $R^{2}$ is easily seen to be equal to the statistics of the
sum of correlated Gamma RVs. Thus, the MGF of $R^{2}$ is represented by
[References, Eq. (11)]

$M_{R^{2}}(s)=\det\left({\bf{I}}-s\frac{\Omega_{R}}{m_{R}}{\bf{\Lambda}}\right)^{-m_{R}}=\prod_{k=1}^{L}\left(1-s\frac{\Omega_{R}}{m_{R}}\lambda_{k}\right)^{-m_{R}}$
(5)

where ${\bf{I}}$ is the $L\times L$ identity matrix and ${\bf{\Lambda}}$ is
the $L\times L$ positive definite matrix (denoted as the correlation matrix)
whose elements are the square roots of the power correlation coefficients,

$\displaystyle{\bf{\Lambda}}=\left[\begin{array}[]{ccccc}1&\sqrt{\rho_{12}}&\cdots&\sqrt{\rho_{1L}}\\\
\sqrt{\rho_{21}}&1&\cdots&\sqrt{\rho_{2L}}\\\ .&.&\cdots&.\\\
\sqrt{\rho_{L1}}&\sqrt{\rho_{L2}}&\cdots&1\\\ \end{array}\right]\,.$ (10)

The $L$ eigenvalues of the correlation matrix ${\bf{\Lambda}}$ are denoted by
$\lambda_{k}$, $1\leq k\leq L$.

Throughout literature, the PDF of $R^{2}$ is determined by using several
different approaches that result in alternative closed-form solutions, two of
which are given by [References, Eq. (29)] and [References, Eq. (10)]. After a
simple RV transformation, these two alternatives for the PDF of $R$ are
expressed as

$f_{R}(r)=\frac{2r}{\pi}\int_{0}^{\infty}\frac{\cos\left[m_{R}\sum_{k=0}^{L-1}\arctan\left(t\frac{\Omega_{R}\lambda_{k}}{m_{R}}\right)-tr^{2}\right]}{\prod_{k=0}^{L-1}\left[1+\left(t\frac{\Omega_{R}\lambda_{k}}{m_{R}}\right)^{2}\right]^{m_{R}/2}}dt$
(11)
$\displaystyle=\frac{2r^{2Lm_{R}-1}}{\Gamma(Lm_{R})}\left(\frac{m_{R}}{\Omega_{R}}\right)^{Lm_{R}}\left(\frac{1}{\det({\bf{\Lambda}})}\right)^{m_{R}}\qquad\qquad\qquad$
$\displaystyle\times\Phi_{2}^{(L)}\left(m_{R},m_{R},\dots,m_{R};Lm_{R};-\frac{m_{R}}{\Omega_{R}}\frac{r^{2}}{\lambda_{1}},\dots,-\frac{m_{R}}{\Omega_{R}}\frac{r^{2}}{\lambda_{L}}\right),$
(12)

where $\Phi_{2}^{(L)}(\cdot)$ is the confluent Lauricella hypergeometric
function of $L$ variables, defined in [33] and [References, Eqs. (9)-(10)].
Note that (II) is here presented to demonstrate existence of an exact closed-
form solution, whereas (11) is much more convenient for accurate and efficient
numerical integration. For example, the PDF may be obtained using the Gauss-
Legendre quadrature rule [References, Eq. (25.4.29)] over (11) [References].

Next, we apply the moment matching method to determine the parameters
$\Omega_{R}$ and $m_{R}$ of the proposed approximation (11)-(II) to the PDF of
$Z$. In wireless communications, moment matching methods are most typically
applied to approximate distributions of the sum of log-normal RVs [29]. Most
recently, a variant of moment matching, matching of the normalized first and
second moments, had been applied to arrive at an improved approximation to the
sum of independent Nakagami-$m$ RVs via the $\alpha$-$\mu$ distribution
[20]-[21].

We arrive at required approximation by matching the first and the second
moments of the powers of $Z$ and $R$, i.e., the second and fourth moments of
the envelopes $Z$ and $R$,

$E[Z^{2}]=E[R^{2}],\quad E[Z^{4}]=E[R^{4}]\,.$ (13)

Matching the first and the second moments of the powers aids the analytical
tractability of the proposed approximation due to the availability of the MGF
of $R^{2}$ in closed form, given by (5). The second and the fourth moments of
$R$ are determined straightforwardly by applying the moment theorem over (5),
yielding

$\displaystyle
E[R^{2}]=\frac{dM_{R^{2}}(s)}{ds}\Bigg{|}_{s=0}=\Omega_{R}\sum_{l=1}^{L}\lambda_{l}=\Omega_{R}L\,,\qquad\quad$
(14) $\displaystyle
E[R^{4}]=\frac{d^{2}M_{R^{2}}(s)}{ds^{2}}\Bigg{|}_{s=0}=\frac{\Omega_{R}^{2}}{m_{R}}\left[\sum_{l=1}^{L}\lambda_{l}^{2}+m_{R}L^{2}\right]\,.$
(15)

Introducing (14) and (15) into (13), one obtains the unknown parameters for
the statistics of $R$

$\Omega_{R}=\frac{E[Z^{2}]}{L}\,,\quad
m_{R}=\frac{\sum_{l=1}^{L}\lambda_{l}^{2}}{L^{2}}\frac{(E[Z^{2}])^{2}}{E[Z^{4}]-(E[Z^{2}))^{2}}\,.$
(16)

Using the multinomial theorem and [References, Eq. (137)], the second and the
fourth moments of $Z$ are determined as

$\displaystyle
E[Z^{2}]=\sum_{k=1}^{L}\Omega_{k}+\frac{2\Gamma^{2}(m_{z}+1/2)}{m_{z}\Gamma^{2}(m_{z})}\qquad\qquad\qquad\qquad$
$\displaystyle\times\sum_{i=1}^{L}\sum_{j=i+1}^{L}\sqrt{\Omega_{i}\Omega_{j}}\,{}_{2}F_{1}\left(-1/2,-1/2;m_{z};\rho_{ij}\right),$
(17) $\displaystyle
E[Z^{4}]=\frac{m_{z}+1}{m_{z}}\sum_{m=1}^{L}\Omega_{m}^{2}+\frac{6\Gamma^{2}(m_{z}+1)}{m_{z}^{2}\Gamma^{2}(m_{z})}\sum_{i=1}^{L}\sum_{j=i+1}^{L}\Omega_{i}\Omega_{j}\qquad$
$\displaystyle\times{}_{2}F_{1}\left(-1,-1;m_{z};\rho_{ij}\right)+\frac{4\Gamma(m_{z}+3/2)\Gamma(m_{z}+1/2)}{m_{z}^{2}\Gamma^{2}(m_{z})}\qquad\quad$
$\displaystyle\times\sum_{i=1}^{L}\sum_{j=i+1}^{L}(\Omega_{i}^{3/2}\Omega_{j}^{1/2}+\Omega_{i}^{1/2}\Omega_{j}^{3/2}){}_{2}F_{1}\left(-\frac{3}{2},-\frac{1}{2};m_{z};\rho_{ij}\right)\,\,\,\,$
$\displaystyle+12\sum_{m=1}^{L}\sum_{i=m+1}^{L}\sum_{j=i+1}^{L}\Omega_{m}\sqrt{\Omega_{i}\Omega_{j}}E[Z_{m}^{2}Z_{i}Z_{j}]\qquad\qquad\qquad\quad$
$\displaystyle+12\sum_{m=1}^{L}\sum_{i=m+1}^{L}\sum_{j=i+1}^{L}\sqrt{\Omega_{m}}\Omega_{i}\sqrt{\Omega_{j}}E[Z_{m}Z_{i}^{2}Z_{j}]\qquad\qquad\quad\quad$
$\displaystyle+12\sum_{m=1}^{L}\sum_{i=m+1}^{L}\sum_{j=i+1}^{L}\sqrt{\Omega_{m}\Omega_{i}}\Omega_{j}E[Z_{m}Z_{i}Z_{j}^{2}]\qquad\qquad\qquad\quad$
$+\,\,\,24\sum_{m=1}^{L}\sum_{n=m+1}^{L}\sum_{i=n+1}^{L}\sum_{j=i+1}^{L}\sqrt{\Omega_{m}\Omega_{n}\Omega_{i}\Omega_{j}}E[Z_{m}Z_{n}Z_{i}Z_{j}]\,,\qquad\qquad\qquad\qquad\quad\,\,\,$
(18)

where ${}_{2}F_{1}(\cdot)$ is the Gauss hypergeometric function [References].
The joint moments $E[Z_{m}^{2}Z_{i}Z_{j}]$, $E[Z_{m}Z_{i}^{2}Z_{j}]$,
$E[Z_{m}Z_{i}Z_{j}^{2}]$ and $E[Z_{m}Z_{n}Z_{i}Z_{j}]$ are not known in
closed-form for arbitrary branch correlation. Exact closed-form expressions
are available only for some particular correlation models, such as the
exponential and the equal correlation models. For the case or arbitrary
correlation, we utilize the method presented in [27], where an arbitrary
correlation matrix ${\bf{\Lambda}}$ is approximated by its respective Green’s
matrix, followed by the application of the available joint moments of the
exponential correlation model.

TABLE I: Fading parameter $m_{R}$ for some feasible scenarios with equal correlation | $m_{z}=1$ | $m_{z}=2$ | $m_{z}=3$
---|---|---|---
$\rho$ | $L=2$ | $L=3$ | $L=4$ | $L=2$ | $L=3$ | $L=4$ | $L=2$ | $L=3$ | $L=4$
| $m_{R}$ | $m_{R}$ | $m_{R}$ | $m_{R}$ | $m_{R}$ | $m_{R}$ | $m_{R}$ | $m_{R}$ | $m_{R}$
0 | $0.9552$ | $0.9411$ | $0.9343$ | $1.947$ | $1.93$ | $1.9217$ | $2.943$ | $2.9258$ | $2.9168$
0.2 | $0.9195$ | $0.8884$ | $0.8709$ | $1.9102$ | $1.876$ | $1.8569$ | $2.9068$ | $2.8715$ | $2.8518$
0.4 | $0.9156$ | $0.8841$ | $0.8672$ | $1.907$ | $1.8722$ | $1.8535$ | $2.9039$ | $2.868$ | $2.8487$
0.6 | $0.9304$ | $0.9056$ | $0.8929$ | $1.9242$ | $1.8971$ | $1.8831$ | $2.9222$ | $2.8944$ | $2.8799$
0.8 | $0.9587$ | $0.9445$ | $0.9374$ | $1.956$ | $1.9409$ | $1.9333$ | $2.9553$ | $2.9399$ | $2.9321$
TABLE II: Fading parameter $m_{R}$ for some feasible scenarios with exponential correlation | $m_{z}=1$ | $m_{z}=2$ | $m_{z}=3$
---|---|---|---
$\rho$ | $L=2$ | $L=3$ | $L=4$ | $L=2$ | $L=3$ | $L=4$ | $L=2$ | $L=3$ | $L=4$
| $m_{R}$ | $m_{R}$ | $m_{R}$ | $m_{R}$ | $m_{R}$ | $m_{R}$ | $m_{R}$ | $m_{R}$ | $m_{R}$
0 | $0.9552$ | $0.9411$ | $0.9343$ | $1.947$ | $1.93$ | $1.9217$ | $2.943$ | $2.9258$ | $2.9168$
0.2 | $0.9195$ | $0.9033$ | $0.9015$ | $1.9102$ | $1.892$ | $1.8897$ | $2.9068$ | $2.8878$ | $2.8852$
0.4 | $0.9156$ | $0.8887$ | $0.88$ | $1.907$ | $1.877$ | $1.8675$ | $2.9039$ | $2.8728$ | $2.8629$
0.6 | $0.9304$ | $0.8988$ | $0.8817$ | $1.9242$ | $1.889$ | $1.87$ | $2.9222$ | $2.8858$ | $2.866$
0.8 | $0.9587$ | $0.934$ | $0.9162$ | $1.956$ | $1.9291$ | $1.9093$ | $2.9553$ | $2.9277$ | $2.9072$

###  II-A Equal correlation model

Equal correlation typically corresponds to the scenario of multichannel
reception from closely spaced diversity antennas (e.g., three antennas placed
on an equilateral triangle). This model may be employed as a worst case
correlation scenario, since the impact of correlation on system performance
for other correlation models typically will be less severe [22], [30].

For this correlation model, the power correlation coefficients are all equal,

$\rho_{ij}=\rho,\quad i\neq j,\quad 0\leq\rho\leq 1\,.$ (19)

When $m_{z}$ is assumed to be integer, the unknown joint moments in (II) can
be expressed in closed-form as [References, Eq. (43)]

$\displaystyle
E[Z_{m}^{2}Z_{i}Z_{j}]=E[Z_{m}Z_{i}^{2}Z_{j}]=E[Z_{m}Z_{i}Z_{j}^{2}]\qquad\qquad$
$\displaystyle=\left(\frac{1-\sqrt{\rho}}{m_{z}}\right)^{2}W(2,1,1)\,\,,\qquad\qquad$
(20) $\displaystyle
E[Z_{m}Z_{n}Z_{i}Z_{j}]=\left(\frac{1-\sqrt{\rho}}{m_{z}}\right)^{2}W(1,1,1,1)\,,\qquad\quad$
(21)

where the coefficients $W(\cdots)$ are determined as

$\displaystyle
W(k_{1},...,k_{N})=\left(\prod_{j=1}^{N}\frac{\Gamma(m_{z}+k_{j}/2)}{\Gamma(m_{z})}\right)\left(\frac{1-\sqrt{\rho}}{1+(N-1)\sqrt{\rho}}\right)^{m_{z}}$
$\displaystyle\times
F_{A}^{(N)}\left(m_{z};m_{z}+\frac{k_{1}}{2},\cdots,m_{z}+\frac{k_{N}}{2};m_{z},\cdots,m_{z};\right.\qquad\qquad$
$\displaystyle\left.\frac{\sqrt{\rho}}{1+(N-1)\sqrt{\rho}},\cdots,\frac{\sqrt{\rho}}{1+(N-1)\sqrt{\rho}}\right)\,,$
(22)

with $F_{A}^{(N)}(\cdot)$ denoting the Lauricella $F_{A}$ hypergeometric
function of $N$ variables, defined by [References, Eq. (9.19)] and
[References, Eqs. (11)-(13)].

Note that the coefficient $W(2,1,1)$ needs to be evaluated when $L\geq 3$,
whereas the coefficient $W(1,1,1,1)$ needs to be evaluated when $L\geq 4$. In
Appendix A, $W(2,1,1)$ is reduced to the more familiar hypergeometric
functions, attaining the form given by (A). $W(1,1,1,1)$ requires numerical
evaluation of the Lauricella $F_{A}$ function of 4 variables, which can be
computed with desired accuracy by using one of the two numerical methods
presented in [References, Section IV.A].

The assumption of equal average powers, $\Omega_{k}=\Omega_{Z}$, $1\leq k\leq
L$, yields independence of $m_{R}$ from $\Omega_{Z}$. For this case, Table I
gives the values of $m_{R}$ for several combinations of $\rho$, $L$ and
$m_{Z}$. The use of Table I aids the practical applicability of our approach
for the case of equal average powers.

For the equal correlation model, the eigenvalues of ${\bf{\Lambda}}$ are
exactly found as $\lambda_{1}=1+(L-1)\sqrt{\rho}$ and
$\lambda_{k}=(1-\sqrt{\rho})$ for $2\leq k\leq L$, so the statistics of
$R^{2}$ is identical to that of the sum of a pair of independent Nakagami RVs.
Thus, the MGF of $R^{2}$ is given by [References, Eq. (9.213)], whereas the
PDF of $R$ is given by [References, Eq. (9.208)]

$\displaystyle
f_{R}(r)=\left(\frac{m_{R}}{\Omega_{R}}\right)^{m_{R}L}\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\times\frac{2r^{2m_{R}L-1}\exp\left({-m_{R}r^{2}}/{((1-\sqrt{\rho})\Omega_{R}})\right)}{\Gamma(m_{R}L)(1-\sqrt{\rho})^{m_{R}(L-1)}(1+(L-1)\sqrt{\rho})^{m_{R}}}\qquad$
$\displaystyle\times{}_{1}F_{1}\left(m_{R};m_{R}L;\frac{m_{R}L\sqrt{\rho}}{(1-\sqrt{\rho})(1+(L-1)\sqrt{\rho})\Omega_{R}}r^{2}\right),$
(23)

where ${}_{1}F_{1}(\cdot)$ is the Kummer confluent hypergeometric function
[References, Eq. (9.210)].

###  II-B Exponential correlation model

Exponential correlation typically corresponds to the scenario of multichannel
reception from equispaced diversity antennas in which the correlation between
the pairs of combined signals decays as the spacing between the antennas
increases [22], [30].

For this correlation model, the power correlation coefficients are determined
as

$\rho_{ij}=\rho^{|i-j|},\qquad 0\leq\rho\leq 1\,.$ (24)

The unknown joint moments in (II), $E[Z_{m}^{2}Z_{i}Z_{j}]$,
$E[Z_{m}Z_{i}^{2}Z_{j}]$, $E[Z_{m}Z_{i}Z_{j}^{2}]$ and
$E[Z_{m}Z_{n}Z_{i}Z_{j}]$, can be calculated from [References, Eqs. (11) and
(12)]. The Appendix B derives simpler alternatives to [References, Eqs. (11)
and (12)], which involve a single infinite sum and a familiar hypergeometric
function,

$\displaystyle
E[Z_{m}^{n_{1}}Z_{i}^{n_{2}}Z_{j}^{n_{3}}]=\frac{|{\bf{\Delta}}|^{m_{z}}}{\delta_{11}^{m_{z}+n_{1}/2}\delta_{22}^{m_{z}+n_{2}/2}\delta_{33}^{m_{z}+n_{3}/2}}\qquad\qquad$
$\displaystyle\times\,\frac{\Gamma(m_{z}+n_{3}/2)}{\Gamma^{2}(m_{z})m_{z}^{(n_{1}+n_{2}+n_{3})/2}}\,\sum_{k=0}^{\infty}\left(\frac{\delta_{12}^{2}}{\delta_{11}\delta_{22}}\right)^{k}\qquad\,$
$\displaystyle\times\,\frac{\Gamma(m_{z}+k+n_{1}/2)\Gamma(m_{z}+k+n_{2}/2)}{\Gamma(m_{z}+k)k!}\qquad\quad$
$\displaystyle\times{}_{2}F_{1}\left(m_{z}+k+\frac{n_{2}}{2},m_{z}+\frac{n_{3}}{2},m_{z},\frac{\delta_{23}^{2}}{\delta_{22}\delta_{33}}\right),\,$
(25) $\displaystyle
E[Z_{m}Z_{n}Z_{i}Z_{j}]=\frac{|{\bf{\Psi}}|^{m_{z}}}{(\psi_{11}\psi_{22}\psi_{33}\psi_{44})^{m_{z}+1/2}}\frac{\Gamma^{2}(m_{z}+1/2)}{\Gamma^{3}(m_{z})m_{z}^{2}}$
$\displaystyle\times\sum_{k=0}^{\infty}\frac{\Gamma^{2}(m_{z}+k+1/2)}{k!\Gamma(m_{z}+k)}\left(\frac{\psi_{23}^{2}}{\psi_{22}\psi_{33}}\right)^{k}\qquad\,\,\,$
$\displaystyle\times\,{}_{2}F_{1}\left(m_{z}+\frac{1}{2},k+m_{z}+\frac{1}{2},m_{z},\frac{\psi_{12}^{2}}{\psi_{11}\psi_{22}}\right)$
$\displaystyle\times{}_{2}F_{1}\left(m_{z}+\frac{1}{2},k+m_{z}+\frac{1}{2},m_{z},\frac{\psi_{34}^{2}}{\psi_{33}\psi_{44}}\right).$
(26)

In (II-B), $(n_{1},n_{2},n_{3})$ = $(2,1,1)$ for the calculation of
$E[Z_{m}^{2}Z_{i}Z_{j}]$, $(n_{1},n_{2},n_{3})$ = $(1,2,1)$ for the
calculation of $E[Z_{m}Z_{i}^{2}Z_{j}]$ and $(n_{1},n_{2},n_{3})$ = $(1,1,2)$
for the calculation of $E[Z_{m}Z_{i}Z_{j}^{2}]$. The matrix
${\bf{\Delta}}=[\delta_{i,j}]$ is the inverse of ${\bf{\Lambda}}$’s principal
submatrix composed of the $m$-th, $i$-th and $j$-th rows and columns of
${\bf{\Lambda}}$, whereas the matrix ${\bf{\Psi}}=[\psi_{i,j}]$ is the inverse
of ${\bf{\Lambda}}$’s principal submatrix composed of the $m$-th, $n$-th,
$i$-th and $j$-th rows and columns of ${\bf{\Lambda}}$,

$\displaystyle{\bf{\Delta}}=\left[\begin{array}[]{ccccc}1&\sqrt{\rho_{mi}}&\sqrt{\rho_{mj}}\\\
\sqrt{\rho_{im}}&1&\sqrt{\rho_{ij}}\\\ \sqrt{\rho_{jm}}&\sqrt{\rho_{ji}}&1\\\
\end{array}\right]^{-1},$
$\displaystyle{\bf{\Psi}}=\left[\begin{array}[]{ccccc}1&\sqrt{\rho_{mn}}&\sqrt{\rho_{mi}}&\sqrt{\rho_{mj}}\\\
\sqrt{\rho_{nm}}&1&\sqrt{\rho_{ni}}&\sqrt{\rho_{nj}}\\\
\sqrt{\rho_{im}}&\sqrt{\rho_{in}}&1&\sqrt{\rho_{ij}}\\\
\sqrt{\rho_{jm}}&\sqrt{\rho_{jm}}&\sqrt{\rho_{ji}}&1\\\
\end{array}\right]^{-1}\,.$ (32)

The exactness of (II-B)-(II-B) arise from the fact that both matrices
${\bf{\Delta}}$ and ${\bf{\Psi}}$ are tridiagonal matrices due to (24)
[References, Section IV]. Introducing (II-B)-(II-B) into (II), one obtains the
closed-form expression for $E[Z^{4}]$, which is omitted here for brevity.
Combining (II)-(II) into (16), one obtains the unknown parameters $\Omega_{R}$
and $m_{R}$ for the statistics of $R$.

The assumption of equal average powers, $\Omega_{k}=\Omega_{Z}$, $1\leq k\leq
L$, again renders independence of $m_{R}$ from $\Omega_{Z}$ for the
exponential correlation model. Under such assumptions, Table II displays the
values of $m_{R}$ for several illustrative combinations of $\rho$, $L$ and
$m_{Z}$.

###  II-C Arbitrary correlation model

In the general case of arbitrary branch correlations, the correlation matrix
${\bf{\Lambda}}$ is approximated by its appropriate Green’s matrix,
${\bf{C}}$, utilizing the method presented in [References, Section IV]. Since
principal submatrices of Green’s matrices are also Green’s matrices, the
matrices ${\bf{\Delta}}$ and ${\bf{\Psi}}$, defined by (II-B), are determined
to be tridiagonal, yielding direct applicability of the results presented in
Section II.B to determine the unknown parameters $\Omega_{R}$ and $m_{R}$ for
the statistics of $R$. Thus, the statistics of $Z$ are approximated by the
statistics of $R$, whose arbitrary correlation matrix ${\bf{\Lambda}}$ is
approximated by the Green’s matrix ${\bf{C}}$. In the following subsection, we
illustrate the highly accurate approximation to the PDF of $Z$ facilitated by
this approach.

TABLE III:  Significance levels of C-S and K-S tests for goodness
of fit between the exact and the approximative distributions of Fig. 1 | $m_{z}=1$ | $m_{z}=3$
---|---|---
$\rho$ | $L=2$ | $L=5$ | $L=2$ | $L=5$
| $\alpha_{CS}$ | $\alpha_{KS}$ | $\alpha_{CS}$ | $\alpha_{KS}$ | $\alpha_{CS}$ | $\alpha_{KS}$ | $\alpha_{CS}$ | $\alpha_{KS}$
0.2 | $0.004$ | $0.02$ | $0.17$ | $0.06$ | $<0.001$ | $<0.001$ | $<0.001$ | $<0.001$
0.7 | $0.04$ | $0.03$ | $0.2$ | $0.18$ | $<0.001$ | $<0.001$ | $<0.001$ | $<0.001$
TABLE IV:  Significance levels of C-S and K-S tests for goodness
of fit between the exact and the approximative distributions of Fig. 2 | $m_{z}=1$ | $m_{z}=3$
---|---|---
$\rho$ | $L=2$ | $L=5$ | $L=2$ | $L=5$
| $\alpha_{CS}$ | $\alpha_{KS}$ | $\alpha_{CS}$ | $\alpha_{KS}$ | $\alpha_{CS}$ | $\alpha_{KS}$ | $\alpha_{CS}$ | $\alpha_{KS}$
0.2 | $<0.001$ | $0.02$ | $<0.001$ | $<0.001$ | $<0.001$ | $<0.001$ | $<0.001$ | $<0.001$
0.7 | $0.08$ | $0.04$ | $0.02$ | $0.17$ | $<0.001$ | $<0.001$ | $<0.001$ | $<0.001$
Figure 1: Exact (obtained by simulation) and the approximative analytical PDFs
to the sum of equally correlated Nakagami-$m$ RVs with equal average powers,
when $\Omega_{Z}=1$

###  II-D Validation via statistical goodness-of-fit tests

We now statistically validate the proposed PDF approximations for equal,
exponential and arbitrary branch correlation by using two different goodness-
of-fit tests. The Chi-square (C-S) and Kolmogorov-Smirnov (K-S) tests provide
two different statistical metrics, $\chi_{n}^{2}$ and $D_{n}$, which describe
the discrepancy between the observed samples of $Z$ and the samples expected
under the analytical distribution (11)-(II).

Each metric is averaged over 100 statistical samples, where each statistical
sample comprises of 10000 independent random samples of $Z$. The random
samples of $Z$ are generated by computer simulations of correlated
Nakagami-$m$ RVs based on the method proposed in [References, Section VII].

For each metric, we calculate the significance level $\alpha$ from the C-S and
K-S distributions, respectively denoted as $\alpha_{CS}$ and $\alpha_{KS}$.
The significance level $\alpha$ represents the probability of rejecting the
tested null hypothesis ($H_{0}$: ”the random samples of $Z$, obtained from
(1), belong to the distribution given by (11)-(II)”), when it is actually
true. The small values of $\alpha$ indicate a good fit.

Note that, significance levels $\alpha$ less then $0.2$ still indicate a good
fit, due to the rigourousness of both C-S and K-S tests in accepting the null
hypothesis $H_{0}$.

Figure 2: Exact (obtained by simulation) and the approximative analytical PDFs
to the sum of exponentially correlated Nakagami-$m$ RVs with equal average
powers, when $\Omega_{Z}=1$

####  II-D1 Equal and exponential correlation

For the equal and exponential correlation models, the goodness-of-fit testing
is conducted for combinations of the followings input parameters: $L=2$ and
$5$, $m_{Z}=1$ and $3$, $\rho=0.2$ and $0.7$, whereas the average powers of
$Z_{k}$ are assumed equal to unity ($\Omega_{Z}=1$). The needed fading
parameter $m_{R}$ of distribution (11)-(II) is obtained directly from Tables I
and II, whereas the average power $\Omega_{R}$ is calculated from (16).

Figs. 1 and 2 depict the excellent (visual) match between the histogram
obtained from generated samples of $Z$ and the proposed approximation, for the
cases of equal and exponential correlation models, respectively. Tables III
and IV complement Figs. 1 and 2, by presenting the significance levels
$\alpha$ for the corresponding input parameters’ combinations. The Table III
and the Table IV entries reveal the very low significance levels $\alpha$ for
all input parameters’ combinations, thus proving an excellent goodness of fit
in statistical sense.

####  II-D2 Arbitrary correlation

For illustrative purposes, we use same two example correlation matrices from
[References, Sections V.B and V.D], $\Sigma_{3\\_\rm{lin}}$ and
$\Sigma_{4\\_\rm{circ}}$, here denoted as ${\bf{\Lambda_{1}}}$ and
${\bf{\Lambda_{2}}}$, respectively. They are approximated by their Green’s
matrices $C_{3\\_\rm{lin}}$ and $C_{4\\_\rm{circ}}$, here denoted as
${\bf{C_{1}}}$ and ${\bf{C_{2}}}$, respectively.

Using ${\bf{C_{1}}}$ and ${\bf{C_{2}}}$, one obtains the needed tridiagonal
matrices ${\bf{\Delta}}$ and ${\bf{\Psi}}$ from their definitions given by
(II-B). The required joint moments are then calculated from (II-B) and (II-B),
which are then substituted into (II) and (II) to calculate $E[Z^{2}]$ and
$E[Z^{4}]$, and then (16) is used to describe the statistics of $R$.

TABLE V: Significance levels of C-S and K-S tests for goodness of fit between exact and approximative distributions of Fig. 3 | $m_{z}=1$ | $m_{z}=3$
---|---|---
| $\alpha_{CS}$ | $\alpha_{KS}$ | $\alpha_{CS}$ | $\alpha_{KS}$
$\Lambda_{1}$ | $0.19$ | $0.18$ | $<0.001$ | $<0.001$
$\Lambda_{2}$ | $0.12$ | $0.11$ | $<0.001$ | $<0.001$
Figure 3: Exact (obtained by simulation) and the approximative analytical PDFs
to the sum of correlated Nakagami-$m$ RVs with equal average powers, when
$\Omega_{Z}=1$), and correlation matrices ${\bf{\Lambda_{1}}}$ and
${\bf{\Lambda_{2}}}$

Fig. 3 depicts the excellent (visual) match between the histogram obtained
from generated samples of $Z$ and the proposed approximation (11)-(II), for
the two example correlation matrices $\Lambda_{1}$ and $\Lambda_{2}$. Table V
complements Figs. 3, by revealing the very low significance levels
$\alpha_{s}$, thus again proving an excellent goodness of fit.

###  II-E Validation in case of maximal correlation

We now consider the case of maximal correlation coefficient between any pair
of Nakagami-$m$ envelopes $Z_{i}$ and $Z_{j}$, i.e., $\rho_{ij}=1$. It
indicates a perfect linear relationship between these pairs, which, after
applying the model from [References, Eq. (37)], can be defined as
$Z_{i}=\sqrt{\Omega_{i}}\,Z_{0}$ for $1\leq i\leq L$, where $Z_{0}$ is an
auxiliary Nakagami-$m$ RV with unity average power and same fading parameter
$m_{z}$. After replacing the latter expression into (1), $Z$ is transformed
into a Nakagami-$m$ RV with fading parameter $m_{z}$ and average power of

$E[Z^{2}]=\left(\sum_{i=1}^{L}\sqrt{\Omega_{i}}\right)^{2}=\sum_{i=1}^{L}\sum_{j=1}^{L}\sqrt{\Omega_{i}\Omega_{j}}\,,$
(33)

which agrees with (II) when $\rho_{ij}\rightarrow 1$.

Replacing $\rho_{ij}=1$ into (10), the $L-1$ eigenvalues of the matrix
$\bf{\Lambda}$ turn up equal to 0, except $\lambda_{1}=L$. After plugging
these eigenvalues into (5), $R$ is transformed into a Nakagami-$m$ RV with
fading parameter $m_{R}$ and average power $L\Omega_{R}$. After the moment
matching, $\Omega_{R}$ and $m_{R}$ can be obtained from (16), as

$\Omega_{R}=\frac{1}{L}\sum_{i=1}^{L}\sum_{j=1}^{L}\sqrt{\Omega_{i}\Omega_{j}},\quad
m_{R}=\frac{(E[Z^{2}])^{2}}{E[Z^{4}]-(E[Z^{2}])^{2}}=m_{z}\,,$ (34)

respectively, where the latter equality is attributed to the definition of the
Nakagami-$m$ fading parameter, given by [References, Eq. (4)].

Thus, maximal correlation yields (11)-(II) as an accurate distribution of $Z$,
when our moment matching approach is applied. This conclusion further
validates our approach.

##  III Application to the performance analysis of EGC receivers

We now consider a typical $L$-branch EGC diversity receiver exposed to slow
and flat Nakagami-$m$ fading. The envelopes of the branch signals $Z_{k}$ are
non-identical correlated Nakagami-$m$ random processes with PDFs given by (2),
whereas their respective phases are i.i.d. uniform random processes. Each
branch is also corrupted by additive white Gaussian noise (AWGN) with power
spectral density $N_{0}/2$, which is added to the useful branch signal. In the
EGC receiver, the random phases of the branch signals are compensated (co-
phased), equally weighted and then summed together to produce the decision
variable. The envelope of the composite useful signal, denoted by $Z$, is
given by (1), whereas the composite noise power is given by
$\sigma_{EGC}^{2}=LN_{0}/2$, resulting in the instantaneous output SNR given
by

$\gamma_{EGC}=\frac{Z^{2}}{2\sigma_{EGC}^{2}}=\frac{1}{LN_{0}}\left(\sum_{k=1}^{L}Z_{k}\right)^{2}=\left(\sum_{k=1}^{L}G_{k}\right)^{2}$
(35)

where RVs $G_{k}=Z_{k}/\sqrt{LN_{0}}$, $1\leq k\leq L$, form a set of $L$ non-
identical equally correlated Nakagami-$m$ RVs with
$E[G_{k}^{2}]=\bar{\gamma}_{k}/L$, same fading parameters $m_{Z}$ and
correlation coefficient $\rho_{ij}$ between branch pair $(i,j)$. Note that
$\bar{\gamma}_{k}=\Omega_{k}/N_{0}$ denotes the average SNR in $k$-th branch.

Using the results from Section II, the MGF and the PDF of (35) can be
approximated using (5) and (11)-(II), respectively, when $\Omega_{R}$ is
replaced by $\bar{\gamma}_{R}=\Omega_{R}/(LN_{0})$. These approximations are
then used to determine the outage probability $F_{\gamma_{EGC}}$ and the error
probability $\bar{P}_{EGC}$ of an $L$-branch EGC systems in correlated
Nakagami-$m$ fading with high accuracy.

Figure 4: Exact and approximate error probabilities of an EGC receiver with
correlated Nakagami-$m$ branches, when $m_{z}=2$, $\mu=0$ and $\rho=0.7$

###  III-A Outage probability

The outage probability of the EGC system with arbitrary correlated
Nakagami-$m$ fading branches, whose output SNR drops below threshold $t$, is
approximated by the known outage probability expressions of an equivalent MRC
system [References, Eq. (28)], [References, Eq. (13)],

$\displaystyle F_{\gamma_{EGC}}(t)\approx
F_{\gamma_{MRC}}(t)\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle=\frac{1}{2}-\frac{1}{\pi}\int_{0}^{\infty}\frac{\sin\left[m_{R}\sum_{k=0}^{L-1}\arctan\left(x\frac{\Omega_{R}\lambda_{k}}{m_{R}}\right)-xt\right]}{\prod_{k=0}^{L-1}\left[1+\left(x\frac{\Omega_{R}\lambda_{k}}{m_{R}}\right)^{2}\right]^{m_{R}/2}}\frac{dx}{x}$
$\displaystyle=\frac{1}{\Gamma(1+Lm_{R})}\left(\frac{m_{R}}{\Omega_{R}}t\right)^{Lm_{R}}\frac{1}{\det(\Lambda)}\qquad\qquad\qquad\qquad$
$\displaystyle\times\Phi_{2}^{(L)}\Big{(}m_{R},m_{R},\dots,m_{R};1+Lm_{R};\qquad\qquad\qquad$
$\displaystyle\left.-\frac{m_{R}}{\Omega_{R}\lambda_{1}}t,-\frac{m_{R}}{\Omega_{R}\lambda_{2}}t,\dots,-\frac{m_{R}}{\Omega_{R}\lambda_{L}}t\right)\,.$
(36)

For the equal correlation model, (III-A) can be simplified using [References,
Eq. (2.1.3(1))].

###  III-B Average error probability

The average error probability of the correlated Nakagami-$m$ EGC system with
BPSK modulation / coherent demodulation is approximated using the available
expressions for the average error probability of the equivalent MRC systems.
Based on [References, Eq. (9.11)] and [References, Eq. (17)], the error
performance of this EGC system is alternatively approximated as

$\bar{P}_{EGC-BPSK}\approx\bar{P}_{MRC-
BPSK}=\frac{1}{\pi}\int_{0}^{\pi/2}M_{R^{2}}\left(\frac{-1}{\sin^{2}\theta}\right)d\theta$
(37)
$\displaystyle=\frac{\Gamma(Lm_{R}+1/2)}{2\sqrt{\pi}\Gamma(Lm_{R}+1)}\left(\frac{m_{R}}{\Omega_{R}}\right)^{Lm_{R}}\left(\frac{1}{\det(\Lambda)}\right)^{m_{R}}\qquad\quad$
$\displaystyle\times\,F_{D}^{(L)}\Big{(}Lm_{R}+1/2,m_{R},\dots,m_{R};Lm_{R}+1;$
$\displaystyle\left.-\frac{m_{R}}{\Omega_{R}\lambda_{1}},-\frac{m_{R}}{\Omega_{R}\lambda_{2}},\dots,-\frac{m_{R}}{\Omega_{R}\lambda_{L}}\right)\,.$
(38)

Figure 5: Exact and approximate error performance of an EGC receiver with
correlated Nakagami-$m$ branches, when $m_{z}=2$, $\mu=0.3$ and $\rho=0.7$
Figure 6: Exact and approximate error performance of an EGC receiver with
correlated Nakagami-$m$ branches, when correlation is described by correlation
matrices ${\bf{\Lambda_{1}}}$ and ${\bf{\Lambda_{2}}}$, $m_{z}=2$ and $\mu=0$

In (37), $M_{R^{2}}(\cdot)$ is replaced with the MGF given by (5). In (38),
$F_{D}^{(L)}(\cdot)$ denotes the Lauricella $F_{D}$ hypergeometric function of
$L$ variables, defined in [33] and [References, Eq. (18)]. For the equal
correlation model, the average error probability can be calculated using
[References, Eq. (32)], which is a special case of (38).

Note that (38) is here presented to demonstrate existence of an exact closed-
form solution, whereas (37) is much more convenient for accurate and efficient
numerical integration. For example, the average error probability may be
obtained by applying the Gauss-Chebyshev quadrature rule [References, Eq.
(25.4.38)] over (37). In the case of the balanced diversity branches with
equal or exponential correlation, the combination of this quadrature rule with
Tables I and II allows efficient and extremely accurate evaluation of the EGC
performance.

The average error probability of correlated Nakagami-$m$ EGC system with BFSK
modulation / non-coherent demodulation is approximated by known expression of
the equivalent MRC system [References, Eq. (16)], $\bar{P}_{EGC-
BFSK}\approx\bar{P}_{MRC-BFSK}=\frac{1}{2}\,M_{R^{2}}(-\frac{1}{2})$, where
$M_{R^{2}}(\cdot)$ is given by (5).

###  III-C Validation via Monte-Carlo simulations

Next, we illustrate the tightness of the error performance of an correlated
Nakagami-$m$ EGC system with BPSK modulation / coherent demodulation to that
of the equivalent MRC system. The results for the actual EGC system are
obtained by Monte-Carlo simulations, whereas those of the equivalent MRC
system are obtained using (37).

####  III-C1 Equal and exponential branch correlation

Figs. 4 and 5 displays the comparative error performance of the actual EGC and
the equivalent MRC systems, for several combinations of
$(\rho,L,m_{Z},\Omega_{k})$. In order to accommodate unequal average branch
powers (thus, unequal average branch SNRs), we used the exponentially decaying
profile, modelled as

$\Omega_{k}=\Omega_{1}\exp(-\mu(k-1)),\quad 1\leq k\leq L\,,$ (39)

where $\Omega_{1}$ is the average power of branch 1 and $\mu$ is the decaying
exponent, with $\mu=0$ denoting the case of branches with equal power (i.e.,
the balanced branches).

####  III-C2 Arbitrary branch correlation

Fig. 6 depicts the comparative error performances of the EGC with same
correlation matrices from Section II.D, ${\bf{\Lambda_{1}}}$ and
${\bf{\Lambda_{2}}}$, and the equivalent MRC system with respective Green’s
matrices ${\bf{C_{1}}}$ and ${\bf{C_{2}}}$. The high accuracy of our approach
is maintained for arbitrary branch correlations.

##  IV Conclusions

A tight closed-form approximation to the distribution of the sum of correlated
Nakagami-$m$ RVs was introduced for the case of identical and integer fading
parameters. The proposed method approximates this distribution by using the
statistics of the square-root of the sum of statistically independent Gamma
RVs. Examples indicate that the new approximation is highly accurate over the
entire range of abscissas. To demonstrate this more rigorously, the proposed
distribution is tested against the computer generated data by the use of the
Chi-square and the Kologorov-Smirnov goodness-of-fit tests. In case of maximal
correlation, the proposed distribution becomes the exact distribution.

The presented approach allowed to successfully tackle the famous problem of
analytical performance evaluation of an EGC system with arbitrarily correlated
and unbalanced Nakagami-$m$ branches. The significance of the presented
results is underpinned by the existence of a large body of literature dealing
with MRC performance analysis, which permits highly accurate and efficient EGC
performance evaluation.

## Appendix A

Using [References, Eqs. (9.212 (1)) and (7.622 (1))], one has the following
identity

$\displaystyle
J(m,a,p,q)=\frac{1}{\Gamma(m)}\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad$
$\displaystyle\times\int_{0}^{\infty}u^{m-1}e^{-u}{}_{1}F_{1}\left(-\frac{p}{2};m;-au\right){}_{1}F_{1}\left(-\frac{q}{2};m;-au\right)du$
$=(1+a)^{\frac{p}{2}}\left(\frac{1+2a}{1+a}\right)^{\frac{q}{2}}{}_{2}F_{1}\left(m+\frac{p}{2},-\frac{q}{2};m;-\frac{a^{2}}{1+2a}\right)$
(A.1)

Using [References, Eq. (9.212 (3)), pp. 1023] with some simple algebraic
manipulations, the general form (II-A) of the coefficient $W(2,1,1)$ can be
simplified as

$\displaystyle
W(2,1,1)=m_{z}\left(\frac{\Gamma(m_{z}+1/2)}{\Gamma(m_{z})}\right)^{2}\left(\frac{1-\sqrt{\rho}}{1+(N-1)\sqrt{\rho}}\right)^{m_{z}}$
$\displaystyle\times\bigg{[}J(m_{z},a,1,1)+\frac{a(m_{z}+1/2)^{2}}{m_{z}^{2}}\qquad\qquad\qquad\qquad\quad$
$\displaystyle\times\,J(m_{z}+1,a,1,1)+\frac{a}{4m_{z}^{2}}J(m_{z}+1,a,-1,-1)$
$\displaystyle-\frac{a(m_{z}+1/2)^{2}}{m_{z}^{2}}J(m_{z}+1,a,-1,1)\bigg{]}$
(A.2)

where $a=\sqrt{\rho}/(1+(N-1)\sqrt{\rho})$

## Appendix B

The unknown joint moments in (II), $E[Z_{m}^{2}Z_{i}Z_{j}]$,
$E[Z_{m}Z_{i}^{2}Z_{j}]$, $E[Z_{m}Z_{i}Z_{j}^{2}]$ and
$E[Z_{m}Z_{n}Z_{i}Z_{j}]$ can be calculated from [References, Eqs. (11) and
(12)]. Here we derive their simpler and computationally more efficient
alternatives. The alternative to [References, Eq. (12)] is derived directly
from the definition of the joint moment $E[Z_{1}Z_{2}Z_{3}Z_{4}]$,

$\displaystyle
E[Z_{m}Z_{n}Z_{i}Z_{j}]=\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}z_{m}z_{n}z_{i}z_{j}\qquad$
$\displaystyle\times
f_{Z_{m}Z_{n}Z_{i}Z_{j}}(z_{m},z_{n},z_{i},z_{j})dz_{m}dz_{n}dz_{i}dz_{j}\,,$
(B.1)

where joint pdf of four exponentially correlated Nakagami-$m$ RVs is expressed
as [References, Eq. (9)]

$\displaystyle
f_{Z_{m}Z_{n}Z_{i}Z_{j}}(z_{m},z_{n},z_{i},z_{j})\qquad\qquad\qquad\qquad\qquad\quad$
$\displaystyle=\frac{2^{4}m_{z}^{m_{z}+3}|{\bf{\Psi}}|^{m_{z}}}{\Gamma(m_{z})}\frac{z_{m}^{m_{z}}z_{n}z_{i}z_{j}^{m_{z}}}{|\psi_{12}\psi_{23}\psi_{34}|^{m_{z}-1}}\qquad\qquad\quad$
$\displaystyle\times
I_{m_{z}-1}\left(2m_{z}|\psi_{12}|z_{m}z_{n}\right)I_{m_{z}-1}\left(2m_{z}|\psi_{23}|z_{n}z_{i}\right)$
$\displaystyle\times
I_{m_{z}-1}(2m_{z}|\psi_{34}|z_{i}z_{j})\,\exp\Big{(}-m_{z}\big{(}\psi_{11}z_{m}^{2}\qquad$
$\displaystyle+\psi_{22}z_{n}^{2}+\psi_{33}z_{i}^{2}+\psi_{44}z_{j}^{2}\big{)}\Big{)},$
(B.2)

where ${\bf{\Psi}}=[\psi_{i,j}]$ is defined by (32). Now, we integrate
[References, Eq. (11)] over $z_{m}$ and $z_{j}$, respectively obtaining

$\displaystyle\int_{0}^{\infty}z_{m}^{m_{z}+1}\exp\left(-m_{z}\psi_{11}z_{m}^{2}\right)I_{m_{z}-1}\left(2m_{z}\psi_{12}z_{m}z_{n}\right)$
$\displaystyle=\frac{1}{2}\left(m_{z}\psi_{12}z_{n}\right)^{m_{z}-1}\left(m_{z}\psi_{11}\right)^{-(m_{z}+1/2)}\qquad\qquad$
$\displaystyle\times\frac{\Gamma(m_{z}+1/2)}{\Gamma(m_{z})}{}_{1}F_{1}\left(m_{z}+\frac{1}{2};m_{z};m_{z}\frac{\psi_{12}^{2}}{\psi_{11}}z_{n}^{2}\right),$
(B.3)

and

$\displaystyle\int_{0}^{\infty}z_{j}^{m_{z}+1}\exp\left(-m_{z}\psi_{44}z_{j}^{2}\right)I_{m_{z}-1}\left(2m_{z}\psi_{34}z_{i}z_{j}\right)$
$\displaystyle=\frac{1}{2}\left(m_{z}\psi_{34}z_{n}\right)^{m_{z}-1}\left(m_{z}\psi_{44}\right)^{-(m_{z}+1/2)}\qquad\qquad$
$\displaystyle\times\frac{\Gamma(m_{z}+1/2)}{\Gamma(m_{z})}{}_{1}F_{1}\left(m_{z}+\frac{1}{2};m_{z};m_{z}\frac{\psi_{34}^{2}}{\psi_{44}}z_{i}^{2}\right)\,.$
(B.4)

We then use the series expansion of the modified Bessel function of first kind
[References, Eq. (8.445)] that allows to separate the integrations per
variables $z_{n}$ and $z_{i}$, yielding (II-B). A similar procedure yields to
an alternative of [References, Eq. (11)] given by (II-B).

## Acknowledgment

The authors would like to thank the Editor and the anonymous reviewers for
their valuable comments that considerably improved the quality of this paper.

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