We consider the *L*-branch spatial diversity combining system shown in Figure 1, in which it is assumed that the received signals *x*_{
l
} (*t*) (*l* = 1, 2, ..., *L*) at the combiner input experience flat fading in all branches. The transmitted signal is represented by *s*(*t*), while the total transmitted power per symbol is denoted by *P*_{
s
} . The complex random channel gain of the *l* th diversity branch is denoted by {\u0125}_{l}\left(t\right) and *n*_{
l
} (*t*) designates the corresponding additive white Gaussian noise (AWGN) component with variance *N*_{0}. The relationship between the transmitted signal *s*(*t*) and the received signals *x*_{
l
} (*t*) at the combiner input can be expressed as

x\left(t\right)=\widehat{h}\left(t\right)s\left(t\right)+n\left(t\right)

(1)

where **x**(*t*), \widehat{h}\left(t\right), and **n**(*t*) are *L ×* 1 vectors with entries corresponding to the *l* th (*l* = 1, 2, ..., *L*) diversity branch denoted by *x*_{
l
} (*t*), {\u0125}_{l}\left(t\right), and *n*_{
l
} (*t*), respectively. The spatial correlation between the diversity branches arises due to the spatial correlation between closely located receiver antennas in the antenna array. The correlation matrix **R**, describing the correlation between diversity branches, is given by R=E\left[\widehat{h}\left(t\right){\widehat{h}}^{H}\left(t\right)\right], where (·) ^{H} represents the Hermitian operator. Using the Kronecker model, the channel vector \widehat{h}\left(t\right) can be expressed as \widehat{h}\left(t\right)={R}^{\frac{1}{2}}h\left(t\right)[34]. Here, the entries of the *L ×* 1 vector **h**(*t*) are mutually uncorrelated with amplitudes and phases given by *|h*_{
l
} (*t*)*|* and *ϕ*_{
l
} , respectively. We have assumed that the phases *ϕ*_{
l
} (*l* = 1, 2, ..., *L*) are uniformly distributed over (0, 2*π*], while the envelopes *ζ*_{
l
} (*t*) = *|h*_{
l
} (*t*)*|* (*l* = 1, 2, ..., *L*) follow the Nakagami-*m* distribution {p}_{{\zeta}_{l}}\left(z\right) given by [13]

{p}_{\zeta \iota}\left(z\right)=\frac{2{m}_{l}^{{m}_{l}}{z}^{2{m}_{l}-1}}{\Gamma \left({m}_{l}\right){\Omega}_{l}^{{m}_{l}}}{\mathsf{\text{e}}}^{-\frac{{m}_{l}{z}^{2}}{{\Omega}_{l}}},\phantom{\rule{1em}{0ex}}z\ge 0

(2)

where {\Omega}_{l}=E\left\{{\zeta}_{l}^{2}\left(t\right)\right\}, {m}_{l}={\Omega}_{l}^{2}\u2215\mathsf{\text{Var}}\left\{{\zeta}_{l}^{2}\left(t\right)\right\}, and Γ(·) represents the gamma function [35]. Here, *E*{·} and Var{·} denote the mean (or the statistical expectation) and variance operators, respectively. The parameter *m*_{
l
} controls the severity of the fading. Increasing the value of *m*_{
l
} decreases the severity of fading associated with the *l* th branch and vice versa.

The eigenvalue decomposition of the correlation matrix **R** can be expressed as **R** = **UΛU**^{H} . Here, **U** consists of the eigenbasis vectors at the receiver and the diagonal matrix **Λ** comprise the eigenvalues *λ*_{
l
} (*l* = 1, 2, ..., *L*) of the correlation matrix **R**. The receiver antenna correlations *ρ*_{
p
}_{,}_{
q
} (*p*, *q* = 1, 2, ..., *L*) under isotropic scattering conditions can be expressed as *ρ*_{
p
}_{,}_{
q
} = *J*_{0} (*b*_{
pq
} ) [36], where *J*_{0}(·) is the Bessel function of the first kind of order zero [35] and *b*_{
pq
} = 2*πδ*_{
pq
} /*λ*. Here, *λ* is the wavelength of the transmitted signal, whereas *δ*_{
pq
} represents the spacing between the *p* th and *q* th receiver antennas. In this article, we have considered a uniform linear array with adjacent receiver antennas separation represented by *δ*_{
R
} . Increasing the value of *δ*_{
R
} decreases the spatial correlation between the diversity branches and vice versa. It is worth mentioning here that the analysis presented in this article is not restricted to any specific receiver antenna correlation model, such as given by *J*_{0}(·), for the description of the correlation matrix **R**. Therefore, any receiver antenna correlation model can be used as long as the resulting correlation matrix **R** has the eigenvalues *λ*_{
l
} (*l* = 1, 2, ..., *L*).

### 2.1 Spatially correlated Nakagami-*m* channels with MRC

In MRC, the combiner computes y\left(t\right)={\widehat{h}}^{H}\left(t\right)x\left(t\right), and hence, the instantaneous SNR *γ*(*t*) at the combiner output in an MRC diversity system with correlated diversity branches can be expressed as [1, 22]

\gamma \left(t\right)=\frac{{P}_{s}}{{N}_{0}}{\hat{h}}^{H}\left(t\right)\hat{h}\left(t\right)=\frac{{P}_{s}}{{N}_{0}}\sum _{l=1}^{L}{\lambda}_{l}{\zeta}_{l}^{2}\left(t\right)={\gamma}_{s}\Xi \left(t\right)

(3)

where *γ*_{
s
} = *P*_{
s
} /*N*_{0} can be termed as the average SNR of each branch, \Xi \left(t\right)={\sum}_{l=1}^{L}{\stackrel{\u0301}{\zeta}}_{l}^{2}\left(t\right), and {\stackrel{\u0301}{\zeta}}_{l}\left(t\right)=\sqrt{{\lambda}_{l}}{\zeta}_{l}\left(t\right). It is worth mentioning that although we have employed the Kronecker model, the study in [22] reports that (3) holds for any arbitrary correlation model, as long as the correlation matrix **R** is non-negative definite. It is also shown in [22] that despite the diversity branches are spatially correlated, the instantaneous SNR *γ*(*t*) at the combiner output of an MRC system can be expressed as a sum of weighted statistically independent gamma variates {\zeta}_{l}^{2}\left(t\right), as given in (3). The PDF {p}_{{\stackrel{\u0301}{\zeta}}_{l}^{2}}\left(z\right) of processes {\stackrel{\u0301}{\zeta}}_{l}^{2}\left(t\right) follows the gamma distribution with parameters *α*_{
l
} = *m*_{
l
} and {\stackrel{\u0301}{\beta}}_{l}={\lambda}_{l}{\Omega}_{l}\u2215{m}_{l} [[37], Equation 1]. Therefore, the process Ξ(*t*) can be considered as a sum of weighted independent gamma variates. As a result, the PDF *p*_{Ξ} (*z*) of the process Ξ(*t*) can be expressed using [[37], Equation 2] as

{p}_{\Xi}\left(z\right)=\prod _{l=1}^{L}{\left(\frac{{\stackrel{\u0301}{\beta}}_{1}}{{\beta}_{l}}\right)}^{{\alpha}_{l}}\sum _{k=0}^{\infty}\frac{{\epsilon}_{k}{z}^{{\sum}_{l=1}^{L}{\alpha}_{l}+k-1}{\mathsf{\text{e}}}^{-z\u2215{\stackrel{\u0301}{\beta}}_{1}}}{{{\stackrel{\u0301}{\beta}}_{1}}^{{\sum}_{l=1}^{L}{\alpha}_{l}+k}\Gamma \begin{array}{c}\hfill \left({\sum}_{l=1}^{L}{\alpha}_{l}+k\right)\hfill \\ \hfill z\ge 0\hfill \end{array}},

(4)

where

\begin{array}{ll}\hfill {\epsilon}_{k+1}=\frac{1}{k+1}\sum _{i=1}^{k+1}\left[\sum _{l=1}^{L}{\alpha}_{l}{\left(1-\frac{{\stackrel{\u0301}{\beta}}_{1}}{{\stackrel{\u0301}{\beta}}_{l}}\right)}^{l}\right]& \phantom{\rule{2.77695pt}{0ex}}{\epsilon}_{k+1-l},\phantom{\rule{2em}{0ex}}\\ k=0,1,2...\phantom{\rule{2em}{0ex}}\end{array}

(5)

*ε*_{0} = 1, and {\stackrel{\u0301}{\beta}}_{1}=\underset{l}{min}\left\{{\stackrel{\u0301}{\beta}}_{l}\right\}\left(l=1,2,\dots ,L\right).

When using MRC, if the diversity branches are uncorrelated having identical Nakagami-*m* parameters (i.e., when in (3) *λ*_{
l
} = 1 (*l* = 1, 2, ..., *L*), *α*_{1} = *α*_{2} = ⋯ = *α*_{
L
} = *α*, and {\stackrel{\u0301}{\beta}}_{1}={\stackrel{\u0301}{\beta}}_{2}=\cdots ={\stackrel{\u0301}{\beta}}_{L}=\beta ), it is shown in [16] that the joint PDF {p}_{\Xi \dot{\Xi}}\left(z,\u017c\right) of Ξ(*t*) and its time derivative \dot{\Xi}\left(t\right) at the same time *t*, under the assumption of isotropic scattering, can be written with the help of the result reported in [[16], Equation 35] as

{p}_{\Xi \dot{\Xi}}\left(z,\u017c\right)={p}_{\Xi}\left(z\right)\frac{1}{\sqrt{2\pi {\sigma}_{\dot{\Xi}}^{2}}}{\mathsf{\text{e}}}^{-\frac{{\u017c}^{2}}{2{\sigma}_{\dot{\Xi}}^{2}}},\phantom{\rule{1em}{0ex}}z\ge 0,\phantom{\rule{2.77695pt}{0ex}}|\u017c|<\infty

(6)

where {\sigma}_{\dot{\Xi}}^{2}=4{\beta}_{x}z{\left(\pi {f}_{\mathsf{\text{max}}}\right)}^{2}, *f*_{max} is the maximum Doppler frequency, and *β*_{
x
} can be expressed as a ratio of the variance and the mean of the sum process Ξ(*t*), i.e., *β*_{
x
} = Var{Ξ(*t*)}/*E*{Ξ(*t*)}. Therefore, for uncorrelated diversity branches with identical parameters {*α* = *m*, *β* = Ω*/m*}, *β*_{
x
} = *β*. On the other hand, when the diversity branches are spatially correlated, *λ*_{
l
} ≠ 1 (*l* = 1, 2, ..., *L*) as well as the eigenvalues are all distinct. Moreover, we have also considered that the parameters {*m*_{
l
} , Ω_{
l
}} (and therefore \{{\alpha}_{l},{\dot{\beta}}_{l}\}) are non-identical. However, as given by (3), even when the diversity branches are spatially correlated and have non-identical parameters, the process Ξ(*t*) is still expressed using a sum of statistically independent gamma variates, similar to the uncorrelated scenario considered in [16] to obtain (6). Hence, in our case, we follow a similar approach as in [16], i.e., by assuming that (6) is also valid for the process \Xi \left(t\right)={\sum}_{l=1}^{L}{\stackrel{\u0301}{\zeta}}_{l}^{2}\left(t\right) with parameters \{{\alpha}_{l},{\stackrel{\u0301}{\beta}}_{l}\}) and finding appropriate value of {\sigma}_{\dot{\Xi}}^{2}. The results show that (6) holds for the process Ξ(*t*) if the parameter *β*_{
x
} in {\sigma}_{\dot{\Xi}}^{2} is chosen according to {\beta}_{x}={\sum}_{l=1}^{L}\left({\alpha}_{l}{\stackrel{\u0301}{\beta}}_{l}^{2}\right)\u2215{\sum}_{l=1}^{L}\left({\alpha}_{l}{\stackrel{\u0301}{\beta}}_{l}\right). In Section 3, we will use the results presented in (4) and (6) to obtain the statistical properties of the capacity of Nakagami-*m* channels with MRC.

### 2.2 Spatially correlated Nakagami-*m* channels with EGC

In EGC, the combiner computes **y**(*t*) = *ϕ*^{H}**x**(*t*) [4], where *ϕ* = [*ϕ*_{1}*ϕ*_{2}, ..., *ϕ*_{
L
} ]^{T} and (·)^{T} denotes the vector transpose operator. Therefore, the instantaneous SNR *γ* (*t*) at the combiner output in an *L*-branch EGC diversity system with correlated diversity branches can be expressed as [1, 4, 38]

\gamma \left(t\right)=\frac{{P}_{s}}{L{N}_{0}}{\left(\sum _{l=1}^{L}\sqrt{{\lambda}_{l}}{\zeta}_{l}\left(t\right)\right)}^{2}=\frac{{\gamma}_{s}}{L}\Psi \left(t\right)

(7)

where \Psi \left(t\right)={\left({\sum}_{l=1}^{L}{\stackrel{\u0301}{\zeta}}_{l}\left(t\right)\right)}^{2}, while here the processes {\stackrel{\u0301}{\zeta}}_{l}\left(t\right) follow the Nakagami-*m* distribution with parameters *m*_{
l
} and {\stackrel{\u0301}{\Omega}}_{l}={\lambda}_{l}\phantom{\rule{0.3em}{0ex}}{\Omega}_{l}. Again we proceed by first finding the PDF *p*_{Ψ}(*z*) of the process Ψ(*t*) as well as the joint PGF {p}_{\Psi \dot{\Psi}}\left(z,\u017c\right) of the process Ψ(*t*) and its time derivative \dot{\Psi}\left(t\right). However, the exact solution for the PDF of a sum of Nakagami-*m* processes {\sum}_{l=1}^{L}{\stackrel{\u0301}{\zeta}}_{l}\left(t\right) cannot be obtained. One of the solutions to this problem is to use an appropriate approximation to the sum {\sum}_{l=1}^{L}{\stackrel{\u0301}{\zeta}}_{l}\left(t\right) to find the PDF *p*_{Ψ}(*z*) (see, e.g., [13] and [39]). In this article, we have approximated the sum of Nakagami-*m* processes {\sum}_{l=1}^{L}{\stackrel{\u0301}{\zeta}}_{l}\left(t\right) by another Nakagami-*m* process *S*(*t*) with parameters *m*_{
S
} and Ω_{
S
}, as suggested in [39]. Hence, the PDF *p*_{
S
} (*z*) of *S*(*t*) can be obtained by replacing *m*_{
l
} and Ω _{
l
} in (2) by *m*_{
S
} and Ω_{
S
}, respectively, where Ω _{
S
} = *E*{*S*^{2}(*t*)} and {m}_{S}={\Omega}_{S}^{2}\u2215\left(E\left\{{S}^{4}\left(t\right)\right\}-{\Omega}_{S}^{2}\right). The *n* th-order moment *E* {*S*^{n} (*t*)} can be calculated using [39]

\begin{array}{ll}\hfill E\left\{{S}^{n}\left(t\right)\right\}=& \sum _{{n}_{1}=0}^{n}\sum _{{n}_{2}=0}^{{n}_{1}}\cdot \cdot \cdot \sum _{{n}_{L-1}=0}^{{n}_{L-2}}\left(\begin{array}{c}\hfill n\hfill \\ \hfill {n}_{1}\hfill \end{array}\right)\left(\begin{array}{c}\hfill {n}_{1}\hfill \\ \hfill {n}_{2}\hfill \end{array}\right)\dots \left(\begin{array}{c}\hfill {n}_{L-2}\hfill \\ \hfill {n}_{L-1}\hfill \end{array}\right)\phantom{\rule{2em}{0ex}}\\ \times E\left\{{\stackrel{\u0301}{\zeta}}_{1}^{n-{n}_{1}}\left(t\right)\right\}E\left\{{\stackrel{\u0301}{\zeta}}_{2}^{{n}_{1}-{n}_{2}}\left(t\right)\right\}\dots E\left\{{\stackrel{\u0301}{\zeta}}_{L}^{{n}_{L-1}}\left(t\right)\right\}\phantom{\rule{2em}{0ex}}\end{array}

(8)

where \left(\begin{array}{c}\hfill {n}_{i}\hfill \\ \hfill {n}_{j}\hfill \end{array}\right), for *n*_{
j
} ≤ *n*_{
i
} , denotes the binomial coefficient and

E\left\{{\stackrel{\u0301}{\zeta}}_{l}^{n}\left(t\right)\right\}=\frac{\Gamma \left({m}_{l}+n\u22152\right)}{\Gamma \left({m}_{l}\right)}{\left(\frac{{\stackrel{\u0301}{\Omega}}_{l}}{{m}_{l}}\right)}^{n\u22152},\phantom{\rule{1em}{0ex}}l=1,2,\dots ,L.

(9)

By using this approximation for the PDF of a sum {\sum}_{l=1}^{L}{\stackrel{\u0301}{\zeta}}_{l}\left(t\right) of Nakagami-*m* processes and applying the concept of transformation of random variables [[40], Equations 7-8], the PDF *p*_{Ψ}(*z*) of the squared sum of Nakagami-*m* processes Ψ(*t*) can be expressed using {p}_{\Psi}\left(z\right)=1\u2215\left(2\sqrt{z}\right)\phantom{\rule{0.3em}{0ex}}{p}_{S}\phantom{\rule{0.3em}{0ex}}\left(\sqrt{z}\right) as

{p}_{\Psi}\left(z\right)\approx \frac{{m}_{S}^{{m}_{S}}{z}^{{m}_{S}-1}}{\Gamma \left({m}_{S}\right){\Omega}_{S}^{{m}_{S}}}{\mathsf{\text{e}}}^{-\frac{{m}_{{S}^{Z}}}{{\Omega}_{S}}},\phantom{\rule{1em}{0ex}}z\ge 0.

(10)

The joint PDF {p}_{\Psi \dot{\Psi}}\left(z,\u017c\right) can now be expressed with the help of [[16], Equation 19], (10) and by using the concept of transformation or random variables [[40], Equations 7-8] as

{p}_{\Psi \dot{\Psi}}\left(z,\u017c\right)\approx \frac{{\mathsf{\text{e}}}^{-\frac{{\u017c}^{2}}{2{\sigma}_{\dot{\Psi}}^{2}}}}{\sqrt{2\pi {\sigma}_{\dot{\Psi}}^{2}}}{p}_{\Psi}\left(z\right),\phantom{\rule{1em}{0ex}}z\ge 0,\phantom{\rule{2.77695pt}{0ex}}|\u017c|<\infty

(11)

where {\sigma}_{\dot{\Psi}}^{2}=4z{\left(\pi {f}_{\mathsf{\text{max}}}\right)}^{2}{\sum}_{l=1}^{L}\left({\stackrel{\u0301}{\Omega}}_{l}\u2215{m}_{l}\right). Using (10) and (11), the statistical properties of the capacity of Nakagami-*m* channels with EGC will be obtained in the next section.