The *κ*-*μ* distribution fading model corresponds to a signal composed of clusters of multipath waves, propagating in a nonhomogeneous environment. The phases of the scattered waves are random and have similar delay times, within a single cluster, while delay-time spreads of different clusters are relatively large. It is assumed that the clusters of multipath waves have scattered waves with identical powers, and that each cluster has a dominant component with arbitrary power. This distribution is well suited for LoS applications, since every cluster of multipath waves has a dominant component (with arbitrary power). The *κ*-*μ* distribution is a general physical fading model which includes Rician and Nakagami-*m* fading models as special cases (as the one-sided Gaussian and the Rayleigh distributions) since they also constitute special cases of Nakagami-*m*. *κ* parameter represents the ratio between the total power of dominant components and the total power of scattered components. Parameter *μ* is related to multipath clustering. As *μ* decreases, fading severity increases. For the case of *κ* = 0, the *κ*-*μ* distribution is equivalent to the Nakagami-*m* distribution. When *μ* = 1, the *κ*-*μ* distribution becomes the Rician distribution with *κ* as the Rice factor. Moreover, the *κ*-*μ* distribution fully describes the characteristics of the fading signal in terms of measurable physical parameters [2].

Let us consider macro-diversity system of SC type which consists of two micro-diversity systems with switching between the base stations based on their output signal power values. Each micro-diversity system is of MRC type with an arbitrary number of branches in the presence of *κ*-*μ* fading. The optimal combining technique is MRC [4]. This combining technique involves co-phasing of the useful signal in all branches, multiplication of the received signal in each branch by a weight factor that is proportional to the estimated ratio of the envelope and the power of that particular signal and the summing of the received signals from all antennas. By co-phasing, all the random phase fluctuations of the signal that emerged during the transmission are eliminated. For this process, it is necessary to estimate the phase of the received signal, so this technique requires all the amount of the channel state information of the received signal, and separate receiver chain for each branch of the diversity system, which increases the complexity of the system. In [2, 10], it is shown that the sum of *κ*-*μ* powers is *κ*-*μ* power distributed as well (but with different parameters), which is an ideal choice for MRC analysis. The expression for the pdf of the outputs of MRC micro-diversity systems is as follows [10]:

\begin{array}{ll}\hfill p\left({z}_{i}|{\Omega}_{i}\right)=& \frac{{L}_{i}{\mu}_{i}{\left(1+{\kappa}_{i}\right)}^{\frac{{L}_{i}{\mu}_{i}+1}{2}}}{{\kappa}_{i}^{\frac{{L}_{i}{\mu}_{i}-1}{2}}exp\left({L}_{i}{\mu}_{i}{\kappa}_{i}\right){\left({L}_{i}{\Omega}_{i}\right)}^{\frac{{L}_{i}{\mu}_{i}+1}{2}}}{{z}_{i}}^{\frac{{L}_{i}{\mu}_{i}-1}{2}}exp\left(-\frac{{\mu}_{i}\left(1+{\kappa}_{i}\right){z}_{i}}{{\Omega}_{i}}\right)\phantom{\rule{2em}{0ex}}\\ {I}_{{L}_{i}{\mu}_{i}-1}\left(2{L}_{i}{\mu}_{i}\sqrt{\frac{{\kappa}_{i}\left(1+{\kappa}_{i}\right){z}_{i}}{{L}_{i}{\Omega}_{i}}}\right).\end{array}

(1)

In the previous equation, *I*_{
r
} (·)denotes the *r* th-order modified Bessel function of first kind [[17], eq. 8.445], *μ*_{
i
} and *κ*_{
i
} are well-known *κ*-*μ* fading parameters of each micro-diversity system, while *L*_{
i
} denotes the number of channels at each micro-level.

Since the outputs of a MRC system and their derivatives follow [9]:

{z}_{i}^{2}=\sum _{k=1}^{{L}_{i}}{z}_{ik}^{2}\phantom{\rule{1em}{0ex}}\mathsf{\text{and}}\phantom{\rule{1em}{0ex}}{\u017c}_{i}=\sum _{k=1}^{{L}_{i}}\frac{{z}_{ik}}{{z}_{i}}{\u017c}_{ik},

(2)

then {\u017c}_{i} is a Gaussian random variable with a zero mean:

p({\dot{z}}_{i})=\frac{1}{\sqrt{2\pi}{\dot{\sigma}}_{{z}_{i}}}\mathrm{exp}\left(-\frac{{\dot{z}}_{i}{}^{2}}{2{\dot{\sigma}}_{{z}_{i}}{}^{2}}\right)

(3)

and the variance given with [13]

{{\dot{\sigma}}_{{z}_{i}}}^{2}=\frac{{\sum}_{k=1}^{{N}_{i}}{z}_{ik}^{2}{{\dot{\sigma}}_{{z}_{ik}}}^{2}}{{\sum}_{k=1}^{{N}_{i}}{z}_{ik}^{2}}.

(4)

For the case of equivalently assumed channels {{\dot{\sigma}}_{{z}_{i1}}}^{2}={{\dot{\sigma}}_{{z}_{i2}}}^{2}=\cdots ={{\dot{\sigma}}_{{z}_{ik}}}^{2}, *k* = 1, . . . , *N*, previous reduces into [18]

{{\dot{\sigma}}_{{z}_{i}}}^{2}={{\dot{\sigma}}_{{z}_{ik}}}^{2}=2{\pi}^{2}{f}_{d}^{2}{\Omega}_{i},

(5)

where *f*_{
d
} is a Doppler shift frequency.

Conditioned on Ω_{
i
}, the joint PDF {p}_{{z}_{i},{\u017c}_{i}|{\Omega}_{i}}\left({z}_{i},{\u017c}_{i}|{\Omega}_{i}\right) can be calculated as

\begin{array}{ll}\hfill {p}_{{z}_{i},{\u017c}_{i}|{\Omega}_{i}}\left({z}_{i},{\u017c}_{i}|{\Omega}_{i}\right)& =\frac{{L}_{i}{\mu}_{i}\left(1+{\kappa}_{i}\right)\left(\frac{{L}_{i}{\mu}_{i}+1}{2}\right)}{{\kappa}_{i}^{\frac{\left({L}_{i}{\mu}_{i}-1\right)}{2}}exp\left({L}_{i}{\mu}_{i}{\kappa}_{i}\right){\left({L}_{i}{\Omega}_{i}\right)}^{\frac{{L}_{i}{\mu}_{i}+1}{2}}}{{z}_{i}}^{\frac{{L}_{i}{\mu}_{i}-1}{2}}\\ \times exp\left(-\frac{{\mu}_{i}\left({\kappa}_{i}+1\right){z}_{i}}{{\Omega}_{i}}\right){I}_{{L}_{i}{\mu}_{i}-1}\left(2{L}_{i}{\mu}_{i}\sqrt{\frac{{\kappa}_{i}\left(1+{\kappa}_{i}\right){z}_{i}}{{L}_{i}{\Omega}_{i}}}\right)\\ \phantom{\rule{1em}{0ex}}\times \frac{1}{\sqrt{2\pi}{\dot{\sigma}}_{{z}_{i}}}exp\left(-\frac{{{\u017c}_{i}}^{2}}{2{{\dot{\sigma}}_{{z}_{i}}}^{2}}\right);\phantom{\rule{1em}{0ex}}i=1,2.\end{array}

(6)

It is already quoted that our macro-diversity system is of SC type and that the selection is based on the micro-combiners output signal power values. At the macro-level, this type of selection is used as handoff mechanism, that chooses the best base station to serve mobile, based on the signal power received. The joint probability density of the *Z* and \u017b conditioned on Ω_{1} and Ω_{2}, equals the density of *Z*_{1} and {\u017b}_{1} at *Z* and \u017b for the case when Ω_{1} > Ω_{2}, and equivalently the density of *Z*_{2} and {\u017b}_{2} at *Z* and \u017b for the case when Ω_{2} > Ω_{1}. Now the unconditional joint probability density of the *Z* and \u017b is then obtained by averaging over the joint pdf {p}_{{\Omega}_{1},{\Omega}_{2}}\left({\Omega}_{1},{\Omega}_{2}\right) as

\begin{array}{ll}\hfill {p}_{z,\u017c}\left(z,\u017c\right)=& \underset{0}{\overset{\infty}{\int}}d{\Omega}_{1}\underset{0}{\overset{{\Omega}_{1}}{\int}}{p}_{{z}_{1},{\u017c}_{1}|{\Omega}_{1}}\left(z,\u017c|{\Omega}_{1}\right)\times {p}_{{\Omega}_{1},{\Omega}_{2}}\left({\Omega}_{1},{\Omega}_{2}\right)d{\Omega}_{2}\\ +\phantom{\rule{1em}{0ex}}\underset{0}{\overset{\infty}{\int}}d{\Omega}_{2}\underset{0}{\overset{{\Omega}_{2}}{\int}}{p}_{{z}_{2},{\u017c}_{2}|{\Omega}_{2}}\left(z,\u017c|{\Omega}_{2}\right){p}_{{\Omega}_{1},{\Omega}_{2}}\left({\Omega}_{1},{\Omega}_{2}\right)d{\Omega}_{1}.\end{array}

(7)

Similarly, this selection can be written through the cumulative distribution function (CDF) at the macro-diversity output in the form of

\begin{array}{c}{F}_{z}\left(z\right)\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}\underset{0}{\overset{\infty}{\int}}d{\Omega}_{1}\underset{0}{\overset{{\Omega}_{1}}{\int}}{F}_{{z}_{1}|{\Omega}_{1}}\left(z|{\Omega}_{1}\right)\times {p}_{{\Omega}_{1},{\Omega}_{2}}\left({\Omega}_{1},{\Omega}_{2}\right)d{\Omega}_{2}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\phantom{\rule{1em}{0ex}}\underset{0}{\overset{\infty}{\int}}d{\Omega}_{2}\underset{0}{\overset{{\Omega}_{2}}{\int}}{F}_{{z}_{2}|{\Omega}_{2}}\left(z|{\Omega}_{2}\right)\times {p}_{{\Omega}_{1},{\Omega}_{2}}\left({\Omega}_{1},{\Omega}_{2}\right)d{\Omega}_{1}.\end{array}

(8)

Here *F*(*z*_{
i
} |Ω_{
i
}) defines the CDF of the SNR at the outputs of microdiversity systems given with

F\left({z}_{i}|{\Omega}_{i}\right)=\underset{0}{\overset{{z}_{i}}{\int}}p\left({t}_{i}|{\Omega}_{i}\right)d{t}_{i}.

(9)

Since base stations at the macro-diversity level are widely located, due to sufficient spacing between antennas, signal powers at the outputs of the base stations are modelled as statistically independent. Here long-term fading is as in [7] described with Gamma distributions, which are, as above mentioned, independent as

\begin{array}{c}{p}_{{\Omega}_{1},{\Omega}_{2}}\left({\Omega}_{1},{\Omega}_{2}\right)\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}{p}_{{\Omega}_{1}}\left({\Omega}_{1}\right)\times {p}_{{\Omega}_{2}}\left({\Omega}_{2}\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}\frac{1}{\Gamma \left({c}_{1}\right)}\frac{{\Omega}_{1}^{{c}_{1}-1}}{{\Omega}_{01}^{{c}_{1}}}exp\left(-\frac{{\Omega}_{1}}{{\Omega}_{01}}\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\times \frac{1}{\Gamma \left({c}_{2}\right)}\frac{{\Omega}_{2}^{{c}_{2}-1}}{{\Omega}_{02}^{{c}_{2}}}exp\left(-\frac{{\Omega}_{2}}{{\Omega}_{02}}\right).\end{array}

(10)

In the previous equation, *c*_{1} and *c*_{2} denote the order of Gamma distribution, the measure of the shadowing present in the channels. Ω_{01} and Ω_{02} are related to the average powers of the Gamma long-term fading distributions.