**1.1.1** *κ***-***μ* **channel and system model**

The multipath fading in wireless communications is modelled by several distributions including Nakagami-*m,* Hoyt, Rayleigh, and Rice. By considering important phenomena inherent to radio propagation, *κ-μ* fading model was recently proposed in [14] as a fading model which describes the short-term signal variation in the presence of line-of-sight (LOS) components. This distribution is more realistic than other special distributions, since its derivation is completely based on a non-homogeneous scattering environment. Also *κ-μ* as general physical fading model includes Rayleigh, Rician, and Nakagami-*m* fading models, as special cases [14]. It is written in terms of two physical parameters, *κ* and *μ*. The parameter *κ* is related to the multipath clustering and the parameter *μ* is the ratio between the total power of the dominant components and the total power of the scattered waves. In 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.

The SNR in a *κ*-*μ* fading channel follows the probability density function (pdf) given by [15]:

$$p\_{\boldsymbol{\gamma}}\left(\boldsymbol{\gamma}\right) = \frac{\mu}{k^{(\mu-1)/2}} \frac{(1+k)^{(\mu+1)/2}}{e^{\mu k}} \int\_{\boldsymbol{\gamma}}^{(\mu+1)/2} \boldsymbol{\eta}^{(\mu-1)/2} \, e^{-\mu(1+k)\boldsymbol{\gamma}\cdot\boldsymbol{\bar{\gamma}}} I\_{\mu-1} \left(2\mu \sqrt{\frac{(1+k)k\boldsymbol{\gamma}}{\boldsymbol{\bar{\gamma}}}}\right) . \tag{1.1}$$

In the previous equation, is the corresponding average SNR, while *In*(*x*) denotes the *n*-th order modified Bessel function of first kind [16], and *κ* and *μ* are well-known *κ*-*μ* fading parameters. Using the series representation of Bessel function [16, eq. 8.445]:

$$I\_n(\mathbf{x}) = \sum\_{k=0}^{+n} \frac{\mathbf{x}^{2k+n}}{\mathfrak{Z}^{2k+n} \Gamma(k+n+1)k!},\tag{1.2}$$

the cumulative distribution function (cdf) of *γ* can be written in the form of:

$$F\_{\gamma}(\gamma) = \sum\_{p=0}^{+\pi} \frac{\mu^p \kappa^p}{e^{\mu \kappa} \Gamma(p + \mu)} \Lambda \left( p + \mu, \frac{\mu (1 + \kappa) \gamma}{\bar{\gamma}} \right) \tag{1.3}$$

with Γ(*x*) and Λ(*a*,*x*) denoting Gamma and lower incomplete Gamma function, respectively [16, eqs. 8.310.1, 8.350.1].

It is shown in [15], that the sum of *κ*-*μ* squares is *κ*-*μ* square as well (but with different parameters), which is an ideal choice for MRC analysis. Then the expression for the pdf of the outputs of MRC diversity systems follows [15, eq.11]:

$$p\_{\gamma}^{\text{MRC}}\left(\boldsymbol{\chi}\right) = \frac{L\boldsymbol{\mu}}{k^{(L\mu-1)/2}} \frac{\left(1+k\right)^{(L\mu+1)/2}}{L\boldsymbol{\chi}} \bigg| \qquad \gamma^{(L\mu-1)/2} \; e^{-\mu(1+k)\boldsymbol{\gamma}\cdot\boldsymbol{\Omega}} I\_{L\mu-1}\left(2\boldsymbol{\mu}L\sqrt{\frac{(1+k)k\boldsymbol{\gamma}}{L\boldsymbol{\gamma}}}\right) \tag{1.4}$$

with *L* denoting the number of diversity branches.

The expression for the pdf of the outputs of SC diversity systems can be obtained by substituting expressions (1.1) and (1.3) into:

$$\mathbb{P}\_{\mathcal{I}}^{\rm SC}\left(\boldsymbol{\mathcal{Y}}\right) = \sum\_{i=1}^{L} p\_{\boldsymbol{\gamma}\_{i}}\left(\boldsymbol{\mathcal{Y}}\right) \prod\_{j=1 \atop j \neq i}^{L} F\_{\boldsymbol{\gamma}\_{j}}\left(\boldsymbol{\mathcal{Y}}\right) \tag{1.5}$$

where *pγi*(*γ*) and *Fγi*(*γ*) define pdf and cdf of SNR at input branches respectively and *L*  denotes the number of diversity branches.

#### **1.1.2 Weibull channel and system model**

The above mentioned well-known fading distributions are derived assuming a homogeneous diffuse scattering field, resulting from randomly distributed point scatterers. The assumption of a homogeneous diffuse scattering field is certainly an approximation, because the surfaces are spatially correlated characterizing a nonlinear environment. With the aim to explore the nonlinearity of the propagation medium, a general fading distribution, the Weibull distribution, was proposed. The nonlinearity is manifested in terms of a power parameter *β* > 0, such that the resulting signal intensity is obtained not simply as the modulus of the multipath component, but as the modulus to a certain given power. As *β* increases, the fading severity decreases, while for the special case of *β* = 2 reduces to the

order modified Bessel function of first kind [16], and *κ* and *μ* are well-known *κ*-*μ* fading

( ) 2 ( 1) ! 

2 0

( ) , ( )

*p eIL*

*L*

*MRC L k*

 1 /2

*L L k L*

1 /2 1

The expression for the pdf of the outputs of SC diversity systems can be obtained by

 1 1

*j i*

 

where *pγi*(*γ*) and *Fγi*(*γ*) define pdf and cdf of SNR at input branches respectively and *L* 

The above mentioned well-known fading distributions are derived assuming a homogeneous diffuse scattering field, resulting from randomly distributed point scatterers. The assumption of a homogeneous diffuse scattering field is certainly an approximation, because the surfaces are spatially correlated characterizing a nonlinear environment. With the aim to explore the nonlinearity of the propagation medium, a general fading distribution, the Weibull distribution, was proposed. The nonlinearity is manifested in terms of a power parameter *β* > 0, such that the resulting signal intensity is obtained not simply as the modulus of the multipath component, but as the modulus to a certain given power. As *β* increases, the fading severity decreases, while for the special case of *β* = 2 reduces to the

 *i j <sup>L</sup> <sup>L</sup> SC i j*

*p pF*

with Γ(*x*) and Λ(*a*,*x*) denoting Gamma and lower incomplete Gamma function, respectively

It is shown in [15], that the sum of *κ*-*μ* squares is *κ*-*μ* square as well (but with different parameters), which is an ideal choice for MRC analysis. Then the expression for the pdf of

*n k n k <sup>x</sup> I x*

the cumulative distribution function (cdf) of *γ* can be written in the form of:

 

*p p*

*F p e p*

0

*p*

2

*k n*

parameters. Using the series representation of Bessel function [16, eq. 8.445]:

is the corresponding average SNR, while *In*(*x*) denotes the *n*-th

2

 

1

 

1 /2 1 /

 

 

1 1

*L k k k*

*k e L L*

*kn k* , (1.2)

(1.3)

(1.4)

(1.5) 

 

In the previous equation,

[16, eqs. 8.310.1, 8.350.1].

 substituting expressions (1.1) and (1.3) into:

denotes the number of diversity branches.

**1.1.2 Weibull channel and system model** 

with *L* denoting the number of diversity branches.

 the outputs of MRC diversity systems follows [15, eq.11]:

 well-known Rayleigh distribution. Weibull distribution seems to exhibit good fit to experimental fading channel measurements, for both indoor and outdoor environments.

The SNR in a Weibull fading channel follows the pdf given by [17, eq.14]:

$$p\left(\boldsymbol{\chi}\right) = \frac{\mathcal{B}}{2a\overline{\boldsymbol{\chi}}} \left(\frac{\boldsymbol{\chi}}{a\overline{\boldsymbol{\chi}}}\right)^{\frac{\mathcal{B}}{2}-1} e^{-\left(\frac{\boldsymbol{\chi}}{a\overline{\boldsymbol{\chi}}}\right)^{\mathbb{R}^{12}}} \tag{1.6}$$

In the previous equation, is the corresponding average SNR, *β* is well-known Weibull fading parameter, and *a*= 1/Γ(1+2/*β*).

It is shown in [18,19], that the expression for the pdf of the outputs of MRC diversity systems follows [19, eq.1]:

$$p\_{\gamma}^{\text{MRC}}\left(\boldsymbol{\gamma}\right) = \frac{\beta \boldsymbol{\gamma}^{L\boldsymbol{\beta}/2 - 1}}{2\boldsymbol{\Gamma}\left(L\right)\left(\boldsymbol{\Xi}\overline{\boldsymbol{\gamma}}\right)^{L\boldsymbol{\beta}/2}} e^{-\left(\frac{\boldsymbol{\Gamma}}{\boldsymbol{\Xi}\overline{\boldsymbol{\gamma}}}\right)^{\boldsymbol{\beta}/2}}; \qquad \boldsymbol{\Xi} = \frac{\boldsymbol{\Gamma}\left(L\right)}{\boldsymbol{\Gamma}\left(L + 2\boldsymbol{\beta}/\boldsymbol{\beta}\right)}\tag{1.7}$$

with *L* denoting the number of diversity branches.

Similary, expression for the pdf of the outputs of SC diversity systems can be obtained as (1.5)

### **2. Optimal power and rate adaptation**

In the OPRA protocol the power level and rate parameters vary in response to the changing channel conditions. It achieves the ergodic capacity of the system, i. e. the maximum achievable average rate by use of adaptive transmission. However, OPRA is not suitable for all applications because for some of them it requires fixed rate.

During our analysis it is assumed that the variation in the combined output SNR over *κ*-*μ* fading channels *γ* is tracked perfectly by the receiver and that variation of *γ* is sent back to the transmitter via an error-fee feedback path. Comparing to the rate of channel variation, the time delay in this feedback is negligible. These assumptions allow the transmitter to adopt its power and rate correspondingly to the actual channel state. Channel capacity of the fading channel with received SNR distribution, *pγ*(*γ*), under optimal power and rate adaptation policy, for the case of constant average transmit power is given by [8]:

$$\_{\rho a} = B \bigcap\_{\gamma\_0}^{\alpha} \log\_2 \left( \frac{\gamma}{\gamma\_0} \right) p\_\gamma \left( \gamma \right) d\gamma,\tag{1.8}$$

where *B* (Hz) denotes the channel bandwidth and *γ0* is the SNR cut-off level bellow which transmission of data is suspended. This cut-off level must satisfy the following equation:

$$\int\_{\gamma\_0}^{\gamma} \left( \frac{1}{\gamma\_0} - \frac{1}{\gamma} \right) p\_\gamma \left( \gamma \right) d\gamma = 1,\tag{1.9}$$

Since no data is sent when *γ* < *γ*0, the optimal policy suffers a probability of outage *Pout* equal to the probability of no transmission, given by:

$$P\_{out} = \int\_0^{\gamma\_0} p\_\gamma(\gamma) d\gamma = 1 - \int\_{\gamma\_0}^\alpha p\_\gamma(\gamma) d\gamma \tag{1.10}$$

#### **2.1** *κ***-***μ* **fading channels**

To achieve the capacity in (1.8), the channel fading level must be attended at the receiver as well as at the transmitter. The transmitter has to adapt its power and rate to the actual channel state; when *γ* is large, high power levels and rates are allocated for good channel conditions and lower power levels and rates for unfavourable channel conditions when *γ* is small. Substituting (1.1) into (1.9), we found that the cut-off level must satisfy:

$$\begin{aligned} \sum\_{i=0}^{\sigma} \frac{\left(kL\mu\right)^{i}}{e^{L\mu k} \Gamma(i+L\mu) i!} \left(\frac{1}{\gamma\_{0}} \Lambda\left(L\mu+i, \frac{\mu(1+k)\gamma\_{0}}{\bar{\gamma}}\right) - \\ \frac{\mu(1+k)}{\bar{\gamma}} \Lambda\left(L\mu+i-1, \frac{\mu(1+k)\gamma\_{0}}{\bar{\gamma}}\right) \right) - 1 = 0 \end{aligned} \tag{1.11}$$

Substituting (1.1) into (1.8), we obtain the capacity per unit bandwidth, <*C*>*opra*/*B*, as:

$$\frac{{\left\{C\right\}}\_{\alpha\mu}^{MIC}}{B} = \sum\_{i=0}^{n} \frac{L\,\mu}{k^{(L\mu-1)/2}e^{L\mu k}} \left(\frac{1+k}{L\,\bar{\chi}}\right)^{(L\mu+1)/2} \int\_{\gamma\_{0}}^{\gamma} \log\_{2}\left(\frac{\chi}{\chi\_{0}}\right) \mathcal{I}^{L\mu+i-1} e^{-\mu(1+k)\chi\cdot\bar{\chi}} d\chi \tag{1.12}$$

Now, by making change of variables, , <*C*>*opra*/*B* can be obtained as:

$$\begin{split} \frac{\left\{C\right\}\_{\boldsymbol{\eta}^{\text{nuc}}}^{\text{MRC}} &= \sum\_{i=0}^{n} \frac{\left(L\,\mu k\right)^{i}}{\Gamma\left(i+L\,\mu\right)i!e^{L\mu k}} \left(\int\_{0}^{\upsilon} \log\_{2}\left(\frac{t\,\bar{\boldsymbol{\chi}}}{\mu\left(1+k\right)\boldsymbol{\chi}\_{0}}\right) t^{L\mu+i-1} e^{-t} dt - \\ & \int\_{0}^{\upsilon\_{0}\mu\left(1+k\right)\bar{\boldsymbol{\chi}}\_{0}} \log\_{2}\left(\frac{t\,\bar{\boldsymbol{\chi}}}{\mu\left(1+k\right)\boldsymbol{\chi}\_{0}}\right) t^{L\mu+i-1} e^{-t} dt \right) = \sum\_{i=0}^{n} \frac{\left(L\,\mu k\right)^{i}}{\Gamma\left(i+L\,\mu\right)i!e^{L\mu k}} \left(I\_{1} - I\_{2}\right) \end{split} \tag{1.13}$$

Integral *I*1 can be solved by applying Gauss-Laguerre quadrature formulae:

$$I\_1 = \bigcap\_{0}^{\alpha} f\_1\left(t\right) e^{-t} dt \equiv \sum\_{k=1}^{R} A\_k f\_1\left(t\_k\right); \quad f\_1\left(t\right) = \log\_2 \left(\frac{t\,\bar{\gamma}}{\mu\left(1+k\right)\chi\_0}\right) t^{L\mu + i - 1} \tag{1.14}$$

In the previous equation *Ak* and *tk*, *k*=1,2,…,*R*, are respectively weights and nodes of Laguerre polynomials [20, pp. 875-924].

Similarly, integral *I*2 can be solved by applying Gauss-Legendre quadrature formulae:

$$I\_2 = \left(\frac{\chi\_o \mu (1+k)}{2\bar{\gamma}}\right)^{L\mu + i} \int\_{-1}^{1} f\_2(u) du \equiv \left(\frac{\chi\_o \mu (1+k)}{2\bar{\gamma}}\right)^{L\mu + i} \sum\_{k=1}^{R} B\_k f\_2\left(u\_k\right) \tag{1.15}$$

where *Bk* and *uk*, *k*=1,2,…,*R*, are respectively weights and nodes of Legendre polynomials.

Convergence of infinite series expressions in (1.13) is rapid since we need about 10 terms to be summed in order to achieve accuracy at the 5th significant digit for corresponding values of system parameters.

#### **2.2 Weibull fading channels**

286 Wireless Communications and Networks – Recent Advances

Since no data is sent when *γ* < *γ*0, the optimal policy suffers a probability of outage *Pout* equal

<sup>0</sup>

To achieve the capacity in (1.8), the channel fading level must be attended at the receiver as well as at the transmitter. The transmitter has to adapt its power and rate to the actual channel state; when *γ* is large, high power levels and rates are allocated for good channel conditions and lower power levels and rates for unfavourable channel conditions when *γ* is

1

<sup>1</sup> <sup>1</sup> , !

 

*kL k*

  *L i*

*out P pd pd* (1.10)

<sup>0</sup>

1, 1 0

 

*t e dt I I*

 

(1.14)

(1.13)

1 1 /

*L i k*

1

*Li t*

*L k*

*k*

 

(1.11)

 

 

0

*<sup>B</sup> k e <sup>L</sup>* (1.12)

<sup>0</sup> 0

small. Substituting (1.1) into (1.9), we found that the cut-off level must satisfy:

*e iL i*

*i*

0 0 0

*B i L ie k*

 

Now, by making change of variables, , <*C*>*opra*/*B* can be obtained as:

*L k*

*i*

*MRC <sup>i</sup>*

*MRC L*

*opra*

*<sup>L</sup> L k <sup>i</sup>*

1 /

*opra*

 

*i*

<sup>0</sup> 1 1

*k k L i*

Substituting (1.1) into (1.8), we obtain the capacity per unit bandwidth, <*C*>*opra*/*B*, as:

1 /2 2 0 0

*<sup>C</sup> L k <sup>t</sup> t e dt*

log ! 1

1

log 1 !

0 1 0

*<sup>t</sup> I f t e dt A f t f t <sup>t</sup>*

0 0 0

*L k*

Integral *I*1 can be solved by applying Gauss-Laguerre quadrature formulae:

1 1 11 2

*k k*

*k*

*R*

*k i Li t*

<sup>0</sup>

*t L k*

2 1 2

*k i L ie*

<sup>1</sup>

*t L i*

; log <sup>1</sup>

*i*

2

<sup>1</sup> log 

*<sup>C</sup> L k e d*

1 /2

0 0

to the probability of no transmission, given by:

**2.1** *κ***-***μ* **fading channels** 

Substituting (1.7) in (1.8) integral of the following form need to be solved

$$I = \frac{1}{\ln 2} \int\_{\gamma\_0}^{\gamma} \mathcal{V}^{L\beta/2 - 1} \ln \left( \frac{\mathcal{V}}{\mathcal{I}\_0} \right) e^{-\left(\frac{\mathcal{V}}{\Xi \overline{\gamma}}\right)^{\beta/2}} d\mathcal{V} \cdot \tag{1.16}$$

After making a change of variables *<sup>t</sup>* /2 / <sup>0</sup> and some simple mathematical manipulations, we get:

$$I = \frac{4\chi\_0^{L\beta/2}}{\beta^2 \ln 2} \Big|\_{1}^{\circ} t^{L-1} \ln \left( t \right) e^{-\left(\frac{\gamma\_0}{\Xi \overline{\tau}}\right)^{\theta/2} t} dt \\ \tag{1.17}$$

Furthermore, this integral can be evaluated using partial integration:

$$\int\_{1}^{\alpha} \mu \, d\nu = \lim\_{t \to \nu \nu} (\mu \nu) - \lim\_{t \to 1} (\mu \nu) - \int\_{1}^{\alpha} \nu \, d\mu \tag{1.18}$$

with respect to:

$$d\mu = \ln t; \quad d\upsilon = t^{L-1} e^{-\left(\frac{\mathcal{V}\_0}{\Xi \overline{\mathcal{V}}}\right)^{\delta/2} t} dt \,. \tag{1.19}$$

Performing *L*-1 successive integration by parts [16, eq. 2.321.2], we get

$$\psi = -e^{-mt} \sum\_{p=1}^{L} \frac{(L-1)!}{(L-p)!} \frac{t^{L-p}}{m^p} \tag{1.20}$$

denoting /2 0 / *m* . Substituting (1.20) in (1.18), we see that first two terms tend to zero. Hence, the integral in (1.17) can be solved in closed form using [16, eq 3.381.3]

$$I = \frac{(L-1)!}{m^L} \sum\_{p=0}^{L-1} \frac{\Gamma\left(p, m\right)}{p!} \tag{1.21}$$

with Γ(*a*, *x*) higher incomplete Gamma function [16]. Finaly, <*C*>*opra*/*B* using *L*-branch MRC diversity receiver over Weibull fading channels has this form

$$\frac{\left\langle C \right\rangle\_{opru}^{MRC}}{B} = \frac{2}{\beta \ln 2} \sum\_{p=0}^{L-1} \frac{\Gamma \left(p, m \right)}{p \, !} \, \, \, \, \tag{1.22}$$

#### **3. Constant power with optimal rate adaptation**

With ORA protocol, the transmitter adapts its rate only while maintaining a fixed power level. Thus, this protocol can be implemented at reduced complexity and is more practical than that of optimal simultaneous power and rate adaptation.

The channel capacity, <*C*>*ora* (bits/s) with constant transmit power policy is given by [1]:

$$\left\{\mathbb{C}\right\}\_{ora} = B \bigwedge\_{0}^{v} \log\_{2}{(1+\gamma)} p\_{\gamma}\left(\gamma\right) d\gamma \tag{1.22}$$

#### **3.1** *κ***-***μ* **fading channels**

To achieve the capacity in (1.22), the channel fading level must be attended at the receiver as well as at the transmitter.

After substituting (1.1) into (1.22), by using partial integration:

$$\int\_0^\pi \mu \, d\nu = \lim\_{\gamma \to \infty} (\mu \nu) - \lim\_{\gamma \to 0} (\mu \nu) - \int\_0^\pi \nu \, d\mu \tag{1.23}$$

with respect to:

$$du = \ln(1+\gamma); \qquad du = \frac{d\gamma}{1+\gamma}; \quad d\upsilon = \gamma^{\rho+\mu-1}e^{-\frac{\mu(1+k)\gamma}{\bar{\gamma}}};\tag{1.24}$$

and performing successive integration by parts [16 , eq. 2.321.2], we get

$$\mathbf{v} = e^{-\frac{\mu(1+k)\gamma}{\bar{\gamma}}} \sum\_{q=1}^{p+\mu} \frac{(p+\mu-1)! \gamma^{p+\mu-k}}{(p+\mu-q)!} \left(\frac{\bar{\gamma}}{\mu(1+k)}\right)^q \tag{1.25}$$

By substituting (1.25) in (1.23), we see that first two terms tend to zero. Hence, the integral in (1.23) can be solved in closed form using [16, eq. 3.381.3]. Finaly, <*C*>*ora*/*B* over *κ*-*μ* fading channels has this form:

<sup>1</sup>

 *L*

*L p <sup>L</sup> p m <sup>I</sup>*

diversity receiver over Weibull fading channels has this form

**3. Constant power with optimal rate adaptation** 

**3.1** *κ***-***μ* **fading channels** 

well as at the transmitter.

with respect to:

channels has this form:

than that of optimal simultaneous power and rate adaptation.

After substituting (1.1) into (1.22), by using partial integration:

(1 )

*<sup>p</sup> v e*

 

and performing successive integration by parts [16 , eq. 2.321.2], we get

1

*q*

1! ,

0

with Γ(*a*, *x*) higher incomplete Gamma function [16]. Finaly, <*C*>*opra*/*B* using *L*-branch MRC

 *MRC <sup>L</sup> opra*

With ORA protocol, the transmitter adapts its rate only while maintaining a fixed power level. Thus, this protocol can be implemented at reduced complexity and is more practical

The channel capacity, <*C*>*ora* (bits/s) with constant transmit power policy is given by [1]:

 *ora C B pd* <sup>2</sup> 0 log 1

To achieve the capacity in (1.22), the channel fading level must be attended at the receiver as

0 0 lim ( ) lim( )

 

( 1)!

0

*udv uv uv vdu* (1.23)

> 

 

 

( )! (1 )

*pq k*

*<sup>d</sup> <sup>p</sup> <sup>u</sup> du dv e* (1.24)

(1 )

*k*

(1.25)

 

 

 

<sup>1</sup> ln(1 ); ; ; <sup>1</sup>

*<sup>q</sup> <sup>k</sup> <sup>p</sup> p k*

By substituting (1.25) in (1.23), we see that first two terms tend to zero. Hence, the integral in (1.23) can be solved in closed form using [16, eq. 3.381.3]. Finaly, <*C*>*ora*/*B* over *κ*-*μ* fading

 

 

*C p m*

!

<sup>1</sup>

0 2 ,

 ln 2 ! 

*p*

*m p* (1.21)

*B p* . (1.22)

(1.22)

$$\left\langle C \right\rangle\_{ora} = \frac{B}{\ln 2} \sum\_{p=0}^{\infty} \sum\_{q=1}^{p+\mu} \frac{\mu^{2+\mu-q} \kappa^p \left(1+\kappa\right)^{p+\mu-q} (n-1)!}{e^{\mu \kappa} \gamma} e^{\frac{\mu(1+\kappa)}{\gamma}}$$

$$\Gamma \left( -n+p+\mu, \frac{\mu(1+\kappa)}{\gamma} \right)$$

On the other hand, substituting (1.4) into (1.22) and applying similar procedure, the expression for the <*C*>*ora*/*B* with MRC diversity receiver is derived as:

$$\left\{C\right\}\_{\nu\kappa}^{\text{MRC}} = \frac{B}{\ln 2} \sum\_{p=0}^{\alpha} \sum\_{q=1}^{p+\mu} \frac{\mu^{2p+\mu-q} \kappa^p \left(1+\kappa\right)^{p+\mu-q} (n-1)! L^p}{e^{L\mu\kappa} \bar{\gamma}^{p+\mu-q}} e^{\frac{\mu(1+\kappa)}{\bar{\gamma}}} \tag{1.27}$$
 
$$\Gamma\left(-n+p+\mu L, \frac{\mu(1+\kappa)}{\bar{\gamma}}\right)$$

Convergence of infinite series expressions in (1.26) and (1.27) is rapid, since we need 5-10 terms to be summed in order to achieve accuracy at the 5th significant digit for corresponding values of system parameters.

#### **3.2 Weibull fading channels**

After substituting (1.6) into (1.22), when MRC reception is applied over Weibull fading channel, we can obtain expression for the ORA channel capacity, in the form of:

$$\frac{\left\langle C \right\rangle\_{\rm on}}{B} = \frac{\beta}{2\Gamma\left(L\right)\left(\Xi\overline{\gamma}\right)^{L\beta/2}\ln 2} \int\_0^\eta \gamma^{L\beta/2} \ln\left(1+\gamma\right) e^{-\left(\frac{\overline{\gamma}}{\Xi\overline{\gamma}}\right)^{\beta/2}} d\gamma. \tag{1.28}$$

By expressing the logarithmic and exponential integrands as Meijer's G- functions [21, eqs. 11] and using [22, eq. 07.34.21.0012.01], integral in (1.28) is solved in closed-form:

$$\frac{\left\langle C \right\rangle\_{\rm out}}{B} = \frac{\beta}{2\Gamma\left(L\right)\left(\Xi\overline{\gamma}\right)^{L\beta/2}\ln 2} H\_{2,3}^{3,1}\left(\left(\Xi\overline{\gamma}\right)^{-\beta/2} \begin{vmatrix} (-L\beta/2, \beta/2), (1 - L\beta/2, \beta/2) \\ (0, 1), (-L\beta/2, \beta/2), (-L\beta/2, \beta/2) \end{vmatrix}\right) \tag{1.29}$$

with:

$$H\_{p,q}^{m,n}\left(\mathbf{x}\middle|\frac{(a\_1,\alpha\_1)...(a\_p,\alpha\_p)}{(b\_1,\beta\_q)...(b\_p,\beta\_q)}\right)\tag{1.30}$$

denoting the Fox H function [23].

### **4. Channel inversion with fixed rate**

Channel inversion with fixed rate policy (CIFR protocol) is quite different than the first two protocols as it maintains constant rate and adapts its power to the inverse of the channels fading. CIFR protocol achieves what is known as the outage capacity of the system; that is the maximum constant data rate that can be supported for all channel conditions with some probability of outage. However, the capacity of channel inversion is always less than the capacity of the previous two protocols as the transmission rate is fixed. On the other hand, constant rate transmission is required in some applications and is worth the loss in achievable capacity. CIFR is adaptation technique based on inverting the channel fading. It is the least complex technique to implement assuming that the transmitter on this way adapts its power to maintain a constant SNR at the receiver. Since a large amount of the transmitted power is required to compensate for the deep channel fades, channel inversion with fixed rate suffers a certain capacity penalty compared to the other techniques.

The channel capacity with this technique is derived from the capacity of an AWGN channel and is given in [8]:

$$\left\{C\right\}\_{cjr} = B\log\_2\left(1 + \bigvee\_{0}^{n}\left(p\_\gamma\left(\boldsymbol{\gamma}\right)/\boldsymbol{\gamma}\right)d\boldsymbol{\gamma}\right).\tag{1.31}$$

#### **4.1** *κ***-***μ* **fading channels**

After substituting (1.1) into (1.31), and by using [16, eq. 6.643.2]:

$$\int\_0^{\pi} x^{\mu - \frac{1}{2}} e^{-ax} I\_{2\nu}(2\beta\sqrt{x}) dx = \frac{\Gamma\left(\mu + \nu + \frac{1}{2}\right)}{\Gamma(2\nu + 1)} \beta^{-1} e^{-\frac{\beta^2}{2a}} a^{-\nu} M\_{-\mu, \nu} \left(\frac{\beta^2}{a}\right) \text{(1.32)}$$

where M*<sup>k</sup>*,*<sup>m</sup>*(*z*) is the Wittaker's function, we can obtained expression for the CIFR channel capacity in the form of:

$$\{\mathbb{C}\}\_{cir} = B \log\_2 \left( 1 + \frac{\left(\mu - 1\right)}{e^{-\frac{\mu k}{2}} \left(\frac{1 + k}{k\overline{\gamma}}\right)^{\frac{\mu}{2}} \mathbf{M}\_{1 - \frac{\mu}{2}, \frac{\mu - 1}{2}}(\mu k)}\right). \tag{1.33}$$

Case when MRC diversity is applied can be modelled by:

$$\left\{C\right\}\_{c\neq\tilde{r}}^{MRC} = B\log\_2\left(1 + \frac{\left(L\,\mu - 1\right)}{e^{-\frac{\mu k L}{2}} \left(\frac{1 + k}{kL\tilde{\sqrt{\cdot}}}\right)^{\frac{L\mu}{2}} \mathbf{M}\_{1-\frac{L\mu}{2},\frac{L\mu - 1}{2}}\left(\mu k L\right)}\right). \tag{1.34}$$

Channel inversion with fixed rate policy (CIFR protocol) is quite different than the first two protocols as it maintains constant rate and adapts its power to the inverse of the channels fading. CIFR protocol achieves what is known as the outage capacity of the system; that is the maximum constant data rate that can be supported for all channel conditions with some probability of outage. However, the capacity of channel inversion is always less than the capacity of the previous two protocols as the transmission rate is fixed. On the other hand, constant rate transmission is required in some applications and is worth the loss in achievable capacity. CIFR is adaptation technique based on inverting the channel fading. It is the least complex technique to implement assuming that the transmitter on this way adapts its power to maintain a constant SNR at the receiver. Since a large amount of the transmitted power is required to compensate for the deep channel fades, channel inversion

with fixed rate suffers a certain capacity penalty compared to the other techniques.

After substituting (1.1) into (1.31), and by using [16, eq. 6.643.2]:

<sup>2</sup> (2 ) (2 1)

*cifr <sup>k</sup>*

2

log 1

2

The channel capacity with this technique is derived from the capacity of an AWGN channel

<sup>2</sup> 1 2 <sup>2</sup> <sup>1</sup> <sup>2</sup> <sup>2</sup> ,

where M*<sup>k</sup>*,*<sup>m</sup>*(*z*) is the Wittaker's function, we can obtained expression for the CIFR channel

*k*

 

1

1

*kL*

 

*k*

*cifr <sup>L</sup> kL*

<sup>1</sup> log 1

 *<sup>x</sup> <sup>v</sup> <sup>v</sup> x e I x dx e M* (1.32)

 

<sup>2</sup> 0 log 1 1 /

1

 

*<sup>k</sup> e k*

1

 

*L L*

<sup>2</sup> <sup>2</sup> <sup>1</sup> 1 , 2 2

*e kL*

*L*

 

 

<sup>2</sup> <sup>2</sup> <sup>1</sup> 1 , 2 2

 

 

 

> >

 

. (1.33)

. (1. 34)

*cifr CB p d* . (1.31)

**4. Channel inversion with fixed rate** 

and is given in [8]:

**4.1** *κ***-***μ* **fading channels** 

capacity in the form of:

*C B*

*MRC*

*C B*

Case when MRC diversity is applied can be modelled by:

0

 Similarly, after substituting (1.5) into (1.31), with respect to [16, eqs. 8.531, 7.552.5, 9.14]:

$$
\Lambda\left(a,\mathbf{x}\right) = \frac{\mathbf{x}^a}{a} e^{-\mathbf{x}}\,\_1F\_1(\mathbf{l}; \mathbf{l}+a; \mathbf{x})\tag{1.35}
$$

$$\int\_0^n e^{-x} x^{z-1} \,\_pF\_q\left(a\_1, \dots, a\_p; b\_1, \dots, b\_q; ax\right) d\mathbf{x} = \Gamma(\mathbf{s}) \,\_{p+1}F\_q\left(\mathbf{s}, a\_1, \dots, a\_p; b\_1, \dots, b\_q; ax\right) \tag{1.36}$$

$$\,\_1F\_1(a;b;\boldsymbol{\chi}) = \sum\_{k=0}^{n} \frac{(a)\_k \,\boldsymbol{\chi}^k}{(b)\_k \,k\,\,!\,} \tag{1.37}$$

expressions for the CIFR channel capacity over *κ*-*μ* fading with SC diversity applied for dual and triple branch combining at the receiver can be obtained in the form of:

$$\begin{aligned} \left\langle C \right\rangle\_{cip}^{SC-2} &= B \log\_2 \left( 1 + \sqrt{\sum\_{p=0}^{\infty} \sum\_{q=0}^{\infty} f\_1} \right) \\ f\_1 &= \frac{\mu^{P+q+1} \kappa^{P+q} (1+\kappa) \Gamma(p+q+2\mu-1)}{2^{P+q+2\mu-2} e^{2\mu\kappa} \sqrt{\Gamma(p+\mu)} p! \Gamma(q+\mu) q! (q+\mu)} \\ &\quad \,\_2F\_1 \left( \begin{matrix} p+q+2\mu-1, 1; 1+q+\mu; \frac{1}{2} \\ p+q+2\mu-1, 1; 1+q+\mu; \frac{1}{2} \end{matrix} \right) \end{aligned} \tag{1.38}$$
 
$$\begin{aligned} \left\langle C \right\rangle\_{cip}^{SC-3} &= B \log\_2 \left( 1 + \sqrt{\sum\_{p=0}^{\infty} \sum\_{q=0}^{\infty} f\_2} \right) \\ f\_2 &= \frac{\mu^{P+q+r+1} \kappa^{P+q+r} (1+\kappa) \Gamma(p+q+r+s+3\mu-1)}{3^{P+q+r+s+3\mu-1} \zeta^{\lambda p} \gamma \Gamma(p+\mu) p! (q+\mu) \Gamma(r+\mu) \Gamma(r+\mu) (1+r+\mu)\_s} \\ &\quad \times \,\_2F\_1 \left[ \begin{matrix} p+q+r+s+3\mu-1, 1+q+\mu \end{matrix} \right] \end{aligned} \tag{1.39}$$

Number of terms that need to be summed in (1.38) and (1.39) to achieve accuracy at 5th significant digit for some values of system parameters is presented in Table 1 in the section Numerical results.

#### **4.2 Weibull fading channels**

After substituting (1.6) into (1.31) we can obtain expression for the CIFR channel capacity when MRC diversity is applied in the form of:

$$\frac{\left\langle C \right\rangle\_{cjr}^{\text{MRC}}}{B} = \log\_2 \left( 1 + \frac{\left( \Xi \overline{\gamma} \right) \Gamma \left( L \right)}{\Gamma \left( L - 2 \mid \beta \right)} \right)\_\cdot \tag{1.40}$$
