**5. Truncated channel inversion with fixed rate**

The channel inversion and truncated inversion policies use codes designed for AWGN channels, and are therefore the least complex to implement, but in severe fading conditions they exhibit large capacity losses relative to the other techniques.

The truncated channel inversion policy inverts the channel fading only above a fixed cutoff fade depth *γ*0. The capacity with this truncated channel inversion and fixed rate policy <*C*>*tifr*/*B* is derived in [8]:

$$\left\{ C \right\}\_{\text{njr}} = B \log\_2 \left( 1 + 1 \right) \left\langle \prod\_{\mathcal{I}\_0}^n \left( p\_\gamma \left( \mathcal{\gamma} \right) / \mathcal{\gamma} \right) d \, \mathcal{\gamma} \right\rangle \left( 1 - P\_{out} \right) \,. \tag{1.41}$$

#### **5.1** *κ***-***μ* **fading channels**

After substituting (1.2) into (1.40) we can obtain expression for the CIFR channel capacity over *κ*-*μ* fading channel in the following form:

$$\begin{split} \left\{ \left< C \right>\_{ijr} &= B \log\_2 \left( 1 + \mathbf{l} \Big/ \sum\_{p=0}^{\nu} f\_3 \right) \bigg| \left( 1 - \sum\_{i=0}^{\nu} \frac{\left( k \, \mu \right)^i}{e^{\mu k} \Gamma \left( i + \mu \right) i!} \Lambda \left( \mu + i, \frac{\mu \left( 1 + k \right) \chi\_0}{\gamma} \right) \right) \\ &= \sum\_{p=0}^{\nu} \frac{\mu^{p+1} \kappa^p \left( 1 + \kappa \right) \Lambda \left( p + \mu - 1, \frac{\mu \left( 1 + k \right) \chi\_0}{\gamma} \right)}{e^{\mu \kappa} \, \bar{\chi} \Gamma (p + \mu) p!} \end{split} \tag{1.42}$$

Case when MRC diversity is applied can be modelled by:

$$\begin{split} \left\{ C \right\}\_{\boldsymbol{\nu}}^{\boldsymbol{M} \text{RC}} &= B \log\_{2} \left( 1 + 1 \Big/ \sum\_{\nu=0}^{\boldsymbol{\nu}} f\_{4} \right) \Big| \left( 1 - \sum\_{\nu=0}^{\boldsymbol{\nu}} \frac{\left( kL \boldsymbol{\mu} \right)^{\boldsymbol{\ell}}}{e^{L\boldsymbol{\mu} k} \Gamma \left( \boldsymbol{i} + L\boldsymbol{\mu} \right) i!} \Lambda \left( \mu L + i, \frac{\mu L \left( \boldsymbol{1} + \boldsymbol{k} \right) \boldsymbol{\nu}\_{0}}{\bar{\boldsymbol{\gamma}}} \right) \right) \\ \boldsymbol{f}\_{4} &= \sum\_{p=0}^{\boldsymbol{\nu}} \frac{\left( \mu^{p+1} \kappa^{p} \left( 1 + \kappa \right) L^{p} \Lambda \left( \bar{\boldsymbol{p}} + \mu L \left( \boldsymbol{1} + \boldsymbol{k} \right) \boldsymbol{\nu}\_{0} \right) \right)}{\bar{\boldsymbol{\gamma}}} \\ &\leq \frac{e^{\boldsymbol{\mu} \boldsymbol{\nu} L} \bar{\boldsymbol{\gamma}}^{\boldsymbol{\nu}} \Gamma \left( \boldsymbol{p} + \mu L \right) p!}{e^{\boldsymbol{\mu} \boldsymbol{\nu} L} \bar{\boldsymbol{\gamma}}^{\boldsymbol{\ell}} \Gamma \left( \boldsymbol{p} + \mu L \right) p!} \end{split} (1.43)$$

Convergence of infinite series expressions in (1.42) and (1.43) is rapid, since we need about 10-15 terms to be summed in order to achieve accuracy at the 5th significant digit.

#### **5.2 Weibull fading channels**

After substituting (1.6) into (1.41) we can obtained expression for the CIFR channel capacity over Weibull fading channels when MRC diversity is applied in the form of:

$$\frac{{\{C\}}\_{\boldsymbol{\eta}\boldsymbol{\tau}}^{\rm{MRC}}}{B} = \log\_{2}\left(1 + \frac{\Xi\overline{\boldsymbol{\gamma}}\boldsymbol{\Gamma}(L)}{\Gamma\left(L - 2\left(\boldsymbol{\beta}, \left(\boldsymbol{\gamma}\_{0}/\Xi\overline{\boldsymbol{\gamma}}\right)^{\boldsymbol{\beta}/2}\right)\right)}\right) \frac{\Gamma\left(L, \left(\boldsymbol{\gamma}\_{0}/\Xi\overline{\boldsymbol{\gamma}}\right)^{\boldsymbol{\beta}/2}\right)}{\Gamma(L)}\cdot\tag{1.44}$$
