## 3 Transformation to Equivalent Concave Programming  
The fractional programming problem in Equation (3) can be expressed in the following generalized form [36]:  
\[\max_{p\in\mathcal{P}} \frac{f(p)}{g(p)}\] (8) subject to: \[h_{i}(p)\leq 0,\ \forall i\in\mathcal{M}\]  
Problem (8) is a concave fractional program (CFP) [36] since it satisfies the following properties: (i) the function \(f(p)\), \(g(p)\), and \(h_{i}(p),\ \forall i\in\mathcal{M}\) are all real-valued functions that are defined on the set \(\mathcal{P}\) of \(\mathbb{R}^{n}\); (ii) the \(f(p)\) and \(g(p)\) are concave and affine on \(\mathcal{P}\) respectively; and (iii) \(f(p)\) is positive on \(S\) if \(g(p)\) is not affine, where \(S=\{p\in\mathcal{P}\ :\ h_{i}(p)\leq 0,\ \forall i\in\mathcal{M}\}\). The objective function of a CFP is quasiconcave, and a local optimal solution in quasiconcave problem cannot be guaranteed as a global maximum [37]. Only in a special case, CFP problem has one maximum solution point when \(f(p,z)\) is strictly concave or \(g(p)\) is strictly convex [36]. However, using Charnes-Cooper transformation (CCT) [36,38] a CFP can be transformed into a concave program (CP) problem, and for CP we can guarantee any local maximum point as global maximum point. In a differentiable equivalent CP, a solution of the Karush Kuhn Tucker (KKT) conditions provides the optimal solution.  
A CFP with affine denominator can be transformed to an equivalent concave program with Charnes-Cooper Transformation (CCT) with the following transformations: \(y=tp\) and \(t=\frac{1}{g(p)}\). With CCT, we can write equivalent concave program for Equation (3) as follows:\[\max_{\mathbf{y},t} t\sum_{j=1}^{N}\log_{2}\left(1+\frac{y_{j}\gamma_{j}^{ss}}{t(\sigma^{2}+ \sum_{i=1}^{M}J_{ij})}\right)\] (9) subject to: \[C_{1}: \sum_{j=1}^{N}y_{j}K_{ji}\leq\frac{tI_{i}^{th}}{\bar{\gamma}_{i}^{ sp}(-\ln(1-I_{i}^{P}))},\;\forall i\in\mathcal{M}\] \[C_{2}: \sum_{j=1}^{N}\log\left(1+\frac{y_{j}\gamma_{j}^{ss}}{t(\sigma^{ 2}+\sum_{i=1}^{M}J_{ij})}\right)\geq C^{th}\] \[C_{3}: \sum_{j=1}^{N}y_{j}\leq tP_{T}\] \[C_{4}: y_{j}\geq 0,\;\forall j\in\mathcal{N}\] \[C_{5}: tp_{c}+\sum_{j=1}^{N}y_{j}=1\] \[C_{6}: t>0\]  
The concave program (9) can be solved using any standard optimization techniques and/or using mathematical programming software (e.g., MATLAB, CPLEX, MINOS, _etc._). When the constraint \(C_{2}\) is a strictly inequality, we show that the power allocation in the subcarriers has water-filling type policy. We express this by the following theorem:  
**Theorem 1**: _The optimal power allocation rule that maximizes the total energy efficiency in Equation (9) is given by the following relation when the capacity is greater than the threshold in constraint \(C_{2}\):_  
\[p_{j}^{*}=\left(\frac{y_{j}^{*}}{t^{*}}\right)=\left(\frac{1}{\beta_{j}}-\frac {1}{\xi_{j}}\right)^{+},\forall j\in\mathcal{N} \tag{10}\]  
_where \(\beta_{j}=\ln 2\left(\upsilon-\alpha-\lambda_{j}+\sum_{i=1}^{M}(\phi_{i}+\eta_{ i})K_{ji}\right)\), \(\upsilon\), \(\alpha\), \(\lambda_{j}\), \(\phi_{i}\) and \(\eta_{i}\) are Lagrange multipliers of the Lagrangian function \(L(\boldsymbol{y},t,\boldsymbol{\phi},\upsilon,\boldsymbol{\lambda},\mu, \boldsymbol{\eta},\psi,\alpha)\), and \(\xi_{j}\)=\(\frac{\gamma_{j}^{ss}}{\sigma^{2}+\sum_{i=1}^{M}J_{ij}}\), \(y_{j}^{*}=t^{*}\left(\frac{1}{\beta_{j}}-\frac{1}{\xi_{j}}\right)^{+}\), and \(t^{*}=\frac{1}{p_{c}+\sum_{j=1}^{N}\left(\frac{1}{\beta_{j}}-\frac{1}{\xi_{j} }\right)^{+}}\)._  
The proof of Theorem 1 is given in Appendix B.