## 2 System Model
### Problem Formulation  
The performance of the green radio systems are usually measured in terms of energy-efficiency metric, which is defined as the total achievable rate throughput per unit total transmitter power. Let \(\Gamma(\mathbf{p})\) denote the energy-efficiency in information bits per Joule as,  
\[\Gamma(\mathbf{p})=\frac{\sum_{j=1}^{N}C_{j}}{p_{c}+\sum_{j=1}^{N}p_{j}} \tag{2}\]  
where \(p_{c}\) is the static circuit power of the source in the transmit mode. Our objective is to maximize energy-efficiency of the green cognitive radio systems with the constraints on interference probability limit, sum throughput over all subcarriers and allocated powers. That is,\[\begin{array}{ccc}\underset{\textbf{p}}{\max}&\Gamma(\textbf{p})\\ \text{subject to:}&\\ C_{1}:&Pr\left(\sum_{j=1}^{N}I_{ji}\leq I_{i}^{th}\right)\geq I_{i}^{P},\; \forall i\in\mathcal{M}\\ C_{2}:&\sum_{j=1}^{N}C_{j}\geq C^{th}\\ C_{3}:&\sum_{j=1}^{N}p_{j}\leq P_{T}\\ C_{4}:&p_{j}\geq 0,\;\forall j\in\mathcal{N}\end{array}\]  
where \(I_{ji}\) is the interference power introduced by the \(j^{\text{th}}\) subcarrier to the \(i^{\text{th}}\) PU; \(I_{i}^{th}\) is the interference threshold (upper limit of the permissible interference) for \(i^{\text{th}}\) PU, \(I_{i}^{P},\;\forall i\in\mathcal{M}\) are the interference probability thresholds (lower limit of the interference threshold maintaining probability); \(P_{T}\) is the total transmitter power threshold and \(C^{th}\) is the capacity threshold (lower limit of the total rate as determined by QoS requirement). Constraint \(C_{1}\) ensures that the total interference to a PU is below a specified threshold by a specified probability margin, \(C_{2}\) ensures the sum rate constraints over all the subcarriers and \(C_{3}\) ensures that the total allocated power does not exceed the power threshold, \(P_{T}\) and \(C_{4}\) ensures nonnegative power values in the optimization process.  
We assume that the fading channel between the SU transmitter and \(i^{\text{th}}\) PU receiver can be described by Rayleigh distribution with fading gain denoted as \(h_{i}^{sp}\). The probability density function (pdf) of the channel power gain \(\gamma_{i}^{sp}=|h_{i}^{sp}|^{2}\) is exponentially distributed and can be written as,  
\[f(\gamma_{i}^{sp})=\frac{1}{\bar{\gamma}_{i}^{sp}}\exp\left(-\frac{\gamma_{i}^ {sp}}{\bar{\gamma}_{i}^{sp}}\right) \tag{4}\]  
where \(\bar{\gamma}_{i}^{sp}\) is the average channel power gain. The interference power \(I_{ji}\) can be expressed as a function of \(d_{ji}\) and \(p_{j}\) as follows:  
\[I_{ji}=\gamma_{i}^{sp}\int_{d_{ji}-B_{i}/2}^{d_{ji}+B_{i}/2}\phi_{j}(f)df \tag{5}\]  
where \(\phi_{j}(f)=p_{j}T_{s}\left(\frac{\sin(\pi fT_{s})}{\pi fT_{s}}\right)^{2}\) is the power density spectrum (PDS) of the \(j^{\text{th}}\) subcarrier in the CR user band assuming an ideal Nyquist pulse; \(d_{ji}\) is the spectral distance between the \(j^{th}\) subcarrier and \(i^{th}\) PU, and \(T_{s}\) is the OFDM symbol duration. Suppose \(K_{ji}=T_{s}\int_{d_{ji}-B_{i}/2}^{d_{ji}+B_{i}/2}\left(\frac{\sin(\pi fT_{s})} {\pi fT_{s}}\right)^{2}df\), then Equation (5) becomes,  
\[I_{ji}=\gamma_{i}^{sp}K_{ji}p_{j} \tag{6}\]Note that the terms \(K_{ji},\ \forall j\in\mathcal{N},\ \forall i\in\mathcal{M}\) are constants for a given set of carriers and PUs. Substituting the value of \(I_{ji}\) from Equation (6) into constraint \(C_{1}\) of Equation (3), we can write  
\[\sum_{j=1}^{N}p_{j}K_{ji}\leq\frac{I_{i}^{th}}{\bar{\gamma}_{i}^{sp}(-\ln(1-I_{ i}^{P}))},\ \forall i\in\mathcal{M} \tag{7}\]  
The detailed derivation of Equation (7) is given in Appendix A. Equation (7) means that the weighted sum of all the subcarrier powers is upper bounded by the channel between the secondary transmitter and the primary receiver, and PU's QoS requirement in terms of required interference thresholds.  
The problem in Equation (3) is a non-linear and non-concave problem. Below, via suitable transformation, we transform it into a concave optimization problem that can be solved easily.