## 2 System Model  
We consider a cognitive radio network as shown in a simplified Figure 1, where a cognitive transmitter (base station) is transmitting to its cognitive users in the downlink direction over pooled wireless spectrum from the primary users (who are also communicating in the adjacent spectrum at the same time). The primary users are the licensed owners of the spectrum and secondary users have the capability to use the unused spectrum opportunistically in time and/or frequency. When the SUs transmit on the adjacent pooled spectra, it inherently causes the mutual interference to the PUs. PUs cause mutual interference to the SUs as well. We assume that the interference due to PU's transmission can be estimated perfectly. Unless specified otherwise, we use \(\mathbf{A}\), \(\mathbf{a}\), and \(a\) to represent matrix, vector and an element of a vector respectively. When \(a_{i}\geq 0\) for all components \(i\) of a vector \(\mathbf{a}\), we use \(\mathbf{a}\geq 0\). The notations for most common variables are given at the end of this paper.  
Figure 1: A typical scenario of primary and secondary user communications in a geographic area, where primary and secondary users co-exist in side-by-side bands. They cause adjacent channel interference to each other.  
Let \(M\) denote the number of PUs that co-exist side-by-side with the CR user pair and \(B_{i},\ \forall i\in\mathcal{M}\) denote the bandwidth of the frequency band of the \(i^{th}\) PU pair, where \(\mathcal{M}=\{1,2,3,\cdots,M\}\). We assume that the SUs use orthogonal frequency division multiplexing (OFDM) technique for transmission of data. The SUs divide the acquired bandwidth into \(N\) subcarriers, where \(\Delta f\) is the bandwidth of each subcarrier. Let us denote the set of subcarriers by \(\mathcal{N}=\{1,2,3,\cdots,N\}\).  
The channel state between the CR transmitter and the CR receiver varies randomly in both time and frequency domain due to fading, and \(h_{j}^{ss}\) denotes the channel fading gain of the associated SUs in the \(j^{\text{th}}\) subcarrier. We assume that the channel state information (CSI) of all subcarriers are perfectly estimated at the receiver and are fed back to the transmitter for power loading on the subcarriers. Without losing generality, we assume that the set of achievable rate points for the broadcast channel can be computed using any methods, such as, frequency division, time-division, superposition coding (SPC), _etc._[35]. For example, in SPC, the users are ordered based on their channel gains, from strong to weak and the first user is encoded last. For simplicity, we carry out our discussions using last user rate, which gives us the lower bound of the energy-efficiency that must be maintained across all the users for QoS guarantee. If \(p_{j},\ j=1,2,3...,N\), where \(p_{j}\in\mathcal{P}\in\mathbb{R}^{n}\) denotes the power allocated for the \(j^{\text{th}}\) subcarrier, its rate throughput, \(C_{j}\) in bits/sec can be written by following Shannon capacity formula,  
\[C_{j}=\Delta f\log_{2}\left(1+\frac{\gamma_{j}^{ss}p_{j}}{\sigma^{2}+\sum_{i= 1}^{M}J_{ij}}\right),\ \forall j\in\mathcal{N} \tag{1}\]  
where \(\gamma_{j}^{ss}=|h_{j}^{ss}|^{2}\) is the channel power gain; \(\sigma^{2}\) is the additive white Gaussian noise (AWGN) variance and \(J_{ij}\) is the interference contributed by the \(i^{\text{th}}\) PU transmitter to the \(j^{\text{th}}\) subcarrier. We assume that the randomly varying interference can be perfectly estimated at the SU receiver and also the estimated interference can be fed back to the transmitter.