## 4 Iterative Algorithm using Parametric Optimization
For Figure 2, the total transmitter power threshold, \(P_{T}=1\) W and the rate threshold, \(C^{th}=0.8\) Mbits/sec. The optimal sum of subcarrier powers, the optimal rate and the optimal energy efficiency are found to be \(0.6675\) W, \(0.8\) Mbits/sec and \(1.1985\) Mb/Joule respectively by calculating the optimal point. Note that sum rate constraint is the active constraint (i.e., the optimal point lies on the sum rate constraint) here. In Figure 3, the rate threshold is increased to \(C^{th}=1.0\) Mbits/sec for the same power threshold. The resulting optimal sum of subcarrier powers, the optimal rate and the optimal energy efficiency are \(1.0\) W, \(1.1754\) Mbits/sec and \(1.1754\) Mb/Joule respectively. It can be seen that the power constraint is the active constraint here. Although the value of optimal rate is increased, the energy efficiency is decreased. Therefore, it is evident from the results that maximizing energy efficiency does not maximize the capacity. In fact, they may be sometimes conflicting goals in wireless networks, as was discussed in [26] also for general wireless network. In Figures 4-6, we have shown the effect of number of PUs on the channel access probability, average total subcarrier power and average energy efficiency for different power thresholds. For a particular value of power threshold, the channel access probability decreases with the increase of number of PUs. Because more PUs causes more interference to the SU's receiver. Also, the SU transmitter need to maintain interference constraints of additional PUs. As a results, the average numbers of channel realization for which all the constraints, given other bounds remain the same, need to satisfied decreases. Thus, the SU need to raise the power to counteract the increased number of PUs, as can be seen from

Figure 3: Pictorial view of the optimization objective and all the constraints for different rate constraint. The feasible region and the optimal solution point are also shown in this case. Unlike Figure 2, in this case, the power and the interference constraints are active. Figure 2: Pictorial view of the optimization objective and all the constraints. The feasible region (common region created by the constraints) and the optimal solution point (solid circle) are also shown. Interference and sum rate constraints are active for this case. Figure 6. Since for the same rate threshold the power increases, the energy efficiency decreases with the increase of number of PUs as in Figure 5. It is also seen that as the power threshold decreases, the channel access probability also decreases. For \(P_{T}=5\) mW, the channel access probability is zero for \(M=6\) or more PUs. The effect of rate threshold on the channel access probability is shown in Figure 7. It is seen that for a particular number of PUs, the channel access probability decreases as the rate threshold increases. Because in order to achieve higher rate, the SU transmitter need to use higher transmitter power, which in turns disqualify many sets of channel samples for which there existed a feasible region before. Figure 4: The variation of channel access probability with number of primary users for different power thresholds. Figure 5: The variation of average energy efficiency with number of primary users for different power thresholds. The effect of interference probability threshold on the channel access probability, total power and energy efficiency are shown in Figures 8-10 respectively for different interference thresholds. Note that these two interference parameters, namely the interference threshold and interference probability threshold, are specified by the PU networks depending on its QoS requirements. These curves show the performance of SU for different pairs of these parameters. It can be seen from the Figure 8 that the channel access probability decreases with the interference threshold for a particular interference probability threshold.