## 4 Iterative Algorithm using Parametric Optimization
It makes sense because as the interference threshold gets tighter, less channel realization can give a feasible region and hence the channel access probability decreases. The SU

Figure 6: The variation of average total subcarrier power with number of primary users for different power thresholds. Figure 7: The effect of number of primary users on channel access probability for different rate thresholds. transmitter need to use less power in order to meet the interference constraints (7) for the tighter interference threshold as can be seen in Figure 9. Since the total power decreases, the energy efficiency in Figure 10 increases. For a particular value of interference threshold, similar effect is seen for tighter interference probability threshold. It can be explained similarly using Equation (7) since tighter bound requires use of less power in order to satisfy interference constraint, which makes rate thresholds infeasible for greater number of sets of channel realizations. Figure 2:0 Channel access probability _vs._ interference probability threshold for different interference thresholds. Figure 2:1 Total power _vs._ interference probability threshold for different interference thresholds. ## 6 Conclusions

In this paper, we studied energy-efficient downlink power allocation techniques for OFDM-based green cognitive radio systems. We assumed that the primary and secondary users coexist in adjacent bands. In this scenario, the mutual interference is very important to control. We formulated the problem using a fraction programming technique, where the objective is to maximize the energy efficiency that is defined as the ratio of capacity and transmitted power. We first provided the optimal solution of the problem through Charne-Cooper transformation that gives us concave form of the problem from fractional form. Although the original fractional program is quasi-concave for which global optimal point is not guaranteed, the equivalent concave problem is easier to solve using standard optimization technique and also has only one global optimal point. Therefore, finding a local optimal point is enough. Then we investigated the Dinkelbach method, where the fractional objective function is converted into parametric objective function. We provided the proof of one-to-one relationship of the solution between the fractional programming method and the Dinkelbach method. The resulting parametric problem is solved using iterative techniques and \(\varepsilon\)-optimal solution is found. It is found that Dinkelbach method provides optimal solution very quickly, it converges mostly in \(6\)-\(10\) iterations. We also provided numerical simulation results to show the channel access probability, average total subcarrier power and average energy efficiency performance for different system parameters, such as, interference limits, number of primary users, rate and power thresholds. Figure 2:2 The convergence of Dinkelbach method. Energy efficient _vs._ number of iterations for transformed concave and parametric Dinkelbach formulations. While transformed concave problem gives the optimal solution directly (in one iteration), Dinkelbach method provides solution iteratively over several iterations. #### Notations