Abstract
In this paper, we analytically quantify the optimal power allocations for two-users spectrum sharing cognitive radio, in which two secondary users (SUs) share the licensed spectrum resource of a primary user (PU) for data transmissions. Specifically, the SUs aim at maximizing their total throughput while taking into account both their respective throughput requirements and more particularly, an interference limit constraint imposed by the PU. To derive the optimal power allocations, we categorize the feasible region of proposed problem into three different cases, and for each case, we derive the optimal power allocations in analytical expressions. Different from previous works showing that the optimal power allocations only resided on one of the vertexes of the feasible region, our results reveal that the optimal power allocations might reside on the boundary of the feasible region that corresponds to the PU’s interference limit constraint. Numerical results are performed to validate our analytical results.
Keywords
This work is supported in part by the National Natural Science Foundation of China (61303235 and 61402416), ZJNSF-LQ13F010006, and the Macau Science and Technology Development Fund under Grant 104/2014/A3.
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- 1.
Different from that for Case I, the two points \(Y_1\) and \(Y_2\) in Case II have the same horizontal-coordinate, and thus there only exist three (instead of four) subcases in Case II.
References
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Appendices
Appendix I: Proof of Lemma 1
The proof is similar to that in [4], which considered the feasible region comprised of each user’s peak transmit power constraint, but without considering the PU’s interference limit constraint (represented by line segment \(l_0\) between points \(X_0\) and \(Y_0\) as shown in Fig. 2(a)). Thus, using the similar technique as in [4], we can prove that \(X_0\), \(X_1\), \(Y_0\), and \(Y_2\) are the candidates for the optimal power allocations of Problem (P1). Moreover, since (i) for some \(\lambda >1\), there exists
i.e., the total throughput of the two SUs increases when the power allocation pair \((p_1,p_2)\) increases, and (ii) \(p_1\) and \(p_2\) are coupled by function \(g_{1B}p_1+g_{2B}p_2=\varGamma \) on line \(l_0\). Therefore, the points on the line segment of \(l_0\) between \(X_0\) and \(Y_0\) are also the possible candidates for the optimal power allocations.
Thus, the optimal power allocations of Problem (P1) under Subcase I.1 must reside within the vertexes \(X_0\), \(X_1\), \(Y_0\), and \(Y_2\), or on the line segment of \(l_0\) between points \(X_0\) and \(Y_0\).
Appendix II: Proof of Lemma 2
To derive the possible extremal points on line segment \(l_0\) between points \(X_0\) and \(Y_0\), we calculate the first order derivative of \(\hat{F}(p_1)\) as follows:
where parameters A, B, and C have given in Eqs. (11), (12) and (13), respectively. Meanwhile, parameter D is given by:
In particular, \(D>0\) always holds, since the four items (in the product form) in D correspond to the numerators and denominators in the \(\log (\cdot )\) expression in Eq. (10). By imposing \(\frac{d \hat{F}(p_1)}{d p_1}=0\), we obtain a quadratic equation \(A p_1^2+B p_1+C=0\), whose roots can be analytically given by
which correspond to the extremal points on line segment \(l_0\) that might maximize \(\hat{F}(p_1)\) (or equivalently, \(F(p_1,p_2)\)).
Appendix III: Proof of Theorem 1
For the case of \(A\ne 0\), there exist two extremal points \(O_1^{I.1}=(p_1^{o_1},f_{l_0}(p_1^{o_1}))\), \(O_2^{I.1}=(p_1^{o_2},f_{l_0}(p_1^{o_2}))\) on the line segment of \(l_0\) according to Lemma 2. In companion with the previous four points \(X_0, X_1, Y_0\), and \(Y_2\) (which have been proved in Lemma 1), the optimal power allocations \((p_1^*,p_2^*)\) of Problem (P1) under Subcase I.1 can thus be given by
Similarly, for the case of \(A=0\), there exists one extremal point \(O_3^{I.1}=(p_1^{o_3},f_{l_0}(p_1^{o_3}))\) on the line segment of \(l_0\). Again, in companion with the previous four points \(X_0, X_1, Y_0\), and \(Y_2\), the optimal power allocations \((p_1^*,p_2^*)\) under Subcase I.1 can thus be given by
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He, Y., Wu, Y., Chen, J., Zhao, Q., Lu, W. (2015). Optimal Power Allocations for Two-Users Spectrum Sharing Cognitive Radio with Interference Limit. In: Xu, K., Zhu, H. (eds) Wireless Algorithms, Systems, and Applications. WASA 2015. Lecture Notes in Computer Science(), vol 9204. Springer, Cham. https://doi.org/10.1007/978-3-319-21837-3_18
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