Mutual interference considered power allocation in OFDM-based cognitive networks: the multiple SUs case

Abstract

Power allocation for secondary users (SUs) in cognitive networks is an important issue to ensure the SUs’ quality of service. When the mutual interference between the primary users (PUs) and the SUs is taken into consideration, it is wanted to achieve the conflict-free power allocation while synchronously maximizing the capacity of the secondary network. In this paper, the optimal power allocation problem is considered in orthogonal frequency division multiplexing cognitive networks. The single SU case is primarily formulated as a constrained optimization problem. On this basis, the multiple SUs case is then studied and simulated in detail. During the analysis, the mutual interference among the PUs and the SUs is comprehensively formulated as the restrictions on the SU’s transmission power and the optimization problems are finally resolved by iterative water-filling algorithms. Consequently, the proposed power allocation scheme restrains the interference to the primary network, as well as maximizing the capacity of the secondary network. Specifying the multiple-SUs case, simulation results are exhibited in a simplified scenario to confirm the efficiency of the proposed water-filling algorithm, and the influence of the mutual interference on the power allocation and the system capacity is further illustrated.

Appendix

We would like to append a novel proof of water-filling theorem in this part, which is more compact and straightforward.

$$ \min \biggl( -\sum_{n=1}^{N}lg{(1+E_n\cdot {\rm CGNR}_n)}\biggl) $$

$$ \left\{ \begin{array}{lll} -E_n\leq0\\ \sum_{n=1}^{N}{E_n}-E_{\rm budget}\leq0\\ E_b-E_n^{\rm max}\leq0\\ \end{array}\right. $$

where n ∈ {1...N}. The constraint conditions obviously satisfy Slater’s conditions, so the Karush–Kuhn–Tucker (KKT) conditions are sufficient and necessary for the optimal vector [20]. Thus, we obtain

$$ \label{eq:A.1} \frac{-1}{E_n+{\rm CGNR}_n^{-1}}-\lambda_n+\varpi+\mu_n=0 $$

(20)

$$ \label{eq:A.2.1} \lambda_n\cdot E_n=0 $$

(21)

$$ \label{eq:A.2.2} \varpi\left(\sum_{n=1}^{N}{E_n}-E_{\rm budget}\right)=0 $$

(22)

$$ \label{eq:A.2.3} \mu_n(E_n-E_n^{\rm max})=0, $$

(23)

where

$$ \label{eq:A.3.1} \lambda_n\geq0\qquad \forall \ n\in\{1\ldots N\} $$

(24)

$$ \label{eq:A.3.2} \varpi \geq0 $$

(25)

$$ \label{eq:A.3.3} \mu_n\geq0\qquad \forall \ n\in\{1\ldots N\} $$

(26)

If λ _n  > 0, we achieve E _n  = 0 using Eq. 21 and CGNR _n  − λ _n  + ϖ + μ _n  = 0 using Eq. 20. Since the power allocation is zero in this case, we will focus on the power allocation in the case of λ _n  = 0.

If λ _n  = 0, using Eq. 21, we obtain

$$ \label{eq:A.4.1} E_n\geq0, $$

(27)

and using Eq. 20, we achieve

$$ \label{eq:A.4.2} \varpi+\mu_n=\frac{1}{E_n+{\rm CGNR}_n^{-1}} \leq {\rm CGNR}_n $$

(28)

If λ _n  = 0 and μ _n  > 0, using Eq. 23, we attain

$$ \label{eq:A.5.1} E_n=E_n^{\rm max} $$

(29)

and using Eq. 28, we obtain

$$ \label{eq:A.5.2} \varpi+\mu_n=\frac{1}{E_n^{\rm max}+{\rm CGNR}_n^{-1}} $$

(30)

If λ _n  = 0 and μ _n  = 0, using Eq. 23, we attain

$$ \label{eq:A.6.1} E_n\leq E_n^{\rm max} $$

(31)

and using Eq. 28, we obtain

$$ \label{eq:A.6.2} \varpi=\frac{1}{E_n+{\rm CGNR}_n^{-1}}\geq\frac{1}{E_n^{\rm max}+{\rm CGNR}_n^{-1}} $$

(32)

Colligating the results of Eqs. 27–32, we achieve Fig. 10, where λ _n  = 0. From the figure, we obtain that the power allocation satisfies

$$ \label{eq:A.7} E_{n}=\biggl[[\varpi^{-1}-{\rm CGNR}_n^{-1},0]^+,E_n^{\rm max}\biggl]^- $$

(33)

and allocated power is zero in other cases.

Fig. 10

Feasible region of Lagrange multipliers where λ _n  = 0. Only when Lagrange multipliers locate in the area indicated by ::: is the allocated power not zero

If we set $E_0=\varpi^{-1}$, E ₀ is the water level for this water-filling case.