A Spectral Distance Based Power Control Scheme for Capacity Enhancement of OFDM Cognitive Radio

Abstract

This work investigates power allocation algorithms for capacity maximization of a non-contiguous orthogonal frequency division multiplexing (NC-OFDM) based cognitive radio (CR) user coexisting with multiple active primary users (PU), under total power budget, sub-channel power and overall interference constraints. Based on the spectral distance between specific subcarriers and PU bands, the scheme allocates power among a minimum number of PU adjacent subcarriers with a water level that is different from the ‘far-away’ subcarriers. The proposed ‘n-adjacent’ approach is subjected to both aggregate and individual PU band interference constraint. Unlike some of the existing methods, the proposed method is capable of meeting the constraints including the contribution of ‘good quality’ subcarriers even when they are close to PU bands. Comparison of results for a wide range of power budget shows that the improved method, using the same complexity, can achieve higher cognitive user capacity with greater power allocation efficiency.

Appendix: Proof of Theorem 1

We use the fact that maximizing a concave function (3) is nothing but a minimizing of its negative value. Considering the Lagrange parameters λ, β _j , δ and μ _i for (i), (ii), (iii) and (iv) inequality constraints of (3) respectively. The KKT conditions are written as [28]

$$\lambda \ge 0$$

(10)

$$\beta_{j} \ge 0,\quad \forall j \in \left\{ {1,2, \ldots ,L} \right\}$$

(11)

$$\delta \ge 0$$

(12)

$$\mu_{i} \ge 0,\quad \forall i \in \left\{ {1,2, \ldots ,N} \right\}$$

(13)

$$\lambda \left( { \, \sum\limits_{i = 1}^{N} {P_{i} } - P_{T} } \right) = 0$$

(14)

$$\beta_{j} \left( { \, P_{j}^{T} - P_{j}^{sc} } \right) = 0,\quad \forall j \in \left\{ {1,2, \ldots ,L} \right\}$$

(15)

$$\delta \left( { \, \sum\limits_{i = 1}^{N} {\alpha_{i} P_{i} - I_{th} } } \right) = 0$$

(16)

and

$$\mu_{i} P_{i} = 0,\quad \forall i \in \left\{ {1,2, \ldots ,N} \right\}$$

(17)

$$- \frac{1}{{h_{i}^{ - 1} + P_{i} }} - \mu_{i} + \lambda + \beta_{j,\phi \left( i \right)} + \delta \alpha_{i}= 0,\quad \forall i \in \left\{ {1,2, \ldots ,N} \right\}$$

(18)

where $h_{i} = \frac{{\left| {G_{i}^{ss} } \right|^{2} }}{{\sigma_{2} + \gamma_{i} }}$. From (18) it can be written that

$$\mu_{i} = \lambda + \beta_{j,\phi \left( i \right)} + \delta \alpha_{i}- \frac{1}{{h_{i}^{ - 1} + P_{i} }}$$

(19)

Now, substituting (19) into (17)

$$P_{i} \left( {\lambda + \beta_{j,\phi \left( i \right)} + \delta \alpha_{i}- \frac{1}{{h_{i}^{ - 1} + P_{i} }}} \right) = 0,\quad \forall i \in \left\{ {1,2, \ldots ,N} \right\}$$

(20)

and substituting (19) into (13)

$$P_{i} \ge \frac{1}{{\lambda + \beta_{j,\phi \left( i \right)} + \delta \alpha_{i}}} - h_{i}^{ - 1} ,\quad \forall i$$

(21)

From (21), P _i  > 0 if λ + β _j,ϕ(i) + δα _i  < h ⁻¹ _i and from (20) P _i expressed as $P_{i} = \frac{1}{{\lambda + \beta_{j,\phi \left( i \right)} + \delta \alpha_{i}}} - h_{i}^{ - 1}$. On the other hand, if λ + β _j,ϕ(i) + δα _i  ≥ h ⁻¹ _i then due to violation of (20), P _i  > 0 is impossible and the only solution is P _i  = 0. Combining these two results, the solution is expressed as

$$P_{i} = \left[ {\frac{1}{{\lambda + \beta_{j,\phi \left( i \right)} + \delta \alpha_{i}}} - \frac{{\sigma_{2} + \gamma_{i} }}{{\left| {G_{i}^{ss} } \right|^{2} }}} \right]^{ + } ,\quad \forall i$$

(22)

The theorem is proved.