Robust Probabilistic Distributed Power Control Algorithm for Underlay Cognitive Radio Networks under Channel Uncertainties

Abstract

Due to limited cooperation among users and erratic nature of wireless channel, it is difficult for secondary users (SUs) to obtain exact values of system parameters, which may lead to severe interference to primary users (PUs) and cause communication interruption for SUs. In this paper, we study robust power control problem for spectrum underlay cognitive radio networks with multiple SUs and PUs under channel uncertainties. Precisely, our objective is to minimize total transmit power of SUs under the constraints that the satisfaction probabilities of both interference temperature of PUs and signal-to-interference-plus-noise ratio of SUs exceed some thresholds. With knowledge of statistical distribution of fading channel, probabilistic constraints are transformed into closed forms. Under a weighted interference temperature constraint, a globally distributed power control iterative algorithm with forgetting factor to increase convergence speed is obtained by dual decomposition methods. Numerical results show that our proposed algorithm outperforms worst case method and non-robust method.

Appendix

1.1 Proof of Equation (9)

For simplicity, we define two independent random variables $A=\frac{p_i g_{ii} }{n_i }$ and $B_j =\frac{p_j g_{ji} }{n_i }$. According to probability distribution, $A$ and $B_j$ follow exponential distribution with mean $\updelta _i =\frac{p_i \bar{{g}}_{ii} }{n_i }$ and $\updelta _j =\frac{p_j \bar{{g}}_{ji} }{n_i }$ respectively. The probabilistic SINR constraint is expressed as

$$\begin{aligned} \Pr \left( {\frac{p_i g_{ii} }{{\sum }_{j=1,j\ne i}^M {p_j g_{ji} }+n_i }\ge {\upgamma }_i^d } \right)&= \Pr \left( {\frac{A}{1+\sum _{j=1,j\ne i}^M {B_j } }\ge {\upgamma } _i^d } \right) \nonumber \\&= \Pr \left( {A\ge {\upgamma } _i^d +{\upgamma } _i^d \sum \nolimits _{j=1,j\ne i}^M {B_j } } \right) \nonumber \\&= \int \limits _{\,0}^{\,+\infty } {\cdots \int \limits _{\,0}^{\,+\infty } {\left( {\int \limits _{{\upgamma } _i^d +{\upgamma } _i^d {\sum }_{j=1,j\ne i}^M {b_j } }^{\,+\infty } {\frac{1}{\updelta _i }\exp \left( {-\frac{a}{\updelta _i }} \right) da} } \right) } }\nonumber \\&\quad \times \left\{ {\prod _{j=1,j\ne i}^M {\frac{1}{\updelta _j }\exp \left( {-\frac{b_j }{\updelta _j }} \right) } } \right\} db_1 \cdots db_M\nonumber \\&= \int \limits _{\,0}^{\,+\infty } {\cdots \int \limits _{\,0}^{\,+\infty } {\exp \left( {-\frac{{\upgamma } _i^d \left( {1+{\sum }_{j=1,j\ne i}^M {b_j } } \right) }{\updelta _i }} \right) } }\nonumber \\&\quad \times \left\{ {\prod _{j=1,j\ne i}^M {\frac{1}{\updelta _j }\exp \left( {-\frac{b_j }{\updelta _j }} \right) } } \right\} db_1 \cdots db_M\nonumber \\&= \exp \left( {-\frac{{\upgamma } _i^d }{\updelta _i }} \right) \int \limits _{\,0}^{\,+\infty } \cdots \int \limits _{\,0}^{\,+\infty } \left( \prod _{j=1,j\ne i}^M \frac{1}{\updelta _j }\right. \nonumber \\&\quad \times \left. \exp \left( {-\left( {\frac{{\upgamma } _i^d {\sum }_{j=1,j\ne i}^M {b_j } }{\updelta _i }+\frac{b_j }{\updelta _j }} \right) } \right) \right) db_1 \cdots db_M\nonumber \\&= \exp \left( {\!-\frac{{\upgamma } _i^d }{\updelta _i }} \right) \prod _{j\!=\!1,j\ne i}^M {\int \limits _{\,0}^{\!+\!\infty } {\frac{1}{\updelta _j }\exp \left( {\!-\!\left( {\frac{1}{\updelta _j }\!+\!\frac{{\upgamma } _i^d }{\updelta _i }} \right) b_j } \right) } } db_j\nonumber \\&= \exp \left( {-\frac{{\upgamma } _i^d }{\updelta _i }} \right) \prod _{j=1,j\ne i}^M {\frac{1}{\updelta _j }\left( {\frac{1}{\updelta _j }+\frac{{\upgamma } _i^d }{\updelta _i }} \right) ^{-1}}\nonumber \\&= \exp \left( {-\frac{{\upgamma } _i^d }{\updelta _i }} \right) \prod _{j=1,j\ne i}^M {\left( {1+\frac{\updelta _j {\upgamma } _i^d }{\updelta _i }} \right) ^{-1}} \end{aligned}$$

(26)

Substituting the variables $\updelta _i =\frac{p_i \bar{{g}}_{ii} }{n_i }$ and $\updelta _j =\frac{p_j \bar{{g}}_{ji} }{n_i }$ into (26), we get

$$\begin{aligned} \Pr \left\{ {\frac{p_i g_{ii} }{{\sum }_{j=1,j\ne i}^M {p_j g_{ji} } +n_i }\ge {\upgamma }_i^d } \right\} =\exp \left( {-\frac{{\upgamma }_i^d n_i }{p_i \bar{{g}}_{ii} }} \right) \prod _{j=1,j\ne i}^M {\left( {1+\frac{p_j \bar{{g}}_{ji} }{p_i \bar{{g}}_{ii} }{\upgamma }_i^d } \right) } ^{-1} \end{aligned}$$

(27)

The proof is completed.