QoS-Driven Power Allocation Under Peak and Average Interference Constraints in Cognitive Radio Networks

Abstract

Efficient radio spectrum utilization can be improved using cognitive radio technology. In this work, we consider a spectrum underlay cognitive radio system operating in a fading environment. We propose an efficient power control scheme that maximizes the effective capacity of the secondary user, provisioning quality of service while on the same time the communication of the primary user is guaranteed through interference constraints. The specific power allocation scheme uses a policy in which the outage events of the primary user are exploited leading to a significant increase of the secondary user’s effective capacity. Moreover, the interference of the primary link to the secondary is taken into account so as to study a more realistic scenario. In order to safeguard primary user’s communication, two types of restrictions are considered: the traditional interference power constraint and the proposed inverse signal to interference plus noise ratio constraint. Different scenarios depending on the nature of the constraints (peak/average) are studied and their impact on the performance of the primary and secondary users is investigated. The superiority of the proposed schemes is demonstrated through their comparison with two reference power control schemes. Finally, numerical calculations, validated with simulation results, confirm the theoretical analysis and evaluate the performance of the proposed scheme for all the different scenarios.

Appendix

The solution of (15) for $P_s\epsilon \varOmega _1$ can be found by using the Lagrange function that is defined as:

$$\begin{aligned} L&= \int \limits _{h_{out}}^\infty \int \limits _0^\infty \int \limits _0^\infty \int \limits _0^\infty \left( 1+ \frac{P_s(a_{SU},h_{sp},h_s,h_{ps}, h_p)h_s}{P_ph_{ps}+N_0B}\right) ^{-a_{SU}}f_{ch}(h_s,h_{ps},h_{sp},h_p)dh_sdh_{ps}dh_{sp}dh_p \nonumber \\&\quad +\,\lambda _0 \left\{ \, \int \limits _{h_{out}}^\infty \int \limits _0^\infty \int \limits _0^\infty \int \limits _0^\infty h_{sp}P_s(a_{SU},h_{sp},h_s,h_{ps}, h_p)f_{ch}(h_s,h_{ps},h_{sp},h_p)dh_sdh_{ps}dh_{sp}dh_p -Q_{I,av} \right\} \nonumber \\ \end{aligned}$$

(29)

where $f_{ch}(h_s,h_{ps},h_{sp},h_p)=f_{h_s}(h_s)f_{h_{ps}}(h_{ps})f_{h_{sp}}(h_{sp})f_{h_p}(h_p)$ denotes the joint pdf of the uncorrelated fading channel power gains. Giving the fact that the optimal solution should satisfy the Lagrange–Euler equation [34, 35] we get the following equation:

$$\begin{aligned} -\frac{a_{SU}h_s}{P_ph_{ps}+N_0B}\left( 1+ \frac{P_s(a_{SU},h_{sp},h_s,h_{ps}, h_p)h_s}{P_ph_{ps}+N_0B}\right) ^{-a_{SU}-1}+\lambda _0 h_{sp}=0 \end{aligned}$$

(30)

Solving the above with respect to $P_s$, we found the optimal power allocation defined in (16). The value of the parameter $\lambda _0$ can be found from the satisfaction of the average interference power constraint:

$$\begin{aligned} \int \limits _{h_{out}}^\infty {\int \limits _0^\infty }{\int \limits _0^\infty }{\int \limits _0^\infty }h_{sp} \cdot P_s^*(h_s,h_{sp},h_{ps}, h_p) \cdot f_{ch}(h_s,h_{ps},h_{sp},h_p)dh_sdh_{sp}dh_{ps}dh_p=Q_{I,av}\nonumber \\ \end{aligned}$$

(31)

In the following, we simplify the integration part by finding the pdf of the random variable $\frac{h_s}{h_{ps}P_p+N_0B}$ where $h_s$ and $h_{ps}$ follow the Gamma distribution with parameters $m_s$ and $m_{ps}$, respectively. In order to do that, at first we define two new random variables $g_0=h_{ps}P_p+N_0B$ and $g_1=h_s$. Taking into account that $dh_sdh_{ps}=\frac{1}{P_p}dg_0dg_1$, the joint pdf $f_{h_s,h_{ps}}(h_s, h_{ps})$ of the uncorrelated channel power gains can be easily transformed to $f_{g_0,g_1}(g_0, g_1)$ with the simple substitution of the variables.

Afterwards, we define the variables $x$ and $y$ as $x=g_1/g_0$ and $y=g_0+g_1$ and we find the Jacobian determinant [36], which is given by

$$\begin{aligned} J=-\frac{(1+x)^2}{y}. \end{aligned}$$

Thus, the joint pdf of $x$ and $y$ can be expressed as follows:

$$\begin{aligned} f_{x,y}(x,y)&= \frac{m_s^{m_s}m_{ps}^{m_{ps}}P_p^{-m_{ps}} e^{N_0Bm_{ps}/P_p}}{\varGamma (m_s)\varGamma (m_{ps})}x^{m_s-1}(1+x)^{-m_{ps}-m_s} \nonumber \\&\quad \times \, y^{m_s}\left( y-N_0B(1+x)\right) ^{m_{ps}-1}\cdot \exp \left\{ -y\left( \frac{m_s P_p x+m_{ps}}{P_p(1+x)}\right) \right\} \end{aligned}$$

(32)

Integrating $f_{x,y}(x,y)$ with respect to $y$, we can find the marginal distribution of $x$, which is given by:

$$\begin{aligned} f_x(x)&= \int \limits _{N_0B(1+x)}^\infty f_{x,y}(x,y)dy=\frac{m_s^{m_s}m_{ps}^{m_{ps}}P_p^{-m_{ps}} e^{N_0Bm_{ps}/P_p}}{\varGamma (m_s)\varGamma (m_{ps})}x^{m_s-1}(1+x)^{-m_{ps}-m_s}\nonumber \\&\cdot \int \limits _{N_0B(1+x)}^\infty y^{m_s}\left( y-N_0B(1+x)\right) ^{m_{ps}-1}\exp \left\{ -y\left( \frac{m_s P_p x+m_{ps}}{P_p(1+x)}\right) \right\} dy \end{aligned}$$

(33)

Using (4.11) from [15, p.348], we get that

$$\begin{aligned} f_x(x)&= \frac{m_s^{m_s}m_{ps}^{m_{ps}}P_p^{\frac{1+m_s-m_{ps}}{2}} e^{\frac{N_0Bm_{ps}}{P_p}}(N_0B)^{\frac{m_{ps}+m_s-1}{2}}}{\varGamma (m_s)}x^{m_s-1}(m_s x P_p+m_{ps})^{-\frac{m_s+m_{ps}+1}{2}} \nonumber \\&\quad \cdot \exp \left\{ -\frac{N_0B(m_s x P_p+m_{ps})}{2P_p}\right\} W \left( \frac{1+m_s-m_{ps}}{2},\right. \nonumber \\&\left. \frac{-(m_s+m_{ps})}{2},\frac{N_0B(m_s x P_p+m_{ps})}{P_p}\right) \end{aligned}$$

(34)

where $W(\cdot )$ is the Whitakker function [15, p.1024]. Consequently, the average interference power constraint can be simplified to the following:

$$\begin{aligned} Q_{I,av}&= \frac{\varGamma (m_p,m_p h_{out})}{\varGamma (m_p)} \left\{ P_{max}\int \limits _0^\infty \left( 1-\frac{m_{sp}^{-1}}{\varGamma (m_{sp})}\varGamma \Big (m_{sp}+1,\right. \right. \nonumber \\&\left. \left. m_{sp}\lambda _0^{-1}a_{SU}x(P_{max}x+1)^{-(a_{SU}+1)}\right) \right) f_x(x)dx + \frac{m_{sp}^{-\frac{a_{SU}}{a_{SU}+1}}}{\varGamma (m_{sp})}\left( \frac{\lambda _0}{a_{SU}}\right) ^{-\frac{1}{a_{SU}+1}}\nonumber \\&\left. \times \,\int \limits _0^\infty x^{-\frac{a_{SU}}{a_{SU}+1}}\left[ \varGamma \left( m_{sp}+\frac{a_{SU}}{a_{SU}+1},m_{sp}\lambda _0^{-1}a_{SU}x(P_{max}x+1)^{-(a_{SU}+1)}\right) \right. \right. \nonumber \\&\left. -\,\varGamma \left( m_{sp}+\frac{a_{SU}}{a_{SU}+1},m_{sp}\lambda _0^{-1}a_{SU}x\right) \right] f_x(x)dx\nonumber \\&-\,\frac{m_{sp}^{-1}}{\varGamma (m_{sp})}\int \limits _0^\infty x^{-1}\left[ -\varGamma \left( m_{sp}+1 ,m_{sp}\lambda _0^{-1}a_{SU}x\right) \right. \nonumber \\&\left. \left. +\,\varGamma \left( m_{sp}+1 ,m_{sp}\lambda _0^{-1}a_{SU}x(P_{max}x+1)^{-(a_{SU}+1)}\right) \right] f_x(x)dx\right\} \end{aligned}$$

(35)

We must note here that the computation of the infinite integrals can be easily performed due to the decreasing behavior of the integrand functions.