Novel sensing and joint beam and null steering based resource allocation for cross-tier interference mitigation in cognitive femtocell networks

Abstract

Cognitive radios and femtocell networks are gaining much popularity due to the formers ability to carry out unlicensed transmission in licensed bands and the latter’s ability to increase the indoor capacity by bringing the transmitter and receiver closer to each other. The combination of the two provides us with a joint benefit of unlicensed transmission and increased system capacity. However, deployments of such cognitive radio based femtocell networks (CRFN) can pose some severe cross-tier interference towards the primary system. In this paper, we propose a novel spectrum sensing technique based on Uniform Linear array (ULA) combined with Genetic Algorithm to detect not only the presence of a Primary user but also estimate its parameters such as range, frequency, angle of arrival and amplitude. Furthermore, a joint beamforming and null steering technique is proposed to maximize the signal to interference plus noise ratio (SINR) towards the secondary user equipment (SUE) while minimizing the interference towards the primary Base Station (PBS). We are reutilizing the sensed uplink (UL) frequency of the Primary users for the downlink (DL) transmission; the cognitive femtocell base station (CFBS) calculates such weight vector that aims at sending the maximum signal power towards the femto user and null in the direction of PBS.

Appendices

Appendix 1: derivation of PDF of a ratio of two lognormal RVs

Suppose we define a random variable Z that is a function of two random variables, as follows

$$Z = \frac{X}{Y}$$

where both $X$ and $Y$ are independent log- normal distributed random variables. The PDF of $Z$ is given by

$$f_{Z} (z) = \int y f_{X} (yz)f_{Y} (y)dy$$

Mean value of X and Y are denoted by $\mu_{X}$ and $\mu_{Y}$ respectively. We suppose that both X and Y have same variance denoted by $\sigma$, that is $\sigma_{X}^{2} = \sigma_{Y}^{2} = \sigma^{2} .$

$$\begin{aligned} f_{Z} (z) = \int {y\frac{1}{{\sqrt {2\pi } yz\sigma }}\text{e}^{{\frac{{ - (\ln yz - \mu_{X} )^{2} }}{{2\sigma^{2} }}}} \times \frac{1}{{\sqrt {2\pi } y\sigma }}\text{e}^{{\frac{{ - (\ln y - \mu_{Y} )^{2} }}{{2\sigma^{2} }}}} dy} \hfill \\ \quad let\ln y = v\;{\text{and}}\;\ln z = u \hfill \\ \end{aligned}$$

Thus after changing the variables, the PDF is written as

$$\begin{aligned} f_{Z} (z) = & \frac{1}{{2\pi z\sigma^{2} }}\int {\text{e}^{{\frac{{ - (v + u - \mu_{X} )^{2} }}{{2\sigma^{2} }}}} \text{e}^{{\frac{{ - (v - \mu_{Y} )^{2} }}{{2\sigma^{2} }}}} dv} \\ \, = & \frac{1}{{2\pi z\sigma^{2} }}\int {\text{e}^{{\frac{ - 1}{{2\sigma^{2} }}(u^{2} + \mu_{X}^{2} + \mu_{Y}^{2} - 2\mu_{X} u)}} e^{{\frac{ - 1}{{2\sigma^{2} }}(2v^{2} + 2v(u - (\mu_{X} + \mu_{Y} ))}} } dv\quad let \, u - (\mu_{X} + \mu_{Y} ) = \alpha \\ f_{Z} (z) = & \frac{1}{{2\pi z\sigma^{2} }}\int {\text{e}^{{\frac{ - 1}{{2\sigma^{2} }}(u^{2} + \mu_{X}^{2} + \mu_{Y}^{2} - 2\mu_{X} u)}} e^{{\frac{ - 1}{{2\sigma^{2} }}(2v^{2} + 2v\alpha )}} } dv \\ \end{aligned}$$

Taking independent terms outside the integral we get,

$$f_{Z} (z) = \frac{1}{{2\pi z\sigma^{2} }}\text{e}^{{\frac{ - 1}{{2\sigma^{2} }}(u^{2} + \mu_{X}^{2} + \mu_{Y}^{2} - 2\mu_{X} u)}} \int {e^{{\frac{ - 1}{{\sigma^{2} }}((v + \alpha /2)^{2} }} e^{{\frac{{\alpha^{2} }}{{4\sigma^{2} }}}} } dv.$$

Again applying the change of variable,

$$\begin{aligned} \, \left( {\frac{v + \alpha /2}{\sigma }} \right) = & \frac{s}{\sqrt 2 } \\ f_{Z} (z) = & \frac{1}{2\sqrt \pi z\sigma }\text{e}^{{\frac{ - 1}{{2\sigma^{2} }}(u^{2} + \mu_{X}^{2} + \mu_{Y}^{2} - 2\mu_{X} u)}} e^{{\frac{{\alpha^{2} }}{{4\sigma^{2} }}}} \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^{\infty } {e^{{ - s^{2} /2}} } ds \\ \end{aligned}$$

$$\begin{aligned} \, = & \frac{1}{2\sqrt \pi z\sigma }\text{e}^{{\frac{ - 1}{{4\sigma^{2} }}(2u^{2} + 2\mu_{X}^{2} + 2\mu_{Y}^{2} - 4\mu_{X} u - \alpha^{2} )}} \, \because \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^{\infty } {e^{{ - s^{2} /2}} } ds = 1 \\ \, = & \frac{1}{2\sqrt \pi z\sigma }\text{e}^{{\frac{ - 1}{{4\sigma^{2} }}(u^{2} + \mu_{X}^{2} + \mu_{Y}^{2} - 2\mu_{X} u - 2\mu_{X} \mu_{Y} + 2\mu_{Y} u)}} \\ \, = & \frac{1}{2\sqrt \pi z\sigma }\text{e}^{{\frac{ - 1}{{4\sigma^{2} }}(u - (\mu_{X} - \mu_{Y} ))^{2} }} \\ \, = & \frac{1}{{\sqrt {2\pi } z(\sqrt 2 \sigma )}}\text{e}^{{\frac{{ - (\ln z - (\mu_{X} - \mu_{Y} ))^{2} }}{{2(\sqrt 2 \sigma )^{2} }}}} . \\ \end{aligned}$$

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As it can be seen that it is log-normal distribution with mean value $\mu_{X} - \mu_{Y}$ and variance $\sigma^{{{\prime }2}} = 2\sigma^{2} = \sigma_{X}^{2} = \sigma_{Y}^{2}$

Appendix 2: derivation of average rate

The expected value of a random variable is given as

$$E[X] = \int\limits_{0}^{\infty } {x \, f_{X} } (x)dx.$$

Since rate is given by $\log_{2} (1 + SINR)$. Where SNR is denoted by $\gamma$. Hence the average rate will be given as

$$R_{av} = \int\limits_{0}^{\infty } {\log_{2} } (1 + \gamma ) \, f_{\gamma } (\gamma ) \, d\gamma$$

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