Novel adaptive collaboration sensing for efficient acquisition of spectrum opportunities in cognitive radio networks

Abstract

In this paper, we aim to acquire more spectrum opportunities in a limited sensing time. We construct a novel MAC-layer sensing framework for efficient acquisition of spectrum opportunities. Specifically, we employ the sequential probability ratio test and develop a new collaboration sensing scheme for multiuser to collaborate during multi-slot, in which we effectively utilize the resources of secondary users to sense the channels for efficient acquisition of spectrum opportunities. Subsequently, we formulate the proposed scheme as an optimization problem and derive an optimal solution with low complexity, which is based on dynamic programming theory. Simulation results show that the proposed scheme could offer good performance with efficient acquisition of spectrum opportunities in cognitive radio networks.

Appendices

Appendix 1: Proof of L(t _ik )

Based on (2), the probability density function (PDF) of t _ik can be obtained by

$$ f \left( t_{ik} |H_{i0} \right) = \frac{1}{\sqrt {2\pi \sigma _0^2 } }\exp \left( \frac{- (t_{ik} - u_0 )^2 } {2\sigma _0^2 } \right), $$

(15)

$$ f \left( t_{ik} |H_{i1}\right) = \frac{1}{\sqrt {2\pi \sigma _1^2 }}\exp \left( \frac{- (t_{ik} - u_1 )^2 } {2\sigma _1^2} \right), $$

(16)

where u ₀ and σ ²₀ are the mean and variance of t _ik under H _i0, and $u_0 = \sigma _v^2, \sigma _0^2 =\frac{\sigma _v^4 } {S}$, u ₁ and σ ²₁ are the mean and variance of t _ik under H _i1, and u ₁ = σ ² _v (1 + γ _k ), $\sigma _1^2 = \frac{\sigma _v^4} {S}(1 + 2\gamma_k )$.

Then, the LLR of t _ik is obtained by

$$ \begin{aligned} L \left( t_{ik} \right) &= \ln \frac{f \left( t_{ik} |H_{i1} \right)}{f \left( t_{ik} |H_{i0}\right)}\\ &= \ln \frac{\sigma _0}{\sigma _1} + \frac{(t_{ik} - u_0 )^2} {2\sigma _0^2} - \frac{(t_{ik} - u_1 )^2}{2\sigma _1^2 }\\ &= \ln \frac{\sigma _0} {\sigma _1 } + \left( \frac{1} {2\sigma _0^2 } - \frac{1}{2\sigma _1^2} \right)t_{ik}^2 \\ &\quad- \left( \frac{u_0}{\sigma _0^2} - \frac{u_1}{\sigma_1^2 } \right)t_{ik} + \left( \frac{u_0^2}{2\sigma _0^2} - \frac{u_1^2 } {2\sigma _1^2} \right) \\ &= A_kt_{ik}^2 - B_kt_{ik} + C_k \\ &= A_k\left( t_{ik} -\frac{B_k} {2A_k} \right)^2 - \frac{B_k^2} {4A_k} + C_k, \end{aligned} $$

(17)

where $A_k = \frac{1}{2\sigma _0^2} - \frac{1}{2\sigma _1^2 } = \frac{S\gamma_k}{\sigma _v^4 \left( 1 + 2\gamma_k\right)},\; B_k = \frac{u_0}{\sigma _0^2} - \frac{u_1}{\sigma _1^2 } = \frac{S\gamma_k}{\sigma _v^2 \left( 1 +2\gamma_k \right)},\; C_k = \ln \frac{\sigma_0}{\sigma _1} + \left( \frac{u_0^2 } {2\sigma_0^2} - \frac{u_1^2} {2\sigma_1^2} \right) = - \frac{1}{2}\ln \left( 1 + 2\gamma_k \right) - \frac{S\gamma_k^2} {2\left( 1 + 2\gamma_k \right)}$.

$A_k(t_{ik} - \frac{B_k}{2A_k}) ^2$ has the non-central chi-square distribution since t _ik has the normal distribution and A _k ,   B _k ,   C _k are constant for certain SU. The mean and variance of the corresponding normal distribution are

$$ \left\{ \begin{array}{lll} m_{0k} = \frac{\sqrt {S\gamma_k } } {2\sqrt {\left( 1 + 2\gamma_k \right)} },& \sigma _{0k}^2 = \frac{\gamma_k }{\left( 1 + 2\gamma_k \right)}, & \hbox{under}\; H_{i0} \\ m_{1k} = \frac{\sqrt {S\gamma_k \left( 1 + 2\gamma_k \right)} }{2}, & \sigma _{1k}^2 = \gamma_k , & \hbox{under}\; H_{i1} \\ \end{array} \right. $$

(18)

Hence, the proof is completed.$\square$

Appendix 2: Derivation of $P\left( \sum\nolimits_{k = 1}^{U_i } {T_{ik} } < \ln \eta _0 - \xi_i \right)$

Let F denote $P\left( \sum\nolimits_{k = 1}^{U_i } {T_{ik} } < \ln \eta _0 - \xi_i \right).$ From (17), we can obtain

$$ F= P\left( \sum\limits_{k = 1}^{U_i } {A_k \left( t_{ik} - \frac{B_k} {2A_k} \right)^2 } < D_i \right), $$

(19)

where $D_i=\ln \eta _0 - \xi_i + \sum\limits_{k = 1}^{U_i } {\left( \frac{B_k^2} {4A_k} - C_k \right)}.$ Let X _k denote $A_k \left( t_{ik} - \frac{B_k} {2A_k} \right)^2$ and Y _i denote $\sum\limits_{k = 1}^{U_i } {X_k}.$ From Appendix 1, we know X _k has the non-central chi-square distribution with freedom 1. Hence, we can easily obtain the characteristic functions of random variable Y _i when we assume the random variables X _k are statistically independent

$$ \begin{aligned} &\phi _{Y_i} \left( jv \right)\\ &= E\left( e^{jvY_i} \right)\\ &= E\left[ \hbox{exp} \left( jv\sum\limits_{k = 1}^{U_i } {X_k } \right) \right]\\ &= \int\limits_{ - \infty }^\infty { \cdots \int_{ - \infty }^\infty {\left(\prod\limits_{k = 1}^{U_i } {e^{jvx_k } } \right)p\left( x_1 , \ldots, x_{U_i }\right)dx_1, \ldots, dx_{U_i } } }\\ &= \prod\limits_{k = 1}^{U_i } {\phi _{X_k } \left(jv \right)}, \end{aligned} $$

(20)

where $\phi _{X_k } \left( jv\right) = \left(1 - j2v\sigma _{k}^2 \right)^{ - \frac{1}{2}} \exp \left( \frac{jvm_{k}^2 }{\left( 1 - j2v\sigma _{k}^2 \right)} \right)$ is the characteristic function of X _k , σ ² _k and m ² _k can be obtained by (18).

Then, the probability density function (PDF) of Y _i is obtained by

$$ f_i\left( y \right) = \frac{1}{2\pi}\int\limits_{ - \infty }^\infty {\left(\prod\limits_{k = 1}^{U_i } {\phi _{X_k } (jv)} \right)e^{ - jvy} dv}, $$

(21)

and F can be obtained by

$$ \begin{aligned} F &= P\left( \sum\limits_{k = 1}^{U_i } {A_k \left( t_{ik} - \frac{B_k}{2A_k} \right)^2 } < D_i \right)\\ &= P\left( Y_i < D_i \right)\\ &= \int\limits_{ - \infty }^{D_i } {f_i \left( y \right)dy}\\ &= \frac{1} {2\pi}\int\limits_{ - \infty }^{D_i } {\int\limits_{ - \infty }^\infty {\left(\prod\limits_{k = 1}^{U_i } {\phi _{X_k } (jv)} \right)e^{ - jvy} dv} dy}. \end{aligned} $$

(22)

For a Rayleigh fading channel, γ _k has an exponential probability density function (PDF) with the parameter γ which is the average of γ _k . As illustrated in Fig. 1, the distances among SUs are much smaller than the distances between SUs and PU in our considered scenario. Hence, it is reasonable to assume that all SUs have the same average SNR γ [25]. However, it is still difficult to compute F in (22). We try to approximate F in (22) by using the average SNR γ instead of the SNR of the kth SU $\gamma_k,\, k=1, 2,\ldots, M.$ Then, the approximate Y _i , denoted as $\bar{Y}_i,$ has the non-central chi-square distribution with freedom U _i , and $\bar{F}$ can be obtained by

$$ \begin{aligned} \bar{F} &= P\left( \sum\limits_{k = 1}^{U_i } {\bar{A}_k \left( \bar{t}_{ik} - \frac{\bar{B}_k}{2\bar{A}_k} \right)^2} < \bar{D}_i \right)\\ &= P\left( \bar{Y}_i < \bar{D}_i \right). \end{aligned} $$

(23)

To evaluate the gap between F and $\bar{F},$ we obtain the cumulative distribution function (CDF) of F via Monte Carlo searching, and $\bar{F}$ via the CDF of non-central chi-square distribution, respectively. In this simulation, we consider one idle channel and the average SNR γ = −10 dB. As shown in Fig. 13, the CDFs of F and $\bar{F}$ match well under U _i  = 10, 30, which illustrates that $\bar{F}$ could approximate F well.

Fig. 13

The cumulative distribution function of F and $\bar{F}$ with 100 SUs, P _m  = P _fa  = 10⁻²

In addition, when the channel has not been determined, it is reasonable to assume that the channel is idle with probability P _a and busy with probability 1 − P _a , where P _a is the probability of each channel being available. We also evaluate the performance loss if P _a cannot be estimated exactly in Sect. 5. Then, $\bar{F}$ can be obtained by

$$ \begin{aligned} \bar{F} &= P\left( \bar{Y}_i < \bar{D}_i \right)\\ &= P_aP\left( \bar{Y}_i < \bar{D}_i |H_{i0} \right)+ (1-P_a)P\left( \bar{Y}_i < \bar{D}_i |H_{i1} \right), \end{aligned} $$

(24)

where $P\left( \bar{Y}_i < \bar{D}_i |H_{i0} \right)$ and $P\left( \bar{Y}_i < \bar{D}_i |H_{i1}\right)$ could be easily obtained by the CDF of non-central chi-square distribution (the corresponding parameters of the non-central chi-square distribution could be obtained by (18)).

Hence, the expression of $P\left( \sum\nolimits_{k = 1}^{U_i} {T_{ik} } < \ln \eta _0 - \xi_i \right)$ in (9) is obtained.