计算机应用   2017, Vol. 37 Issue (4): 960-964  DOI: 10.11772/j.issn.1001-9081.2017.04.0960 0

### 引用本文

WANG Sixiu, GUO Wenqiang, WANG Xiaojie. Joint spectrum sensing algorithm for multi-user based on coherent multiple-access channels in cognitive radio[J]. Journal of Computer Applications, 2017, 37(4): 960-964. DOI: 10.11772/j.issn.1001-9081.2017.04.0960.

### 文章历史

Joint spectrum sensing algorithm for multi-user based on coherent multiple-access channels in cognitive radio
WANG Sixiu, GUO Wenqiang, WANG Xiaojie
College of Computer Science and Engineering, Xinjiang University of Finance and Economics, Urumqi Xinjiang 830012, China
Abstract: For joint sensing of multiple Cognitive Users (CUs), considering the case of fading channels between the CU and the decision center, a joint spectrum sensing algorithm based on Multiple-Access Channels (MAC) was proposed. On the basis of the system structure and signal modeling, the asymptotic behavior and outage probability of the traditional MAC algorithm were analyzed. Under the constraint of the average transmit power of the CU, the transmit gain of the MAC algorithm was optimized to maximize the detection probability; and the problem of minimizing the number of CUs was also studied in the case of certain Quality of Service (QoS). Simulation results show that the proposed MAC algorithm can ensure good detection performance; in particular, it achieves exponential performance improvement in detection error probability.
Key words: Cognitive Radio (CR)    spectrum sensing    Multiple-Access Channels (MAC)    multi-user
0 引言

1 基于MAC联合检测算法分析 1.1 系统模型

 图 1 认知无线电系统模型 Figure 1 System model of CR

 图 2 系统结构 Figure 2 System architecture

 $r(i) = \left\{ \begin{array}{l} {n_i},{\rm{ }}{H_0}{\rm{ }}\\ \sqrt {{\gamma _i}} s + {n_i},{\rm{ }}{H_1} \end{array} \right.{\rm{ }}$ (1)

 ${x_i} = {\left| {r(i)} \right|^2},i = 1,2,...,N{\rm{ }}$ (2)

 ${x_i} = \left\{ \begin{array}{l} E(1),{\rm{ }}{H_0}\\ E(1 + {\gamma _i}),{\rm{ }}{H_1} \end{array} \right.$ (3)

 $\sum\limits_{i = 1}^N {{P_i}} = \sum\limits_{i = 1}^N {2{\omega _i}{{(1 + {\gamma _i})}^2}} \le P$ (4)

 $y = \sum\limits_{i = 1}^N {\sqrt {{h_i}} \sqrt {{\omega _i}} {x_i}} + e$ (5)

 ${\mu _y} = \left\{ \begin{array}{l} \sum\limits_{i = 1}^N {\sqrt {{h_i}{\omega _i}} {\rm{, }}{H_0}} \\ \sum\limits_{i = 1}^N {\sqrt {{h_i}{\omega _i}} (1 + {\gamma _i}),{\rm{ }}{H_1}} \end{array} \right.$ (6)
 $\sigma _y^2 = \left\{ \begin{array}{l} 1 + \sum\limits_{i = 1}^N {{h_i}{\omega _i}{\rm{, }}{H_0}} \\ 1 + \sum\limits_{i = 1}^N {{h_i}{\omega _i}{{(1 + {\gamma _i})}^2},{\rm{ }}{H_1}} \end{array} \right.$ (7)

 ${P_{\rm{f}}} = Q\left( {\frac{{\eta - \sum\limits_{i = 1}^N {\sqrt {{h_i}{\omega _i}} } }}{{1 + \sum\limits_{i = 1}^N {\sqrt {{h_i}{\omega _i}} } }}} \right)$ (8)
 ${P_{\rm{d}}} = Q\left( {\frac{{\eta - \sum\limits_{i = 1}^N {\sqrt {{h_i}{\omega _i}} (1 + {\gamma _i})} }}{{1 + \sum\limits_{i = 1}^N {\sqrt {{h_i}{\omega _i}} {{(1 + {\gamma _i})}^2}} }}} \right)$ (9)

Pd可以表示为Pf的函数如下:

 ${P_{\rm{d}}} = Q\left( {\frac{{{Q^{ - 1}}\left( {{P_{\rm{f}}}} \right)\sqrt {1 + \sum\limits_{i = 1}^N {{h_i}{\omega _i}} } - \sum\limits_{i = 1}^N {\sqrt {{h_i}{\omega _i}} {\gamma _i}} }}{{\sqrt {1 + \sum\limits_{i = 1}^N {{h_i}{\omega _i}{{(1 + {\gamma _i})}^2}} } }}} \right)$ (10)
1.2 性能分析

1) 渐近性能 (asymptotic behavior) 分析。

 ${\omega _i} = \frac{P}{{2N{{(1 + {\gamma _i})}^2}}},i = 1,2,...,N$ (11)

 $\mathop {\lim }\limits_{N \to \infty } \frac{{\sqrt {1 + \sum\limits_{i = 1}^N {{h_i}{\omega _i}} } }}{{\sqrt {1 + \sum\limits_{i = 1}^N {{h_i}{\omega _i}{{(1 + {\gamma _i})}^2}} } }} = \frac{{\sqrt {1 + \frac{{hP}}{{2{{(1 + \gamma )}^2}}}} }}{{\sqrt {1 + \frac{{hP}}{2}} }}$ (12)
 $\mathop {\lim }\limits_{N \to \infty } \frac{{\sum\limits_{i = 1}^N {\sqrt {{h_i}{\omega _i}} {\gamma _i}} }}{{\sqrt {1 + \sum\limits_{i = 1}^N {{h_i}{\omega _i}{{(1 + {\gamma _i})}^2}} } }} = \infty$ (13)

 $\mathop {\lim }\limits_{N \to \infty } {P_{\rm{d}}} = 1$ (14)

2) 中断概率 (outage probability)。

 $I(y,s) < \min \left( {I(s,r),I(x,y)} \right)$ (15)

 $I(\mathit{\boldsymbol{Y}},\mathit{\boldsymbol{X}}) = \ln {\rm{ det}}\left( {\mathit{\boldsymbol{I}}{\rm{ + }}\mathit{\boldsymbol{H}}E\left[ {\mathit{\boldsymbol{X}}{\mathit{\boldsymbol{X}}^H}} \right]E{{\left[ {\mathit{\boldsymbol{n}}{\mathit{\boldsymbol{n}}^H}} \right]}^{ - 1}}{\mathit{\boldsymbol{H}}^H}} \right)$ (16)

 $I(s,r) = \sum\limits_{i = 1}^N {\ln (1 + {\gamma _i})}$ (17)
 $I(y,x) = \ln \left( {1 + \sum\limits_{i = 1}^N {\frac{{P{h_i}}}{N}} } \right)$ (18)

 $\begin{array}{l} \Pr (I < R) = 1 - \Pr (I(s,r) > R) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;1 - \Pr \left[ {\sum\limits_{i = 1}^N {\frac{{P{h_i}}}{N}} > {2^R} - 1} \right] \end{array}$ (19)

 $\Pr (I < R) = 1 - \sum\limits_{n = 1}^N {\frac{{{a^{N - n}}{e^{ - a}}}}{{N - n}}} \Pr \left[ {\sum\limits_{i = 1}^N {\ln (1 + {\gamma _i})} > R} \right]$ (20)

2 MAC算法的优化 2.1 参数优化求解

 $\mathop {\max }\limits_{{\omega _i} \ge 0} \;\;\;\;\;{P_{\rm{d}}}$ (21)

 ${\rm{ }}\sum\limits_{i = 1}^N {{P_i}} \le P;{\rm{ }}\forall {P_i} \ge 0$ (22)

 ${\mathop {\max }\limits_{{\omega _i} \ge 0} \;\;\;\;Q\left( {\frac{{{Q^{ - 1}}\left( {{P_{\rm{f}}}} \right)\sqrt {1 + \sum\limits_{i = 1}^N {{h_i}{\omega _i}} } - \sum\limits_{i = 1}^N {\sqrt {{h_i}{\omega _i}} {\gamma _i}} }}{{\sqrt {1 + \sum\limits_{i = 1}^N {{h_i}{\omega _i}{{(1 + {\gamma _i})}^2}} } }}} \right)}$ (23)
 $\mathop {{\rm{ }}\sum\limits_{i = 1}^N {2{\omega _i}{{(1 + {\gamma _i})}^2}} \le P;{\rm{ }}\forall {P_i} \ge 0}\limits_{{\rm{ }}}$ (24)

 $\mathop {\mathop {\max }\limits_{{\omega _i} \ge 0} {\rm{ }}\frac{{\left( {\sum\limits_{i = 1}^N {\sqrt {{h_i}{\omega _i}} {\gamma _i}} } \right){}^2}}{{1 + \sum\limits_{i = 1}^N {{h_i}{\omega _i}{{(1 + {\gamma _i})}^2}} }}}\limits_{{\rm{ }}}$ (25)
 $\sum\limits_{i = 1}^N {2{\omega _i}{{(1 + {\gamma _i})}^2}} \le P;{\rm{ }}\forall {P_i} \ge 0$ (26)

 ${g_i} = \sqrt {\frac{{3{\omega _i}{{\left( {1 + {\gamma _i}} \right)}^2}}}{P}}$ (27)

 ${\mathop {\max }\limits_{{\omega _i} \ge 0} {\rm{ }}\frac{{\left( {\sum\limits_{i = 1}^N {{g_i}\sqrt {\frac{{{h_i}P{\gamma _i}^2}}{{2{{(1 + {\gamma _i})}^2}}}} } } \right){}^2}}{{1 + \sum\limits_{i = 1}^N {\frac{1}{2}{h_i}{g_i}^2P} }}}$ (28)
 $\mathop {\sum\limits_{i = 1}^N {g_i^2} \le 1;{\rm{ }}\forall {g_i} \ge 0}\limits_{{\rm{ }}}$ (29)

 $\mathit{\boldsymbol{g}} = {\left[ {{g_1},{g_2},...,{g_N}} \right]^{\rm{T}}}$ (30)
 $\mathit{\boldsymbol{m}} = {\left[ {\sqrt {\frac{{{h_1}P{\gamma _1}^2}}{{2{{(1 + {\gamma _1})}^2}}}} ,\sqrt {\frac{{{h_2}P{\gamma _2}^2}}{{2{{(1 + {\gamma _2})}^2}}}} ,...,\sqrt {\frac{{{h_N}P{\gamma _N}^2}}{{2{{(1 + {\gamma _N})}^2}}}} } \right]^{\rm{T}}}$ (31)
 $\mathit{\boldsymbol{D}} = {\rm{diag}}\left( {\left[ {1 + \frac{{{h_1}P}}{2},1 + \frac{{{h_2}P}}{2},...,1 + \frac{{{h_N}P}}{2}} \right]} \right)$ (32)

 $\mathop {\mathop {\max }\limits_\mathit{\boldsymbol{g}} {\rm{ }}\frac{{\left( {\mathit{\boldsymbol{mg}}} \right){}^2}}{{{\mathit{\boldsymbol{g}}^T}\mathit{\boldsymbol{Dg}}}}}\limits_{{\rm{ }}}$ (33)
 $\mathit{\boldsymbol{g}}{\mathit{\boldsymbol{g}}^{\rm{T}}} \le {\bf{1}},{\rm{ }}\mathit{\boldsymbol{g}} \ge {\bf{0}}$ (34)

 $\mathop {\mathop {\max }\limits_\mathit{\boldsymbol{q}} {\rm{ }}\frac{{{\mathit{\boldsymbol{q}}^{\rm{T}}}{\mathit{\boldsymbol{D}}^{ - \frac{1}{2}}}{\mathit{\boldsymbol{m}}^{\rm{T}}}\mathit{\boldsymbol{mD}}{\mathit{\boldsymbol{D}}^{ - \frac{1}{2}}}\mathit{\boldsymbol{q}}}}{{{\mathit{\boldsymbol{q}}^{\rm{T}}}\mathit{\boldsymbol{q}}}}}\limits_{{\rm{ }}}$ (35)
 ${\rm{ }}{\mathit{\boldsymbol{D}}^{ - \frac{1}{2}}}\mathit{\boldsymbol{q}}{\mathit{\boldsymbol{q}}^{\rm{T}}}{\mathit{\boldsymbol{D}}^{ - \frac{{\rm{T}}}{{\rm{2}}}}} \le {\bf{1}},{\rm{ }}\mathit{\boldsymbol{q}} \ge {\bf{0}}$ (36)

 $\begin{array}{l} {\omega _i} = \\ \frac{{{P^2}{h_i}\gamma _i^2}}{{{{\left( {1 + {\gamma _i}} \right)}^4}{{\left( {1 + \frac{{{h_i}P}}{2}} \right)}^2}\left( {\sum\limits_{i = 1}^N {\frac{{P{h_i}\gamma _i^2}}{{2{{\left( {1 + {\gamma _i}} \right)}^2}{{\left( {1 + {h_i}P/2} \right)}^2}}}} } \right)}} \end{array}$ (37)

2.2 最小次用户集合分析

 $\mathop {\min {\rm{ }}C{\rm{ = }}\sum\limits_{i = 1}^N {{C_i}} }\limits_{{\rm{ }}}$ (38)
 $\begin{array}{l} {\rm{s}}.{\rm{t}}.\\ \;\;\;\;\left\{ \begin{array}{l} \sum\limits_{i = 1}^N {{C_i}{P_i}} \le P,{\rm{ }}\forall {P_i} \ge 0\\ {P_d} \ge {P_{d0}}\\ {P_f} \le {P_{f0}} \end{array} \right. \end{array}$ (39)

 ${\xi _i} = \frac{{{h_i}{\gamma _i}^2}}{{{{(1 + {\gamma _i})}^2}{{(1 + \frac{{{h_i}P}}{2})}^2}}}$ (40)

 ${\xi _1} \ge {\xi _2} \ge ... \ge {\xi _N}$ (41)

1) 对次用户按照式 (41) 进行排序。

2) 从k=1到N, 执行:

2.1) 根据式 (37) 求得最优的ωi, 设置Pf=Pf0, 并代入至式 (10), 求得Pd;

2.2) 如果k=N或者PdPd0, 则C=k, 相应的前k个次用户为参与传输次用户, 退出循环。

C=1时, 即只需要一个次用户即可达到需要的检测概率, 从ξi的定义可得, 该节点应该在决策中心和主用户附近; 当C=N时, 表明需要所有次用户参与决策, 但可能也无法满足相应的检测指标。

3 数值仿真

 图 3 决策中心错误概率关于次用户数目的变化曲线 Figure 3 Error probability of decision center versus number of CUs
 $\Pr (e) = \mathop {\min }\limits_\eta {\rm{ }}\left[ {{\rm{1 - }}{P_{\rm{d}}}(\eta ) + {P_{\rm{f}}}(\eta )} \right]$ (42)

 图 4 决策中心中断概率关于次用户数目的变化曲线 Figure 4 Outage probability of decision center versus number of CUs

 图 5 次用户数目比例关于检测概率的变化曲线 Figure 5 Proportion of CUs in detection versus detection probability of decision center

 图 6 决策中心检测概率关于虚警概率的变化曲线 Figure 6 Detection probability of decision center versus false alarm probability
4 结语

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