A Cluster-Based Cooperative Spectrum Sensing Strategy to Maximize Achievable Throughput

Abstract

In this paper, a novel cooperative spectrum sensing (CSS) strategy is proposed for cognitive radio networks (CRN) with imperfect reporting channels. This CSS strategy uses simultaneously four techniques to overcome undesirable effects of reporting channels, which are errors and overhead traffic. First, it uses an energy efficient clustering algorithm to maximize the CRN lifetime. Second, in each cluster, an incremental weighing fusion rule is used to improve the accuracy of local sensing performed by secondary users. Third, it selects more reliable improved decisions for sending to the fusion center, to decrease overhead traffic. Fourth, it employs a space–time block code to reduce the probability of errors in reporting channels. We determine the optimal settings of the proposed strategy, such as number of clusters, and their corresponding members by maximizing the achievable throughput of the CRN. Numerical and simulation results will prove the proposed CSS strategy yields the highest throughput for the CRN, while it guarantees the maximum lifetime of CRN, and maximum protection of primary users.

Appendices

Appendix 1

This appendix provides the proof of (7). Toward this, we solve the optimization problem in (1) for cluster m, $1 \le m \le M$, then we extend it to other clusters.

Consider an arbitrary cluster, m, whose number of members has the following relationship with numbers of other clusters’ members,

$$n_{m} = N - n_{q} - \mathop \sum \limits_{{\begin{array}{*{20}c} {i = 1} \\ {i \ne m,q} \\ \end{array} }}^{M} n_{i} ,\quad 1 \le q \le M,\;q \ne m.$$

(45)

In this case, MSDE in (1) can be rewritten in the following form

$$MSDE(\vec{N}_{M} ) = \frac{2}{{\left( {M^{2} - M} \right)}}\left\{ {\mathop \sum \limits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne m,q} \\ \end{array} }}^{M} \left[ {E_{q} \left( {n_{q} } \right) - E_{j} \left( {n_{j} } \right)} \right]^{2} + \mathop \sum \limits_{{\begin{array}{*{20}c} {i = 1} \\ {i \ne m,q} \\ \end{array} }}^{M} \left[ {E_{m} \left( {n_{m} } \right) - E_{i} \left( {n_{i} } \right)} \right]^{2} + \left[ {E_{q} \left( {n_{q} } \right) - E_{m} \left( {n_{m} } \right)} \right]^{2} + X} \right\}.$$

(46)

where $X = \mathop \sum \nolimits_{{\begin{array}{*{20}c} {i = 1} \\ {i \ne m,q} \\ \end{array} }}^{M - 1} \mathop \sum \nolimits_{{\begin{array}{*{20}c} {j = i + 1} \\ {j \ne m,q} \\ \end{array} }}^{M} \left[ {E_{i} \left( {n_{i} } \right) - E_{j} \left( {n_{j} } \right)} \right]^{2}$. Substituting (45) into (46), the constraint of (1) is satisfied (i.e. $\mathop \sum \nolimits_{j = 1}^{M} n_{j} = N$). Therefore, (1) converts to the following unconstrained case:

$$\mathop {\hbox{min} }\limits_{{\begin{array}{*{20}c} {n_{q} } \\ {1 \le q \le M,q \ne m} \\ \end{array} }} MSDE\left( {n_{q} } \right) = \frac{2}{{\left( {M^{2} - M} \right)}}\left\{ {\mathop \sum \limits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne m,q} \\ \end{array} }}^{M} \left[ {E_{q} \left( {n_{q} } \right) - E_{j} \left( {n_{j} } \right)} \right]^{2} + \mathop \sum \limits_{{\begin{array}{*{20}c} {i = 1} \\ {i \ne m,q} \\ \end{array} }}^{M} \left[ {E_{m} \left( {N - \mathop \sum \limits_{{\begin{array}{*{20}c} {i = 1} \\ {i \ne m,q} \\ \end{array} }}^{M} n_{i} - n_{q} } \right) - E_{i} \left( {n_{i} } \right)} \right]^{2} + \left[ {E_{q} \left( {n_{q} } \right) - E_{m} \left( {N - \mathop \sum \limits_{{\begin{array}{*{20}c} {i = 1} \\ {i \ne m,q} \\ \end{array} }}^{M} n_{i} - n_{q} } \right)} \right]^{2} +\, X} \right\}$$

(47)

In (47), $MSDE\left( {n_{q} } \right)$ is a function with discrete variable $n_{q}$. Hence, we define

$$D\left( {n_{q} } \right) = MSDE\left( {n_{q} + 1} \right) - MSDE\left( {n_{q} } \right).$$

(48)

Substituting $MSDE\left( {n_{q} } \right)$ into (48), and using (45), we obtain

$$D\left( {n_{q} } \right) = \left( {E_{q} \left( {n_{q} + 1} \right) - E_{q} \left( {n_{q} } \right)} \right)\left( {H_{1} \left( {n_{q} } \right) + H_{1} \left( {n_{q} + 1} \right)} \right) + \left( {E_{m} \left( {n_{m} - 1} \right) - E_{m} \left( {n_{m} } \right)} \right)\left( {H_{2} \left( {n_{q} } \right) + H_{2} \left( {n_{q} + 1} \right)} \right),$$

(49)

where

$$H_{1} \left( {n_{q} } \right) = \left( {m - 1} \right)E_{q} \left( {n_{q} } \right) - E_{m} \left( {n_{m} } \right) - \mathop \sum \limits_{{\begin{array}{*{20}c} {i = 1} \\ {i \ne m,q} \\ \end{array} }}^{M} E_{i} \left( {n_{i} } \right),$$

(50)

$$H_{2} \left( {n_{q} } \right) = \left( {m - 1} \right)E_{m} \left( {n_{m} } \right) - E_{q} \left( {n_{q} } \right) - \mathop \sum \limits_{{\begin{array}{*{20}c} {i = 1} \\ {i \ne m,q} \\ \end{array} }}^{M} E_{i} \left( {n_{i} } \right).$$

(51)

According to (2)–(6), we know that $E_{q} \left( {n_{q} } \right)$ increases by increasing $n_{q}$. So, we can define the following two cases:

Case 1

If $E_{q} \left( {n_{q} + 1} \right) \le E_{m} \left( {n_{m} - 1} \right)$ for $\forall q,\forall m,$ $q \ne m$, then $H_{1} \left( {n_{q} + 1} \right) \le 0$ and $H_{2} \left( {n_{q} + 1} \right) \ge 0$. Furthermore, from (50) and (51), we have $H_{1} \left( {n_{q} } \right) < H_{1} \left( {n_{q} + 1} \right)$ and $H_{2} \left( {n_{q} + 1} \right) < H_{2} \left( {n_{q} } \right)$. Hence, in this case, by looking at (49), we have $D\left( {n_{q} } \right) < 0$.

Case 2

If $E_{q} \left( {n_{q} - 1} \right) \ge E_{m} \left( {n_{m} + 1} \right)$ for $\forall q,\forall m,$ $q \ne m$, then $H_{1} \left( {n_{q} - 1} \right) \ge 0$ and $H_{2} \left( {n_{q} - 1} \right) \le 0$. Furthermore, from (50) and (51), $H_{1} \left( {n_{q} - 1} \right) < H_{1} \left( {n_{q} } \right)$ and $H_{2} \left( {n_{q} } \right) < H_{2} \left( {n_{q} - 1} \right)$. Hence, in this case, by looking at (49), we have $D\left( {n_{q} - 1} \right) > 0$.

Furthermore, by using the properties of discrete-variable functions, solution of (47), $\hat{n}_{q} , q = 1, \ldots ,m - 1,m + 1, \ldots ,M,$ must satisfy the following inequalities

$$D\left( {\hat{n}_{q} } \right) \ge 0,$$

(52)

$$D\left( {\hat{n}_{q} - 1} \right) \le 0.$$

(53)

Therefore, $E_{q} \left( {\hat{n}_{q} + 1} \right) > E_{m} \left( {\hat{n}_{m} - 1} \right)$ and $E_{q} \left( {\hat{n}_{q} - 1} \right) < E_{m} \left( {\hat{n}_{m} + 1} \right)$ for $\forall q,\forall m,$ $q \ne m$ are the necessary conditions for solution of (47), which can rewrite as

$$\left\lfloor {E_{q}^{ - 1} \left( {E_{m} \left( {\hat{n}_{m} - 1} \right)} \right)} \right\rfloor \le \hat{n}_{q} \le \left\lceil {E_{q}^{ - 1} \left( {E_{m} \left( {\hat{n}_{m} + 1} \right)} \right)} \right\rceil , 1 \le q \le M, q \ne m.$$

(54)

where $\hat{n}_{m} = N - \mathop \sum \nolimits_{{\begin{array}{*{20}c} {i = 1} \\ {i \ne m,q} \\ \end{array} }}^{M} \hat{n}_{i} - \hat{n}_{q}$. To obtain the exact value of $\hat{n}_{m}$, we add M-1 inequalities (54) together as follows

$$\mathop \sum \limits_{{\begin{array}{*{20}c} {q = 1} \\ {q \ne m} \\ \end{array} }}^{M} \left\lfloor {E_{q}^{ - 1} \left( {E_{m} \left( {\hat{n}_{m} - 1} \right)} \right)} \right\rfloor \le N - \hat{n}_{m} \le \mathop \sum \limits_{{\begin{array}{*{20}c} {q = 1} \\ {q \ne m} \\ \end{array} }}^{M} \left\lceil {E_{q}^{ - 1} \left( {E_{m} \left( {\hat{n}_{m} + 1} \right)} \right)} \right\rceil , 1 \le m \le M.$$

(55)

Hence, we get (7) that we wanted.

Appendix 2

In this appendix, we derive (23) and (24). By using $\vec{\varvec{X}}_{k}^{m} ,$ $\vec{\varvec{A}}_{k}^{a} ,$ $\vec{\varvec{B}}_{k}^{b}$ and $\vec{\varvec{C}}_{k}^{c}$ defined in Theorem 1, the probability of $x_{k}^{m} = 1$ for $y \in \left\{ {0,1} \right\}$ is given as

$$\Pr \left\{ {x_{k}^{m} = 1|H_{y} } \right\} = { \Pr }\left\{ {\left. {w_{m} \mathop \sum \limits_{i = 1}^{k - 1} \hat{x}_{i,k}^{m} + d_{k}^{m} \ge \psi_{IW} \left( {m,k} \right)} \right|H_{y} } \right\} = \mathop \sum \limits_{a = 0}^{{2^{k - 1} - 1}} { \Pr }\left\{ {\left. {w_{m} \mathop \sum \limits_{i = 1}^{k - 1} A_{i}^{a} + d_{k}^{m} \ge \psi_{IW} \left( {m,k} \right)} \right|\vec{\varvec{X}}_{k}^{m} = \vec{\varvec{A}}_{k}^{a} } \right\}{ \Pr }\left\{ {\vec{\varvec{X}}_{k}^{m} = \left. {\vec{\varvec{A}}_{k}^{a} } \right|H_{y} } \right\} = \mathop \sum \limits_{a = 0}^{{2^{k - 1} - 1}} \left( {1 - F_{k} \left( {\left. {\vec{\varvec{A}}_{k}^{a} } \right|H_{y} } \right)} \right){ \Pr }\left\{ {\vec{\varvec{X}}_{k}^{m} = \left. {\vec{\varvec{A}}_{k}^{a} } \right|H_{y} } \right\},$$

(56)

where $F_{k} \left( {\left. {\vec{\varvec{A}}_{k}^{a} } \right|H_{y} } \right)$ is the CDF of random variable $d_{j}^{m} ,$ which is defined in (27).

Now, we evaluate the pmf of $\vec{\varvec{X}}_{k}^{m} ,$ ${ \Pr }\left\{ {\vec{\varvec{X}}_{k}^{m} = \left. {\vec{\varvec{A}}_{k}^{a} } \right|H_{y} } \right\},$ as

$${ \Pr }\left\{ {\vec{\varvec{X}}_{k}^{m} = \left. {\vec{\varvec{A}}_{k}^{a} } \right|H_{y} } \right\} = \mathop \sum \limits_{b = 0}^{{2^{k - 1} - 1}} { \Pr }\left\{ {\left. {\vec{\varvec{X}}_{k}^{m} = \vec{\varvec{A}}_{k}^{a} } \right|\vec{\varvec{Z}}_{k}^{m} = \vec{\varvec{B}}_{k}^{b} } \right\}{ \Pr }\left\{ {\vec{\varvec{Z}}_{k}^{m} = \left. {\vec{\varvec{B}}_{k}^{b} } \right|H_{y} } \right\},$$

(57)

where $\vec{\varvec{Z}}_{k}^{m} = \left\{ {x_{1}^{m} ,x_{2}^{m} , \ldots ,x_{k - 1}^{m} } \right\}$ is the vector of decisions that transmitted from $k - 1$ SUs. The elements of $\vec{\varvec{X}}_{k}^{m}$ are conditionally independent for a given $\vec{\varvec{Z}}_{k}^{m}$. So,

$$\begin{aligned} { \Pr }\left\{ {\vec{\varvec{X}}_{k}^{m} = \left. {\vec{\varvec{A}}_{k}^{a} } \right|\vec{\varvec{Z}}_{k}^{m} = \vec{\varvec{B}}_{k}^{b} } \right\} & = \prod\limits_{j = 2}^{k - 1} {{ \Pr }\left\{ {\hat{x}_{j,k}^{m} = A_{j}^{a} |x_{j}^{m} = B_{j}^{b} } \right\}} \\ & = \prod\limits_{j = 1}^{k - 1} { ( {\text{P}}_{e1} )^{{\left| {A_{j}^{a} - B_{j}^{b} } \right|}} (1 - P_{e1} )^{{\left| {1 - A_{j}^{a} - B_{j}^{b} } \right|}} } , \\ \end{aligned}$$

(58)

where $P_{e1} = { \Pr }\left\{ {\left. {A_{j}^{a} \ne B_{j}^{b} } \right|x_{j}^{m} = B_{j}^{b} ,\hat{x}_{j,k}^{m} = A_{j}^{a} } \right\} = P_{e}^{in} \left( {SNR_{in}^{m} } \right).$ Furthermore, $x_{j}^{m}$ depends on $j - 1$ previous SUs’ decisions, $\vec{\varvec{Z}}_{j}^{m}$. So, we have

$${ \Pr }\left\{ {\left. {\vec{\varvec{Z}}_{k}^{m} = \vec{\varvec{B}}_{k}^{b} } \right|H_{y} } \right\} = { \Pr }\left\{ {\left. {x_{1}^{m} = B_{1}^{b} } \right|H_{y} } \right\}\mathop \prod \limits_{j = 2}^{k - 1} { \Pr }\left\{ {\left. {x_{j}^{m} = B_{j}^{b} } \right|\vec{\varvec{Z}}_{j}^{m} = \vec{\varvec{B}}_{j}^{b} ,H_{y} } \right\}.$$

(59)

Now,

$$\begin{aligned} { \Pr }\left\{ {\left. {x_{j}^{m} = 1} \right|\vec{\varvec{Z}}_{j}^{m} = \vec{\varvec{B}}_{j}^{b} ,H_{y} } \right\} & = \mathop \sum \limits_{c = 0}^{{2^{j - 1} - 1}} { \Pr }\left\{ {\left. {x_{j}^{m} = 1} \right|\vec{\varvec{X}}_{j}^{m} = \vec{\varvec{C}}_{j}^{c} ,\vec{\varvec{Z}}_{j}^{m} = \vec{\varvec{B}}_{j}^{b} ,H_{y} } \right\}{ \Pr }\left\{ {\left. {\vec{\varvec{X}}_{j}^{m} = \vec{\varvec{C}}_{j}^{c} } \right|\vec{\varvec{Z}}_{j}^{m} = \vec{\varvec{B}}_{j}^{b} } \right\} \\ & = \mathop \sum \limits_{c = 0}^{{2^{j - 1} - 1}} \left\{ {{ \Pr }\left\{ {\left. {x_{j}^{m} = 1} \right|\vec{\varvec{X}}_{j}^{m} = \vec{\varvec{C}}_{j}^{c} ,H_{y} } \right\}\mathop \prod \limits_{i = 1}^{j - 1} \left( {P_{e2} } \right)^{{\left| {B_{i}^{b} - C_{i}^{c} } \right|}} \left( {1 - P_{e2} } \right)^{{1 - \left| {B_{i}^{b} - C_{i}^{c} } \right|}} } \right\} \\ & = \mathop \sum \limits_{c = 0}^{{2^{j - 1} - 1}} \left\{ {\left( {1 - F_{j} \left( {\left. {\vec{\varvec{C}}_{j}^{c} } \right|H_{y} } \right)} \right)\left( {P_{e2} } \right)^{{\mathop \sum \limits_{i = 1}^{j - 1} \left| {B_{i}^{b} - C_{i}^{c} } \right|}} \left( {1 - P_{e2} } \right)^{{j - 1 - \mathop \sum \limits_{i = 1}^{j - 1} \left| {B_{i}^{b} - C_{i}^{c} } \right|}} } \right\} \\ \end{aligned}$$

(60)

and

$$\begin{aligned} { \Pr }\left\{ {\left. {x_{j}^{m} = 0} \right|\vec{\varvec{Z}}_{j}^{m} = \vec{\varvec{B}}_{j}^{b} ,H_{y} } \right\} & = \mathop \sum \limits_{c = 0}^{{2^{j - 1} - 1}} \left\{ {\left( {1 - { \Pr }\left\{ {\left. {x_{j}^{m} = 1} \right|\vec{\varvec{X}}_{j}^{m} = \vec{\varvec{C}}_{j}^{c} ,H_{y} } \right\}} \right)\mathop \prod \limits_{i = 1}^{j - 1} \left( {P_{e2} } \right)^{{\left| {B_{i}^{b} - C_{i}^{c} } \right|}} \left( {1 - P_{e2} } \right)^{{1 - \left| {B_{i}^{b} - C_{i}^{c} } \right|}} } \right\} \\ & = 1 - \mathop \sum \limits_{c = 0}^{{2^{j - 1} - 1}} \left\{ {\left( {1 - F_{j} \left( {\left. {\vec{\varvec{C}}_{j}^{c} } \right|H_{y} } \right)} \right)\left( {P_{e2} } \right)^{{\mathop \sum \limits_{i = 1}^{j - 1} \left| {B_{i}^{b} - C_{i}^{c} } \right|}} \left( {1 - P_{e2} } \right)^{{j - 1 - \mathop \sum \limits_{i = 1}^{j - 1} \left| {B_{i}^{b} - C_{i}^{c} } \right|}} } \right\} \\ & = 1 - { \Pr }\left\{ {\left. {x_{j}^{m} = 1} \right|\vec{\varvec{Z}}_{j}^{m} = \vec{\varvec{B}}_{j}^{b} ,H_{y} } \right\} \\ \end{aligned}$$

(61)

Using $p_{j} \left( {\left. {\vec{\varvec{B}}_{k}^{b} } \right|H_{y} } \right)$ defined in (26), and (60) and (61), we can rewrite (59) as

$${ \Pr }\left\{ {\left. {\vec{\varvec{Z}}_{k}^{m} = \vec{\varvec{B}}_{k}^{b} } \right|H_{y} } \right\} = \mathop \prod \limits_{j = 1}^{k - 1} p_{j} \left( {\left. {\vec{\varvec{B}}_{j}^{b} } \right|H_{y} } \right)^{{B_{j}^{b} }} \left( {1 - p_{j} \left( {\left. {\vec{\varvec{B}}_{j}^{b} } \right|H_{y} } \right)} \right)^{{1 - B_{j}^{b} }} .$$

(62)

Substituting (62) into (56), then we get the same formula for $y = 0$ and $y = 1$ as (23) and (24) that we wanted.