Orthogonal Signaling Enabled Cooperative Cognitive Radio Networking

Abstract

In this chapter, a cross-layer two-phase time division multiple access (TDMA) cooperation framework for primary users (PUs) and secondary user (SUs) in a cooperative cognitive radio network (CCRN) is proposed and analyzed. Specifically, the cooperation framework in which the SU uses the two-dimensional orthogonal modulation for leveraging two degrees of freedom to relay the PU’s packet and transmit its own data orthogonally in the same time slot is firstly explored. To evaluate the cooperation performance of the proposed framework, a weighted sum throughput maximization problem is then formulated. With the help of primal-dual sub-gradient algorithms, the optimization problem is solved to obtain closed-form solutions to the optimal powers and allocation of the PU and the SU for both the amplify-and-forward (AF) and decode-and-forward (DF) relaying modes. Cooperative regions based on channel state information are given and discussed, and a cross-layer multi-user coordination for a PU to select a relaying SU for both AF and DF are presented. Extensive simulation results validate the theoretical analysis and show that the proposed two-phase TDMA cooperation framework can achieve mutual benefit in the CCRN.

Appendix

2.1.1 Derivation of Optimal Solutions for AF Cooperation

Using the Lagrange multiplier method, the objective function in (2.4) can be expressed as

$$\displaystyle\begin{array}{rcl} \mathcal{L}(\alpha,P_{P},P_{S})& =& (1-\zeta )C_{PB\_AF} +\zeta C_{SB\_AF} \\ & & +\lambda _{1}(C_{Pd} - C_{PB\_AF}) +\lambda _{2}(C_{ST} - C_{SB\_AF}) \\ & & +\lambda _{3}(P_{P} + P_{S} - P_{M}) +\lambda _{4}(\alpha -1) -\lambda _{5}\alpha {}\end{array}$$

(2.21)

where $\lambda _{i} \geq 0$ (i = 1, . . , 5) are Lagrange multipliers.

By applying the KKT conditions, a set of equations can be obtained as

$$\displaystyle{ P_{P} > 0,P_{S} > 0,\text{and}\quad \lambda _{i} \geq 0,\text{for}\quad i = 1,\ldots 5 }$$

(2.22a)

$$\displaystyle{ \lambda _{1}[C_{Pd} -\frac{1} {2}\log _{2}(1 + SNR_{PB\_AF})] = 0 }$$

(2.22b)

$$\displaystyle{ \lambda _{2}[C_{ST} -\frac{1} {2}\log _{2}(1 + SNR_{SB\_AF})] = 0 }$$

(2.22c)

$$\displaystyle{ \lambda _{3}(P_{P} + P_{S} - P_{M}) = 0 }$$

(2.22d)

$$\displaystyle{ \lambda _{4}(\alpha -1) = 0 }$$

(2.22e)

$$\displaystyle{ \lambda _{5}\alpha = 0 }$$

(2.22f)

$$\displaystyle{ \frac{\partial \mathcal{L}} {\partial \alpha } = 0, \frac{\partial \mathcal{L}} {\partial P_{P}} = 0\quad \text{and}\quad \frac{\partial \mathcal{L}} {\partial P_{S}} = 0. }$$

(2.22g)

Solving the equation set yields

$$\displaystyle{ \alpha ^{{\ast}} = \frac{M(P_{P}\gamma _{\mathit{PS}} +\gamma _{\mathit{SB}}P_{S} + 1)} {N\gamma _{\mathit{SP}}P_{S}} }$$

(2.23)

where M and N are given by

$$\displaystyle\begin{array}{rcl} M& =& (\lambda _{4}\zeta -\lambda _{1})P_{P}^{2}\gamma _{ \mathit{PB}}\gamma _{\mathit{PS}} + (2\zeta -\lambda _{4})P_{P}\gamma _{\mathit{PS}} {}\\ & \ \ & -(\lambda _{1} + P_{S}\gamma _{\mathit{SP}})P_{P}\gamma _{\mathit{PS}} + (\zeta -\lambda _{1})(P_{P}\gamma _{\mathit{PB}} + 1) {}\\ & \ \ & +(\zeta \lambda _{5}\gamma _{\mathit{PS}}\gamma _{\mathit{SB}} -\lambda _{1}\gamma _{\mathit{PB}}\gamma _{\mathit{SP}})P_{S}P_{P} {}\\ & \ \ & +(\zeta P_{P}\gamma _{\mathit{PB}}+\zeta )P_{S}\gamma _{\mathit{SB}} -\lambda _{1}\gamma _{\mathit{SP}}P_{S} {}\\ N& =& (\zeta \lambda _{4}\gamma _{\mathit{SP}} -\lambda _{1}\gamma _{\mathit{SB}})P_{P}\gamma _{\mathit{PS}}P_{S} + (\zeta -\lambda _{1})P_{P}\gamma _{\mathit{PB}}\gamma _{\mathit{SB}}P_{S} {}\\ & \ \ & -\lambda _{1}\gamma _{\mathit{SP}}P_{S} +\gamma _{\mathit{SB}}P_{S} + (\lambda _{5}\zeta -\lambda _{1})P_{P}\gamma _{\mathit{PB}} {}\\ & \ \ & +(\zeta -\lambda _{1})P_{P}^{2}\gamma _{ \mathit{PB}}\gamma _{\mathit{PS}} -\lambda _{1} + 2(\zeta -\lambda _{1})P_{P}\gamma _{\mathit{PS}} {}\\ & \ \ & +(1 -\lambda _{1})P_{P}^{2}\gamma _{ \mathit{PS}}^{2} +\zeta. {}\\ \end{array}$$

The quantity P _P ^∗ is the root of the quadratic equation

$$\displaystyle{ a_{4}P_{P}^{4} + a_{ 3}P_{P}^{3} + a_{ 2}P_{P}^{2} + a_{ 1}P_{P} + a_{0} = 0 }$$

(2.24)

where the coefficients of the equation are

$$\displaystyle\begin{array}{rcl} a_{4}& =& \ln 2\gamma _{\mathit{PB}}\gamma _{\mathit{PS}}^{2}\lambda _{ 2} {}\\ a_{3}& =& 2\ln 2\gamma _{\mathit{PB}}\gamma _{\mathit{PS}}\lambda _{2} -\lambda _{1}\gamma _{\mathit{PS}}^{2}\gamma _{ \mathit{PB}} +\lambda _{3}\gamma _{\mathit{PS}}^{2}\gamma _{ \mathit{PB}} {}\\ & \ \ & +2\ln 2\gamma _{\mathit{PS}}^{2}\lambda _{ 2} + 2\ln 2P_{S}\gamma _{\mathit{PB}}\gamma _{\mathit{PS}}\gamma _{\mathit{SP}}\lambda _{2} {}\\ a_{2}& =& (\gamma _{\mathit{SB}}P_{S} + 1)(\ln 2P_{S}\gamma _{\mathit{PS}}\gamma _{\mathit{SB}}\lambda _{2} +\ln 2P_{S}\gamma _{\mathit{PB}}\gamma _{\mathit{SP}}\lambda _{2} {}\\ & \ \ & +3\lambda _{3}\ln 2\gamma _{\mathit{PS}}\lambda _{2} + 2\gamma _{\mathit{PB}}\gamma _{\mathit{PS}} -\lambda _{1}\gamma _{\mathit{PB}}\gamma _{\mathit{PS}}) {}\\ a_{1}& =& (\gamma _{\mathit{SB}}P_{S} + 1)(-\gamma _{\mathit{PS}}\zeta \gamma _{\mathit{SP}}P_{S} -\zeta \gamma _{\mathit{SP}}P_{S} +\gamma _{\mathit{PB}}\gamma _{\mathit{SB}}P_{S} {}\\ & \ \ & +\ln 2P_{S}\gamma _{\mathit{SB}}\lambda _{2} -\zeta \gamma _{\mathit{PB}} +\lambda _{2}\ln 2 - 2\zeta \gamma _{\mathit{PS}} +\lambda _{3}\gamma _{\mathit{PB}}) {}\\ a_{0}& =& -\lambda _{3}\zeta + 2\lambda _{1}\gamma _{\mathit{SP}}P_{S} - 2\lambda _{3}\zeta \gamma _{\mathit{SB}}P_{S} {}\\ & \ \ & -\zeta \gamma _{\mathit{SB}}^{2}P_{ S}^{2} +\lambda _{ 1}\gamma _{\mathit{SP}}^{2}P_{ S}^{2} +\lambda _{ 1}. {}\\ \end{array}$$

Also,

$$\displaystyle{ P_{S}^{{\ast}} = \left [ \frac{\sqrt{E} - D} {2\lambda _{2}\lambda _{3}\ln 2(\gamma _{\mathit{PS}} + 1)\gamma _{\mathit{SB}}},0\right ]^{+} }$$

(2.25)

where

$$\displaystyle\begin{array}{rcl} D& =& \lambda _{2}\ln 2(2\lambda _{3} + 3\gamma _{\mathit{PS}} + 2\lambda _{3}\gamma _{\mathit{PB}}\gamma _{\mathit{PS}} +\gamma _{\mathit{PB}}) {}\\ E& =& \ln 2\lambda _{2}(\gamma _{\mathit{PB}} -\lambda _{3}\gamma _{\mathit{PS}})(\ln 2\gamma _{\mathit{PB}}\lambda _{2} -\lambda _{3}\ln 2\gamma _{\mathit{PS}}\lambda _{2} {}\\ & \ \ & +4\lambda _{3}\gamma _{\mathit{PS}}\gamma _{\mathit{SB}} - 4\gamma _{\mathit{SB}}\lambda _{1}\gamma _{\mathit{PS}} + 4\gamma _{\mathit{SB}} - 4\lambda _{1}\gamma _{\mathit{SP}}). {}\\ \end{array}$$

This quadratic equation can be solved analytically or numerically, and according to the power constraint of the PU, the optimal P _P can be obtained.

Using the Lagrangian relaxation iterative algorithm, we can further iteratively compute $\lambda _{i}$ by

$$\displaystyle\begin{array}{rcl} \lambda _{1}^{(n+1)}& =& [\lambda _{ 1}^{(n)} +\mu ^{(n)}(C_{ PB} - C_{PB\_AF}^{(n)})]^{+}{}\end{array}$$

(2.26)

$$\displaystyle\begin{array}{rcl} \lambda _{2}^{(n+1)}& =& [\lambda _{ 2}^{(n)} +\mu ^{(n)}(C_{ ST} - C_{SB\_AF}^{(n)})]^{+}{}\end{array}$$

(2.27)

$$\displaystyle\begin{array}{rcl} \lambda _{3}^{(n+1)}& =& [\lambda _{ 3}^{(n)} +\mu ^{(n)}(P_{ P}^{(n)} + P_{ S}^{(n)} - P_{ M})]^{+},{}\end{array}$$

(2.28)

$$\displaystyle\begin{array}{rcl} \lambda _{4}^{(n+1)}& =& [\lambda _{ 4}^{(n)} +\mu ^{(n)}(\alpha ^{(n)} - 1)]^{+}{}\end{array}$$

(2.29)

$$\displaystyle\begin{array}{rcl} \lambda _{5}^{(n+1)}& =& [\lambda _{ 5}^{(n)} +\mu ^{(n)}\alpha ^{(n)}]^{+}{}\end{array}$$

(2.30)

where $[x]^{+} =\max (x,0)$, n is the iteration index and μ ⁽ⁿ⁾ is a sequence of scalar step sizes. Substituting (2.26)–(2.30) into (2.23)–(2.25), we can obtain α ^∗, P _P ^∗, and P _S ^∗.