Modeling of spectrum sharing using ITLinQ scheme in device-to-device networks with full-duplex relays

Abstract

The wide utilization of frequency spectrum sharing causes problems such as interference as the result of active links’ effects on each other. One of the proposed methods to decrease the interference in device-to-device networks is information-theoretic links scheduling (ITLinQ), where it schedules links by using information theory optimally conditions. In this paper, the idea of simultaneous using frequency sharing and reliable full-duplex relaying (FDR) is proposed. Although FDR causes self-interference when the same frequency spectrum is used for reception and transmission at the relays simultaneously, it is shown that the proposed system can increase the transmission rate and coverage area. Moreover, a relay selection scheme to select relays, and a power allocation mechanism between sources and relays to allocate appropriate power are presented. This is investigated by an analytical study to validate the superiority of the proposed method over the ITLinQ scheme in improving the total rate in the network. In addition, the simulation results confirm that by using FDR with the ITLinQ scheme, the better achievable sum-rate of users is obtained.

Appendix

In the proposed scheme, the objective is to maximize the sum of the SINR of links according to the optimal e variables (28). There is one constraint to acquire the optimal coefficients of power dedication to transmitters and relays (29). This constraint is between the ${e}_{min}$ and${e}_{max}$, which $P_{{S_{j} }}$ and $P_{{R_{j} }}$ are defined based on the ${e}_{min}$ and ${e}_{max}$ (30, 31). At first by substituting $and$ according to the ${e}_{j},$ also, $P_{{S_{i} }}$ and $P_{{R_{i} }}$ according to the ${e}_{i},$ the following equations are concluded (39–44).

Then we calculate the gradient of main formula of SINR. Derivative of SINR is conducted. according ${e}_{j}$ and${e}_{i}$. For convenience of calculations, the following formulas are equivalent to $\varphi ,\varphi_{1} ,\tau ,\tau_{1} ,\vartheta ,\vartheta_{1} ,\ell ,\ell_{1} ,\xi ,\xi_{1}$ sequentially. Here, φ and $\varphi_{1}$ denote to the $G_{j}^{ - 2}$ and $G_{i}^{ - 2}$ which mean the inverse of the second power of coefficient of j-th relay amplification and i-th relay amplification, respectively. Also, $\xi$ and $\xi_{1}$ are SI coefficient of j-th relay and i-th relay. The power of j-th source and the power of i-th source is shown by p-$\tau$ and p-$\tau_{1}$. Also, the power of j-th relay and the power of i-th relay is shown by $\tau$ and $\tau_{1}$, respectively.

$$ \begin{gathered} \varphi = \frac{{e_{j} p}}{{\mathop d\nolimits_{{1_{jj} }}^{m} }}\left| {h_{{1_{jj} }} } \right|^{2} + (1 - e_{j} )p\left| {h_{{2_{jj} }} } \right|^{2} + \sigma_{n}^{2} ,\varphi_{1} = \frac{{e_{i} p}}{{\mathop d\nolimits_{{1_{ii} }}^{m} }}\left| {h_{{1_{ii} }} } \right|^{2} + (1 - e_{i} )p\left| {h_{{2_{ij} }} } \right|^{2} + \sigma_{n}^{2} , \hfill \\ p_{{R_{j} }} = (1 - e_{j} )p = \tau ,p_{{S_{j} }} = (e_{j} p) = p - \tau ,p_{{R_{i} }} = (1 - e_{i} )p = \tau_{1} ,p_{{S_{i} }} = (e_{i} p) = p - \tau_{1} , \hfill \\ \frac{{h_{{1_{ij} }} }}{{\sqrt {d_{{1_{ij} }}^{m} } }} = \ell ,\frac{{h_{{2_{ii} }} }}{{\sqrt {d_{{2_{ii} }}^{m} } }} = \ell ,_{1} \frac{{h_{{1_{jj} }} }}{{\sqrt {d_{{1_{jj} }}^{m} } }} = \vartheta ,\frac{{h_{{2_{jj} }} }}{{\sqrt {d_{{2_{jj} }}^{m} } }} = \vartheta_{1} ,\mathop h\limits^{\sim}_{{R_{j} }} = \xi ,\mathop h\limits^{\sim}_{{R_{i} }} = \xi_{1} , \hfill \\ \end{gathered} $$

$$ {\text{S}}_{j} = \vartheta \vartheta_{1} x_{{S_{j} }} \frac{{\tau (\sqrt {(p - \tau )} )}}{\sqrt \varphi },\frac{{\partial {\text{S}}_{j} }}{{\partial e_{j} }} = \vartheta \vartheta_{1} x_{{S_{j} }} p\frac{{((\tau - 0.5p)\phi - (0.5(\vartheta^{2} - \left| {h_{{2_{jj} }} } \right|^{2} )\tau (p - \tau ))}}{{\varphi \sqrt \varphi \sqrt {(p - \tau )\tau } }},\frac{{\partial {\text{S}}_{j} }}{{\partial e_{i} }} = 0, $$

(39)

$$ \begin{gathered} \text{IN}_{{R_{jj} }} = \vartheta_{1} \xi \frac{{\tau (\sqrt {(p - \tau )} \vartheta x_{{S_{j} }} + N_{{R_{j} }} )}}{{\varphi (1 - \xi \frac{\sqrt \tau }{{\sqrt \varphi }})}} = \vartheta_{1} \xi \frac{{\tau (\sqrt {(p - \tau )} \vartheta x_{{S_{j} }} + N_{{R_{j} }} )}}{{\varphi (\frac{\sqrt \varphi - \xi \sqrt \tau }{{\sqrt \varphi }})}} = \vartheta_{1} \xi \frac{{\tau (\sqrt {(p - \tau )} \vartheta x_{{S_{j} }} + N_{{R_{j} }} )}}{{(\varphi - \xi \sqrt {\varphi \tau } )}},\frac{{\partial {\text{IN}}_{{R_{jj} }} }}{{\partial e_{i} }} = 0 \hfill \\ \frac{{\text{IN}_{{R_{jj} }} }}{{\partial e_{j} }} = \vartheta_{1} \xi (\frac{{(p(\frac{(1.5\tau - p)}{{\sqrt {(p - \tau )} }}\vartheta x_{{_{{S_{j} }} }} )(\varphi - \xi \tau \sqrt \varphi )) - ((\tau \sqrt {(p - \tau )} \vartheta x_{{_{{S_{j} }} }} + N_{{R_{j} }} )((p\vartheta^{2} - p\left| {h_{{2_{jj} }} } \right|^{2} )}}{{(\varphi - \xi \tau \sqrt \varphi )^{2} }} - \hfill \\ \frac{{\xi (\frac{{ - p\sqrt \varphi + \tau (\vartheta^{2} p - p\left| {h_{{2_{jj} }} } \right|^{2} )}}{{2\sqrt {\varphi \tau } }})))}}{{(\varphi - \xi \tau \sqrt \varphi )^{2} }}), \hfill \\ \end{gathered} $$

(40)

$$ {\text{IN}}_{{N_{jj} }} = \vartheta_{1} N_{{R_{j} }} \sqrt {\frac{\tau }{\varphi }} ,\frac{{\partial {\text{IN}}_{{N_{jj} }} }}{{\partial e_{j} }} = \vartheta_{1} N_{{R_{j} }} p\frac{{( - \varphi + \tau (\left| {h_{{2_{jj} }} } \right|^{2} - \vartheta^{2} ))}}{{2\varphi \sqrt {\varphi \tau } }},\frac{{\partial {\text{IN}}_{{N_{jj} }} }}{{\partial e_{i} }} = 0, $$

(41)

$$ \text{IN}_{{N_{ij} }} = \sum\limits_{\begin{subarray}{l} i = 1, \\ i \ne j \end{subarray} }^{i = j - 1} {\ell_{1} } N_{{R_{i} }} \sqrt {\frac{{\tau_{1} }}{{\varphi_{1} }}} ,\frac{{\partial {\text{IN}}_{{N_{ij} }} }}{{\partial e_{i} }} = \sum\limits_{\begin{subarray}{l} i = 1, \\ i \ne j \end{subarray} }^{i = j - 1} {\ell_{1} } N_{{R_{i} }} p\frac{{(\varphi_{1} + (\ell^{2} - \left| {h_{{2_{ij} }} } \right|^{2} )\tau_{1} )}}{{ - 2\varphi_{1} \sqrt {\varphi_{1} \tau_{1} } }},\frac{{\partial {\text{IN}}_{{N_{ij} }} }}{{\partial e_{j} }} = 0, $$

(42)

$$ \begin{gathered} {\text{IN}}_{{R_{ij} }} = \sum\limits_{\begin{subarray}{l} i = 1 \\ j \ne i \end{subarray} }^{i = j - 1} {\ell_{1} \xi_{1} } \frac{{(\tau_{1} \sqrt {(p - \tau_{1} )} \ell x_{{S_{i} }} + N_{{R_{j} }} )}}{{(\varphi_{1} - \xi_{1} \sqrt {\tau_{1} \varphi_{1} } )}},\frac{{\partial {\text{IN}}_{{R_{ij} }} }}{{\partial e_{j} }} = 0, \hfill \\ \frac{{\partial {\text{IN}}_{{R_{ij} }} }}{{\partial e_{i} }} = \sum\limits_{\begin{subarray}{l} i = 1 \\ j \ne i \end{subarray} }^{i = j - 1} {\ell_{1} \xi_{1} } (\frac{{(p\frac{{(1.5\tau_{1} - p)}}{{\sqrt {(p - \tau_{1} )} }}\ell x_{{S_{i} }} )(\varphi_{1} - \xi_{1} \tau_{1} \sqrt {\varphi_{1} } ) - (\tau_{1} \sqrt {(p - \tau_{1} )} \ell x_{{S_{i} }} + N_{{R_{j} }} )(p(\ell_{1}^{2} - \left| {h_{{2_{ij} }} } \right|^{2} ) - \xi_{1} (\frac{{ - p\sqrt {\varphi_{1} } + p\tau_{1} (\ell_{1}^{2} - \left| {h_{{2_{ij} }} } \right|^{2} )}}{{2\sqrt {\varphi_{1} \tau_{1} } }}))}}{{(\varphi_{1} - \xi_{1} \tau_{1} \sqrt {\varphi_{1} } )^{2} }}), \hfill \\ \end{gathered} $$

(43)

$$ {\text{IN}}_{{RD_{ij} }} = \sum\limits_{\begin{subarray}{l} i = 1, \\ i \ne j \end{subarray} }^{i = j - 1} {\ell \ell_{1} } x_{{s_{i} }} \sqrt {\frac{{\tau_{1} }}{{\varphi_{1} }}} ,\frac{{\partial {\text{IN}}_{{RD_{ij} }} }}{{\partial e_{i} }} = \sum\limits_{\begin{subarray}{l} i = 1, \\ i \ne j \end{subarray} }^{i = j - 1} {\ell \ell_{1} } x_{{s_{i} }} p\frac{{(\varphi_{1} + 0.5\tau_{1} p(\ell_{1}^{2} - \left| {h_{{2_{ij} }} } \right|^{2} ))}}{{ - 2\varphi_{1} \sqrt {\varphi_{1} \tau_{1} } }},\frac{{\partial {\text{IN}}_{{RD_{ij} }} }}{{\partial e_{j} }} = 0. $$

(44)

After calculating the SINR of links based on the ${e}_{i}$ and ${e}_{j}$, the partial derivative of the SINR of links respect to the ${e}_{i}$ and ${e}_{j}$ are obtained as follow:

$$ \begin{gathered} \frac{\begin{gathered} \frac{{\partial {\text{SINR}}_{j} }}{{\partial e_{j} }} = ((2\vartheta^{2} \vartheta_{1}^{2} x_{{S_{j} }}^{2} p\frac{{((\tau - 0.5p) - {(}0.5(\vartheta^{2} - \left| {h_{{2_{jj} }} } \right|^{2} )\tau (p - \tau ))}}{{\varphi^{2} }}((\vartheta \xi \frac{{\tau (\sqrt {(p - \tau )} \vartheta x_{s} + N_{{R_{j} }} }}{{(\varphi - \xi \sqrt {\varphi \tau } )}})^{2} + (\vartheta_{1} N_{{R_{j} }} \sqrt {\frac{\tau }{\varphi }} )^{2} + \sigma_{{N_{j} }}^{2} )) \hfill \\ - 2((\vartheta \vartheta_{1} x_{{S_{j} }} \frac{{\sqrt {\tau (p - \tau )} }}{\sqrt \varphi })^{2} )(\vartheta_{1} \xi \frac{{\tau (\sqrt {(p - \tau )} \vartheta x_{{S_{j} }} + N_{{R_{j} }} }}{{{(}\vartheta - \xi \sqrt {\varphi \tau } )}} + \vartheta_{1} N_{{R_{j} }} \sqrt {\frac{\tau }{\varphi }} ) \hfill \\ ((\vartheta_{1} N_{{R_{j} }} p\frac{{( - \varphi + \tau (\left| {h_{{2_{jj} }} } \right|^{2} - \vartheta^{2} )}}{{2\varphi \sqrt {\varphi \tau } }}) + \vartheta_{1} \xi (\frac{{(p(\frac{(1.5\tau - p)}{{\sqrt {(p - \tau )} }}\vartheta x_{{S_{j} }} )(\varphi - \xi \tau \sqrt \varphi )) - ((\tau \sqrt {(p - \tau )} \vartheta x_{{S_{j} }} + N_{{R_{j} }} )((p\vartheta^{2} - p\left| {h_{{2_{jj} }} } \right|^{2} )}}{{(\vartheta - \xi \tau \sqrt \varphi )^{2} }} \hfill \\ \end{gathered} }{{((\vartheta_{1} \xi \frac{{\tau (\sqrt {(p - \tau )} \vartheta x_{{S_{j} }} + N_{{R_{j} }} }}{{(\varphi - \xi \sqrt {\varphi \tau } )}})^{2} + (\vartheta_{1} N_{{R_{j} }} \sqrt {\frac{\tau }{\varphi }} )^{2} + \sigma_{N}^{2} ))^{2} }} - \hfill \\ \frac{{\frac{{ - \xi (\frac{{ - p\sqrt \varphi + \tau (\vartheta^{2} p - p\left| {h_{{2_{jj} }} } \right|^{2} )}}{{2\sqrt {\varphi \tau } }})))}}{{(\vartheta - \xi \tau \sqrt \varphi )^{2} }}}}{{((\vartheta_{1} \xi \frac{{\tau (\sqrt {(p - \tau )} \vartheta x_{{S_{j} }} + N_{{R_{j} }} }}{{(\varphi - \xi \sqrt {\varphi \tau } )}})^{2} + (\vartheta_{1} N_{{R_{j} }} \sqrt {\frac{\tau }{\varphi }} )^{2} + \sigma_{N}^{2} ))^{2} }}))), \hfill \\ \end{gathered} $$

(45)

$$ \frac{\begin{gathered} \frac{{\partial {\text{SINR}}_{j} }}{{\partial e_{i} }} = ( - 2(\vartheta_{1} \xi \frac{{\tau (\sqrt {(p - \tau )} \vartheta x_{{S_{j} }} + N_{{R_{j} }} )}}{{(\varphi - \xi \sqrt {\varphi \tau } )}} + \vartheta_{1} N_{{R_{j} }} \sqrt {\frac{\tau }{\varphi }} )(\sum\limits_{\begin{subarray}{l} i = 1, \\ i \ne j \end{subarray} }^{i = j - 1} {\ell_{1} } N_{{R_{i} }} p\frac{{\varphi_{1} + (\ell_{1}^{2} - \left| {h_{{2_{ij} }} } \right|^{2} )\tau_{1} }}{{ - 2\varphi_{1} \sqrt {\varphi_{1} \tau_{1} } }} + \hfill \\ \sum\limits_{\begin{subarray}{l} i = 1 \\ j \ne i \end{subarray} }^{i = j - 1} {\xi_{1} \ell_{1} } (\frac{{(p\frac{{(1.5\tau_{1} - p)}}{{\sqrt {(p - \tau_{1} )} }}\ell_{1} x_{{S_{i} }} )(\varphi_{1} - \xi_{1} \tau_{1} \sqrt {\varphi_{1} } ) - (\tau_{1} \sqrt {(p - \tau_{1} )} \ell x_{{S_{i} }} + N_{{R_{j} }} )(p(\ell_{1}^{2} - \left| {h_{{2_{ij} }} } \right|^{2} ) - \xi_{1} (\frac{{ - p\sqrt {\varphi_{1} } + p\tau_{1} (\ell_{1}^{2} - \left| {h_{{2_{ij} }} } \right|^{2} )}}{{2\sqrt {\varphi_{1} \tau_{1} } }}))}}{{(\varphi_{1} - \xi_{1} \tau_{1} \sqrt {\varphi_{1} } )^{2} }} + \hfill \\ \sum\limits_{\begin{subarray}{l} i = 1, \\ i \ne j \end{subarray} }^{i = j - 1} {\ell_{1} \ell } x_{{s_{i} }} p\frac{{{(}\varphi_{1} + 0.5\tau_{1} p(\ell_{1}^{2} - \left| {h_{{2_{ij} }} } \right|^{2} )}}{{ - 2\varphi_{1} \sqrt {\varphi_{1} \tau_{1} } }})(\vartheta \vartheta_{1} x_{{S_{j} }} \frac{{\sqrt {\tau (p - \tau )} }}{\sqrt \varphi })^{2} )) \hfill \\ \end{gathered} }{{((\vartheta_{1} \xi \frac{{\tau_{1} (\sqrt {(p - \tau )} \vartheta x_{{S_{j} }} + N_{{R_{j} }} }}{{(\varphi - \xi \sqrt {\varphi \tau } )}})^{2} + (\vartheta_{1} N_{{R_{j} }} \sqrt {\frac{\tau }{\varphi }} )^{2} + \sigma_{N}^{2} ))^{2} }}. $$

(46)

The optimal variable value ${e}_{j}$ and ${e}_{i}$ are obtained by setting the gradient of SINR to zero. Then by applying these achieved values, power is allocated to users in the network.