Power control of D2D communication based on quality of service assurance under imperfect channel information

Abstract

In cellular networks, proximity users establish device-to-device(D2D) links directly under the control of base stations (BSs) for data exchange, and no base station forwarding is required. For the advantages such as lower transmission power, higher data transmission rate, D2D communication has been paid more and more attention. However, the environmental impact of introducing D2D shared spectrum resources has become a major challenge. This paper studies a robust downlink power control scheme with imperfect channel state information (CSI) in D2D communication system. In order to improve the system sum rates, this paper optimizes the transmitting power of D2D users (DUEs) and the base stations through geometric analysis and convex optimization while guaranteeing the quality-of-service (QoS) requirements for both DUEs and cellular users (CUEs). Because of the dynamic characteristics of wireless channel, it is difficult and expensive to acquire CSI without direct communication with BS. On this account, we consider channel uncertainty rather than perfect or instantaneous known CSI. Aiming at enhancing the system robustness, we formulate an outage-based robust optimization problem. The original deterministic the signal-to-interference-noise-ratio (SINR) constraints are substituted with the transformed probability constraints, which transforms the random channel gain into a deterministic one with outage threshold. The simulation results show that the power allocation based on the proposed algorithm can adapt to the complex channel environment and are with better robustness.

Appendices

Appendix A: Proof of (11)

The detailed proof process of Eq. 11 is given as follows. Considering the constraint (10b),

$$ \begin{array}{@{}rcl@{}} \begin{array}{ll} Pr\left[\frac{P_{Bi}K_{Bi}L_{Bi}^{-\beta}}{P_{ji}(1-\alpha)|\hat{h}_{ji}|^{2}L_{ji}^{-\delta}+P_{ji}\alpha \tilde{h}_{ji}^{2}L_{ji}^{-\delta}+N_{0}}>\gamma_{B}\right]\geq1-\varepsilon_{c}. \end{array} \end{array} $$

Let $Y_{1}=\gamma _{B}\alpha L_{ji}^{-\delta }\tilde {h}_{ji}^{2}$, where $\tilde {h}_{ji}^{2}\ \thicksim E(1)$. Based on the conclusion from Appendix B of [24], we can conclude that the probability density function of Y₁ is

$$ \begin{array}{@{}rcl@{}} f(y_{1})=\frac{L_{ji}^{\delta}}{\gamma_{B}\alpha}e^{-\frac{L_{ji}^{\delta}}{\gamma_{B}\alpha}y_{1}}. \end{array} $$

(35)

Then the constraint (10b) is equivalent to the following form,

$$ \begin{array}{@{}rcl@{}} \begin{array}{ll} Pr\left[Y_{1}<\frac{P_{Bi}K_{Bi}L_{Bi}^{-\beta}-\gamma_{B}P_{ji}(1-\alpha)|\hat{h}_{ji}|^{2}L_{ji}^{-\delta}-\gamma_{B}N_{0}}{P_{ji}}\right] \geq1-\varepsilon_{c}. \end{array} \end{array} $$

According to Eq. 35, it is obtained

$$ \begin{array}{@{}rcl@{}} \begin{array}{ll} {\int}_{0}^{\frac{P_{B}K_{Bi}L_{Bi}^{-\beta}-\gamma_{B}[P_{ji}(1-\alpha)|\hat{h}_{ji}|^{2}L_{ji}^{-\delta}+N_{0}]}{P_{ji}}}\frac{{L_{ji}}^{\delta}}{\gamma_{B}\alpha}e^{-\frac{L_{ji}^{\delta}}{\gamma_{B}\alpha}y_{1}}dy_{1} \geq 1 - \varepsilon_{c}. \end{array} \end{array} $$

Then, it holds that

$$ \begin{array}{@{}rcl@{}} e^{-\frac{P_{Bi}K_{Bi}{L_{Bi}}^{-\beta}-\gamma_{B}N_{0}L_{ji}^{-\delta}}{P_{ji}\gamma_{B}\alpha}}+e^{\frac{(1-\alpha)|\hat{h}_{ji}|^{2}}{\alpha}}\leq1+\varepsilon_{c}. \end{array} $$

That is

$$ \begin{array}{@{}rcl@{}} \ln\left( 1+\varepsilon_{c}-e^{\frac{(1-\alpha)|\hat{h}_{ji}|^{2}}{\alpha}}\right)\frac{P_{ji}\gamma_{B}\alpha}{L_{ji}^{\delta}}\geq\gamma_{B}N_{0}-P_{Bi}K_{Bi}L_{Bi}^{-\beta}. \end{array} $$

Finally,

$$ \begin{array}{@{}rcl@{}} \frac{P_{Bi}K_{Bi}L_{Bi}^{-\beta}}{\ln\left( \frac{1}{1+\varepsilon_{c}-e^{\frac{(1-\alpha)|\hat{h}_{ji}|^{2}}{\alpha}}}\right)\frac{\alpha}{L_{ji}^{\delta}}P_{ji}+N_{0}}\geq\gamma_{B}. \end{array} $$

The proof is completed.

Appendix B: Proof of (13)

The proof process of Eq. 13 is similar to Appendix A. Let $Y_{2}=\alpha L_{jj}^{-\rho }\tilde {h}_{jj}^{2}$, where $\tilde {h}_{jj}^{2}\ \thicksim E(1)$. The density function of Y₂ is $f(y_{2})=\frac {L_{jj}^{\rho }}{\alpha }e^{-\frac {L_{jj}^{\rho }}{\alpha }y_{2}}$. Then the constraint (10c),

$$ \begin{array}{@{}rcl@{}} Pr \left[\frac{P_{ji}L_{jj}^{-\rho}(1 - \alpha)|\hat{h}_{jj}|^{2}+P_{j,i}\alpha L_{jj}^{-\rho}\tilde{h}_{jj}^{2}}{P_{Bi}K_{Bj}L_{Bj}^{-\kappa}+N_{0}} > \gamma_{D}\right] \geq {1-\varepsilon_{d}}, \end{array} $$

can be transform to the deterministic form as follows,

$$ \begin{array}{@{}rcl@{}} e^{-\frac{(1-\alpha)|\hat{h}_{jj}|^{2}}{\alpha}}-e^{-\frac{(\gamma_{D}P_{Bi}K_{Bj}L_{Bj}^{-\kappa}+\gamma_{D}N_{0})L_{jj}^{\rho}}{P_{ji}\alpha}}\leq\varepsilon_{d}. \end{array} $$

That is,

$$ \begin{array}{@{}rcl@{}} \ln\left( \frac{1}{e^{-\frac{(1-\alpha)|\hat{h}_{jj}|^{2}}\alpha}-\varepsilon_{d}}\right)\frac{\alpha P_{ji}}{L_{jj}^{\rho}} - (\gamma_{D}P_{Bi}K_{Bj}L_{Bj}^{-\kappa} + \gamma_{D}N_{0})\geq0. \end{array} $$

Finally,

$$ \begin{array}{@{}rcl@{}} \frac{{\ln\left( \frac{1}{e^{-\frac{(1-\alpha)|\hat{h}_{jj}|^{2}}\alpha}-\varepsilon_{d}}\right)\frac{\alpha }{L_{jj}^{\rho}}}P_{ji}}{P_{Bi}K_{Bj}L_{Bj}^{-\kappa}+N_{0}}\geq\gamma_{D}. \end{array} $$

The proof is completed.