Energy-spectral efficient resource allocation and power control in heterogeneous networks with D2D communication

Abstract

Heterogeneous networks (HetNets) provide the demand for high data rates. In this study, we analyze the coexistence of femtocells and device-to-device (D2D) communication with macrocells. Interference management and decreasing energy consumption are two important issues in HetNets. To this end, we propose an efficient fractional frequency reuse (FFR)-based spectrum partitioning scheme to reduce the cross-tier interference. We also propose to use different optimization problems for resource allocation in different tiers. For this purpose, an energy efficient optimization problem is applied to D2D user equipment. Further, an optimization problem based on the spectral efficiency, i.e., throughput, is considered for macrocell and femtocell tiers. These problems are modeled as a non-cooperative game that results in low computational complexity. Iterative algorithms with fast convergence are used to solve the optimization problems. It is shown that applying different optimizations on different tiers leads to better performance than considering the same optimization for all tiers. The results indicate that the proposed FFR structure and optimization problems improve system performance. We also analyze the tradeoff between energy efficiency and spectral efficiency of the introduced structure.

Appendix

Lemma

$U_{EE,o}^{c}$is quasi-concave, that is, by increasing the value of$P_{TD}^{n,c}$, $U_{EE,o}^{c}$first increases and then decreases [25].

Proof

We obtain the derivation of $R_{do}^{n,c} \left( {P_{TD}^{n,c} } \right) = \log_{2} \left( {1 + \gamma_{do}^{n,c} } \right)$ ($\gamma_{do}^{n,c}$ is given in (12)) with respect to $P_{TD}^{n,c}$ as follows

$$\frac{{\partial R_{do}^{n,c} \left( {P_{TD}^{n,c} } \right)}}{{\partial P_{TD}^{n,c} }} = \frac{{K_{d}^{n,c} }}{a\ln \left( 2 \right)}$$

(41)

where

$$a = P_{TD}^{n,c} \, K_{d}^{n,c} + \sigma_{N}^{2} + P_{TM}^{c} \,K_{md}^{n,c} + \sum\limits_{j = 1}^{{N_{F} }} {P_{TF}^{j,c} \,K_{fd}^{j,n,c} } + \sum\limits_{\begin{subarray}{l} l = 1 \\ l \ne n \end{subarray} }^{{N_{D} }} {P_{TD}^{l,c} \,K_{dd}^{l,n,c} } + I_{do} .$$

It is apparent that $\frac{{\partial R_{do}^{n,c} \left( {P_{TD}^{n,c} } \right)}}{{\partial P_{TD}^{n,c} }} > 0$. Thus $R_{do}^{n,c} \left( {P_{TD}^{n,c} } \right)$ increases by increment of $P_{TD}^{n,c}$.

Also taking the derivation of $U_{EE,o}^{c} = \frac{{\sum\nolimits_{l = 1}^{{N_{D} }} {R_{do}^{l,c} \left( {P_{TD}^{l,c} } \right)} }}{{\sum\nolimits_{l = 1}^{{N_{D} }} {P_{T,do}^{l,c} } + 2P_{cir} }}$ (eq. (22)) with respect to $P_{TD}^{\,n,c}$ yields

$$\frac{{\partial U_{EE,o}^{c} }}{{\partial P_{TD}^{\,n,c} }} = {{\left( {\frac{{bK_{d}^{n,c} }}{a\ln \left( 2 \right)} - \sum\limits_{l = 1}^{{N_{D} }} {R_{do}^{l,c} \left( {P_{TD}^{\,l,c} } \right)} } \right)} \mathord{\left/ {\vphantom {{\left( {\frac{{bK_{d}^{n,c} }}{a\ln \left( 2 \right)} - \sum\limits_{l = 1}^{{N_{D} }} {R_{do}^{l,c} \left( {P_{TD}^{\,l,c} } \right)} } \right)} {b^{2} }}} \right. \kern-0pt} {b^{2} }}$$

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where $b = \sum\nolimits_{l = 1}^{{N_{D} }} {P_{\,T,do}^{\,l,c} } + 2P_{cir}$. The positive value of denominator can be ignored so the shortened equation is defined as:

$$N(P_{TD}^{\,n,c} ) = \frac{{bK_{d}^{n,c} }}{a\ln \left( 2 \right)} - \sum\limits_{l = 1}^{{N_{D} }} {R_{do}^{l,c} \left( {P_{TD}^{\,l,c} } \right)}$$

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In this way, $N(\infty ) = \mathop {\lim }\limits_{{P_{TD}^{n,c} \to \infty }} \,N(P_{TD}^{n,c} ) = \frac{1}{\ln \left( 2 \right)} - \infty < 0$ and $N(0) = \mathop {\lim }\limits_{{P_{TD}^{n,c} \to 0}} \, N(P_{TD}^{\,n,c} ) = \frac{{2K_{d}^{\,n,c} P_{cir} }}{\ln \left( 2 \right)a} > 0$

Taking the first-order derivation of $N(P_{TD}^{n,c} )$ results in

$$\frac{{\partial N(P_{TD}^{n,c} )}}{{\partial P_{TD}^{n,c} }} = \left[ {\frac{{a - b\left( {K_{d}^{n,c} } \right)^{2} }}{{a^{2} \ln \left( 2 \right)}}} \right. - \left. {\frac{{K_{d}^{n,c} }}{a\ln \left( 2 \right)}} \right]\,\,\, < 0$$

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Therefore, it is concluded that $N\left( \infty \right) < N(P_{TD}^{n,c} ) < N\left( 0 \right).$ Consequently, when $P_{\,TD}^{\,n,c} < P_{\,TD,o}^{\, * ,c}$, we have $\frac{{\partial U_{EE,o}^{\,c} }}{{\partial P_{\,TD}^{\,n,c} }} > 0$ and $\frac{{\partial U_{EE,o}^{\,c} }}{{\partial P_{\,TD}^{\,n,c} }} < 0$ when $P_{\,TD}^{\,n,c} > P_{\,TD,o}^{\, * ,c}$. Hence, the increment and then decrement of $U_{EE,o}^{\,c}$ by increasing the value of $P_{\,TD}^{\,n,c}$ is proved. As a result, the concaveness of numerator and denominator of $U_{EE,o}^{\,c}$ results in the quasi concaveness of $U_{EE,o}^{\,c}$. Similar results hold for $U_{EE,\,i}^{\,k} .$