Fairness constrained diffusion adaptive power control for dense small cell network

Abstract

Small cell is an emerging and promising technology for improving hotspots coverage and capacity, which tends to be densely deployed in populated areas. However, in a dense small cell network, the performances of users differ vastly due to the random deployments and the interferences. To guarantee fair performance among users in different cells, we propose a new distributed strategy for fairness constrained power control, referred to as the diffusion adaptive power control (DAPC). DAPC achieves overall network fairness in a distributed manner, in which each base station optimizes a local fairness with little information exchanged with neighboring cells. We study several adaptive algorithms to implement the proposed DAPC strategy. To improve the efficiency of the standard least mean square algorithm (LMS), we derive an adaptive step-size logarithm LMS algorithm, and discuss its convergence properties. Simulation results confirm the efficiency of the proposed methods.

Appendix

1.1 Proof of (13)

We define the transmit power in dB, $p\_dB_{FAP_{i,t} }$, as

$$\begin{aligned} p\_dB_{FAP_{i,t} } =10\log p_{FAP_i ,t} \end{aligned}$$

(A-1)

Since the update is in logarithm, the gradient should also be changed as the gradient of $p\_dB_{FAP_{i,t} }$.

$$\begin{aligned} p\_dB_{FAP_{i,t+1} }= & {} p\_dB_{FAP_{i,t} }\nonumber \\&+\, \mu _i \left( -\frac{dF_{FAP_i ,t} }{dp\_dB_{FAP_{i,t} } }\right) \end{aligned}$$

(A-2)

$$\begin{aligned} \Rightarrow \quad p\_dB_{FAP_{i,t+1} }= & {} p\_dB_{FAP_{i,t} } \nonumber \\&+\, \mu _i \left( -\frac{dF_{FAP_i ,t} }{dp_{FAP_{i,t} } }\frac{dp_{FAP_{i,t} } }{dp\_dB_{FAP_{i,t} } }\right) \nonumber \\ \end{aligned}$$

(A-3)

$$\begin{aligned} \Rightarrow \quad p\_dB_{FAP_{i,t+1} }= & {} p\_dB_{FAP_{i,t} }\nonumber \\&+\, \mu _i \underbrace{\left( -\frac{dF_{FAP_i ,t} }{dp_{FAP_{i,t} } }\frac{p_{FAP_{i,t} } }{10}\ln 10\right) }_{gradient} \nonumber \\ \end{aligned}$$

(A-4)

where $\mu$ is a fixed value and $\frac{10}{p_{FAP_{i,t} } \ln 10}$ is always positive. In this way, the update is adaptive step-size and it is redefined as

$$\begin{aligned} 10\log p_{FAP_i ,t+1} =10\log p_{FAP_i ,t} +\mu \left( {-\frac{dF_{FAP_i ,t} }{dp_{FAP_{i,t} } }} \right) \nonumber \\ \end{aligned}$$

(A-5)

which completes the proof. $\square$