Characterization of sparse beamforming for energy efficiency in cloud radio access networks using Gauss–Poisson process

Abstract

In this paper, we propose an energy efficient Sparse Beamforming Strategy (SBS) in Cloud Radio Access Networks (C-RAN) to achieve an optimimum energy efficiency. The stochastic geometry method is used to derive some expressions for ergodic rate and coverage probability in downlink transmission. In this system model, Remote Radio Heads (RRHs) are coordinated by a Baseband Unit to transmit data toward users. We assume using RRH clusters where each cluster includes one or two RRHs. To investigate this system, we have used Gauss–Poisson process (GPP). The GPP well describes this clustering scenarios. Considering the intra-cell interference power control, we propose SBS to gain the best performance subject to the energy efficiency (EE) metric. SBS which is based on sparse selections of RRHs, is characterized by introducing analytical expressions and simulation of a C-RAN scenario which the allocation of the RRHs follows the GPP distribution. The numerical results demonstrate that the proposed SBS method improves the overall EE of C-RAN scenario up to 15% with respect to full RRHs coordination, at the high intra-cell interference conditions, and about 30% with respect to no coordination state between RRHs in low intra-cell interference regime.

Appendix: A detailed proof of (12) and (13)

The left side of (12) is expanded to

$$\begin{aligned} {\mathbb {E}}\left[ {{\rm Pr}} \left( \ln (1+{{\rm SIR}})> \tau \right) \right] {\mathop{\approx }\limits^{(a)}}\, {\mathbb{E}}\left[ {{\rm Pr}} \left( {{\rm ln}} ({{\rm SIR}})> \tau \right) \right] = \int _{\tau>0}\underbrace{ {{\rm Pr}}\left[ {{\rm ln(SIR)}} > \tau \right] }_T {{\rm d}}\tau, \end{aligned}$$

(26)

where (a) is from considering high SIR in the C-RAN, and T is expressed as

$$\begin{aligned} T = \int _{\tau>0} {\text {Pr}}\left[ \frac{P_1h_0b^{-\alpha }}{\underbrace{\sum \limits _{i\in {\varPhi }} P_i h_x \Vert x \Vert ^{-\alpha }}_I} \right] {\text {d}}\tau {\mathop {=}\limits ^{(b)}} \int _{\tau >0} {\mathcal {L}}_I \underbrace{\left( \frac{{\text {e}}^\tau b^\alpha }{P_1} \right) }_s, \end{aligned}$$

(27)

where (b) results from exponential distribution of h, i.e. $h \sim \exp (1)$. (27) can be continued as

$$\begin{aligned} \begin{aligned} {\mathcal {L}}_I \left( s \right)&= {\mathbb {E}}_{\phi ,\{h\}} \left[ {\text {exp}} \left( -\sum (P_i s h_x \Vert x \Vert ^{-\alpha }\right) \right] {\mathop {=}\limits ^{(c)}} {\mathbb {E}} \left[ \prod _{x\in {\varPhi }}{\mathbb {E}}_h ({\text {exp}} (-P s h \Vert x \Vert ^{-\alpha }) \right] \\&{\mathop {=}\limits ^{(d)}} {\text {exp}} \left( \lambda _p \int _{{\mathbb {R}}^2} \right. \left. \left[ (1-p) W_1(x) + pW_2(x) \int _{{\mathbb {R}}^2} W_2(x+z) f_u(\Vert z \Vert ) {\text {d}} z-1 \right] {\text {d}}x \right) , \end{aligned} \end{aligned}$$

(28)

where (c) is because of i.i.d distribution of h and its independency from ${\varPhi }$. (d) is PGFL of ${\varPhi }$ [24]. Moreover, $W_1(x)$ and $W_2(x)$ can be described as

$$\begin{aligned} \begin{aligned} W_1(x) = {\mathbb {E}}_h \left( {\text {exp}} \left( -P_1 s h \Vert x \Vert ^{-\alpha }\right) \right) = \frac{1}{1+P_1 s \Vert x \Vert ^ {-\alpha }}, \\ \end{aligned} \end{aligned}$$

(29)

after some manipulation, $W_1(x)$ and $W_2(x)$ can be calculated as follow

$$\begin{aligned} \begin{aligned} W_1(x) = \frac{1}{1+{\text {e}}^\tau b^\alpha \Vert x \Vert ^ {-\alpha }}, \; W_2(x) = \frac{1}{1+K {\text {e}}^\tau b^\alpha \Vert x \Vert ^ {-\alpha }}. \end{aligned} \end{aligned}$$

(30)

where $K=P_2/P_1$. Therefore, the proof is completed.