Average power allocation based sum-rate optimization in massive MIMO systems

Abstract

This paper exploits variations in the average channel gains in multi-cell multi-user massive multiple input multiple output (MIMO) systems. An average transmit power-control-based sum-rate optimization scheme is presented for the uplink of the system. The matched filtering (MF) and the zero forcing (ZF) processors are considered with perfect and imperfect channel state information at receiver (CSIR) under frequency flat Rayleigh fading channel. An average power-control-based system model is constructed for analyzing the sum-rate and formulating an optimization problem. A discrete level combinatorial optimization is performed for MF and ZF sum-rate under perfect and imperfect CSIR. The numerical results show a significant improvement in the sum-rate and power consumption. A low complexity algorithm for numerical optimization of the sum-rate is proposed. The performance of algorithm is quantified with different scenarios including different number of users, macro cells, and micro cells with low and high inter-cell interference powers. The evaluation results show that the improvement in sum-rate and energy efficiency increases with inter-cell interference power and the number of MTs.

Appendices

Appendix A

1.1 Appendix A.1

Let us find the ρ for which $R|{~}_{\gamma _{2}= 0}=R|{~}_{\gamma _{2}= 1}\Rightarrow \log _{2}\left \{1+\frac {(N-1)\beta _{1}}{\rho } \right \}=$

$$ \log_{2}\left\{\left( 1+\frac{(N-1)\beta_{1}}{\beta_{2}+\rho}\right)\left( 1+\frac{(N-1)\beta_{2}}{\beta_{1}+\rho}\right) \right\} $$

$$\Rightarrow1+\frac{(N-1)\beta_{1}}{\rho}=\left( 1+\frac{(N-1)\beta_{1}}{\beta_{2}+\rho}\right)\!\left( 1+\frac{(N-1)\beta_{2}}{\beta_{1}+\rho}\right)$$

$$ \Rightarrow \rho^{2}+\{(N-2)\beta_{1}+\beta_{2}\}\rho-{\beta_{1}^{2}}= 0. $$

(32)

The positive root of above equation (ρ > 0) is:

$$ \rho_{c}=\frac{1}{2}\left[\sqrt{\{(N-2)\beta_{1}+\beta_{2}\}^{2}+ 4{\beta_{1}^{2}}}-(N-2)\beta_{1}-\beta_{2}\right]. $$

(33)

The ρ_c is the critical value of ρ where sum-rate is same whether weak user is transmitting full power or not transmitting at all.

1.2 Appendix A.2

By setting $\frac {d}{d\gamma _{2}}(R)= 0$ we obtain

$${\beta_{2}^{2}}{\gamma_{2}^{2}}+ 2\rho_{c}\beta_{2}\gamma_{2}+(N-2)\rho_{c}\beta_{1}-{\beta_{2}^{2}}+{\rho_{c}^{2}}= 0.$$

Solving the above equation for positive γ₂,

$$ \gamma_{c}=\frac{1}{\beta_{2}}\left\{-\rho_{c}+\sqrt{{\beta_{1}^{2}}-(N-2)\rho_{c}\beta_{1}}\right\}. $$

(34)

Since the γ_c is unique thus sum-rate has a maxima or minima at γ₂ = γ_c. Numerical results show that for two users it is minima but for multi-user case, there exists a global maxima for 0 < γ_k < 1;1 ≤ k ≤ K for certain range of ρ.

Appendix B

Path loss is modeled as follows[16, eq:1-6]:

$$\begin{array}{@{}rcl@{}} PL &=&A + 10B\log_{10}\left( \frac{d}{d_{o}}\right)+C; d\geq d_{o}\\ A &\triangleq& 20\log_{10}(4\pi{d_{o}}/\lambda)\\ B &\triangleq& a-bh_{BS}+\frac{c}{h_{BS}}+x_{B}\sigma_{B}; 10m\leq h_{BS} \leq 80m\\ C &\triangleq& x_{C}\sigma_{C} \text{ and} \sigma_{C}=\mu_{\sigma}+y_{C}\sigma_{\sigma} \end{array} $$

(35)

Path loss equation can be written in terms of fixed and varying components as follows:

$$\begin{array}{@{}rcl@{}} PL(d)\!&=&\! \left\{20\log_{10}(4\pi{d_{o}}/\lambda)\!+ 10\left( \!a-bh_{BS}\,+\,\frac{c}{h_{BS}}\!\right)\log_{10}\!\left( \!\frac{d}{d_{o}}\!\right)\right\} \\ &&+\left\{x_{B}\sigma_{B}\log_{10}\left( \frac{d}{d_{o}}\right)+x_{C}(\mu_{\sigma}+y_{C}\sigma_{\sigma})\right\} \end{array} $$

x_B, x_C and y_C are distributed as $\mathbb {N}(0,1)$.

Table 5 Numerical values of parameters for path loss modeling (Source: [16, Table(1)])