Performance Analysis of Downlink Linear Precoding in Massive MIMO Systems Under Imperfect CSI

Abstract

Demands of wireless data traffic, throughput, the number of wireless mobile connections and devices will always increase. In addition, the concern about energy consumption is also growing for wireless communication systems. Massive MIMO system is a new emerging research area to resolve these issues. In this paper, the performance of Massive MIMO downlink including linear precoding is evaluated. Spectral efficiency through achievable rate and energy efficiency through transmit power of ZF and MRT linear precoding is investigated under practical limitations, such as imperfect CSI, less complexity processing and inter user interference. Since ZF and MRT precoding can balance the system performance and complexity. Different channel estimation values are considered in order to compare the performance of these precoding techniques in the given system. The achievable rate and the downlink transmit power of ZF and MRT precoding techniques are derived, analyzed and compared under the same conditions and assumptions. Several scenarios are considered to investigate these performance parameters. It is found that when the ratio of BS antennas and number of users is large, ZF is better than MRT while when the ratio is quite small it makes MRT better than ZF for the same conditions.

Appendices

Appendix 1

The desired signal power $S_k$, interference $I_k$ and Noise $n_k$ is

$$\begin{aligned} |S_k|^2= & {} \frac{P_d}{tr(\bar{{\mathbf {A}}} \bar{{\mathbf {A}}^H)}}\\ |I_k|^2= & {} P_d\sigma _e^2 \end{aligned}$$

and

$$\begin{aligned} |n_k|^2 = n = 1 \end{aligned}$$

By substituting the valuse of $S_k, I_k$ and $n_k$ in (13), the SINR of the kth user is given as

$$\begin{aligned} SINR_{k}^{zf_{csi}} = \frac{P_d}{tr(\bar{{\mathbf {A}}} \bar{{\mathbf {A}}^H)}} \left( \frac{1}{P\sigma _e^2 + 1}\right) \end{aligned}$$

(25)

$$\begin{aligned} SINR_{k}^{zf_{csi}} = \frac{P_d}{({P\sigma _e^2 + 1}){tr(\bar{{\mathbf {A}}}\bar{{\mathbf {A}}}^H)}} \end{aligned}$$

(26)

When the value of M and K is large [23], then

$$\begin{aligned} \frac{1}{tr({\mathbf {A}}{\mathbf {A}}^H)}\approx {\text{Diversity}}\, {\text{order}}\, {\text{of}} \,{\text{ZF}}\, {\text{precoding}} \end{aligned}$$

The diversity order measures the number of independent paths over which the data is received [23]

$$\begin{aligned} {\text{Diversity}}\, {\text{order}}\, {\text{of}} \,{\text{ZF}}\, {\text{precoding}} = {\alpha - 1} \end{aligned}$$

where $\alpha = \frac{M}{K}$

$$\begin{aligned} \therefore \frac{P_d}{tr(\bar{{\mathbf {A}}}\bar{{\mathbf {A}}^H)}} = \frac{(M - K) (1 - \sigma _e^2)P_d}{K} \end{aligned}$$

(27)

Putting the values of (27) in (26) and after some manipulations, we obtain the result of (9).

Appendix 2

Received Signal at the k th user is given as

$$\begin{aligned} y_k = \beta \bar{{\mathbf {h}}_k} \bar{{\mathbf {h}}_k^H} x_k + \sum ^k_{i=1,i\ne k} \beta \bar{{\mathbf {h}}_k} \bar{{\mathbf {h}}_i^H} x_i + \sum ^k_{i=1}\beta \bar{{\mathbf {h}}_i^H} {\mathbf {E}}_i x_i + n_k \end{aligned}$$

(28)

Therefore, SINR of MRT precoding under imperfect CSI is

$$\begin{aligned} SINR_k^{mrt_{csi}} = \frac{\beta ^{2} |{{\mathbf {h}}_k}{{\mathbf {h}}_k^H}|^2 (1-\sigma ^2) }{\beta ^2\left[ \sum ^k_{i=1,i\ne k} |{{\mathbf {h}}_k} {{\mathbf {h}}_i^H}|^2 (1-\sigma ^2) + \sigma ^2\right] + 1} \end{aligned}$$

(29)

where $'\beta '$ for imperfct CSI is given as

$$\begin{aligned} \beta = \sqrt{\frac{P_d}{MK (1-\sigma ^2)}} \end{aligned}$$

(30)

But from [36]

$$\begin{aligned} \frac{1}{K} \sum ^k_{i=1,i\ne k}|{\mathbf {h}}_k {\mathbf {h}}_i^H|^2 = {\mathbf {E}} {|{\mathbf {h}}_k {\mathbf {h}}_i^H|^2} = M \end{aligned}$$

(31)

and

$$\begin{aligned} {\mathbf {h}}_k {\mathbf {h}}_k^H = M \end{aligned}$$

(32)

Substituting (30), (31) and (32) in (29) and after some manipulations, we obtain the result of (11)