Achieving Bandwidth Efficiency by Improved Zero-Forcing Combining Algorithm in Massive MIMO

Abstract

Massive multiple-input multiple-output (MIMO) or large scale MIMO (LS-MIMO) systems indicate the usage of very large number of antennas at Base Stations to communicate with comparatively small number of user terminals. The concept of LS-MIMO systems has gained so much popularity in recent years because of two main benefits, sufficient improvement in Energy Efficiency and Bandwidth Efficiency (BE). Certain factors such as length of coherence block, hardware design, pilot contamination, receive/transmit combining techniques and other design parameters limit the performance of LS-MIMO systems. In this paper our goal is to improve BE of LS-MIMO systems in multi-cell scenarios. We present an improved version of zero-forcing algorithm and based on this algorithm, we derive new equations for achievable rates and signal-to-interference-plus-noise ratio in uplink and downlink. We compare our algorithm with the two fundamental receive combining algorithms such as maximum ratio combining and zero-forcing, results show sufficient improvement in the performance of our LS-MIMO system in terms of BE. Then we change different system parameters such as coherence block length, signal-to-noise ratio and reveal their impact on the performance of system .

Appendix

It has been observed from [31] that MMSE estimator for Gaussian channel can be derived as follows:

$$\begin{aligned} {\mathbb {E}}\left\{ {\mathbf {H}}_{alu}\mathrm {vec}\left( {\mathbf {Y}}_{a}\right) ^{H} \right\} \left( {\mathbb {E}}\left\{ \mathrm {vec}\left( {\mathbf {Y}}_{a}\right) \mathrm {vec}\left( {\mathbf {Y}}_{a}\right) ^{H} \right\} \right) ^{-1}\mathrm {vec}\left( {\mathbf {Y}}_{a}\right) \end{aligned}$$

(27)

We use vectorization rule $\left( {\mathbf {c}}^{T}\otimes {\mathbf {a}}\right) \mathrm {vec}\left( {\mathbf {b}}\right) =\mathrm {vec}\left( {\mathbf {abc}} \right)$ here $\otimes$ represents Kronecker product, and for our scenario

$$\begin{aligned} {\mathbb {E}}\left\{ {\mathbf {H}}_{alu}\mathrm {vec}\left( {\mathbf {Y}}_{a}\right) ^{H} \right\} = {\mathbb {E}}\left\{ {\mathbf {H}}_{alu}{\mathbf {H}}_{alu}^{H}\left( {\mathbf {q}}_{d_{au}}^{{H}}\otimes {\mathbf {I}}_{N}\right) \right\} =\left( {\mathbf {q}}_{d_{au}}^{{H}}\otimes \rho \frac{d_{a}({\mathbf {z}}_{lu})}{d_{l}({\mathbf {z}}_{lu})}\right) {\mathbf {I}}_{N} \end{aligned}$$

(28)

Because of mutually independent channels we can write as

$$\begin{aligned} {\mathbb {E}}\left\{ \mathrm {vec}\left( {\mathbf {Y}}_{a}\right) \mathrm {vec}\left( {\mathbf {Y}}_{a}\right) ^{H}\right\}&=\sigma ^{2}{\mathbf {I}}_{ND} +\sum \limits _{l\in L}^{}\sum \limits _{u=1}^{K}{\mathbb {E}}\left\{ \mathrm {vec}\left( {\mathbf {H}}_{alu}{\mathbf {q}}_{d_{au}}^{T}\right) \mathrm {vec}\left( {\mathbf {H}}_{alu}{\mathbf {q}}_{d_{au}}^{T}\right) ^{H}\right\} \nonumber \\&= \left( \sum \limits _{l \in L}^{}\sum \limits _{u=1}^{K} \rho \frac{d_{a}({\mathbf {z}}_{lu})}{d_{l}({\mathbf {z}}_{lu})}{\mathbf {q}}_{d_{au}}{\mathbf {q}}_{d_{au}}^{H}+\sigma ^{2}{\mathbf {I}}_{D}\right) \otimes {\mathbf {I}}_{N} \end{aligned}$$

(29)

(7) can be obtained by putting (28) and (29) into (27).