Performance evaluation and implementation complexity analysis framework for ZF based linear massive MIMO detection

Abstract

This paper discusses a framework for algorithm-architecture synergy for (1) performance evaluation and (2) FPGA implementation complexity analysis of linear massive MIMO detection techniques. Three low complexity implementation techniques of the zero-forcing (ZF) based linear detection are evaluated, namely, Neumann series expansion (NSE), Gauss–Seidel (GS) and a proposed recursive Gram matrix inversion update (RGMIU) techniques. The performance analysis framework is based on software-defined radio platform. By extrapolating the real data measured average error vector magnitude versus a number of served single-antenna user terminals, GS and RGMIU are showing no performance degradation with respect to ZF with direct matrix inversion. It is shown that under high load regime NSE and GS require more processing iterations at the expense of increased processing latency. We, therefore, consider a unified approach for field-programmable gate array based implementation complexity analysis and discuss the required baseband processing resources for real-time transmission. Due to the wide differences of NSE, GS and RGMIU in terms of performance, processing complexity and latency, practical deployment and real-time implementation insights are derived.

Appendix

To make the paper self-contained this appendix derives the matrix inversion of a large $K \times K$ matrix of the form ${\mathbf{H}}^{H} {\mathbf{H}}$. We propose to apply the matrix inversion update lemma when a new column is added [16]. Assume we have the inverse of a $\left( {K - 1} \right) \times \left( {K - 1} \right)$ matrix ${\varvec{\Delta}}_{K - 1}^{ - 1} = {\mathbf{H}}_{1:K - 1}^{H} {\mathbf{H}}_{1:K - 1}^{{}}$ (note that we have adopted the Matlab index notation $1:K - 1$ in ${\mathbf{H}}_{1:K - 1}^{{}}$ to designate columns 1 to $K - 1$ of ${\mathbf{H}}$). Therefore, the inverse of a $K \times K$ matrix ${\varvec{\Delta}}_{K}^{ - 1} = {\mathbf{H}}_{1:K}^{H} {\mathbf{H}}_{1:K}^{{}}$ can be computed as follow

$$ \begin{aligned} {\varvec{\Delta}}_{K}^{{}} & = \left( {{\mathbf{H}}_{1:K}^{H} {\mathbf{H}}_{1:K}^{{}} } \right)^{ - 1} \\ & = \left( {\left[ {\begin{array}{*{20}c} {{\mathbf{H}}_{1:K - 1}^{H} } \\ {{\mathbf{H}}_{K:K}^{H} } \\ \end{array} } \right]{\kern 1pt} {\kern 1pt} \left[ {\begin{array}{*{20}c} {{\mathbf{H}}_{1:K - 1}^{{}} } & {{\mathbf{H}}_{K:K}^{{}} } \\ \end{array} } \right]} \right)^{ - 1} \\ & = \left[ {\begin{array}{*{20}c} {{\mathbf{H}}_{1:K - 1}^{H} {\mathbf{H}}_{1:K - 1}^{{}} } & {{\mathbf{H}}_{1:K - 1}^{H} {\mathbf{H}}_{K:K}^{{}} } \\ {{\mathbf{H}}_{K:K}^{H} {\mathbf{H}}_{1:K - 1}^{{}} } & {{\mathbf{H}}_{K:K}^{H} {\mathbf{H}}_{K:K}^{{}} } \\ \end{array} } \right]^{ - 1} \\ & = \left[ {\begin{array}{*{20}c} {\varvec{\Gamma}} & { - c{\kern 1pt} {\varvec{\Delta}}_{K - 1}^{H} {\mathbf{H}}_{1:K - 1}^{H} {\mathbf{H}}_{K:K}^{{}} } \\ { - c{\kern 1pt} {\mathbf{H}}_{K:K}^{H} {\mathbf{H}}_{1:K - 1}^{{}} {\varvec{\Delta}}_{K - 1}^{H} } & c \\ \end{array} } \right]^{{}} \\ \end{aligned} $$

(6)

where $c = \frac{1}{{\left( {{\mathbf{H}}_{K:K}^{H} {\mathbf{H}}_{K:K}^{{}} } \right) - \left( {{\mathbf{H}}_{1:K - 1}^{H} {\mathbf{H}}_{K:K}^{{}} } \right)^{H} {\varvec{\Delta}}_{K - 1}^{{}} \left( {{\mathbf{H}}_{1:K - 1}^{H} {\mathbf{H}}_{K:K}^{{}} } \right)}}$ and ${\varvec{\Gamma}} = {\varvec{\Delta}}_{K - 1}^{{}} + c{\kern 1pt} {\varvec{\Delta}}_{K - 1}^{{}} \left( {{\mathbf{H}}_{K:K}^{H} {\mathbf{H}}_{1:K - 1}^{{}} } \right)^{H} \left( {{\mathbf{H}}_{K:K}^{H} {\mathbf{H}}_{1:K - 1}^{{}} } \right){\varvec{\Delta}}_{K - 1}^{H}$.

If we set ${\mathbf{z}} \triangleq {\mathbf{H}}_{K:K}^{{}}$, ${\mathbf{y}}_{1} \triangleq {\mathbf{H}}_{1:K - 1}^{H} {\mathbf{z}}$,${\mathbf{y}}_{2} \triangleq {\varvec{\Delta}}_{K - 1}^{{}} {\mathbf{y}}_{1}$, and ${\mathbf{y}}_{3} \triangleq c{\kern 1pt} {\kern 1pt} {\mathbf{y}}_{2}$ then $c = {1 \mathord{\left/ {\vphantom {1 {\left( {{\mathbf{z}}^{H} {\mathbf{z}} - {\mathbf{y}}_{1}^{H} {\mathbf{y}}_{2}^{{}} } \right)}}} \right. \kern-0pt} {\left( {{\mathbf{z}}^{H} {\mathbf{z}} - {\mathbf{y}}_{1}^{H} {\mathbf{y}}_{2}^{{}} } \right)}}$ and ${\varvec{\Gamma}} = {\varvec{\Delta}}_{K - 1}^{{}} + c{\kern 1pt} {\mathbf{y}}_{2}^{{}} {\mathbf{y}}_{2}^{H}$.

Applying Eq. (6) successively from the second column all the way to the last column $K$, the algorithm, dubbed recursive gram matrix inversion update (RGMIU), is outlined in Table 1.