Implicit vs. Explicit Approximate Matrix Inversion for Wideband Massive MU-MIMO Data Detection

Abstract

Massive multi-user (MU) MIMO wireless technology promises improved spectral efficiency compared to that of traditional cellular systems. While data-detection algorithms that rely on linear equalization achieve near-optimal error-rate performance for massive MU-MIMO systems, they require the solution to large linear systems at high throughput and low latency, which results in excessively high receiver complexity. In this paper, we investigate a variety of exact and approximate equalization schemes that solve the system of linear equations either explicitly (requiring the computation of a matrix inverse) or implicitly (by directly computing the solution vector). We analyze the associated performance/complexity trade-offs, and we show that for small base-station (BS)-to-user-antenna ratios, exact and implicit data detection using the Cholesky decomposition achieves near-optimal performance at low complexity. In contrast, implicit data detection using approximate equalization methods results in the best trade-off for large BS-to-user-antenna ratios. By combining the advantages of exact, approximate, implicit, and explicit matrix inversion, we develop a new frequency-adaptive e qualizer (FADE), which outperforms existing data-detection methods in terms of performance and complexity for wideband massive MU-MIMO systems.

Appendix A: Proofs and Derivations

1.1 A.1 Proof of Lemma 1

We use [33, Thm. 4.20], which establishes that for a given matrix ${\mathbf {P}}\in \mathbb {C}^{{{U}}\times {{U}}}$ for which $\lim _{k\to \infty }{\mathbf {P}}^{k}={\mathbf {0}}_{{{U}}\times {{U}}}$, we have $({\mathbf {I}}_{{{U}}}-{\mathbf {P}})^{-1}=\sum\limits_{k = 0}^{\infty } {\mathbf {P}}^{k}$ and I _U −P is invertible. As a consequence, by defining $\widetilde {{\mathbf {A}}}_{0}^{-1}{\mathbf {A}}={\mathbf {I}}_{{{U}}}-{\mathbf {P}}$ and assuming that $\lim _{k\to \infty } ({\mathbf {I}}_{{{U}}}-\widetilde {{\mathbf {A}}}^{-1}_{0}{\mathbf {A}})^{k}={\mathbf {0}}_{{{U}}\times {{U}}}$, we have

$$ {\mathbf{A}}^{-1}\widetilde{{\mathbf{A}}}_{0} = (\widetilde{{\mathbf{A}}}_{0}^{-1}{\mathbf{A}})^{-1} =\sum\limits_{k = 0}^{\infty} ({\mathbf{I}}_{{{U}}}-\widetilde{{\mathbf{A}}}_{0}^{-1}{\mathbf{A}})^{k}, $$

which can be rewritten to the accelerated Neumann series in Eq. 4, since $\widetilde {{\mathbf {A}}}^{-1}$ was assumed to be full rank.

1.2 A.2 Proof of Lemma 4

We start by rewriting the residual error term as a function of ${\mathbf {A}}_{0}^{-1}$. We have the following identities:

$$\begin{array}{@{}rcl@{}} \mathbf{e}_{K_{\max}} &=& \tilde{{\mathbf{y}}}_{K_{\max}} - {\mathbf{A}}^{-1}{\mathbf{y}}^{\text{MF}} = (\widetilde{{\mathbf{A}}}^{-1}_{K_{\max}}- {\mathbf{A}}^{-1})\mathbf{y}^{\text{MF}}\\ &=& \left( -\sum\limits_{k=K_{\max}+ 1}^{\infty}({\mathbf{I}}_{{{U}}}-\widetilde{{\mathbf{A}}}^{-1}_{0}{\mathbf{A}})^{k}\widetilde{{\mathbf{A}}}^{-1}_{0}\right)\!{\mathbf{y}}^{\text{MF}}\\ &=& -({\mathbf{I}}_{{{U}}}-\widetilde{{\mathbf{A}}}^{-1}_{0}{\mathbf{A}})^{K_{\max}+ 1}{\mathbf{A}}^{-1}{\mathbf{y}}^{\text{MF}}. \end{array} $$

By using basic properties of induced norms, we get the following inequality:

$$\|\mathbf{e}_{K_{\max}}\| \leq \|{\mathbf{I}}_{{{U}}}-\widetilde{{\mathbf{A}}}_{0}^{-1}{\mathbf{A}}\|^{K_{\max}+ 1}\|\tilde{{\mathbf{y}}}\|, $$

where we define $\tilde {{\mathbf {y}}} = {\mathbf {A}}^{-1}{\mathbf {y}}^{\text {MF}}$.

1.3 A.3 Derivation of Initialization 1

We start by noting that squaring both sides in Eq. 5 results in the equivalent sufficient condition $\|{\mathbf {I}}_{{{U}}}-\widetilde {{\mathbf {A}}}^{-1}_{0}{\mathbf {A}}\|^{2}<1$. Furthermore, by assuming the spectral norm (which is a consistent norm), we have $\|{\mathbf {I}}_{{{U}}}-\widetilde {{\mathbf {A}}}^{-1}_{0}{\mathbf {A}}\|^{2}\leq \|{\mathbf {I}}_{{{U}}}-\widetilde {{\mathbf {A}}}^{-1}_{0}{\mathbf {A}}\|^{2}_{F}$, which enables us to obtain a more restrictive sufficient condition that allows the design of efficient initializers:

$$ \|{\mathbf{I}}_{{{U}}}-\widetilde{{\mathbf{A}}}_{0}^{-1}{\mathbf{A}}\|^{2}_{F}<1. $$

(17)

The initialization method developed next⁹ is of the form $\widetilde {{\mathbf {A}}}^{-1}_{0} = {\mathbf {S}}{\mathbf {D}}^{-1}$, where S is a diagonal scaling matrix that is designed to meet condition (17).

Let W contain the diagonal part of D ^− 1 A and Q the off-diagonal part. We define

$$\begin{array}{@{}rcl@{}} f &=& \|{\mathbf{I}}_{{{U}}}-{\mathbf{S}}{\mathbf{D}}^{-1}{\mathbf{A}}\|_{F}^{2} = \|{\mathbf{I}}_{{{U}}}-{\mathbf{S}}({\mathbf{W}}+{\mathbf{Q}})\|_{F}^{2} \\ & =& \|{\mathbf{I}}_{{{U}}}-{\mathbf{S}}{\mathbf{W}}\|^{2}_{F}+\|{\mathbf{S}}{\mathbf{Q}}\|_{F}^{2}, \end{array} $$

(18)

and seek a diagonal scaling matrix S that minimizes f.

We define the diagonal scaling matrix to have the form S = α I, which leads to $f = {\sum }_{i = 1}^{{{U}}}\left |1-\alpha \,W_{i,i}\right |^{2}+|\alpha |^{2}\|{\mathbf {Q}}\|_{F}^{2}$. We now find the optimum scaling parameter α ^opt by computing ∂ f/∂ α ^∗ = 0and solving for α. Standard manipulations yield

$$\alpha^{\text{opt}}= \left( \sum\limits_{i = 1}^{{{U}}}{W_{i,i}^{*}}\right)\|{\mathbf{D}}^{-1}{\mathbf{A}}\|^{-2}_{F}, $$

where $W_{i,i}^{*}$ are the complex conjugates of the diagonal entries of W. Since W = I _U , we get ${\alpha ^{\text {opt}}= {{U}}\|{\mathbf {D}}^{-1}{\mathbf {A}}\|^{-2}_{F}}$. Consequently, the first initialization matrix is $\widetilde {{\mathbf {A}}}_{0}^{-1} = \alpha ^{\text {opt}}{\mathbf {D}}^{-1}$.

1.4 A.4 Derivation of Initialization 2

For the second initialization method, we follow the derivation in Appendix A.3 but with a more general diagonal scaling matrix of the form S = diag(α ₁,…,α _U ). We obtain

$$f = \sum\limits_{i = 1}^{{{U}}}\left|1-\alpha_{i}\,W_{i,i}\right|^{2}+|\alpha_{i}|^{2}\|{\mathbf{r}}_{i}\|_{2}^{2}, $$

where r _i corresponds to the i ^th row of Q. To find the optimal scaling parameters α _i , i = 1,…,U, we set ${\partial f_{i}}/{\partial \alpha _{i}^{*}} = 0$ and solve for α _i . Standard manipulations yield

$$\alpha_{i}^{\text{opt}}=W_{i,i}^{*}/(|W_{i,i}|^{2}+\|{\mathbf{r}}_{i}\|^{2}_{2}), \,\, i = 1,\ldots,{{U}}, $$

and we use the fact that W _i,i = 1, ∀i.

Consequently, the second initialization matrix is

$$\widetilde{{\mathbf{A}}}_{0}^{-1} = \text{diag}(\alpha_{1}^{\text{opt}},\ldots,\alpha_{{{U}}}^{\text{opt}}){\mathbf{D}}^{-1}. $$