On the performance of hybrid beamforming for closely-spaced and randomly located users

Abstract

Hybrid (analog-digital) beamforming has received tremendous attention for realizing multiuser multiple-input-multiple-output systems at millimeter wave frequencies. However, almost all of the hybrid beamforming literature characterizes the spectral and energy efficiency performance for randomly located user terminals. In stark contrast, this article evaluates and compares the multiuser downlink ergodic sum spectral efficiency (ESSE) using different combinations of analog and digital beamforming techniques when users are closely-spaced. For any given combination of hybrid beamforming, we derive an analytical expression characterizing the loss of per-user ergodic spectral efficiency relative to fully digital beamforming. To get the negligible ESSE loss for both hybrid and fully digital beamforming, we quantify the angle-of-departure separation across multiple users, it is the term associated to the channel correlation induced by the user positions. We show the generality of the derived expression by testing it across a wide range of system dimensions, number of radio-frequency (RF) chains, and signal-to-noise ratios. Our results demonstrate that an extra RF chain may be sufficient to compensate most of the loss in ESSE due to closely-spaced terminals.

Appendices

Appendix 1: Derivation of SE bound for hybrid beamforming

Note that expectation in (13) is over the small-scale fading in the propagation channel coefficients. Since $\textbf{H}_{\text {eq}}\textbf{H}_{\text {eq}}^H$ is a positive definite Hermitian matrix, by virtue of eigenvalue decomposition, we can express $\textbf{H}_{\text {eq}}\textbf{H}_{\text {eq}}^H=\textbf{U}\varvec{\Sigma }\textbf{U}^H,$ where $\varvec{\Sigma }\in \mathcal {C}^{K\times {}K}$ is the eigenvalue matrix and $\textbf{U} \in \mathcal {C}^{K \times K}$ is a unitary matrix. Noting that the sum of the eigenvalues in $\varvec{\Sigma }=\text {tr}(\varvec{\Sigma })$, we can rewrite $\tilde{\beta }$ with

$$\begin{aligned} \frac{K}{\textrm{tr}(\textbf{H}_{\text {eq}}\textbf{H}_{\text {eq}}^H)^{-1}}&=K\big [\textrm{tr}(\textbf{U}\Sigma \textbf{U}^H)^{-1}\big ]^{-1} = \bigg [\sum _{i=1}^K{\frac{1}{K}\lambda _i^{-1}}\bigg ]^{-1}. \end{aligned}$$

(17)

Now, our function of interest is in the form $f(x)=x^{-1}$ and $x>0$, which is a strictly decreasing convex function. Exploiting this, we can bound the eigenvalues in $\varvec{\Sigma }$ such that

$$\begin{aligned} \bigg [\sum _{i=1}^K{\frac{1}{K}\lambda _i^{-1}}\bigg ]^{-1}\le \sum _{i=1}^K{\frac{1}{K}(\lambda _i^{-1})^{-1}}=\sum _{i=1}^K{\frac{1}{K}\lambda _i}. \end{aligned}$$

(18)

Based on these manipulations, we have

$$\begin{aligned} \frac{1}{\textrm{tr}\big ((\textbf{H}_{\text {eq}}\textbf{H}_{\text {eq}}^H)^{-1}\big )}\le \sum _{i=1}^K{\frac{1}{K^2}\lambda _i} =\textrm{tr}\big (\textbf{H}_{\text {eq}}\textbf{H}_{\text {eq}}^H\big ). \end{aligned}$$

(19)

Using (19), the expression in (13) can be written as

$$\begin{aligned} \text {R}_k \le \mathop {\mathbb {E}}\bigg [\log _2 \big (1+\frac{1}{K^2 \sigma ^2}\textrm{tr}\big (\textbf{H}_{\text {eq}}\textbf{H}_{\text {eq}}^H\big )\big )\bigg ]. \end{aligned}$$

(20)

Using Jensen’s inequality for strictly convex functions, the expectation is brought inside the logarithmic term, giving the result in Proposition 1

$$\begin{aligned} \text {R}_{k}&\le \log _2 \bigg (1+\frac{1}{K^2\sigma ^2} \mathop {\mathbb {E}}\bigg [ \textrm{tr}\big (\textbf{H}_{\text {eq}}\textbf{H}_{\text {eq}}^H\big )\bigg ] \bigg )\nonumber \\&=\log _2 \bigg (1+\frac{1}{K^2\sigma ^2}\big [N_{\text {t}} \left\| \textbf{F}_{\text {RF}}^{H}\textbf{F}_{\text {RF}}\right\| ^{2}_{F}\big ] \bigg ). \end{aligned}$$

(21)

The expectation is taken over the raw channel entries in $\textbf{H}$, giving rise to the result above.

Appendix 2: Derivation of SE bound for fully digital beamforming

For the fully digital case, when there is no RF precoding, $\textbf{H}_{\text {eq}}=\textbf{H}$. Based on this, $\textbf{F}_{\text {BB}}^{\text {eq}}= ~ \textbf{H}^H(\textbf{H}\textbf{H}^H)^{-1}$ is the fully digital precoding matrix, resulting in the per-user ergodic SE of

$$\begin{aligned} \text {R}_{k}^{\text {digital}}&=\log _2\bigg (1+\frac{1}{\sigma ^2\textrm{tr}\big (\textbf{F}_{\text {BB}}^{\text {eq}}\textbf{F}_{\text {BB}}^{\text {eq}^H}\big )}\bigg ), \nonumber \\&\le \log _2\bigg (1+\frac{1}{K\sigma ^2}\textrm{tr}\big (\textbf{H}^H \textbf{H}\big )\bigg ). \end{aligned}$$

(22)