Channel Correlation Diversity in MU-MIMO Systems -- Analysis and Measurements

TL;DR

This paper analyzes the impact of diverse, physically motivated channel correlation models on MU-MIMO system performance, showing that realistic correlation variations significantly influence spectral efficiency and system metrics.

Contribution

It introduces measurement-based correlation models capturing variations across terminals and derives closed-form expressions for key performance metrics under these models.

Findings

Diverse correlation patterns significantly affect system performance.

Measurement-based models provide more accurate performance predictions.

Closed-form approximations offer insights into correlation impacts.

Abstract

In multiuser multiple-input multiple-output (MU-MIMO) systems, channel correlation is detrimental to system performance. We demonstrate that widely used, yet overly simplified, correlation models that generate identical correlation profiles for each terminal tend to severely underestimate the system performance. In sharp contrast, more physically motivated models that capture variations in the power angular spectra across multiple terminals, generate diverse correlation patterns. This has a significant impact on the system performance. Assuming correlated Rayleigh fading and downlink zero-forcing precoding, tight closed-form approximations for the average signal-to-noise-ratio, and ergodic sum spectral efficiency are derived. Our expressions provide clear insights into the impact of diverse correlation patterns on the above performance metrics. Unlike previous works, the correlation models are parameterized with measured data from a recent 2.53 GHz urban macrocellular campaign in Cologne, Germany. Overall, results from this paper can be treated as a timely re-calibration of performance expectations from practical MU-MIMO systems.

Equations

$$y_{\ell }=\sqrt{\frac{{\rho }_{\text{t}}{\beta }_{\ell }}{\eta }}{\mathbf{h}}_{\ell }{\mathbf{g}}_{\ell }{s}_{\ell }+\sum _{\begin{array}{c}i=1\\ i\ne \ell \end{array}}^L\sqrt{\frac{{\rho }_{\text{t}}{\beta }_{\ell }}{\eta }}{\mathbf{h}}_{\ell }{\mathbf{g}}_i{s}_i+n_{\ell },\text{vspace}-4pt$$

$${\text{SNR}}_{\ell }^{\text{ZF}}=\frac{{\rho }_{\text{t}}{\beta }_{\ell }}{{\sigma }^2\eta }=\frac{{\rho }_{\text{t}}{\beta }_{\ell }}{{\sigma }^2\left\{\frac{1}{L}\left\{\text{Tr}\left[{\left(\mathbf{H}{\mathbf{H}}^H\right)}^{-1}\right]\right\}\right\}},$$

$${\text{R}}_{\text{ZF}}=\mathbb{E}\left\{\sum _{\ell =1}^L{\log }_2\left(1+{\text{SNR}}_{\ell }^{\text{ZF}}\right)\right\},\text{vspace}-2pt$$

$${\left(\mathbf{H}{\mathbf{H}}^H\right)}^{-1}\approx \frac{1}{M}\sum _{n=0}^N{\left(-1\right)}^n{\left(\frac{\mathbf{\Delta }}{M}\right)}^n.\text{vspace}-1pt$$

$$\text{vspace}-7pt{\left(\mathbf{H}{\mathbf{H}}^H\right)}^{-1}\approx$$

$$=$$

$${\text{SNR}}_{\ell }^{\text{ZF}}\approx \frac{{\rho }_{\text{t}}{\beta }_{\ell }}{{\sigma }^2\left\{\frac{1}{L}\left\{\text{Tr}\left[\frac{1}{M}{\sum }_{n=0}^N{\sum }_{q=0}^n\left(\genfrac{}{}{0ex}{}{n}{q}\right)\frac{{\left(-1\right)}^q}{{\left(M\right)}^q}{\left(\mathbf{H}{\mathbf{H}}^H\right)}^q\right]\right\}\right\}}.\text{vspace}3pt$$

$$\mathbb{E}\left\{\textrm{SNR}_{\ell}^{\textrm{ZF}}\right\}\approx\frac{\rho_{\textrm{t}}\hskip 1.0pt\beta_{\ell}}{\sigma^{2}\left\{\frac{1}{L}\left\{\frac{1}{M}\sum\nolimits_{n=0}^{N}\sum\nolimits_{q=0}^{n}{n\choose{}q}\frac{\left(-1\right)^{q}}{\left(M\right)^{q}}\hskip 2.0pt\mathbb{E}\hskip 2.0pt\Big{\{}\textrm{Tr}\left[\left(\mathbf{H}\mathbf{H}^{H}\right)^{q}\right]\Big{\}}\right\}\right\}}.$$

$$\text{vspace}-2pt\mathbb{E}\left\{{\text{SNR}}_{\ell }^{\text{ZF}}\right\}\approx \frac{{\rho }_{\text{t}}{\beta }_{\ell }{M}^3}{{\sigma }^2\left\{L\left[M^2+L\left(\text{Tr}\left[{\bar{\mathbf{R}}}^2\right]\right)\right]\right\}},$$

$${\left(\mathbf{H}{\mathbf{H}}^H\right)}^{-1}\approx \frac{1}{M}\left[3{\mathbf{I}}_L-\frac{3}{M}\mathbf{H}{\mathbf{H}}^H+\frac{1}{M^2}{\left(\mathbf{H}{\mathbf{H}}^H\right)}^2\right].$$

$$\text{Tr}\left[{\left(\mathbf{H}{\mathbf{H}}^H\right)}^{-1}\right]\approx \frac{1}{M}$$

$$+\frac{1}{M^2}\text{Tr}\left[{\left(\mathbf{H}{\mathbf{H}}^H\right)}^2\right]\}.$$

$$\text{Tr}\left[{\left({\mathbf{HH}}^H\right)}^2\right]=\sum _{\ell =1}^L\sum _{j=1}^L{\mathbf{h}}_{\ell }{\mathbf{h}}_j^H{\mathbf{h}}_j{\mathbf{h}}_{\ell }^H.\text{vspace}-4pt$$

$$\mathbb{E}\left\{\text{Tr}\left[{\left({\mathbf{HH}}^H\right)}^2\right]\right\}=\mathbb{E}\{\sum _{\ell =1}^L{\left({\mathbf{h}}_{\ell }{\mathbf{h}}_{\ell }^H\right)}^2$$

$$\displaystyle\hskip 90.0pt+\Bigg{.}\sum\limits_{\ell=1}^{L}\sum\limits_{\begin{subarray}{c}j=1\\ j\neq{}\ell\end{subarray}}^{L}\hskip 2.0pt\mathbf{h}_{\ell}\hskip 1.0pt\mathbf{h}_{j}^{H}\mathbf{h}_{j}\mathbf{h}_{\ell}^{H}\Bigg{\}}$$

$$=\mathbb{E}\left\{\sum _{\ell =1}^L\left\{M^2+\text{Tr}\left[{\mathbf{R}}_{\ell }^2\right]\right\}+\sum _{\ell =1}^L\sum _{\begin{array}{c}j=1\\ j\ne \ell \end{array}}^L\text{Tr}\left[{\mathbf{R}}_{\ell }{\mathbf{R}}_j\right]\right\}.$$

$$\mathbb{E}\left\{\text{Tr}\left[{\left({\mathbf{HH}}^H\right)}^2\right]\right\}=$$

$$=$$

$$\mathbb{E}\left\{\text{Tr}\left[{\left({\mathbf{HH}}^H\right)}^{-1}\right]\right\}\approx \frac{L}{M^3}\left\{M^2+L\left(\text{Tr}\left[{\bar{\mathbf{R}}}^2\right]\right)\right\}.\text{vspace}-2pt$$

$$\mathbb{E}\left\{{\text{R}}_{\text{ZF}}\right\}\approx \sum _{\ell =1}^L{\log }_2\left(1+\mathbb{E}\left\{{\text{SNR}}_{\ell }^{\text{ZF}}\right\}\right).\text{vspace}-4pt$$

Full text

I Introduction

††©  2019 IEEE. This paper was originally accepted for publication at IEEE GLOBECOM 2018, Abu Dhabi, December 2018, where it was not presented, and thus not included in IEEE Xplore. As such, the paper is now accepted for publication in IEEE PIMRC 2019, Istanbul, September 2019, with permission from the IEEE Copyrights Office and from IEEE GLOBECOM 2018 Technical Program Committee Chair.

It is now well understood that multiuser multiple–input multiple–output (MU–MIMO) systems with large antenna arrays at the base station (BS) will form an integral part of fifth–generation cellular [1, 2]. In its most general form, propagation between the BS and user terminals can be characterized by far–field multipath components (MPCs) arriving at the terminals from set of objects (a.k.a. scatterers) interacting with the transmitted waveform [3, 4]. MPCs often arrive at multiple terminals from clusters of scatterers, leading to non–uniform power angular spectra as seen by the BS, resulting in correlation of signals at the BS array elements [5]. Furthermore, depending on the severity of scattering in a given environment, and the relative physical separation of terminals, MPCs to different terminals arrive via the same cluster of scatterers, and hence are correlated [6, 7]. From single–user MIMO literature, it is known that channel correlation has a negative impact on the terminal and system spectral efficiency (see e.g., [3, 6, 7, 8, 9, 27]).

In stark contrast, a different set of investigations has shown that correlation can enhance MU–MIMO performance [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Collectively, a critical observation from these studies is that the departing spread of electromagnetic energy from the BS can arrive at the terminals with substantially different power angular spectra, leading to variations in the channel statistics. Fundamentally, such variations depend on the geometry of scattering, as well as the inter–element spacing at the BS. To encapsulate these variations power angular spectra, models such as one–ring correlation have been proposed [11, 12, 13, 14, 17, 20]. Such models are characterized in terms of the mean direction–of–arrival (DOA) at a terminal, angular spread of departure, as well as the antenna spacing at the BS. The authors of [11, 12] utilized the one–ring model to group the terminals with similar correlation characteristics in order to motivate the joint spatial division multiplexing technique. The study in [14] shows that if correlation matrices span orthogonal subspaces, pilot contamination can vanish, allowing the ergodic spectral efficiency to grow without bound with increasing numbers of BS antennas. The net spectral efficiency of a MU–MIMO system was numerically investigated in [16] as a function of the azimuth angular spread with pilot contamination.

Despite the above efforts, it remains to be seen just how much performance gain is available with correlation diversity, relative to the case when each terminal has an identical correlation pattern. More importantly, almost all of the analytical results predicting MU–MIMO performance with correlation diversity, are left in terms of complex mathematical expressions making it difficult to gain any insights into their behavior (see e.g., [12, 21, 22]). To gain a fundamental understanding of MU–MIMO performance with and without correlation diversity, it is desirable to have an insightful and simple downlink performance measure. This is largely missing from the literature, and hence is the aim of the paper. With this in mind, we derive simple, closed–form approximations to the downlink zero–forcing (ZF) expected signal–to–noise–ratio (SNR) and ergodic sum spectral efficiency. It is noteworthy that our recent work in [17] makes an initial attempt to explore the aforementioned issues with simple matched filter transmission, where we derive approximations for determining the impact of correlation diversity on MU–MIMO performance. Unlike [17], here we consider the more challenging case of ZF transmission, and examine the MU–MIMO gains with diversity in the channel correlation structure. As more physical correlation models rely on the propagation channel’s spatial parameters, for the most accurate parameterization, we extract the required parameters from a recent 2.53 GHz MU–MIMO measurement campaign in Cologne, Germany. To the best of the our knowledge, studies which use measured multipath parameters to investigate the diversity of correlation profiles in multiuser systems are rare.

Contributions. Assuming spatially correlated Rayleigh fading, our ZF approximations provide clear insights into the impact of correlation diversity, as well as other system parameters, such as the number of BS antennas, number of terminals, and the average operational downlink SNR. We provide explicit insights into the fact that fixed correlation profiles tend to amplify the expected ZF noise power, unlike diverse correlation profiles. We therefore argue that fixed correlation profiles can be used as a useful lower bound on the resulting system performance. Our results disclose that the choice of a particular correlation model has a direct relation to the expected ZF SNR and ergodic sum spectral efficiency. More physical models such as one–ring correlation, give superior performance over more simple models, such as the exponential and Clerckx correlation [9, 23]. To parameterize the correlation models, we utilize measured angular spreads and mean DOA distributions at 2.53 GHz from an urban macrocellular (UMa) measurement campaign.

Notation. Upper and lower boldface letters represent matrices and vectors. The $M\times{}M$ identity matrix is denoted as $\mathbf{I}_{M}$. Transpose, Hermitian transpose, inverse and trace operators are denoted by $(\cdot)^{T}$, $\left(\cdot\right)^{H}$, $(\cdot)^{-1}$, and $\textrm{Tr}[\cdot]$, respectively. Moreover, $||\cdot{}||_{F}$ denotes the Frobenius norm. We use $\mathbf{h}\sim\mathcal{CN}\left(\mathbf{m},\mathbf{R}\right)$ to denote a complex Gaussian distribution for $\mathbf{h}$ with mean $\mathbf{m}$ and covariance matrix $\mathbf{R}$. Similarly, $h\sim\mathcal{U}\left[a,b\right]$ is used to denote a uniform random variable for $h$, taking on values from $a$ to $b$. Finally, $\mathbb{E}\{\cdot\}$ denotes the statistical expectation.

II System Model

We consider the downlink of a single–cell MU–MIMO system operating in an UMa environment. The BS is located at the center of a circular cell with radius $R_{\textrm{c}}$, and is equipped with an array of $M$ transmit antennas. The BS serves $L$ single–antenna terminals $\left(M\geq{}L\right)$ in the same time–frequency resource. Channel knowledge is assumed at the BS with narrow–band transmission and uniform power allocation.

Remark 1. At first sight, the assumption of perfect channel knowledge may seem rather naive. However, there are several reasons for this: Firstly, unlike previous studies, the central focus of the work is to devise a simple, yet accurate, performance metric to gain insights into the behavior of correlation diversity in multiuser systems. In contrast to prior studies, measured spatial parameters of the channel are utilized to capture the power angular spectra variations across multiple terminals. Under this heterogeneous scenario, it is extremely difficult to make analytical progress without perfect channel knowledge. Secondly, this assumption allows us to effectively separate the propagation effects from the system related effects, i.e., to study the influence of correlation diversity in isolation. Thirdly, it is worth noting that the results obtained from the subsequent analysis and evaluations can be treated as a useful upper bound on the performance which may be seen in practice with estimated propagation channels.

The $1\times{}M$ propagation channel to terminal $\ell$ from the BS array is denoted by $\mathbf{h}_{\ell}$, which is assumed to follow a spatially correlated Rayleigh fading distribution, i.e., $\mathbf{h}_{\ell}\sim\mathcal{CN}\left(\mathbf{0},\mathbf{R}_{\ell}\right)$. Unlike previously (see e.g., [8, 9]), $\mathbf{R}_{\ell}$, the $M\times{}M$ correlation matrix, is specific to terminal $\ell$. Naturally, $\mathbf{R}_{\ell}$ will be a function of the channel’s spatial parameters [11, 12, 14, 6]. For clarity, further discussion on the possible structures of $\mathbf{R}_{\ell}$ is deferred till Sec. V. The received signal at terminal $\ell$ is

[TABLE]

where $\rho_{\textrm{t}}$ is the average transmit power at the BS and $\beta_{\ell}$ denotes the link gain of terminal $\ell$ (discussed later in the text). Furthermore, $\mathbf{g}_{\ell}$ is the $M\times{}1$ un–normalized downlink precoding vector from the BS array to terminal $\ell$, obtained from $\ell$–th column of $\mathbf{G}$, the composite $M\times{}L$ un–normalized precoding matrix. The data symbol for terminal $\ell$ is denoted by $s_{\ell}$, such that $\mathbb{E}\{|s_{\ell}|^{2}\}=1,\forall{}\ell=1,2,\dots,L$, and $n_{\ell}\sim{}\mathcal{CN}\left(0,\sigma^{2}\right)$ models the additive white Gaussian noise at terminal $\ell$. Note that $\sigma^{2}$ is fixed for all terminals $1,2,\dots,L$. Following [9, 22], $\eta=\|\mathbf{G}\|^{2}_{F}/L$ is the precoding normalization parameter such that $\mathbb{E}\{\|\mathbf{g}_{\ell}\|^{2}\}=1$, for $\ell=1,2,\dots,{}L$ (discussed further in the text). The link gain at terminal $\ell$, $\beta_{\ell}=A\zeta_{\ell}\left(r_{0}/r_{\ell}\right)^{\alpha}$ is composed of the large–scale propagation effects: Particularly, $A$ is the unit–less constant for geometric attenuation at a reference distance $r_{0}$, $r_{\ell}$ is the link distance between the BS and terminal $\ell$, $\alpha$ is the attenuation exponent and $\zeta_{\ell}$ models the effects of shadow fading which follows a lognormal distribution, i.e., $10\log_{10}\left(\zeta_{\ell}\right)\sim{}\mathcal{N}\left(0,\sigma_{\textrm{sh}}^{2}\right)$. For clarity, we delay quoting values for the above parameters to Sec. V. It is well known that ZF precoding nulls multiuser interference (second–term of (1)). This means that the signal–to–interference–plus–noise–ratio translates to a SNR [9]. Note that $\mathbf{g}_{\ell}$ forms the $\ell$–th column of $\mathbf{G}=\mathbf{H}^{H}\left(\mathbf{H}\mathbf{H}^{H}\right)^{-1}$, where $\mathbf{H}=[\hskip 2.0pt\mathbf{h}_{1}^{T},\mathbf{h}_{2}^{T},\dots,\mathbf{h}_{L}^{T}\hskip 2.0pt]^{T}$ is the $L\times{}M$ matrix containing channels for all $L$ terminals. Recognizing that $\mathbf{H}\mathbf{G}=\mathbf{H}\mathbf{H}^{H}\left(\mathbf{H}\mathbf{H}^{H}\right)^{-1}=\mathbf{I}_{L}$, the ZF SNR at terminal $\ell$ is

[TABLE]

since $\eta=\|\mathbf{G}\|^{2}_{F}/L=\textrm{Tr}\hskip 2.0pt[\hskip 1.0pt(\hskip 1.0pt\mathbf{H}\mathbf{H}^{H})^{-1}\hskip 2.0pt]/L$. The ZF SNR in (2) can be used to estimate the ergodic sum spectral efficiency (in bits/sec/Hz) for all $L$ terminals. This is given by

[TABLE]

where the expectation is over small–scale fading. Below, closed–form approximations of (2) and (3) are derived.

III Analytical Results and Implications

III-A Expected SNR and Ergodic Sum Spectral Efficiency

Remark 2. Finding exact moments of the ZF SNR in (2) is an extremely challenging task, since the matrix trace in its denominator is a random function of the inverse, of a complex non–standard semi–correlated central Wishart distribution formed by $\mathbf{H}\mathbf{H}^{H}$. Moreover, $\mathbf{H}$ has a fully heterogeneous structure, since it contains $L$ different correlation patterns and link gains. Due to these reasons, we approximate the inverse in (2) with a finite order Neumann series expansion [24, 25]. To do this, we separate $\mathbf{H}\mathbf{H}^{H}$ into its expected diagonal components and correction terms. That is, $\mathbf{H}\mathbf{H}^{H}=M\hskip 1.0pt\mathbf{I}_{L}\hskip 1.0pt+\hskip 1.0pt\mathbf{\Delta}$, where $\mathbf{\Delta}=\mathbf{H}\mathbf{H}^{H}-M\hskip 1.0pt\mathbf{I}_{L}$ and $\mathbb{E}\left\{\mathbf{\Delta}\right\}=\mathbf{0}$. Then, with an order $N$ Neumann series, we can write $(\mathbf{H}\mathbf{H}^{H})^{-1}$ as

[TABLE]

Substituting the definition of $\mathbf{\Delta}$, and simplifying yields

[TABLE]

Substituting the above into the denominator of (2) allows us to write the ZF SNR for terminal $\ell$ as (6) (on top of the following page for reasons of space).

Remark 3. In what follows, we evaluate the expected value of (6). The expectation is extremely cumbersome to perform, since it needs to be taken over the myriad of small–scale fading in $\mathbf{HH}^{H},$ a term on the denominator of (6). To overcome this difficulty, we employ the univariate special case of the commonly used first–order Laplace approximation [9, 26, 27], allowing us to express (6) as (7) (shown on top of the following page for the reasons of space).

The approximation in (7) is a first–order Laplace expansion and is of the form $\mathbb{E}\{\gamma/X\}\approx\gamma/\mathbb{E}\{X\}$, where $\gamma$ is a scalar value. As shown in [9, 26, 27], the accuracy of such approximations relies on $X$ having a small variance relative to its mean. This can be seen by applying a multivariate Taylor series expansion to $\gamma/X$ around $\gamma/\mathbb{E}\{X\}$. The terms in (7) are well suited to this approximation, especially when $M$ and $L$ start to grow, since the implicit averaging in the denominator gives rise to the variance reduction required [9]. In the following proposition, with a two–term Neumann series (i.e., $N=2$), a closed–form solution to (7) is presented for heterogeneous channels.

Proposition 1. When $\mathbf{h}_{\ell}\sim\mathcal{CN}\left(\mathbf{0},\mathbf{R}_{\ell}\right)$, where $\mathbf{R}_{\ell}$ is the correlation matrix specific to terminal $\ell$, the expected ZF SNR for the $\ell$–th terminal can be approximated as

[TABLE]

where $\bar{\mathbf{R}}=\frac{\sum\nolimits_{\ell=1}^{L}\mathbf{R}_{\ell}}{L}$, and is the average correlation matrix of all terminals in the system.

Proof: From (5), when $N=2$, one can write

[TABLE]

Taking the matrix trace of (9) yields

[TABLE]

Performing the expectation of (10) results in taking the expectation of the individual terms on the right–hand side (RHS) of (10). As $\mathbb{E}\{\textrm{Tr}\hskip 1.0pt[\hskip 1.0pt\mathbf{HH}^{H}]\hskip 1.0pt\}=ML$, the first two–terms on the RHS of (10) result in a cancellation, allowing us to focus on the expectation of the final term of (10). By definition, the final term on the RHS of (10) is given by

[TABLE]

Taking the expectation of (11) over small–scale fading yields

[TABLE]

Further simplifying the above expression allows us to state

[TABLE]

Inserting (14) into the mean of (10), and simplifying gives

[TABLE]

The result in (15) can now be substituted into the denominator of (7) with some routine simplifications to obtain (8). ∎

III-B Implications of Proposition 1

To the best of the authors’ knowledge, the result in (8) is the first, simple, closed–form approximation with ZF precoding and correlation diversity. The structure of (8) readily demonstrates the impact of correlation diversity via the term $\textrm{Tr}[\bar{\mathbf{R}}^{2}]$, which influences the expected noise power of the desired terminal. Fixing all system parameters, one can observe that the expected ZF noise power is maximized when $\textrm{Tr}\left[\bar{\mathbf{R}}^{2}\right]$ is maximized. By definition, $\textrm{Tr}[\bar{\mathbf{R}}^{2}]=M+2\sum\nolimits_{i=1}^{M-1}\sum\nolimits_{j=i+1}^{M}|\bar{r}_{i,j}|^{2}$, where $\bar{r}_{i,j}$ denotes the $(i,j)$–th element of $\bar{\mathbf{R}}$, such that $|\bar{r}_{i,j}|^{2}=|\frac{1}{L}\sum\nolimits_{\ell=1}^{L}(\mathbf{R}_{\ell})_{i,j}|^{2}$. Maximizing $|\bar{r}_{i,j}|^{2}$ requires alignment of all the terms in the modulus. Specifically, the phases of all entries in $\mathbf{R}_{\ell}$ need to align in the $(i,j)$–th position, $\forall{}\ell=1,2,\dots,L$. While such a scenario is generally unlikely to occur in practice, we identify one possible situation when this may take place: Precisely, when each terminal’s correlation matrix is identical (the case for homogeneous channels), all phases will be aligned across all terminals in the $(i,j)$–th position. Though the above condition does not require the amplitudes of each terminal’s correlation matrix to be equal, in the case of identical correlation matrices, the amplitudes will naturally also be equal across all terminals. Hence, identical correlation matrices result in the lowest ZF SNRs, serving as a useful lower bound for the performance of correlated multiuser channels. On the other hand, when each terminal experiences the same angular spread, yet a different mean angle, $\textrm{Tr}[\bar{\mathbf{R}}^{2}]$ yields a smaller value, since diversity is induced via the differences in the mean DOAs. On a similar note, with variations in both the angular spread and mean DOAs, maximum diversity kicks in, where $\textrm{Tr}[\bar{\mathbf{R}}^{2}]$ tends to be even smaller, leading to a higher ZF SNR.

In addition to the above, holding all other propagation and system parameters constant, increasing the number of BS antennas, $M$, causes a linear increase in the expected SNR (due to its numerator containing $M^{3}$ and the denominator containing a $M^{2}$). Meanwhile, increasing the number of user terminals, $L$, while keeping other parameters fixed leads to a quadratic increase in the expected noise power, degrading the ZF SNR. The above insights are difficult to obtain from more complex solutions derived in the literature (see e.g., [12, 21, 22]), which require a linked set of equations, even in the large system regime. In contrast, our analysis poses no such constraints and is general to the operational system dimension, and channel correlation structure. Note that (8) can be used to approximate the ergodic sum spectral efficiency given by

[TABLE]

The accuracy of the derived results in (8) and (16) with their simulated counterparts is presented in Sec. V.

Remark 4. In addition to the results presented in Sec. V, we note that (8) and (16) were explicitly tested with sample simulations based on scattering cluster circles in the propagation channel (one circle per–terminal) with varying center distances from each other. The one–ring correlation model described in Sec. V-A was used for the aforementioned evaluations, where agreement within 1 dB was found from the results derived in (8) and (16) relative to the simulated cases.

IV Channel Measurements

We will evaluate our analytical results on MU–MIMO channel measurements in an urban macrocellular environment. The measurements were performed in the city of Cologne in Germany. The BS acted as TX, and was placed on a high–rise building, and was thus located 30 m above ground (see e.g., [17, 28, 30] for a picture). The buildings in the area of interest were (with the exception of the high rise on which the BS was located, and the Cologne cathedral) of approximately similar height, ranging from 4–8 floors. The terminal, acting as RX, was on the rooftop of a car, roughly 2.5 m above ground (see e.g., [17, 28, 30] for a picture). The RX was placed at 50 distinct locations throughout the cell, distributed across 800 m from the TX. The distribution of RX positions throughout the measurement campaign is depicted in Fig. 1. Note that line-of-sight can be “seen" at multiple positions, as shown in the figure.

Since the impact of moving scatterers was negligible, the channels for the different RX positions can be used to emulate a multiuser propagation measurement campaign with service across 50 terminals. The channel sounding was performed with a MEDAV RUSK channel sounder for wideband MIMO channels. The sounder operated with a center frequency of 2.53 GHz, and a bandwidth of 20 MHz; since the operator (Deutsche Telekom) owning this band switched off the surrounding BSs for the duration of the measurements, no interference was present. The arrays used at both link ends were cylindrical, with 8 vertical and 60 horizontal antenna elements at the TX, and 2 vertical and 8 horizontal elements at the RX; all elements were dual polarized. Due to the cylindrical structures, MPCs from all directions (all azimuths and elevations) could be received and measured. The sounding principles are slightly different at TX and RX: the RX uses the switched antenna array principle [29, 28], such that a single antenna element is measured at a time; an electronic switch is used to quickly cycle through all RX antenna elements and connect them sequentially to the single existing radio–frequency downconversion chain. The TX, on the other hand, uses a combination of this switched sounding (for the vertical and polarization domains) with a virtual array in azimuth, created by mechanical rotation of the physical (vertical) array. Measurement of a MIMO snapshot (all combinations of TX and RX antennas) took about 10 minutes; this long duration was acceptable because the high terminal position mostly eliminated the impact of moving scatterers and the channel was thus essentially static. More details about the measurement setup and the environment can be found in [30].

The channel sounder measures the channel transfer function matrix. $\mathbf{H}(f,t,r)$; multiple snapshots of that matrix are averaged to improve the SNR. Here $f$ is the index of the subcarrier at which $\mathbf{H}$ is evaluated, and $t$ and $r$ denote the indices of the TX and RX antenna elements, respectively. For simplicity, we do not explicitly denote polarization henceforth. From the transfer function matrix, we extract the parameters of the MPCs via RiMAX [31], a high resolution parameter estimation algorithm that can be interpreted as an iterative maximum–likelihood estimator. RiMAX provides a double–directional channel description [32], i.e., an antenna independent characterization of the channel. In particular, it provides the amplitude, delays, DOAs, and direction–of–departures (DODs) of all MPCs. We refer the reader to [28, 30] for a more detailed description of the parameter extraction routines. From the double–directional channel characterization, we can extract the root mean square azimuth angular spread of departure, as well as the mean DOA distributions across all 50 terminal positions. The obtained distributions and functional fits are depicted in fig. 2. Rather interestingly, we can observe variability in the azimuth angular spread of departure and mean DOA, which can be modeled as $\mathcal{N}(14.02^{\circ},(6.45^{2})^{\circ})$ and $\mathcal{U}[-180^{\circ},180^{\circ}]$, primarily reflecting the multipath characteristics at the TX and RX positions, respectively. From these we can obtain the parameterized correlation structures considered in the paper (see Sec. V-A).

V Numerical Results

Unless otherwise specified, the parameters described below are utilized for all numerical results, and are obtained from [33]. The cell radius, $R_{\textrm{c}}=500$ m was chosen with a reference distance $r_{0}=50$ m, such that terminals are randomly located outside $r_{0}$ and inside $R_{c}$, following $\mathcal{U}[-180^{\circ},180^{\circ}]$. Uniform power control is assumed so that the average transmit power is independent of distance. The UMa attenuation exponent of $\alpha=3.67$ was chosen. Furthermore, it is assumed that $\sigma^{2}=1$, and hence the average downlink SNR is equivalent to the average downlink transmit power, $\rho_{\textrm{t}}/\sigma^{2}=\rho_{\textrm{t}}$. The unit–less constant for geometric attenuation, $A$, is chosen such that the fifth percentile of the instantaneous SNR with ZF precoding at terminal $\ell$ is 0 dB, when $\rho_{\textrm{t}}=0$ dB with the baseline system parameters of $M=64$ and $L=6$. Note that the exponential correlation model (described later, with the correlation coefficient $\xi=0.9$ was used to obtain $A$. The shadowing standard deviation, $\sigma_{\textrm{sh}}=6$ dB. For all numerical results, $10^{5}$ Monte–Carlo realizations were generated with an inter–element spacing, $d=0.5\lambda$ at the BS, where applicable with $\lambda$ denoting the wavelength at the desired frequency.

V-A Channel Correlation Models

As a baseline case, we assign a fixed correlation profile to each terminal with the widely used exponential model. Here, the $(i,j)$–th element of $\mathbf{R}_{\ell}$ is expressed as $\left[\mathbf{R}_{\ell}\right]_{i,j}=\xi^{|i-j|}$, for any $i,j$ in $1,2,\dots,M$ with $0\leq\xi\leq{}1$ [9]. With correlation diversity, we employ two models, namely Clerckx [23], and one–ring [12, 14] correlation. For the Clerckx correlation model, $\left[\mathbf{R}_{\ell}\right]_{i,j}=\xi_{\ell}^{|i-j|}$, where $\xi_{\ell}=|\xi|e^{j\Delta_{\ell}}$. Here, $|\xi|=\xi$, as in the exponential model, and is the same for each terminal. However, a terminal specific phase, $\Delta_{\ell}$, is assumed to be $\mathcal{U}[-180^{\circ},180^{\circ}]$. This is used to differentiate the terminal locations. In each result, the range of $\Delta_{\ell}$ is specified. The one–ring model for terminal $\ell$ states $\left[\mathbf{R}_{\ell}\right]_{i,j}=\frac{1}{2\Delta_{\ell}}\int\nolimits_{-\Delta_{\ell}+\phi_{0}^{\ell}}^{\Delta_{\ell}+\phi_{0}^{\ell}}e^{-j2\pi{}d\left(i,j\right)\sin{}\left(\phi_{\ell}\right)}d\phi_{\ell}$, where $\Delta_{\ell}$ denotes the azimuth angular spread for terminal $\ell$, $\phi_{0}^{\ell}$ denotes the mean DOA, and $\phi_{\ell}$ is the actual DOA, uniformly distributed within the angular spread around the mean DOA. Furthermore, $d\left(i,j\right)$ captures the normalized antenna spacing between the $i$–th and $j$–th elements. The precise values of $\Delta_{\ell}$ for the one–ring model are specified in each subsequent result.

V-B Impact of Correlation Diversity

Figure 3 shows the CDFs of the expected ZF SNR with $M=64$, $L=6$, and $\rho_{\textrm{t}}=5$ dB. Each curve is obtained by averaging over the small–scale fading, with the CDFs capturing the randomness from the link gains. Two insights can be drawn: (1) With correlation diversity from the Clerckx model, the larger the spread of the random phases in $\Delta$’s, the higher the expected ZF SNR. Irrespective of the correlation magnitude being as high as $\xi=0.9$, increasing the spread of $\Delta$ to $\mathcal{U}\left[0,14^{\circ}\right]$, $\mathcal{U}\left[0,28^{\circ}\right]$, and $\mathcal{U}\left[0,38^{\circ}\right]$ yields a 0.93, 2, and 3 dB gain in the expected ZF SNR, relative to fixed correlation (exponential model) patterns. This difference is due to the increasing the spread of $\Delta$, consequently increasing the amount of path selectivity across multiple channels, yielding higher composite channel rank. The result demonstrates the sensitivity of MU–MIMO to changes in the phase of the correlation patterns. Secondly, our proposed approximations agree well with the simulated cases, for all values of $\Delta$.

Figure 4 demonstrates the expected ZF SNR with changes in the average operating SNRs. In addition to the cases for exponential and Clerckx correlations, performance with the one–ring model is also evaluated. We can see that even with a fixed angular spread, the one–ring model predicts higher ZF SNRs in comparison to the Clerckx model. This is because both the magnitude and phase of the correlation matrices are variable across each terminal in the case of the one–ring model. Further to this, when evaluating the expected ZF SNRs with the measured angular parameters, a further 3 dB increase in the ZF SNR is seen across all SNRs. This is attributed to the increased diversity brought by the variations in the angular spreads (Gaussian random variables). The proposed approximations are seen to remain tight across all the considered models, and SNR values. Keeping all other parameters constant, Fig. 5 depicts the ZF ergodic sum spectral efficiency as a function of the operating SNR with $M=128$ and $L=10$. While similar trends to Fig. 4 can be observed, it is notable that the remarkably simple approximations remain tight across a wider range of system dimensions and operational SNRs.

VI Conclusions

The paper presents closed–form approximations to the ZF expected SNR and ergodic sum spectral efficiency of a MU–MIMO system. With unequally correlated Rayleigh fading, our analysis is robust to various physical and non–physical channel correlation models, as well as average downlink SNR. More physically motivated models, such as one–ring correlation, consider unequal magnitudes and phases in the correlation matrices for each terminal, and tend to estimate higher MU–MIMO performance. Data from the recent 2.53 GHz UMa propagation measurements was extracted to accurately parameterize correlation models in order to characterize their impact on MU–MIMO performance. Such an evaluation emphasizes the fact that the performance of a MU–MIMO system is ultimately governed by the correlation model, and its parameters in use.