Diversity gain of millimeter-wave massive MIMO systems with distributed antenna arrays

Abstract

This paper is concerned with diversity gain analysis for millimeter-wave (mmWave) massive MIMO systems employing distributed antenna subarray architecture. First, for a single-user mmWave system in which the transmitter and receiver consist of K_t and K_r subarrays, respectively, a diversity gain theorem is established when the numbers of subarray antennas go to infinity. Specifically, assuming that all subchannels have the same number of propagation paths L, the theorem states that by employing such a distributed antenna subarray architecture, a diversity gain of K_rK_tL−N_s+1 can be achieved, where N_s represents the number of data streams. This result means that compared to the co-located antenna architecture, using the distributed antenna subarray architecture can scale up the diversity gain proportionally to K_rK_t. The analysis of diversity gain is then extended to the multiuser scenario as well as the scenario with conventional partially connected radio-frequency structure in the literature. Simulation results obtained with the hybrid analog/digital processing corroborate the analysis results and show that the distributed subarray architecture indeed yields a significantly better diversity performance than the co-located antenna architectures.

1 Introduction

Millimeter-wave (mmWave) communication has recently gained considerable attention as a candidate technology for 5G mobile communication systems and beyond [1–5]. The main reason for this is the availability of vast spectrum in the mmWave band that is very attractive for high data rate communications. However, compared to communication systems operating at lower microwave frequencies (such as those currently used for 4G mobile communications), propagation loss in mmWave frequencies is much higher, in orders of magnitude. Fortunately, given the much smaller carrier wavelengths, mmWave communication systems can make use of compact massive antenna arrays to compensate for the increased propagation loss.

Nevertheless, the large-scale antenna arrays together with high cost and large power consumption of the mixed analog/digital signal components make it difficult to equip a separate radio-frequency (RF) chain for each antenna element and perform all the signal processing in the baseband. Therefore, research on hybrid analog-digital processing of precoder and combiner for mmWave communication systems has attracted considerably strong interests from both academia and industry [6–18]. In particular, a large body of work has been performed to address challenges in employing a limited number of RF chains for massive antenna arrays. For example, the authors in [6] investigated single-user precoding in massive MIMO mmWave systems and obtained the optimality of beam steering for both single-stream and multi-stream transmission scenarios. In [9], the authors showed that the hybrid processing can realize any fully digital processing exactly when the number of RF chains is twice the number of data streams.

Two architectures for connecting the RF chains in the hybrid processing that have been investigated in the literature are fully connected and partially connected. In the former, each RF chain is connected to all the antenna elements, while only a subset of antenna elements is connected to each RF chain in the latter. The partially connected architecture is more energy-efficient and implementation-friendly since the number of required phase shifters can be reduced efficiently without a significant performance loss. In the conventional partially connected architecture [10–14], the antenna array is partitioned into a number of smaller disjoint subarrays, each of which is driven by a single transmission chain. More recently, a more general partially connected architecture, referred to as hybridly connected in [15] and overlapped subarray-based in [16], has been proposed. In such a hybridly connected structure, each subarray is connected to multiple RF chains while each RF chain is connected to all the antennas, which corresponds to the subarray in question. In particular, the authors in [15] show that the spectral efficiency of the hybridly connected structure is higher than that of the partially connected structure and that as the number of RF chains increases, its spectral efficiency can close to that of the fully connected structure.

Nevertheless, due to the facts that the antenna arrays in the abovementioned RF architectures are co-located and mmWave signal propagation has an important feature of multipath sparsity in both the spatial and temporal domains [19, 20], it is expected that the potentially available diversity and multiplexing gains are not large for the co-located antenna deployment. In order to enlarge the diversity gain and/or multiplexing gain in mmWave massive MIMO communication systems, this paper considers a more general antenna array architecture, called distributed antenna subarray architecture, which includes co-located array architecture as special cases. It is pointed out that deploying distributed antennas has shown a promising technique to increase spectral efficiency and expand the coverage of wireless communication networks [21–25]. As such, it is of great interest to consider distributed antenna deployment in the context of mmWave massive MIMO systems.

The diversity-multiplexing trade-off (DMT) is a compact and convenient framework to compare different MIMO systems in terms of the two main and related system indicators: data rate and error performance [26–29]. This trade-off was originally characterized in [26] for MIMO communication systems operating over independent and identically distributed (i.i.d.) Rayleigh fading channels. Then, the framework has ignited a lot of interests in analyzing various communication systems and under different channel models. For a massive MIMO mmWave system, how to quantify the diversity performance and characterize its DMT is a fundamental and open research problem. In particular, to the best of our knowledge, until now, there is no unified diversity gain analysis for massive MIMO mmWave systems that is applicable to both co-located and distributed antenna array architectures.

To fill this gap, this paper investigates the diversity performance of massive MIMO mmWave systems with the proposed distributed subarray architecture. The focus is on the asymptotical diversity gain analysis in order to find out the potential diversity advantage provided by multiple distributed antenna arrays. The obtained analysis can be used conveniently to compare various massive MIMO mmWave systems with different distributed antenna array structures. Our main contributions are summarized as follows: First, for a single-user system employing the proposed distributed subarray architecture, a diversity gain expression is obtained when the number of antennas at each subarray increases without bound. This expression clearly indicates that one can obtain a large diversity gain and/or multiplexing gain by employing the proposed distributed subarray architecture. Second, the diversity gain analysis is extended to the multiuser scenario with downlink and uplink transmission, as well as the single-user system employing the conventional partially connected RF structure based on the distributed subarrays. Simulation results are provided to corroborate the analysis results and show that the distributed subarray architecture yields significantly better diversity performance than the co-located single-array architecture.

The remainder of this paper is organized as follows: Section 2 describes the massive MIMO mmWave system model and hybrid processing with the distributed subarray architecture in mmWave fading channels. Section 3 provides the asymptotical diversity analysis for the single-user mmWave system. In Sections 4 and 5, the diversity gain analysis is extended to the multiuser scenario and the scenario with the conventional partially connected RF architecture, respectively. Numerical results are presented in Section 6. Section 7 finally concludes the paper.

Throughout this paper, the following notations are used. The superscripts (·)^T and (·)^H stand for transpose and conjugate transpose, respectively. Boldface upper and lower case letters denote matrices and column vectors, respectively. diag{a₁,a₂,…,a_N} stands for a diagonal matrix with diagonal elements {a₁,a₂,…,a_N}. The expectation operator is denoted by $\mathbb {E}(\dot)$. [A]_ij gives the (i,j)th entry of matrix A. $\mathbf {A} \bigotimes \mathbf {B}$ is the Kronecker product of A and B. A function a(x) of x is written as o(x) if ${\lim }_{x \to 0}a(x)/x=0$. Finally, $\mathcal {CN}(0, 1)$ represents a circularly symmetric complex Gaussian random variable with zero mean and unit variance.

2 System model

A single-user massive MIMO mmWave system is considered as shown in Fig. 1. The transmitter is equipped with a distributed antenna array to send N_s data streams to a receiver, which is also equipped with a distributed antenna array. Here, a distributed antenna array means an array consisting of several remote antenna units (RAUs) (i.e., antenna subarrays) that are distributively located, as depicted in Fig. 2. Specifically, the antenna array at the transmitter consists of K_t RAUs, each of which has N_t antennas and is connected to a baseband processing unit (BPU) via fiber. Likewise, the distributed antenna array at the receiver consists of K_r RAUs, each having N_r antennas and also being connected to a BPU by fibers. Such a MIMO system shall be referred to as a (K_t,N_t,K_r,N_r)-distributed MIMO (D-MIMO) system. When K_t=K_r=1, the system reduces to a conventional co-located MIMO (C-MIMO) system.

Fig. 1

Block diagram of a mmWave massive MIMO system with distributed antenna arrays

Fig. 2

Illustration of distributed antenna array deployment

The transmitter accepts as its input N_s data streams and is equipped with $N_{t}^{(\text {rf})}$ RF chains, where $N_{s} \leq N_{t}^{\left (\text {rf}\right)} \leq N_{t} K_{t}$. Given $N_{t}^{(\text {rf})}$ transmit RF chains, the transmitter can apply a low-dimension $N_{t}^{(\text {rf})} \times N_{s}$ baseband precoder, W_t, followed by a high-dimension $K_{t}N_{t} \times N_{t}^{(\text {rf})}$ RF precoder, F_t. Note that for the baseband precoder W_t, amplitude and phase modifications are feasible while only phase changes can be made by the RF precoder F_t through the use of variable phase shifters and combiners. The transmitted signal vector can be written as:

$$ \mathbf{x} = \mathbf{F}_{t}\mathbf{W}_{t}\mathbf{s}, $$

(1)

where s is the N_s×1 symbol vector such that $\mathbb {E}\left [\mathbf {s}\mathbf {s}^{H}\right ] = \frac {P}{N_{s}}\mathbf {I}_{N_{s}}$. Thus, P represents the average total input power. Considering a narrowband block fading channel, the K_rN_r×1 received signal vector is:

$$ \mathbf{y} =\mathbf{H}\mathbf{F}_{t}\mathbf{W}_{t}\mathbf{s} + \mathbf{n} $$

(2)

where H is K_rN_r×K_tN_t channel matrix, and n is a K_rN_r×1 vector consisting of i.i.d. $\mathcal {CN}(0, 1)$ noise samples. Throughout this paper, H is assumed known to both the transmitter and receiver. Given that $N_{r}^{(\text {rf})}$ RF chains $\left (\text {where} N_{s} \leq N_{r}^{(\text {rf})} \leq N_{r}K_{r}\right)$ are used at the receiver to detect the N_s data streams, the processed signal vector can be given by:

$$ \mathbf{z} =\mathbf{W}_{r}^{H}\mathbf{F}_{r}^{H}\mathbf{H}\mathbf{F}_{t}\mathbf{W}_{t}\mathbf{s} + \mathbf{W}_{r}^{H}\mathbf{F}_{r}^{H}\mathbf{n} $$

(3)

where W_r is the $N_{r}^{(\text {rf})} \times N_{s}$ baseband combining matrix, and F_r is the $K_{r} N_{r} \times N_{r}^{(\text {rf})}$ RF combining matrix.

Furthermore, according to the architecture of RAUs at the transmitting and receiving ends, H can be written as:

$$ \mathbf{H}=\left[ \begin{array}{lll} \sqrt{g_{11}}\mathbf{H}_{11} & \cdots & \;\;\sqrt{g_{1K_{t}}}\mathbf{H}_{1K_{t}} \\ \;\;\;\;\vdots & \ddots & \;\;\;\;\;\;\;\;\vdots \\ \sqrt{g_{K_{r}1}}\mathbf{H}_{K_{r}1} & \cdots & \sqrt{g_{K_{r}K_{t}}}\mathbf{H}_{K_{r}K_{t}} \end{array}\right]. $$

(4)

In the above expression, g_ij represents the large scale fading parameter between the ith RAU at the receiver and the jth RAU at the transmitter. We assume that g_ij is constant over many coherence time intervals. The normalized subchannel matrix H_ij represents the MIMO channel between the jth RAU at the transmitter and the ith RAU at the receiver.

Based on the extended Saleh-Valenzuela model, a clustered channel model is used often in mmWave channel modeling and standardization [6, 14, 15], and it is also adopted in this paper. As in [6], each scattering cluster is assumed to contribute a single propagation path for simplicity of exposition. Using this model, the subchannel matrix H_ij is given by:

$$ \mathbf{H}_{ij} = \sqrt{\frac{N_{t}N_{r}}{L_{ij}}} \sum\limits_{l=1}^{L_{ij}} \alpha_{ij}^{l} \mathbf{a}_{r} \left(\phi^{rl}_{ij}, \theta^{rl}_{ij}\right) \mathbf{a}_{t}^{H} \left(\phi^{tl}_{ij},\theta^{tl}_{ij}\right), $$

(5)

where L_ij denotes the number of propagation paths, $\alpha _{ij}^{l}$ denotes the complex gain of the lth ray, and $\phi ^{rl}_{ij} \left (\theta ^{rl}_{ij}\right)$ and $\phi ^{tl}_{ij} \left (\theta ^{tl}_{ij}\right)$ denote its random azimuth (elevation) angles of arrival and departure, respectively. Without loss of generality, the complex gains $\alpha _{ij}^{l}$ are assumed to be $\mathcal {CN}(0, 1)$¹. The vectors $\mathbf {a}_{t} \left (\phi ^{tl}_{ij}, \theta ^{tl}_{ij}\right)$ and $\mathbf {a}_{r} \left (\phi ^{rl}_{ij},\theta ^{rl}_{ij} \right)$ stand for the normalized transmit/receive array response vectors at the corresponding angles of departure/arrival. The array response vector of an N-element uniform linear array (ULA) is:

$$ \mathbf{a}^{\text{ULA}} (\phi) = \frac{1}{\sqrt{N}} \left[1,{\mathrm{e}}^{j2 \pi \frac{d} {\lambda} \sin(\phi)}, \ldots, {\mathrm e}^{j2\pi(N-1) \frac{d}{\lambda} \sin(\phi)} \right]^{T} $$

(6)

where λ is the wavelength of the carrier, and d is the inter-element spacing. It is pointed out that the angle θ is not included in the argument of a^ULA since the response for an ULA is independent of the elevation angle. In contrast, for a uniform planar array (UPA), which is composed of N_v and N_h antenna elements in the vertical and horizontal directions, respectively, the array response vector is represented by:

$$ \mathbf{a}^{\text{UPA}}(\phi, \theta)=\mathbf{a}^{\text{ULA}}_{h}(\phi) \otimes \mathbf{a}^{\text{ULA}}_{v}(\theta), $$

(7)

where

$$ \mathbf{a}^{\text{ULA}}_{h}(\phi) = \frac{1}{\sqrt{N_{h}}} \left[ 1, \mathrm{e}^{j2 \pi \frac{d_{h}} {\lambda} \sin (\phi)}, \ldots, \mathrm{e}^{j2 \pi\left(N_{h}-1 \right) \frac{d_{h}} {\lambda} \sin(\phi)} \right]^{T} $$

(8)

and

$$ \mathbf{a}^{\text{ULA}}_{v}(\theta) = \frac{1}{\sqrt{N_{v}}} \left[ 1, \mathrm{e}^{j2\pi\frac{d_{v}} {\lambda} \sin(\theta)}, \ldots, \mathrm{e}^{j2\pi(N^{v}-1) \frac{d_{v}} {\lambda} \sin (\theta)}\right]^{T}. $$

(9)

3 Diversity gain analysis

The most common performance metric of a digital communication system is the error probability, which can be defined either as the probability of symbol error or the probability of bit error (i.e., the bit error rate (BER)). When communicating over a fading channel, errors obviously depend on specific channel realizations. As such, the random behavior of a fading channel needs to be taken into account, which leads to the concept of average error probabilities [30]. Determining the exact expressions for the average error probabilities for a digital communication system operating over a certain fading channel is usually tedious and might not give a clear insight about the system behavior. As such, there is a need to characterize the performance of a communication system in an insightful and simple way. A popular approach is to shift the focus from exact performance analysis to asymptotic performance analysis, i.e., analyzing the performance at the high signal-to-noise (SNR) region, as done in [31]. This is a reasonable approach since the performance of practical interest is in the high SNR region, and in such a region, good approximation can be made on the exact analysis.

For most cases, the average BER function can be approximated in the high SNR region as [31]:

$$ \overline{\text{BER}} \approx \left(G_{c}\cdot \bar{\gamma} \right)^{-G_{d}} $$

(10)

where $\bar {\gamma }$ stands for the average received SNR, and G_d and G_c are just the diversity and coding gains, respectively. At high SNR, the diversity gain determines the slope of the BER curve versus $\bar {\gamma }$ in a log-log scale, whereas the coding gain determines how the curve is shifted along the horizontal axis with respect to a benchmark BER curve $\bar {\gamma }^{-G_{d}}$. Therefore, this yields a simple parameterized average BER characterization for high SNR, which can provide meaningful insights on the system performance behavior.

In this section, the diversity gain is first examined for a generalized selection combining. The main result is then invoked in the diversity analysis of the distributed mmWave massive MIMO system studied in this paper.

3.1 Diversity gain of generalized selection combining

Selection combining (SC) is the most popular low-complexity combining scheme. In selection combining, the receiver estimates the SNRs of all available diversity branches and then select the one with the highest SNR for detection. For generalized selection combining (GSC) considered here, the receiver also estimates the SNRs of all available diversity branches. However, instead of selecting the branch with the highest SNR, it selects a branch with the lth highest SNR for detection. It is pointed out that, while such a GSC scheme has no practical interest in its own right, its diversity analysis can be used in the performance analysis of the mmWave massive MIMO system considered in this paper.

Lemma 1

Consider a GSC system with L receive antennas operating over i.i.d. Rayleigh fading channels. If the receiver selects the branch with the lth highest SNR for detection then the system achieves diversity gain:

$$ G_{d}=L-l+1. $$

(11)

Proof

Let F(γ) and f(γ) be the cumulative distribution function (CDF) and probability density function (PDF) of the instantaneous SNRs in all branches, respectively. Let $\bar {\gamma }$ denote the average receive SNR of each branch. With Rayleigh fading, it follows from [31] that F(γ) and f(γ) can be expressed as:

$$ F(\gamma)=1-\mathrm{e}^{-\frac{\gamma}{\bar{\gamma}}} = \frac{\gamma} {\bar{\gamma}} + o \left(\frac{\gamma} {\bar{\gamma}} \right) $$

(12)

and

$$ f(\gamma) = \frac{1}{\bar{\gamma}} \mathrm{e}^{-\frac{\gamma}{\bar{\gamma}}} = \frac{1}{\bar{\gamma}} - \frac{\gamma} {{\bar{\gamma}}^{2}} + o \left(\frac{\gamma}{\bar{\gamma}} \right). $$

(13)

If the receiver selects the branch with the lth highest SNR for detection, then based on the theory of order statistics [32], the PDF of the instantaneous receive SNR at the receiver, denoted γ_l, is given by:

$$ \begin{aligned} &f_{l:L}(\gamma_{l})=\frac{L!}{(L-l)!(l-1)!} \left[F(\gamma_{l})\right]^{L-l}[1-F(\gamma_{l})]^{l-1}f(\gamma_{l}) \\ &= \frac{L!} {(L-l)!(l-1)!} \frac{1}{\bar{\gamma}} \left(\frac{\gamma_{l}} {\bar{\gamma}} \right)^{L-l} + o \left(\left(\frac{\gamma_{l}} {\bar{\gamma}} \right)^{L-l} \right). \end{aligned} $$

(14)

Applying the above PDF in Proposition 1 in [31] leads to the desired result. □

Lemma 1 can be extended to the case of independent but not identically distributed (i.n.i.d.) Rayleigh fading channels and the result is stated in the next lemma.

Lemma 2

Suppose that the GSC system with L receive antennas operates over the i.n.i.d. Rayleigh fading channels. If it selects the path with the lth highest SNR for detection, then it can achieve diversity gain:

$$ G_{d}=L-l+1. $$

(15)

Proof

Let $\bar {\gamma }_{\text {max}}$ and $\bar {\gamma }_{\text {min}}$ denote the maximum and minimum values of the average receive SNRs of all these L diversity paths, respectively. Furthermore, let $\mathcal {A}$ and $\mathcal {B}$ denote two GSC systems, each equipped with L receive antennas and operating over i.i.d. Rayleigh fading channels such that the average receive SNRs equal to $\bar {\gamma }_{\text {max}}$ and $\bar {\gamma }_{\text {min}}$, respectively. It is known from Lemma 1 that the diversity gains of these two systems are the same and equal to L−l+1 if both systems select the branch with the lth highest instantaneous SNR for detection. Furthermore, since the GSC system under consideration cannot have better diversity performance than system $\mathcal {A}$ and cannot have worse diversity performance than system $\mathcal {B}$, it can then be concluded that the i.n.i.d. system must also achieve the diversity gain of L−l+1. □

3.2 Diversity gain analysis of the distributed mmWave massive MIMO system

From the structure and definition of the channel matrix H in Section 2, there is a total of $L_{s}={\sum \nolimits }_{i=1}^{K_{r}}{\sum \nolimits }_{j=1}^{K_{t}}L_{ij}$ propagation paths. Naturally, H can be decomposed into a sum of L_s rank-one matrices, each corresponding to one propagation path. Specifically, H can be rewritten as:

$$ \mathbf{H} = \sum\limits_{i=1}^{K_{r}} \sum\limits_{j=1}^{K_{t}} \sum\limits_{l=1}^{L_{ij}} \tilde{\alpha}_{ij}^{l} \tilde{\mathbf{a}}_{r} \left(\phi^{rl}_{ij}, \theta^{rl}_{ij} \right) \tilde{\mathbf{a}}_{t}^{H} \left(\phi^{tl}_{ij}, \theta^{tl}_{ij} \right), $$

(16)

where

$$ \tilde{\alpha}_{ij}^{l} =\sqrt{g_{ij}\frac{N_{t}N_{r}}{L_{ij}}}\alpha_{ij}^{l}, $$

(17)

$\tilde {\mathbf {a}}_{r} \left (\phi ^{rl}_{ij},\theta ^{rl}_{ij} \right)$ is a K_rN_r×1 vector whose bth entry is defined as:

$$ \left[ \tilde{\mathbf{a}}_{r} \left(\phi^{rl}_{ij}, \theta^{rl}_{ij} \right)\right]_{b} = \left\{ \begin{array}{ll} \left[\mathbf{a}_{r} \left(\phi^{rl}_{ij},\theta^{rl}_{ij}\right)\right]_{b-(i-1)N_{r}}, & b\in Q_{i}^{r}\\ 0, & b \notin Q_{i}^{r} \end{array} \right. $$

(18)

where $Q_{i}^{r}=((i-1)N_{r}, iN_{r}]$. And $\tilde {\mathbf {a}}_{t} \left (\phi ^{tl}_{ij},\theta ^{tl}_{ij}\right)$ is a K_tN_t×1 vector whose bth entry is defined as:

$$ \left[ \tilde{\mathbf{a}}_{t} \left(\phi^{tl}_{ij},\theta^{tl}_{ij}\right)\right]_{b}=\left\{ \begin{array}{ll} \left[\mathbf{a}_{t} \left(\phi^{tl}_{ij},\theta^{tl}_{ij}\right)\right]_{b-(j-1)N_{t}}, & b\in Q_{j}^{t}\\ 0, & b \notin Q_{j}^{t}. \end{array}\right. $$

(19)

where $Q_{j}^{t}=((j-1)N_{t}, jN_{t}]$.

Lemma 3

Suppose that the antenna configurations at all RAUs are either ULA or UPA. Then, all L_s vectors $\left \{\tilde {\mathbf {a}}_{r} \left (\phi ^{rl}_{ij}, \theta ^{rl}_{ij} \right)\right \}$ are orthogonal to each other when N_r→∞. Likewise, all L_s vectors $\left \{\tilde {\mathbf {a}}_{t} \left (\phi ^{tl}_{ij}, \theta ^{tl}_{ij} \right)\right \}$ are orthogonal to each other when N_t→∞.

Proof

It follows immediately from (18) and (19) that if u≠v, then vectors $\left \{\tilde {\mathbf {a}}_{r} \left (\phi ^{rl}_{up}, \theta ^{rl}_{up} \right)\right \}$ and $\left \{\tilde {\mathbf {a}}_{r} \left (\phi ^{rl}_{vq}, \theta ^{rl}_{vq} \right)\right \}$ are orthogonal. On the other hand, when u=v and p≠q, it is known from Lemma 1 and Corollary 2 in [6] (also see [33]) that vectors $\left \{\tilde {\mathbf {a}}_{r} \left (\phi ^{rl}_{up}, \theta ^{rl}_{up} \right) \right \}$ and $\left \{\tilde {\mathbf {a}}_{r} \left (\phi ^{rl}_{vq}, \theta ^{rl}_{vq} \right)\right \}$ are orthogonal. The proof that $\left \{\tilde {\mathbf {a}}_{t} \left (\phi ^{tl}_{ij}, \theta ^{tl}_{ij} \right) \right \}$ is a set of orthogonal vectors can be shown similarly. □

Theorem 1

Suppose that both sets $\left \{\tilde {\mathbf {a}}_{r} \left (\phi ^{rl}_{ij},\theta ^{rl}_{ij} \right)\right \}$ and $\left \{\tilde {\mathbf {a}}_{t} \left (\phi ^{tl}_{ij}, \theta ^{tl}_{ij} \right)\right \}$ are orthogonal vector sets when N_r→∞ and N_t→∞. Let N_s≤L_s. Then, the distributed massive MIMO system with large N_r and N_t can achieve a diversity gain of:

$$ G_{d}=L_{s}-N_{s}+1. $$

(20)

Remarks 1

When N_t and N_r are large enough, (33) indicates that the system multiplexing gain is at most equal to L_s. This is reasonable since there exist only L_s effective singular values in the channel matrix H. Theorem 1 provides a simple diversity-multiplexing trade-off of a mmWave massive MIMO system: adding one data stream to the system decreases the diversity gain by one, whereas removing one data stream increases the diversity gain by one. Such a trade-off is useful in designing a system to meet requirements on both data rate and error performance.

Remarks 2

Under the case where N_t and N_r are large enough, it can be found from the proof of Theorem 1 that the diversity performance of the mmWave massive MIMO system only depends on the singular value set $\left \{\tilde {\alpha }^{l} \right \}$ and is not influenced by how sub-matrices $\left \{\sqrt {g_{ij}}\mathbf {H}_{ij}\right \}$ are placed in the channel matrix H (see further discussion of Fig. 9 on this point).

Corollary 1

Consider the scenario that the antenna configuration at each RAU is ULA. Also assume that L_ij=L for any i and j. Let N_s≤K_rK_tL. When both N_t and N_r are very large, the distributed massive MIMO system can achieve a diversity gain:

$$ G_{d}=K_{r}K_{t}L-N_{s}+1. $$

(21)

In particular, when K_r=K_t=1, the massive MIMO system with co-located antennas arrays can achieve a diversity gain:

$$ G_{d}=L-N_{s}+1 $$

(22)

Remarks 3

Corollary 1 implies that for a mmWave co-located massive MIMO system, its diversity gain and multiplexing gain are limited and at most equal to the number of paths L. However, these gains can be increased by employing the distributed antenna architecture and can be scaled up proportionally to K_rK_t.

4 Diversity gain analysis with the conventional partially connected structure

The previous section has analyzed the diversity gain for the massive MIMO system with the general fully connected RF architecture. This section focuses on a massive MIMO system employing the conventional partially connected RF architecture as illustrated in Fig. 3. Here, the transmitter equipped with K_t RF chains sends N_s data streams to the receiver equipped with K_r RF chains. Each of RF chains at the transmitter or receiver is connected to only one RAU. It is assumed that N_s≤ min{K_t,K_r}. The numbers of antennas per each RAU at the transmitter and receiver are fixed as N_t and N_r, respectively. Note that N_t≫N_s and N_r≫N_s. Both the transmitter and receiver employ very small digital processors and very large analog processors, represented respectively by W_t and F_t for the transmitter and W_r and F_r for the receiver.

Fig. 3

Block diagram of a mmWave massive MIMO system with the conventional partially connected RF architecture

As before, denote by s the transmitted symbol vector, by H the fading channel matrix, and by n the noise vector. Then, at the receiver, the processed signal vector z is given by (3), whereas H is described as in (4). Due to the partially connected RF architecture, the analog processors F_t and F_r are block diagonal matrices, expressed as:

$$ \mathbf{F}_{t}=\text{diag}\left\{\mathbf{f}_{t1},\mathbf{f}_{t2},\ldots,\mathbf{f}_{tK_{t}} \right\} $$

(23)

and

$$ \mathbf{F}_{r}=\text{diag} \left\{\mathbf{f}_{r1},\mathbf{f}_{r2},\ldots,\mathbf{f}_{rK_{r}} \right\} $$

(24)

where f_ti stands for the N_t×1 steering vector of phases for the ith RAU at the transmitter, and f_rj stands for the N_r×1 steering vector of phases for the jth RAU at the transmitter.

Theorem 2

Consider the case that the antenna array configuration at each RAU is ULA and L_ij=L for any i and j. In the limit of large N_t and N_r, the distributed massive MIMO system with partially connected RF architecture can achieve a diversity gain:

$$ G_{d}=(K_{t}-N_{s}+1)(K_{r}-N_{s}+1)L. $$

(25)

Remarks 4

Comparing the diversity gain given in Corollary 1 with that given in Theorem 2 reveals that when N_s=1, the diversity gains with the two systems under consideration are the same. However, when N_s>1, the proposed distributed antenna system with fully connected RF architecture achieves a higher diversity gain than the system with the partially connected architecture, and the gap between the two diversity gains is (N_s−1)[ (K_r+K_t−N_s+1)L−1].

5 Diversity gain analysis for the multiuser scenario

This section considers the downlink transmission in a multiuser massive MIMO system as illustrated in Fig. 4. Here, the base station (BS) employs K_b RAUs with each having N_b antennas and $N_{b}^{(\text {rf})}$ RF chains to transmit data streams to K_u mobile stations. Each mobile station (MS) is equipped with N_u antennas and $N_{u}^{(\text {rf})}$ RF chains to support the reception of its own N_s data streams. This means that there is a total of K_uN_s data streams transmitted by the BS. The numbers of data streams are constrained as $K_{u}N_{s} \leq N_{b}^{(\text {rf})}\leq K_{b}N_{b}$ for the BS and $N_{s} \leq N_{u}^{(\text {rf})}\leq N_{u}$ for each MS.

Fig. 4

Block diagram of a multiuser mmWave system with distributed antenna arrays

At the BS, denote by F_b the $K_{b}N_{b} \times N_{b}^{(\text {rf})}$ RF precoder and by W_b the $N_{b}^{(\text {rf})} \times N_{s}K_{u}$ baseband precoder. With the narrowband flat fading channel model, then the received signal vector at the ith MS is given by:

$$ \mathbf{y}_{i}=\mathbf{H}_{i} \mathbf{F}_{b}\mathbf{W}_{b}\mathbf{s}+\mathbf{n}_{i}, \; i=1,2,\ldots, K_{u} $$

(26)

where s is the signal vector for all K_u mobile stations, which satisfies $\mathbb {E} \left [ \mathbf {s} \mathbf {s}^{H} \right ] = \frac {P}{K_{u}N_{s}}\mathbf {I}_{K_{u}N_{s}}$, and P is the average transmit power. The N_u×1 vector n_i represents additive white Gaussian noise, whereas the N_u×K_bN_b matrix H_i is the channel matrix corresponding to the ith MS, whose entries H_ij are described as in Section 2. Furthermore, the signal vector after combining can be expressed as:

$$ \mathbf{z}_{i} = \mathbf{W}_{ui}^{H} \mathbf{F}_{ui}^{H} \mathbf{H}_{i} \mathbf{F}_{b} \mathbf{W}_{b} \mathbf{s} + \mathbf{W}_{ui}^{H} \mathbf{F}_{ui}^{H} \mathbf{n}_{i}, \; i=1,2,\ldots, K_{u} $$

(27)

where F_ui is the $N_{u} \times N_{u}^{(\text {rf})}$ RF combining matrix and W_ui is the $N_{u}^{(\text {rf})} \times N_{s}$ baseband combining matrix for the ith MS.

Theorem 3

Consider the case that all antenna array configurations for the downlink transmission are ULA and L_ij=L for any i and j (i.e., all subchannels H_ij have the same number of propagation paths). In the limit of large N_b and N_u, the downlink transmission in a massive MIMO multiuser system can achieve a diversity gain:

$$ G_{d}=K_{b}L-N_{s}+1. $$

(28)

Remarks 5

Theorem 3 implies that when N_b and N_u are large enough, the available diversity gain G_d does not depend on the number of mobile users K_u.

Remarks 6

In a similar fashion, it is easy to prove that the uplink transmission in a massive MIMO multiuser system can also achieve a diversity gain G_d=K_bL−N_s+1. Moreover, it can also be proved that when L=1, the system diversity gain is equal to G_d=K_b for the case N_u=1, i.e., each MS has only one antenna.

6 Simulation results and discussion

For all simulation results presented in this section, it is assumed that each subchannel matrix H_ij consists of L_ij=L=3 paths, each of the large scale fading coefficients g_ij equals to g=−20 dB (except for Fig. 9), and the numbers of transmit and receive RF chains are twice the number of data streams [9] (i.e., $N_{t}^{(\text {rf})}=N_{r}^{(\text {rf})}=2N_{s}$). It is further assumed that the variance of AWGN samples is unity, and hence, the input SNR is the same as the average input power P/N_s. For simplicity, only ULA array configuration with d=0.5 is considered at RAUs and BPSK modulation is employed for each data stream. With such system configurations, the instantaneous BER is given by $Q(\sqrt {2 \gamma })$ [34], where γ denotes the instantaneous receive SNR and the Q function is defined as $Q(x)=\int _{x}^{\infty } \exp { \left (-\frac {y^{2}}{2} \right)} \mathrm {d}y$. For ease of comparison and discussion, introduce the concept of designed SNR as $\overline {\text {SNR}}_{\text {dg}} = PN_{r}N_{t}/(N_{s}L)$. This means that $P= \overline {\text {SNR}}_{\text {dg}} N_{s}L/(N_{r}N_{t})$ for a given designed SNR $\overline {\text {SNR}}_{\text {dg}}$. In fact, there exists a power scaling law for mmWave communications which states that the data transmit power P can be scaled down proportionally to 1/(N_rN_t) in order to maintain a desirable BER performance [35].

In all simulations, unless stated otherwise, there are three main steps for hybrid digital-analog processing as follows:

• Perform the SVD for channel matrix H and find the optimal overall digital precoder and combiner for N_s data streams.

• Form an analog precoder and an analog combiner based on the optimal overall digital precoder and combiner, respectively.

• Perform zero-forcing (ZF) digital detection based on the analog precoder and analog combiner and complete the data detection operation.

First studied is the diversity performance of a massive MIMO mmWave system with distributed antenna arrays. With N_r=N_t=N=50 and K_r=K_t=K=2, Fig. 5 plots BER curves versus the designed SNR for different numbers of data streams, N_s=2,4,6. For comparison, the BER curve obtained in the case of co-located antenna arrays are also plotted for N_s=1,2,3. It can be seen that even for the larger number of data streams, the BER performance with distributed antenna arrays is clearly better than that with co-located antenna arrays. Furthermore, as N_s decreases, the BER performance with either distributed or co-located antenna arrays is improved. These observations are expected and agree with Corollary 1, which states that using distributed antenna arrays yields higher diversity gains than using co-located antenna arrays. To verify exactly the diversity gain result given in Corollary 1, Fig. 6 plots the diversity gain verifying (GDV) curves produced by simulating the GSC systems. It can be seen that a BER curve with either distributed or co-located antenna arrays has the same slope in the high SNR region as the corresponding GDV curve.

Fig. 5

BER versus designed SNR: comparison between distributed and co-located antenna array architectures

Fig. 6

BER versus designed SNR: verifying diversity gain

Illustrated in Fig. 7 is the performance with the conventional partially connected (PC) RF architecture analyzed in Section 4. With this structure, one first carries out the SVDs for subchannel matrices {H_ij} rather than for the whole channel matrix H and then forms the analog precoder and analog combiner. Let K_r=K_t=K. With N_r=N_t=N=50, Fig. 7 plots the BER curves for the following four cases: (K=1,N_s=1), (K=2,N_s=2), (K=3,N_s=3), and (K=4,N_s=4). It is known from Theorem 2 that the diversity gains for the four cases are identical and equal to G_d=L=3. To illustrate this, a DGV curve with diversity gain G_d=3 is also plotted in this figure. It can be seen that the system with the conventional PC structure for the four cases can achieve the full diversity gain 3, while the coding gain increases when both K and N_s increase. For comparison, the BER curve obtained with the general fully connected (FC) RF structure when N_s=4 and K=2 is also plotted. The theoretical limit on the diversity gain in this case is 9, which agrees well with the DGV curve having G_d=9. Observe that in the high SNR region, the general FC structure yields significantly better diversity performance than the conventional PC structure.

Fig. 7

BER versus designed SNR: comparion between the the proposed distributed subarray architecture and the conventional partially connected architecture

Next, when N_s=1, we consider the diversity performance with the multiuser downlink scenario where there are 5 or 10 mobile users, each having 10 antennas and each RAU at the BS is equipped with 50 antennas. Due to the fact that there is no cooperation among the users, one first carries out the SVDs for subchannel matrices {H_i} rather than for the whole channel matrix H and then forms the analog precoder for the BS and analog combiners for the users. Note that the BS needs to carry out ZF digital preprocessing before transmitting data. Figure 8 plots the BER curves versus the designed SNR for different numbers of subarrays at the BS, namely K_b=1,3,5. It can be observed from this figure that as K_b increases, the diversity performance of the multiuser system improves remarkably. This is because, as established in Theorem 3, the diversity gain becomes larger with increasing K_b. Furthermore, it can be seen from Fig. 8 that the system has the same diversity gain for different numbers of users while the coding gain increases as K_u decreases. This observation agrees with Remark 5.

Fig. 8

BER versus designed SNR: multiuser scenario with different numbers of subarrays

Fig. 9

BER versus designed SNR: comparison between homogeneous distributions and inhomogeneous distributions for large scale fading coefficients {g_ij}

Finally, the diversity performance of the single-user massive MIMO mmWave system is examined under the scenario that the distributions of large scale fading coefficients, {g_ij}, are inhomogeneous. To this end, let G=[g_ij (dB)] denote the large scale fading coefficient matrix. When N_r=N_t=N=50 and K_r=K_t=K=2, simulation is performed for the following six inhomogeneous G:

$$\begin{array}{*{20}l} \mathbf{G}_{1}=\left[ \begin{array}{ll} -25 & -20 \\ -20 & -25 \end{array} \right], \; \mathbf{G}_{2}=\left[ \begin{array}{ll} -20 & -20 \\ -25 & -25 \end{array} \right],\\ \mathbf{G}_{3}=\left[ \begin{array}{ll} -20 & -25 \\ -25 & -20 \end{array} \right], \; \mathbf{G}_{4}=\left[ \begin{array}{ll} -20 & -25 \\ -20 & -25 \end{array} \right],\\ \mathbf{G}_{5}=\left[ \begin{array}{ll} -25 & -25 \\ -20 & -20 \end{array} \right],\; \mathbf{G}_{6}=\left[ \begin{array}{ll} -25 & -20 \\ -25 & -20 \end{array} \right]. \end{array} $$

It can be found that the diversity performance for the six inhomogeneous cases are almost the same (see Remark 2). In order to illustrate this interesting phenomenon, Fig. 9 plots the BER curves versus the designed SNR with G₁ and G₂, respectively. For comparison, the two BER curves for the homogeneous distributions with g=−20 dB and g=−25 dB are also plotted. As expected, the BER curves with the inhomogeneous coefficient distributions are between the two BER curves with homogeneous coefficient distributions. It can be concluded from this figure that the case of inhomogeneous coefficient distributions has the same diversity gain as in the case of homogeneous coefficient distributions.

7 Conclusions

This paper has provided asymptotical diversity analysis for massive MIMO mmWave systems with co-located and distributed antenna architectures when the number of antennas at each subarray goes to infinity. Theoretical analysis shows that with a co-located massive antenna array, scaling up the number of antennas of the array can increase the coding gain but not the diversity gain. However, if the array is built from distributed subarrays (RAUs), each having a very large number of antennas, then increasing the number of RAUs does increase the diversity gain and/or multiplexing gain [36, 37]. As such, the analysis leads to a novel approach to improve the diversity and multiplexing gains of massive MIMO mmWave systems. It is acknowledged that the asymptotical diversity analysis obtained in this paper is under the idealistic assumption of having perfect CSI. Performing the diversity analysis for massive MIMO mmWave systems under imperfect CSI is important and deserves further research.

8 Appendix 1: Proof of Theorem 1

Proof

The distributed massive MIMO system can be considered as such a co-located system with L_s paths that have complex gains $\left \{\tilde {\alpha }_{ij}^{l} \right \}$, receive array response vectors $\left \{\tilde {\mathbf {a}}_{r} \left (\phi ^{rl}_{ij}, \theta ^{rl}_{ij}\right)\right \}$, and transmit response vectors $\left \{\tilde {\mathbf {a}}_{t} \left (\phi ^{tl}_{ij}, \theta ^{tl}_{ij} \right)\right \}$. Furthermore, order all paths in a decreasing order of the absolute values of the complex gains $\left \{\tilde {\alpha }_{ij}^{l} \right \}$. Then, the channel matrix can be written as:

$$ \mathbf{H} = \sum\limits_{l=1}^{L_{s}} \tilde{\alpha}^{l} \tilde{\mathbf{a}}_{r} \left(\phi^{rl}, \theta^{rl} \right) \tilde{\mathbf{a}}_{t} \left(\phi^{tl}, \theta^{tl} \right)^{H}, $$

(29)

where $\tilde {\alpha }^{1}\geq \tilde {\alpha }^{2}\geq \cdots \geq \tilde {\alpha }^{L_{s}}$. One can rewrite H in a matrix form as:

$$ \mathbf{H}=\mathbf{A}_{r}\mathbf{D}\mathbf{A}_{t}^{H} $$

(30)

where D denotes a L_s×L_s diagonal matrix with $[\mathbf {D}]_{ll}=\tilde {\alpha }^{l}$, and A_r and A_t are defined as follows:

$$ \mathbf{A}_{r} = \left[ \tilde{\mathbf{a}}_{r} \left(\phi^{r1}, \theta^{r1}\right), \tilde{\mathbf{a}}_{r} \left(\phi^{r2}, \theta^{r2} \right), \ldots, \tilde{\mathbf{a}}_{r} \left(\phi^{rL_{s}}, \theta^{rL_{s}} \right)\right] $$

(31)

and

$$ \mathbf{A}_{t} = \left[ \tilde{\mathbf{a}}_{t} \left(\phi^{t1}, \theta^{t1} \right), \tilde{\mathbf{a}}_{t} \left(\phi^{t2},\theta^{t2} \right), \ldots, \tilde{\mathbf{a}}_{t} \left(\phi^{tL_{s}}, \theta^{tL_{s}}\right)\right]. $$

(32)

Since both $\left \{\tilde {\mathbf {a}}_{r} \left (\phi ^{rl}, \theta ^{rl} \right)\right \}$ and $\left \{\tilde {\mathbf {a}}_{t} \left (\phi ^{tl},\theta ^{tl}\right)\right \}$ are orthogonal vector sets when N_r→∞ and N_t→∞, A_r and A_t are asymptotically unitary matrices. Then, one can form a singular value decomposition (SVD) of matrix H as:

$$ \mathbf{H} = \mathbf{U}\mathbf{\Sigma} \mathbf{V}^{H} = \left[\mathbf{A}_{r}| \mathbf{A}_{r}^{\bot} \right] \mathbf{\Sigma} \left[\tilde{\mathbf{A}}_{t} | \tilde{\mathbf{A}}_{t}^{\bot} \right]^{H} $$

(33)

where Σ is a diagonal matrix containing all singular values on its diagonal, i.e.:

$$ [\mathbf{\Sigma}]_{ll}=\left\{ \begin{array}{ll} \left|\tilde{\alpha}^{l} \right|, & \text{for}\; 1 \leq l\leq L_{s}\\ 0, & \text{for}\; l>L_{s} \end{array} \right. $$

(34)

and the submatrix $\tilde {\mathbf {A}}_{t}$ is defined as:

$$ \tilde{\mathbf{A}}_{t} = \left[ {\mathrm{e}}^{-j\psi_{1}} \tilde{\mathbf{a}}_{t} \left(\phi^{t1}, \theta^{t1} \right), \ldots,{\mathrm{e}}^{-j\psi_{L_{s}}} \tilde{\mathbf{a}}_{t} \left(\phi^{tL_{s}}, \theta^{tL_{s}} \right)\right] $$

(35)

where ψ_l is the phase of complex gain $\tilde {\alpha }^{l}$ corresponding to the lth path. □

Based on (33), the optimal precoder and combiner are chosen respectively as:

$$ [\mathbf{F}_{t} \mathbf{W}_{t}]_{\text{opt}} = \left[\mathrm{e}^{-j\psi_{1}} \tilde{\mathbf{a}}_{t} \left(\phi^{t1}, \ldots, \mathrm{e}^{-j\psi_{L_{s}}} \tilde{\mathbf{a}}_{t} \left(\phi^{tN_{s}}, \theta^{tN_{s}} \right) \right. \right] $$

(36)

and

$$ [\mathbf{F}_{r} \mathbf{W}_{r} ]_{\text{opt}} = \left[ \tilde{\mathbf{a}}_{r} \left(\phi^{r1}, \ldots, \tilde{\mathbf{a}}_{r} \left(\phi^{rN_{s}}, \theta^{rN_{s}} \right) \right. \right]. $$

(37)

To summarize, when N_t and N_r are large enough, the massive MIMO system can employ the optimal precoder and combiner given in (36) and (37), respectively. Then, it follows from the above SVD analysis that the instantaneous SNR of the lth data stream is given by:

$$ \text{SNR}_{l}=\frac{P}{N_{s}}\left|\tilde{\alpha}^{l} \right|^{2}, \;\; l=1,2,\ldots, N_{s}. $$

(38)

Now the detection of the lth data stream is equivalent to the detection in a generalized selection combining system, which selects the path with the lth highest SNR for detection. Therefore, it follows from Lemma 2 that the detection performance of the lth data stream has a diversity gain L_s−l+1. Since the overall BER is the arithmetic mean of individual BERs, i.e., $\overline {\text {BER}}=\frac {1}{N_{s}}\sum _{l=1}^{N_{s}} \overline {\text {BER}}(l)$, the system’s diversity gain equals to the diversity gain in detecting the N_sth data stream, which is the worst among all data streams. Therefore, the result in (20) is obtained. $\square$

9 Appendix 2: Proof of Theorem 2

Proof

When N_t and N_r are very large, the diversity gain analysis is similar to that in Theorem 1. For the first data stream that enjoys the best path, it is simple to see that its diversity gain is the largest and equal to K_rK_tL. This is because the detection of the first data stream is equivalent to a selection combining system operating with K_rK_tL paths. However, for the second data stream, due to the structure of F_t and F_r, its detection is equivalent to a selection combining system operating with (K_r−1)(K_t−1)L paths. Therefore, it can be inferred that its diversity gain is equal to (K_r−1)(K_t−1)L. Similarly, for the last data stream among the N_s data streams, its diversity gain is (K_r−N_s+1)(K_t−N_s+1)L. It then follows that the diversity gain of the whole system is just (K_r−N_s+1)(K_t−N_s+1)L. □

10 Appendix 3: Proof of Theorem 3

Proof

For the downlink transmission in a massive MIMO multiuser system, the overall equivalent multiuser basedband channel can be written as:

$$ \mathbf{H}_{\text{eq}}= \left[ \begin{array}{llll} \mathbf{F}_{u1}^{H} & \mathbf{0} & \cdots & \;\;\mathbf{0} \\ \mathbf{0} & \mathbf{F}_{u2}^{H} & \cdots & \;\;\mathbf{0} \\ \vdots & \vdots & \ddots& \;\;\vdots\\ \mathbf{0} & \mathbf{0} & \cdots & \mathbf{F}_{uK_{u}}^{H} \end{array} \right]\left[\begin{array}{l} \mathbf{H}_{1} \\ \mathbf{H}_{2}\\ \vdots \\ \mathbf{H}_{K_{u}} \end{array} \right] \mathbf{F}_{b}. $$

(39)

On the other hand, when both N_b and N_u are very large, both receive and transmit array response vector sets, $\left \{\tilde {\mathbf {a}}_{r} \left (\phi ^{rl}_{ij}, \theta ^{rl}_{ij} \right)\right \}$ and $\left \{\tilde {\mathbf {a}}_{t} \left (\phi ^{tl}_{ij}, \theta ^{tl}_{ij} \right)\right \}$, are asymptotically orthogonal. Therefore, the diversity performance for the ith user depends only on the subchannel matrix H_i and the choices of F_ui and F_b. The subchannel matrix H_i has a total of K_bL propagation paths. Similar to the proof of Theorem 1, by employing the optimal RF precoder and combiner for the ith user, the user can achieve a maximum diversity gain K_bL−N_s+1. It is then concluded that the downlink transmission can achieve a diversity gain G_d=K_bL−N_s+1. □