Xiaohu Ge _et al._ 5G Multimedia Massive MIMO Communications Systems

Dr. Qiang Li, School of Electronic Information and Communications, Huazhong
University of Science and Technology, Wuhan 430074, Hubei, P. R.China. E-mail:
<EMAIL_ADDRESS>
Part of this work appeared in the IEEE IWCMC 2015 [1], which was granted the
best paper award.

# 5G Multimedia Massive MIMO Communications Systems

Xiaohu Ge  1  Haichao Wang  1  Ran Zi  1  Qiang Li and Qiang Ni  1  2
11affiliationmark:  School of Electronic Information and Communications,
Huazhong University of Science & Technology, Wuhan, China.
22affiliationmark:  School of Computing and Communications, Lancaster
University, Lancaster LA1 4WA, UK.

###### Abstract

In the Fifth generation (5G) wireless communication systems, a majority of the
traffic demands is contributed by various multimedia applications. To support
the future 5G multimedia communication systems, the massive multiple-input
multiple-output (MIMO) technique is recognized as a key enabler due to its
high spectral efficiency. The massive antennas and radio frequency (RF) chains
not only improve the implementation cost of 5G wireless communication systems
but also result in an intense mutual coupling effect among antennas because of
the limited space for deploying antennas. To reduce the cost, an optimal
equivalent precoding matrix with the minimum number of RF chains is proposed
for 5G multimedia massive MIMO communication systems considering the mutual
coupling effect. Moreover, an upper bound of the effective capacity is derived
for 5G multimedia massive MIMO communication systems. Two antenna receive
diversity gain models are built and analyzed. The impacts of the antenna
spacing, the number of antennas, the quality of service (QoS) statistical
exponent, and the number of independent incident directions on the effective
capacity of 5G multimedia massive MIMO communication systems are analyzed.
Comparing with the conventional zero-forcing precoding matrix, simulation
results demonstrate that the proposed optimal equivalent precoding matrix can
achieve a higher achievable rate for 5G multimedia massive MIMO communication
systems.

###### keywords:

mutual coupling, massive MIMO systems, effective capacity, RF chains,
equivalent precoding.

††articletype: RESEARCH ARTICLE

##  1 Introduction

As various wireless multimedia applications are getting more and more popular,
the demand for wireless traffic is increasing rapidly, and the massive multi-
input-multi- output (MIMO) technology has been proposed as a key technology
for the next generation (5G) wireless communication systems [2, 3, 4].
Recently, a number of excellent studies have validated that massive MIMO
systems are specialized in improving the wireless communication capacity
vastly in cellular networks[5]. Apparently the huge antenna arrays have to be
deployed compactly because enough space are not available at not only base
stations (BSs) but also mobile terminals, therefor the interaction of mutual
coupling among antennas gets so strong that it can’t be ignored in massive
MIMO systems[6]. Also, the realistic channel capacity which is subject to the
quality of service (QoS) in multimedia wireless communication systems and the
Shannon capacity are not the same thing [7, 8]. So, exploring a new precoding
solution for the 5G massive MIMO multimedia communication systems is
necessary.

A lot of studies have achieved great achievements about mutual coupling among
multiple antennas on many topics such as antenna propagation, signal
processing and antenna arrays [9, 10, 11, 12]. Utilizing the real measurement
data, the authors of [9] have made a comparison on the antenna array
performance between the systems considering the mutual coupling and the
systems not. It has been proved that mutual coupling has a great influence on
the performance of antenna arrays for not only small but also large inter-
antenna spacing, because that in order to contain the changes in all the
anticipant vectors, the steering vectors of the antenna arrays should be
adjusted not only in amplitude but also in phase[10]. Clerckx _et. al._
studied how the mutual coupling influenced a simple multi-antenna
communication system performance [11]. In order to recover the signals
received by separate antennas without mutual coupling, the authors of [12]
have invented a new technique to make a compensation for mutual coupling in
small antenna arrays.

At practical wireless communication transmission terminals, each data stream
is first passed through the baseband precoding to radio frequency (RF) chains
and then is transmitted to antennas by the RF chains precoding. For MIMO
wireless systems, the precoding technologies are focused on the baseband
precoding, i.e., the first order precoding, and each RF chain corresponds to
an antenna. Utilizing the phase matrix between RF chains and antennas, the
joint precoding of baseband and RF chains was proposed for massive MIMO
systems with limited RF chains [13]. However, it is still a great challenge to
reduce the number of RF chains for saving the cost of massive MIMO wireless
communication systems.

Lots of excellent studies in the field of wireless multimedia communication
have emerged [14, 15, 16, 17, 18]. In order to evaluate the QoS of wireless
multimedia networks, the authors in [14, 15] created a constrained model of
statistical QoS to study the transmission characteristics of data queues. In
[16, 17], the authors referred to the effective capacity of the block fading
channel model and proposed a rate and power adaption scheme in which the power
is driven by QoS. And in [18], the authors further combined the effective
capacity with information theory and developed some rate adaptation and QoS-
driven power schemes which were suitable for the systems of multiplexing and
diversity. Also they concluded that stringent QoS and high throughput can be
achieved by the multi-channel communication systems simultaneously according
to their simulation results. However, rare efforts has been made to study the
effective capacity of massive MIMO multimedia wireless communication systems
which consider the QoS constraint and mutual coupling effect.

Motivated by the above gaps, we propose an optimal equivalent precoding matrix
to reduce the cost of RF chains in 5G massive MIMO multimedia communication
systems and derive the upper bound of effective capacity with QoS constraints.
The main contributions of this paper are listed as follows.

  1. 1.

We define the receive diversity gain to analyze how the mutual coupling
influence the performance of the rectangular antenna arrays in the massive
MIMO wireless communication systems.

  2. 2.

An optimal equivalent precoding matrix is proposed to reduce the cost of RF
chains and satisfy the multimedia data requirements for 5G massive MIMO
multimedia communication systems.

  3. 3.

We refer to the QoS statistical exponent constraint and the mutual coupling
effect, then derive the upper bound of effective capacity for 5G massive MIMO
multimedia communication systems.

  4. 4.

Based on numerical results, the proposed optimal equivalent precoding matrix
is compared with the conventional zero-forcing (ZF) precoding matrix in 5G
massive MIMO multimedia communication systems.

The rest of this paper is summarized as follows. In Section 2, a system model
in which there is a 2D antenna array is described for massive MIMO wireless
communications. In Section 3, the effect of mutual coupling on the massive
MIMO wireless systems is evaluated by the receive diversity gain. Moreover, an
optimal equivalent precoding matrix is proposed to reduce the cost of RF
chains and satisfy the multimedia data requirements for 5G massive MIMO
multimedia communication systems. Furthermore, the upper bound of effective
capacity is derived for 5G massive MIMO multimedia communication systems.
Numerical simulations and analysis are presented in Section 4. Finally,
Section 5 summarizes the paper.

##  2 System model

Figure 1: System model.

A massive MIMO wireless transmission system is illustrated in Fig. 1. The
wireless down-link between a user equipment (UE) with multi-antenna and a BS
with a 2D rectangular antenna array is studied in this paper.

First of all, we define some basic parameters for this model. We define
$\lambda$ as the wavelength of the carrier, $d$ as the antenna spacing of this
antenna array, $a\lambda{\rm{}}\left({a\geq 1}\right)$ as the length of this
antenna array and $b\lambda{\rm{}}\left({b\geq 1}\right)$ as the width of this
antenna array. If we would like to deploy $m$ antennas in each row and $n$
antennas in each column for this antenna array, then we will have the
relationship as listed in (1),

$d=\frac{{a\lambda}}{{m-1}}=\frac{{b\lambda}}{{n-1}},$

and the total number of antennas in this antenna array $M$ can be derived
easily as

$M=mn.$

If we define $SN{R_{BS}}$ as the signal-to-noise ratio (SNR) at the BS,
${\mathbf{H}}$ and $\beta$ stand for the small scale fading matrix and large
scale fading coefficient of the channel in this model respectively, the signal
the BS transmits is defined as ${\mathbf{x}}$, $\mathbf{w}$ means the additive
white Gaussian noise (AWGN) over wireless channels, and the mutual coupling
matrix is configured as ${\mathbf{K}}$, equivalent precoding matrix is
configured as ${{\mathbf{F}}_{eq}}$, ${\mathbf{A}}$ is defined as steering
matrix, then the down-link signal vector received at a UE equipped with $N$
antennas can be expressed as

${\mathbf{y}}=\sqrt{SN{R_{BS}}}{\mathbf{HAK}}{{\mathbf{F}}_{eq}}{\beta^{1/2}}{\mathbf{x}}+{\mathbf{w}},$

in which ${\mathbf{x}}$ is a ${{N_{s}}\times 1}$ vector, $\mathbf{w}$ is a
${{N}\times 1}$ vector.

${\mathbf{H}}\sim\mathcal{C}\mathcal{N}\left({0,{P{\mathbf{I}}}}\right)$ is
governed by a complex Gaussian distribution, and is expressed as

${\mathbf{H}}={\left[{{\mathbf{h}_{1}},...,{\mathbf{h}_{p}},...,{\mathbf{h}_{P}}}\right]^{T}}\in{\mathbb{C}^{N\times
P}},$

in which ${\mathbb{C}^{N\times P}}$ denotes a $N\times P$ matrix, $P$ stands
for the number of the independent incident directions,
${\mathbf{h}_{p}}\sim\mathcal{C}\mathcal{N}\left({0,{{\mathbf{I}}}}\right)$
stands for the complex coefficient vector of small scale fading received from
the $p$th incident direction, which is expressed as

${\mathbf{h}_{p}}=\mathbf{h}_{p}^{(r)}+j\mathbf{h}_{p}^{(i)},$

in which $\mathbf{h}_{p}^{(r)}$ is defined as the real part of
${\mathbf{h}_{p}}$, and $\mathbf{h}_{p}^{(i)}$ is defined as the imaginary
part of ${\mathbf{h}_{p}}$. Furthermore, both of them are Gaussian random
variables distributed independently and identically, whose expectation and
variance are $0$ and $0.5$ respectively.

Definitely, $P$ will be very large if considerable scatterers exist in the
propagation environment. According to [19, 20], we divide the angular domain
into $P$ independent incident directions with $P$ being large but finite.

Here we assume both of the azimuth angle ${\phi_{q}}$($q=1,...,P$) and
elevation angle $\theta$ are within the scope of
$\left[{{{-\pi}\mathord{\left/{\vphantom{{-\pi}2}}\right.\kern-1.2pt}2},{\pi\mathord{\left/{\vphantom{\pi
2}}\right.\kern-1.2pt}2}}\right]$. Each independent incident direction
corresponds to one steering vector
${\mathbf{a}}\left({{\phi_{q}},\theta}\right)\in{\mathbb{C}^{M\times 1}}$, so
all the $P$ steering vectors can constitute the steering matrix ${\bf{A}}$ of
the rectangular antenna array which is expressed as

${\mathbf{A}}=\left[{{\mathbf{a}}\left({{\phi_{1}},\theta}\right),...,{\mathbf{a}}\left({{\phi_{q}},\theta}\right),...,{\mathbf{a}}\left({{\phi_{P}},\theta}\right)}\right].$

If we define ${{\mathbf{A}}^{q}}\in{\mathbb{C}^{n\times m}}$ as the steering
matrix of the $q$th incident direction of the rectangular antenna array, we
will get the following relationship,

${\mathop{\rm
vec}\nolimits}\left({{{\bf{A}}^{q}}}\right)={\bf{a}}\left({{\phi_{q}},\theta}\right),$

in which $\operatorname{vec}\left(\cdot\right)$ is defined as the matrix
vectorization operation.

Without loss of generality, we assume the antenna which locates at the first
place for both the row and the column of the rectangular antenna array as the
reference point of which the phase response is zero. And we normalize
amplitude responses of all the antennas of the antenna array as 1. We define
${\bf{A}}_{ce}^{q}$, $\left({1\leq c\leq m,{\rm{}}1\leq e\leq n}\right)$ as
the element in the steering matrix ${{\bf{A}}^{q}}$ which locates at the $c$th
row and $e$th column, and it is expressed as

${\mathbf{A}}_{ce}^{q}=\exp\left\\{{j\frac{{2\pi}}{\lambda}\left[\begin{gathered}{\text{(}}c-1{\text{)}}dcos{\phi_{q}}sin\theta\hfill\\\
+(e-1)dsin{\phi_{q}}sin\theta\hfill\\\ \end{gathered}\right]}\right\\}.$

For a rectangular antenna array with M elements, we define ${\mathbf{{\bf
K}}}\in{\mathbb{C}^{M\times M}}$ as the corresponding mutual coupling matrix,
which is expressed as [11]

${\bf{{\bf
K}}}={Z_{L}}{\left({{Z_{L}}{{\bf{I}}}+{{\bf{Z}}_{{M}}}}\right)^{-1}},$

in which ${Z_{L}}$ denotes the antenna load impedance that is constant for
each antenna, ${{\bf{Z}}_{{M}}}$ denotes the $M\times M$ mutual impedance
matrix, and ${{\bf{I}}}$ denotes an $M\times M$ unit matrix. From Fig. 1,
${{\bf{Z}}_{{M}}}$ can be constructed by $n\times n$ sub-matrices,
i.e.,${{\bf{Z}}_{{M}}}={\left[{{{\bf{Z}}_{st}}}\right]_{n\times n}}$, where
${{\bf{Z}}_{st}}$, as an $m\times m$ mutual impedance sub-matrix, denotes the
mutual impedances between the $m$ antennas located at the $s$th $(s=1,...,n)$
row and the $m$ antennas located at the $t$th $(t=1,...,n)$ row in the
rectangular antenna array. For ease of exposition, we define $An{t_{su}}$ as
the antenna located at the $s$th row and $u$th $(s=1,...,m;u=1,...,m)$ column
of the rectangular antenna array, and define $An{t_{tv}}$ as the antenna
located at the $t$th row and the $v$th $(t=1,...,m;v=1,...,m)$ column of the
rectangular antenna array, the corresponding distance between which is given
as $d_{uv}^{st}=d\sqrt{{{(t-s)}^{2}}+{{(v-u)}^{2}}}$. Thus ${{\bf{Z}}_{st}}$
can be written as

${{\bf{Z}}_{{{st}}}}=\left[{\begin{array}[]{*{20}{c}}{z_{11}^{st}}&{z_{12}^{st}}&{...}&{z_{1m}^{st}}\\\
{z_{21}^{st}}&{z_{22}^{st}}&{...}&{z_{2m}^{st}}\\\
\vdots&\vdots&\ddots&\vdots\\\
{z_{m1}^{st}}&{z_{m2}^{st}}&{...}&{z_{mm}^{st}}\end{array}}\right],$

Consider a special case where all $M$ antenna elements in the rectangular
antenna array are dipole antennas with the same parameters. Then the mutual
impedance $z_{uv}^{st}$ only depends on the antenna spacing and can be
obtained with the EMF method in [27]. With a fixed antenna spacing $d$, we
have the following properties:

$z_{uv}^{st}=z_{\left({u+1}\right),\left({v+1}\right)}^{st},$
$z_{uv}^{st}=z_{vu}^{st}.{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}$

Similar properties can be derived for ${{\bf{Z}}_{st}}$ as

${{\mathbf{Z}}_{st}}={{\mathbf{Z}}_{\left({s+1}\right),\left({t+1}\right)}},$
${{\mathbf{Z}}_{st}}={{\mathbf{Z}}_{ts}}.{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}$

Together with (10)-(14), the mutual impedance matrix ${{\bf{Z}}_{{M}}}$ can be
readily obtained. It bears noting that with (10)-(14), the computational
complexity can be significantly reduced compared to the direct calculation of
the $M\times M$ entries of ${{\bf{Z}}_{{M}}}$, especially with a large $M$.

The equivalent precoding matrix
${{\mathbf{F}}_{{\text{eq}}}}={{\mathbf{F}}_{{\text{RF}}}}{{\mathbf{F}}_{{\text{BB}}}}$
consists of baseband precoding matrix ${{\mathbf{F}}_{BB}}$ and the RF
precoding matrix${{\mathbf{F}}_{RF}}$.

${N_{s}}$ data streams are transmitted by $N_{RF}^{t}$ radio frequency (RF)
chains and $M$ antennas at the BS. All wireless data is received by
$N_{RF}^{r}$ RF chains and $N$ antennas at the UE. In this case, the detected
wireless signals at the UE is expressed by

$\mathbf{\tilde{y}}={\mathbf{W}}_{eq}^{\dagger}\mathbf{y}={\mathbf{W}}_{BB}^{\dagger}{\mathbf{W}}_{RF}^{\dagger}\mathbf{y},$

in which ${\dagger}$ is a conjugate transpose operation, ${\mathbf{W}}_{eq}$
is a $N\times{N_{s}}$ equivalent signal detection matrix which consists of
baseband detection matrix ${{\mathbf{W}}_{BB}}$ and the RF detection matrix
${{\mathbf{W}}_{RF}}$, $\mathbf{y}$ is the received signal vector at antennas
of the UE. Essentially, ${{\mathbf{F}}_{RF}}$ and ${{\mathbf{W}}_{RF}}$ are
phase shift matrices used for the signal precoding and detection at the RF
chains. Hence, the absolute value of the RF detection matrix
${{\mathbf{F}}_{RF}}$ and the RF precoding matrix ${{\mathbf{F}}_{RF}}$ is
equal to 1.

##  3 Mutual Coupling Effect Modeling

${R_{\max}}=\log\left|{{\mathbf{I}}+\frac{{SN{R_{BS}}}}{{{N_{s}}}}{\mathbf{W}}_{eq}^{\dagger}{\mathbf{HAK}}{{\mathbf{F}}_{eq}}{\mathbf{F}}_{eq}^{\dagger}{{\mathbf{K}}^{\mathbf{{\dagger}}}}{{\mathbf{{A}}}^{\mathbf{{\dagger}}}}{{\mathbf{H}}^{\mathbf{{\dagger}}}}{{\mathbf{W}}_{eq}}{{\mathbf{R}}^{-1}}}\right|,$

with

${\mathbf{R}}={\mathbf{W}}_{eq}^{\dagger}{{\mathbf{W}}_{eq}}.$

###  3.1 Receive Diversity Gain Models

Deployed in a constrained space at the BS, the number of antenna elements is
inversely proportional to the antenna spacing, i.e., a larger number of
antennas lead to a smaller antenna spacing. As concluded in [21], more
antennas lead to a higher receive diversity gain of the massive MIMO system,
whereas the diversity gain can be compromised by the mutual coupling effect
that is caused by decreasing the antenna spacing. Thus, when a number of
antennas are deployed in a fixed constrained area, there exists a tradeoff
between $M$ and $d$, and it is important to analyze the the effect of mutual
coupling on the achievable receive diversity gain of the massive MIMO systems.

Firstly, with a fixed antenna spacing, the antenna number receive diversity
gain ${\mathbb{G}_{M}}$ is defined as

${\mathbb{G}_{M}}=\xi_{M}^{{d_{\min}}}-\xi_{{M_{{\text{min}}}}}^{{d_{\min}}},$

in which $\xi_{M}^{{d_{\min}}}$ denotes the expectation of the receive SNR at
the UE with $M$ antennas and minimum antenna spacing being $d_{\min}$ at the
antenna array of the BS, and $\xi_{{M_{{\text{min}}}}}^{{d_{\min}}}$ denotes
the expectation of the receive SNR at the UE subject to the minimum antenna
spacing $d_{\min}$ and minimum antenna number ${M_{\min}}$ at the antenna
array of the BS.

Secondly, with a fixed number of antennas, the antenna spacing receive
diversity gain ${\mathbb{G}_{d}}$ is defined as

${\mathbb{G}_{d}}=\xi_{{M_{\min}}}^{d}-\xi_{{M_{\min}}}^{{d_{\min}}},$

in which $\xi_{{M_{\min}}}^{d}$ denotes the expectation of the receive SNR at
the UE with an antenna spacing of $d$ and with at least ${M_{\min}}$ antennas
at the antenna array of the BS. $\xi_{{M_{{\text{min}}}}}^{{d_{\min}}}$, which
can be considered as the baseline for $\mathbb{G}_{d}$ and $\mathbb{G}_{M}$,
is the same as that in (16).

In order to obtain the expectation of the received SNR in (16) and (17),
perfect channel state information is assumed to be available at the BS, which
uses maximal-ratio combining (MRC) for signal detection. When there are $M$
antennas and the neighboring antennas are separated with a spacing of $d$, the
received signal after MRC detection at the UE is given as [22]

$\tilde{y}={{\bf{G}}^{\dagger}}{\bf{y}}{\kern
1.0pt}=\sqrt{SN{R_{UE}}}{{\bf{G}}^{\dagger}}{\bf{G}}x{\kern
1.0pt}+{{\bf{G}}^{\dagger}}{\bf{w}},$

in which ${{\mathbf{G}}^{\dagger}}$ denotes the conjugate transpose of
${\mathbf{G}}$ with

${\mathbf{G}}={\mathbf{HA{K}}}{{\mathbf{F}}_{eq}}{\beta^{1/2}}.$

In addition, the average SNR seen at the UE side can be written as

$\begin{gathered}SN{R_{UE}}=\frac{{SN{R_{BS}}{{\left\|{{{\mathbf{G}}^{\mathbf{{\dagger}}}}{\mathbf{G}}}\right\|}^{2}}}}{{\left\|{{{\mathbf{G}}^{\mathbf{{\dagger}}}}{\mathbf{G}}}\right\|}}\hfill\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}=SN{R_{BS}}\left\|{{{\mathbf{G}}^{\mathbf{{\dagger}}}}{\mathbf{G}}}\right\|\hfill\\\
\end{gathered},$

Then with $M$ antenna elements with antenna spacing $d$, the expectation of
the SNR at the UE can be obtained as

$\begin{gathered}\xi_{M}^{d}={E}\left\\{{SN{R_{UE}}}\right\\}\hfill\\\ {\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}=SN{R_{BS}}\mathbb{E}\left\\{{\left\|{{{\mathbf{G}}^{\dagger}}{\mathbf{G}}}\right\|}\right\\}\hfill\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern
1.0pt}=SN{R_{BS}}N\beta\left\|{{\mathbf{F}}_{eq}^{\dagger}{{\mathbf{K}}^{\mathbf{{\dagger}}}}{{\mathbf{{A}}}^{\mathbf{{\dagger}}}}{{\mathbf{H}}^{\mathbf{{\dagger}}}}{\mathbf{HA{K}}}{{\mathbf{F}}_{eq}}}\right\|\hfill\\\
\end{gathered},$

in which $E\left\\{\cdot\right\\}$ denotes the expectation operation. We can
further obtain ${\mathbb{G}_{d}}$ and ${\mathbb{G}_{M}}$ through substituting
(20) into (16) and (17), and replacing $M$ and $d$ with $M_{\min}$ and
$d_{\min}$.

###  3.2 Shannon Capacity with Optimal RF Chains

$\begin{gathered}{R_{\max}}=\log\left|{{\mathbf{I}}+\frac{{SN{R_{BS}}}}{{{N_{s}}}}{\mathbf{HF}}{{\mathbf{F}}^{\dagger}}{{\mathbf{H}}^{\dagger}}{{\mathbf{W}}_{eq}}{{\left({{\mathbf{W}}_{eq}^{\dagger}{{\mathbf{W}}_{eq}}}\right)}^{-1}}{\mathbf{W}}_{eq}^{\dagger}}\right|\hfill\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern
1.0pt}=\log\left|{{\mathbf{I}}+\frac{{SN{R_{BS}}}}{{{N_{s}}}}{{\mathbf{U}}_{\mathbf{H}}}{{\mathbf{\Sigma}}_{\mathbf{H}}}{\mathbf{V}}_{\mathbf{H}}^{\dagger}{{\mathbf{U}}_{\mathbf{F}}}{{\mathbf{\Sigma}}_{\mathbf{F}}}{\mathbf{\Sigma}}_{\mathbf{F}}^{\dagger}{\mathbf{U}}_{\mathbf{F}}^{\dagger}{{\mathbf{V}}_{\mathbf{H}}}{\mathbf{\Sigma}}_{\mathbf{H}}^{\dagger}{\mathbf{U}}_{\mathbf{H}}^{\dagger}}\right|\hfill\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern
1.0pt}=\log\left|{{\mathbf{I}}+\frac{{SN{R_{BS}}}}{{{N_{s}}}}{\mathbf{V}}_{\mathbf{H}}^{\dagger}{{\mathbf{U}}_{\mathbf{F}}}{{\mathbf{\Sigma}}_{\mathbf{F}}}{\mathbf{\Sigma}}_{\mathbf{F}}^{\dagger}{\mathbf{U}}_{\mathbf{F}}^{\dagger}{{\mathbf{V}}_{\mathbf{H}}}{\mathbf{\Sigma}}_{\mathbf{H}}^{\dagger}{{\mathbf{\Sigma}}_{\mathbf{H}}}}\right|\hfill\\\
\end{gathered},$

A phase shift matrix is designed to separate the RF chains and the antennas.
Assume that the relationship among the numbers of data stream, RF chains and
antennas is configured as ${N_{s}}\leqslant N_{RF}^{t}\leqslant M$.
Considering the equivalent precoding matrix
${\left({{{\mathbf{F}}_{eq}}}\right)_{M\times{N_{s}}}}={\left({{{\mathbf{F}}_{RF}}}\right)_{M\times
N_{RF}^{t}}}{\left({{{\mathbf{F}}_{BB}}}\right)_{N_{RF}^{t}\times{N_{s}}}}$,
about it’s rank, we have
$rank\left({{{\mathbf{F}}_{eq}}}\right)\leqslant\min\left({{N_{s}},N_{RF}^{t},M}\right)$.
This result implies that the up-bound of the degree of freedom at the
equivalent precoding matrix is depended on the minimum among the numbers of
data stream, RF chains and antennas. When the number of RF chains is larger
than the number of data stream, a part of number of RF chains, i.e.,
${N_{RF}^{t}}-{N_{s}}$, has not been utilized by the equivalent precoding
matrix. To save the cost, the number of RF chains can be configured as
${{N_{s}}}$ to satisfy the requirement of the equivalent precoding matrix
${{\mathbf{F}}_{eq}}$.

Based on the system model in Fig. 1, the system achievable rate, i.e., the
maximum Shannon capacity is expressed by (21) and normalized on the unit
bandwidth [23]. Let $MIN=\min\left({M,P}\right)$, the eigenvalues of wireless
channels ${\mathbf{H}}$ are ordered by
${\lambda_{1}}\geqslant{\lambda_{2}}\cdots\geqslant{\lambda_{MIN}}$. The
maximum available rate is
${R_{\max}}=\sum\limits_{k=1}^{MIN}{\log\left({1+\frac{{SN{R_{BS}}}}{{{N_{s}}}}{\lambda_{k}}}\right)}$,
where ${SN{R_{BS}}}$ is the SNR value at the BS. The rank of wireless channel
${\mathbf{H}}$ is denoted by $r=rank\left({\mathbf{H}}\right)$. As a
consequence, the maximum available rate is rewritten by
${R_{\max\\_full}}=\sum\limits_{k=1}^{rank\left({\mathbf{H}}\right)}{\log\left({1+\frac{{SNR\\_BS}}{{{N_{s}}}}{\lambda_{k}}}\right)}$.
When the numbers of data stream and RF chains are equal to the rank of
wireless channel $rank\left({\mathbf{H}}\right)$, the wireless channel
capacity has been fully utilized. When the rank of wireless channels
$rank\left({\mathbf{H}}\right)$ is less than the number of data stream, what
should we do to configure the number of RF chains? To utilize the wireless
channel capacity and save the implementation cost of RF chains, the number of
RF chains is configured as the rank of wireless channels in this paper.

To simplify the derivation, the rank of wireless channels is assumed to be
larger than the number of data stream in the following study. The optimal
equivalent detection matrix ${{\mathbf{W}}_{eq}}$ is derived by a singular
value decomposition (SVD) method

${{\mathbf{W}}_{eq}}={{\mathbf{U}}_{\mathbf{W}}}{{\mathbf{\Sigma}}_{\mathbf{W}}}{\mathbf{V}}_{\mathbf{W}}^{\dagger},$

with

${{\mathbf{\Sigma}}_{\mathbf{W}}}=\left(\begin{gathered}{{\mathbf{\Delta}}_{\mathbf{W}}}\hfill\\\
{\mathbf{0}}\hfill\\\ \end{gathered}\right),$
${{\mathbf{\Delta}}_{\mathbf{W}}}=\left({\begin{array}[]{*{20}{c}}{{w_{1}}}&0&\cdots&0\\\
0&{{w_{2}}}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&{{w_{r}}}\end{array}}\right),$

in which ${{\mathbf{U}}_{\mathbf{W}}}$ and
${\mathbf{V}}_{\mathbf{W}}^{\dagger}$ are unitary matrices.

The optimal equivalent precoding matrix ${{\mathbf{F}}_{eq}}$ is derived by a
SVD method

${\mathbf{F}}_{eq}={{\mathbf{U}}_{\mathbf{F}}}{{\mathbf{\Sigma}}_{\mathbf{F}}}{\mathbf{V}}_{\mathbf{F}}^{\dagger},$

with

${{\mathbf{\Sigma}}_{\mathbf{F}}}=\left({\begin{array}[]{*{20}{c}}{{{\mathbf{\Delta}}_{\mathbf{F}}}}&{\mathbf{0}}\\\
{\mathbf{0}}&{\mathbf{0}}\end{array}}\right),$
${{\mathbf{\Delta}}_{\mathbf{F}}}=\left({\begin{array}[]{*{20}{c}}{{f_{1}}}&0&\cdots&0\\\
0&{{f_{2}}}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&{{f_{r}}}\end{array}}\right),$

in which ${{\mathbf{U}}_{\mathbf{F}}}$ and
${\mathbf{V}}_{\mathbf{F}}^{\dagger}$ are unitary matrices.

$\begin{gathered}{R_{\max}}=\log\left|{\frac{{SN{R_{BS}}}}{{{N_{s}}}}\left({\frac{{{N_{s}}}}{{SN{R_{BS}}}}{\mathbf{I}}+{{\mathbf{\Sigma}}_{{{\mathbf{F}}^{2}}}}{{\mathbf{\Sigma}}_{{{\mathbf{H}}^{2}}}}}\right)}\right|\hfill\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}=r\log\left({\frac{{SN{R_{BS}}}}{{{N_{s}}}}}\right)+\log\left({\mathop{\Pi}\limits_{i=1}^{r}\left({\frac{{{N_{s}}}}{{SN{R_{BS}}}}+f_{i}^{2}\lambda_{i}^{2}}\right)}\right)\hfill\\\
\end{gathered},$

$\begin{gathered}L\left({f_{1}^{2},f_{2}^{2}\cdots f_{r}^{2}}\right)\hfill\\\
=\left({\frac{{{N_{s}}}}{{SN{R_{BS}}}}+\lambda_{1}^{2}f_{1}^{2}}\right)\left({\frac{{{N_{s}}}}{{SN{R_{BS}}}}+\lambda_{2}^{2}f_{2}^{2}}\right)\cdots\left({\frac{{{N_{s}}}}{{SN{R_{BS}}}}+\lambda_{r}^{2}f_{r}^{2}}\right)\hfill\\\
+\alpha\left({f_{1}^{2}+f_{2}^{2}+\cdots+f_{r}^{2}-N_{s}^{2}}\right)\hfill\\\
\end{gathered}.$

$\begin{gathered}\frac{{\partial L\left({f_{1}^{2},f_{2}^{2}\cdots
f_{r}^{2}}\right)}}{{\partial f_{i}^{2}}}\hfill\\\
\;\;\;\;=\lambda_{i}^{2}\left({\frac{{{N_{s}}}}{{SN{R_{BS}}}}+\lambda_{1}^{2}f_{1}^{2}}\right)\cdots\left({\frac{{{N_{s}}}}{{SN{R_{BS}}}}+\lambda_{i-1}^{2}f_{i-1}^{2}}\right)\hfill\\\
\;\;\;\;\;\;\;\times\left({\frac{{{N_{s}}}}{{SN{R_{BS}}}}+\lambda_{i+1}^{2}f_{i+1}^{2}}\right)\cdots\left({\frac{{{N_{s}}}}{{SN{R_{BS}}}}+\lambda_{r}^{2}f_{r}^{2}}\right)+\alpha\hfill\\\
\end{gathered},$

When a SVD method is performed over the equivalent channel
${{\mathbf{H}}_{{\text{eq}}}}={\mathbf{HAK}}$, the equivalent channel is
derived by

${{\mathbf{H}}_{{\text{eq}}}}={{\mathbf{U}}_{\mathbf{H}}}{{\mathbf{\Sigma}}_{\mathbf{H}}}{\mathbf{V}}_{\mathbf{H}}^{\dagger},$

with

${{\mathbf{\Sigma}}_{\mathbf{H}}}=\left({\begin{array}[]{*{20}{c}}{{{\mathbf{\Delta}}_{\mathbf{H}}}}&{\mathbf{0}}\\\
{\mathbf{0}}&{\mathbf{0}}\end{array}}\right),$
${{\mathbf{\Delta}}_{\mathbf{H}}}=\left({\begin{array}[]{*{20}{c}}{{\lambda_{1}}}&0&\cdots&0\\\
0&{{\lambda_{2}}}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&{{\lambda_{r}}}\end{array}}\right),$

in which ${{\mathbf{U}}_{\mathbf{H}}}$ and
${\mathbf{V}}_{\mathbf{H}}^{\dagger}$ are unitary matrices.

Based on (22), (23) and (24), the maximum available rate ${R_{\max}}$ is
further derived by a SVD method in (25) when the optimal equivalent detection
matrix ${{\mathbf{W}}_{eq}}$ is assumed to be a non-singular matrix. Let

${{\mathbf{\Sigma}}_{{{\mathbf{F}}^{2}}}}={{\mathbf{\Sigma}}_{\mathbf{F}}}{\mathbf{\Sigma}}_{\mathbf{F}}^{\dagger}=\left({\begin{array}[]{*{20}{c}}{{{\mathbf{\Delta}}_{\mathbf{F}}}{\mathbf{\Delta}}_{\mathbf{F}}^{\dagger}}&{\mathbf{0}}\\\
{\mathbf{0}}&{\mathbf{0}}\end{array}}\right),$

${{\mathbf{\Sigma}}_{{{\mathbf{H}}^{2}}}}={\mathbf{\Sigma}}_{\mathbf{H}}^{\dagger}{{\mathbf{\Sigma}}_{\mathbf{H}}}=\left({\begin{array}[]{*{20}{c}}{{\mathbf{\Delta}}_{\mathbf{H}}^{\dagger}{{\mathbf{\Delta}}_{\mathbf{H}}}}&{\mathbf{0}}\\\
{\mathbf{0}}&{\mathbf{0}}\end{array}}\right),$

${\mathbf{U}}={\mathbf{V}}_{\mathbf{H}}^{\dagger}{{\mathbf{U}}_{\mathbf{F}}}.$

(25) is rewritten by

$\begin{gathered}R=\log\left|{{\mathbf{I}}+\frac{{SN{R_{BS}}}}{{{N_{s}}}}{\mathbf{U}}{{\mathbf{\Sigma}}_{{{\mathbf{F}}^{2}}}}{{\mathbf{U}}^{\dagger}}{{\mathbf{\Sigma}}_{{{\mathbf{H}}^{2}}}}}\right|\hfill\\\
\end{gathered},$

in which ${{\mathbf{\Sigma}}_{{{\mathbf{F}}^{2}}}}$ and
${{\mathbf{\Sigma}}_{{{\mathbf{H}}^{2}}}}$ are diagonal matrices and the
values of elements at diagonal line are larger than zero. Furthermore, the
eigenvalues of ${{\mathbf{\Sigma}}_{{{\mathbf{F}}^{2}}}}$ and
${{\mathbf{\Sigma}}_{{{\mathbf{H}}^{2}}}}$ are the elements at the diagonal
lines, respectively. ${\mathbf{U}}$ is a unitary matrix, i.e.,
${\left\|{\mathbf{U}}\right\|_{2}}=M$. When ${\mathbf{U}}$ is configured as a
diagonal matrix, the maximum available rate is achieved by

$\begin{gathered}R_{\max}=\log\left|{\mathbf{I}+\frac{{SN{R_{BS}}}}{{{N_{s}}}}{{\mathbf{\Sigma}}_{{{\mathbf{F}}^{2}}}}{{\mathbf{\Sigma}}_{{{\mathbf{H}}^{2}}}}}\right|\hfill\\\
\end{gathered}.$

Assume that the transmission power at the BSs is independent of the equivalent
precoding matrix. This assumption implies that
${\left\|{{{\mathbf{F}}_{eq}}}\right\|_{F}}{\text{ = }}{N_{s}}$, i.e.,

$\sum\limits_{i=1}^{r}{f_{i}^{2}}=N_{s}^{2}.$

The maximum available rate $R_{max}$ can be simplified as (32).

To achieve the maximum achievable rate, the optimal solution of the equivalent
precoding is derived by a Lagrange multiplier method in the following. A
function is first constructed in (33) with Lagrange factor $\alpha$. And then
take the derivative of $L\left({f_{1}^{2},f_{2}^{2}\cdots f_{r}^{2}}\right)$
with respect to $f_{i}^{2},\left({i=1,2...r}\right)$ in (34).

${{\mathbf{F}}_{eq}}=\underbrace{\left[{\begin{array}[]{*{20}{c}}{{e^{j{\vartheta_{1,1}}}}}&\cdots&{{e^{j{\vartheta_{1,2r}}}}}&0&\cdots&0\\\
\vdots&\cdots&\vdots&\vdots&\cdots&\vdots\\\
{{e^{j{\vartheta_{M,1}}}}}&\cdots&{{e^{j{\vartheta_{M,2r}}}}}&0&\cdots&0\end{array}}\right]}_{{{\left({{{\mathbf{F}}_{RF}}}\right)}_{M\times
2{N_{s}}}}}\underbrace{\left[{\begin{array}[]{*{20}{c}}{{b_{1}}}&0&\cdots&0&0&{\begin{array}[]{*{20}{c}}\cdots&0\end{array}}\\\
{{b_{1}}}&0&\cdots&0&0&{\begin{array}[]{*{20}{c}}\cdots&0\end{array}}\\\
0&{{b_{2}}}&\cdots&0&0&{\begin{array}[]{*{20}{c}}\cdots&0\end{array}}\\\
0&{{b_{2}}}&\cdots&0&0&{\begin{array}[]{*{20}{c}}\cdots&0\end{array}}\\\
\vdots&\vdots&\ddots&\vdots&\vdots&{\begin{array}[]{*{20}{c}}\cdots&\vdots\end{array}}\\\
0&0&\cdots&{{b_{r}}}&0&{\begin{array}[]{*{20}{c}}\cdots&0\end{array}}\\\
0&0&\cdots&{{b_{r}}}&0&{\begin{array}[]{*{20}{c}}\cdots&0\end{array}}\\\
0&0&\cdots&0&0&{\begin{array}[]{*{20}{c}}\cdots&0\end{array}}\\\
\vdots&\vdots&\ddots&\vdots&\vdots&{\begin{array}[]{*{20}{c}}\ddots&\vdots\end{array}}\\\
0&0&\cdots&0&0&{\begin{array}[]{*{20}{c}}\cdots&0\end{array}}\end{array}}\right]}_{{{\left({{{\mathbf{F}}_{BB}}}\right)}_{2{N_{s}}\times{N_{s}}}}},$

$\begin{gathered}{C_{E}}\left(\theta\right)=-\frac{1}{{\theta
T}}\ln\left({E\left\\{{{e^{-\theta
TB\left({\log\left({\frac{{SN{R_{BS}}}}{{{N_{s}}}}}\right)+\log\left({\mathop{\Pi}\limits_{i=1}^{r}\left({\frac{{{N_{s}}}}{{SN{R_{BS}}}}+f_{i}^{2}\lambda_{i}^{2}}\right)}\right)}\right)}}}\right\\}}\right)\hfill\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}=-\frac{1}{{\theta T}}\ln\left({{e^{-\theta
TB\log\left({\frac{{SN{R_{BS}}}}{{{N_{s}}}}}\right)}}E\left\\{{{{\left({{e^{\log\left({\mathop{\Pi}\limits_{i=1}^{r}\left({\frac{{{N_{s}}}}{{SN{R_{BS}}}}+f_{i}^{2}\lambda_{i}^{2}}\right)}\right)}}}\right)}^{-\theta
TB}}}\right\\}}\right)\hfill\\\ {\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 3.0pt}=-\frac{1}{{\theta T}}\ln\left({{e^{-\theta
TB\log\left({\frac{{SN{R_{BS}}}}{{{N_{s}}}}}\right)}}E\left\\{{{{\left({\mathop{\Pi}\limits_{i=1}^{r}\left({\frac{{{N_{s}}}}{{SN{R_{BS}}}}+f_{i}^{2}\lambda_{i}^{2}}\right)}\right)}^{-\frac{{\theta
TB}}{{\ln\left(2\right)}}}}}\right\\}}\right)\hfill\\\ \end{gathered}.$

Let

$\frac{{\partial L\left({f_{1}^{2},f_{2}^{2}\cdots
f_{r}^{2}}\right)}}{{\partial f_{i}^{2}}}=0,$

in which $\left({i=1,2...r}\right)$, we can further derive the following
result

$\begin{gathered}\frac{{\frac{{{N_{s}}}}{{SN{R_{BS}}}}}}{{\lambda_{i}^{2}}}+f_{i}^{2}=\frac{{\frac{{{N_{s}}}}{{SN{R_{BS}}}}}}{{\lambda_{j}^{2}}}+f_{j}^{2}\hfill\\\
\Rightarrow{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}f_{i}^{2}-f_{j}^{2}=\frac{{{N_{s}}}}{{SN{R_{BS}}}}\left({\frac{1}{{\lambda_{j}^{2}}}-\frac{1}{{\lambda_{i}^{2}}}}\right)\hfill\\\
\end{gathered},$

in which $\left({i,j=1,2...r}\right)$.

Based on (31), the square of eigenvalues at the equivalent precoding matrix
${{\mathbf{F}}_{eq}}$ is derived by

$\begin{gathered}f_{i}^{2}=\frac{{N_{s}}{\left({\frac{1}{{\lambda_{1}^{2}}}+\frac{1}{{\lambda_{2}^{2}}}+\cdots+\frac{1}{{\lambda_{r}^{2}}}}\right)}}{{SN{R_{BS}}r}}\hfill\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}+\frac{{N_{s}^{2}}}{r}-\frac{{{N_{s}}}}{{SN{R_{BS}}\lambda_{i}^{2}}}\hfill\\\
\end{gathered}.$

According to (25) and (30), we know that ${{\mathbf{W}}_{eq}}$,
${{\mathbf{V}}_{\mathbf{F}}}$, ${{\mathbf{U}}_{\mathbf{H}}}$ and
${\mathbf{U}}$ are removed in the simplification process of ${R_{\max}}$, so
${{\mathbf{V}}_{\mathbf{F}}}$, ${{\mathbf{U}}_{\mathbf{H}}}$ and
${\mathbf{U}}$ can be unit matrices. And furthermore,
${{\mathbf{U}}_{\mathbf{F}}}$ and ${{\mathbf{V}}_{\mathbf{H}}}$ can also be
unit matrices, and ${{\mathbf{U}}_{\mathbf{F}}}={{\mathbf{V}}_{\mathbf{H}}}$.
Considering the ${\mathbf{W}}_{eq}$ is a $N\times{N_{s}}$ non-singular matrix,
${\mathbf{W}}_{eq}$ can be decomposed by

${{\mathbf{W}}_{eq}}=\left({\begin{array}[]{*{20}{c}}{\mathbf{I}}\\\
{\mathbf{0}}\end{array}}\right),$

In this case, the optimal equivalent precoding matrix is simplified by

$\begin{gathered}{{\mathbf{F}}_{eq}}={{\mathbf{U}}_{\mathbf{F}}}{{\mathbf{\Sigma}}_{\mathbf{F}}}{\mathbf{V}}_{\mathbf{F}}^{\dagger}\hfill\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}={{\mathbf{V}}_{\mathbf{H}}}{{\mathbf{\Sigma}}_{\mathbf{F}}}\hfill\\\
\end{gathered}.$

Based on the method in [23], the optimal equivalent precoding matrix
${{\mathbf{F}}_{eq}}$ is composed of ${{\mathbf{F}}_{RF}}$ and
${{\mathbf{F}}_{BB}}$, which are designed in (40) with

${b_{j}}=\frac{1}{2}\mathop{\max}\limits_{1\leqslant i\leqslant
M}\left|{{f_{i,j}}}\right|,$
${\vartheta_{i,\left({2j-1}\right)}}=\measuredangle{f_{i,j}}-{\cos^{-1}}\frac{{\left|{{f_{i,j}}}\right|}}{{2{b_{j}}}},$
${\kern 11.0pt}{\kern 1.0pt}{\kern
3.0pt}{\vartheta_{i,2j}}=\measuredangle{f_{i,j}}+{\cos^{-1}}\frac{{\left|{{f_{i,j}}}\right|}}{{2{b_{j}}}},$

in which ${{f_{i,j}}}$ is the element of ${{\mathbf{F}}_{eq}}$ located at the
$i$th row and the $j$th column $\measuredangle{f_{i,j}}$ is the corresponding
angle.

Based on the result in (40), the number of RF chains $2{N_{s}}$ can satisfy
the requirement of the optimal equivalent precoding matrix. In general, the
number of antennas $M$ is larger than the number of RF chains $2{N_{s}}$ in 5G
massive MIMO wireless systems. Hence, our proposed optimal equivalent
precoding matrix can save the number of RF chains $M-2{N_{s}}$.

###  3.3 Effective Capacity with Mutual Coupling Effect

From [28], we define the effective capacity under multimedia constraints as

${C_{E}}\left(\theta\right)=-\frac{1}{{\theta T}}\ln\left({E\\{{e^{-\theta
TBR}}\\}}\right),$

in which $\theta$ and B denote the QoS statistical exponent and bandwidth
respectively, ${\emph{E}\\{\cdot\\}}$ is the expectation operation. Without
losing generality, we consider independent fading channels that keep static
within a frame duration $T$.

$\begin{gathered}E\left\\{{{R_{\max}}}\right\\}=E\left\\{{r\log\left({\frac{{SN{R_{BS}}}}{{{N_{s}}}}}\right)+\log\left({\mathop{\Pi}\limits_{i=1}^{r}\left({\frac{{{N_{s}}}}{{SN{R_{BS}}}}+f_{i}^{2}\lambda_{i}^{2}}\right)}\right)}\right\\}\hfill\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}\leqslant{r\log\left({\frac{{SN{R_{BS}}}}{{{N_{s}}}}}\right)+\sum\limits_{i=1}^{r}{\log\left({E\left\\{{\frac{{{N_{s}}}}{{SN{R_{BS}}}}+f_{i}^{2}\lambda_{i}^{2}}\right\\}}\right)}}\hfill\\\
\end{gathered},$

$\begin{gathered}E\\{{R_{\max}}\\}\leqslant{r\log\left({\frac{{SN{R_{BS}}}}{{{N_{s}}}}}\right)+\sum\limits_{i=1}^{r}{\log\left({\frac{{{N_{s}}}}{r}E\left\\{{\left({\lambda_{i}^{2}{N_{s}}+\lambda_{i}^{2}\frac{{\sum\limits_{i=1}^{r}{\frac{1}{{\lambda_{i}^{2}}}}}}{{SN{R_{BS}}}}}\right)}\right\\}}\right)}}\hfill\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern
1.0pt}\leqslant{r\log\left({\frac{{SN{R_{BS}}}}{{{N_{s}}}}}\right)+\sum\limits_{i=1}^{r}{\log\left({\frac{{{N_{s}}}}{{{r^{2}}}}\left({{N_{s}}+\frac{r}{{SN{R_{BS}}}}}\right)E\left\\{{\sum\limits_{i=1}^{r}{\lambda_{i}^{2}}}\right\\}}\right)}}\hfill\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}={r\log\left({\frac{{SN{R_{BS}}}}{{{N_{s}}}}}\right)+\sum\limits_{i=1}^{r}{\log\left({\frac{{{N_{s}}}}{{{r^{2}}}}\left({{N_{s}}+\frac{r}{{SN{R_{BS}}}}}\right)E\left\\{{tr\left({{{\mathbf{H}}^{\dagger}}{\mathbf{HAK}}{{\mathbf{K}}^{\dagger}}{{\mathbf{A}}^{\dagger}}}\right)}\right\\}}\right)}}\hfill\\\
\end{gathered},$

$\begin{gathered}E\\{{R_{\max}}\\}\leqslant{r\log\left({\frac{{SN{R_{BS}}}}{{{N_{s}}}}}\right)+r\log\left({\frac{{{N_{s}}}}{{{r^{2}}}}\left({{N_{s}}+\frac{r}{{SN{R_{BS}}}}}\right)\left({tr\left({{\mathbf{AK}}{{\mathbf{K}}^{\dagger}}{{\mathbf{A}}^{\dagger}}}\right)+E\left\\{{tr\left({{{\mathbf{H}}^{\dagger}}{\mathbf{H}}}\right)}\right\\}}\right)}\right)}\hfill\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}={r\log\left({\frac{{SN{R_{BS}}}}{{{N_{s}}}}}\right)+r\log\left({\frac{{{N_{s}}}}{{{r^{2}}}}\left({{N_{s}}+\frac{r}{{SN{R_{BS}}}}}\right)\left({tr\left({{\mathbf{AK}}{{\mathbf{K}}^{\dagger}}{{\mathbf{A}}^{\dagger}}}\right)+\Pr}\right)}\right)}\hfill\\\
\end{gathered},$

${C_{E}}\left(\theta\right)\leqslant
B\left({r\log\left({\frac{{SN{R_{BS}}}}{{{N_{s}}}}}\right)+r\log\left({\frac{{{N_{s}}}}{{{r^{2}}}}\left({{N_{s}}+\frac{r}{{SN{R_{BS}}}}}\right)\left({tr\left({{\mathbf{AK}}{{\mathbf{K}}^{\dagger}}{{\mathbf{A}}^{\dagger}}}\right)+\Pr}\right)}\right)}\right).$

Considering the maximum available rate in (32), (41) can be extended as (42),
and it is clear that $f\left(x\right)={x^{-a}},\left({a>0}\right)$ is a convex
function. Then an upper bound of the effective capacity can be obtained using
Jensen’s inequality,

$\begin{gathered}{C_{E}}\left(\theta\right)=-\frac{1}{{\theta
T}}\ln\left({E\left\\{{{e^{-\theta TBR}}}\right\\}}\right)\hfill\\\ {\kern
27.0pt}\leqslant-\frac{1}{{\theta T}}\ln\left({{e^{-\theta
TBE\left\\{R\right\\}}}}\right)\hfill\\\ {\kern
27.0pt}=BE\left\\{R\right\\}\hfill\\\ \end{gathered}.$

Based on (32), the upper bound of the maximum available rate is derived in
(44). When the optimal equivalent precoding matrix is used for massive MIMO
wireless systems, the upper bound of the maximum available rate is further
derived in (45). Considering the lemma 2.9 in [29], the upper bound of the
maximum available rate is finally expressed in (46). As a consequence, the
upper bound of the effective capacity in 5G multimedia communication systems
is given by (47).

##  4 Numerical Results and Analysis

In this section, we demonstrate the performance of the multimedia oriented
massive MIMO communication systems in terms of the receive diversity gain as
well as the effective capacity, where both effects of the QoS statistical
exponent and mutual coupling are evaluated. For ease of illustration, we
consider a rectangular antenna array with the length-width ratio of
${a\mathord{\left/{\vphantom{ab}}\right.\kern-1.2pt}b}=2$. There are
altogether 128 dipole antenna elements in the rectangular antenna array [24],
each with length and diameter of $0.5\lambda$ and $0.001\lambda$,
respectively. Then, a reasonable minimum antenna spacing is
$d_{min}=0.1\lambda$. The large scale fading factor is normalized to $\beta=1$
[20, 21, 26], with a load impedance at each antenna as ${Z_{L}}=50$ Ohms [25].
Without loss of generality, we assume that the BS is located in rich
scattering environment where the incident directions can arrive at an
arbitrary angle uniformly. Thus it is reasonable to assume that the elevation
angle $\theta$ and azimuth ${\phi_{q}}$ follow i.i.d. uniform distributions
within $\left[{-\pi/2,\pi/2}\right]$. For ease of demonstration, the default
number of independent incident directions $P=70$ is configured, with frame
duration $T=1$ms and bandwidth $B=1$MHz [28].

Figure 2: Antenna spacing receive diversity gain with respect to the antenna
spacing considering different SNRs.

In Fig. 2, the antenna spacing receive diversity gain ${\mathbb{G}_{d}}$ with
respect to the antenna spacing is investigated. ${M_{\min}}$ is configured as
$1$. When the antenna spacing $d$ is fixed, the antenna spacing receive
diversity gain increases with the increase of SNR. But if we fix the SNR at
the BS, it is shown that there is almost no correlation between the antenna
spacing receive diversity gain and the antenna spacing.

Figure 3: Antenna number receive diversity gain with respect to the antenna
number considering different SNRs.

Fig. 3 illustrates the correlation between ${\mathbb{G}_{M}}$ and the antenna
number considering different SNRs. $d_{\min}$ is configured as $0.1\lambda$.
And it is shown that the antenna number receive diversity gain has a positive
correlation with the antenna number and SNR.

Figure 4: Effective capacity with respect to the antenna number considering
different SNRs.

In Fig. 4, the effective capacity is illustrated with varying values of the
antenna number and SNR. For ease of illustration, the antenna spacing is set
as 0.5$\lambda$ and $\theta$ is set as 0.01. With a fixed SNR value, it is
observed that a higher effective capacity is obtained by increasing the
antenna number. In addition, with a fixed number of antennas, a higher
effective capacity is obtained with a higher SNR.

Figure 5: Effective capacity with respect to the QoS statistical exponent
$\theta$ and the antenna spacing.

Fig. 5 shows the effective capacity with varying values of QoS statistical
exponent and the antenna spacing. With a fixed antenna spacing, it is observed
that a higher effective capacity is reached by decreasing the QoS statistical
exponent. On the other hand, for a fixed QoS statistical exponent and
increasing of antenna spacing, the effective capacity almost keeps stationary.

Figure 6: Shannon Capacity with respect to the antenna number considering
different precoding matrices.

When the number of user and the baseband data stream are configured as one,
Fig. 6 compares the Shannon capacity with respect to the antenna number
considering different precoding matrices. the proposed optimal equivalent
precoding matrix labeled as “Feq” and the zero-forcing precoding matrix
labeled as “ZF” are compared in Fig. 6. When the number of antennas and the
SNR are fixed, the Shannon capacities with the proposed optimal equivalent
precoding matrix are greater than the Shannon capacities with the zero-forcing
precoding matrix. Moreover, the Shannon capacities with the proposed optimal
equivalent precoding matrix has a positive correlation with the the number of
antennas. However, the Shannon capacities with the zero-forcing precoding
matrix almost keeps stationary with the increase of the number of antennas.
This result indicates that our proposed optimal equivalent precoding matrix
can improve the Shannon capacity, i.e., the available rate in massive MIMO
wireless communication systems.

Figure 7: Effective capacity with respect to the QoS statistical exponent
$\theta$ considering different SNRs.

Fig. 7 analyzes the effective capacity and the upper bound of effective
capacity with respect to the QoS statistical exponent considering different
SNRs, in which “EC_SNR” labels the effective capacity results and
“EC_upper_bound_SNR” represents the upper bound of the effective capacity
results. When the SNR is fixed, there is a positive correlation between the
effective capacity and the QoS statistical exponent. Moreover, the more the
the QoS statistical exponent decrease, the closer the upper bound of effective
capacity gets to the effective capacity.

Figure 8: Effective capacity with respect to the independent incident
directions $P$ considering different SNRs.

When the number of antennas is configured as 128, Fig. 8 describes the
effective capacity with respect to the number of independent incident
directions. When the SNR is fixed, there is a positive correlation between the
effective capacity and the independent incident directions.

##  5 Conclusions

Based on the mutual coupling effect, an optimal equivalent precoding matrix
has been proposed to maximize the available rate and save the cost of RF
chains for 5G massive MIMO multimedia communication systems. Considering the
requirements of multimedia wireless communications, the upper bound of the
effective capacity has been derived for 5G massive MIMO multimedia
communication systems with the QoS statistical exponent constraint. Compared
with the conventional ZF precoding matrix, numerical results show that the
proposed optimal equivalent precoding matrix can obviously improve the
available rate for 5G massive MIMO multimedia communication systems. In the
future work, taking into account the QoS statistical exponent constraints, a
more efficient signal detection precoding algorithm is worth exploring towards
better performance of the multimedia massive MIMO communication systems.

The authors would like to acknowledge the support from the National Natural
Science Foundation of China (NSFC) under the grants 60872007, 61301128,
61461136004 and 61271224, NFSC Major International Joint Research Project
under the grant 61210002, the Ministry of Science and Technology (MOST) of
China under the grants 2015FDG12580 and 2014DFA11640, the Fundamental Research
Funds for the Central Universities under the grant 2015XJGH011 and
2013ZZGH009. This research is partially supported by the EU FP7-PEOPLE-IRSES,
project acronym S2EuNet (grant no. 247083), project acronym WiNDOW (grant no.
318992) and project acronym CROWN (grant no. 610524).

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