# Capacity Scaling Law in Massive MIMO with Antenna Selection

Chongjun Ouyang, Hao Xu, Xujie Zang, and Hongwen Yang School of Information
and Communication Engineering
Beijing University of Posts and Telecommunications, Beijing, 100876, China
{DragonAim, Xu_Hao, zangxj<EMAIL_ADDRESS>
###### Abstract

Antenna selection is capable of handling the cost and complexity issues in
massive multiple-input multiple-output (MIMO) channels. The sum-rate capacity
of a multiuser massive MIMO uplink channel is characterized under the Nakagami
fading. A mathematically tractable sum-rate capacity upper bound is derived
for the considered system. Moreover, for a sufficiently large base station
(BS) antenna number, a deterministic equivalent (DE) of the sum-rate bound is
derived. Based on this DE, the sum-rate capacity is shown to grow double
logarithmically with the number of BS antennas. The validity of the analytical
result is confirmed by numerical experiments.

###### Index Terms:

Antenna selection, capacity scaling law, massive multi-input multi-output
(MIMO), sum-rate capacity.

##  I Introduction

To meet the immense demand for mobile data traffic, massive multiple-input
multiple-output (MIMO) has emerged as a key enabling technology in wireless
networks due to its promising spectral efficiency performance [1]. By
deploying a large-scale antenna array at the base station (BS), massive MIMO
offers unprecedented degrees of freedom (DoFs) to the wireless channel. Yet,
to reap the maximal transmission rate, each BS antenna element needs to be
accompanied with a dedicated radio frequency (RF) chain, which significantly
increases the hardware cost, power consumption, and signal processing
complexity of the system [2]. To alleviate these issues, numerous methods have
been recently proposed as cost-effective alternatives. Among them is antenna
selection (AS) [3, 4, 5], a technique to activate an antenna subset for
communications. It is proved that AS is capable of reducing the number of RF
chains while simultaneously preserving the DoFs offered by the large-scale
antenna array [3].

In AS-aided MIMO systems, an intriguing question to ask is “ _how the channel
capacity scales as the BS antenna number grows large?_ ”, namely the capacity
scaling law (CSL) or the capacity scaling rate. The CSL plays a critical role
in defining the capacity limits of AS in MIMO channels, which, thus, provides
important system design insights. Because of its importance and extensive
applications, the CSL has been one of the research focuses in AS for recent
years [6, 8, 9, 7, 10, 11, 12].

###  I-A Related Works

In a single-user (SU) MIMO downlink channel where a single transmit antenna
with the strongest channel gain is selected, the capacity is shown to scale
with the BS antenna number at a double logarithmic rate [6]. This work is
extended to another downlink case with a single-antenna receiver served by a
multi-antenna BS, where a number of BS antennas with strongest channel
coefficients are selected for communications [7]. Essentially, these two works
characterize the CSL achieved by AS in large-scale multiple-input single-input
(MISO) systems. As a further advance, with the aid of a well-known capacity
bound [8], the CSL is unveiled for SU-MIMO uplink systems using the sub-array
switching (SAS) architecture, where multiple receive antennas are selected via
an exhaustive search (ES). The so-called SAS architecture means that each RF
chain is connected to one disjoint sub-array of the BS, which is in contrast
to the full-array switching (FAS) architecture where each RF chain is
connected to all the BS antennas. Moreover, the CSL achieved by the greedy
search-based method [10] and the channel gains-based method [11] were also
studied for SU-MIMO channels. It is noteworthy that the aforementioned works
did not consider the CSL in multiuser systems. In an effort to fill this gap,
the CSL was discussed for multiuser MIMO multicast channels where multiple
antennas are selected via the ES-based method [12].

###  I-B Motivations and Main Contributions

Although the above works have laid a solid foundation for understanding the
CSL in massive MIMO systems, they can be improved from three aspects. Firstly,
multiuser unicast transmission is one of the most typical applications of
massive MIMO, while the CSL therein has not yet been fully understood when
multiple antennas are selected. Secondly, some existing CSLs were derived for
either specific switching architectures [9] or specific selection algorithms
[10, 11], which makes the analytical results lack sufficient generality. Last
but not least, most existing works assumed the channels suffer from Rayleigh
fading [6, 8, 9, 7, 10, 11, 12]. Although Rayleigh is the most popular fading
model, a more accurate fading model for practical radio environment is the
Nakagami-$m$ fading that encompasses Rayleigh as a special case [13].
Nevertheless, the CSL under this fading model has received little attention.

In an effort to overcome these limitations, this article investigates the CSL
in multiuser massive MIMO uplink channels when multiple BS antennas are
selected by the ES-based method. Under the Nakagami-$m$ fading model, we
derive an analytically tractable bound to characterize the sum-rate capacity.
By setting the BS antenna number to infinity, we obtain a deterministic
equivalent of the sum-rate capacity bound. On this basis, we show that the
sum-rate capacity of the considered system grows double logarithmically with
the number of BS antennas. It is worth mentioning that the derived CSL is
based on the capacity-optimal FAS architecture and ES-based method [9], which,
hence, serves as an upper bound for the CSLs achieved by other switching
architectures and selection methods.

###  I-C Notation

Scalars, vectors, and matrices are denoted by non-bold, bold lower-case, and
bold upper-case letters, respectively; $\mathbbmss{C}$ stands for the complex
plane. The Hermitian of matrix $\mathbf{A}$ is indicated with
${\mathbf{A}}^{\mathsf{H}}$ and ${\mathbf{I}}_{N}$ is the $N{\times}N$
identity matrix; $\det(\cdot)$, ${\mathbbmss{E}}\left\\{\cdot\right\\}$, and
${\mathbbmss{V}}\left\\{\cdot\right\\}$ denote the determinant, expectation,
and variance operator, respectively;
${\rm{erf}}\left(x\right)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}\exp\left(-t^{2}\right){\rm
d}t$ denotes the Gauss error function. In addition, the real Gaussian
distribution having mean $\eta$ and variance $\sigma^{2}$ is represented by
${\mathcal{N}}(\eta,\sigma^{2})$. Finally,
${\mathbf{x}}\sim{\mathcal{CN}}\left({\mathbf{0}},{\mathbf{X}}\right)$ denotes
a circularly symmetric complex Gaussian vector having zero mean and covariance
matrix $\mathbf{X}$.

##  II System Model

###  II-A Uplink Massive MIMO Channel

The considered massive MIMO uplink communication system consists of $K$ users
and one BS. We assume that each user
$k\in{\mathcal{K}}=\left\\{1,\ldots,K\right\\}$ is equipped with a single
transmit antenna to convey signals while the BS has $N\gg K$ antennas for
receiving. The received vector at the BS is given by

$\displaystyle{{\mathbf{y}}}=\sqrt{p_{u}}{{\mathbf{G}}}\mathbf{x}+{{\mathbf{n}}}=\sqrt{p_{u}}{\mathbf{H}}{\mathbf{D}}^{{1}/{2}}\mathbf{x}+{{\mathbf{n}}},$
(1)

where ${\mathbf{x}}\in{\mathbbmss{C}}^{K\times 1}$ is the vector of symbols
transmitted by all users and the input covariance matrix is given by
${\mathbf{R}}_{x}={\mathbbmss{E}}\left\\{{\mathbf{x}}{\mathbf{x}}^{\mathsf{H}}\right\\}={\mathbf{I}}_{K}$;
$p_{u}$ is the average transmit power of each user terminal;
${\mathbf{H}}=\left[{h}_{n,k}\right]\in{{\mathbbmss{C}}^{{N}{\times}{K}}}$ and
$h_{n,k}$ models the fast fading from user $k$ to the $n$th antenna of the BS;
${\mathbf{D}}={\mathsf{diag}}\left\\{{\beta_{1}},\ldots,{\beta_{K}}\right\\}\in{\mathbbmss{C}}^{K\times
K}$ and $\beta_{k}$ models both geometric attenuation and shadowing from user
$k$ to the BS;
${\mathbf{G}}={\mathbf{H}}{\mathbf{D}}^{{1}/{2}}=\left[g_{n,k}\right]\in{\mathbbmss{C}}^{N\times
K}$ is the channel matrix;
${{\mathbf{n}}}\sim{\mathcal{CN}}{({\mathbf{0}},\sigma^{2}{{\mathbf{I}}_{N}})}$
is the additive white Gaussian noise (AWGN) with $\sigma^{2}$ being the noise
power.

The diagonal elements of $\mathbf{D}$, i.e., $\\{\beta_{k}\\}_{k=1}^{K}$, are
assumed to be constant across the antenna array, whereas the elements of
${\mathbf{H}}$ are assumed to be independent but not identically distributed
(i.n.i.d.) with Nakagami-$m$ distributed magnitude and uniformly distributed
phase on $[0,2\pi)$. Particularly, the probability density function (PDF) of
$\left|h_{n,k}\right|^{2}$ is given by

$f_{n,k}\left(x\right)=\frac{1}{\Gamma\left(m_{k}\right)}m_{k}^{m_{j}}x^{m_{k}-1}\exp\left(-m_{k}x\right),x\geq
0,$ (2)

where $m_{k}\geq 0.5$ [13] and
$\Gamma\left(z\right)=\int_{0}^{\infty}t^{z-1}\exp\left(-t\right){\rm d}t$ is
the complete gamma function. To evaluate the theoretical system performance
upper bound, we assume that the CSI is perfectly known at the BS. By denoting
$\rho_{u}=\frac{\rho_{u}}{\sigma^{2}}$, the sum-rate capacity of the
considered system is given by

$\displaystyle{\mathcal{R}}=\log_{2}\det\left({{\mathbf{I}}_{N}}+\rho_{u}{\mathbf{G}}{{\mathbf{G}}^{\mathsf{H}}}\right).$
(3)

###  II-B Receive Antenna Selection

Since the BS has full channel information of $\mathbf{G}$, it can exploit
receive antenna selection to reduce the hardware implementation complexity. To
guarantee spatial multiplexing gain for the $K$ users, we assume that $L\geq
K$ antennas are selected and each antenna is fed by an independent RF chain.
Moreover, we assume that an FAS-based architecture is adopted by the BS, where
each RF chain is connected to all the BS antennas via a switching network [9].
Let ${{\overline{\mathbf{G}}}}\in{{\mathbbmss{C}}^{{L}{\times}{K}}}$ denote
the resulting sub-matrix of $\mathbf{G}$ after receive antenna selection.
Without loss of generality, we resort to the sum-rate maximization as the
criterion of antenna selection. As a result, the optimal channel sub-matrix
satisfies

$\displaystyle{\overline{\mathbf{G}}}_{\rm{opt}}=\arg\max\nolimits_{{\overline{\mathbf{G}}}{\in}{\mathcal{S}}}\log_{2}\det\left({{\mathbf{I}}_{L}}+{\rho_{u}}{\overline{\mathbf{G}}}{{\overline{\mathbf{G}}}}^{\mathsf{H}}\right),$
(4)

where $\mathcal{S}$ denotes the full set of the candidate channel sub-matrices
with $\left|\mathcal{S}\right|=\binom{N}{L}$. For simplicity, we define
$\mathcal{R}_{\rm
o}^{(K)}\triangleq\log_{2}\det\left({{\mathbf{I}}_{L}}+{\rho_{u}}{\overline{\mathbf{G}}}_{\rm{opt}}{\overline{\mathbf{G}}}{{}_{\rm{opt}}^{\mathsf{H}}}\right)$.
It is worth noting that problem (4) is NP-hard and
${\overline{\mathbf{G}}}_{\rm{opt}}$ can be obtained via the ES-based method
with exponential searching complexity ${\mathcal{O}}\left(N^{L}\right)$.

##  III Sum-Rate Capacity Upper Bound

While the previous section has established the fundamental model of the
massive MIMO systems with receive antenna selection, in the following sections
we discuss more properties of the sum-rate capacity therein in order to unveil
more system insights. It is worth noting that the calculation of
$\mathcal{R}_{\rm o}^{(K)}$ requires an ES, which makes the subsequent
analyses of $\mathcal{R}_{\rm o}^{(K)}$ intractable. As a compromise, our
efforts will focus on an upper bound of $\mathcal{R}_{\rm o}^{(K)}$. This sum-
rate capacity bound is obtained by firstly relaxing the system model described
in (1) to the aggregation of $K$ independent single-input multiple-output sub-
channels and then selecting the best $L$ out of $N$ antennas for maximal-ratio
combining in each sub-channel. Afterwards, $\mathcal{R}_{\rm o}^{(K)}$ will be
upper bounded by the summation of the capacity of the $K$ sub-channels [8].
Particularly, let $\left\\{|\bar{h}_{n,k}|\right\\}_{n=1}^{N}$ denote the
ordered set of $\left\\{|{h}_{n,k}|\right\\}_{n=1}^{N}$, i.e.,
$|\bar{h}_{1,k}|\geq|\bar{h}_{2,k}|\geq\cdots\geq|\bar{h}_{N,k}|$. Then, the
sum-rate capacity upper bound can be expressed as

$\displaystyle{\mathcal{R}}_{\rm{u}}\triangleq\sum\nolimits_{k=1}^{K}\log_{2}\left(1+{\rho_{u}\beta_{k}}{\sum\nolimits_{l=1}^{L}|\bar{h}_{l,k}|^{2}}\right)\overset{\star}{\geq}\mathcal{R}_{\rm
o}^{(K)},$ (5)

where the equality in $\star$ holds for $K=1$.

Having defining the upper bound of $\mathcal{R}_{\rm o}^{(K)}$, we now move to
characterizing its statistical properties by analyzing its mean for the sake
of gleaning further insights. As stated before, the PDF of
$\left|h_{n,k}\right|^{2}$ is given by
$f_{n,k}\left(x\right)=\frac{1}{\Gamma\left(m_{k}\right)}m_{k}^{m_{j}}x^{m_{k}-1}\exp\left(-m_{k}x\right)$,
$x\geq 0$. Thus, the joint PDF of the ordered variables
$\left\\{x_{n,k}=|\bar{h}_{n,k}|^{2}\right\\}_{n=1}^{N}$ is given by

$\displaystyle
f_{\bar{h}_{k}}\left(x_{1,k},\cdots,x_{N,k}\right)=N!\prod\nolimits_{n=1}^{N}f_{n,k}\left(x_{n,k}\right).$
(6)

Accordingly, the characteristic function of
${\mathcal{R}}_{k}\triangleq\log_{2}\left(1+{\rho_{u}\beta_{k}}\sum_{l=1}^{L}{x_{l,k}}\right)$
can be expressed as

$\begin{split}\Phi_{k}\left(\omega\right)=&\int_{0}^{\infty}{\rm
d}x_{1,k}\int_{0}^{x_{1,k}}{\rm d}x_{2,k}\cdots\int_{0}^{x_{N-1,k}}{\rm
d}x_{N,k}\\\ &\times\exp\left({\rm
j}\omega{\mathcal{R}}_{k}\right)N!\prod\nolimits_{n=1}^{N}f_{n,k}\left(x_{n,k}\right).\end{split}$
(7)

With the characteristic function at hand, the mean of ${\mathcal{R}}_{\rm{u}}$
can be numerically calculated, which yields

$\displaystyle{\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\}=\sum\nolimits_{k=1}^{K}\left.\frac{1}{\rm
j}\frac{{\rm d}}{{\rm d}\omega}\Phi_{k}\left(\omega\right)\right|_{\omega=0}.$
(8)

Nevertheless, we notice that although (8) can be used to calculate
${\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\}$, the $(NK)$-fold
integrations therein involve a huge computation burden, especially in massive
MIMO settings where $N$ is a large value. Furthermore, due to the mathematical
intractability of (8), it is challenging to use this characteristic function
to unveil more system insights. To handle this difficulty, we intend to derive
more concise expressions to approximate
${\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\}$, based on which the
sum-rate capacity scaling law is explored.

Let us introduce some key preliminary results that will be useful in
constructing the approximation of
${\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\}$.

######  Lemma 1.

Let $x_{1},x_{2},\cdots,x_{N}$ be independent and identically distributed,
each with the cumulative distribution function (CDF) $F\left(x\right)$ that
satisfies: 1)
$\lim_{x\rightarrow+\infty}[F\left(x\right)-F\left(-x\right)]=1$, and 2)
$\forall\alpha>0$,
$\sup\left\\{x:F\left(x\right)\leq\alpha\right\\}=\inf\left\\{x:F\left(x\right)\geq\alpha\right\\}$.
Here, $\sup{\mathcal{A}}$ and $\inf{\mathcal{A}}$ return the supremum and
infimum of set $\mathcal{A}$, respectively. Let $x_{\left(1\right)}\geq
x_{\left(2\right)}\geq\cdots\geq x_{\left(N\right)}$ denote the order
statistics of the sample $\left\\{x_{l}\right\\}_{l=1,2,\cdots,N}$. Then as
$N\rightarrow+\infty$ and for fixed L ($L<N$),
$Y_{N}^{(L)}=\sum_{l=1}^{L}x_{\left(l\right)}$ converges in distribution to
$Y\sim{\mathcal{N}}\left(\mu_{y},\sigma_{y}^{2}\right)$, where

$\displaystyle\mu_{y}=N_{\text{r}}\int_{\upsilon}^{\infty}x{\rm
d}F\left(x\right),$ (9)
$\displaystyle\sigma_{y}^{2}=L\left(\sigma^{2}+\left(\upsilon-\frac{\mu_{{y}}}{L}\right)^{2}\left(1-\frac{L}{N}\right)\right),$
(10)

and where

$\displaystyle
F\left(\upsilon\right)=1-\frac{L}{N},\quad\sigma^{2}=\frac{N}{L}\int_{\upsilon}^{\infty}x^{2}{\rm
d}F\left(x\right)-\frac{\mu{{}_{y}^{2}}}{L^{2}}.$ (11)

###### Proof:

Please refer to [14, pp. 473–475]. ∎

We now intend to leverage the asymptotic properties stated in Lemma 1 to
investigate the statistical features of ${\mathcal{R}}_{\rm{u}}$ in the large
limit of the array size, $N$. Particularly, based on Lemma 1,
$\sum_{l=1}^{L}{x_{l,k}}$ converges in distribution to a Gaussian random
variable $X_{k}\sim\mathcal{N}\left(\mu_{k},\sigma_{k}^{2}\right)$ as $N$
approaches infinity, where

$\displaystyle{\sigma_{k}^{2}}=L\left({\bar{\sigma}}_{k}^{2}+\left({\upsilon_{k}}-\frac{\mu_{k}}{L}\right)^{2}\left(1-\frac{L}{N}\right)\right),$
(12)
$\displaystyle{\bar{\sigma}}_{k}^{2}=\frac{N}{L}\int_{\upsilon_{k}}^{\infty}x^{2}{f_{n,k}\left(x\right)}{\rm{d}}x-\frac{\mu{{}_{k}^{2}}}{L^{2}}$
(13)
$\displaystyle\quad=\frac{N}{Lm_{k}^{2}\Gamma\left(m_{k}\right)}\Upsilon\left(m_{k}+2,m_{k}{\upsilon_{k}}\right)-\frac{\mu{{}_{k}^{2}}}{L^{2}},$
(14)
$\displaystyle\mu_{k}=N\int_{\upsilon_{k}}^{\infty}x{f_{n,k}\left(x\right)}{\rm{d}}x=\frac{N\Upsilon\left(m_{k}+1,m_{k}{\upsilon_{k}}\right)}{m_{k}\Gamma\left(m_{k}\right)},$
(15)
$\displaystyle\int_{{\upsilon_{k}}}^{\infty}f_{n,k}\left(x\right){\rm{d}}x=\frac{1}{\Gamma\left(m_{k}\right)}\Upsilon\left(m_{k},m_{k}\upsilon_{k}\right)=\frac{L}{N},$
(16)

and where
$\Upsilon\left(s,x\right)=\int_{x}^{+\infty}t^{s-1}\exp\left(-t\right){\rm
d}t$ is the upper incomplete gamma function. By defining
${\mathcal{R}}_{\rm{a}}\triangleq\sum_{k=1}^{K}\log_{2}\left(1+{\rho_{u}\beta_{k}}X_{k}\right)$,
we can obtain

$\lim\nolimits_{N\rightarrow\infty}{\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\}={\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{a}}\right\\}.$
(17)

Based on Lemma 1,
${\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\}={\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{a}}\right\\}$
holds in the large limit of $N$. However, as will be shown in Section V,
${\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{a}}\right\\}$ can provide an
accurate approximation of
${\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\}$ even for finite
system dimensions, such as $N=128$. As a result, the sum-rate capacity upper
bound can be approximated as
${\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\}\approx{\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{a}}\right\\}$
[15, 16], which suggests that

$\displaystyle{\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\}\approx\sum\nolimits_{k=1}^{K}\int_{-\infty}^{\infty}{\log_{2}\left(1+{\rho_{u}\beta_{k}}x\right)f_{X_{k}}\left(x\right){\rm{d}}x},$
(18)

where
$f_{X_{k}}\left(x\right)=\frac{1}{\sqrt{2\pi\sigma_{k}^{2}}}\exp\left({-\frac{\left(x-\mu_{k}\right)^{2}}{2\sigma_{k}^{2}}}\right)$.

It is worth mentioning that the random variable
$\sum_{l=1}^{L}{x_{l,k}}=\sum_{l=1}^{L}|\bar{h}_{l,k}|^{2}$ takes positive
values for sure. Yet, its approximation,
$X_{k}\sim\mathcal{N}\left(\mu_{k},\sigma_{k}^{2}\right)$, can take negative
values for a small probability. This is due to the fact that $X_{k}$ is only
an approximation of $\sum_{l=1}^{L}{x_{l,k}}$ when $N<\infty$. As will be
detailed in Appendix, $\lim_{N\rightarrow\infty}\mu_{k}=\infty$ and
$\lim_{N\rightarrow\infty}\frac{\sigma_{k}^{2}}{\mu_{k}^{2}}=0$, which
indicates that

$\displaystyle\lim_{N\rightarrow\infty}\Pr\left(X_{k}<0\right)$
$\displaystyle=\lim_{N\rightarrow\infty}\left[1+{\rm{erf}}\left(-{\mu_{k}^{2}}/{\sigma_{k}^{2}}\right)\right]/2$
(19) $\displaystyle=\left[1+{\rm{erf}}\left(-\infty\right)\right]/2=0.$ (20)

The above arguments imply that the deviation of the approximated distribution
$\mathcal{N}\left(\mu_{k},\sigma_{k}^{2}\right)$ from the exact distribution
of $\sum_{l=1}^{L}{x_{l,k}}$ vanishes as $N$ grows infinitely large.
Considering this fact, we can also approximate
${\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\}$ as

$\displaystyle{\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\}\approx\sum\nolimits_{k=1}^{K}\int_{-\infty}^{\infty}{\log_{2}\left(1+{\rho_{u}\beta_{k}}|x|\right)f_{X_{k}}\left(x\right){\rm{d}}x}.$
(21)

We notice that both (18) and (21) only involve $K$-fold integrations, which,
thus, are much more computationally tractable than the characteristic function
presented in (7).

(a) Sum-rate capacity versus the transmit power $p_{u}$.

(b) The mean of the sum-rate capacity bound.

(c) The SCV of the capacity bound.

Figure 1: Sum-rate capacity and its upper bound of the considered system. The
simulation parameters are given as follows: (a) $N=128$ and $K=8$; (b) $K=4$
and $L=8$; (c) $K=4$, $L=8$, and $p_{u}=10$ dBm.

##  IV Sum-Rate Capacity Scaling Law

The fact of
$\lim\nolimits_{N\rightarrow\infty}{\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\}={\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{a}}\right\\}$
suggests that ${\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{a}}\right\\}$
presents the same scaling law as
${\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\}$ with the increment of
$N$. Accordingly, it makes sense to rely on
${\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{a}}\right\\}$ to explore the
scaling law of ${\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\}$. By
setting $N$ as infinity, the following theorem can be found.

######  Theorem 1.

For fixed $L$ ($L<N$),
$\lim\limits_{N\rightarrow\infty}\frac{{\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\}}{\sum_{k=1}^{K}\log_{2}\left(1+{L}\frac{\rho_{u}\beta_{k}}{m_{k}}{\log{{N}}}\right)}=1$
and
$\lim\limits_{N\rightarrow\infty}\frac{{\mathbbmss{V}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\}}{({\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\})^{2}}=0$.

###### Proof:

The proof is detailed in Appendix. ∎

Note that in statistics,
$\frac{{\mathbbmss{V}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\}}{({\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\})^{2}}$
is also termed as the of the squared coefficient of variation (SCV) of the
random variable ${\mathcal{R}}_{\rm{u}}$. Based on Theorem 1, we can write
${\mathcal{R}}_{\rm{u}}$ as

$\displaystyle{\mathcal{R}}_{\rm{u}}={\mathcal{X}}\sum\nolimits_{k=1}^{K}\log_{2}\left(1+{\rho_{u}\beta_{k}}\frac{L}{m_{k}}{\log{{N}}}\right)$
(22)

with the random variable $\mathcal{X}$ satisfying
$\lim_{N\rightarrow\infty}\mathbbmss{E}\\{{\mathcal{X}}\\}=1$ and
$\lim_{N\rightarrow\infty}\mathbbmss{V}\\{{\mathcal{X}}\\}=0$. Therefore, it
can be easily shown that as $N\rightarrow\infty$,

$\displaystyle{\mathcal{R}}_{\rm{u}}-\sum\nolimits_{k=1}^{K}\log_{2}\left(1+{\rho_{u}\beta_{k}}\frac{L}{m_{k}}{\log{{N}}}\right)\overset{a.s.}{\longrightarrow}0,$
(23)

where -a.s. denotes almost sure convergence. Note that
$\sum_{k=1}^{K}\log_{2}\left(1+{\rho_{u}\beta_{k}}\frac{L}{m_{k}}{\log{{N}}}\right)$
serves as a deterministic equivalent for ${\mathcal{R}}_{\rm{u}}$.

######  Corollary 1.

The sum-rate capacity, $\mathcal{R}_{\rm o}^{(K)}$, scales with $N$ at a
double logarithmic rate.

###### Proof:

The results in (23) indicate that $\mathcal{R}_{\rm
u}=\mathcal{O}\left(\log(\log{N})\right)$ as $N\rightarrow\infty$. Let
$\mathcal{R}_{1}$ denote the capacity bound for $K=1$. Thus, we have
$\mathcal{R}_{1}=\mathcal{O}\left(\log(\log{N})\right)$ and $\mathcal{R}_{\rm
u}\geq\mathcal{R}_{\rm o}^{(K)}\geq\mathcal{R}_{\rm
o}^{(1)}={\mathcal{R}}_{1}$. Using the Sandwich Theorem, we get
$\mathcal{R}_{\rm o}^{(K)}=\mathcal{O}\left(\log(\log{N})\right)$ and the
proof is completed. ∎

######  Remark 1.

The results in (23) indicate that ${\mathcal{R}}_{\rm{u}}$ converges to
$\sum_{k=1}^{K}\log_{2}\left(1+{\rho_{u}\beta_{k}}\frac{L}{m_{k}}{\log{{N}}}\right)$
with probability 1 in the large system limit, which can be treated as a
consequence of the channel hardening [17].

######  Remark 2.

The results in (23) also indicate that a larger value of the sum-rate capacity
bound can be attained in a richer scattering environment with a smaller
Nakagami parameter $m$ [13]. This is because that more gains are possible due
to antenna selection for smaller values of $m$. More specifically, due to the
larger channel variations involved by richer scattering, the system sum-rate
can be more improved by properly choosing a candidate subsect of antennas, and
thus a smaller value of $m$ yields a larger selection diversity gain. When $m$
tends to infinity, the channel becomes the AWGN channel and no selection
diversity gain can be exploited because there is no channel variation.

######  Remark 3.

Note that ${\mathcal{R}}_{\rm{u}}$ is achieved by the FAS-based architecture
and the ES-based selection method, both being capacity-optimal. Hence, it can
be concluded that the sum-rate achieved by other switching architectures and
selection methods will grow no faster than double logarithmically with the BS
antenna number.

##  V Numerical Results

In this section, numerical results are provided to verify the correctness of
the above theoretical analyses. During the simulations, the user terminals are
assumed to be distributed uniformly over a hexagonal cell with radius of 1000
m. The free-space path loss model is
$-10\log_{10}\beta_{k}=92.5+20\log_{10}\left[f_{0}\left(\text{GHz}\right)\right]+20\log_{10}\left[d_{k}\left(\text{km}\right)\right]$,
where $f_{0}=4$ GHz is the carrier frequency and $d_{k}$ is the distance
between the BS and user $k$. Besides, we set $\sigma^{2}=-100$ dBm. All the
simulation results are obtained via $5\times 10^{5}$ Monte-Carlo trials.

To verify the correctness of (18), Figure 1(a) plots the exact sum-rate
capacity, the simulated capacity bound, and the asymptotically approximated
capacity bound (calculated by (18)) in terms of the transmit power for
selected numbers of $L$. It can be observed that the asymptotically
approximated results match perfectly with the simulations, which, thus,
verifies our previous derivations. Moreover, the capacity bound becomes
tighter for a larger value of $L$. As Figure 1(a) shows, the sum-rate capacity
upper bound is tight and presents a similar changing trend to the exact
capacity. Accordingly, one can conclude that this bound serves as a good
alternative for the exact capacity in terms of system sum-rate performance
evaluation. Actually, the Gaussian-distribution approximation stated in Lemma
1 is based on the assumption of $N\rightarrow\infty$. Yet, it can be seen from
Figure 1(a) that this approximation works well even for a limited value of
$N$, i.e., $N=128$, which indicates the robustness of Lemma 1 as well as the
tightness of (18).

Figure 1(b) shows the mean of the capacity bound versus $N$ for a fixed value
of $L$. For reference, the curves representing the exact sum-rate capacity and
$\sum_{k=1}^{K}\log_{2}\left(1+{\rho_{u}\beta_{k}}\frac{L}{m_{k}}{\log{{N}}}\right)$
(labeled as ‘${\mathcal{O}}\left(\log\left(\log N\right)\right)$’) are also
plotted. As shown, both the capacity and its upper bound grow like
${\mathcal{O}}\left(\log\left(\log N\right)\right)$, which supports our
conclusions in Corollary 1. Then, we assume $m_{k}=m$, $\forall
k\in\mathcal{K}$, and plot the SCV of ${\mathcal{R}}_{\rm{u}}$, i.e.,
$\frac{{\mathbbmss{V}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\}}{({\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\})^{2}}$,
versus $N$ for selected values of $m$ in Figure 1(c). As this graph shows, the
SCV tends towards zero as $N$ increases and this consists with the conclusion
drawn in Theorem 1. Moreover, as stated before, a smaller Nakagami-$m$ fading
parameter corresponds to larger channel variations, which yields a larger
value of the SCV as well as a slower channel hardening speed. This is also
validated by Figure 1(c).

##  VI Conclusion

The sum-rate capacity achieved by receive antenna selection in massive MIMO
uplink systems have been discussed. For the sake of mathematical tractability,
a low-complexity sum-rate capacity upper bound has been proposed. A
deterministic equivalent of this bound has been found to characterize its
asymptotic behavior in the large limit of the BS antenna number. On this
basis, the sum-rate capacity is shown to sacle with the BS antenna number at a
double logarithmic rate.

The results of this study can be used to investigate antenna selection-aided
massive MIMO systems in various respects, which will be addressed in the
extended version of this work.

It follows from (16) that
$\lim_{N\rightarrow\infty}\int_{{\upsilon_{k}}}^{\infty}f_{n,k}\left(x\right){\rm{d}}x=\lim_{N\rightarrow\infty}\frac{L}{N}=0$
and thus $\lim_{N\rightarrow\infty}\upsilon_{k}=\infty$. Using [18, eq.
(8.11.2)], we have

$\displaystyle\lim_{N\rightarrow\infty}\frac{\int_{{\upsilon_{k}}}^{\infty}f_{n,k}\left(x\right){\rm{d}}x}{L/N}=\lim_{N\rightarrow\infty}\frac{{\frac{\left(m_{k}{\upsilon_{k}}\right)^{m_{k}-1}}{\Gamma(m_{k})\exp\left({m_{k}{\upsilon_{k}}}\right)}}}{L/N}=1,$
(24)

which yields

$\displaystyle\lim_{N\rightarrow\infty}\frac{\log{\frac{\left(m_{k}{\upsilon_{k}}\right)^{m_{k}-1}}{\Gamma(m_{k})\exp\left({m_{k}{\upsilon_{k}}}\right)}}}{\log{N}-\log{L}}=\lim_{N\rightarrow\infty}\frac{m_{k}{\upsilon_{k}}}{\log{N}}=1.$
(25)

Similarly, based on (15), we can get

$\displaystyle\lim\limits_{N\rightarrow\infty}\frac{\mu_{k}}{L{\upsilon_{k}}}$
$\displaystyle=\lim\limits_{N\rightarrow\infty}\frac{\frac{Nm_{k}^{m_{k}-1}{\upsilon_{k}}^{m_{k}}}{\Gamma\left(m_{k}\right)\exp\left({m_{k}{\upsilon_{k}}}\right)}}{{\upsilon_{k}}L}\overset{\diamond}{=}\lim\limits_{N\rightarrow\infty}\frac{N{\upsilon_{k}}\frac{L}{N}}{{\upsilon_{k}}L}=1,$

where the equality in $\diamond$ holds for (24). By continuously following the
above steps, we have
$\lim_{N\rightarrow\infty}\frac{\sigma_{k}^{2}}{L^{2}{\upsilon_{k}}^{2}}=0$.
As stated before,
$\lim_{N\rightarrow\infty}\frac{\mathbbmss{E}\\{\mathcal{R}_{k}\\}}{\log_{2}\left(1+{\rho_{u}\beta_{k}}X_{k}\right)}=1$,
where $X_{k}\sim\mathcal{N}\left(\mu_{k},\sigma_{k}^{2}\right)$, which
together with the fact of
$\lim_{N\rightarrow\infty}\frac{\mu_{k}}{L{\upsilon_{k}}}=1$, suggests that
$\mathbbmss{E}\\{\mathcal{R}_{k}\\}$ satisfies

$\displaystyle\lim_{N\rightarrow\infty}\frac{\mathbbmss{E}\\{\mathcal{R}_{k}\\}}{{\mathbbmss{E}}\\{\log_{2}\left(1+{\rho_{u}\beta_{k}}L{\upsilon_{k}}W_{k}\right)\\}}=1.$
(26)

Here,
$W_{k}=\frac{X_{k}}{L{\upsilon_{k}}}\sim\mathcal{N}\left(\frac{\mu_{k}}{L{\upsilon_{k}}},\frac{\sigma_{k}^{2}}{L^{2}{\upsilon_{k}^{2}}}\right)$
with $\lim_{N\rightarrow\infty}{{\mathbbmss{E}}\left\\{W_{k}\right\\}}=1$ and
$\lim_{N\rightarrow\infty}{{\mathbbmss{V}}\left\\{W_{k}\right\\}}=0$. Hence,
the random variable $W_{k}$ converges to 1 with probability one as
$N\rightarrow\infty$, which together with (26) and the fact of
$\lim_{N\rightarrow\infty}\frac{{\upsilon_{k}}}{\frac{1}{m_{k}}\log{N}}=1$,
suggests that
$\lim_{N\rightarrow\infty}\frac{\mathbbmss{E}\\{\mathcal{R}_{k}\\}}{\log_{2}\left(1+{\rho_{u}\beta_{k}}L\frac{1}{m_{k}}\log{N}\right)}=1$.
Note that for $\forall a,b>0$, it has
$\lim_{x\rightarrow\infty}\frac{\log\left(1+ax\right)}{\log\left(1+bx\right)}=1$.
Therefore, we can obtain

$\displaystyle\lim_{N\rightarrow\infty}\frac{\sum_{k=1}^{K}\mathbbmss{E}\\{\mathcal{R}_{k}\\}}{\sum_{k=1}^{K}\log_{2}\left(1+{\rho_{u}\beta_{k}}L\frac{1}{m_{k}}\log{N}\right)}=1,$
(27)

which completes the proof of the first part of Theorem 1. Following a similar
approach as that in obtaining (27), we find that
${\mathcal{P}}_{k}=\frac{{\mathbbmss{V}}\left\\{{\mathcal{R}}_{k}\right\\}}{({\mathbbmss{E}}\left\\{{\mathcal{R}}_{k}\right\\})^{2}}$
satisfies

$\displaystyle\lim\limits_{N\rightarrow\infty}{\mathcal{P}}_{k}=\lim\limits_{N\rightarrow\infty}\frac{{\mathbbmss{E}}\left\\{\left[\log_{2}\left(1+{\rho_{u}\beta_{k}}L{\upsilon_{k}}W_{k}\right)\right]^{2}\right\\}}{({\mathbbmss{E}}\left\\{{\mathcal{R}}_{k}\right\\})^{2}}-1.$

Since $\lim_{N\rightarrow\infty}{\mathbbmss{E}}\left\\{W_{k}\right\\}=1$,
$\lim_{N\rightarrow\infty}{\mathbbmss{V}}\left\\{W_{k}\right\\}=0$, and
$\lim_{N\rightarrow\infty}\frac{\mathbbmss{E}\\{\mathcal{R}_{k}\\}}{{\mathbbmss{E}}\\{\log_{2}\left(1+{\rho_{u}\beta_{k}}L{\upsilon_{k}}\right)\\}}=1$,
we get $\lim_{N\rightarrow\infty}{\mathcal{P}}_{k}=0$. Based on this and the
fact of $\lim_{x\rightarrow\infty}\frac{(\log(1+ax))^{2}}{(\log(1+bx))^{2}}=1$
($\forall a,b>0$), we arrive at the following relationship:

$\begin{split}&0\leq\lim\limits_{N\rightarrow\infty}\frac{{\mathbbmss{V}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\}}{({\mathbbmss{E}}\left\\{{\mathcal{R}}_{\rm{u}}\right\\})^{2}}=\lim\limits_{N_{\text{r}}\rightarrow\infty}\frac{\sum_{k=1}^{K}{\mathbbmss{V}}\left\\{{\mathcal{R}}_{k}\right\\}}{\left(\sum_{k=1}^{K}{\mathbbmss{E}}\left\\{{\mathcal{R}}_{k}\right\\}\right)^{2}}\\\
&\leq\lim\limits_{N\rightarrow\infty}\frac{\sum_{k=1}^{K}{\mathbbmss{V}}\left\\{{\mathcal{R}}_{k}\right\\}}{\sum_{k=1}^{K}({\mathbbmss{E}}\left\\{{\mathcal{R}}_{k}\right\\})^{2}}=0.\end{split}$
(28)

Thus, the second part of Theorem 1 is proved.

## References

  * [1] E. Björnson _et al._ , “Massive MIMO: Ten myths and one critical question,” _IEEE Commun. Mag._ , vol. 54, no. 2, pp. 114–123, Feb. 2016.
  * [2] L. Lu _et al._ , “An overview of massive MIMO: Benefits and challenges,” _IEEE J. Sel. Top. Signal Process._ , vol. 8, no. 5, pp. 742–758, Oct. 2014.
  * [3] A. F. Molisch and M. Z. Win, “MIMO systems with antenna selection,” _IEEE Microw. Mag._ , vol. 5, no. 1, pp. 46–56, Mar. 2004.
  * [4] S. Sanayei and A. Nosratinia, “Antenna selection in MIMO systems,” _IEEE Commun. Mag._ , vol. 42, no. 10, pp. 68–73, Oct. 2004.
  * [5] X. Gao _et al._ , “Low RF-complexity technologies to enable millimeterwave MIMO with large antenna array for 5G wireless communications,” _IEEE Commun. Mag._ , vol. 56, no. 4, pp. 211–217, Apr. 2018
  * [6] D. Bai _et al._ , “Rate of channel hardening of antenna selection diversity schemes and its implication on scheduling,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 10, pp. 4353–4365, Oct. 2009.
  * [7] H. Li, L. Song, and M. Debbah, “Energy efficiency of large-scale multiple antenna systems with transmit antenna selection,” _IEEE Trans. Commun._ , vol. 62, no. 2, pp. 638–647, Feb. 2014.
  * [8] A. F. Molisch, M. Z. Win, Y. S. Choi, and J. H. Winters, “Capacity of MIMO systems with antenna selection,” _IEEE Trans. Wireless Commun._ , vol. 4, no. 4, pp. 1759–1772, Jul. 2005.
  * [9] Y. Gao _et al._ , “Massive MIMO antenna selection: Switching architectures, capacity bounds, and optimal antenna selection algorithms,” _IEEE Trans. Signal Process._ , vol. 66, no. 5, pp. 1346–1360, Mar. 2018.
  * [10] S. Sanayei _et al._ , “Capacity of MIMO channels with antenna selection,” _IEEE Trans. Inf. Theory_ , vol. 53, no. 11, pp. 4356–4362, Nov. 2007.
  * [11] S. Asaad _et al._ , “Massive MIMO with antenna selection: Fundamental limits and applications,” _IEEE Trans. Wireless Commun._ , vol. 17, no. 12, pp. 8502–8516, Dec. 2018.
  * [12] S. Y. Park and D. J. Love, “Capacity limits of multiple antenna multicasting using antenna subset selection,” _IEEE Trans. Signal Process._ , vol. 56, no. 6, pp. 2524–2534, Jun. 2008.
  * [13] A. Glodsmith, _Wireless Communications_. Cambridge, U.K.: Cambridge Univ. Press, 2005.
  * [14] S. M. Stigler, “The asymptotic distribution of the trimmed mean,” _The Annals of Statistics_ , pp. 472–477, 1973.
  * [15] C. Ouyang _et al._ , “Asymptotic upper capacity bound for receive antenna selection in massive MIMO systems,” in _Proc. IEEE Int. Conf. Commun._ , May. 20–24, 2019, pp. 1–6.
  * [16] C. Ouyang _et al._ , “Receive antenna selection under discrete inputs: Approximation and applications,” _IEEE Trans. Commun._ , vol. 68, no. 4, pp. 2634–2647, Apr. 2020.
  * [17] B. M. Hochwald _et al._ , “Multi-antenna channel hardening and its implications for rate feedback and scheduling,” _IEEE Trans. Inf. Theory_ , vol. 50, no. 9, pp. 1893–1909, Sep. 2004.
  * [18] R. B. Paris, _NIST Handbook of Mathematical Functions_ , Cambridge University Press, New York, 2010.