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arXiv:1904.00453v1 [cs.IT] 31 Mar 2019
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Performance Analysis of Large Intelligent
Surfaces (LISs): Uplink Spectral Efficiency and
Pilot Training
Minchae Jung, Member, IEEE, Walid Saad, Fellow, IEEE, and Gyuyeol Kong
Abstract
Large intelligent surfaces (LISs) constitute a new and promising wireless communication paradigm
that relies on the integration of a massive number of antenna elements over the entire surfaces of man-
made structures. The LIS concept provides many advantages, such as the capability to provide reliable
and space-intensive communications by effectively establishing line-of-sight (LOS) channels. In this
paper, the system spectral efficiency (SSE) of an uplink LIS system is asymptotically analyzed under a
practical LIS environment with a well-defined uplink frame structure. In order to verify the impact on
the SSE of pilot contamination, the SSE of a multi-LIS system is asymptotically studied and a theoretical
bound on its performance is derived. Given this performance bound, an optimal pilot training length for
multi-LIS systems subjected to pilot contamination is characterized and, subsequently, the performance-
maximizing number of devices that the LIS system must service is derived. Simulation results show
that the derived analyses are in close agreement with the exact mutual information in presence of a
large number of antennas, and the achievable SSE is limited by the effect of pilot contamination and
intra/inter-LIS interference through the LOS path, even if the LIS is equipped with an infinite number
of antennas. Additionally, the SSE obtained with the proposed pilot training length and number of
scheduled devices is shown to reach the one obtained via a brute-force search for the optimal solution.
A preliminary version of this work was submitted to IEEE GLOBECOM 2019 [1].
M. Jung and G. Kong are with Advanced Communications Laboratory, School of Electrical Electronic Engineering, Yonsei
University, Seoul 03722, Korea (e-mail: hosaly, gykong@yonsei.ac.kr).
W. Saad is with Wireless@VT, Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA 24061
USA (e-mail: walids@vt.edu).
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF)
funded by the Ministry of Education (NRF-2016R1A6A3A11936259) and by the U.S. National Science Foundation under Grants
CNS-1836802 and OAC-1638283.
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Index Terms
Large intelligent surface (LIS), large system analysis, performance analysis, pilot contamination,
system spectral efficiency,
I. INT RODUCTI O N
The notion of a large intelligent surface (LIS) that relies on equipping man-made structures,
such as buildings, with massive antenna arrays is rapidly emerging as a key enabler of tomorrow’s
Internet of Things (IoT) and 6G applications [2]–[15]. An LIS system can potentially provide
pervasive and reliable wireless connectivity by exploiting the fact that pervasive city structures,
such as buildings, roads, and walls, will become electromagnetically active in the near future. If
properly operated and deployed, the entire environment is expected to be active in wireless trans-
mission providing near-field communications. In contrast, conventional massive multiple-input
multiple-output (MIMO) systems is essentially regarded as far-field communications generating
non-line-of-sight (NLOS) channels with a high probability. Indeed, the wireless channels of an
LIS can become nearly line-of-sight (LOS) channels, resulting in several advantages compared
to conventional massive MIMO system. First, noise and inter-user interference through a NLOS
path become negligible as the number of antenna arrays on LIS increases [8]. Also, the inter-user
interference through a LOS path is negligible providing an interference-free environment, when
the distances between adjacent devices are larger than half the wavelength [9], [10]. Moreover, an
LIS offers more reliable and space-intensive communications compared to conventional massive
MIMO systems as clearly explained in [8] and [11].
A. Prior Art
Owing to these advantages, LISs have recently been receiving significant attention in the
literature [8]–[15]. In particular, the works in [8] and [9] provided an analysis of the uplink data
rate to evaluate LIS performance considering channel estimation errors, and studied the space-
normalized capacity achieved by an optimal receiver and a matched filter (MF), respectively.
Moreover, in [10] and [11], the authors proposed an optimal user assignment scheme to select
the best LIS unit and analyzed the reliability of an LIS system in terms of the outage probability,
respectively. Meanwhile, the works in [12] and [13] derived, respectively, the Fisher-information
and Cramer–Rao lower bound for user positions exploiting the LIS uplink signal and the uplink
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capacity considering hardware impairments. By enabling an LIS to reflect signals from conven-
tional transmitters, such as base stations (BS) and access points, to desired devices, the authors in
[14] and [15] designed beamformer and LIS phase shifter that maximize the ergodic rate and the
energy efficiency, respectively. However, these recent works in [8]–[15] have not considered the
effects on spectral efficiency (SE) resulting from the use of a practical uplink frame structure in
which the pilot training and data transmission period are jointly considered. Given that statistical
channel state information (CSI) is typically acquired by pilot signaling, and because the channel
uses for data transmission are closely related to the length of the pilot sequence [16], [17], an
uplink frame structure that includes pilot training strongly impacts the achievable SE of LIS
systems. Moreover, this pilot signal will be contaminated by inter-LIS interference, similar to
inter-cell interference in multi-cell MIMO environment (e.g., see [17]–[19]). Therefore, accurate
CSI estimations with an optimal pilot training lengths constitute an important challenge in multi-
LIS systems where the pilot sequences are reused in adjacent LISs. In fact, prior studies on
massive MIMO [16]–[19] do not directly apply to LIS, because the channel model of LIS is
significantly different from the one used in these prior studies. For densely located LISs, all
channels will be modeled by device-specific spatially correlated Rician fading depending on the
distance between each LIS and device, however, the massive MIMO works in [16]–[19] rely on
a Rayleigh fading channel considering far-field communications. Moreover, in LIS, each area of
the large surface constitutes one of the key parameters that determine the performance of an LIS
system [8]–[10], however, in existing massive MIMO works [8]–[10], this notion of an area is
not applicable.
B. Contributions
The main contribution of this paper is an asymptotic analysis of the uplink system SE (SSE)
in a multi-LIS environment that considers a practical uplink frame structure based on the 3GPP
model in [20]. The SSE is typically measured as the data rates that can be simultaneously
supported by a limited bandwidth in a defined geographic area [21]. Given an LIS serving
multiple devices, we define the SSE as the sum of the individual SE of each LIS device.
Then, we analyze the asymptotic SSE including its ergodic value, channel hardening effect,
and performance bound, under pilot contamination considerations, relying on a scaling law for a
large number of antennas. The devised approximation allows for accurate estimations of the SSE,
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deterministically, and it also allows verifying the reliability of an LIS system. Subsequently, we
analyze the effect of pilot training in a realistic channel fading scenario in which channel states
are limited by the channel coherence block and are exclusively static within limited time and
frequency blocks. The pilot training analysis provides insights on how the optimal pilot training
length scales with the various parameters of LIS systems in a deterministic way. It also reveals a
particular operating characteristic of an LIS, whereby the optimal pilot training length converges
to the number of devices located within an LIS area, as the number of antennas increases without
bound. Given the derived pilot training length, we finally derive the optimal number of devices
that each LIS must schedule, to maximize the SSE. Simulation results show that an LIS with the
proposed operating parameters, including the pilot training length and the number of scheduled
devices, can achieve a maximum SSE performance both in single- and multi-LIS environments,
regardless of the effect of pilot contamination and inter-LIS interference. Moreover, the impact
on an LIS system of pilot contamination can be negligible when inter-LIS interference channels
are generated from spatially correlated Rayleigh fading, which highlights a significant difference
from conventional massive MIMO.
The rest of this paper is organized as follows. Section II presents the LIS-based system
model. Section III describes the asymptotic analysis of the SSE, and Section IV describes the
performance bound and optimal pilot training length based on the results of Section III. The
optimal number of scheduled devices is also discussed in Section IV. Simulation results are
provided in Section V to support and verify the analyses, and Section VI concludes the paper.
Notations: Hereinafter, boldface upper- and lower-case symbols represent matrices and vectors
respectively, and IMdenotes a size-Midentity matrix. µXand σ2
Xdenote the mean and variance
of a random variable X, respectively. The conjugate, transpose, and Hermitian transpose operators
are denoted by (·)∗,(·)T, and (·)H, respectively. The norm of a vector ais denoted by |a|and
the Frobenious norm of a matrix Ais kAkF.E [·]and O(·)denote the expectation operator and
big O notation, respectively. CN (m, σ2)is a complex Gaussian distribution with mean mand
variance σ2.
II. SY STEM MOD EL
Consider an uplink LIS system with N≥1LISs sharing the same frequency band. Each LIS
is located in two-dimensional Cartesian space along the xy-plane, serving Kdevices, as shown
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(a)
(b)
Fig. 1. Illustrative system model of the uplink LIS under consideration of (a) indoor case with single LIS, (b) outdoor case
with multiple LISs, with Kdevices per each LIS.
in Fig. 1. Each LIS is composed of KLIS units, each of which serves a single-antenna device
occupying a 2L×2Lsquare shaped subarea of the entire LIS. A large number of antennas, M,
are deployed on the surface of each LIS unit with ∆Lspacing, arranged in a rectangular lattice
centered on the (x, y)coordinates of the corresponding device. Considering the location of device
kat LIS nas (xnk, ynk , znk ), antenna mof LIS unit kat LIS nwill be located at (xLIS
nkm, yLIS
nkm,0)
where xLIS
nkm ∈[xnk −L, xnk +L]and yLIS
nkm ∈[ynk −L, ynk +L]. Fig. 1 illustrates our system
model for an indoor case with a single LIS and an outdoor case with multiple LISs. In case
of single LIS, as shown in Fig. 1(a), the desired signal is affected, exclusively, by intra-LIS
interference which is defined as the interference generated by multiple devices located within
the same LIS area. On the other hand, for the case of multiple LISs, as shown in Fig. 1(b),
the desired signal can be affected by both intra-LIS and inter-LIS interference simultaneously.
Here, inter-LIS interference corresponds to the interference generated by devices serviced by
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other LISs.
Some LIS units may overlap depending on the locations of their associated devices, resulting
in severe performance degradation. To prevent this problem, we assume that each LIS consists
of Knon-overlapping LIS units that use an orthogonal multiple access scheme among devices
with similar locations. Moreover, each device controls its transmission power toward the center
of its LIS unit according to a target signal-to-noise-ratio (SNR), to avoid the near-far problem.
A. Wireless Channel Model
In LIS systems, entire man-made structures are electromagnetically active and can be used
for wireless communication. We then consider the LIS channel hL
nnkk ∈CMbetween device k
at LIS nand LIS unit kpart of LIS nas a LOS path defined by:
hL
nnkk =βL
nnkk1hnnkk1,···, βL
nnkkM hnnk kM T,(1)
where βL
nnkkm =αL
nnkkm lL
nnkkm and hnnkkm = exp (−j2πdnnkkm/λ)denote a LOS channel gain
and state, respectively, between device kat LIS nand antenna mof LIS unit kpart of LIS n
[22]. The terms aL
nnkkm =pznk /dnnkkm and lL
nnkkm = 1/p4πd2
nnkkm represent, respectively, the
antenna gain and free space path loss attenuation, where dnnkkm is the distance between device
kat LIS nand antenna mof LIS unit kpart of LIS n.λis the wavelength of a signal. We
model the interference channel hlnjk ∈CMbetween device jat LIS land LIS unit kpart of
LIS nas a Rician fading channel with Rician factor κlnjk, given by:
hlnjk =¯
hlnjk +˜
hlnjk =rκlnjk
κlnjk + 1 hL
lnjk +s1
κlnjk + 1 hNL
lnjk ,(2)
where hL
lnjk ∈CM=βL
lnjk1hlnjk1,···, βL
lnjk M hlnjk M Tand hNL
lnjk ∈CM=R1/2
lnjk glnj k denote
the deterministic LOS and the correlated NLOS component, respectively. Here, if l=nand
j6=k, then hlnjk indicates the intra-LIS interference channel, otherwise, if l6=n∀j, k, then
hlnjk indicates the inter-LIS interference channel. Considering Pdominant paths among all
NLOS paths, we define Rlnjk ∈CM×Pand glnjk = [glnjk1,···, glnj kP ]T∼ CN (0,IP)to
be the deterministic correlation matrix and an independent fast-fading channel vector between
device jat LIS land LIS unit kpart of LIS n, respectively. Since the LIS is deployed on the
horizontal plane, as shown in Fig. 1, we can model it as a uniform planar array [23]. Then, the
correlation matrix can be defined as R1/2
lnjk =lNL
lnjk Dlnj k, where lNL
lnjk = diag lNL
lnjk1···, lNL
lnjk M
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Fig. 2. Illustrative uplink frame structure with a pilot training period tand a data transmission period T−t.
is a diagonal matrix that includes the path loss attenuation factors lNL
lnjkm =d−βPL/2
lnjkm with a
path loss exponent βPL and Dlnjk =αNL
lnjk1dφv
lnjk1, φh
lnjk1,···, αNL
lnjk P dφv
lnjk P , φh
lnjk P .
dφv
lnjkp, φh
lnjkp∈CMis the NLOS path pat given angles of φv
lnjkp, φh
lnjkpdefined as:
dφv
lnjkp, φh
lnjkp=1
√Mdvφv
lnjkp⊗dhφh
lnjkp,(3)
dvφv
lnjkp=h1, ej2π∆L
λφv
lnjkp ,···, ej2π∆L
λ(√M−1)φv
lnjkp iT,(4)
dhφh
lnjkp=h1, ej2π∆L
λφh
lnjkp ,···, ej2π∆L
λ(√M−1)φh
lnjkp iT,(5)
where φv
lnjkp = sin θv
lnjkp and φh
lnjkp = sin θh
lnjkp cos θh
lnjkp when the elevation and azimuth angles
of path pbetween device jat LIS land LIS unit kpart of LIS nare θv
lnjkp and θh
lnjkp, respectively
[24]. Further, αNL
lnjkp =qcos θv
lnjkp cos θh
lnjkp denotes the antenna gain of path pwith θlnjkp ∈
−π
2,π
2and θlnjkp ∈θv
lnjkp, θh
lnjkp.
B. Uplink Pilot Training
We consider that an MF is used at the LIS to amplify the desired signals and suppress
interfering signals. This MF receiver requires CSI which can be estimated by pilot signaling
with known pilot signals being transmitted from the device to the LIS. The device transmits its
data signals immediately after sending the pilot signals within the channel coherence time Tin
which the uplink channel is approximately static. We consider the uplink frame structure shown
in Fig. 2, in which the total duration of Tchannel uses is divided into a tperiod used for pilot
training and a T−tperiod used for data transmission. Every device simultaneously transmits
t≥Korthogonal pilot sequence over the uplink channel to the LIS, so that the required CSI
can be acquired. Given that those Kpilot sequences are pairwise orthogonal to each other, we
have ΨHΨ=IK, where Ψ= [ψ1, ..., ψK]and ψkis the t×1pilot sequence for device k.
For the multi-LIS scenario in which the same frequency band is shared by all LISs and adjacent
LISs reuse the pilot sequences, the pilot symbols between adjacent LISs are no longer orthogonal
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to each other and this non-orthogonality causes pilot contamination. In large antenna-array
systems such as massive MIMO and LIS, the performance can be dominantly limited by residual
interference from pilot contamination as explained in [17]–[19]. Since LISs will be located more
densely than BSs, the LIS channels associated with pilot contamination will be significantly
different than those of massive MIMO, and hence, prior studies on pilot contamination for
massive MIMO [16]–[19] do not directly apply to LIS. In order to verify the effect of pilot
contamination in an LIS system theoretically, we consider such multi-LIS scenario in which a
total of NLISs share the same frequency band and each LIS reuses Kpilot sequences. Moreover,
all LISs are assumed to use the same uplink frame structure shown in Fig. 2, whereby a pilot
sequence kis allocated to device kfor all 1≤k≤K. The uplink pilot sequence received from
all devices at LIS unit kpart of LIS nduring period twill be:
Yp
nk =ptρpnk hL
nnkkψH
k+
K
X
j6=kptρpnj hnnjk ψH
j+
N
X
l6=n
K
X
j=1 ptρplj hlnjk ψH
j+Nnk,(6)
where ρpnk ,ρpnj , and ρplj are the transmit SNRs for the pilot symbols of device kat LIS n, device
jat LIS n, device jat LIS l, respectively, and Nnk ∈CM×t∼ CN (0,IM)is a noise matrix at
LIS unit kpart of LIS n. We assume that the target SNR for a pilot symbol is assumed to be
equal to ρpand each device controls its pilot power toward the center of the corresponding LIS
unit. On the basis of orthogonal characteristic of the pilot sequences, each LIS unit kmultiplies
the received pilot signal by ψkfor channel estimation. After multiplying ψkat both sides of
(6), we have
Yp
nkψk=ptρpnk hL
nnkk +
N
X
l6=nptρplk hlnkk +Nnk ψk
=ptρpnk hL
nnkk +
N
X
l6=nptρplk ¯
hlnkk +˜
hlnkk +Nnkψk.(7)
In most prior research on pilot contamination in large antenna-array systems such as in [18] and
[25], the minimum mean square error (MMSE) channel estimator is assumed to estimate a desired
channel given that the BS has knowledge of every correlation matrix between itself and interfering
users located in adjacent cells. However, this assumption is impractical for LIS systems because
a massive number of devices will be connected to an LIS, and thus, processing complexity will
increase tremendously when estimating and sharing device information. Therefore, we consider
a simple least square (LS) estimator which does not require such information as a practical
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alternative [26]. The LS estimate of the deterministic desired channel hL
nnkk is then obtained as
follows:
ˆ
hnnkk =hL
nnkk +enk ,(8)
where enk indicates the estimation error vector given by enk =PN
l6=nqρplk
ρpnk ¯
hlnkk +˜
hlnkk +
1
√tρpnk
wnk and wnk = [wnk1,···, wnkM ]T∈CM∼ CN (0,IM).
C. Uplink SSE
The uplink signal received from all devices at LIS unit kpart of LIS nis given by:
ynk =√ρnkhL
nnkkxnk +
K
X
j6=k
√ρnjhnnjk xnj +
N
X
l6=n
K
X
j=1
√ρljhlnjkxlj +nnk,(9)
where xnk,xnj , and xlj are uplink transmit signals of device kat LIS n, device jat LIS
n, and device jat LIS l, respectively, and ρnk ,ρnj , and ρlj are their transmit SNRs. Also,
nnk ∈CM∼ CN (0,IM)is the noise vector at LIS unit kpart of LIS n. Given a linear receiver
fH
nk for signal detection, we will have
fH
nkynk =√ρnkfH
nkhL
nnkkxnk +
K
X
j6=k
√ρnjfH
nkhnnj kxnj +
N
X
l6=n
K
X
j=1
√ρlj fH
nkhlnjk xlj +fH
nknnk,(10)
We consider an MF receiver such that fnk =ˆ
hnnkk. Under the imperfect CSI results from an
LS estimator, fnk can be obtained from (8) as fnk =hL
nnkk +enk where ekis the estimation
error vector uncorrelated with nnk [27]. Therefore, the received signal-to-interference-plus-noise
ratio (SINR) at LIS unit kpart of LIS nwill be:
γnk =ρnkhL
nnkk4
ρnkeH
nkhL
nnkk2+
K
P
j6=k
ρnj ˆ
hH
nnkkhnnjk
2+
N
P
l6=n
K
P
j=1
ρlj ˆ
hH
nnkkhlnjk
2+hL
nnkkH+eH
nk
2,
(11)
For notational simplicity, we define γnk =ρnk Snk /Ink , where Snk =hL
nnkk4,and
Ink =ρnkXnk +
K
X
j6=k
ρnj Ynjk +
N
X
l6=n
K
X
j=1
ρljYlnjk +Znk,(12)
where Xnk=eH
nkhL
nnkk2, Ynjk=ˆ
hH
nnkkhnnjk
2, Ylnjk =ˆ
hH
nnkkhlnjk
2,and Znk=hL
nnkkH+eH
nk
2.
Considering tand T−tperiods used for pilot training and data transmission, respectively, the
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SSE can be obtained as follows:
RSSE
n=1−t
TK
X
k=1
Rnk =1−t
TK
X
k=1
log (1 + γnk ).(13)
Given this SSE, we will be able to analyze the asymptotic value of the SSE and devise an optimal
pilot training length tthat maximizes the asymptotic SSE as Mincreases without bound. Note
that in the following sections, we use a generalized value of N≥1in order to analyze both
single- and multi-LIS cases, simultaneously (i.e., N= 1 and N≥2indicate a single-LIS and
N-LIS cases, respectively).
III. ASYMP TOTI C SSE ANALYSIS
We analyze the asymptotic value of the SSE under consideration of the pilot contamination
as Mincreases to infinity. In an uplink LIS system with MF receiver, the desired signal power,
Snk, converges to a deterministic value as Mincreases to infinity as proved in [8] and [9]:
Snk −¯pnk −−−−→
M→∞ 0,where ¯pnk =M2p2
nk
16π2L4and pnk = tan−1L2/znkp2L2+z2
nk. Given the
definition of γnk =ρnkSnk/Ink, we have γnk −¯γnk −−−−→
M→∞ 0, where
¯γnk =ρnk p2
nk
16π2L4Ink/M2,(14)
We can observe from (14) that the distribution of ¯γnk depends exclusively on the distribution of
Ink. In order to analyze the distribution of Ink theoretically, we derive the following lemmas.
Here, we define R1/2
lnjk = [clnjk1,···,clnjk P ] = rH
lnjk1,···,rH
lnjk M H, where clnjkp ∈CM×1and
rlnjkm ∈C1×P.
Lemma 1. The mean of Xnk is obtained by µXnk =σ2
xnk +|µxnk |2, where
µxnk =
N
X
l6=nrρplk
ρpnk
¯
hH
lnkk hL
nnkk,(15)
σ2
xnk =
N
X
l6=n
ρplk
ρpnk (κlnkk + 1)
P
X
p=1 cH
lnkkphL
nnkk2+1
tρpnk
M
X
m=1
β2
nnkkm .(16)
Proof: The detailed proof is presented in Appendix A.
Lemma 2. The mean values of Ynj k and Ylnjk follow µYnjk −¯µYnjk −−−−→
M→∞ 0and µYlnjk −
¯µYlnjk −−−−→
M→∞ 0, respectively, where ¯µYnjk and ¯µYlnj k are given, respectively, in (52) and (53).
Proof: The detailed proof is presented in Appendix B.
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Lemma 3. The mean of Znk is obtained by µZnk =
M
P
m=1 σ2
znkm +|µznkm |2, where
µznkm =βL
nnkkmh∗
nnkkm +
N
X
l6=nrρplk κlnkk
ρpnk (κlnkk + 1)βL
lnkkm h∗
lnkkm ,(17)
σ2
znkm =
N
X
l6=n
ρplk |rlnkkm|2
ρpnk (κlnkk + 1) +1
tρpnk
.(18)
Proof: The detailed proof is presented in Appendix C.
In Lemmas 1–3, the variables, µXnk ,¯µYnjk ,¯µYlnj k , and µZnk, are obtained by the deterministic
information such as the locations of the devices and covariance matrices. On the basis of Lemmas
1–3, we can asymptotically derive the mean of Ink from (12) as follows: µInk −¯µInk −−−→
M→∞ 0,
where
¯µInk =ρnkµXnk +
K
X
j6=k
ρnj ¯µYnjk +
N
X
l6=n
K
X
j=1
ρlj ¯µYlnjk +µZnk .(19)
Since the variance of ¯γnk exclusively depends on the variance of Ink /M2from (14), the following
Lemma 4 is used to obtain σ2
Ink /M4based on the scaling law for M.
Lemma 4. According to the scaling law for M, the variance of Ink /M2asymptotically follows
σ2
Ink /M4−−−−→
M→∞ 0.
Proof: The detailed proof is presented in Appendix D.
Lemma 4 shows that Ink /M2converges to the deterministic value ¯µInk /M2without any
variance, as Mincreases. Then, ¯γnk converges to a deterministic value as Mincreases, and
finally, we have the following Theorem 1 related to the asymptotic convergence of RSSE
n.
Theorem 1. As Mincreases to infinity, we have the following asymptotic convergence of
SSE: RSSE
n−¯µSSE
n−−−−→
M→∞ 0,where
¯µSSE
n=1−t
TK
X
k=1
log 1 + ρnkp2
nk
16π2L4¯µInk /M2.(20)
Proof: The detailed proof is presented in Appendix E.
Theorem 1 shows that the multi-LIS system will experience a channel hardening effect result-
ing in the deterministic SSE. This deterministic SSE provides the improved system reliability
and a low latency. Moreover, we can observe from (19) and (20) that the asymptotic SSE
can be obtained from the deterministic information such as the locations of the devices and
correlation matrices. Therefore, this asymptotic approximation enables accurate estimation of
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the SSE without the need for extensive simulations. Next, we use this asymptotic SSE to derive
a performance bound on the SSE, asymptotically, by analyzing ¯µSSE
nvia a scaling law for an
infinite M.
IV. SSE PE RFO RMA NCE BOUND FOR LISS AND OPTIM AL SYS TEM PARAMETER S
We now analyze the performance bound of the SSE for a large (infinite) value for M. As
Mincreases, the SSE converges to ¯µSSE
nwhich depends on the value of ¯µInk /M2, as seen from
(20). Hence, in an LIS system equipped with an infinite number of antennas, it is important
to obtain its limiting value, lim
M→∞ ¯µInk /M2, and this can provide the performance bound of the
SSE, asymptotically. In this section, we derive the asymptotic SSE performance bound for an
infinite Musing a scaling law of ¯µSSE
n, and propose an optimal pilot training length based
on that asymptotic bound. We consider the uplink frame structure shown in Fig. 2, in which
all devices simultaneously transmit their orthogonal pilot sequences before transmitting their
own data signals like a 3GPP model in [20]. Due to the pilot training overhead t, there is a
fundamental tradeoff between the received SINR enhancement and loss of uplink channel uses
for the data signal. Therefore, it is imperative to optimize the pilot training length to achieve
the maximum SSE and define how the optimal pilot training length deterministically scales with
the various parameters of LIS system.
In order to derive the performance bound of LIS system, we first determine the scaling law of
¯µInk /M2according to M. Then, we have the following result related to the performance bound
of SSE.
Theorem 2. As Mincreases, ¯µSSE
nasymptotically converges to its performance bound, ˆµSSE
n,
as given by: ¯µSSE
n−ˆµSSE
n−−−−→
M→∞ 0,where
ˆµSSE
n=1−t
TK
X
k=1
log 1 + M2ρnk p2
nk
16π2L4ˆµInk ,(21)
ˆµInk =ρnk|µxnk |2+
K
X
j6=k
ρnjµynjk 2+
N
X
l6=n
K
X
j=1
ρlj µylnjk 2.(22)
Proof: From (19), ¯µInk /M2is obtained as follows:
¯µInk
M2=ρnk
µXnk
M2+
K
X
j6=k
ρnj
¯µYnjk
M2+
N
X
l6=n
K
X
j=1
ρlj
¯µYlnjk
M2+µZnk
M2.(23)
13
On the basis of Lemmas 1–3, we determine the scaling laws of the terms µXnk /M2,¯µYnjk /M2,
¯µYlnjk /M2, and µZnk /M 2in (23) according to M. As proved in Appendix D, the terms σ2
xnk /M2,
σ2
ynjk /M2,σ2
ylnjk /M2, and PM
m=1 σ2
znkm /M2converge to zero as Mincreases. Similarly, the terms
|µxnk |2/M2,µynj k 2/M2, and µylnj k 2/M 2follow O(1), and PM
m=1 |µznkm |2/M2decreases
with O(1/M)as Mincreases, based on the scaling laws for large M. Therefore, we have the
following asymptotic convergence:
µXnk
M2−|µxnk |2
M2−−−−→
M→∞ 0,(24)
¯µYnjk
M2−µynjk 2
M2−−−−→
M→∞ 0,(25)
¯µYlnjk
M2−µylnjk 2
M2−−−−→
M→∞ 0,(26)
µZnk
M2−−−−→
M→∞ 0,(27)
which completes the proof.
The terms µxnk ,µynjk , and µylnjk in (22) are determined by the LOS channels depending on the
locations of the devices, as shown, respectively, in (36), (50), and (54). Therefore, the asymptotic
SSE performance bound can be obtained, deterministically, and that deterministic bound leads
to several important implications when evaluating an LIS system, that significantly differ from
conventional massive MIMO. First, an LIS system has a particular operating characteristic
whereby the pilot contamination and intra/inter-LIS interference through the NLOS path and
noise become negligible as Mincreases. If all of the inter-LIS interference is generated from the
NLOS path, the pilot contamination and inter-LIS interference will vanish lead to a performance
convergence between the SSE of single- and multi-LIS system. Moreover, unlike conventional
massive MIMO in which the performance is dominantly limited by pilot contamination, the
channel estimation error including pilot contamination gradually loses its effect on the SSE, and
eventually, the SSE of a multi-LIS system will reach that of a single-LIS system with perfect CSI.
More practically, even if all of the intra/inter-LIS interference channels are generated by device-
specific spatially correlated Rician fading, an LIS system also has a particular characteristic
whereby its SSE performance is bounded by three factors that include pilot contamination,
intra-, and inter-LIS interference through the LOS path,
14
Next, we formulate an optimization problem whose goal is to maximize the SSE with respect
to the pilot training length tby using the asymptotic SSE, ¯µSSE
n, from Theorem 1. Since ¯µInk is
a function of t, we have
max
t1−t
TK
X
k=1
log 1 + M2ρnkp2
nk
16π2L4¯µInk (t),(28)
s.t. K ≤t≤T, (28a)
¯µInk (t) = ρnk σ2
xnk (t) + |µxnk |2+
K
X
j6=k
ρnj σ2
ynjk (t) + µynjk 2
+
N
X
l6=n
K
X
j=1
ρlj σ2
ylnjk (t) + µylnjk 2+
M
X
m=1 σ2
znkm (t) + |µznkm |2.(28b)
In (28b), the terms, σ2
xnk (t),σ2
ynjk (t),σ2
ylnjk (t), and σ2
znkm (t), are monotonically decreasing
functions with respect to tas observed from (37), (51), (55), and (60), respectively. Thus,
¯µInk (t)is also a monotonically decreasing function and log 1 + M2ρnkp2
nk
16π2L4¯µInk (t)is thus a positive
concave increasing function with respect to t. From the operations that preserve the concavity of
functions [28], the product of a positive decreasing function and a positive concave increasing
function is concave. Thus, (28) is a convex optimization problem and we can obtain the globally
optimal result, topt, using a simple gradient method. Moreover, we can observe from the objective
function in (28) that topt changes according to deterministic values such as the locations of the
devices and correlation matrices. From Theorem 2, ¯µInk(t)asymptotically converges to ˆµInk as
Mincreases, resulting in the received SINR independent with t. Therefore, in the following
Corollary 1, we obtain an asymptotic value of topt, independent of the locations of the devices
and correlation matrices, using the asymptotic bound of the SSE from Theorem 2.
Corollary 1. As Mincreases, the optimal pilot training length can be asymptotically obtained
as follows: topt −K−−−−→
M→∞ 0.
Proof: As proved in Theorem 2, the SSE asymptotically converges to its performance
bound ˆµSSE
nas Mincreases, and ˆµInk is independent with tas seen from (22). Then, ˆµSSE
nis
a monotonically decreasing function with respect to t, and therefore, the optimal pilot training
length asymptotically converges to K, which completes the proof.
In conventional massive MIMO system, the pilot training length affects the received SINR
and its optimal value is determined by various system parameters such as the number of users,
uplink transmission period T, and transmit SNR, as proved in [16]. Unlike conventional massive
15
TABLE I
SIM UL ATI ON PA RA M ET E RS
Parameter Value
Carrier frequency 3GHz
Minimum distance 1 m
between LIS & device
Target SNR for uplink pilot 0dB
Target SNR for uplink data 3dB
Coherence block length (T) 500 symbols
Length of LIS unit (2L) 0.5 m
Rician factor (κ[dB]) [29] 13 −0.03d[m]
LOS path loss model [22] 11 + 20log10 d[m]
NLOS path loss model [18] 37log10d[m] (βPL = 3.7)
MIMO, Corollary 1 shows that, for both the single- and multi-LIS cases, the asymptotic bound
of the received SINR does not increase with the increase in the pilot training length and the
maximum SSE can be achieved through a minimum pilot training length such as t=K(i.e., one
pilot symbol per device), despite the pilot contamination effect. With the proposed pilot training
length t=K, we then formulate an optimization problem that asymptotically maximizes the
sum of SSE for neighboring NLISs with respect to K, as follows:
max
K1−K
TN
X
n=1
K
X
k=1
log (1 + ˆγnk (K)) ,(29)
s.t.1≤K≤T, (29a)
ˆγnk (K) = M2ρnkp2
nk
16π2L4ˆµInk (K).(29b)
Consider that each device has its own prioritization coefficients for scheduling and each LIS
schedules devices in order of their priority. According to this priority order, the objective function
in (29) can be calculated from K= 1 to K=T, deterministically, given that the value of ˆγnk(K)
depends on the locations of the devices. Therefore, by comparing those values over the entire
ranges of K, the optimal number of scheduled devices can be asymptotically derived, without
16
Fig. 3. Illustrative system model of the multiple LISs considered when N= 4.
the need for extensive simulations. Note that an LIS has the potential for estimating the locations
of serving devices from their uplink signal [12] and the Rician factor is calculated according to
the distance between the LIS and device [29]. By cooperating across adjacent multi-LISs like a
network LIS, an LIS is able to share those information without the heavy burden of backhaul load
and perform joint scheduling to maximize the network SE (NSE). Therefore, the asymptotically
optimal number of scheduled devices, Kopt , can be calculated at each LIS, practically, when
adjacent multi-LISs cooperate with each other as a network LIS and share the information about
the locations of their own serving devices.
V. SIMULATION RESULTS AND ANALYSIS
We present Monte Carlo simulation results for the uplink SSE in an LIS system, and compare
them with the results of the asymptotic analyses. All simulations are statistically averaged over a
large number of independent runs. The simulation parameters are based on the LTE specifications,
presented in Table I. In accordance with the LTE specifications [30], the target SNR for uplink
power control is semi-statically configured by upper-layer signaling in the LTE system. The
range of target SNRs for the uplink data and pilot signals are -8 dB to 7 dB and -8 dB to 23 dB,
respectively [30]. The uplink target SNRs presented in Table I satisfy the constraints of practical
target SNRs. Furthermore, the minimum scheduling unit is defined as 1 ms in the time domain,
17
0 200 400 600 800 900
Number of antennas on each LIS unit (M)
0
0.02
0.04
0.06
0.08
0.1
Variance of individual SE
Single-LIS
Multi-LIS
K=40
K=10
Fig. 4. Variance of individual SE of an LIS system as a function of the number of antennas on the LIS unit.
and 180 kHz in the frequency domain, which is the so-called physical resource block (PRB)
in the LTE specifications [20]. Each PRB consists of a total of 168 symbols (14 orthogonal
frequency-division multiplexing (OFDM) symbols in the time domain, and 12 subcarriers in the
frequency domain) including the cyclic prefix overhead of the OFDM symbols. Since the value
of T= 500 presented in Table I corresponds to approximately 3 PRBs (500 ≈168 ×3), we
consider the uplink frame as one of two frames such as 1 ms ×540 kHz or 3 ms ×180 kHz.
The value of T= 500 used in performance evaluation may be regarded as a moderate coherence
block length, given that the generalized coherence lengths are T= 200 for high-mobility or
high-delay spread scenarios, and T= 5000 for low-mobility or low-delay spread scenarios [31].
In the simulations, we consider both single- and multi-LIS cases. In both cases, we consider a
scenario in which the devices are randomly and uniformly distributed within a three-dimensional
space of 4m×4m×2m in front of each LIS. For the single-LIS case, N= 1 and only a
single target LIS is located in two-dimensional Cartesian space along the xy-plane. For the
multi-LIS case, to be able to consider the effect of pilot contamination, we assume a total of
N= 4 LISs consist of one target LIS and three neighboring LISs, located on both sides and
in front of the target LIS, as shown in Fig. 3. The parameters presented in Fig. 3 are such that
xL=yL= 4,dx= 4, and dz= 6. All LISs are assumed to share the same frequency band,
18
0 200 400 600 800 900
Number of antennas on each LIS unit (M)
40
60
80
100
120
Ergodic SSE (bps/Hz)
Simulation (Single-LIS)
Simulation (Multi-LIS)
Estimation
Performance bound
Single-LIS
Multi-LIS
Fig. 5. Uplink ergodic SSE of an LIS system with Rician fading interference as a function of the number of antennas on the
LIS unit when K= 20.
each of which serves Kdevices and reuses Kpilot sequences. According to the 3GPP model
in [29], the existence of a LOS path depends on the distance from the transmitter and receiver.
The probability of a LOS is then given by
PLOS
lnjk =
(dC−dlnjk )/dC,0< dlnjk < dC,
0, dljk > dC,
(30)
where dlnjk is the distance in meters between device jat LIS land the center of the LIS unit
kpart of LIS n, and dCdenotes the cutoff point, which is assumed to be 10 m, as in [8]. The
Rician factor, κlnjk, is calculated according to dlnjk, as per Table I.
Fig. 4 shows the channel hardening effect of an LIS system whereby the variances of individual
SE in both single- and multi-LIS cases converge to zero as Mincreases, despite the pilot
contamination effect. Here, the individual SE of device Kat LIS nis derived from (13) as
follows:
RSE
nk =1−t
Tlog 1 + ρnk Snk
Ink .(31)
The variance convergence of RSE
nk verifies the asymptotic convergence of Snk/Ink, and Lemma
4 is then verified given that Snk converges to a deterministic value as Mincreases.
In Figs. 5–7, Theorems 1 and 2 are verified in the following scenario. All intra-LIS interference
19
36 100 200 300 400
Number of antennas on each LIS unit (M)
60
110
160
210
240
Ergodic SSE (bps/Hz)
Simulation (Single-LIS)
Simulation (Multi-LIS)
Estimation
Performance bound
K=20
K=80
Multi-LIS
K=40
Single-LIS
Fig. 6. Uplink ergodic SSE of an LIS system with NLOS inter-LIS interference as a function of the number of antennas on
the LIS unit.
channels are generated by device-specific spatially correlated Rician fading. In Fig. 5, all inter-
LIS interference channels are also generated by that Rician fading, however, in Figs. 6 and 7,
those channels are generated entirely from the NLOS path such as spatially correlated Rayleigh
fading. In both Figs. 5 and 6, the asymptotic results from Theorem 1 become close to the results
of our simulations and these results gradually approach to their performance bounds obtained
from Theorem 2, as Mincreases. Moreover, those performance bounds also converge to the
limiting values resulting from the intra/inter-LIS interference through the LOS path. In Fig. 5,
the performance gap between the results of the single- and multi-LIS is roughly 33 bps/Hz at
M= 900, and it is expected to converge to 36 bps/Hz from the bound gap between the two
systems, as Mincreases. This performance gap between the two systems results from pilot
contamination and inter-LIS interference generated from the LOS path, as proved in Theorem
2. In Fig. 6, the performance gap between the results of the single- and multi-LIS increases as
Kincreases from K= 20 to K= 80 because of the increase in the inter-LIS interference.
However, the performance gap between the two systems converges to zero even at K= 80,
as Mincreases, and their bounds achieve an equal performance over the entire range of M.
Since the the pilot contamination and inter-LIS interference generated from the NLOS path
become negligible compared to the intra-LIS interference through the LOS path, this results
20
50 100 150 200 250 300 350 400
Number of antennas on each LIS unit (M)
85
95
105
115
Ergodic SSE (bps/Hz)
Imperfect CSI (Single-LIS)
Imperfect CSI (Multi-LIS)
Perfect CSI (Single-LIS)
Perfect CSI (Multi-LIS)
Perfect CSI
Imperfect CSI
Fig. 7. Performance comparison of the ergodic SSE resulting from scenarios with perfect CSI and imperfect CSI when K= 20
with NLOS inter-LIS interference.
in the performance convergence between the two systems and eventually the multi-LIS system
becomes an inter-LIS interference-free environment.
Fig. 7 compares the ergodic SSE resulting from cases with perfect CSI and imperfect CSI,
when K= 20 and all inter-LIS interference channels are generated by spatially correlated
Rayleigh fading. We can observe that all ergodic SSE converge to same value of roughly 110
bps/Hz. Hence, despite the pilot contamination in the multi-LIS case, the ergodic SSE of the
multi-LIS system with the imperfect CSI converges to that with the perfect CSI, and it eventually
reaches the single-LIS performance with perfect CSI, as Mincreases. This clearly shows a
particular characteristic of LIS systems whereby pilot contamination and inter-LIS interference
become negligible, representing a significant difference from conventional massive MIMO.
Fig. 8 compares the ergodic SSE resulting from the optimal training lengths obtained from
Corollary 1 and a brute-force search, as a function of M. As shown in Fig. 8, the optimal
performance obtained by a brute-force search is nearly achieved by the proposed pilot training
length, t=K, over the entire range of M. Although the ergodic SSE of a multi-LIS system is
affected by pilot contamination and inter-LIS interference through the LOS path, thus resulting
in performance degradation compared with the single-LIS case, the minimum training length
21
0 200 400 600 800 900
Number of antennas on each LIS unit (M)
0
40
80
120
Ergodic SSE (bps/Hz)
Brute-force (Single-LIS)
Brute-force (Multi-LIS)
Proposed (t=K)
K=20
K=10
K=5
Single-LIS Multi-LIS
Fig. 8. Performance comparison between ergodic SSE resulting from the proposed pilot training length and brute-force search
as a function of the number of antennas on the LIS unit.
always achieves the optimal performance, regardless of the number of neighboring LISs and
devices located within their serving area. This result shows a particular characteristic of an LIS
that the accurate CSI is not an important system parameter in both single- and multi-LIS cases,
unlike conventional massive MIMO.
Fig. 9 shows the average NSE with the proposed pilot training length as a function of K,
when T= 50, considering very high-mobility scenarios. The average NSE is defined by the sum
of ergodic SSE for NLISs divided by N. Since the pilot training length tdoes not affect the
asymptotic received SINR as proved in Corollary 1, the uplink channel uses for the data signal
and its SINR decrease as Kincreases due to the the increase in the pilot training overhead and
the intra/inter-LIS interference, respectively. Meanwhile, the NSE improves as Kincreases given
that it stems from the sum rate of NK devices located within the serving area of a network LIS.
Therefore, a fundamental tradeoff exists in terms of the average NSE according to the value of
K. Due to the logarithmic nature of the mutual information, the average NSE increases with
Kwhen Kis small, and starts decreasing with Kwhen Kexceeds a given threshold point, as
shown in Fig. 9. The maximum NSE can be achieved, statistically, by the optimal value of K,
which could be obtained experimentally as K= 20.
Fig. 10 compares the total NSE with the proposed number of scheduled devices, Kopt, and
22
10 15 20 25 30
Number of devices in each LIS (K)
35
45
55
65
75
Average NSE (bps/Hz/LIS)
Single-LIS
Multi-LIS
Optimal point
M=100
M=400
M=900
Fig. 9. Average NSE of an LIS system as a function of the number of scheduled devices in each LIS when T= 50 and
t=K.
that with a fixed number of scheduled devices, as a function of M. This fixed number is obtained
experimentally as K= 20 from Fig. 9, and Kopt can be obtained deterministically according
to each device distribution. The optimal performance is also presented in Fig. 10. This optimal
value is obtained, experimentally, by comparing every NSE over the entire ranges of Kfor
each device distribution. Fig. 10 shows that the NSE with the proposed Kopt is always higher
than that with K= 20 over the entire ranges of Mand their performance gap increases as M
increases. Moreover, the NSE with Kopt nearly achieves the optimal performance obtained from
our extensive simulations.
VI. CO NCLUSIONS
In this paper, we have asymptotically analyzed the performance of an LIS system under
practical LIS environments with a well-defined uplink frame structure and the pilot contaminaion.
In particular, we have derived the asymptotic SSE by considering a practical LIS environment
in which the interference channels are generated by device-specific spatially correlated Rician
fading and channel estimation errors can be caused by pilot contamination based on a practical
uplink frame structure. We have shown that the asymptotic results can accurately and analytically
determine the performance of an LIS without the need for extensive simulations. Moreover, we
23
100 300 500 700 900
Number of antennas on each LIS unit (M)
50
100
150
200
250
300
NSE (bps/Hz)
Optimal
Proposed
Fixed K (K=20)
Single-LIS
Multi-LIS
Fig. 10. Performance comparison between NSE resulting from the optimal K, proposed Kopt , and fixed K, as a function of
the number of antennas on the LIS unit, when T= 50 and t=K.
have studied the performance bound of SSE from the derived asymptotic SSE and obtained
the optimal pilot training length to maximize the SSE, showing that the maximum SSE can be
achieved with a minimum pilot training length of t=K, regardless of the pilot contamination
effect. Furthermore, we have proved that the SSE of a multi-LIS system is bounded by three
factors: pilot contamination, intra-LIS interference, and inter-LIS interference generated from the
LOS path. On the other hand, the pilot contamination and intra/inter-LIS interference generated
from the NLOS path and noise become negligible as Mincreases. Simulation results have
shown that our analytical results are in close agreement with the results arising from extensive
simulations. Our results also show that, unlike conventional massive MIMO system, the effect
of pilot contamination has been shown to become negligible when the inter-LIS interference is
generated from NLOS path. Moreover, we have observed that the SSE of the proposed pilot
training lengths achieve those obtained with the optimal lengths determined by a brute-force
search, both in single- and multi-LIS environments. Furthermore, the maximum value of the
NSE has been shown to be achievable, practically, by using the proposed number of scheduled
devices based on a network LIS. In summary, in order to properly conduct the standardization
process for LIS systems, it will be necessary to take into account the need for an adequate frame
24
structure including the proposed pilot training length and the number of scheduled devices.
APPEN DIX A
PROO F O F LEMM A 1
Given the definition of Xnk =eH
nkhL
nnkk2, we define
xnk =eH
nkhL
nnkk =XN
l6=nrρplk
ρpnk ¯
hH
lnkk hL
nnkk +˜
hH
lnkkhL
nnkk+1
√tρpnk
wH
nkhL
nnkk.(32)
Since ¯
hH
lnkk hL
nnkk is deterministic value without any variance, the terms ˜
hH
lnkk hL
nnkk and wH
nkhL
nnkk
determine the distribution of xnk . From [8], the terms ˜
hH
lnkk hL
nnkk and wH
nkhL
nnkk follow a complex
Gaussian distribution as follows:
˜
hH
lnkk hL
nnkk ∼ CN 0,1
κlnkk + 1 XP
p=1 cH
lnkkphL
nnkk2,(33)
wH
nkhL
nnkk ∼ CN 0,XM
m=1 β2
nnkkm,(34)
Since ˜
hH
lnkk hL
nnkk and wH
nkhL
nnkk are independent random variables, we have
xnk ∼ CN µxnk, σ2
xnk ,(35)
where
µxnk =XN
l6=nrρplk
ρpnk
¯
hH
lnkk hL
nnkk,(36)
σ2
xnk =XN
l6=n
ρplk
ρpnk (κlnkk + 1) XP
p=1 cH
lnkkphL
nnkk2+1
tρpnk XM
m=1 β2
nnkkm .(37)
From the definition of Xnk =|xnk |2, the mean of Xnk can be obtained by µXnk =σ2
xnk +|µxnk |2,
which completes the proof.
APPEN DIX B
PROO F O F LEMM A 2
Given the definition of Ynjk =ˆ
hH
nnkkhnnjk
2
, we define
ynjk =ˆ
hH
nnkkhnnjk = (hL
nnkk)H¯
hnnjk +eH
nk¯
hnnjk +eH
nk + (hL
nnkk)H˜
hnnjk .(38)
25
For notational simplicity, we define that yLL
njk = (hL
nnkk)H¯
hnnjk ,yEL
njk =eH
nk ¯
hnnjk , and yEN
njk =
eH
nk + (hL
nnkk)H˜
hnnjk . Then, yLL
njk is the deterministic value depending on the locations of the
devices. Also, yEL
njk is obtained similarly as (35) as follows: yEL
njk ∼ CNµxEL
njk , σ2
xEL
njk ,where
µyEL
njk =XN
l6=nrρplk
ρpnk
¯
hH
lnkk ¯
hnnjk ,(39)
σ2
yEL
njk =XN
l6=n
ρplk
ρpnk (κlnkk + 1) XP
p=1 cH
lnkkp ¯
hnnjk 2+1
tρpnk XM
m=1 β2
nnjkm.(40)
Next, a random variable yEN
njk can be expressed as follows:
yEN
njk =¯qH
nk ˜
hnnjk +
N
X
l6=nrρplk
ρpnk
˜
hH
lnkk ˜
hnnjk +wH
nk ˜
hnnjk
√tρpnk
,(41)
where
¯qnk =hL
nnkk +
N
X
l6=nrρplk
ρpnk
¯
hlnkk .(42)
The terms ¯qH
nk˜
hnnjk and wH
nk˜
hnnjk in (41) can be obtained, respectively, similarly as (33) and
using the Lyapunov central limit theorem from [8], as follows:
¯qH
nk ˜
hnnjk ∼ CN 0,1
κnnjk + 1
P
X
p=1 ¯qH
nkcnnjkp2!,(43)
sMtρpnk (κnnjk + 1)
PM,P
m,p αNL
nnjkplNL
nnjkm2wH
nk˜
hnnjk
d
−−−→
M→∞ CN (0,1) ,(44)
where “ d
−−−→
M→∞ ” denotes the convergence in distribution. Also, ˜
hH
lnkk ˜
hnnjk in (41) is given by
˜
hH
lnkk ˜
hnnjk =
gH
lnkk R1/2
lnkk HR1/2
nnjk gnnjk
p(κlnkk + 1) (κnnjk + 1) .(45)
Given random vectors glnkk and gnnjk , those elements are independent each other and identically
follow CN (0,1). Similarly as (44), we have
p(κlnkk + 1) (κnnj k + 1)
R1/2
lnkk HR1/2
nnjk
F
˜
hH
lnkk ˜
hnnjk
d
−−−→
M→∞ CN (0,1) .(46)
Since the terms ¯qH
nk˜
hnnjk ,˜
hH
lnkk ˜
hnnjk , and wH
nk˜
hnnjk in (41) are independent of each other, we
have the following convergence in distribution:
1
σyEN
njk
yEN
njk
d
−−−→
M→∞ CN (0,1) ,(47)
26
where
σ2
yEN
njk =1
κnnjk + 1
P
X
p=1 ¯qH
nkcnnj kp 2+
N
X
l6=n
ρplk
R1/2
lnkk HR1/2
nnjk
2
F
ρpnk (κlnkk + 1) +
M,P
X
m,p αNL
nnjkplNL
nnjkm2
Mtρpnk
.
(48)
We can observe from (38) that yLL
njk ,yEL
njk , and yEN
njk are independent of each other. Thus, ynjk
asymptotically follows
1
σynjk ynjk −µynj k d
−−−→
M→∞ CN (0,1) ,(49)
where
µynjk = (hL
nnkk)H¯
hnnjk +
N
X
l6=nrρplk
ρpnk
¯
hH
lnkk ¯
hnnjk ,(50)
σ2
ynjk =σ2
yEL
njk +σ2
yEN
njk .(51)
From the definition of Ynj k =|ynjk|2, the mean of Ynjk follows µYnjk −¯µYnjk −−−→
M→∞ 0, where
¯µYnjk =σ2
ynjk +µynjk 2.(52)
Similarly, given that Ylnjk =ˆ
hH
nnkkhlnjk
2
and ylnjk =ˆ
hH
nnkkhlnjk, the mean of Ylnjk follows
µYlnjk −¯µYlnjk −−−→
M→∞ 0, where
¯µYlnjk =σ2
ylnjk +µylnjk 2,(53)
and
µylnjk = (hL
nnkk)H¯
hlnjk +
N
X
l6=nrρplk
ρpnk
¯
hH
lnkk ¯
hlnjk ,(54)
σ2
ylnjk =σ2
yEL
lnjk +σ2
yEN
lnjk ,(55)
σ2
yEL
lnjk =
N
X
l6=n
ρplk
ρpnk(κlnkk +1)
P
X
p=1 cH
lnkkp ¯
hlnjk 2+1
tρpnk
M
X
m=1
β2
lnjkm,(56)
σ2
yEN
njk =1
κlnjk + 1
P
X
p=1 ¯qH
nkclnjkp2+
N
X
l6=n
ρplk
R1/2
lnkk HR1/2
lnjk
2
F
ρpnk (κlnkk + 1) +
M,P
X
m,p αNL
lnjkplNL
lnjkm2
Mtρpnk
,
(57)
which completes the proof.
27
APPEN DIX C
PROO F O F LEMM A 3
Given the definition of Znk =(hL
nnkk)H+eH
nk2, we define
znk = (hL
nnkk)H+eH
nk = (hL
nnkk)H+
N
X
l6=nrρplk
ρpnk ¯
hH
lnkk +˜
hH
lnkk +1
√tρpnk
wH
nk,(58)
where znk ∈CM= [znk1,·· · , znkM ]and
znkm =βL
nnkkm h∗
nnkkm +
N
X
l6=nrρplk κlnkk
ρpnk (κlnkk + 1) βL
lnkkm h∗
lnkkm +gH
lnkkrH
lnkkm
√κlnkk +1
√tρpnk
w∗
nkm.
Since gH
lnkk rH
lnkkm is calculated by the sum of Pindependent complex Gaussian random variables,
gH
lnkk rH
lnkkm is also a complex Gaussian random variable. Thus, we have znkm ∼ CN µznkm , σ2
znkm ,
where
µznkm =βL
nnkkm h∗
nnkkm +
N
X
l6=nrρplk κlnkk
ρpnk (κlnkk + 1) βL
lnkkm h∗
lnkkm (59)
σ2
znkm =
N
X
l6=n
ρplk |rlnkkm|2
ρpnk (κlnkk + 1) +1
tρpnk
.(60)
From the definition of Znk =|znk |2, we finally have µZnk =PM
m=1 σ2
znkm +|µznkm |2,which
completes the proof.
APPEN DIX D
PROO F O F LEMM A 4
Given the definition of Ink from (12), we have
Ink
M2=ρnk
eH
nkhL
nnkk
M
2
+
K
X
j6=k
ρnj
ˆ
hH
nnkkhnnjk
M
2
+
N
X
l6=n
K
X
j=1
ρlj
ˆ
hH
nnkkhlnjk
M
2
+
(hL
nnkk)H+eH
nk
M
2
=ρnk
xnk
M
2+
K
X
j6=k
ρnj
ynjk
M
2+
N
X
l6=n
K
X
j=1
ρlj
ylnjk
M
2+
znk
M
2.(61)
In order to analyze the scaling law of σ2
Ink /M4, we determine the scaling laws of σ2
xnk /M2,
σ2
ynjk /M2,σ2
ylnjk /M2, and σ2
znkm /M2according to M. First, we determine the scaling law of
σ2
xnk /M2from (37). From (3), the correlation vector clnkkp is normalized by √Mand cH
lnkkp hL
nnkk
in (37) is calculated by the sum of Melements. Then, both cH
lnkkphL
nnkk2and PM
m=1 β2
nnkkm in
(37) increase with O(M)and thus, σ2
xnk /M2decreases with O(1/M)as Mincreases. Therefore,
28
we have σ2
xnk /M2−−−→
M→∞ 0. Next, we analyze σ2
ynjk /M2from (51) for large M. From (40), σ2
yEL
njk
follows O(M)similarly as σ2
xnk , and therefore, σ2
yEL
njk
/M2follows O(1/M)as Mincreases. To
determine the scaling law of σ2
yEN
njk
/M2, we analyze the terms ¯qH
nkcnnj kp 2,
R1/2
lnkk HR1/2
nnjk
2
F,
and PM,P
m,p αNL
nnjkplNL
nnjkm2/M in (48) for large M. Given that the correlation vector, cnnjkp, and
matrices, R1/2
lnkk and R1/2
nnjk , are normalized by √M, the terms ¯qH
nkcnnj kp 2,
R1/2
lnkk HR1/2
nnjk
2
F,
and PM,P
m,p αNL
nnjkplNL
nnjkm2/M follow, respectively, O(M),O(1), and O(1) as Mincreases.
Consequently, σ2
yEN
njk
/M2decreases with O(1/M)and we have σ2
ynjk /M2−−−→
M→∞ 0. Similarly,
σ2
ylnjk /M2from (55) also converges to zero as M→ ∞. Finally, we determine the scaling law
of σ2
znkm /M2from (60). Given that rlnkkm is a correlation vector normalized by √M,|rlnkkm|2is
calculated by the sum of Pelements divided by M. Hence, σ2
znkm /M2decreases with O(1/M3)
as Mincreases, and the variance of |znk /M|2eventually converges to zero as M→ ∞. In
conclusion, we have σ2
Ink /M4−−−→
M→∞ 0, which completes the proof.
APPEN DIX E
PROO F O F THEO REM 1
We begin with the definition of Rnk as follows:
Rnk = log 1 + ρnkSnk
Ink = log (ρnkSnk +Ink )
|{z }
RL
nk
−log Ink
|{z }
RR
nk
.(62)
Here, RL
nk can be expressed as
RL
nk = log 1 + Ink −¯µInk
ρnkSnk + ¯µInk + log (ρnk Snk + ¯µInk )
=Ink −¯µInk
M2log1 + (Ink −¯µInk )/M2
(ρnkSnk + ¯µInk )/M2M2
Ink−¯µInk + log (ρnk Snk + ¯µInk ).(63)
Since Ink/M2−¯µInk /M2−−−−→
M→∞ 0from Lemma 4 and Snk −¯pnk −−−−→
M→∞ 0where ¯pnk =M2p2
nk
16π2L4,
we have the following asymptotic convergence using the exponential function definition ex=
lim
n→∞ (1 + x/n)n:RL
nk −¯
RL
nk −−−→
M→∞ 0, where
¯
RL
nk =Ink −¯µInk
ρnk ¯pnk + ¯µInk + log (ρnk ¯pnk + ¯µInk ).(64)
29
Similarly, we have RR
nk −¯
RR
nk −−−→
M→∞ 0, where
RR
nk =Ink −¯µInk
M2log1 + (Ink −¯µInk )/M2
¯µInk /M2M2
Ink−¯µInk + log ¯µInk,
¯
RR
nk =Ink −¯µInk
¯µInk
+ log ¯µInk .(65)
From (64) and (65), we thus have
¯
RL
nk −¯
RR
nk =1−Ink
¯µInk ρnk ¯pnk
ρnk ¯pnk + ¯µInk
+ log 1 + ρnk ¯pnk
¯µInk .(66)
Given that Ink/M 2−¯µInk /M2−−−−→
M→∞ 0,¯
Rnk can be derived as follows:
¯
Rnk = log 1 + ρnk ¯pnk
¯µInk = log 1 + M2ρnkp2
nk
16π2L4¯µInk ,(67)
where Rnk −¯
Rnk −−−→
M→∞ 0. Since the pilot training period consumes part of uplink channels for
channel estimation, the uplink SSE can be ultimately obtained from (13) as
RSSE
n−1−t
TK
X
k=1
log 1 + M2ρnk p2
nk
16π2L4¯µInk −−−−→
M→∞ 0,(68)
which completes the proof.
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