In this paper, we investigate the performance evaluation of three dimensional (3D) multipleinput multipleoutput (MIMO) systems with an adjustable base station (BS) antenna tilt angle and zeroforcing (ZF) receivers in Ricean/Lognormal fading channels. In particular, we take the lognormal shadow fading, 3D antenna gain with antenna tilt angle and pathloss into account. First, we derive a closedform lower bound on the sum rate, then we obtain the optimal BS antenna tilt angle based on the derived lower bound, and finally we present linear approximations for the sum rate in high and lowSNR regimes, respectively. Based on our analytical results, we gain valuable insights into the impact of key system parameters, such as the BS antenna tilt angle, the Ricean
K
factor and the radius of cell, on the sum rate performance of 3D MIMO with ZF receivers.
1. Introduction
M
ultipleinput multipleoutput (MIMO) technology can significantly improve the transmission data rate and reliability of wireless communication systems without requiring extra bandwidth or transmit power
[1]
. In particular, multiuser MIMO (MUMIMO) systems, where a multiantenna base station (BS) serves a number of users in the same timefrequency resource, have gained substantial interest for the spatial multiplexing gains
[2]
. Currently, most existing studies on MUMIMO techniques mainly focus on twodimensional (2D) channel model only consider the horizontal dimension of the BS antenna pattern. They ignore the effect of elevation in the vertical dimension. To make the channel model more practical, threedimensional (3D) channel is introduced. In the existing 3DMIMO models, an approximated 3D radiation pattern is used to address the antenna tilt angle in vertical planes
[3

5]
.
An important topic in MUMIMO communication theory is to obtain theoretical results of the system sum rate. Unfortunately, many of the existing works focus only on the simple Rayleigh fading channels
[6

8]
. However, many realistic channels are characterized by a deterministic or lineofsight (LoS) component. In such scenarios, Ricean fading channel model is more useful and practical
[9]
. The sum rate of MIMO system with Ricean fading has been discussed in
[10

13]
. An analytical framework of the capacity was provided for uncorrelated Ricean fading MIMO channels in
[10]
. In
[11
,
12]
, several upper and lower bounds were provided for uncorrelated Ricean fading MIMO system, where the transmitter has knowledge of statistical properties of the fading channel but not the instantaneous channel state information (CSI). In
[13]
, the sum rate analysis of MIMO systems with minimum meansquared error (MMSE) was performed, some simplified closedform expressions for the achievable sum rate were derived in the asymptotic regimes of high and low signaltonoise ratios (SNR). Unfortunately, in above mentioned studies, only smallscale fading was considered; little attention was paid to the realistic effect of largescale fading, mainly due to the difficulty in analyzing the statistical distribution of largescale fading. Furthermore, in practical scenarios, the distribution of the largescale fading might largely vary in different scenarios, such as urban and open areas. The study on the effect of the largescale fading in MIMO system is still open. Motivated by this fact, in this paper we focus on investigating the performance analysis of the 3DMIMO system over composite fading channels including the pathloss, the lognormal (LN) shadow fading and 3D antenna gain.
In this paper, we introduce a general analytical framework for investigating the sum rate of Ricean/lognormal (RCLN) 3DMIMO systems with ZF receivers by exploiting BS antenna tilt angle. We derive a lower bound sum rate expression for the whole SNR regime, and further study the asymptotic approximation of the sum rate to gain more insights. Moreover, the optimal BS antenna tilt angle is investigated for maximizing the system sum rate. Simulation results show that the derived lower bound and high/lowSNR approximations are applicable to arbitrary Ricean
K
factor and remain relatively tight across the entire BS antenna tilt angle range and SNR regimes. Our analytical results are quite informative and insightful to characterize the impact of the shadow fading, the Ricean
K
factor, the BS antenna tilt angle and the pathloss on the system sum rate.
The rest of the paper is organized as follows. In Section 2, we specify the 3DMIMO system model. In Section 3, we derive the lower bounds, low/highSNR approximations of the system sum rate and the optimal BS antenna tilt angle. In Section 4, the numerical results are shown with discussion. Finally, we conclude in Section 5.
Notations:
For a matrix
A
,
tr(A)
,
A
^{H}
,
A
^{T}
, and
A
^{†}
denote the trace, conjugate transpose, transpose, and pseudo inverse of
A
, respectively. The symbol
CN
(
M
,
∑
) denotes a complex Gaussian matrix with mean
M
and covariance
∑
. The (
i
,
j
)
^{th}
entry of matrix
A
is denoted by
[A]
_{i, j}
, while
A
_{i}
is A with
i
^{th}
column removed. The symbol
E[ ]
stands for the expectation operation. The symbol
Γ( )
stands for the Gamma function,
Ψ
( )
is Euler's digamma function, and
E_{i}(x)
=
is the exponential integral function. Finally,
_{P}F_{q}
( ) is the generalized hypergeometric function with
p
,
q
nonnegative integers
[14]
.
2. 3DMIMO System Model
 2.1 Channel Model
Considering an uplink singlecell MUMIMO system depicted in
Fig. 1
, the system consists of a BS equipped with
M
antennas that receive data from
N
singleantenna users. It is assumed that
M
≥
N
. The users transmit their data in the same timefrequency resource. All users are distributed in an
L
floor building. For the sake of effective analysis, the floor penetration loss and the surface reflection loss are not taken into account. Assuming no CSI at the transmitters, the available average transmit power
P
is distributed uniformly amongst all data streams. Then, the
M
×1 received signal vector at the BS is
where
G
∈
C^{M×N}
is the MIMO channel matrix between the BS and the
N
users,
s
∈
C
^{N×1}
is the vector containing the transmitted symbols of the
N
users which are draw from a zeromean Gaussian codebook with unit average power. And
n
∈
C
^{N×1}
is the complex additional white Gaussian noise (AWGN) vector, such that
CN
(0,
N
_{0}
I
_{N}
).
A 3DMIMO system with N users located in an Lfloor building.
The channel matrix
G
includes independent small and largescale fading, it can be expressed by
where
H
∈
C^{M×N}
is the smallscale fading between the
N
users and the BS. In Ricean fading channel, the entries of
H
are nonzero mean complex Gaussian random variables (RVs), it consists of two parts, namely, a deterministic component
H
_{L}
corresponding to the LoS signal and a Rayleigh distributed random component
H
_{ω}
accounting for the scattered signals.
H
can be written as
where
K
stands for the Ricean
K
factor, it denotes the ratio between the deterministic (specular) and the random (scattered) energies. The term
H
_{L}
is typically associated with a LoS or a diffracted component and thus, assuming far field transmission, it is can expressed as
where
a
_{r}
(
Θ_{r}
) =
and
a
_{t}
(
Θ_{t}
) =
are the specular array response at the receiver and transmitter, respectively, and
d
is the antenna spacing in wavelengths,
Θ_{r}
,
Θ_{t}
are the angles of arrival and departure of the specular component, respectively. The scattered component of
H
is denoted by
H
_{ω}
, where the entries of
H
_{ω}
are independent and identically distributed (i.i.d)
CN
(0,1) RVs.
The diagonal matrix
∈
R^{N×N}
represents the largescale fading and can be written as
where
,
g_{n}
(
Θ
_{tilt}
) and
ξ_{n}
are the pathloss, the 3D antenna gain and the shadow fading coefficient corresponding to the
n
^{th}
entry, respectively. The shadow fading coefficient
ξ_{n}
is modeled as an independent LN random variable (RV), namely,
ξ_{n}
~LN
, or
where
η
=10/ln10, while
μ_{n}
and
σ_{n}
are the mean and standard deviation (both in dB) of the RV 10lg
ξ_{n}
, respectively.
The considered channel model in (2) is a simplified 3DMIMO model of the commonly used kathrein antenna 742215
[3]
. To model the channel between the users and the BS, we assume that the height of the BS is much larger than that of the user. The antenna gain of the
n
^{th}
user,
g_{n}
(
Θ
_{tilt}
) depends on the relative angles between the direct line from the user to the BS and the main lobe of the antenna pattern, both in horizontal (azimuth,
ϕ_{n}
) and vertical (elevation,
θ_{n}
) directions. (
x_{B}
,
y_{B}
,
z_{B}
) and (
x_{n}
,
y_{n}
,
z_{n}
) denote the coordinates of the BS and the
n
^{th}
user, respectively. We denote Δ
x_{n}
=
x_{n}

x_{B}
and Δ
y_{n}
=
y_{n}

y_{B}
as relative distances between the
n
^{th}
user and the BS in the x and y coordinate, respectively. Similarly, Δ
z_{n}
=
z_{n}

z_{B}
is the height difference between the
n
^{th}
user and the BS.
The distance between the
n
^{th}
user and the BS is denoted as
d_{n}
, it can be calculated as
According to this model, the horizontal antenna radiation attenuation adopted by the 3GPP [3] is expressed in dB scale as
Similarly, the vertical antenna attenuation can be expressed in dB scale as
where
Φ_{n}
=arctan(Δ
y_{n}
/Δ
x_{n}
) denotes the horizontal angle between the BS antennas boresight and the
n
^{th}
user in the horizontal plane, and
Θ_{n}
=arctan
indicates the vertical angle between the horizon and the line connecting the BS to the
n
^{th}
user. In addition,
θ
_{tilt}
denotes the BS antenna tilt angle which is adjustable. Moreover,
A_{m}
represents the maximum attenuation of the BS antennas. The halfpower beamwidth (HPBW) in the horizontal and vertical planes are denoted as
ϕ
_{3dB}
and
θ
_{3dB}
, respectively.
Let us denote
G_{m}
(in dB) as the maximum antenna gain at the antenna boresight. Then, after combining the antenna attenuation and the maximum antenna gain, the resultant antenna gain in dB scale for the
n
^{th}
user with horizontal angle
ϕ_{n}
and the vertical angle
θ_{n}
can be formulated as
The resultant antenna gain in the linear scale can be approximated as
The approximation is valid when
G_{H}
(
ϕ_{n}
)+
G_{V}
(
θ_{n}
,
θ_{tilt}
)≤
A_{m}
or
A_{m}
is large enough. For example, for the antenna model in
[15]
,
A_{m}
is given as
A_{m}
= 20dB. In this case, the difference between 10
^{0.1Gn(θtilt)}
and the approximated value
g_{n}
(
θ
_{tilt}
) is less than 1%. The condition corresponds to the typical cell deployments.
With the LN shadow fading coefficient
ξ_{n}
, 3D antenna gain
g_{n}
(
θ_{tilt}
), and distance dependent pathloss
, the largescale fading coefficient is finally expressed as
It is noteworthy that the largescale fading
β_{n}
(
θ_{tilt}
) is treated as a random variable instead of a given value and in the following we will analyze the ergodic sum rate over
β_{n}
(
θ_{tilt}
).
 2.2 User Distribution Model
In the considered 3DMIMO system, to analyze the collective behavior of users in the
L
floor building, the building is approximated as a cylinder with a radius
R
, the height of floortofloor is set to be
h_{f}
. The radius of cell (i.e. the distance between BS and the center of the
L
floor building) is
D
. We consider the horizontal distribution in each floor and vertical distribution in different floors.
For vertical distribution, it is modeled that users on different floors follow some rules. The user distribution is discrete. We assume the user number in the
l
^{th}
floor is
N_{l}
, for
l
=1,…,
L
, and
For horizontal distribution, we consider uniform distribution, i.e., the
N_{l}
users in the
l
^{th}
floor are assumed to be i.i.d on the circular floor. The distribution of the users along the radius of the floor can be modeled as
[16]
3. Achievable Sum Rate of 3DMIMO System with ZF Receivers
In this section, we firstly derive a closedform lower bound on the sum rate of 3DMIMO with ZF receivers. Furthermore, the optimal BS antenna tilt angle
is obtained to achieve maximum sum rate. Finally, the linear approximations are presented in high and lowSNR regimes, respectively.
We assume that BS has perfect CSI, i.e., it knows
G
, then the ZF filter is expressed as
[17]
. The instantaneous received SNR at the
n
^{th}
ZF output (1≤
n
≤
N
) is
[18]
where
is the average transmit SNR. Note that the second equation follows from the fact that
and
is a diagonal matrix. The achievable sum rate is then determined as
where the expectation E[ ] is taken over all channel realizations of
H
,
. Due to randomness of smallscale fading
H
and largescale fading
, it is difficult to obtain the exact expression of R(
θ
_{tilt}
,
γ
). We circumvent this problem in the following by deriving some tractable bounds and approximations on the sum rate of 3DMIMO system with ZF receivers.
 3.1 Closedform Bounds on the Achievable Sum Rate
In this subsection, we turn to derive a novel closedform lower bound of the achievable sum rate of 3DMIMO with ZF receivers. The key result is summarized in the following proposition.
Proposition 1.
The achievable sum rate of 3DMIMO ZF receivers in RCLN fading channels is lower bounded by R
_{L}
(
θ
_{tilt}
,
γ
)
where Δ=
KMN
, Δ
_{1}
=
KM
(
N
1), and
Proof
: Please see Appendix
A
.
Remark 1
: It is easy to see that the lower bound of the sum rate monotonically grows with the mean of the LN shadow fading, and 3D antenna gain. We can get the maximum 3D antenna gain by optimizing the BS antenna tilt angle, and then obtain the maximum sum rate of 3DMIMO system.
 3.2 Optimization for the BS Antenna Tilt Angle
In this subsection, we aim to derive the optimal BS antenna tilt angle to maximize the sum rate in (14). Note that there are multiple random variables in (14), it is difficult to obtain a closedform expression for (14) with clear physical insights. For tractability, we turn to optimize the BS antenna tilt angle regarding the lower bound of the sum rate in (15). The optimal BS antenna tilt angle is given by the following theorem.
Theorem 1.
For 3DMIMO systems with ZF receivers in RCLN fading channel, the optimal BS antenna tilt angle
(regarding the lower bound) is the mean value of vertical angles of the
N
users, i.e.,
Proof
: Please see Appendix
B
.
Remark 2:
The optimal antenna tilt angle
can be easily obtained by averaging the vertical angles of the all users. No complicated calculation is required.
 3.3 High SNR Analysis
In order to derive the diversity order of the system, we now analyze the sum rate performance in the highSNR regime. We can invoke the affine sum rate expansion in the analysis of MIMO systems
[19]
as follow:
where S
_{∞}
is the highSNR slope in bits/s/Hz per 3dB units, and L
_{∞}
is the highSNR power offset, in 3dB units, given by
Proposition 2.
At the high SNR regime, the sum rate of 3DMIMO with ZF receivers in RCLN fading channel can be expressed with parameters in the general form (18) as follows
Proof
: Please see Appendix
C
.
Remark 3:
It can be observed that the S
_{∞}
in (19) verifies that the highSNR sum rate increases linearly with the minimun number of antennas, which agrees with
[18]
. From the ∞L in (20), we can infer that the small and largescale fading terms are decoupled in the high SNR regime. Furthermore, the greater the distances between the BS and users
d_{i}
, the much more effectively reduce the system sum rates due to the increased pathloss.
 3.4 Low SNR Analysis
A wide variety of digital communication systems operate at low power where both spectral efficiency and the energyperbit can be very low. The low SNR analysis can provide a useful reference in understanding the system performance at low SNR regime. In this subsection, we examine the achievable sum rate at the low SNR regime. At low SNR, it has proved useful to investigate the sum rate of MIMO systems in terms of the normalized transmit energy per information bit
, rather than the persymbol SNR
At lowSNR, the sum rate of MIMO systems can be well approximated for low
by the following expression
[19]
where
_{min}
and S
_{0}
are the minimum normalized energy per information bit required to convey any positive rate reliably and wideband slope, respectively. According to
[20]
, these two key parameters can be obtained from R(
γ
) via
where
(0) and
(0) denote the first and secondorder derivatives of the sum rate in (14) with respect to SNR (i.e.,
γ
), respectively.
Proposition 3.
For 3DMIMO systems with ZF receivers in RCLN fading channels, the minimum energy per information bit and the wideband slope are respectively given by
where
.
Proof
: Please see Appendix
D
.
Remark
4:
It is clear to see that
_{min}
in (23) depends on the number of BS antennas
M
, the number of users
N
, the largescale fading mean parameter
c_{n}
and the covariance matrix of
H
,
. For fixed
M
, having more users is not beneficial for ZF receivers since the
increases due to the additional power that is required to cancel out the exact interference. Note that the S
_{0}
is by definition greater than one.
4. Numerical Results
In this section, we present various simulations to furtherly verify the derived analytical results. The BS is located at the origin of spatial coordinates. We assume that the uses are located in the 3floor building, the horizontal distribution is modeled as (12) and the vertical distribution is uniform, namely, the user number in each floor is equal,
N_{l}
=
N/L
for
l
=1,…,
L
. We set the height of floortofloor
h_{f}
=5m, the height of BS
h_{B}
=30m, the radius of the floor
R
=50m, the distance between the BS and the Lfloor building
D
is variable. All the MonteCarlo simulation results were obtained by averaging over 1×10
^{4}
independent channel realizations. For rank1 Ricean fading MIMO channels, we assume
θ_{r}
=
θ_{t}
=
,
d
=0.5. The antenna parameters are set to be
θ
_{3dB}
=6.2°,
φ
_{3dB}
=65°,
G
_{m}
=20dB. Other channel parameters used in the simulations are set to be as follows: the pathloss exponent
υ
=4, the standard deviation of
ξ_{n}
,
σ_{n}
=2dB, the mean of
ξ_{n}
,
μ_{n}
=4dB.
We firstly assess the sum rate performance of 3DMIMO system against different parameters (the radius of cell
D
, the BS antenna tilt angle
θ
_{tilt}
).
In
Fig. 2
and
Fig. 3
, the simulated sum rates of 3DMIMO systems with ZF receivers are compared with their lower bounds, highSNR approximations and lowSNR approximations, respectively. Results are presented for three different the radius of cell. In all cases, we can clearly see a precise match between the simulated results and the analytical results. At the same time, the radius of cell
D
also shows impact on the optimal BS antenna tilt angle
, which achieves the maximum sum rate of the system. More specifically, the sum rate increases with
θ
_{tilt}
before the optimal BS antenna tilt angle
, and then decreases with the
θ
_{tilt}
further increases since the radiation angle of the BS deviates from users.
Simulated sum rate, lower bound and highSNR approximation versus the BS antenna tilt angle θ_{tilt} in the high SNR regime (M = 10, N = 6, K = 1, γ= 20dB).
Simulated sum rate, lower bound and lowSNR approximation versus the BS antenna tilt angle θ_{tilt} in the low SNR regime (M = 10, N = 6, K = 1, = 20dB).
To capture the effect of Ricean
K
factor on the system sum rate, we further compared the simulated sum rate with their lower bounds, high and lowSNR approximations over different Ricean
K
factor in the high and lowSNR regime in
Fig. 4
and
Fig. 5
, respectively. A quite good match between the simulated results and analytical results can be observed with various Ricean
K
factors. The main observation is that a higher Ricean
K
factor will decrease the system sum rate, since the specular components of channel increase, especially
K
=0 corresponds to the Rayleighfading. From the results of
Fig. 4
, we can observe that the simulated results are extremely accurate in comparison with the lower bounds and highSNR approximations in the entire high SNR regime. In
Fig. 5
, it can be seen that the match of simulated results, lower bounds and lowSNR approximation is very good if the SNR of interest is sufficiently low (i.e., below 60 bits/s/Hz of sum rate).
Simulated sum rate, lower bound and highSNR approximation versus the transmit SNR γ in the high SNR regime (M = 10, N = 6, D = 400, θ_{tilt}=4°).
Simulated sum rate, lower bound and lowSNR approximation versus the transmit energy per bit in the low SNR regime (M = 10, N = 6, D = 400, θ_{tilt}=4°)
5. Conclusion
This paper presented a detailed sum rate analysis of 3DMIMO ZF receivers utilizing the BS antenna tilt angle in RCLN fading channels. Specifically, closedform lower bounds, high/lowSNR approximations on the sum rate were derived, which is applicable to the scenarios with arbitrary Ricean
K
factor and is sufficiently tight across the entire BS antenna tilt angle and SNR regimes. In particular, the optimal BS antenna tilt angle to maximize the sum rate of the 3DMIMO system was also obtained. More importantly, these analytical results encompass the small and largescale fading (include LN shadow fading, 3D antenna gain and pathloss) models of practical interest. Finally, we examined in detail (both theoretically and via numerical simulations) the impact of the Ricean
K
factor, the radius of cell
D
and the BS antenna tilt angle
θ
_{tilt}
on the performance of system.
BIO
Fangqing Tan received the B.E degree in Electronics and Information Engineering from Hebei Polytechnic University in 2009 and received the M.S. degree in communication and information system from Chongqing University of Post and Telecommunications in 2012. He is currently pursuing the Ph.D. degree in Beijing University of Post and Telecommunications, Beijing, China. His research interests include Massive MIMO systems and digital signal processing in wireless communications.
Hui Gao received his B. Eng. degree in Information Engineering and Ph.D. degree in Signal and Information Processing from Beijing University of Posts and Telecommunications (BUPT), Beijing, China, in July 2007 and July 2012, respectively. From May 2009 to June 2012, he also served as a research assistant for the Wireless and Mobile Communications Technology R&D Center, Tsinghua University, Beijing, China. From Apr. 2012 to June 2012, he visited Singapore University of Technology and Design (SUTD), Singapore, as a research assistant. From July 2012 to Feb. 2014, he was a Postdoc Researcher with SUTD. He is now with the School of Information and Communication Engineering, Beijing University of Posts and Telecommunications (BUPT), as an assistant professor. His research interests include massive MIMO systems, cooperative communications, ultrawideband wireless communications.
Xin Su received the M.S. and Ph.D. degrees in Electronic Engineering from UESTC (University of Electronic Science and Technology of China) in 1996 and 1999, respectively. Currently he is a full professor of the Research Institute of Information Technology in Tsinghua University. He is also the chairman of IMT2020(5G) wireless technology work group in MIIT (Ministry of Industry and Information Technology of People’s Republic of China) and vice chairman of the Innovative Wireless Technology Work Group of CCSA (China Communications Standards Association). His research interests include broadband wireless access, wireless and mobile network architecture, selforganizing network, software defined radio, and cooperative communications. Dr. Su has published over 100 papers in the core journals and important conferences, and owned more than 30 patents.
Tiejun Lv received the M.S. and Ph.D. degrees in electronic engineering from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 1997 and 2000, respectively. From January 2001 to December 2002, he was a Postdoctoral Fellow with Tsinghua University, Beijing, China. From September 2008 to March 2009, he was a Visiting Professor with the Department of Electrical Engineering, Stanford University, Stanford, CA. He is currently a Full Professor with the School of Information and Communication Engineering, Beijing University of Posts and Telecommunications (BUPT). He is the author of more than 100 published technical papers on the physical layer of wireless mobile communications. His current research interests include signal processing, communications theory and networking. Dr. Lv is also a Senior Member of the Chinese Electronics Association. He was the recipient of the Program for New Century Excellent Talents in University Award from the Ministry of Education, China, in 2006.
Gesbert D.
,
Kountouris M.
,
H. R. W.
,
Chae C. B.
,
Salzer T.
2007
“Shifting the MIMO Paradigm,”
IEEE Signal Processing Magazine
24
36 
46
DOI : 10.1109/MSP.2007.904815
Vishwanath S.
,
Jindal N.
,
Goldsmith A.
2003
“Duality, achievable rates, and sumrate capacity of Gaussian MIMO broadcast channels,”
IEEE Transactions on Information Theory
49
2658 
2668
DOI : 10.1109/TIT.2003.817421
2010
3GPP TR 36.814 V9.0.0, “Further advancements for EUTRA physical layer aspects,”Technology Report
Li X. W.
,
Li L. H.
,
Xie L.
2014
“Achievable Sum Rate Analysis of ZF Receivers in 3D MIMO Systems,”
KSII Transactions on Internet and Information Systems
8
1368 
1389
DOI : 10.3837/tiis.2014.04.012
Liu H. J.
,
Gao H.
,
Zhang C.
,
Lv T. J.
2015
“LowComplexity Joint Antenna Tilting and User Scheduling for LargeScale ZF Relaying,”
IEEE Signal Processing Letter
22
361 
365
DOI : 10.1109/LSP.2014.2360521
Zhang Q. T.
,
Cui X. W.
,
Li X. M.
2005
“Very tight capacity bounds for MIMOcorrelated Rayleighfading channels,”
IEEE Transactions on Wireless Communications
4
681 
688
DOI : 10.1109/TWC.2004.842959
Tan F. Q.
,
Gao H.
,
Lv T. J.
,
Zeng J.
2014
“Achievable Sum Rate Analysis of ZF Receivers in 3D MIMO with Rayleigh/Lognormal Fading Channels,”
in Proc. of the IEEE GLOBECOMWorkshop
Austin, TX, USA
900 
905
Wang D.
,
Wang J.
,
You Xiaohu
,
Wang Yan
2013
“Spectral efficiency of distributed MIMO systems,”
IEEE Journal on Selected Areas in Communications
31
2112 
2127
DOI : 10.1109/JSAC.2013.131012
Matthaiou M.
,
de Kerret P.
,
Karagiannidis G. K.
2007
“Mutual Information Statistics and Beamforming Performance Analysis of Optimized LoS MIMO Systems,”
IEEE Transactions on Communications
58
3316 
3329
DOI : 10.1109/TCOMM.2010.091710.090770
Alfano G.
,
Lozano A.
,
Tulino A. M.
“Mutual information and eigenvalue distribution of MIMO Ricean channels,”
in Proc. Of the IEEE ISITA
Parma, Italy
2004
Jayaweera S. K.
,
Poor H. V.
“MIMO capacity results for Rician fading channels,”
in Proc. of the IEEE GLOBECOM
San Francisco, CA, USA
2003
1806 
1810
Jayaweera S. K.
,
Poor H. V.
2005
“On the capacity of multipleantenna systems in rician fading,”
IEEE Transactions on Wireless Communications
4
1102 
1111
DOI : 10.1109/TWC.2005.846970
McKay M. R.
,
Collings I. B.
,
Tulino A. M.
2010
“Achievable Sum Rate of MIMO MMSE Receivers: A General Analytic Framework,”
IEEE Transactions on Information Theory
56
396 
410
DOI : 10.1109/TIT.2009.2034893
Gradshteyn I. S.
,
Ryzhik I. M.
2007
“Table of Integrals, Series, and Products,”
seventh
Academic Press
2009
“Guidelines for evaluation of radio interface technologies for IMTadvanced,”
Geneva, Switzerland
Rep. M.21351
Yang A.
,
He Z. W.
,
Xing C. W.
,
Fei Z. S.
,
Kuang J. M.
2015
“The Role of LargeScale Fading in Uplink Massive MIMO Systems,”
IEEE Transactions on Vehicular Technology
Clerckx Bruno
,
Oestges Claude
2013
MIMO Wireless Networks: Channels, Techniques and Standards for Multiantenna, Multiuser and Multicell Systems
Academic Press
Paulraj Arogyaswami
,
Nabar Rohit
,
Gore Dhananjay
2003
Introduction to spacetime wireless communications
Cambridge university press
Shitz S. S.
,
Verdú S.
2001
“The impact of frequencyflat fading on the spectral efficiency of CDMA,”
IEEE Transactions on Information Theory
47
1302 
1327
DOI : 10.1109/18.923717
Lozano A.
,
Tulino A. M.
,
Verdú S.
2003
“Multipleantenna capacity in the lowpower regime,”
IEEE Transactions on Information Theory
49
2527 
2544
DOI : 10.1109/TIT.2003.817429
Couillet R
,
Debbah M
2011
Random matrix methods for wireless communications
Cambridge University Press
Cambridge, MA
Jin S.
,
Gao X.
,
You X.
2007
“On the ergodic capacity of rank1 Ricean fading MIMO channels,”
IEEE Transactions on Information Theory
53
(2)
502 
517
DOI : 10.1109/TIT.2006.889707
Simon M K
,
Alouini M S
2005
Digital communication over fading channels.
John Wiley & Sons
McKay Matthew R.
,
Collings Iain B.
2005
“Statistical properties of complex noncentral Wishart matrices and MIMO capacity,”
in Proc. of the IEEE ISIT
Adelaide, SA
785 
789
Zhang Q.
,
Jin S.
,
Wong K. K.
,
Zhu H. B.
,
Mattaiou M.
2014
“Power Scaling of Uplink Massive MIMO Systems with ArbitraryRank Channel Means,”
IEEE Journal of Selected Topics in Signal Processing
8
966 
981
DOI : 10.1109/JSTSP.2014.2324534
Gore D. A.
,
Heath R. W.
,
Paulraj A. J.
2002
“Transmit selection in spatial multiplexing systems,”
IEEE Communications Letters
6
491 
493
DOI : 10.1109/LCOMM.2002.805517