1. Introduction
Multiple-input multiple-output (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, multi-user MIMO (MU-MIMO) systems, where a multi-antenna base station (BS) serves a number of users in the same time-frequency resource, have gained substantial interest for the spatial multiplexing gains [2]. Currently, most existing studies on MU-MIMO techniques mainly focus on two-dimensional (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, three-dimensional (3D) channel is introduced. In the existing 3D-MIMO models, an approximated 3D radiation pattern is used to address the antenna tilt angle in vertical planes [3-5].
An important topic in MU-MIMO 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 line-of-sight (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 mean-squared error (MMSE) was performed, some simplified closed-form expressions for the achievable sum rate were derived in the asymptotic regimes of high and low signal-to-noise ratios (SNR). Unfortunately, in above mentioned studies, only small-scale fading was considered; little attention was paid to the realistic effect of large-scale fading, mainly due to the difficulty in analyzing the statistical distribution of large-scale fading. Furthermore, in practical scenarios, the distribution of the large-scale fading might largely vary in different scenarios, such as urban and open areas. The study on the effect of the large-scale fading in MIMO system is still open. Motivated by this fact, in this paper we focus on investigating the performance analysis of the 3D-MIMO system over composite fading channels including the path-loss, 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 (RC-LN) 3D-MIMO 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/low-SNR 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 path-loss on the system sum rate.
The rest of the paper is organized as follows. In Section 2, we specify the 3D-MIMO system model. In Section 3, we derive the lower bounds, low/high-SNR 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), AH, AT, 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 Ai is A with ith 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 Ei(x)= is the exponential integral function. Finally, PFq( ) is the generalized hyper-geometric function with p, q non-negative integers [14].
2. 3D-MIMO System Model
2.1 Channel Model
Considering an uplink single-cell MU-MIMO system depicted in Fig. 1, the system consists of a BS equipped with M antennas that receive data from N single-antenna users. It is assumed that M ≥ N. The users transmit their data in the same time-frequency 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∈CM×N is the MIMO channel matrix between the BS and the N users, s∈CN×1 is the vector containing the transmitted symbols of the N users which are draw from a zero-mean Gaussian codebook with unit average power. And n∈CN×1 is the complex additional white Gaussian noise (AWGN) vector, such that CN (0, N0IN).
Fig. 1.A 3D-MIMO system with N users located in an L-floor building.
The channel matrix G includes independent small- and large-scale fading, it can be expressed by
where H∈CM×N is the small-scale 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 HL corresponding to the LoS signal and a Rayleigh distributed random component Hω 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 HL is typically associated with a LoS or a diffracted component and thus, assuming far field transmission, it is can expressed as
where ar(Θr) = and at(Θt) = are the specular array response at the receiver and transmitter, respectively, and d is the antenna spacing in wavelengths, Θr, Θt are the angles of arrival and departure of the specular component, respectively. The scattered component of H is denoted by Hω, where the entries of Hω are independent and identically distributed (i.i.d) CN (0,1) RVs.
The diagonal matrix ∈ RN×N represents the large-scale fading and can be written as where , gn(Θtilt) and ξn are the path-loss, the 3D antenna gain and the shadow fading coefficient corresponding to the nth entry, respectively. The shadow fading coefficient ξn is modeled as an independent LN random variable (RV), namely, ξn~LN, or
where η=10/ln10, while μn and σn are the mean and standard deviation (both in dB) of the RV 10lgξn, respectively.
The considered channel model in (2) is a simplified 3D-MIMO 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 nth user, gn(Θtilt) 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, ϕn) and vertical (elevation, θn) directions. (xB, yB, zB) and (xn, yn, zn) denote the coordinates of the BS and the nth user, respectively. We denote Δxn = xn - xB and Δyn = yn - yB as relative distances between the nth user and the BS in the x and y coordinate, respectively. Similarly, Δzn = zn - zB is the height difference between the nth user and the BS.
The distance between the nth user and the BS is denoted as dn, 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 Φn =arctan(Δyn/Δxn) denotes the horizontal angle between the BS antennas boresight and the nth user in the horizontal plane, and Θn =arctan indicates the vertical angle between the horizon and the line connecting the BS to the nth user. In addition, θtilt denotes the BS antenna tilt angle which is adjustable. Moreover, Am represents the maximum attenuation of the BS antennas. The half-power beamwidth (HPBW) in the horizontal and vertical planes are denoted as ϕ3dB and θ3dB, respectively.
Let us denote Gm (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 nth user with horizontal angle ϕn and the vertical angle θn can be formulated as
The resultant antenna gain in the linear scale can be approximated as
The approximation is valid when GH(ϕn)+GV(θn,θtilt)≤Am or Am is large enough. For example, for the antenna model in [15], Am is given as Am = 20dB. In this case, the difference between 100.1Gn(θtilt) and the approximated value gn(θtilt) is less than 1%. The condition corresponds to the typical cell deployments.
With the LN shadow fading coefficient ξn, 3D antenna gain gn(θtilt), and distance dependent path-loss , the large-scale fading coefficient is finally expressed as
It is noteworthy that the large-scale fading βn(θtilt) is treated as a random variable instead of a given value and in the following we will analyze the ergodic sum rate over βn(θtilt).
2.2 User Distribution Model
In the considered 3D-MIMO 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 floor-to-floor is set to be hf. 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 lth floor is Nl, for l=1,…,L, and
For horizontal distribution, we consider uniform distribution, i.e., the Nl users in the lth 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 3D-MIMO System with ZF Receivers
In this section, we firstly derive a closed-form lower bound on the sum rate of 3D-MIMO 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 low-SNR 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 nth 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 small-scale fading H and large-scale fading , it is difficult to obtain the exact expression of R(θtilt,γ). We circumvent this problem in the following by deriving some tractable bounds and approximations on the sum rate of 3D-MIMO system with ZF receivers.
3.1 Closed-form Bounds on the Achievable Sum Rate
In this subsection, we turn to derive a novel closed-form lower bound of the achievable sum rate of 3D-MIMO with ZF receivers. The key result is summarized in the following proposition.
Proposition 1. The achievable sum rate of 3D-MIMO ZF receivers in RC-LN fading channels is lower bounded by RL(θtilt,γ)
where Δ=KMN, Δ1=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 3D-MIMO 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 closed-form 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 3D-MIMO systems with ZF receivers in RC-LN 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 high-SNR regime. We can invoke the affine sum rate expansion in the analysis of MIMO systems [19] as follow:
where S∞ is the high-SNR slope in bits/s/Hz per 3-dB units, and L∞ is the high-SNR power offset, in 3-dB units, given by
Proposition 2. At the high SNR regime, the sum rate of 3D-MIMO with ZF receivers in RC-LN 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 high-SNR 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 large-scale fading terms are decoupled in the high SNR regime. Furthermore, the greater the distances between the BS and users di, the much more effectively reduce the system sum rates due to the increased path-loss.
3.4 Low SNR Analysis
A wide variety of digital communication systems operate at low power where both spectral efficiency and the energy-per-bit 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 per-symbol SNR At low-SNR, the sum rate of MIMO systems can be well approximated for low by the following expression [19]
where min and S0 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 second-order derivatives of the sum rate in (14) with respect to SNR (i.e., γ), respectively.
Proposition 3. For 3D-MIMO systems with ZF receivers in RC-LN 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 large-scale fading mean parameter cn 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 S0 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 3-floor building, the horizontal distribution is modeled as (12) and the vertical distribution is uniform, namely, the user number in each floor is equal, Nl=N/L for l=1,…,L. We set the height of floor-to-floor hf=5m, the height of BS hB=30m, the radius of the floor R=50m, the distance between the BS and the L-floor building D is variable. All the Monte-Carlo simulation results were obtained by averaging over 1×104 independent channel realizations. For rank-1 Ricean fading MIMO channels, we assume θr=θt=, d=0.5. The antenna parameters are set to be θ3dB=6.2°, φ3dB=65°, Gm=20dB. Other channel parameters used in the simulations are set to be as follows: the path-loss exponent υ=4, the standard deviation of ξn, σn=2dB, the mean of ξn, μn=4dB.
We firstly assess the sum rate performance of 3D-MIMO system against different parameters (the radius of cell D, the BS antenna tilt angle θtilt).
In Fig. 2 and Fig. 3, the simulated sum rates of 3D-MIMO systems with ZF receivers are compared with their lower bounds, high-SNR approximations and low-SNR 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 θtilt before the optimal BS antenna tilt angle , and then decreases with the θtilt further increases since the radiation angle of the BS deviates from users.
Fig. 2.Simulated sum rate, lower bound and high-SNR approximation versus the BS antenna tilt angle θtilt in the high SNR regime (M = 10, N = 6, K = 1, γ= 20dB).
Fig. 3.Simulated sum rate, lower bound and low-SNR approximation versus the BS antenna tilt angle θtilt 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 low-SNR approximations over different Ricean K-factor in the high and low-SNR 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 Rayleigh-fading. From the results of Fig. 4, we can observe that the simulated results are extremely accurate in comparison with the lower bounds and high-SNR approximations in the entire high SNR regime. In Fig. 5, it can be seen that the match of simulated results, lower bounds and low-SNR approximation is very good if the SNR of interest is sufficiently low (i.e., below 60 bits/s/Hz of sum rate).
Fig. 4.Simulated sum rate, lower bound and high-SNR approximation versus the transmit SNR γ in the high SNR regime (M = 10, N = 6, D = 400, θtilt=4°).
Fig. 5.Simulated sum rate, lower bound and low-SNR approximation versus the transmit energy per bit in the low SNR regime (M = 10, N = 6, D = 400, θtilt=4°)
5. Conclusion
This paper presented a detailed sum rate analysis of 3D-MIMO ZF receivers utilizing the BS antenna tilt angle in RC-LN fading channels. Specifically, closed-form lower bounds, high/low-SNR 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 3D-MIMO system was also obtained. More importantly, these analytical results encompass the small- and large-scale fading (include LN shadow fading, 3D antenna gain and path-loss) 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 θtilt on the performance of system.