Numerical Evaluation of Massive MIMO Effects

In this work, no approximation or large system assumption is considered due to the practical number of antennas considered and to show that even then a part of the massive MIMO gains can be utilized.

Especially network vendors and operators are interested in cost-efficient operating points, meaning the trade-off between more performance on one side and more capital expenditure and operational expenditure

costs on the other side. This so-called “sweet spot” may not be in the large system regime. Another reason is that in realistic wireless systems, MSs do not have the same SNR as considered in a lot of massive MIMO literature [HtBD13, RPL⁺13, ANAC13], instead the opposite is the case. The SNR of MSs depends on the effective path loss² and other factors resulting in a large variance in SNR, especially in cellular outdoor systems with cell radii in the range of several hundred meters. Additional different multiple-user and inter-cell interference received at each MS further increases the variance in SINR. Also, the definition of SNR is not clear in a multiple-user system, e.g. by adding an additional user to a set of already scheduled users, assuming constant transmit power, the SNR of all the users decreases. Due to the availability of a high performance computation cluster and “massive storage”, the numerical result in this work are obtained by processing of all channel coefficients without any relaxation.

In system level simulations the abstraction level is SINR and the task is to evaluate the performance of new techniques under various conditions, e.g. to ensure that algorithms work not only under favorable channel conditions. Therefore, the placement of MSs is subject to a random process over the Monte-Carlo simulations runs, also referred to as “drops” due to the MSs “dropping” in a certain region. In the example of Fig. 2.2a K = 500 MSs are “dropped” in the “virtual sector” region obtained from the hexagonal deployment with 300 m inter-site distance (ISD) [3GP17e]. In order to allow comparison of results obtained with system-level simulation between different entities most of the parameters are taken from system system-level simulation assumptions in 3GPP standardization.

In Fig. 2.2b the spectral efficiency forT_l= 2 layers over the number of antennasN is given and the expected logarithmic increase is observed for MRT over the whole antenna range and for the other precoders until 50 antennas. The saturation of the other precoders is caused by the bounded SINR according to (2.9) reflecting limitations of realistic systems due to modulation and coding. Thus, also the difference between MRT and ZF precoder decreases, however in the practical regime a significant gap remains. The performance of ZF, SLNR, and MMSE precoder is similar and the curves overlap in Fig. 2.2b. The maximum power per antenna in the precoder is limited by the PAPC P_(PAPC) = ^P(RB)_N resulting in a power normalization loss shown in Fig. 2.3a for N = 2 with approximately the same behaviour for all precoders. The power normalization loss is caused by the maximum allowed transmit power for each antenna as explained in Eq. (2.39) and the subsequent paragraph, see also Section 3.2 in [MGZF13]. Thus, the [5, 50, 95] %-ile of the MMSE precoder as a representative is given in Fig. 2.3b. The power normalization loss due to the PAPC is increasing up to more than 70 % for N ≥100 with a variance<0.02.

The comparison between PAPC and without PAPC in terms of median sum spectral efficiency ˜C^(sum) is given in Table 2.2 where the corresponding result without PAPC is shown in Fig. 2.4a and loss due to the PAPC constraint is ≤ 0.5 bit/s/Hz. Note that according to Eq. (2.9) for T_l = 2 the theoretical maximum of ˜C^(sum) ≈26.6 bit/s/Hz, but in Table 2.2 the sum spectral efficiency saturates at 24.4 bit/s/Hz. This is caused by the cyclic prefix overhead of 7 % taken into account as in LTE OFDM systems.

The corresponding cumulative distribution function (CDF) of the row-norm spread according to Eq. (2.42) that can be interpreted as the power transmitted by the antennas assuming a power budget of one is shown in Fig. 2.4b forN = 2 andN = 300 antennas. WithN = 2 the maximum row-norm spreadd^(row−norm)_(spread) ≈1, which means that in some cases all transmit power is radiated by one of the antennas, so both amplifiers have to be designed for the maximum power budget, whereas withN = 300 the maximum row-norm spread d^(row−norm)_(spread) = 0.018. This means, that the larger the number of antennas the higher the probability that antennas transmit approximately with equal power and the PAPC as required in low-antenna systems can be dropped. In both cases, the four investigated precoder schemes yield approximately the same performance, therefore the results over the number of antennas N are shown for MMSE precoder only in Fig. 2.5.

Up to N = 300 antennas are considered in the above investigations which results in approximately 22.43 m aperture size assuming ^λ(c)₂ spacing between elements. It is obvious that such antenna arrays can hardly be deployed in dense urban environments where space is precious and expensive. Another disadvantage of

2Effective path loss means distance depended path loss plus effects from antenna patterns and multipath propagation.

Table 2.1.: Simulation assumptions for numerical evaluation of downlink massive MIMO in TDD.

Parameter Value

Simulation type Monte Carlo 500 realizations

Channel model QuaDRiGa version 2.0 [JRB⁺17]

Scenario 3GPP 3D Urban Macro NLoS, [3GP17e]

Center frequency 4 GHz

Number of multi-path componentsL_(MPC) 21

OFDM RB Bandwidth 180 kHz

Number of RBs 50

Utilized bandwidth 9 MHz

Cyclic prefix overhead 7 %

BS antenna distribution Horizontal ULA or UPA

Number of BS antenna elementsN [2,300]

Antenna element spacing λ_(c)/2

Antenna element type Patch

Horizontal, vertical HPBW 65^◦,65^◦

Element directive gain ≈9.4 dBi

BS height 25 m

Total transmit power 40 dBm

SINR bounds,γ^(min), γ^(max) γ^(min)=−5 dB ,γ^(max)= 40 dB x^(C)−y^(C) MS distribution Random uniform i.i.d. in hexagonal

layout [3GP17e]

Number of users per hexagonal sector 20

z^(C) coordinate of MSs 1.5 m

Minimum distance MS-BS 25 m

Table 2.2.: Median sum spectral efficiency comparison with and without PAPC. The numbers correspond to Fig. 2.2b and Fig. 2.4a. MMSE precoder.

N 2 10 50 100 300

with PAPC C˜^(sum) in [bit/s/Hz] 17.4 22.2 23.8 24.1 24.4

without PAPC 17.9 23.2 24.3 24.4 24.4

(a) Distribution of MSs in x^(C)-y^(C) plane with the vir-tual sector boarder following 3GPP assumptions. Ex-ample with 500 MSs.

(b) Sum spectral efficiency (SE) for Tl = 2 over num-ber of antennasNfor horizontal uniform linear array (ULA) with PAPC constraintP_(PAPC)=^P(RB)_N . The curves for MMSE, SLNR, and ZF overlap.

Figure 2.2.: Massive MIMO downlink deployment and performance evaluation.

(a) Distribution of power normalization loss s_(PAPC) in [%] according to (2.40) in the non-massive MIMO regime forN = 2 antennas. The curves for MMSE, MRT, SLNR, and ZF overlap.

(b) Power normalization factors_(PAPC) in [%] according to (2.40) over antennas for the MMSE precoder.

Figure 2.3.: Power normalization factor due to PAPC constraint withP_(PAPC)= ^P(RB)_N .

(a) Sum spectral efficiency (SE) for T_l= 2 over number of antennas N for horizontal ULA without PAPC.

The curves for MMSE, SLNR, and ZF overlap.

(b) Cumulative distribution of the row-norm spread ac-cording to Eq. (2.42) forN = 2 andN = 300 anten-nas. The curves for MMSE, SLNR, and ZF overlap.

Figure 2.4.: Performance without PAPC constraint.

Figure 2.5.: Precoder row-power spread according to Eq. (2.42) over the number of antennasN for horizontal ULA andT_l= 2.

(a) Sum spectral efficiency (SE). (b) User spectral efficiency (SE). The large variance is caused by the equal power per stream assumption in combination with a large variance in path loss among users due to random dropping within the sector.

Figure 2.6.: Performance of downlink precoding for T_l= 2 over the number of antennasN =N_(α)N_(β) with N_(α)=N_(β)for a UPA without PAPC constraint. The curves of MMSE, SLNR, and ZF overlap.

ULAs is that they have only one degree of freedom (DoF) for beam steering, however users in cities are not only distributed in the x^(C)-y^(C) plane also in elevation e.g. due large buildings or non-flat surface.

Therefore, standardization adopted 2D uniform planar arrays in [3GP15a] to allow for 3D also called full-dimension beamforming. An example of a UPA is shown in Fig. 3.3, where the number of elements in horizontal/azimuth domain is defined by N_(α), the number of elements in vertical/elevation domain asN_(β), such that N =N_(α)N_(β). The sum spectral efficiency over the number of antennas is given in Fig. 2.6a and the trends are the same as for the ULA above. In the corresponding user spectral efficiency in Fig. 2.6b a large variance can be observed due to the equal power per stream but large variance in user path losses.

This is caused by the random dropping of users within the sector according to Fig. 2.2a. However, it can be observed that sum spectral efficiency gain of the UPA is similar compared to the ULA, see Fig. 2.4a and Fig. 2.6. In combination with the advantage that the maximum aperture size of the UPA is reduced by a factor of √

N compared to the ULA the focus in the remaining part of the chapter is on UPA. For example withN = 100 the edge length of the UPA is≈0.68 m compared to 7.5 m of the ULA. The UPA assumption also follows current assumption in 5G 3GPP standardization, see Section A.2 in [3GP17g].

The main motivation to use massive MIMO is the achievable spatial multiplexing gain, e.g. in areas with a high user density. By utilizing the DoFs and with a full rank channel the sum spectral efficiency scales linearly with the number of spatial multiplexed users or corresponding in this work to the number of effective MISO channels T_l [RPL⁺13]. With an unlimited number of antennas limN → ∞ the channel vectors of users become orthogonal, then MRT precoding also achieves maximum capacity. However, due to the limited number of antennas in practical systems, this property is not fulfilled and the sum spectral efficiency of MRT precoding is significantly less than the other precoders, see Fig. 2.7a. Therein, MRT achieves a maximum of 30 bit/s/Hz at T = 40 which corresponds to a multiplexing gain of ≈ 3 compared to T = 2, where 11 bit/s/Hz are achieved. The number of devices that can be spatially multiplexed depends on the spatial correlation and user SNRs. Spatial correlation means that due to the non-orthogonality of channels inter-user interference is generated which limits the spectral efficiency, see Eq. (2.2) and Eq. (2.3). Regarding the SNR, independent of channel orthogonality among users, the more users are served the lower the power per beam, see Eq. (2.15). Even in the massive MIMO regime atN T, splitting the power to two users results in the optimal case to a SINR reduction of 3 dB compared to the case that one user is served. Gain in sum rate, e.g. for the 2 users case, is only achieved on condition log₂(1 +γ1)>log₂(1+γ1/2)+log₂(1+γ2), where

(a) Sum spectral efficiency (SE). (b) User spectral efficiency (SE).

Figure 2.7.: Performance of spatial multiplexing with a 10×10 UPA over the number of served streamsTl. The curves of MMSE, SLNR, and ZF partly overlap.

γ₁ and γ₂ are the SINRs of user 1 and user 2, respectively, according to Eq. (2.3). In contrast to this the sum spectral efficiency of ZF has a maximum at T = 60 with a gain of≈18 compared to T = 2. A further increase ofT results in a significant sum spectral efficiency loss due to noise enhancement and normalization loss caused by the zero interference constraint in Eq. (2.26), see [KRTT13]. While the sum spectral efficiency of SLNR and MMSE overlap with ZF up to T = 60 in Fig. 2.6a, their maximum is achieved atT = 70 and remains approximately constant up to T = 100. The reason for that is, that in both precoders the noise variance is taken into account as a regularization factor, see Eq. (2.32) and Eq. (2.36). Note that in the single BS the interference term for MMSE in Eq. (2.37) is zero. The corresponding user spectral efficiency is given in Fig. 2.7 and it can be observed that due to the non-orthogonal user channels SLNR and MMSE scale with∝ _K¹. For the study in Fig. 2.7 the assumption was that ˜K_l=K_lmeaning that all users connected to the BS that request downlink data are served and due to the random user placement this corresponds to round robin scheduling. All the precoders in Fig. 2.7 have in common that adding more users to the system decreases the sum spectral efficiency at some point. This result also shows that in practical systems user scheduling is still required for massive MIMO. The expected complexity increase of user grouping due to large number of antennas and users in the system can be compensated according to [BLM16] by the frequency flat effective channels of massive MIMO also called channel hardening. This means that the scheduling has to be performed only once for a given time slot and can be applied to all RBs in the frequency domain. In contrast to this in LTE systems with a lower number of antennas channels are highly frequency selective and thus scheduling has to be performed on a RBs or group of RBs level [STW⁺09]. Therefore, the impact by user grouping and the assumption of frequency flat channels is investigated in Section 2.3.4.

From the results in this section, it can be concluded that requirements on power amplifiers relax with an increasing number of antennas due to decreased power variance over antennas. It is also shown that with a low number of served streams, e.g. T N most of the spectral efficiency is already achieved with a moderate number of antennas, e.g. in Fig. 2.4, 90 % of the performance with N = 300 is already achieved at N = 10. Thus, the deployment of massive MIMO has to be considered carefully. From a sum spectral efficiency perspective the best operating point of massive MIMO is with a high number of users to select users with orthogonal channels [BLD16], where “high” means more than half the antenna number. As a drawback, user selection is required in order to achieve the maximum sum throughput.