Capacity Bounds for Dense Massive MIMO in a Line-of-Sight

Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000.

Digital Object Identifier 10.1109/ACCESS.2017.DOI

Capacity Bounds for Dense MassiveMIMO in a Line-of-Sight PropagationEnvironmentFELIPE A. P. DE FIGUEIREDO12, CLAUDIO F. DIAS4, EDUARDO R. DE LIMA3, ANDGUSTAVO FRAIDENRAICH41Instituto Nacional de Telecomunicações – INATEL Santa Rita do Sapucaí, MG, Brazil.2Ghent University–imec, IDLab, Department of Information Technology, Ghent, Belgium. (e-mail: [email protected])3Eldorado Research Institute, Campinas, Brazil. ([email protected])4DECOM/FEEC–State University of Campinas (UNICAMP), Campinas Brazil. ([aplnx, gf]@decom.fee.unicamp.br)

Corresponding author: Felipe A. P. de Figueiredo (e-mail: [email protected]).

ABSTRACT The use of large-scale antenna arrays grants considerable benefits in energy and spectralefficiency to wireless systems due to spatial resolution and array gain techniques. By assuming a dominantline-of-sight environment in a massive MIMO scenario, we derive analytical expressions for the sum-capacity. Then, we show that convenient simplifications on the sum-capacity expressions are possible whenworking at low and high SNR regimes. Furthermore, in the case of a high SNR regime, it is demonstratedthat the Gamma PDF can approximate the PDF of the instantaneous channel sum-capacity as the number ofBS antennas grows. A second important demonstration presented in this work is that a Gamma PDF can alsobe used to approximate the PDF of the summation of the channel’s singular values as the number of devicesincreases. Finally, it is important to highlight that the presented framework is useful for a massive numberof Internet of Things devices as we show that the transmit power of each device can be made inverselyproportional to the number of BS antennas.

INDEX TERMS Massive MIMO, channel capacity, dense networks, outage probability.

I. INTRODUCTION

During the past years, we have been witnessing MassiveMultiple-Input Multiple-Output (MIMO) becoming an effi-cient and indispensable sub-6 GHz physical-layer technologyfor wireless and mobile networks. The embodiment of suchtechnology was vital for the current 5G New Radio (NR) in-terface [1]. The central concept behind Massive MIMO is theuse of a large antenna array deployed at base stations (BS) tosimultaneously serve a large number of devices over the sametime-frequency resources. In this way, the technique allowsexploiting differences among the propagation signatures ofthe devices in order to perform spatial multiplexing [1]. Eventhough the Massive MIMO technology looks quite mature,being adopted by new standards, it does not mean an end forresearch in this subject but just the beginning of unforeseenpossibilities [2].

Current requirements for next-generation networks such ashigh bit rates, very low latency, high energy efficiency, andlink robustness are not wholly met even by the current 5Gsolutions [3]. Thus, there are still several open challenges that

need to be addressed by researchers.

One of the approaches used to increase throughput is byincreasing the network density, that is, decreasing the sizeand increasing the number of cells in the same coveragearea. As a result, the size of the cells is becoming smallerand smaller, and consequently, it is quite probable thatwireless channels will be predominantly line-of-sight (LOS)[4]. Furthermore, the LOS characteristic becomes even moreapparent as technology moves to millimeter-wave bands inorder to fulfill the requirements for wider bandwidths (i.e.,micro and femtocells) [5], [6].

Densification is a natural process for wireless communi-cation networks as the demands for connectivity increases.Thanks to smartphones, tablets, and the internet of things,wireless subscribers are using more network resources withno sign of a decrease in the demand rate. Operators need toadd more capacity to their networks to continue handling allthe traffic while providing the network speeds those usersexpect. An arbitrary coverage region can be expected tofollow three different degrees of BS densification [2], [7]:

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Felipe A. P. de Figueiredo et al.: Capacity Bounds for Dense Massive MIMO in a Line-of-Sight Propagation Environment

low-density, dense and ultra-dense.Starting from low-density, the smart farming and rural

broadband services provision (i.e., distant areas with lowpopulation density) stewards a significant gap in the researchbody of Massive MIMO nowadays [8], [9]. In the sense ofwireless services, it is essential to observe that most eco-nomically viable areas for agriculture are the plain terrainswith few obstacles [10]. Thus, since the multipath-channelstatistics of rural and distant areas are different from thosefound in urban centers, the wireless channel will likely to bepredominantly LOS in such areas [11], [12].

For regions where high value goods are constantly undersurveillance, a dense networks is required to communicatewith unmanned aerial vehicles (UAVs) or also known asdrones [13]. As with the other examples, these communica-tions will probably be LOS-based as well. Additionally, inwireless and cellular communications, the probability thatthe devices have LoS to the BS is highly probable in manyscenarios [14].

Among the approaches and opportunities cited here, theIoT for drones, sensors, automated processes, etc. [15] poseseveral unsolved research challenges. For example, in thenear future, swarms of drones, sensors and actuators willbe omnipresent adding up to billions of devices [16]–[18].In this ultra-dense scenario, these devices will be alwaysmoving around the cell at different speeds and positions witha dominant LOS link to the BS. The sum-capacity achievedby a BS serving a massive number of drones, which havea dominant LOS link to the BS and are constantly movingaround the cell is still an open issue.

Therefore, in this work, we assume a dominant LOSenvironment in a massive MIMO scenario with favorablepropagation serving a massive number of devices constantlymoving within the cell. With this aspect in mind, the objectiveof this investigation is to find capacity limits concerning thenumber of devices, number of base station antennas, andSNR. More specifically, the contributions of this work areas follows.• Derive an analytical expression for the channel sum-

capacity.• Show that the transmit power of each device can be

made inversely proportional to the number of BS an-tennas.

• Find analytical expressions for the upper and lower-bound channel sum-capacities.

• Present expressions for the channel sum-capacity in lowand high SNR regimes.

• Demonstrate that the Gamma PDF can approximate thePDF of the instantaneous channel sum-capacity in lowand high SNR regimes.

• Demonstrate that a Gamma PDF can also approximatethe PDF of the summation of the channel’s singularvalues (also known as total power gain of the channelmatrix) as the number of devices increases.

The remaining of this paper is organised as follows:Section II discuss some related works. Section III presents

R

Rmin

dk,l

FIGURE 1. Illustration of the adopted system model.

the system model adopted in this work while Section IVdiscusses aspects of favorable propagation in the currentinvestigated conditions. Section V reviews the concept offavorable propagation. Section VI presents the results withparameters from reference systems. Finally, we close ourdiscussion in the Section VII summarising our conclusions.

II. RELATED WORKIn [14] the authors analyse the performance of the outagecapacity for the uplink of Massive MIMO systems consid-ering a random LoS scenario. The authors of [19] prove thatthe distribution of the interference term in the expression ofthe uplink signal-to-interference-plus-noise ratio (SINR) canbe approximated as a Beta-mixture when LoS channels andelement spacing of half-wavelength are considered. In [20],the authors assess the distribution of the interference and theprobability of outage for the downlink of massive MIMOsystems adopting MRC precoding in rich multi-path scenar-ios. In [21], the authors demonstrate that in massive MIMOsystems the LoS channels are asymptotically orthogonal asthe number of antennas increases. They also demonstrate thatwhen a finite number of antennas is considered, the channelswill not be orthogonal. The impact of antenna element spac-ing on the capacity of fixed point-to-point massive MIMOsystems considering LoS channels was evaluated in [22].

III. SYSTEM MODELHere we present the channel model adopted in [1]. Weassume a channel model with only free-space non-fading(i.e., pure) LOS propagation between the BS and the devicesand that the BS is equipped with an uniform linear array(ULA) with the M antenna elements spaced of λ/2, whereλ is the signal’s wavelength. There are K single-antennadevices deployed randomly within the cell that simultane-ously transmit data towards the BS through the same time-frequency resources. Additionally, we also consider that allthe K devices being served by the BS are located in the far-field of the antenna array at the angle θk as measured relativeto the array bore-sight [1]. Therefore,

G = HBD1/2, (1)

2 VOLUME 4, 2016

Author et al.: Capacity Bounds for Dense Massive MIMO in a Line-of-Sight Propagation Environment

where the elements of the matrix H are defined as hmk =e−j(m−1) sin(θk), m is the antenna index, k is the deviceindex, θk models the devices’ locations and is uniformlydistributed in the interval [−π, π], the elements of the diag-onal matrix B are defined as bk = ejφk , φk is a uniformlydistributed random variable defined in the range [−π, π]that models the phase shift associated with a random rangebetween the array and the k-th device, and the elements ofthe diagonal matrix D that defined as dk = βk are the large-scale fading coefficients [1].

The free space path-loss coefficients, βkm, is modeled asdescribed in [23],

βkm =η

d2k,m, (2)

where η is a constant equal to(λ4π

)2[24], dk,m is the

distance between them-th BS antenna and the k-th terminal’santenna, and the path-loss exponent is equal to 2, which isthe value used for free-space propagation. As the distancebetween the k-th device and the antenna array is dk,m �λ, then dk,m ≈ dk, and consequently βkm ≈ βk, i.e.,βkm does not depend on the antenna index as the distancebetween the k-th device and the BS is much greater thanthe distance between the antennas. We consider that dk isan uniformly distributed random variable distributed in theinterval [Rmin, R], where Rmin is the minimum distance adevice can be from the BS and R is the cell radius. Theadopted system model is depicted in Figure 1.

IV. CHANNEL SUM-CAPACITY IN LOS AND FAVORABLEPROPAGATION CONDITIONSThe distance between the device’s antennas and the BS’santennas is not static as the devices are always moving withinthe cell (e.g., drones, cars, etc.). Therefore, the distance hasto be treated as a random variable, resulting in a differentchannel realization for every time instant. As it is known[1], the instantaneous sum-capacity is not a meaningful per-formance metric under such random conditions. Here, weare interested in the capacity performance averaged overall different positions a device might be within the cell.Therefore, in order to assess the performance in this scenario,we need to employ the notion of ergodic capacity, whichresults in the following uplink sum link capacity [25]

C = E[log2|IM + ρGGH |

]= E

[log2|IK + ρGHG|

].

(3)

where ρ is the average signal-to-interference ratio (SNR),also known as average transmitted power of each device,and the expectation is taken over the joint distribution ofall possible positions of the devices. As the additive whitenoise is assumed to have mean and variance equal to 0 and 1respectively, therefore, ρ has, consequently, the interpretationof normalized transmit SNR and is therefore dimensionless[26]. It is important to highlight that the expectation in (3) istaken in relation with devices’ channels, more specifically inrelation with their positions within the cell (i.e., distances

and angles of arrival), which are considered as randomvariables. The second line of (3) is found by applying theSylvester’s determinant theorem. Furthermore, we assumethat the base station has perfect knowledge of the channelmatrix G. The rationale behind the assumption of perfectknowledge of the channel matrix is that the results obtainedwith this assumption are readily comprehended and theybound the performance of massive MIMO systems.

Finding the exact ergodic capacity given by (3) is a quitecomplex task that involves finding the distribution of theeigenvalues of GHG [27]. In this work, as we will shownext, we are concerned with finding the sum-capacity con-sidering a Massive MIMO scenario and asymptotically favor-able propagation [1]. In asymptotically favorable propagationscenarios, the channel vectors of different users become mu-tually orthogonal as the number of antennas, M , increases,i.e.,

gHi gjM

M → ∞→ 0, ∀ i 6= j. (4)

The environment is said to offer asymptotically favorablepropagation when (4) is satisfied. The mutual orthogonalityoffered by environments exhibiting the asymptotic favorablepropagation condition is the most beneficial situation fromthe perspective of sum-capacity maximization. Then, thesum-capacity in (3) can be re-expressed as

C = E[log2 |IK + ρGHG|

]≤ E

[log2

(K∏k=1

[IK + ρGHG

]k,k

)]

= E

[K∑k=1

log2

([IK + ρGHG

]k,k

)]

= E

[K∑k=1

log2

(1 + ρ‖gk‖2

)].

(5)

The inequality in the second line of (5) is found applyingthe Hadamard inequality and assuming that ‖gk‖2,∀k isknown, where gk,∀k, are the columns of the channel matrixG. As will be shown latter, this bound will be proven to bevery tight. The equality in the second line of (5) holds if andonly if GHG is a diagonal matrix (i.e., the channel matrixG has mutually orthogonal columns) that must satisfy [28],

gHi gj =

{0, i, j = 1, . . . ,K, for i 6= j

‖gk‖2 = Mβk, k = 1, . . . ,K, for i = j,(6)

which is the case when the channel exhibits favorable prop-agation [29]. The equality ‖gk‖2 = Mβk is detailed inAppendix A.

Given the assumption of channels exhibiting asymptoti-cally favorable propagation, we know from [1] that the spatialsignature vectors, denoted by ejφkhk, become asymptoti-cally mutually orthogonal and consequently, it can be shownthat

BHHHHB

M

M → ∞≈ IK , (7)

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and therefore,

GHG

M= D1/2B

HHHHB

MD1/2 M → ∞

≈ D, (8)

hence, the results presented in this work hold for any typeof antenna array, i.e.., ULA and uniform rectangular array(URA), as in the asymptotic regime (8) will always tend tothe matrix containing the large-scale fading coefficients.

Now using ‖gk‖2 = Mβk and βk = ηd2k

in the last line of(5), the sum-capacity upper bound can be re-written as

C =

K∑k=1

E[log2

(1 +

ρMη

d2k

)]= KE

[log2

(1 +

ρMη

d2k

)],

(9)

where the last equality is due to the fact that dk is an i.i.d.random variable for all k.

Next, considering z = 1d2k

as a random variable denoted byZ with probability density function (PDF) given by

fZ(z) =1

2(R−Rmin)z√z,

1

R2≤ z ≤ 1

R2min

, (10)

then, (9) can be re-expressed as

C = KE[log2

(1 +

ρMη

d2k

)]= KE [log2 (1 + ρMηz)]

=K

2(R−Rmin)

∫ 1/R2min

1/R2

log2(1 + ρMηz)1

z√zdz,

(11)

where the proof of (10) is given in Appendix B. Next, solving(11) with an integral solver [30], we find an exact closed-formexpression given in (12) for the capacity when the channeloffers favorable propagation.

Remark 1. After analyzing (12), we see that if we make thetransmit power of each device equal to P/Mα, where α > 1,then the sum-capacity, C, will go to zero as M → ∞. Whenα < 1 the sum-capacity grows without bound as M → ∞.This means that 1/M (i.e., α = 1) is the fastest rate at whichwe can decrease the transmit power of each device and stillhave a fixed capacity as M →∞.

Remark 1 clearly shows that as M grows without bound,the transmit power of each device can be reduced proportion-ally to 1/M and that the spectral efficiency increases by afactor of K, meaning that the BS can simultaneously serveK devices over the same time-frequency resources. Thisreduction of the transmit power per device is very importantto power-constrained devices such as IoT devices.

A. APPROXIMATED DISTRIBUTION OF THE TOTALPOWER GAINThe sum-capacity given by the first line of (5) can be ex-pressed in terms of the singular values {λk} of the channel

matrix G and, therefore, if λ1 ≥ λ2 ≥ · · ·λK are the randomordered singular values of G, then (5) can be re-written as [1]

C = E

[K∑k=1

log2

(1 + ρλ2k

)]. (14)

By comparing (5) and (14), it is possible to conclude that

λ2k = ‖gk‖2 = Mβk. (15)

The singular values represent the channel gains of the par-allel channels after Singular Value Decomposition (SVD) ofthe system [25]. Next we present the squared Frobenius normof G when the environment offers favorable propagation, andconsequently, G is full rank

‖G‖2F =

M∑m=1

K∑k=1

|gmk|2 = Tr(GHG) =

K∑k=1

λ2k. (16)

The last term of (16) can be interpreted as the total powergain of the channel matrix if one spreads the energy equallybetween all the antennas [25]. Therefore, comparing (15) and(16), we clearly see that the singular values have the samedistribution as the free space path-loss coefficients, βk,∀k.The summation of the K free space path-loss coefficients, isa random variable that can be approximated by the Gammadistribution as showed in Appendix D.

B. A LOWER-BOUND FOR THE CAPACITYFirst we define the following Jensen’s Inequality [31]

E [log2(1 + z)] ≥ log2

(1 +

1

E[1z

]) , (17)

where z = 1u for z > 0. Therefore, (9) can be re-written as

C = KE[log2

(1 +

ρMη

d2k

)]≥ K log2

(1 +

ρMη

E [d2k]

).

(18)

Remembering that fd(r) = 1R−Rmin

, Rmin ≤ r ≤ R, then

E[d2k]

=R3 −R3

min

3(R−Rmin). (19)

For proof of (19), see Appendix C. Therefore, (18) can bere-expressed as

C ≥ K log2

(1 +

3ρMη(R−Rmin)

R3 −R3min

). (20)

C. AN UPPER-BOUND FOR THE CAPACITYAgain, we can use the Jensen’s Inequality as

E [log2(1 + z)] ≤ log2 (1 + E [z]) , (21)

for z > 0. Therefore, (9) can be re-written as

C = KE[log2

(1 +

ρMη

d2k

)]≤ K log2

(1 + ρMηE

[1

d2k

]).

(22)

4 VOLUME 4, 2016


C =K

(R−Rmin)

2√ρMη

(tan−1

(√ρMηR2min

)− tan−1

(√ρMηR2

))+R log

(ρMηR2 + 1

)−Rmin log

(ρMηR2min

+ 1)

log(2).

(12)

C ≈ K

(R−Rmin)

log(ρMη)(R−Rmin) + 2 [R−Rmin −R log(R) +Rmin log(Rmin)]

log(2). (13)

Remembering that fd(r) = 1R−Rmin

, Rmin ≤ r ≤ R, then

E[

1

d2k

]=

1

RRmin. (23)

For proof of (23), see Appendix C. Therefore, (22) can bere-expressed as

C ≤ K log2

(1 +

ρMη

RRmin

). (24)

D. LOW SNR REGIMEFor the low SNR regime, we have the following approxima-tion: log2(1 + x) ≈ x log2(e), when x � 1, then (9) can beexpressed as

C ≈ KE[ρMη

d2klog2 (e)

]= KρMη log2 (e)E

[1

d2k

]=KρMη log2 (e)

RRmin.

(25)

In (25), we see that the sum-capacity linearly increaseswith the average SNR, ρ, and/or with the number of antennas,M .

E. HIGH SNR REGIMEFor the high SNR regime, we have the following approxima-tion: log2(1 + x) ≈ log2(x), when x � 1, then (9) can beexpressed as

C ≈ KE[log2

(ρMη

d2k

)]=

K

2(R−Rmin)

∫ 1/R2min

1/R2

log2(ρMηz)1

z√zdz,

(26)

where z = 1d2k

. Solving (26) with an integral solver [30], wefind an exact closed-form expression for the capacity whenthe channel offers favorable propagation in the high SNRregime. The exact closed-form expression for the approxi-mated sum-capacity in (26) is given by (13).

In (13), differently from (25), we see that the sum-capacitylogarithmically increases with the average SNR, ρ, and/orwith the number of antennas, M . Therefore, in the highSNR regime, an increase in the transmit power and/or in thenumber of antennas, M , is much less impressive than in thelow SNR regime case.

F. INSTANTANEOUS SUM-CAPACITY PDF IN LOW SNRREGIME AND FAVORABLE PROPAGATION CONDITIONThe instantaneous sum-capacity in low SNR regime andfavorable propagation condition is given by

Cinst. =

K∑k=1

log2

(1 +

ρMη

d2k

)ρMη

d2k

� 1

≈ ρMη log2 (e)

K∑k=1

1

d2k,

(27)

which is a random variable that depends on the distance ofthe k-th device to the BS, denoted by dk.

Lemma 1. By empirically comparing the normalised his-togram of the random variable given by (27) against thetheoretical PDF of a Gamma random variable we notice thatasK increases thatCinst. can be approximated by the GammaPDF with parameters κ and θ given by

κ =3KRRmin

(R−Rmin)2, (28)

θ =ρMη log2 (e) (R−Rmin)2

3R2R2min

. (29)

This comparison is shown in section VI. The parametersκ and θ are found following the same rationale used inAppendix D.

G. INSTANTANEOUS SUM-CAPACITY PDF IN HIGH SNRREGIME AND FAVORABLE PROPAGATION CONDITIONThe instantaneous sum-capacity in high SNR regime andfavorable propagation condition is given by

Cinst. =

K∑k=1

log2

(1 +

ρMη

d2k

)ρMη

d2k

� 1

≈K∑k=1

log2

(ρMη

d2k

),

(30)

which is a random variable that depends on the distance ofthe k-th device to the BS, denoted by dk.

Lemma 2. By empirically comparing the normalised his-togram of the random variable given by (30) against thetheoretical PDF of a Gamma random variable we noticethat as M increases that Cinst. can be approximated by theGamma PDF with parameters κ and θ given by (33) and(34), respectively, where a = log(ρMη), b = log

(ρMηR2

min

),

VOLUME 4, 2016 5


c = log(ρMηR2

), d = log

(1R2

min

), and e = log

(1R2

). This

comparison is shown in section VI. The parameters κ and θare found as described in Appendix E.

H. OUTAGE PROBABILITYTherefore, based on the knowledge of the approximatedPDFs, it is possible to use the Gamma’s CDF as a way toanalyse the outage probability, Cout, in low and high SNRregimes. Results comparing the PDF and CDF of the actualrandom variable and those of a Gamma random variable arepresented and discussed in section VI. Outage probability isthe probability that a certain sum-capacity cannot be reachedand is defined as

Pout = Pr{Cinst. < Cout} (31)M → ∞≈ 1

Γ(κ)γ

(κ,Cout

θ

), (32)

where Γ(.) is the gamma function, γ(., .) is the incompletegamma function, and κ and θ are given by (33) and (34),respectively.

V. AVERAGE DISTANCE FROM FAVORABLEPROPAGATIONThe favorable propagation is a important metric which isdefined as mutual orthogonality among the vector-valuedchannels to the terminals [21]. The measure is one of the keyproperties of the radio channel that is exploited in MassiveMIMO. From [21], we can further characterize favorablepropagation as

∆C =KE [log2(1 + ρMβk)]− E

[log2|IK + ρGHG|

]E [log2|IK + ρGHG|]

.

(35)As can be seen from (35), when ∆C = 0, the channel

offers favorable propagation. The measure defined in (35)is an extension of the one presented in [1] for the ergodiccapacity case assumed in this work.

VI. SIMULATION RESULTS AND DISCUSSIONIn this section we present the results of several simulationsthat were designed to assess the findings we have reportedin this work. All results presented here have the followingsimulation setup parameters R = 100 [m], Rmin = 10 [m],and λ = 0.375 [m], which is equivalent to a carrier frequencyof 800 MHz.

Figure 2 has the following simulation setup parametersK = 10 and ρ = 50 [dB]. As can be seen, the simulatedcapacity is within the lower and upper bound ranges asexpected. It is also possible to see that the simulated andanalytical capacities in favorable propagation match eachother, showing that the derived analytical closed-form is tightfor the favorable propagation case. Moreover, the simulatedcapacity asymptotically approaches the capacity in favorablepropagation as the number of antennas, M , grows, provingthat favorable propagation is asymptotically achieved as Mgrows. It is also important to highlight that the analytical

capacity in favorable propagation, given by (12), providesa good approximation for the capacity. For example, forM = 100 the simulated capacity is equal to 23.59 bits/s/Hzand the analytical capacity in favorable propagation is equalto 24.44 bits/s/Hz.

Figure 2 also presents on the right x-axis the averagedistance from favorable propagation as defined in (35). Ascan be noticed, the average distance asymptotically decreasesas the number of antennas grows, starting at 0.14 forM = 10and decreasing to 0.0049 for M = 1000. This result is an-other indication that favorable propagation is asymptoticallyachieved as M grows.

Figure 3 compares the sum-capacity over the variationof the average SNR, ρ, for the same simulation parametersused for the results in Figure 2 and M constant and equal to300 antennas. As expected, the sum-capacity, simulated andanalytical, stay within the lower and upper capacity bounds.For low SNR values the sum-capacity is closer to the upperbound and as the SNR increases, we see that both lower andupper bounds converge to the sum-capacity.

In Figure 4 we present the results of the sum-capacityfor low and high SNR regimes in favorable propagationcondition versus the average signal-to-interference ratio, ρ,with M constant and equal to 300 antennas. As can be seen,(25) and (13) represent the sum-capacity fairly well for thelow and high SNR regimes, respectively. In the upper part ofthe figure, we show the sum-capacity for the low SNR regimeand it is possible to see that the approximated expressiongiven by (25) closely follows the sum-capacity until around30 [dB]. In the lower part of the figure, we show the sum-capacity for the high SNR regime and we also see thatthe approximated expression given by (13) closely followsthe sum-capacity for SNR values greater than 40 [dB]. Thefigure also shows that the sum-capacity grows linearly andlogarithmically with the average SNR for the low and highSNR regimes, respectively, confirming what was discussedin subsections IV-D and IV-E

Figure 5 shows how the spectral efficiency behaves asρ = P/Mα, where α = 1/2, 1 and 3/2, respectively. Thefollowing simulation setup parameters were used: K = 10and P = 50 [dB]. As expected and stated in Remark 1, whenα = 1 and M increases, the capacity becomes constant nomatter the number of antennas. However, when α = 1/2 thecapacity grows logarithmically fast with M when M → ∞and tends to zero when α = 3/2 and M →∞. These resultsattest that the transmit power of each device can be reducedproportionally to M .

In Figure 6 we show the required transmit power per devicethat is needed to achieve fixed capacities of 1 and 2 bits/s/Hzrespectively. As expected and predicted by Remark 1, thetransmit power can be reduced by approximately 3 [dB] bydoubling the number of antennas, M , for both cases, i.e., 1and 2 bits/s/Hz.

Figure 7 shows the comparison between the normalizedhistogram of the random variable Z, where Z is defined inLemma 3, and the Gamma PDF for number of devices equal

6 VOLUME 4, 2016


κ =

K(R−Rmin)

(b+2R −

c+2Rmin

)21

RRmin

(2e(a+2)+e2+a(a+4)+8

Rmin− 2d(a+2)+d2+a(a+4)+8

R

)− 1

(R−Rmin)

(b+2R −

c+2Rmin

)2 . (33)

θ =

RRmin(R−Rmin)

(b+2R −

c+2Rmin

)2− 2e(a+2)+e2+a(a+4)+8

Rmin+ 2d(a+2)+d2+a(a+4)+8

R

log(2)(b+2R −

c+2Rmin

) . (34)

100 200 300 400 500 600 700 800 900 1000

M

10 1

10 2

Sum

-Cap

acity

[bits

/s/H

z]

10 -3

10 -2

10 -1

10 0

Ave

rage

dis

tanc

e fr

om fa

vora

ble

prop

agat

ion

Simulated Sum-Capacity - Eq. (3)Simulated Sum-Capacity (FP) - Eq. (10)Analytical Sum-Capacity (FP) - Eq. (13)Analytical Lower-bound Sum-Capacity (FP) - Eq. (18)Analytical Upper-bound Sum-Capacity (FP) - Eq. (22)Average distance from FP

FIGURE 2. Simulation considering K = 10, ρ = 50 [dB], R = 100 [m],Rmin = 10 [m], and λ = 0.375 [m].

to K = 10, 20, and 50 respectively. These results werealso obtained with the following simulation setup parametersM = 100 and ρ = 50 [dB]. The histogram, showed in blue,is the histogram of the random variable Z =

∑Kk=1Mβk. As

can be seen, the histogram approaches the Gamma PDF asthe number of devices increases.

Figure 8 comparison of the instantaneous sum-capacity inhigh SNR regime and favorable propagation condition and itsapproximation with the Gamma PDF. This comparison wasobtained with the following parameters K = 10 and ρ = 60[dB]. As can be seen, for high SNR, the Gamma PDF fits thePDF of the instantaneous sum-capacity random variable asthe number of antennas grows.

Figure 9 presents the comparison between the approxi-mated and simulated outage probabilities in the high SNRregime and favorable propagation condition. The simulationparameters are the same used for generating the results inFigure 8. As can be also seen, for high SNR, the Gamma CDFfits the simulated outage probability of the instantaneoussum-capacity random variable as the number of antennasgrows.

Figure 10 comparison of the instantaneous sum-capacity inlow SNR regime and favorable propagation condition and its

20 30 40 50 60 70 80 90 100

[dB]

10-1

100

101

102

Sum

-Cap

acity

[bits

/s/H

z]

Simulated Sum-Capacity - Eq. (3)Simulated Sum-Capacity (FP) - Eq. (10)Analytical Sum-Capacity (FP) - Eq. (13)Analytical Lower-bound Sum-Capacity (FP) - Eq. (18)Analytical Upper-bound Sum-Capacity (FP) - Eq. (22)

FIGURE 3. Simulated and analytical sum-capacity considering M = 300.

-100 -80 -60 -40 -20 0 20 40

[dB]

10 -15

10 -10

10 -5

10 0

10 5

Sum

-Cap

acity

[bits

/s/H

z]

Low SNR Regime

Simulated Sum-Capacity - Eq. (3)Simulated Sum-Capacity (FP) - Eq. (10)Analytical Sum-Capacity (FP) - Eq. (13)Low SNR Regime - Eq. (23)

40 60 80 100 120 140 160 180 200

[dB]

10 0

10 1

10 2

10 3

Sum

-Cap

acity

[bits

/s/H

z]

High SNR Regime

Simulated Sum-Capacity - Eq. (3)Simulated Sum-Capacity (FP) - Eq. (10)Analytical Sum-Capacity (FP) - Eq. (13)High SNR Regime - Eq. (14)

FIGURE 4. Sum-capacity for low and high SNR regimes versus averagesignal-to-interference ratio, ρ.

VOLUME 4, 2016 7


100 200 300 400 500 600 700 800 900 1000

M

10 -1

10 0

10 1

Sum

-Cap

acity

[bits

/s/H

z]

FIGURE 5. Demonstration of the power scaling law for different α values.

100 200 300 400 500 600 700 800 900 1000

M

18

19

20

25

30

35

40

44

Req

uire

d T

x P

ower

[dB

]

C = 1 [bit/s/Hz]C = 2 [bit/s/Hz]

FIGURE 6. Required transmit power to achieve 1 and 2 bits/s/Hz as a functionof the number of antennas.

approximation with the Gamma PDF. This comparison wasobtained with the following parameters M = 100 and ρ =0 [dB]. As can be seen, for low SNR, the Gamma PDF fitsthe PDF of the instantaneous sum-capacity random variableas the number of devices simultaneously served through thesame time-frequency resources grows.

Figure 11 presents the comparison between the approxi-mated and simulated outage probabilities in the low SNRregime and favorable propagation condition. This compari-son was obtained with the following parameters M = 300and ρ = 0 [dB]. As can be also seen, for low SNR, theGamma CDF fits the simulated outage probability of theinstantaneous sum-capacity random variable as the numberof served devices grows.

K = 10

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5z 10 -3

0

200

400

600

800

1000

PD

F

K = 20

0 1 2 3 4 5 6 7 8z 10 -3

0

200

400

600

800

PD

F

K = 50

0 0.002 0.004 0.006 0.008 0.01 0.012z

0

100

200

300

400

PD

F

FIGURE 7. Gamma PDF for number of devices equal to K = 10, 20, and 50respectively.

M = 50

25 30 35 40 45 50 55 60 65

Sum-Capacity [bits/s/Hz]

0

0.02

0.04

0.06

0.08

PD

F

High SNR and favorable-propagationGamma approximation

M = 100

35 40 45 50 55 60 65 70 75


0

0.02

0.04

0.06

0.08

PD

F


M = 200

45 50 55 60 65 70 75 80 85


0

0.02

0.04

0.06

0.08

PD

F


FIGURE 8. Comparison of the approximated PDF for the instantaneoussum-capacity in high SNR regime and favorable propagation condition.

VII. CONCLUSIONSIn this work, we investigated ways to find capacity limitsconcerning the number of users, number of BS antennas,and SNR. By assuming a dominant LOS environment in amassive MIMO scenario, it was possible to derive analyticalexpressions for the channel capacity. Convenient simplifi-cations on expressions were possible working at low andhigh SNR regimes. Furthermore, it is demonstrated that theGamma PDF can approximate the PDF of the instantaneouschannel sum-capacity as the number of antennas grows forboth cases of low and high SNR regimes. A second importantdemonstration is that a Gamma PDF can also approximatethe PDF of the summation of the channel’s singular values as

8 VOLUME 4, 2016


20 25 30 35 40 45 50 55 60 65 70Sum-Capacity [bits/s/Hz]

0

0.2

0.4

0.6

0.8

1

Out

age

Pro

babi

lity

M = 50

Simulated OutageGamma CDF

30 35 40 45 50 55 60 65 70 75Sum-Capacity [bits/s/Hz]

0

0.2

0.4

0.6

0.8

1

Out

age

Pro

babi

lity

M = 100


45 50 55 60 65 70 75 80 85Sum-Capacity [bits/s/Hz]

0

0.2

0.4

0.6

0.8

1

Out

age

Pro

babi

lity

M = 200


FIGURE 9. Comparison between the approximated and simulated outageprobabilities in high SNR regime and favorable propagation condition.

K = 10

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Sum-Capacity [bits/s/Hz] 10-3

0

200

400

600

800

PD

F

Low SNR and favorable-propagationGamma approximation

K = 25

0 1 2 3 4 5 6 7 8Sum-Capacity [bits/s/Hz] 10-3

0

100

200

300

400

PD

F


K = 50

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014Sum-Capacity [bits/s/Hz]

0

100

200

300

PD

F


FIGURE 10. Comparison of the approximated PDF for the instantaneoussum-capacity in low SNR regime and favorable propagation condition.

the number of devices increases. Finally, the utility of such aframework is useful for a massive number of IoT devices aswe show that the transmit power of each device can be madeinversely proportional to the number of BS antennas.

.

APPENDIX A DERIVATION OF ‖gK‖2 =MβK

From (1), we can write each element gk,m of gk as

gk,m =√βke

jφke−j(m−1) sin(θk). (36)

0 0.002 0.004 0.006 0.008 0.01 0.012Sum-Capacity [bits/s/Hz]

0

0.2

0.4

0.6

0.8

1

Out

age

Pro

babi

lity

K = 10


0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022Sum-Capacity [bits/s/Hz]

0

0.2

0.4

0.6

0.8

1

Out

age

Pro

babi

lity

K = 25


0.005 0.01 0.015 0.02 0.025 0.03 0.035Sum-Capacity [bits/s/Hz]

0

0.2

0.4

0.6

0.8

1

Out

age

Pro

babi

lity

K = 50


FIGURE 11. Comparison between the approximated and simulated outageprobabilities in low SNR regime and favorable propagation condition.

Therefore, using the very definition of the product of twomatrices [32], we have

‖gk‖2 =

M∑m=1

∣∣∣√βkejφke−j(m−1) sin(θk)∣∣∣2=

M∑m=1

∣∣∣√βk∣∣∣2 ∣∣ejφk ∣∣2 ∣∣∣e−j(m−1) sin(θk)∣∣∣2 ,(37)

where∣∣ejφk ∣∣2 =

∣∣e−j(m−1) sin(θk)∣∣2 = 1 and∣∣√βk∣∣2 = βk,

therefore

‖gk‖2 =

M∑m=1

βk = Mβk. (38)

APPENDIX B DERIVATION OF THE PDF OF THEDISTANCE RANDOM VARIABLEGiven that the PDF of Y is defined as

fY (y) =1

R−Rmin, Rmin ≤ y ≤ R, (39)

and the strictly monotonic differentiable function Z =g(Y ) = 1/Y 2, then the PDF of Z is given by

fZ(z) =fY (g−1(Z))

|g′(Y )|, (40)

where g′(Y ) = −2/Y 3 and g−1(Z) = 1/√Z. Therefore,

fZ(z) =fY (g−1(Z))

|g′(Y )|=

1R−Rmin|−2y3 |

=1

2(R−Rmin)z√z,

(41)

where in the last equality we used the fact that y = 1/√z.

VOLUME 4, 2016 9


APPENDIX C USEFUL RESULTSA. CALCULATION OF E

[d2k]

E[d2k]

=

∫ R

Rmin

r2fd(r)dr =

∫ R

Rmin

r2

R−Rmindr =

R3 −R3min

3(R−Rmin).

(42)

B. CALCULATION OF E[

1d2k

]E[

1

d2k

]=

∫ R

Rmin

1

r2fd(r)dr =

∫ R

Rmin

1

r2(R−Rmin)dr =

1

RRmin.

(43)

C. CALCULATION OF E[

1d4k

]E[

1

d4k

]=

∫ R

Rmin

1

r4fd(r)dr =

∫ R

Rmin

1

r4(R−Rmin)dr =

R2 +RRmin +R2min

3R3R3min

.

(44)

D. CALCULATION OF E[log2

(cd2k

)]E[log2

(c

d2k

)]=

∫ R

Rmin

log2

( cr2

) 1

R−Rmindr

=log(c) + 2(R−Rmin−R log(R)+Rmin log(Rmin))

R−Rminlog(2)

(45)

APPENDIX D APPROXIMATED DISTRIBUTION OF THETOTAL POWER GAIN OF THE CHANNEL MATRIXLemma 3. Let βk = η/d2k, where dk,∀k are i.i.d. randomvariables following the uniform distribution with PDF givenby 1/(R−Rmin), therefore, as the number of served devicesincreases, the PDF of Z =

∑Kk=1Mβk can be approximated

by the Gamma distribution with parameters κ = 3KRRmin(R−Rmin)2

and θ = Mη(R−Rmin)23R2R2

min, i.e., Γ(κ, θ).

Proof. This is empirically proved by comparing the nor-malised histogram of Z against the theoretical PDF of aGamma random variable with the parameters defined earlier.

The parameters κ and θ are found by using the mean andvariance statistics of Z, where κ =

µ2Z

σ2Z

and θ =σ2Z

µZ. These

relations are derived based on the definitions of mean andvariance of a Gamma random variable. The mean of Z isgiven by

µZ = Mη

K∑k=1

E[

1

d2k

]= MKηE

[1

d2k

]=

MKη

RRmin,

(46)

where we used the assumption that dk,∀k are i.i.d. randomvariables and the results in Appendix C. Next, the variance ofZ is given by

σ2Z = E[Z2]− µ2

Z , (47)

where

E[Z2] = M2K{E[β2] + (K − 1)E[βk]2}

= M2η2K

{E[

1

d4k

]+ (K − 1)E

[1

d2k

]2}= M2η2K

[R2 +RRmin +R2

min

3R3R3min

− (K − 1)

R2R2min

],

(48)

where E[

1d4k

]is calculated in Appendix C. Therefore, plug-

ging (48) into (47) we have

σ2Z =

Kη2M2(R−Rmin)2

3R3R3min

. (49)

The proof is concluded by replacing (46) and (49) into thedefinitions of κ and θ.

APPENDIX E APPROXIMATED DISTRIBUTION OF THEINSTANTANEOUS SUM-CAPACITY IN HIGH SNR REGIMEAND FAVORABLE PROPAGATIONIn this appendix we demonstrate how the parameters κ and θare derived. These parameters, which are used to approximatethe instantaneous sum-capacity PDF in high SNR regimeand favorable propagation, are found by using the meanand variance statistics of the random variable given by (30),

where κ =µ2Cinst.σ2Cinst.

and θ =σ2Cinst.µCinst.

. These relations are derivedbased on the definitions of mean and variance of a Gammarandom variable. The mean of (30) is given by

µCinst. = E[Cinst.]

= E

[K∑k=1

log2

(ρMη

d2k

)]

=

K∑k=1

E[log2

(ρMη

d2k

)]= KE

[log2

(ρMη

d2k

)]= K

log(ρMη) + 2(R−Rmin−R log(R)+Rmin log(Rmin))R−Rmin

log(2),

(50)

where we used the assumption that dk,∀k are i.i.d. randomvariables and the result in Appendix C-D. Next, the varianceof (30) is given by

σ2Cinst.

= E[C2inst.]− E[Cinst.]

2, (51)

where E[C2inst.] is given by (52) with E

[log2

2

(ρMηd2k

)]being

defined by (53) with a = log(ρMη), b = R − Rmin, c =log(R), d = log(Rmin). Therefore, plugging (50) and (52)into (51) we have (54). The proof is concluded by replacing(50) and (54) into the definitions of κ and θ.

10 VOLUME 4, 2016


E[C2inst.] = K

{E[log2

2

(ρMη

d2k

)]+ (K − 1)E

[log2

(ρMη

d2k

)]2}

= K

{E[log2

2

(ρMη

d2k

)]− E

[log2

(ρMη

d2k

)]2}

=

K

(a2b+4a(R−cR−Rmin+dRmin)+8b+4cR(c−2)−4d(d−2)Rmin

b −K2(a+ 2(R−cR−Rmin+dRmin)

b

)2)log2(2)

.

(52)

E[log2

2

(ρMη

d2k

)]=

∫ R

Rmin

log22

(ρMη

d2k

)1

R−Rmindr

=a2b+ 4a(R− cR−Rmin + dRmin) + 8b+ 4cR(c− 2)− 4d(d− 2)Rmin

b log2(2).

(53)

σ2Cinst.

=

K

(b log2(c)+4a(R−cR−Rmin+dRmin)+8b+4c(c−2)R−4d(d−2)Rmin

b −K(K + 1)(a+ 2(R−cR−Rmin+dRmin)

b

)2)log2(2)

(54)

ACKNOWLEDGMENTThis work was funded by the European Union’s Hori-zon 2020 research and innovation programme under grantagreement No. 732174 (ORCA project). Additionally, thiswork was partially supported by R&D ANEEL - PROJECTCOPEL 2866-0366/2013. This work was partially fundedby EMBRAPII - Empresa Brasileira de Pesquisa e InovacaoIndustrial. E. R. Lima was supported in part by CNPq underGrant 313239/2017-7. G. Fraidenraich was supported in partby CNPq under Grant 304946/2016-8.

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12 VOLUME 4, 2016

Documents

Capacity Bounds for Dense Massive MIMO in a Line-of-Sight