ieee transactions on communications, vol. 65, no. 1...

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 1, JANUARY 2017 403

Modeling and Analyzing MillimeterWave Cellular Systems

Jeffrey G. Andrews, Fellow, IEEE, Tianyang Bai, Student Member, IEEE,Mandar N. Kulkarni, Student Member, IEEE, Ahmed Alkhateeb, Student Member, IEEE,

Abhishek K. Gupta, and Robert W. Heath, Jr., Fellow, IEEE

(Invited Paper)

Abstract— We provide a comprehensive overview ofmathematical models and analytical techniques for millimeterwave (mmWave) cellular systems. The two fundamental physicaldifferences from conventional sub-6-GHz cellular systems are:1) vulnerability to blocking and 2) the need for significantdirectionality at the transmitter and/or receiver, which isachieved through the use of large antenna arrays of smallindividual elements. We overview and compare models for bothof these factors, and present a baseline analytical approachbased on stochastic geometry that allows the computation of thestatistical distributions of the downlink signal-to-interference-plus-noise ratio (SINR) and also the per link data rate, whichdepends on the SINR as well as the average load. There aremany implications of the models and analysis: 1) mmWavesystems are significantly more noise-limited than at sub-6 GHzfor most parameter configurations; 2) initial access is much moredifficult in mmWave; 3) self-backhauling is more viable thanin sub-6-GHz systems, which makes ultra-dense deploymentsmore viable, but this leads to increasingly interference-limitedbehavior; and 4) in sharp contrast to sub-6-GHz systems cellularoperators can mutually benefit by sharing their spectrum licensesdespite the uncontrolled interference that results from doing so.We conclude by outlining several important extensions of thebaseline model, many of which are promising avenues for futureresearch.

Index Terms— 5G, cellular systems, millimeter wave commu-nications, stochastic geometry.

I. INTRODUCTION

UNTIL recently, millimeter wave (mmWave) frequencies –spanning from 30-300 GHz – were not considered useful

for dynamic communication environments such as cellularsystems. Millimeter waves have been used extensively forlong-distance point-to-point communication in satellite andterrestrial applications, now they are being investigatedand developed for commercial cellular systems. This newapplication is much more challenging due to unpredictablepropagation environments and strict constraints on size, cost,and power consumption (particularly in the mobile handset).

Manuscript received May 13, 2016; revised September 2, 2016; acceptedOctober 9, 2016. Date of publication October 19, 2016; date of current versionJanuary 13, 2017. This research has been supported by the National ScienceFoundation, CIF-1514275. The associate editor coordinating the review ofthis paper and approving it for publication was M. DiRenzo. (Correspondingauthor: Jeffrey G. Andrews.)

The authors are with The University of Texas at Austin, Austin,TX 78712 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCOMM.2016.2618794

Given the extreme shortage of available spectrum at traditionalcellular frequencies – often referred to in the industry asSub-6GHz – along with a booming demand for broadbandand other wireless data services, the possibility of usingmmWaves for cellular has generated intense interest startingabout five years ago [1].

A. Millimeter Wave: What’s New?

The misconception that mmWave frequencies do not prop-agate well in free space stems from the λ2

c = (c/ fc)2

dependence in the well-known Friis equation, where λc isthe carrier wavelength, fc is the carrier frequency, and c thespeed of light. The baseline Friis equation, however, appliesto omnidirectional transmission and reception with a specifictype of antenna where the effective antenna area is λ2

c/4π ,which implies that a great deal of energy is lost simply becausethe antennas have a small effective area and cannot radiateor capture much energy. The key observation is that for afixed two dimensional antenna area, the number of antennaelements – each proportional in length and/or width to λc –increases as λ2

c . Thus, the small effective area of each antennacan be overcome by a moderately sized 2-D array of smallantenna elements. With such 2-D arrays at both the transmitterand receiver, this aggregate loss of λ2

c turns into a theoreticalaggregate gain of λ2

c due to the gain of λ2c at each end.

This simple observation has been known long before therecent excitement about mmWave cellular. For example, apaper [2] in 1956 on “Millimeter waves and their applica-tions” makes many of the same points. Its abstract reads“Investigations in the vast 30,000- to 300,000-mc [MHz]frequency range is proving that it can accommodate many ofthe communications services, especially where there is needfor high-gain, high-directional antennas, and large bandwidth.”This one sixty year old sentence summarizes the basic ideaeven today: that with sufficient directionality, millimeter wavescan be used in cellular communications as well, although suchenvironments are usually very different than free space. Thisrequired directionality stemming from large antenna arrays isthe key distinguishing feature of mmWave cellular systems,and it has far-reaching implications on how to model, analyze,design, and implement them.

Another important trait of mmWave cellular systems is theirvulnerability to blocking. Although Sub-6GHz cellular systems

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404 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 1, JANUARY 2017

also suffer from blocking, the effects are much more severefor mmWave. Millimeter waves are particularly sensitive toblocking for three main reasons. First, they suffer much higherpenetration losses when passing through many common mate-rials (including concrete, tinted glass, and water [3]), owingto their smaller wavelength. Second, mmWave frequencies donot diffract well in terrestrial environments because the wave-length is much smaller than the objects it would preferablybend around. This makes blocking objects effectively larger.Third, because of the aforementioned required directionality,both the transmitter and receiver beam patterns are focusedover a more narrow beamwidth, which affords millimeterwave signals fewer chances to avoid strong blocking thanin a nearly omnidirectional transmit/receive scenario whereenergy is radiated and collected over much wider angles.Fourth, mmWave systems usually have large bandwidths andrelatively low transmit powers (due to both regulations andpower amplifier efficiencies), as well as various other hard-ware constraints that erode SNR such as high cabling losses.Therefore received signal-to-noise ratio (SNR) is already at adistinct disadvantage irrespective of any antenna gains, and sothere is not much margin for tolerating blocking.

Along with the strong required directionality, mmWavecellular’s susceptibility to blocking requires important changesto the cellular network architecture and deployment. This inturn requires nontrivial changes to their modeling and analysis.Providing a comprehensive overview of how to adapt to thesechanges from a theoretical perspective is the main focus ofthis paper.

B. Scope and Organization

This overview paper focuses on the communication theoryaspects of mmWave cellular systems, attempting to unifyseveral recent works in this area. As such we focus on themodeling and analysis at the physical layer, with implica-tions on the network architecture and higher layer protocols.Specifically, this paper covers the following topics, which alsocorrespond to the subsequent sections:

• History and state of the art. We provide a brief surveyof the history of mmWave cellular, including recentdevelopments in both theory and practice.

• Blocking. We overview several proposed blocking mod-els, and discuss their relative merits and accuracy.

• Directionality via large antenna arrays. We discussdifferent antenna architectures and their tradeoffs, inparticular analog and hybrid beamforming, single userspatial multiplexing, and multiuser MIMO.

• Analytical tools and approaches. In this, the maintechnical section, we show how to analyze key metricslike the signal-to-interference-plus-noise ratio (SINR) andper link data rate in a mmWave cellular system. Specificcontents are:

– Define metrics of SINR and rate– Define and describe a baseline mmWave cellular

system model– Derive the SINR distribution– Approximate the per link rate distribution

• Design Implications. Based on the analysis, we discusskey considerations that are distinct in mmWave systemsincluding the need for novel initial access techniques;noise vs. interference limited behavior in the context ofdensification; the viability of self-backhauling; and novelspectrum licensing paradigms.

• Extensions of Baseline Model. A great many extensionsare possible, and we overview a few key ones. Theseinclude the uplink; joint coverage with the more robustSub-6GHz network including outdoor-to-indoor cover-age; and MIMO techniques beyond directional analogbeamforming.

II. A BRIEF HISTORY ON MILLIMETER WAVE SYSTEMS

Millimeter wave frequencies have been in use for variousapplications for a long time; it is only recently that theyhave been seriously considered for use in commercial cellularsystems, notably 5G. In this section, we provide a concisechronology of how and why mmWave is now viable forcellular.

A. Pre-Cellular mmWave

Millimeter wave frequencies have been considered andstudied for cellular-like systems as early as 1985 [4], whereinthe use of “fan antennas” to provide directionality gains at a60 GHz carrier frequency were claimed to be able to reach arange of 500 meters, assuming the use of spread spectrum andtargeting a very low data rate (tens of kbps). However, withthe possible exception of other obscure outliers, very littleconsideration was given to the use of mmWave frequencies incellular applications over the subsequent 25 years.

During the interim, mmWave frequencies were leveragedin a host of non-communication applications like radar sens-ing [5], automotive navigation [6], [7], and medical imaging[8], [9]. As far as communication applications, mmWavewas mostly considered early for two diametrically opposedapplications. The first being medium to long-range LOS com-munication using large direction (e.g. dish) antennas, includingbackhaul over a few km and satellite communications.The second main application was vehicular communi-cation [10]–[13], allowing cars to internetwork directlyor through the infrastructure. Though there was an ISOstandard [14], dedicated short range communication at5.9 GHz has become the defacto standard for communicationbetween cars [15].

The consumer revolution in mmWave came with therelease of the the large unlicensed band around 60 GHz[16], [17], which has now culminated in wireless personalarea network (WPAN) and WLAN standards [18]–[20]. Themain target application was for very short range “cablereplacement” type applications. The most successful productsthus far has been the proprietary standard WirelessHD, thoughIEEE 802.11ad [18], also known also as “WiGig”, is nowgaining commercial traction. Both achieve data rates on theorder of several Gbps over short ranges (within a room).The commercial advance of these short-range standards isan important tangential development for mmWave cellular,

ANDREWS et al.: MODELING AND ANALYZING mmWAVE CELLULAR SYSTEMS 405

because it has produced considerable consumer grade device-side innovation, for example establishing the viability of smalladaptive arrays and advancing the low power capabilities ofRF and mixed-signal circuitry [21].

B. Understanding the mmWave Channel

The case for mmWave cellular systems relies on an accu-rate understanding of their signal propagation and channelcharacteristics. While indoor mmWave channels have beenextensively studied, especially at the 60 GHz unlicensedband [22]–[35], thorough measurements of outdoor mmWavechannels began much more recently, after [1].

The measurement and characterization effort for indoormmWave channels started in the late 1980’s at The Univer-sity of Bristol [22]. Inside university rooms, channel mea-surements for the 60 GHz band were performed in bothLOS and NLOS conditions [22]. In [23], some widebandchannel characteristics, such as the excess delay and RMSdelay spread, of the 1.78 GHz and 60 GHz bands werecompared, and the impact of some propagation characteristicslike the atmospheric absorption was illustrated. Later in [24],the multi-path propagation characteristics in a modern officebuilding were measured at 60GHz. Similar studies in differentroom settings were then conducted in [25] and [26]. In [27],the impact of antenna polarization and radiation pattern on theindoor mmWave signal propagation was characterized. Withthe increased interest in defining a 60 GHz WLAN standard,more measurement work has been conducted for the sakeof accurately modeling the channel and signal propagationcharacteristics in this band [28]–[35], with an interferenceanalysis in [36].

Outdoor mmWave channel measurement data increasedgreatly in the last few years [3], [37]–[41]. In [37] and [38],38 GHz outdoor urban cellular channels were measured acrossthe campus of UT Austin using directional transmit anten-nas of 25 dBi power gain and 7° beamwidth, and with atransmit power of 21 dBm. These measurements showed thatacceptable SNR can be achieved in outdoor mmWave linksup to a distance of approximately 200m, with a bandwidth of800 MHz. In [3], measurements at 28 GHz and 38 GHz foroutdoor urban environments around UT Austin and New YorkUniversity provided data on the angles of arrival/departure,RMS delay spread, path loss, and building penetration andreflection coefficients, leading to further models for cellularmmWave channels [39]. Following [38], outdoor mmWavechannel measurements by different groups were also con-ducted [40], [41].

These extensive measurements in [3] and [37]–[41] havedemonstrated that although mmWave signals share basicpropagation characteristics (like power law path loss) withtheir lower frequency counterparts, they also have some veryimportant differences. It is also important to note that the useof directional antennas changes the effective channel seen atthe receiver. For example, directional antennas reduce delayspread [42] and Doppler spread [43], but introduce otherimpairments such as pointing (beam misalignment) errors.Another example is the classic two-ray ground reflection

model, which results in the path loss exponent changing fromα = 2 to α = 4 even for LOS [44]. Directional antennasmake such a model even more questionable, since the ground(or other) reflection will likely not occur to do the focusedbeam pattern.

We summarize the key takeaways to date from these mea-surement campaigns as follows:

• There is sharp difference between line-of-sight (LOS) andnon-line-of-sight (NLOS) propagation for mmWave.

• Because of poor diffraction (due to a smaller Fresnelzone, as discussed earlier), NLOS conditions in mmWaveare due to reflections and scattering.

• There is usually more attenuation on NLOS paths whencompared to Sub-6GHz, due to high penetration lossesand energy losses due to scattering.

• Indoor-to-outdoor (and vice versa) penetration losses aremuch higher at mmWave in most materials, to the extentthat it usually will not be possible to serve indoor userswith outdoor base stations.

• Delay spread is generally much lower at mmWave, but thesymbol time is also much smaller due to the large band-width. Therefore, equalization requirements may even behigher at mmWave.

• mmWave channels are often sparse in the angular domain,with a few scattering clusters, each with several rays, inaddition to a dominant LOS path.

These differences are important to bring into any mathemat-ical model for a mmWave cellular system.

C. The Recent Push for mmWave Cellular

Around the start of this decade, Jerry Pi and Farooq Khanin Samsung’s Dallas Technology Lab were the first to publiclymake the case for mmWave cellular, providing a detailed linkbudget analysis and other persuasive arguments in [1]. Theirlink budget showed that with high gain antennas at both thetransmitter and the receiver (about 15−30 dB), the propagationlosses can be overcome and Gbps-type data rates can beobtained in a cellular architecture; at least theoretically. Thiswas followed by the propagation studies in Ted Rappaport’sgroup at UT Austin that developed extensive channel mea-surements for outdoor mmWave communication, culminatingin [3], which triumphantly (although perhaps prematurely)declared the viability of mmWave cellular, up to cell radiion the order of 200 m.

Notable early prototypes and feasibility studies were carriedout by Nokia [45]–[47] and Samsung Electronics [48], [49]shortly thereafter. For example, in [46], Nokia presentedan experimental system operating at 73.5 GHz with a1 GHz bandwidth, with the BS having a steerable dielectriclens antenna offering 28 dB gain over a narrow 3 degreebeamwidth. The mobile station (MS) had an open endedwave guide antenna with a 60 degree beamwidth. Samsung’sprototype [48] instead offered transmit and receive arrays eachwith 32 antenna elements arranged in an 8 ×4 uniform planararray, in a compact area of 6 cm ×3 cm. The antennas weregrouped into 4 subarrays of 8 antennas each, with one RF unitper subarray, known as hybrid beamforming. The resulting


beamwidth was approximately 10° horizontally and 20°vertically with an overall beamforming gain of 18 dB. Thereported peak data rate with no mobility was about 1 Gbps,over a range up to 1.7 km (LOS) or 200 meters (NLOS).

Universal coverage and the support of mobility are arguablythe key distinguishing features of cellular networks: simplysupporting a link budget (especially outdoors-to-outdoors) isnot sufficient. Links need to be able to be set up quicklyregardless of the mobile’s location, and mobile users need tobe tracked and communicated to on demand. The mobilitystudy in [48] claims that users moving at about 8 km/hr canachieve 500 Mbps with 1% block error rates using steerableantennas. Although this is much less mobility than LTE cansupport, performed under specific conditions, it is hopefully afirst step towards supporting the many dynamics inherent tocellular networks. We will not model or analyze mobility inthis paper either, but the difficulty in supporting mobility anddynamic on-demand connectivity should be kept in mind.

To test the feasibility of realizing large antenna arrays atmmWave mobile terminals and its biological implications,[49] prepared a prototype for a mmWave 5G cellular phoneequipped with a pair of 16-element antenna arrays. This studyfound that the electromagnetic filed absorbed by a user at28 GHz is more localized compared to that at 1.9 GHz. Theskin penetration depth, however, at 28 GHz is much less—around 3 mm compared to 45 mm at 1.9 GHz. This impliesthat most of the absorbed energy is limited to the epidermis atmmWave communications. The biological impact of mmWaveradiation has also been further studied in [50] and [51].

D. Performance Analysis

The highly motivating link budget analysis in [1] wasfollowed up in parallel by several simulation and analysisefforts, e.g. [45], [47], [52]–[57]. As far as the simulation-based studies, in [54] and [55], a measurement-based mmWavechannel model that incorporated blockage effects and anglespread was proposed and further used to simulate the mmWavecellular network capacity. It was found that the achievable ratein mmWave networks outperforms conventional cellular net-works by an order-of-magnitude owing to the large availablebandwidth. It was also observed that the impact of thermalnoise on coverage dominates that of out of cell interference inmmWave networks. In [45], a systematic ray tracing studyincluding roads, sidewalks, and rectangular buildings withoutdoor users showed that mean throughput and cell edge ratescan be improved from 3 to 5.8 Gbps and 25 to 1400 Mbps(a factor of 56!), respectively, by increasing number of basestations in the 0.72km2 region under consideration from36 to 96 (corresponds to increasing base station density from50/km2 to 133/km2). This shows the importance of density inmmWave cellular networks for enhancing throughput, espe-cially the cell edge throughput, which is strongly noise-limited.Around the same time, similar observations were also reportedin [47] and [57].

The impact of the number of antennas on system per-formance was reported in [47] and [56]. In [56], it wasreported that the mean rates improve from about 500 Mbps

TABLE I

SUMMARY OF NOTATION

to more than 4 Gbps when the antenna configuration ischanged from (4,2) to (32,8), where the first number in thebracket indicates the number of antennas at the base stationand the second number indicates number of user antennas.Similarly, cell edge rates increase from about 50 Mbps to200 Mbps. Similar observations were reported in [47]. Hybridanalog/digital beamforming was used in [56] to tackle theanalog to digital converter (ADC) power consumption issue inlarge antenna mmWave networks. The importance of enablingas many number of radio frequency (RF) chains as possiblegiven the power constraints was highlighted in this work.


Although these simulation results appear encouraging, theclaimed rates and outage probabilities are not transparentlyrelated to the many simulation parameters. As with anycommunication system, a mathematical model approximatingthe key features of the system is desirable. A mathematicalanalysis of a mmWave cellular system can help expose keydependencies and bottlenecks in the system, and provide amechanism for incubating and comparing new ideas and differ-ent design approaches without building and running a system-level simulation to test each hypothesis. Although generallytheorists use simulations to validate analysis, the reverse canalso be helpful for system engineers: analysis can provide away to sanity check complex simulations that could have anynumber of bugs.

In parallel to the excitement over mmWave cellular systems,a new analytical approach to cellular systems has gatheredmomentum. This approach, typified by [58], provides a mech-anism for mathematically deriving the SINR distribution in adownlink cellular system. This framework relies on stochasticgeometry [59]–[61], which is an increasingly sophisticatedsubfield of applied probability, wherein the BS locations areassumed to follow a stochastic point process, most com-monly the Poisson point process [62], rather than taking updeterministic grid-like locations. Such an approach has beenshown by an increasing body of evidence to be quite accuratefor Sub-6GHz macrocell-based cellular networks, at least asaccurate as the conventional hexagonal grid model in typicalcircumstances, and typically being pessimistic by a nearlyfixed horizontal SINR shift (i.e. independent of the actualSINR or coverage probability) of 1-3 dB [63], [64].

Stochastic geometry was first applied to analyze the SINRand rate in mmWave cellular in 2012 in [52], where theresults indicated that when the link budget is satisfiedusing large arrays in mmWave systems, mmWave couldprovide comparable SINR coverage and much higher ratecompared with conventional cellular networks. The criticaleffect of blockages was first incorporated in [65], and thenextended to mmWave specifically in [53], and subsequently[66]. We will provide a more detailed description of theseand related contributions in the next several sections froma communication system theory point of view. We nowbegin our attempt to mathematically model and analyzemmWave cellular systems with an in depth discussion ofone of their key differentiating traits: its susceptibility toblocking.

III. NOVEL MODELING ASPECTS: BLOCKING

Obstacles in the environment affect wireless communi-cation channels owing to reflection, diffraction, scattering,absorption, and refraction. These effects are complicated andenvironment-specific, and so the received signal power froma transmitter is often modeled statistically, as a function ofdistance. The traditional first-order approach to incorporaterandomness is to introduce a shadowing random variable,most often log-normal distributed, on top of the averagereceived signal, which is modeled as a function of the distance,e.g. the deterministic power law path loss model �(d) ∝ d−α.

The log-normal distribution for shadowing has a classicalinterpretation in terms of the central limit theorem in viewof many independent obstructions [44]. Shadowing, however,does not accurately capture blocking in dense networks. Forinstance, blockage not only adds randomness to the averagepath loss, but also can dramatically change the effective pathloss exponent [21]. Though the 3GPP standards [67], [68]suggest different channel statistics for LOS and NLOS linksin simulations, blockage seems to be a secondary effect inmacrocell Sub-6GHz networks mostly due to the fact that thelinks are long and thus mostly NLOS anyway. Besides, in Sub-6 GHz bands, the path loss exponent α (typically α > 3), fittedfrom measurements using omni-directional antennas, alreadytakes account for the blocking effects, as well as other effects,including diffractions and ground reflections.

Recent experimental investigations have shown a high sen-sitivity of the mmWave channel to blockage effects. To beginwith, penetration losses through buildings can be as high as40 − 80 dB [69], which is usually insurmountable, and soindoor and outdoor mmWave systems can be considered tobe isolated from one another. Moreover, even focusing on thescenario of outdoor-to-outdoor communication, measurementsshow that static blockages like buildings lead to a largedifference in the path loss laws, usually modeled via differentpath loss exponents, between LOS and NLOS mmWave links[21], [69]. In the presence of blocking, the path loss in theNLOS links can be much higher, as diffractions are weak[21], [70], and a larger fraction of signal energy is scatteredin the mmWave bands [71]. On the positive side, it shouldbe noted that blocking also applies to interfering signals buteven more so, since interferers are typically farther than thedesired transmitter and thus more likely to be blocked. Besidesbuildings and other static objects, mmWave signals are alsoattenuated by smaller objects of smaller sizes, e.g. the humanbody and trees. At mmWave frequencies, the penetration lossthrough the human body is as high as 20-40 dB [72], [73].Given that most use cases for mmWave involve human usersinteracting with the device (as well as other humans frequentlybeing nearby), this is a particularly important type of intermit-tent and severe blocking that changes on a much smaller timescale.

Recent theoretical work in [53], [66], and [74]–[76] hasshown that the coverage and rate trends with blockage switch-ing the path loss exponents can be substantially different fromthe prior results assuming a conventional power law pathloss with a single α value. This will be discussed further inSection VI. The experimental investigations as well as systemcapacity results together signify the importance of accurateyet tractable blockage models for analysis of mmWave cellularnetworks. In this section, we first describe the empirical 3GPPblockage model in Section III-A. Then, we introduce theanalytical blockage models: the random shape theory model inSection III-B, LOS ball model in Section III-C, and Poissonline model in Section III-D. We discussed the model for bodyand foliage blocking in Section III-E. In the end, we presentsome comparisons between them using real geographic datain Section III-F.


A. 3GPP Model for Incorporating Blockages

The 3GPP standards [67], [68], suggest modeling buildingblockages by differentiating the LOS and NLOS links using astochastic model. A function PLOS(d) is a deterministic non-increasing function of d that takes values in [0, 1] and isinterpreted as the probability that an arbitrary link of length dis LOS. Although 3GPP refers to PLOS(d) as the “LOSprobabilty function”, it should be understood that it is nota traditional probability function (such as a PDF, CDF, orCCDF) but rather just a mapping from a positive distance dto a probability of being LOS in [0, 1]. The function PLOS(d)is modeled differently for varying environments, e.g. urban,suburban and rural areas. For instance, in urban areas withregular street layouts,

PLOS(d) = min

(A

d, 1

) (1 − e− d

B

)+ e− d

B , (1)

where A = 18 m, and B = 63 m [67]. In suburban areas,

PLOS(d) = e−d/C , (2)

where C = 200 m. Note that when d is large, the urban LOSprobability in (1) has a heavier tail than in (2). Intuitively, ina regular urban street grid, users are fairly likely to receiveLOS signals from far-away base stations on the same street.

The specific values taken for A, B, C in the 3GPP blockagemodel are based on a relatively limited number of measure-ments from the WINNER II 2007 document, which pre-datesthe deployment of LTE [77]. In [68], the parameter values weremodified to incorporate building heights in the 3D channelmodel; in [55] and [78], the urban LOS probabilities werere-fitted using measurement data in the New York city. Forareas with irregular building deployments, one analyticalapproach is to fit the parameters based on a few first-orderstatistics of buildings, e.g the average size and perimeter [65].This last approach will be discussed in the next section.

It is essential to classify the links into the LOS and NLOStype, where different path loss laws are applied. Clearance ofblockages from the first Fresnel zone of a link has been knownto be a good indicator for LOS links [21], [79], [80]. Fresnelzones are frequency dependent and thus, links that are LOSat 73 GHz need not be LOS at 3 GHz, which has a largerFresnel clearance zone. A recent white paper [69] writtenjointly by Nokia, Qualcomm, Docomo, Huawei, Samsung,Intel, Ericsson and others proposes a 3GPP UMi-like LOSprobability function for static blockages which is frequencyindependent for all bands up to 100 GHz. Evaluations ofLOS probability incorporating the Fresnel effects in [80],alternatively, suggested significant variations between Sub-6GHz and fc > 15 GHz networks, but smaller variations inthe 16-63 GHz range. This suggests that across the mmWavebands it should be possible to use a frequency independentbuilding blockage model, since the Fresnel zone above 15 GHzis narrow. A different model for Sub-6GHz, though, with asignificantly wider Fresnel zone, is likely needed.

The blocking models we will now present, in addition to thebaseline 3GPP model in (1), are all frequency independent.Studies to develop frequency-dependent blockage models arestill in a nascent stage [80].

Fig. 1. Analytical models for building blockages. In (a), the irregular LOSregion to the typical user determined by nearby buildings is approximated bya ball in the LOS ball model.

B. Random Shape Theory Model

To model irregular building deployments, one stochasticblockage model was proposed in [65], based on random shapetheory. The Boolean germ grain model is the simplest processof objects in random shape theory [81], where the centers ofobjects form a Poisson point process (PPP), and each objectis allowed to have independent shape, size, and orientationaccording to certain distributions. As shown in Fig. 1(a),the randomly located buildings are modeled as a Booleanmodel of rectangles. Interestingly, the analysis in [65] showedthat the derived LOS probability function has the same formas the 3GPP suburban function in (2), which is a negativeexponential function of the link length d . More importantly,based on the random shape model, the parameter C in (2) canbe analytically computed using the statistics of the buildingsin the area. For example, assuming the orientation of thebuildings are uniformly distributed in space,

C = π

λbldgE[Lbldg

] , (3)

where λbldg is the average number of buildings in a unit area,and E

[Lbldg

]is the average perimeter of the buildings in the


Fig. 2. 2-D building locations used for comparing blockage models.

investigated region. Another way to obtain C is to choose

C = − πE[Abldg

]ln(1 − κ)E

[Lbldg

] , (4)

where E[Abldg

]is the average area of the buildings in

the investigated region and κ is the fraction of area underbuildings. The results in (3) and (4) provide a quick way toapproximate the parameters of the LOS probability functionwithout performing extensive simulations and measurements.Since the buildings in a geographical region are not necessarilyrectangles, (3) and (4) lead to slightly different estimatesin general. For example, the UT Austin building topologyin Fig. 2 corresponds to C = 100m with (3) and C = 85mwith (4).

C. LOS Ball Model

To simplify the mathematical derivation in the system-levelanalysis, a LOS ball model was proposed [53], [82], where theLOS probability function is modeled as a simple step function

PLOS(d) = 1(d < RB), (5)

1(·) is the indicator function, and RB is the maximum lengthof a LOS link. As shown in Fig. 1 (a), in the LOS ball model,the LOS area, defined as the area that is LOS to a typicaluser, is characterized by a ball of radius RB. Consequently,the maximum LOS length RB can be determined by fitting theaverage size of the LOS area, from either the LOS probabilityfunctions derived from other models or geographic datasets.For instance, to have the same average LOS area with therandom shape theory model,

RB =√

2λbldgE(Lbldg)

π. (6)

When the base station density is high, only a minor gap interms of the SINR distributions is observed using a fitted LOSball model versus the random shape theory model, which isimpressive given its simplicity.

In [66], a generalized LOS ball model was proposed andvalidated using 2-D real building locations in Manhattan andChicago downtown regions. The probability of a link beingLOS in this model is:

PLOS(d) = pl1(d < RB), (7)

where the LOS fraction constant pl ∈ [0, 1] represents theaverage fraction of the LOS area in the ball of radius RB.Clearly, for pl = 1, this reverts to the previous LOS ballmodel. MATLAB code to extract and process building data,and differentiate between LOS and NLOS links has been madeavailable online [83].

The LOS ball models were further extended to a multi-ballmodel in [84]–[87] and an intensity matching based numericaloptimization method was proposed to determine the modelparameters. The idea is to match the intensity of the propa-gation point process (see the references for the definition) ofthe multi-ball model with PPP base stations with that obtainedby using real building locations or a more complicated LOSprobability model. A validation of the proposed model withreal building locations in London was provided for mmWavenetworks in [87].

D. Poisson Line Model

To model a dense urban environment, a Poisson line modelwas proposed in [88]. As shown in Fig. 1(b), the streetsare abstracted as a grid of Poisson lines; the intersectionsalong one line are assumed to be randomly distributed as aPoisson process. The users and base stations located on thelines are considered outdoor, where the locations inside theblocks are indoor; two outdoor locations are considered tobe LOS if and only if they are on the same line. In [88],it was shown that the Poisson line model offers a tractableway to incorporate the correlations in the LOS probabilitiesbetween different links, which was ignored in prior analysisand simulations. The results in [88] show that the tail behaviorof the SINR distributions can be different when incorporatingthe correlations in the LOS probability.

E. Human Body and Foliage Blockage Models

The models discussed thus far are primarily motivated frommacroscopic rigid obstructions like buildings. There are some


recent attempts to model blockage effects due to smallerobjects like trees or the human body. In [89], the foliage loss indB is found to be a linear function of the path length throughtree canopies. In [90], ray tracing was used to come up witha distance-based blocking probability function by other usersand foliage was fitted from ray-tracing simulations as a linearfunction of the link length x . The LOS probability was foundto be of the form min(ax +b, c), where the parameters a, b, care deployment dependent. In [75], a cone-blocking modelwas proposed to model the probability of self-body blockingin outdoor mmWave cellular networks, where all the signalsfrom a cone in the angle space are assumed to be blockedby the user’s self-body, and the fraction of the blocking conecan be estimated based on the position and size of the user.In [91], a human body blocking model was proposed forindoor mmWave wearable networks, where the bodies ofboth the self-user and other users are modeled as cylindersof certain sizes, and the blocking probability of a link wascomputed as a function of the relative locations.

Most of the current analysis focuses on static blockages andusers. In the future, it would be interesting to incorporate timedynamics to study the impact of penetration losses on coveragefrom mobile obstacles and the resulting impact on handoverrates. For example, the dynamics of self-body blocking can bemodeled as a shift of the blocking cone over time [75].

F. Comparison and Conclusions on Blockage Models

We have overviewed several blockage models, each withtheir own set of modeling assumptions. An obvious question iswhen to use which blockage model? We attempt to answer thisquestion using simulation methodology similar to [66], basedon 2-D real building data in the UT Austin and downtown LAregions as shown in Fig. 2. Though, every different environ-ment will experience different blocking behavior, observationsbased on these two environments, along with our previousstudies for NYC and Chicago, share a few common points.

We consider a 28 GHz carrier frequency with 200 MHz ofbandwidth operating in the downlink. Path loss exponents arechosen to be 2 for LOS and 3.3 for NLOS, and lognormalshadow fading has standard deviation of 3.1 dB for LOS and8.2 dB for NLOS [69]. The noise figure is 10 dB and thetransmit power is 30 dBm. UEs are assumed to be omnidirec-tional and BSs have a step beam pattern (refer Fig. 6) with10 degree 3 dB beamwidth, 18 dB maximum gain and 20 dBfront to back ratio. For simulations with actual buildings,we consider a dense network with an average 30 BSs/km2

distributed randomly in the outdoor region. If the urbanregion has dimensions X × Y, then the user location whoseperformance is to be evaluated is placed outdoors randomlyin the central X/2 × Y/2 rectangle. We now summarize somekey observations and suggest methodology for choosing theparameters for the blockage models.

1) 3GPP-Like Model: We curve fit the LOS probabilityobtained using the building locations with that in (1). As canbe seen from Figure 3, the fit is good with root mean squarederror 2.18% for the Austin region and 1.45% for LA region.This matches the insight in [69], that 3GPP-like models could

Fig. 3. Fitting LOS probability for 3GPP-like UMi model.

be sufficient to fit the LOS probability in urban regions.The parameters obtained for Austin and LA are as follows.A = 6.659m and B = 129.9m for Austin and A = 13.89mand B = 63.76m for LA. However, as shown in Figure 4,fitting the LOS probability but neglecting the correlation doesnot necessarily mean a good fit to metrics of interest, likesignal-to-interference-plus-noise ratio (SINR) or rate coverage.

2) Random Shape Theory Model: Conditioned on the usersand BSs being outdoors, a simple upper bound on the LOSprobability is PLOS(d) = min (exp(−d/C + δ), 1), where hereδ = − ln(1 − κ), where κ is the fraction of area underbuildings. Using (4), C = 85m for Austin and C = 42mfor Los Angeles. Also, κ = 0.27 for Austin and κ = 0.42 forLA. From Fig. 4 it can be seen that this LOS probability givesa reasonably tight upper bound to the SINR coverage obtainedusing real building locations near UT Austin, similar to [65].However for LA, this model underestimates the coverage.

3) Generalized LOS Ball Model: Similar to [66], pl iscomputed as the average LOS fractional area in a ball ofradius RB from the region under consideration. Since weconsider only outdoor deployments for users and BSs, thefractional area is computed as the ratio of LOS area in ballof radius RB centered at a user location and the total outdoorarea in that ball averaged over several such user locations. Thechoice of RB is flexible in this model, but it should be largeenough to make sure that with high probability, the serving linkand dominant interfering base stations fall within the ball ofradius RB centered at the user. Generally, mmWave networksare envisioned to be dense with inter-site distance less than orequal to 200m: we choose RB = 200m. The correspondingpl = 0.3027 for Austin and pl = 0.2419 for LA. Thismodel accurately fits the SINR coverage obtained using boththe urban regions under consideration, which is surprisingconsidering the simplicity of the model. The observations onthis model until now have suggested that the choice of the ballradius between 150-300m gives a good fit for dense randomdeployment of BSs (with cell radius typically lesser than RB)in Manhattan, Chicago, LA and Austin regions consideredin [66] and this paper. Further empirical studies, including


Fig. 4. Comparison of blocking models.

possible joint optimization of RB and pl to optimize the SINRcurve fit, would be useful. A recent effort in this directionis the intensity matching based approach in [84]–[87] foroptimizing the blockage modeling parameters, as discussedin Section III-C.

All the blockage models mentioned in this comparisonsection neglect the correlation of two links being blocked bythe same obstruction. The Poisson line model [88] can handlecorrelations, but is difficult to validate because it assumes avery specific street and user geometry quite different than allthe other models (or most real cities). The LOS ball and therandom shape theory models are simple to incorporate in theanalysis, wherein the above observations imply that an appro-priate choice of the blockage parameters potentially reflectsreal world blockage scenarios. The 3GPP-like urban micro-cellular model is more complex to incorporate in analysis,and it was observed that fitting the empirical LOS probabilityfunction does not guarantee a good fit to the coverage esti-mates, in fact it in most cases has a much poorer fit that theLOS ball or the random shape theory blocking model.

We conclude by noting that the analytical approachdeveloped starting in Sect. V depends on the blocking modelonly through the use of a generic PLOS(d) function so anarbitrary blocking model can be used, subject to the caveatthat spatial correlation amongst blocking objects is ignored.The analytical results, however, can be reduced to more simpleforms when certain LOS probability models, e.g. the LOS ballmodel, are applied in the derivation.

IV. NOVEL MODELING ASPECTS:LARGE ANTENNA ARRAYS

The use of large – in terms of the number of elements,not the physical size – antenna arrays at the base station andmobile users is a key feature of mmWave cellular systems. Theways these antennas are used at mmWave differs from lowerfrequencies owing to hardware limitations on MIMO trans-ceiver architectures. In this section, we discuss how mmWavesingle-user/multi-user MIMO transmission techniques differfrom their counterparts at lower frequencies. Understandingthese large antenna array aspects is essential for proper mod-eling and analysis of mmWave cellular systems.

A. Hardware Constraints and the Need forDifferent Transceiver Architectures

Initial mmWave research and prototypes suggest array sizesof 32−256 antennas at the base station and 4−16 antennas atthe mobile users [48], [49], [94], [95]. Realizing these numbersof antennas in a small package is feasible thanks to the recentdevelopments in antenna circuit design [49], [96]–[100]. Thelarge arrays, though, can not be used at mmWave in the sameway they are used at lower frequencies due to the high powerconsumption of the mixed-signal components.

In conventional cellular systems, precoding and combiningis performed at baseband using digital signal processing. Thisallows better control over the precoding/combining matrices,which in turn facilitates the implementation of sophisticatedsingle user, multiple user, and multi-cell precoding algorithms.Performing such baseband precoding/combining processingassumes that the transceiver dedicates an RF chain per antennaas shown in Fig. 5(a). This fully-digital processing is hardto realize at mmWave frequencies with wide bandwidths andlarge antenna arrays. This is mainly due to the high cost andpower consumption of mixed-signal components, like high-resolution analog-to-digital converters (ADCs) [101], [102].For example, it is presently infeasible for mmWave receiversto have 32-256 full-resolution ADCs, so traditional MIMOtransceiver architectures that allocate an RF chain for eachantenna are very difficult to realize. Different transceiverarchitectures that comply to these hardware constraints havetherefore been proposed [95], [103]–[107]. We now overviewsome key candidate transceiver architectures for mmWavewireless systems.

1) Analog Beamforming: An immediate solution to over-come the limitation on the number of RF chains is to performbeamforming entirely in the RF domain using analog process-ing. Analog beamforming is normally implemented usingnetworks of phase shifters as shown in Fig. 5(b) [108], [109].


Fig. 5. This figure shows a tranmsitter having Ntx antennas with afully-digital, analog-only, or hybrid analog/digital architecture. In the hybridarchitecture, NRF � Ntx RF chains are deployed.

The weights of these phase shifters are tuned to shape and steerthe transmit and receive beams along the dominant propagation

directions. Mathematically, if the transmitter wants to transmita symbol s, with the Ntx × 1 beamforming vector fRF, thenthe transmitted vector x can be written as

x = fRFs, (8)

where the entries of the RF beamforming vector are subject toa constant modulus constraint due to the implementation withphase shifters. Therefore, these entries can be expressed as(fRF)n = e jθn , n = 1, 2, ..., Ntx. Depending on the channeland the antenna array geometry, these phases {θn}Ntx

n=1 aredesigned normally to maximize the beamforming gain at thereceiver. To avoid the overhead of explicitly estimating thelarge mmWave channel, analog beamforming weights canbe directly trained using beam training [108]. One commonapproach for beam training is to use a codebook of beampatterns at different resolutions, and iteratively find the thebest beamforming vector codeword from this codebook [18],[108], [110]. Despite its simplicity, analog beamforming issubject to hardware constraints such as the phase shifter quan-tization, which make analog beamforming/combining solu-tions limited to single-stream transmission and difficult toextend to multi-stream or multi-user MIMO communication.Analog beamforming is available already in Wireless HD andIEEE 802.11ad products, therefore it is seen as commerciallyviable in the near-term. Much of the analysis in Section Vassumes analog beamforming, which is approximated using asectored antenna model.

2) Hybrid Precoding: Hybrid analog/digital architecturesprovide a flexible compromise between hardware complexityand system performance [95], [103], [111]–[116]. In hybridarchitectures, the precoding/combining is divided between theanalog and digital domains as illustrated in Fig. 5(c). Thisallows the use of a number of RF chains NRF much less thanthe number of antennas, i.e. NRF � Ntx. One key advantage ofhybrid precoding is that it permits the transmitter and receiverto communicate via several independent data streams, andhence achieve spatial multiplexing gains [105]. Consider aBS transmitting NS data streams to a mobile user, and bothof them employing hybrid architectures with NRF RF chains.Let the NRF × NS matrix FBB, and the Ntx × NRF matrixFRF denote the baseband and RF precoders at the BS, andthe NRF × NS matrix WBB, and the Nrx × NRF matrix WRFrepresent the baseband and RF combiners at the mobile user.Then, the received signal after processing can be written as

y = W∗BBW∗

RFHFRFFBBs + W∗BBW∗

RFn, (9)

where the RF precoders/combiners are subject to a similarimplementation constraints as those discussed in the analogbeamforming section.

Despite the much smaller number of RF chains comparedto the number of antennas, hybrid architectures were shownto achieve near-optimal performance compared to fully-digitaltransceivers in [95], [103], and [112]–[116]. To further reducethe power consumption, [106] and [117] proposed to replacethe phase shifter networks in the hybrid architectures witha network of switches. The RF precoding matrices can alsobe realized using lens antennas, which compute the spatialFourier transform, and can work as analog beamforming


vectors with a DFT structure [107]. As the power consumptionin the full-resolution ADCs may be a challenge at mmWave,[101] and [104] proposed using low-resolution ADCs. Hybridarchitectures with few-bit ADC receivers have also beenrecently investigated in [118]. Extending the system analysisin Section V to include all the facets of hybrid precoding orother architectures is largely a topic for future work.

B. Spatial Channel Modeling

Measurements of outdoor mmWave channels show that theynormally have a small number of dominant scattering clusters[3], [55], [119]. Therefore, geometric channel models witha few clusters are commonly adopted to describe mmWavechannels for system capacity analysis or precoder design[55], [103], [120]. Most studies assume a channel whichis non-selective in both time and frequency for simplicity,although there is some recent work on designing precoders andcombiners for frequency selective mmWave channels [113].Let the Nrx × Ntx matrix Hb denote the downlink channelfrom the bth BS at Xb to a typical user at origin. Then,Hb can be written as

Hb = 1√�(|Xb|)

ηb∑p=1

hb,p aMS(θb,p

)a∗

BS

(φb,p

), (10)

where hb,p is the small-scale fading of the pth path, �(|Xb|)is the path loss, ηb is the total number of paths betweenthe BS and user, wherein each path is a representative of acluster of paths due to a scatterer in the environment. Theangles θb,p and φb,p denote the pth path spatial angles ofarrival and departure (AoA/AoD) at the user and the BS.Finally, aMS

(θb,p

)and aBS

(φb,p

)are the array response

vectors at the MS and BS, respectively. The spatial anglesare a function of the physical AoA/AoD as well as the arraygeometry. For a uniform linear array (ULA) with N antennas,where N ∈ {Ntx, Nrx}, inter-antenna spacing d and steered atsome physical AoA/AoD given by ϕ, the corresponding spatialangle is θ = 2πd sin(ϕ)/λc and the array response vector isgiven by

a(θ) = [1 exp(− jθ) exp(−2 jθ) . . . exp(− j (N − 1)θ)

].

(11)

The distribution of the AoAs/AoDs can be modeled using theempirically observed power angular spectrum [39], [121].

For uniform arrays, a useful representation of the channelin (10) can be obtained by characterizing the channel responseat the spatial quantized angles 0, 2π/N . . . , 2π(N − 1)/N .This is particularly useful for network level analysis withhybrid or analog precoders/combiners using phase shifters orlenses and a large number of antennas at the BSs and MSs, asit gives rise to the ON/OFF nature of interference [112], [122].The reason is that each array response vector becomes nowequivalent to a column of the N-point DFT matrix. Thischannel characterization, which is called the virtual channelrepresentation [123], is defined as

Hb = ARHbA∗T =

Ntx∑k=1

Nrx∑l=1

[Hb]k,laMS (θk) a∗BS (φl) (12)

Fig. 6. Approximated sectored-pattern antenna model with main-lobegain Ms, side-lobe gain ms, and main-lobe beamwidth �s.

where AR and AT contain the array response vectors forthe receiver and transmitter with spatial AoAs (AoDs) takenover a uniform grid of size Nrx (Ntx), and Hb is a matrixwith each entry representing the channel gain correspondingto a different combination of the permissible AoAs/AoDs.Exploiting the sparseness of the mmWave channel in thespatial domain, most of the terms in the double summationwill be zero and the above representation can be equivalentlyrepresented as a single summation over the distinct pathsbetween the BS and user, as given in (10) but with quantizedspatial AoA/AoD.

C. Single Stream Analog Beamforming

In single stream beamforming, the BS and mobile useruse the antenna arrays to transmit/receive one data stream.Let f and w denote the beamforming/combining vectors, thereceive SNR can be expressed as

SNR = |w∗Hf |2σ 2 . (13)

The design objective for the beamforming/combining vec-tors is usually to maximize this SNR. When the channel isdominated with a LOS path or when the number of scatterersis small, it becomes reasonable to design the beamformingvectors to maximize the beamforming gain in a certain desireddirection θd , which is called beamsteering. One way to dothat is by adjusting the beamforming weights to match thearray response vector in the desired direction, i.e., to setf = a (θd). This results in a beam pattern with a main lobe inthe desired direction. Other beam designs that trade-off mainlobe and side lobe levels are also possible. For analyticaltractability, it is common to approximate the actual arraybeam pattern by a step function with a constant main-lobeover the beamwidth and a constant side-lobe otherwise, shownin Fig. 6. Such a model has been adopted in [53], [66], [84],and [124] for tractable coverage and rate analysis of mmWavecellular networks, and its accuracy has been validated in [125](see Fig. 3) and [124] (see Fig. 9).

Thanks to its digital processing layer, hybrid architecturesoffer more degrees of freedom in beamforming design thanpurely analog beamforming. This can be used, for example,to realize beam patterns with better characteristics [126]. Forsteering the beam in the azimuthal as well as the verticaldirections, it is desirable to have a uniform planar array (UPA).Most industry papers assume a uniform planar array for singlestream beamforming [47], [48]. Existing analysis of mmWave


cellular networks has been focused on deployments of basestations and UEs on a 2-D plane [53], [66]. In this case,the step beam pattern can be modeled with an antenna gaincorresponding to the entire 2-D UPA whereas the 3 dBbeamwidth corresponds to only the number of antenna ele-ments of the UPA in the azimuth direction. Omni-directionalantenna arrays give rise to an image beam in a non-desirabledirection. This back lobe gain is equal to the gain in the desireddirection. Therefore, using antenna elements which themselveshave a non-omnidirectional pattern makes sense. To provideomni-directional coverage with directional antenna elements,each access point may need to employ several antenna arrayswith each one serving a different sector [47], [54]–[56]. Densenetworks are desirable at mmWave and one cheap (but possiblysuboptimal) way of densifying is having multiple sectors persite that reuse time-frequency resources, as the operators donot need to lease more sites or spectrum. Thus, unlike inSub-6GHz networks, using the same time-frequency resourcesacross all sectors of an access point could be feasible atmmWave [47].

D. SU-MIMO

To improve the spectral efficiency in single-user MIMOsystems, spatial multiplexing – where multiple streams aresimultaneously transmitted – is an obvious solution. In con-ventional single user MIMO systems with fully-digital trans-ceivers and perfect channel knowledge, channel capacity isachieved with singular value decomposition (SVD) precoding/combining and a water-filling power allocation [127], [128].Mathematically, let H = U�V∗ denote the singular valuedecomposition (SVD) of the channel matrix H, then set thetransmitter precoding matrix as F = V�, with � being adiagonal water-filling power allocation matrix, and the receivercombining matrix as W = U. LTE systems operating in closedloop spatial multiplexing (transmission mode 4) can be viewedas performing a crudely quantized approximation of thisSVD-based procedure motivated by information theory [129].It also does not work particularly well due to excessivequantization of the channel state information. At mmWave,the hardware constraints on the entries and dimensions of theprecoding and combining matrices makes using an approxi-mation of SVD precoding even more dubious. This motivatesresearch to develop new precoding solutions for SU-MIMOmmWave systems.

Exploiting the sparsity of mmWave channels, low-complexity hybrid precoding algorithms were proposedin [103] to approximate the spectral efficiency achieved withSVD and fully-digital precoding. With some approximations,the hybrid precoding design problem was formulated as{

F�RF, F�

BB

} = arg minFRF∈A

‖FRFFBB‖2F=NRF

∥∥Fopt − FRFFBB∥∥2

F , (14)

where the first constraint is due to the hardware constraintson the RF precoding matrix, which limits it to a certainset of precoding matrices A, and the second constraint isa power constraint. If the mmWave channel has ηb pathswith known angles of departure at the transmitter, then

Fig. 7. A multi-user mmWave downlink system model, in which a BS useshybrid analog/digital precoding and a large antenna array to serve U Mobileusers.

[103] develops a matching pursuit variant to greedily designthe RF and baseband precoding matrices. Following [103],the work in [115] and [130]–[132] used matrix decomposition,alternative minimization, and other techniques to design thehybrid precoders adopting the same optimization problemin (14). In terms of modeling, the solution in [103] can beinterpreted as a number of NRF beam patterns, that can beapproximated as that in Fig. 6, representing the column of theRF precoding matrix FRF, with additional processing done inthe baseband using the FBB. Other hybrid precoding designsthat do not directly rely on the approximation in (14) have beendeveloped in [113], [116], and [133] with the same objectiveof maximizing the system spectral efficiency. The solutionsin [103], [113], [115], [116], and [130]–[133] showed thathybrid precoding can generally achieve very good spectralefficiencies compared to the fully-digital SVD solution inmmWave systems, specially when the number of RF chains isclose to the number of dominant channel paths.

E. MU-MIMO

The antenna arrays can also be used to support multi-user MIMO, where users share the same time/frequencyresources. To enable efficient multi-user precoding processingin mmWave systems, [112] proposed a two stage hybridprecoding technique. The first stage assigns a different analogbeam to each user to maximize the received signal power,as illustrated in Fig. 7. Considering the effective channels,further baseband processing is performed to cancel the inter-user interference. This simple precoding strategy was shownto achieve very close results to the unconstrained digitalsolutions, despite its requirement of low training and feedbackoverhead. Consider the multi-user hybrid precoding systemmodel in Fig. 7, with a BS employing hybrid analog/digitalarchitecture and serving U users that use analog-only combin-ing. Then for single-path channels in a single cell setup, theSINRu of user u can be lower bounded by [112]

SINRu ≥ SNRu G (U, Ntx, ηb) , (15)

where G (U, Ntx, ηb), is a constant that depends only onU, Ntx, ηb, and represents the signal power penalty resultedfrom canceling the multi-user interference. Note that SNRu

is the SNR of user u without inter-user interference, i.e.,when the BS serves only this user. Similar expressions can bederived for the multi-path case by using the virtual channelapproximation in Section IV-B. One advantage of the dis-cussed multi-user hybrid precoding technique is its relativeanalytical tractability in the stochastic geometry framework,


as the distribution of the SINRu in (15) can be easilycharacterized [122]. Similar multi-user mmWave beamformingalgorithms have been proposed based on lens antenna arrays[107], [134], where the DFT properties of the lens antennas areexploited to dedicate orthogonal directions to different users.Multi-user mmWave combining has also been studied for theuplink system model with hybrid architectures [135].

V. DOWNLINK SINR AND RATE DISTRIBUTION

Having considered the ways that mmWave cellular systemsdiverge from conventional ones, we now turn our attention totechniques to analyze their performance.

A. Performance MetricsWe focus on two fundamental performance metrics which

are both random variables and thus characterized by theirdistributions: the SINR and the per user rate.

1) The SINR and Coverage Probability: The post process-ing SINR after the receiver combining operations in boththe downlink and uplink is the fundamental metric to under-standing how mmWave cellular systems perform. We focus onthe downlink SINR, which is a complicated random variabledepending on many constituent random variables including(i) the distance separating the desired transmitter and receiver,(ii) the relative distances from all interfering transmissions,(iii) the random channel effects in both the desired and inter-fering links, including blocking, fading, and possibly shad-owing, and (iv) the beam patterns appear randomly oriented,particularly for the interferers, and (v) the thermal noise.

The SINR is most commonly characterized as the coverageprobability defined as S(T ) = P[SINR > T ] relative to anSINR threshold T . It is precisely the CCDF of SINR anddescribes the fraction of users that will achieve SINR > Tin the network (averaged over time and space). Unlike inSub-6GHz networks, the SNR, which is a special case ofSINR with the interference being negligible, is also a usefulmetric. In many cases mmWave cellular systems will benoise-limited (or equivalently, power-limited), due to the pathloss and blocking effects discussed earlier, in conjunctionwith the large bandwidth which brings in much more noisepower. In such cases, the SNR distribution can be used asan approximation of the SINR, allowing much simplification.This is in stark contrast to Sub-6GHz cellular networks, whichare often interference-limited, meaning SIR ≈ SINR instead.

2) Data Rate, Characterized by the Rate Coverage Proba-bility R (τ ) = P[R > τ ]: This metric builds upon SINR andis the most important metric from a performance standpoint,since it most directly affects the perceived experience. HereR is the (random) data rate per active user, in units ofeither bits per second (bps) or bps/Hz if normalized by thebandwidth. However note that this is not simply the systemspectral efficiency, which is usually interpreted to be theaggregate rate (not the per user rate R here) divided bythe bandwidth. Rate is also a random variable depending ontwo complicated underlying random variables: (i) the SINRthrough the usual log(1 + SINR) relation assuming Gaussiansignaling and (ii) the fraction of resources that a given userreceives over a fairly short time-frame (fractions of a second).

To analyze these two metrics, we now describe a tractablemodel for downlink mmWave cellular network, based on thephysical characteristics of mmWave systems as described inthe preceding sections.

B. Baseline mmWave Cellular System Model

The model is based on the traditional analytical frameworkfor Sub-6GHz cellular networks [58], [136], but incorpo-rates blockage effects and directional beamforming. The mainaspects of the baseline model are now enumerated.

1) Base station locations. We assume the BSs are alloutdoors, but still independently distributed according toa homogeneous PPP � = {Xb : b ≥ 0} of density λ on aplane, where Xb is the location of the b-th base station.The impact of indoor mmWave base stations is ignored,due to the large penetration losses. Conceptually, thismeans we ignore the locations of buildings as far asdetermining BS locations (or equivalently assume theyare mounted on the buildings when they happen to be“dropped” there). We discuss this model’s applicabilityfurther below.

2) User locations. The users are also assumed to be out-doors, and form an independent PPP �u on the sameplane, with density λu . Each user associates with thebase station that has the smallest path loss. Analysisis conducted for a user at the origin for mathematicalsimplicity. Because of the stationarity of the PPP, thisuser can be considered “typical” and thus has the sameaverage performance as a user at any location. Theserving base station for this user is denoted as X0. In thissection, we assume the user density is sufficiently highthat all base stations are transmitting constantly, whichis pessimistic for SINR [137]. An extension to consideruser loads can be found in the subsequent discussion onrate distributions.

3) Blocking. Recall that both BSs and users are assumedto be outdoors. For this scenario, mmWave base sta-tions can be divided into two sub-processes: the LOSbase stations (unblocked) and the NLOS base stations(blocked). The probability that a link of length d isLOS is given by PLOS(d) as in Section III. The eventsthat any two distinct links in the network are LOSare assumed to be independent. We leave PLOS(d) asa general expression in our analysis, i.e. it can followany of the models discussed in Section III that result inindependent blocking for all links. Therefore, the LOSand NLOS base stations form two independent non-homogeneous PPPs �L and �N, to which different pathloss laws will be applied. Note that the non-homogeneityof the LOS and NLOS base stations processes are due tothe distance-dependence of the LOS probability functionPLOS(d).

4) Beamforming: Analog beamforming is applied atboth base stations and mobile stations. The exten-sions to hybrid beamforming will be discussed inSection VI. The typical user and its associated basestation are assumed to have perfect channel knowledge,


TABLE II

PROBABILITY MASS FUNCTION OF Gb (b > 0)

and adjust their steering orientation to achieve the maxi-mum directionality gain. The steering angles of the inter-fering base stations are uniformly distributed in space.We approximate the actual array pattern by the sectoredmodel as shown in Fig. 6. Let Gb be the total directivitygain (of both transmitter and receiver beamforming) inthe link from the typical user to base station Xb. Then,for the interfering link, i.e. for b > 0, the directivity gainGb is a (discrete) random variable with the probabilitydistribution as Gb = ak with probability bk (k ∈{1, 2, 3, 4}); ak and bk are constants defined in Table II;cBS = �BS

2π , and cMS = �MS2π ; and for s ∈ {MS, BS}, Ms ,

ms , and �s are the main lobe gain, side lobe gain, andmain lobe beamwidth for the base stations and mobilestations (as plotted in Fig. 6). For the desired signal link,perfect beam pointing is assumed with G0 = MBS MMS.

5) Path loss model: Different path loss exponents areapplied to the cases of LOS and NLOS links. Given alink has length d , its path loss �(d) is

�(d) ={

CLd−αL w.p. PLOS(d)

CNd−αN w.p. 1 − PLOS(d),(16)

where αN are the LOS and NLOS path loss exponentsand CL, CN are the intercepts of the LOS andNLOS path loss formulas. The intercepts CL and CNare the same for LOS and NLOS links, when thesame closed-in reference distance dref is used, e.g.dref = 1 meter in [39]. Typical values for mmWave pathloss exponents can be found in measurement results, e.g.in [3], [39], and [138], and for simplicity we use αL = 2and αN = 4 as default values.

6) Association Rule: This paper uses the minimum pathloss association rule, meaning that each user associateswith a single BS, which is the one which has theminimum average power loss �(d) to the UE. Note thatthis may not be the closest BS, since we distinguishbetween LOS and NLOS propagation as just discussed.Such an association rule will result in the maximumaverage SINR, but not necessarily the maximum rate,as discussed in Sect. V-D.

7) Small-scaling fading: Measurements shows that small-scale fading has a relatively minor impact in mmWavecellular systems [3]. The Rayleigh fading model for thesub-6GHz band, which is predicated on a large amountof local scattering, does not apply in principle formmWave bands, especially when directional beamform-ing is applied [3]. Therefore, we assume independentNakagami fading for each link, which is more generalbut still tractable. Different parameters of Nakagamifading νL and νN are assumed for LOS and NLOS links.Let hb be the small-scale fading in signal power in the

b-th link. Then under the Nakagami fading assumption,hb is a normalized Gamma random variable. Forsimplicity, we assume νL and νN are positive integers,and ignore the frequency selectivity in fading.Shadowing is ignored in our baseline model, but can beincorporated using the approach in [66] at some cost intractability.

Based on this model, the downlink SINR can be expressedas

SINR = h0G0� (|X0|)σ 2 + ∑

b>0:Xb∈� hbGb� (|Xb|) , (17)

where σ 2 is the noise power and |Xb| is the norm of thelocation Xb, which denotes the distance of the link from Xb

to the typical user at the origin.As many readers will be aware, significant progress has

been recently made in the characterization of SINR distri-butions of similar form to (17). The key underlying toolsetdescends from stochastic geometry [59]–[61], which relies onthe base stations following a random spatial distribution likewe have here. Although the PPP is an idealized distributionthat assumes independent BS locations, it has been foundto describe important SINR trends observed under actual BSdistributions [58], and is typically accurate to within a 2-3 dBfixed SINR gap (and is pessimistic versus real BS locations)[63], [64]. Since there are not any actual mmWave cellulardeployments at the time of writing to compare against, anymodel – tractable or not – is speculative, but given that thePPP results in similar SINR curves to a very large class ofBS locations (including grids), it seems quite reasonable for abaseline model.

A contemporary entry-level tutorial cellular analysis usingstochastic geometry for both rate and SINR for Sub-6GHzsystems can be found in the recent survey [139] as well as therecent tutorial [136], which includes a step-by-step descriptionof how to get the SINR distribution for downlink, uplinkand multi-tier downlink cellular networks at conventionalfrequencies. Thus, we do not repeat this background here, butinstead extend and generalize these now well-known results tothe mmWave cellular case.

C. SINR Downlink Coverage Probability

We now derive the SINR coverage probability for mmWavecellular systems. The new technical difficulties comparedwith the Sub-6GHz analysis are: (i) the small-scale fadingis modeled as a Nakagami random variable, not Rayleighdistributed, such that the SINR coverage probability cannotbe directly obtained from computing the moment generationfunction (MGF) of the interference as in [58]; (ii) the basestation locations are seen as a superposition of inhomogeneous


TABLE III

ANALYTICAL METHODS FOR NAKAGAMI FADING

PPPs representing LOS and NLOS base stations from a typicaluser in the network; and (iii) the directivity gain from beam-forming introduces additional randomness in the interference,compared with the omni-antenna case.

To overcome the first difficulty, several approaches havebeen proposed to compute the SINR and rate distributionunder the Nakagami fading assumption, as listed in TableIII. One direction is to compute the SINR and rate coverageprobability from the MGF of interference, based on advancedmathematical tools, including the Parseval theorem [60], [140],Gil-Pelaez inversion theorem [141], Hamdi’s lemma [142], Faadi Bruno’s Lemma [143], or numerically inverting the MGF[66]. These types of methods (except Faa di Bruno’s Lemma)generally apply to arbitrary fading distributions, and requirecomputing numerical integrals on the complex plane. Toreduce the computation complexity, approximate expressionsto compute the SINR and rate distributions were proposed forthe Nakagami fading in [141].

Another direction is to apply certain approximations, e.g.,based on Alzer’s Lemma [53], [144], the intensity matchingapproach [84]–[87], or the assumption to treat the small-scalefading as Rayleigh [145], to simplify the SINR and rate com-putation. The results in [145] showed that though not produc-ing the exact SINR and rate distributions as general Nakagamifading, treating the small-scaling fading as Rayleigh largelysimplified the analytical expressions for further analysis,andmaintained certain key design insights, e.g. the stochasticordering of the SINR distributions among different scenarios.In [84]–[87], the authors proposed an intensity matchingapproach that replaces the intensity measure of the interferenceprocess in the received power domain with certain approximatefunctions of simple forms; the parameters of the simplifiedintensity function are determined by solving a numericaloptimization problem minimizing the approximation error. Asillustrated below, the Alzer’s Lemma [144] approximates theCDF of a gamma random variable into an weighted sum ofthe CDFs of exponential random variables. The approximationis shown to be generally tight in numerical simulations withdifferent system parameters, and was used in prior analysis ofSub-6GHz MIMO networks in [146]. To provide a concreteexample of the solutions to overcoming the Nakagami fadingassumption, we present the derivation using Alzer’s Lemma

in this section. Before that, the Alzer’s Lemma is provided asfollows.

Lemma 1 (Alzer’s Lemma [144]): Let h be a normalizedgamma random variable with parameter ν. For a constantγ > 0, the probability P(h < γ ) can be tightly upper boundedby

P(h > γ ) ≤ 1 − [1 − e−ηγ

]ν =ν∑

n=1

(−1)n+1(

ν

n

)e−ηnγ ,

where η = ν(ν!)− 1ν , and the equality holds when ν = 1.

Then, we can compute the SINR coverage probability asthe Laplace functional of the interference as:

S(T ) = P(SINR > T )

= P

(h0 >

T (σ 2 + I )

G0� (|X0|))

(a)≈ν∑

n=1

(−1)n+1(

ν

n

)E

[e− ηnT (σ2+I )

G0�(|X0|)]

(b)=ν∑

n=1

(−1)n+1(

ν

n

)E

[e−nT μσ 2

]E

[e−nT μI

]

(c)=ν∑

n=1

(−1)n+1(

ν

n

)E

[e−nT μσ 2

]LI (nTμ) (18)

where I = ∑b>0:Xb∈� hbGb� (|Xb|) is the total interference;

σ is the normalized noise power by the transmit power;step (a) follows from Lemma 1; in step (b), we denoteμ = η

G0�(|X0|) , and in (c) we denote the Laplace functionalof the interference I as LI (s) = E

[e−s I

].

For the second difficulty on base station inhomogeneity, weconsider the LOS and NLOS base stations as two independenttiers of BSs as seen from the typical user at origin, for weignore the correlations between the LOS probabilities betweennearby links. The non-homogeneous PPP representing LOSBSs has density equal to λPLOS(d), where d is the distancefrom origin. Similarly, the NLOS BS process is PPP withdensity λ(1 − PLOS(d)).

Based on the above discussion, we illustrate the method tocompute the Laplace function of the interference. Given thatthe desired signal link is LOS and has a length of |X0| = d ,


based on the minimum path loss association rule, all theLOS interfering base stations are farther than distance d ,and all NLOS interfering base stations are farther than dis-tance dαL/αN from the typical user. The Laplace transformof the interference LI (t) = E

[e−t(IL+IN)

], where t > 0,

IL = ∑b>0:Xb∈�L

hbGb� (|Xb|) represents the LOS inter-ference, and IN = ∑

b>0:Xb∈�NhbGb� (|Xb|) the NLOS

interference. Given that the typical user associates with a LOSBS at distance d , this can be simplified as follows.

LI (t)(a)= E

[e−t IL

]E

[e−t IN

](c)= exp

(−2π

∫ ∞

d

(1 − Eh,G

[e−CLr−αL hGt

])

× λPLOS(r)rdr

)

× exp

(−2π

∫ ∞

dαLαN

(1 − Eh,G

[e−CNr−αN hGt

])

×λ(1 − PLOS(r))rdr

)(19)

where step (a) follows from the independence between�L and �N; and (b) follows from the probability generatingfunctional of a PPP [61]. Here, h and G are dummy randomvariables for the small-scale fading and directivity gain ininterference channels. The Laplace transform for interferencecan be similarly derived given that the user associates with aNLOS base station at distance d . Here, all NLOS interferersare farther than d from the user at origin and all LOSinterferers are farther than dαN/αL .

For the third difficulty on antenna gain, note that therandomness in the directivity gain is incorporated in therandom variable G in (19). As the actual antenna pattern canbe intractable to incorporate, the sectored antenna approx-imation has been proposed to simplify the computation.In the approximation, the directivity gain G in the interferencechannel is modeled as a discrete random variable with a simplydistribution, as shown in Table II. Therefore, for s ∈ {L, N},the term Eh,G

[e−Csr−αs hGt

]in (19) can be computed as

Eh,G

[e−Csr−αs hGt

](a)=

4∑k=1

bkEh

[e−Csr−αs hak t

](20)

(b)=4∑

k=1

bk(1 + Csr

−αs akt)−νs (21)

where in (a) ak and bk are the constants given in Table II, andstep (b) follows from computing the Laplace transform of thegamma distributed random variable h.

The Laplace functional in (19) can be applied to computethe SINR coverage probability as shown in (18). The remain-ing steps are deconditioning the LOS/NLOS status and thelength of the desired signal link, whose distributions are givenin the following lemma.

Lemma 2 (Association Probability): The probability ALthat the typical user is associated with a LOS base stationis

AL =∫ ∞

0e−2πλ

∫ xαL/αN0 (1−PLOS(t))tdtgL(x)dx, (22)

where gL(x) = πλx PLOS(x)e−2πλ∫ x

0 r PLOS(r)dr , and theprobability that the user is associated with a NLOS basestation is AN = 1 − AL. Given that a user is associated witha LOS base station, the probability density function of thedistance to its serving base station is

fL(x) = gL(x)

ALe−2πλ

∫ xαL/αN0 (1−PLOS(t))tdt, (23)

when x > 0. Given the user is served by a NLOS basestation, the probability density function of the distance to itsserving base station is

fN(x) = gN(x)

ANe−2πλ

∫ xαN/αL0 PLOS(t)tdt, (24)

where x > 0, and gN(x) = 2πλx(1 − PLOS(x))

e−2πλ∫ x

0 r(1−PLOS(r))dr .Now we present the main SINR coverage result in the

following theorem. The detailed proof of Theorem 1 andLemma 2 can be found in [53].

Theorem 1 (SINR Coverage Results): The SINR coverageprobability S(T ) can be computed as

S(T ) = ALSL(T ) + ANSN(T ), (25)

where for s ∈ {L, N}, Ss(T ) is the conditional coverageprobability given that the user is associated with a base stationin �s. Further, Ss(T ) can be evaluated as

SL(T ) ≈νL∑

n=1

(−1)n+1(

νL

n

)

∫ ∞

0e− nηLxαL T σ2

CL MMS MBS−Qn(T ,x)−Vn(T ,x)

fL(x)dx, (26)

and

SN(T ) ≈νN∑

n=1

(−1)n+1(

νN

n

)

∫ ∞

0e− nηNxαN T σ2

CN MMS MBS−Wn(T ,x)−Zn(T ,x)

fN(x)dx, (27)

where

Qn(T, x) = 2πλ

4∑k=1

bk

∫ ∞

xF

(νL,

nηLak T xαL

νLtαL

)·

PLOS(t)tdt, (28)

Vn(T, x) = 2πλ

4∑k=1

bk

∫ ∞

xαL/αNF

(νN,

nCNηLakT xαL

CLνNtαN

)

(1 − PLOS(t))tdt, (29)

Wn(T, x) = 2πλ

4∑k=1

bk

∫ ∞

xαN/αLF

(νL,

nCLηNakT xαN

CNνLtαL

)·

PLOS(t)tdt, (30)

Zn(T, x) = 2πλ

4∑k=1

bk

∫ ∞

xF

(νN,

nηNakT xαN

νNtαN

)·

(1 − PLOS(t))tdt, (31)

and F(ν, x) = 1−1/(1+x)ν. For s ∈ {L, N}, ηs = νs(νs !)−1νs ,

νs are the parameters of the Nakagami small-scale fading;


for k ∈ {1, 2, 3, 4}, ak = akMBS MMS

, ak and bk are definedin Table II.

Note that in a noise-limited network, the SINR distributioncan be replaced by the SNR distribution that has a much sim-pler analytical expression to compute [66], [84]. For example,when the interference is ignored, the expression in Theorem 1is largely simplified, as Wn(·), Qn(·), Vn(·), and Zn(·) allbecome zero. In an interference-limited scenario, substitutingthe noise power σ 2 = 0 gives us the SIR distribution.

We will discuss key extensions of the SINR result inSect. VII and further implications of these results in Sect. VI.We now turn our attention to characterizing the rate distribu-tion for the baseline model.

D. Rate Coverage Probability

The per user rate in bits per second (bps) depends largelyon the user perceived SINR and the amount of time-frequencyresources it can use. Treating interference as noise, the achiev-able per user rate in bps/Hz is close to the point-to-point linkcapacity given by log2(1 + SINR), also called the spectralefficiency. Conventionally, this has been considered to be themetric of interest in evaluating the performance of wirelessnetworks and is still largely used today for analytical purposesgiven the mathematical challenge involved in characterizingthe distribution of amount of per user resources. Once theSINR coverage is known, computing the distribution of thespectral efficiency is straightforward. However, cellular net-works today are becoming increasingly heterogeneous andthus, user association and offloading have been hot topics ofresearch. Further, it is likely that mmWave networks will coex-ist with traditional Sub-6GHz networks. Thus, the problem ofoffloading from one frequency band to another is still relevantfor mmWave cellular networks. For studying such problems,incorporating the impact of load on the rate characterization isessential [147]. The network can then be optimized followinga network optimization framework [148]–[150] or a simplerbut often effective biasing approach whereby the minimumpath loss rule used in this paper is modified to have anaffinity for lower power or more lightly weighted cells, asin [151]–[155]. Some recent work in this direction specificallyfor mmWave includes [156]–[160]. Wireless backhauling anddynamic resource allocation are some other intriguing researchdirections for mmWave cellular [54], [66], [161]. Incorporatingthe impact of load on the rate characterization is essential forstudying such problems as well.

A common assumption in the literature for analyticaltractability has been to assume round robin scheduling, whichis also equivalent to random user selection in each time-frequency resource block. In this case, the per user rate inbps can be modeled as

R = B

�log2(1 + SINR), (32)

where B is the total bandwidth and � is the total numberof users connected to the base station serving the user whoseperformance is being evaluated. Rate coverage is defined asR = P(R(τ ) > τ), where τ is the rate threshold in bps.Recall that association cell of a BS at X ∈ R

2 is a collection

of user locations in space that would associate with X basedon instantaneous channel conditions. In general the SINR andload distribution of serving BS are correlated since larger theassociation cell implies more load and longer link distances(that is smaller SINR). Thus, knowing the joint distributionof association areas and SINR is necessary to find the loaddistribution. As of now, even characterizing the marginaldistribution of the Poisson-Voronoi (PV) tesselations is anopen problem. Voronoi tesselations are association cells inSub-6GHz networks wherein a user connects to the nearestbase station. As was shown in [66], blockage effects lead tovery irregular association cells in mmWave cellular networks,of which PV tesselations is a special case. Given that the asso-ciation scheme is stationary or translation invariant (refer [162]for a formal definition), the mean association area of a typicalcell can be characterized as 1/λ, which is the same as the meanassociation area of a typical cell in a PV tesselation. Note thatthis is different than the mean association area of the cellcontaining the user at origin, which is a size-biased version ofthe area of typical cell [162]. In the case of mmWave path lossmodel in (16), the minimum path loss association rule is sta-tionary since given the stationary point processes for base sta-tions and user locations, the association of a user to a base sta-tion depends only on independent distance dependent randomvariables which are invariant under a translation. Thus, themean number of users in a typical cell is equal to λu/λ [66].

The above observations lead to the following approxima-tions for tractability.

1) Association area distribution: The distribution of associ-ation area of a typical BS in a mmWave cellular networkis assumed to be same as the approximate distribution ofa typical PV cell with the same mean area as proposedin [163].

2) User point process: The point process for UE locationsis a homogeneous PPP with intensity λu .

3) Independence: The load distribution of the BS servingthe typical user at origin is independent of the load ofother BSs in the network, and all these are independentof the user perceived SINR.

The approximation for volume of a typical PV cell proposedin [163] was used to derive and validate the load distributionof the serving as well as the other BSs in the network in[153] and [164]. Such an approximation was subsequentlyverified numerically in [165]–[168] for Sub-6GHz networksand in [66] and [122] for mmWave networks. We provide theformulas for load distribution here, and interested readers canfind the proof in Appendix B of [153].

Lemma 3: The probability mass function (PMF) of thenumber of users �0 associated with the BS at X0 servingthe user at origin is given as

P(�0 = n) = ϒ(n)

= 3.53.5

(n − 1)!�(n + 3.5)

�(3.5)

(λu/λ)n−1 (3.5 + λu/λ)−n−3.5 , (33)

for n ≥ 1 and P(�0 = 0) = 0. The corresponding mean is1 + 1.28λu/λ.


For a typical BS located at Xl , the load distribution isgiven as

P(�l = n) = 3.53.5

n!�(n + 3.5)

�(3.5)(λu/λ)n (3.5 + λu/λ)−n−3.5 ,

(34)

for n ≥ 0. The corresponding mean is λu/λ.Based on the assumptions given above, the rate coverage

can be found from the following theorem.Theorem 2: The rate coverage of a typical user in a

mmWave cellular network for a rate threshold τ is given by

R (τ ) =∑n≥1

ϒ(n)S(

2τn/B − 1)

, (35)

where S(·) is the SINR coverage derived in the Theorem 1.Although the rate coverage expression is an infinite summa-

tion, it can be computed as a finite summation up to nmax termswithout much loss in accuracy [66], [153]. A rule of thumb isto choose nmax as a multiple of λu/λ, where the multiplicativeconstant can be found from numerical investigations. A fasterbut less accurate mean load approximation can also be doneby substituting the random variable representing load by thecorresponding mean given by 1 + 1.28λu/λ.

VI. IMPLICATIONS OF MODELS AND ANALYSIS

In this section we consider the design and deploymentimplications of the baseline model analysis.

A. When Will mmWave Systems be Noise/Power-Limited?

An ongoing major challenge for Sub-6GHz cellular net-works has been the management of interference from neigh-boring cells using the same time-frequency resources. Suchnetworks, especially in urban areas, are typically interference-limited, meaning that SINR ≈ SIR and so increasing transmis-sion power does not increase SINR, on a network-wide basis.Another way to view the interference-limited behavior is thatthe network density is sufficiently high such that further den-sifying the network does not substantially improve the SINRdistribution, because the increase in SNR is counter-acted bythe increase in interference, as shown in [58] and [169]. FormmWave networks, the behavior is somewhat different.

First, the received SNR is nominally very low in mmWavedue to the small per antenna area, the large bandwidth andblocking/penetration effects. As we have discussed, this iscompensated for using large antenna arrays that achieve highlydirectional transmission and reception. Improving the SNRthrough beamforming, as opposed to increasing it via highertransmitter power or base station density, should not increasethe average interference, because the average transmit poweris unchanged. Meanwhile, the interfering signals experienceblocking and typically have misaligned beams. These factorsseem to indicate that mmWave systems are much more likelyto be noise-limited than their Sub-6GHz counterparts. This hasmany important implications on the system design, as will bediscussed in the subsequent sections.

In this section, we explore what circumstances effect thephase transition between noise and interference domination

of the SINR denominator, to better understand when noise orinterference limited behaviors will be observed. Important sys-tem parameters to consider include: the base station and userdensity, antenna gains and beamwidths, operational bandwidth,blockage model, the LOS and NLOS path loss exponents, andthe choice of MIMO technique. For example, our baselineanalysis shows, as discussed in [53], that the SINR coverageprobability exhibits a non-monotonic trend with base stationdensity. This implies that the network eventually transitionsfrom noise to interference limited behavior when there areenough interfering BSs. This of course also holds for Sub-6GHz systems, but occurs at much lower densities. Becausemany parameters simultaneously effect whether the mmWavesystem is described better by the SIR or SNR, it is notpossible to give crisp threshold values where such transitionsoccur. In this section, we first identify specific scenarios wheremmWave networks have been previously observed to follownoise-limited behavior, and then we present some numericalresults to illustrate the dependence of these trends on theaforementioned system parameters.

In the simulations, we compare the interference-to-noiseratio (INR, or I/N), and plot the probability P(I/N > 1) as afunction of the average inter-site distance between neighboringbase stations. We consider two carrier frequencies: 73 GHz inE-band and 28 GHz in LMDS band. The 73 GHz system willlikely have a larger bandwidth which increases noise power,but due to the smaller wavelength also more directionality.In addition, we use the Austin and Los Angeles city, for whichLOS functions have been fitted using real data in Section III,as examples for moderately-dense and ultra-dense buildingenvironments.

As stochastic geometry analysis in [66] and [84] indicated,Fig. 8(a) shows that with a larger bandwidth and more direc-tionality (which reduces the effect of nearby interferers), the73 GHz system tends to be noise-limited even when inter-sitedistance (ISD) is as little as 100 meters, corresponding to abase station density as high as to λ = 200 BSs per squarekm. Similar to [53], with the smaller 200 MHz bandwidth,a larger impact from interference can be observed in the28 GHz band. Moreover, compared with the performance inAustin, the impact of the interference becomes smaller inLos Angeles due to the denser building blockages. In Fig. 8(b),we simulate the distribution of P(I/N > 1) for a 28 GHzsystem in Austin with different system parameters. Numericalresults indicate that the impact of interference increases with asmaller bandwidth, and a large beamwidth in the beamforming(resulted from applying fewer antennas in the ULA), and alarger number of users in the MU MIMO.

The results indicate that it is difficult to provide a generaland crisp answer whether mmWave cellular networks arenoise-limited or noise-limited, but several observations canbe made.

1) Higher carrier frequencies, which typically will allow forconsiderably larger bandwidths and higher directionality,as well as possibly having more sensitivity to blockage(especially penetration into buildings), will be signifi-cantly more noise-limited. For example, to have I ≈ Nwith probability 0.1 requires about a factor of 2 smaller


Fig. 8. Comparison of INR, assuming 200 MHz bandwidth, 32-antennauniform linear array (ULA) at BS and 16-antenna ULA for mobiles for28 GHz systems; whereas at 73 GHz the bandwidth is 1 GHz, and the ULAsare of size 64 and 32 for BS and mobiles, respectively. Path loss followsαL = 2, and αN = 4 and blocking follows random shape theory with LA andAustin fitted parameters as in Section III-F.

ISD at 73 GHz vs. 28 GHz for both Austin and LA.In other words, four times the BS density is requiredfor 73 GHz to experience similar interference (relativeto noise) as at 28 GHz.

2) More blockages make systems more noise-limited, notjust by blocking the desired signal, but also by blocking(in probability) the stronger interfering signals. Thus,the denser the buildings and obstacles, the more noise-limited the system is.

3) The primary concern for initial mmWave cellulardeployments should be achieving sufficient SNR.In large bandwidth and highly directional mmWavenetworks, interference will only become a significantfactor once mmWave cellular has become successful,with extremely dense deployments and heavily loadedcells.1 However, if initial deployment itself requires very

1Note that our results assume the worst-case interference, i.e. neighboringBSs are always transmitting, which is very pessimistic from an SINRviewpoint particularly for a high-rate small-cell mmWave system with fewactive users per cell [137].

dense network deployment (inter-site distance roughlyless than 100 m) to fill in coverage holes, which mighthappen due to use of smaller antenna arrays or highblockage effects, then interference effects need not benegligible.

We now discuss several related implications, in view of thesystem generally being noise-limited.

B. Initial Access in mmWave Systems is Challenging

Initial access is a very important challenging problem formmWave systems, given the often very low SNR before beamsare aligned, which is what prompts the need for large antennagains to close the link budget in the first place [170]–[172].The initial access procedure consists of at least two stages, onefor the downlink referred to as cell search whereby the UEacquires the BS signal, and another for the uplink referred toas random access, whereby the BS acquires the UE signal andlearns from the UE which beamforming direction it can receivedata on. Without successfully completing this procedure, it isnot possible to communicate.

The initial access procedure in LTE is sophisticated [173],but in mmWave it would be more difficult due to the needfor beam alignment, since beams must be searched at boththe BS and UE side to align them at both ends [172], as ourbaseline model assumes. Since it is not possible to beamformuser data in one angular direction over one frequency bandand do a beam search in another direction over a differentfrequency band, this leads to the constraint that the initialaccess procedure must be performed over the entire bandbecause of the necessary use of analog beamforming. Thisraises the overhead cost of initial access and accentuates manydifficult tradeoffs. The more beams that are tested, the betterthe beam alignment will be and the higher directionality canbe achieved, but at the cost of spending a lot of slots testinguseless beam combinations at both ends. The more oftenthe procedure is performed, the more robustly beams can bealigned and new users (or moving users) can quickly attainconnectivity. But this in unwanted overhead that eats into thetime that can be spent transmitting data.

Initial beam training and association is a pressing currenttopic, and recent surveys can be found at [174]–[176], with acomparative study by Qualcomm of different beam formingapproaches for initial access appearing in [177]. Recently,[124] developed a tractable model based on beam sweepingand downlink control pilot reuse using tools similar to thoseoverviewed in this paper. The results showed that unless theemployed beams are very wide or the system coherence blocklength is very small, exhaustive search with full pilot reuseis nearly as good as perfect beam alignment. An early proto-typing effort that leverages unlicensed bands (2.4 and 5 GHz)to assist in the initial access procedure [178]. Exploring andoptimizing initial access further is a key research area for thecoming years.

C. The Promise of Self-Backhauling

Self-backhauling is attractive for mmWave, since it requiresdense deployments and high-speed backhaul (on the order


of Gbps burst rates), which can be difficult to achieve witha wired network. Because mmWave networks typically donot have strong interference, and the backhaul links will bedirectional (and usually not interfering with the access links),self-backhauling is a scalable solution, wherein a fraction ofbase stations with wired backhaul provide in-band wirelessbackhaul to the remaining base stations.

An analytical model and analysis for rate in self-backhauledmmWave cellular networks was developed in [66]. It wasobserved that increasing the fraction of base stations withwired backhaul improves the peak rates in the network.However, the per user rate saturates if the density of BSs isincreased keeping the density of wired backhaul base stationsconstant. The saturation density was found to be proportionalto the density of base stations with wired backhaul. Owingto the subdued interference effects at mmWave, it could beeven feasible for access and in-band backhaul links to operateon the same time resources [1], [54]. This has motivated theinvestigation of whether dynamic time division duplexing isfeasible in self-backhauled mmWave networks [158], [159].

D. Spectrum License Sharing Among CellularOperators is Possible

There is an early proposal by FCC to use the28 GHz (27.5-28.35 GHz), 37 GHz (37-38.6 GHz) and39 GHz (38.6-40 GHz) bands for cellular services [179].There is also large amount of unlicensed mmWave spec-trum (57GHz-71GHz) which will be available for lower-power operations similar to WiGig. Although the mmWavebands have a relatively large amount of available spectrum,effective utilization of the spectrum is always important.Because it will be particularly difficult to deploy a net-work with truly nationwide coverage at mmWave, exclusivelylicensing the spectrum to a single entity seems particularlyinefficient in terms of spectrum utilization and may even byunnecessary.

One way to increase the spectrum utilization is to use autho-rized shared access [156] and licensed shared access [180].These frameworks allow spectrum sharing by a limited numberof parties by letting members define sharing rules that can pro-tect them from interference from each other. Another way toresolve the transmission conflicts between multiple licenseesis by the use of a central database, possibly owned by a thirdparty, which keeps track of transmissions of each licensee toensure fair and reliable services for each licensee [179]. Alongwith these coordinated sharing of spectrum, it is also possiblethat operators can simply share their spectrum licenses withoutany explicit coordination [145] and still achieve higher rateswhen compared to the rate achieved when exclusive licens-ing is used, owing to the low inter-operator interference inmmWave. Moreover, it was shown in [181] that static coor-dination can further improve the overall network performancewhile providing a way to differentiate the spectrum accessbetween the different operators. The work [182], [183] studiedthe gains of spectrum sharing in the presence of inter-operatorcoordination for mmWave cellular networks and observedsimilar trends as observed in [145] and [181].

Fig. 9. Association cells of mmWave (the three circles) and a Sub-6GHzBSs (star) overlaid in a portion of Austin, Texas. The shaded areas show thethree association cells of the mmWave BSs with the macrocell boundariesshowed by the solid lines.

To evaluate various types of spectrum sharing, one impor-tant aspect is to correctly model a multi-operator system withpossible correlation among their locations. It is anticipated formmWave systems (as with the current cellular systems) thatthe entity owing the BS site and the cellular operator usingthat BS may be two independent entities. The site owners canlease the locations to multiple operators which will results inBSs of multiple operators co-located at a single location [184].Our recent work [145] presents two possible cases of a multi-operator system. In the first case, the BS locations of eachoperator are modeled by a PPP and assumed to be independentto BSs locations of other operators. In the second case,consisting of a single PPP representing locations of the siteswhere BSs of all operators are co-located. These can be seenas two extreme cases with a real mmWave deployment (andits corresponding SINR and rate performance) somewhere inbetween. We found that uncoordinated sharing of spectrumlicenses is possible in both the cases as long as the antennabeams are relatively narrow, on the order of about 30 degreesor less.

E. Ultra-Densification in mmWave Networks

Due to high penetration losses and sensitivity to blockages,the coverage areas of mmWave BSs are generally small andirregular, as shown in Fig. 9. This issue can be overcome bydensification of the network. When the BS density is low,BS densification decreases the distance of the serving BSand increases the probability of serving BSs being LOS BS.This results in higher serving power to a typical user. Theinterference power, however, is not affected significantly as theinterferers are still far enough to be NLOS to the typical user.Therefore, densification generally helps mmWave systems andimprove its SINR and rate coverage. After a certain threshold,though there will be enough interferers in the LOS regioncausing the SINR degrade significantly. It was shown in [53]that if LOS path loss exponent αLis below 2, the SINRcoverage will finally become zero as BS density λ → ∞.


Fig. 10. SINR coverage probability targeting 10 dB with different inter-sitedistances, for fc = 28 GHz, and 200 MHz bandwidth. For base stationbeamforming, the main lobe antenna gain is 20 dB regardless of beamwidth;for mobile station, the main lobe gain is 10 dB, and the beamwidth is 45°.

If αL > 2, the SINR coverage converges to a nonzero valueat infinite densification. Therefore, there is an optimal density(termed as “critical density”) after which the performance ofmmWave systems starts degrading [53].

It was also shown in [124] that this critical density isnormally in the order of the LOS range, and gets smalleras narrower beams are employed. To illustrate that, we plotin Fig. 10 the probability P(SINR > 10d B) as a functionof inter-site distances between base stations in a 28 GHzsystem. Numerical results indicate that the critical density isproportional to the average LOS range that is 43 meters forLos Angeles, and 85 meters for Austin, according to the datafitting for the random shape theory model in Section III; andthat the critical density decreases with a smaller beam width,as the interference is reduced.

This behavior is similar to that for Sub-6GHz systems underdual-slope pathloss models [76], wherein a “close-in” path lossexponent is used for distances less than a deterministic cornerdistance Rc, and then changes to larger path loss exponentoutside Rc. These two regions in the dual slope path lossmodel are analogous to the LOS and NLOS regions in themmWave context. The main difference between the mmWaveand the dual-slope case is that the variable which defines theLOS region around a user is not deterministic, but a randomvariable with mean value equal to RB. However, this differencedoes not distinguish the general behavior of mmWave andSub-6GHz systems in the ultra-dense regime.

VII. EXTENSIONS TO THE BASELINE MODEL

The SINR and rate coverage results of the previous sectioncan be extended in many ways, given the complexity, scope,and large design space of mmWave cellular systems. Crucialextensions include the uplink, as well as more realistic andheterogeneous topologies which would at a minimum includespatial overlap with Sub-6GHz macrocells (that can be usedfor fallback coverage and control signaling) as well as han-dling indoor coverage. Other readers may be interested in

the higher layer implications of a mmWave physical layer:there are many, such as scheduling, as discussed in [185]and the survey [186]. Further, given the number of ways thatthe inherently large antenna arrays in mmWave systems canbe used (as discussed in Sect. IV), we will also overviewhow to extend these baseline results to other multi-antennatransmission and reception techniques. To keep the discussionconcise, we point the readers to important references relatedto the extensions and do not delve into the details of how thebaseline mathematical model changes in each case.

A. Uplink

It is natural to extend the downlink SINR analysis to theuplink. Prior analysis in Sub-6GHz cellular networks showsdifferent distributions of interferers between downlink anduplink [166], [187]. For example, when the base stationsare assumed to be distributed as a PPP, then the scheduledusers in all cells do not form a PPP, due to the Voronoi cellstructure [166], [187]. mmWave cellular networks will inheritsuch difference in the network topology between downlinkand uplink. More importantly, based on recent study on elec-tromagnetic field exposure [188], the uplink transmit power inmmWave networks is expected to be even smaller than that ofSub-6GHz system. Consequently, the uplink tends to be evenmore noise-limited than the downlink. Therefore, a somewhattrivial but possibly useful extension for obtaining uplink SINRcoverage is to use SINR ≈ SNR and then use the downlinkcoverage probability for uplink with transmit power replacedwith that of the mobile users [66]. Power control could bepossibly neglected for mmWave networks considering thatthese networks are power-limited, so mobiles will typicallybe transmitting at close to full power (and this will not havemuch effect on other users’ SINRs).

One promising approach to compute the uplink interferencedistribution is to extend the approach in [166] that modelsuplink interferers as a non-homogeneous PPP in Sub-6GHzheterogeneous networks. For example, in [189], the densityfunctions for LOS other-cell user process λu,L(r)and NLOSuser process λu,N(r) are computed as

λu,L(r) = λb pLOS(r)Q(rαL/CL),

λu,N(r) = λb (1 − pLOS(r)) Q(rαN/CN),

where λb is the density of base stations, r is the distance tothe serving base station of the typical user, and

Q(y) = 1 − exp

(−2πλb

(∫ (yCN)1/αN

0s (1 − pLOS(s)) ds

+∫ (yCL)1/αL

0spLOS(s)ds

))

is the probability that a user’s path loss to its associated basestation is smaller than y−1. A similar approach was usedin [161] to study uplink SINR and rate coverage in mmWavecellular networks.


B. Joint Coverage With Sub-6GHz Systems

It seems self-evident that mmWave systems cannot bedeployed stand alone and still achieve a high level of coveragein an urban area, much less nationwide coverage withouta tremendous amount of infrastructure. Rather, a mmWavenetwork will generally be overlaid on an LTE-like networkto provide high-rate hotspots, with the mmWave base stationsbeing used whenever possible to offload from the more con-gested Sub-6GHz network. LTE macrocells can be used inmultiple ways to assist mmWave networks.

The first way is to help with control signaling. As discussedabove, initial access (and thus all control signaling) posesa particularly troublesome “chicken and egg” problem formmWave cellular. It seems very likely that some controlsignaling will happen over the legacy Sub-6GHz spectrum,similar to how some of the variants of unlicensed LTE areusing the licensed spectrum for the control plane. Second, theSub-6GHz BSs may also be used to provide dual connectivityto both systems [47] in which mobile users can be simulta-neously connected to both LTE (or future Sub-6GHz system)and mmWave BSs, and even receive data from both BSs whenpossible. In [66], an offloading technique was proposed wherethe data services are mainly provided from mmWave BSs(whenever available), but when the mmWave link quality dropsbelow a certain threshold, the user reverts to an LTE macrocell.In this sense, mmWave can be viewed as an additional carrieraggregation technique for existing LTE systems.

Recent work [66], [190], [191] has analytically modeledthe coexistence of mmWave and LTE systems to derive per-formance metrics and design insights. In [66] and [191], thelocations of macrocells and mmWave BSs were modeled asindependent PPPs with different densities and the coexistenceis modeled by superposition of these two PPPs. A clusteredpoint process such as Neyman-Scott process can also beused to model the co-existence of LTE and mmWave whereSub-6GHz BS will be located at the center of the clusterswhile mmWave BSs will be spread around the center.

Technically, a coexisting Sub-6GHz and mmWave cellularnetwork is a type of heterogeneous network (HetNet) wherethe tiers do not interfere with each other, and also have a vastdisparity in terms of bandwidth and other parameters. By thelaw of total probability, the rate coverage in such a HetNet isgiven by

R (τ ) = AmmWRmmW(τ ) + AUHFRUHF(τ ), (36)

where AmmW and AUHF are the association probabilities, andRmmW and RUHF are rate coverages conditioned on associationto the respective tiers. If the BSs in both tiers are modeled asPPP, R(.) can be computed from Theorem 2 by replacing λin S(.) and κ(.) expressions with the density of BSs in therespective tier.

It was shown in [160] that incorporating beamforming gainsduring cell association based on downlink/uplink receivedsignal power significantly boosts the probability of connectingto a large bandwidth mmWave BS which is beneficial froma rate standpoint. Due to the bandwidth disparity in the twotiers, it may be desirable to connect to a mmWave BS offering

much lower SNR than an ultra high frequency (UHF) BS.However, there is an optimum value of bias towards mmWaveBSs after which the rate again starts to decrease due toweak SNR over the mmWave link. This insight is in linewith that in [66], wherein it was shown that users shouldbe offloaded to UHF networks only if communication overmmWave link is infeasible. Thus, studying robust modulationand coding schemes that can decode low SNR signals isan intriguing avenue of physical layer research for mmWavecellular networks.

In [160] the BS locations of mmWave and Sub-6GHzbase stations were modeled as independent PPPs for tractableanalysis. In reality, however, some of the mmW andSub-6GHz base stations may be co-located or their locationsmay be correlated depending on hotspot traffic areas. In futurework, examinations that consider the correlations in both basestations locations and load are essential to understand the gainof such multi-band joint-coverage system.

C. Outdoor-to-Indoor Coverage

The framework in Section V mainly focuses on the perfor-mance analysis in outdoor mmWave cellular networks. Dueto the huge penetration losses from outer walls of buildings,indoor users – if not near a window – will unlikely beserved by outdoor mmWave base stations. Therefore, usersinside buildings need to be served either by indoor basestations or by lower carrier frequency systems. This is avery serious limitation of mmWave cellular systems, giventhe prevalence of indoor cellular data usage: over 80% ofall current cellular communications are with indoor usersaccording to [192] and [193], which does not even includeWiFi offloading. One silver lining, as discussed in [194],is that when indoor small cells are deployed, the indoorSINR coverage and rate performance improves with a largerpenetration loss through the outer walls, due to the reductionof the interference from outdoor base stations. Thus, in manycases the mmWave system may serve effectively as a wirelessbackhaul network to a separate indoor wireless data network(which could be mmWave or WiFi) [195].

Extending our baseline outdoor only model to variousindoor scenarios is important given the prevalence of indooruse, but technically challenging. Challenges including thatthe outdoor model does not incorporate (i) the first-orderreflections (from walls) that can have comparable strengthswith the direct paths in certain indoor scenarios [28], and(ii) the partition losses, e.g. from inner walls, that contributeto a large fraction of the overall indoor path loss [30].In [91] and [196], given the semi-specular nature of themmWave signal propagation, the first-order reflections weretaken account by considering the walls as mirrors, and mod-eling the images of transmitters approximately. To simplifythe analysis, the locations of the image transmitters wereapproximated as an independent process of the original trans-mitters with the same density. Numerical results show thatsuch approximation brings in minor losses in accuracy whencomputing the SINR distributions in certain indoor scenarios.Meanwhile, the partition loss has been characterized for the


indoor performance analysis in [197], where the inner wallsof a building were modeled as a Poisson line model. Based onthe proposed model, the distribution of the interference at atypical indoor user was derived, assuming the partition lossesfrom the walls dominate the path loss, i.e., ignoring the free-space path loss.

D. MIMO Techniques Beyond Analog Beamforming

Analog beamforming is the default approach for communi-cation in current commercial systems like IEEE 802.11ad andWirelessHD. The large antenna arrays at mmWave beyondanalog beamforming, allow more sophisticated forms ofMIMO respecting the hardware constraints in Section IV.

Several extensions of the baseline model are possible.We briefly mention some key extensions here.

1) Incorporating a channel with rank greater than one: Theimpact of multiple spatial paths from the transmitterto receiver may affect the interference statistics. Also,enabling multi-stream transmission highly relies on therank of the channel. One possible way to incorporaterank> 1 channels in stochastic geometry analysis isto use the channel model in (10) as done in [122].A discussion on some other approaches to modelmmWave MIMO channel can be found in [120].

2) Incorporating the impact of different MIMO transceiversin network level analysis: Impact of analog beam-forming, hybrid beamforming, low resolution ADCs atreceivers, CAP-MIMO on network-level performancecould be evaluated and compared. This can provideinsight on which MIMO architectures are suitable inwhat scenarios depending on network parameters likeavailable base station density, number of antennas,number of RF chains, power consumption constraints,etc. Recent work in [122] uses the virtual chan-nel representation to develop an analytical model forMU-MIMO based on the hybrid precoding algorithmproposed in [112]. The analysis enabled comparisonwith hybrid precoding based SU-MIMO and single-user analog beamforming, and also highlighted severaldesign insights. For example, (a) the optimum basestation density in terms of SINR coverage decreaseswith increasing degree of multiuser transmission,(b) the cell edge rates suffer with round robin schedulingwhich highlights the importance of user selection whileemploying MU-MIMO, (c) a denser single-user beam-forming network provides higher cell edge rates thana less dense MU-MIMO network that consumes samepower per unit area.

3) Imperfect channel state information: Most existing sto-chastic geometry analysis of mmWave networks assumesperfect channel state information at the transmitter apartfrom a few exceptions like [84] and [124]. Althoughdeveloping analytical models with perfect channel infor-mation is a good first step, studying the impact ofimperfect channel information and channel aging isimportant. Given the importance of narrow beams toenable outdoor mmWave cellular, wrong estimates of

angles of departure or arrival may significantly drop thepost processing SINR.

4) Massive MIMO at mmWave: Another extension isto analyze the performance of mmWave massiveMIMO networks, a natural extension of massiveMIMO networks at lower frequencies [198]. Due tothe aforementioned difference in propagation channelsand hardware constraints, prior stochastic geometrymodels for Sub-6GHz massive MIMO networks, e.g.in [199]–[201], do not directly apply to the mmWavebands. In [189], key features in mmWave channels,including the blockage effects and channel sparsity(in terms of multi-paths), and certain differences inhardware, e.g. the beamforming at mobile stations, wereincorporated to model mmWave massive MIMO cellularnetworks. Based on the model, the asymptotic SINR andrate performance were analyzed, when the number ofbase station antennas goes to infinity. Numerical resultsbased on the analysis show that mmWave massiveMIMO requires a high base station density to achievegood SINR coverage; the SINR distribution in theasymptotic case is a good approximation for the caseswith more than 256 antennas, in certain dense mmWavenetworks. For future work, it would be valuable toincorporate mmWave-compatible channel training intothe system analysis, e.g. the compressed sensing basedapproach in the angular domain [106], [202]–[204]which could potentially reduce training time [198].

VIII. CONCLUSIONS

The upcoming standardization and development ofmmWave cellular systems is one of the largest leaps forwardin wireless communications in the last two decades. Goingto mmWave though introduces novel design challengesand research questions. This paper described the two mostimportant physical challenges – susceptibility to blockingand the need for strong directionality – and has provided abaseline mathematical model and analysis for these systemsaccounting for these factors.

There are many open questions and key extensions remain-ing, some of which we overview in Sects. VI and VII. Forexample, the crucial topic of outdoor-to-indoor coverage isessentially neglected in work to date. How to do load balancingand offloading in light of directionality and blocking, bothbetween mmWave cells and between mmWave and Sub-6GHzcells, is not well understood. The support of mobility is alsonot discussed here, which will require considerable effort(and system overhead) to keep the beams aligned in both thedownlink and uplink directions. Handoffs also will no longerbe based mostly on distance, but blocking (including by one’sown body) may often be the dominant factor, making theneed for handoff more difficult to predict. In short, we expectthis new paradigm for cellular communication to challengewireless engineers for some time. We expect that the modelsdeveloped in this paper will continue to be improved andextended to help aid the understanding and design of thesemmWave cellular systems.


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Jeffrey G. Andrews (S’98–M’02–SM’06–F’13)received the B.S. degree (Hons.) in engineering fromHarvey Mudd College, and the M.S. and Ph.D.degrees in electrical engineering from Stanford Uni-versity. He is currently the Cullen Trust EndowedProfessor (#1) of ECE with The University of Texasat Austin. He developed Code Division MultipleAccess systems at Qualcomm from 1995 to 1997,and has consulted for entities, including Apple,Samsung, Verizon, AT&T, the WiMAX Forum, Intel,Microsoft, Clearwire, Sprint, and NASA. He has

co-authored the books Fundamentals of WiMAX (Prentice-Hall, 2007) andFundamentals of LTE (Prentice-Hall, 2010). He is a member of the TechnicalAdvisory Board of Fastback Networks. He is the Editor-in-Chief of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS.

Dr. Andrews is an elected member of the Board of Governors of the IEEEInformation Theory Society. He received the National Science FoundationCAREER Award in 2007 and the 2015 Terman Award. He has co-authored14 best paper awards, including the 2016 IEEE Communications Society &Information Theory Society Joint Paper Award, the 2011 and 2016 IEEEHeinrich Hertz Prize, the 2014 IEEE Stephen O. Rice Prize, and the 2014IEEE Leonard G. Abraham Prize. He is an ISI Highly Cited Researcher.

Tianyang Bai (S’11) received the B.Eng. degreefrom the Harbin Institute of Technology, Harbin,China, in 2011, and the M.S.E. and Ph.D. degreesfrom The University of Texas at Austin, Austin, TX,USA, in 2013 and 2016, respectively, all in electricalengineering. He is currently a Senior System Engi-neer with Qualcomm Flarion Technologies, Inc.,Bridgewater, NJ, USA. His research interests includeapplications of stochastic geometry, mmWave com-munications, and massive MIMO.

Mandar N. Kulkarni (S’13) received the B.Tech.degree in electronics and communications engineer-ing from IIT Guwahati, Guwahati, in 2013, andthe M.S. degree in electrical engineering from TheUniversity of Texas at Austin in 2015, where heis currently pursuing the Ph.D. degree in electri-cal engineering. He has held internship positionswith Nokia Bell Labs, Murray Hill, NJ, USA, in2016, Nokia Networks, Arlington Heights, IL, USA,in 2014 and 2015, Technical University, Berlin,Germany, in 2012, and IIT Bangalore, Bengaluru,

India, in 2011. His research interests are broadly in wireless communication,with a current focus on modeling and analysis of cellular networks operatingat millimeter wave frequencies. He received the President of India Gold Medalin 2013.

Ahmed Alkhateeb (S’08) received the B.S. (Hons.)and M.S. degrees from Cairo University, Egypt, in2008 and 2012, respectively. He is currently pursu-ing the Ph.D. degree with the Wireless Networkingand Communication Group, Department of Elec-trical and Computer Engineering, The Universityof Texas at Austin, USA. His research interestsare in the broad area of network information the-ory, communication theory, and signal processing.In the context of wireless communication, his inter-ests include mmWave communication and massive

MIMO systems.

Abhishek K. Gupta received the B.Tech. andM.Tech. degrees in electrical engineering from IITKanpur, Kanpur, in 2010. He is currently pursuingthe Ph.D. degree with the Department of Electricaland Computer Engineering, The University of Texasat Austin. He was with Applied MicroelectronicsCircuit Corporation, Pune, Futurewei Technologies,NJ, and Nokia Networks, IL. He has authored thebooks MATLAB by Examples (Finch, 2010) andNumerical Methods in MATLAB (Apress, 2014). Hiscurrent research interests include wireless communi-

cation, stochastic geometry, and numerical methods. He was a recipient of theGE-FS Leadership Award by General Electric Foundation and the Institute ofInternational Education in 2009 and the IITK Academic Excellence Awardfor four consecutive years.

Robert W. Heath, Jr. (S’96–M’01–SM’06–F’11)received the B.S. and M.S. degrees from the Univer-sity of Virginia, Charlottesville, VA, USA, in 1996and 1997, respectively, and the Ph.D. degree fromStanford University, Stanford, CA, USA, in 2002,all in electrical engineering. From 1998 to 2001,he was a Senior Member of the Technical Staffand a Senior Consultant with Iospan Wireless Inc.,San Jose, CA, USA, where he was involved inthe design and implementation of the physical andlink layers of the first commercial MIMO-OFDM

communication system. Since 2002, he has been with the Department ofElectrical and Computer Engineering, The University of Texas at Austin,where he is currently a Cullen Trust for Higher Education Endowed Professorand a member of the Wireless Networking and Communications Group. He isalso the President and the CEO of MIMO Wireless Inc. He has co-authoredthe books Millimeter Wave Wireless Communications (Prentice Hall, 2014)and Digital Wireless Communication: Physical Layer Exploration Lab Usingthe NI USRP (National Technology and Science Press, 2012).

Dr. Heath is also an elected member of the Board of Governors of theIEEE Signal Processing Society, a licensed Amateur Radio Operator, a PrivatePilot, and a registered Professional Engineer in Texas. He has co-authoredseveral award-winning papers, including the 2010 and 2013 EURASIP Journalon Wireless Communications and Networking best paper awards, the 2012Signal Processing Magazine best paper award, the 2013 Signal ProcessingSociety best paper award, the 2014 EURASIP Journal on Advances inSignal Processing best paper award, the 2014 Journal of Communicationsand Networks best paper award, the 2016 IEEE Communications SocietyFred W. Ellersick Prize, and the 2016 IEEE Communications and InformationTheory Societies Joint Paper Award. He was a Distinguished Lecturer in theIEEE Signal Processing Society. He is an ISI Highly Cited Researcher.

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