Coverage Analysis for Dense Millimeter Wave Cellular Networks: The Impact of Array Size

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1 IEEE Wreless Communcatons and Networkng Conference WCNC 26) - Track 2 - MAC and Cross Layer Desgn Coverage Analyss for Dense Mllmeter Wave Cellular Networks: The Impact of Array Sze Xanghao Yu, Jun Zhang, and K. B. Letaef, Fellow, IEEE Dept. of ECE, The Hong Kong Unversty of Scence and Technology, Hamad Bn Khalfa Unversty, Doha, Qatar Emal: {xyuam, eejzhang, eekhaled}@ust.hk, kletaef@hbku.edu.a Abstract Mllmeter wave mmwave) communcatons has been consdered as a promsng technology for 5G cellular networks. Explotng drectonal beamformng usng antenna arrays to combat path loss s one of the defnng features n mmwave cellular networks. However, prevous works on mmwave network analyss usually adopt smplfed antenna patterns for tractablty. In ths paper, we show that there are huge dscrepances between the smplfed and actual antenna patterns when nvestgatng the coverage probablty of mmwave networks. Analytcal expressons for the coverage probabltes are derved usng tools from stochastc geometry, by consderng the actual antenna pattern wth the unform lnear array. Moreover, the mpact of the array sze s nvestgated, whch cannot be revealed from exstng results wth smplfed antenna patterns. Numercal results wll show that large-scale antenna arrays are reured for satsfactory coverage n mmwave cellular networks. Furthermore, dense mmwave cellular networks are shown to acheve much hgher rate coverage than conventonal sub-6 GHz cellular systems. I. INTRODUCTION To meet the ever-ncreasng demands for hgh-data-rate multmeda access, the capacty of next-generaton wreless networks has to ncrease exponentally. One promsng way to boost the capacty s to explot new spectrum bands. Recently, mllmeter wave mmwave) bands from 3 GHz to 3 GHz have been proposed as a promsng canddate for new spectrum n 5G cellular networks, whch prevously are only consdered for ndoor and fxed outdoor scenaros ]. Hghly drectonal antenna arrays at the transcevers can compensate for the addtonal free space path loss caused by the ten-fold ncrease of the carrer freuency n mmwave systems 2]. Recently, channel measurements usng drectonal antenna arrays have revealed some unue propagaton characterstcs of mmwave sgnals 3]. It turns out that mmwave sgnals are senstve to blockages, whch causes totally dfferent path loss laws for the lne-of-sght LOS) and non-lne-of-sght NLOS) mmwave sgnals. In addton, dense deployments are reured to releve the sgnal ntercepton n mmwave cellular networks due to the blockage senstvty, whch, fortunately, fts the evoluton of network densfcaton, e.g., by deployng small cells 4]. More mportantly, combned wth the poor scatterng envronment, drectonal antennas wll dramatcally change the sgnal power, as well as the nterference power. In mmwave cellular networks, the receved sgnal or nterference power s closely related to the angles of departure/arrval AoDs/AoAs). Ths work was supported by the Hong Kong Research Grants Councl under Grant No. 63. In partcular, the antenna array wll provde varous power gans correspondng to dfferent AoDs/AoAs. A slght shft of AoD/AoA may lead to a large array gan varaton when drectonal antenna arrays are used. Therefore, t s necessary and crtcal to ncorporate the mpact of antenna arrays when analyzng mmwave cellular networks. To mantan analytcal tractablty, the antenna patterns are often smplfed n prevous analyss of mmwave systems, wth the flat-top pattern beng a wdely used approxmaton. Snce the drectonal antenna array s one of the dfferentatng features n mmwave cellular systems, t s ntrgung to evaluate ts nfluence on the network performance. However, t s dffcult to depct the mpact of antenna arrays when usng smplfed antenna patterns. In practce, some crtcal parameters of the antenna beam pattern such as beamwdth, the nth mnor lobe maxma, nulls, and front-back rato are all determned by the array sze. Wth smplfed antenna patterns, these parameters can only be determned ualtatvely and naccurately accordng to the array sze. Hence, the smplfcaton of the antenna pattern results n some dffculty and naccuracy when analyzng the performance of mmwave cellular systems. There exst several prevous studes on performance analyss for mmwave networks 5] 9]. In 5] 8], analytcal results on sgnal-to-nterference-plus-nose rato SINR) and rate coverage based on the flat-top pattern were obtaned. Moreover, the channel model was also smplfed and dd not reflect the propagaton characterstcs of mmwave systems. The actual antenna pattern was adopted n 9] for evaluatng the capacty of an nterfered communcaton lnk. Nevertheless, all the nterferers were assumed to use the same array gan, whch weakened the practcalty of the result. In ths paper, we frst demonstrate that the dfference between the smplfed and actual antenna patterns wll ntroduce consderable dscrepances between the theoretcal and practcal results. We wll then nvestgate the coverage probablty n the downlnk mmwave cellular network wth a random spatal network model, where base statons BSs) are modeled as a homogeneous Posson pont process PPP). We assume that the BSs adopt analog beamformng to serve the users and we retan the actual antenna pattern n our work, whch s dfferent from pror works and forms the man challenge for analyss n the meantme. Based on some reasonable approxmatons, we derve a tractable expresson of the coverage probablty for dense mmwave networks, whch contans a sngle ntegral operaton, and can be easly and effcently computed. Based on the analytcal results, the mpacts of antenna array szes are nvestgated. It wll be shown that large /6/$3. 26 IEEE

2 scale antenna arrays are needed n mmwave cellular networks to mantan an acceptable coverage probablty. A comparson between mmwave cellular networks and conventonal sub-6 GHz systems wll also be provded, and shall demonstrate the superorty of mmwave networks. II. SYSTEM MODEL In ths secton, we wll frst descrbe the random spatal network model and blockage model for mmwave cellular networks. Afterwards, we wll present the mmwave channel model. A. Network Model We consder a mmwave cellular network, where BSs and users are dstrbuted accordng to two ndependent homogeneous PPPs ]. Snce the user process s statonary and ndependent of the BS process, the downlnk SINR of the typcal user has the same dstrbuton as the aggregate ones n the network ]. We assume that each user has a sngle receve antenna, and s served by the nearest BS eupped wth a drectonal antenna array composed of N t elements. All BSs are assumed to transmt wth the same transmt power P t, and each BS wll serve the users n a round-robn fashon,.e., ntracell tme dvson multple access TDMA) s adopted n ths paper. We ntroduce the LOS ball to model the blockage as shown n Fg., and t has been shown to be accurate n dense mmwave cellular networks 5]. In ths blockage model, we defne a LOS radus R, whch symbolzes the average dstance between a user and ts nearby blockages, and therefore the LOS probablty of a certan lnk s one wthn R and zero outsde the radus. The ncorporaton of the blockage effects nduces dfferent path loss laws for LOS and NLOS lnks. It turns out that NLOS nterferers are neglgble under dense BS deployments. In other words, dense mmwave networks are LOS nterference lmted. Hence, we wll focus on the nterference brought by the LOS nterferers, whose spatal dstrbuton s a PPP, denoted as Φ wth densty λ, n a fnte area wth radus R. Later we wll justfy ths assumpton through smulaton n Secton V. Drectonal antenna arrays are leveraged to provde sgnfcant beamformng gans to overcome the path loss and to synthesze hghly drectonal beams. The receved sgnal for a typcal user, denoted as the th user, s gven by y = βr α 2 h w Pt s + βr α 2 h w Pt s + n, ) where r s the dstance between the servng BS and the typcal user, whle R s the dstance between th BS and the typcal user. A N t vector h s used to denote the small scale fadng between the th BS and the typcal user, and the path loss exponent and ntercept are α and β, respectvely. In addton, the beamformng vector of the th BS s denoted as w, and n stands for the addtve whte Gaussan nose AWGN). B. Channel Model Due to hgh free-space path loss, the mmwave propagaton envronment s well characterzed by a clustered channel SgnalLnks InterferenceLnks TypcalUser Servng BS of thetypcaluser LOS Interferer NLOS Interferer Blockages Fg.. A sample network model where BSs and users are modeled as two ndependent PPPs, and each user s assocated wth the nearest BS. The LOS ball s used to model the blockages n the network. model,.e., the Saleh-Valenzuela model 3], whch can be depcted as h = N t L l= α l a H t θ l ), 2) where ) H symbolzes the conjugate transpose, and L s the number of clusters. The gan of the lth cluster s denoted as α l, whch follows ndependent Nakagam fadng for each lnk 5]. For mmwave channels contanng LOS components, the effect of NLOS sgnals s margnal snce the channel gans of NLOS paths are typcally 2 db weaker than those of LOS sgnals 3]. Hence, for the remander of ths paper, we wll focus on the LOS path,.e., L =. In addton, a t θ ) represent the transmt array response vectors correspondng to the AoDs θ, whch are ndependent and dentcally dstrbuted..d.) accordng to a unform dstrbuton on the nterval, 2π] 3]. In ths paper, we consder the unform lnear array ULA) wth N t antenna elements. Therefore, the array response vectors can be wrtten as a t θ ) =,, e jkdx cos θ,, e jkdn t ) cos θ ] T, where d and k are the antenna spacng and wavenumber, and x < N t s the antenna ndex. In order to enhance the drectonalty of the beam, the antenna spacng d should be less than half-wavelength to avod gratng lobes ]. III. ANALOG BEAMFORMING AND ANTENNA PATTERN In ths secton, we wll frst ntroduce the optmal analog beamformng strategy and then llustrate the necessty of employng the actual antenna pattern when analyzng mmwave cellular networks. A. Analog Beamformng Whle varous space-tme processng technues can be appled at each mult-antenna mmwave BS, we wll focus on the analog beamformng, whch s able to control the beam drecton va phase shfters. Due to the low cost and power consumpton, analog beamformng has already been used n 3)

3 Normalzed Array Gan Coverage Probablty Angle n Rad. a) Vsualzaton of two dfferent antenna patterns. Fg Actual Pattern Flat-top Pattern SINR Threshold db) b) Coverage probablty evaluaton usng two dfferent antenna patterns. The comparson between the flat-top pattern and the actual pattern. some mmwave systems such as WGg IEEE 82.ad) ]. Assumng the AoD of the channel between the th BS and ts servng user s ϕ, the optmal analog beamformng vector s w = a t ϕ ), 4) whch means the BS should algn the beam drecton exactly wth the AoD of the channel to obtan the maxmum power gan. B. Antenna Pattern Based on the optmal analog beamformng vector 4), for the typcal user, the power gan provded by the small scale fadng and beamformng of the th BS can be expressed as h w 2 = N t α 2 a H t θ )a t ϕ ) 2 = N t g Gθ, ϕ ), 5) where g s the power gan of small scale fadng and Gθ, ϕ ) s the normalzed array gan of the th BS, whch can be expressed as Gθ, ϕ ) = a H t θ )a t ϕ ) 2 = N 2 t 2 e jkdxcos θ cos ϕ ) x= = sn2 N t 2 kdcos θ cos ϕ ) ] 2 sn 2 2 kdcos θ cos ϕ ) ]. The flat-top pattern s often used as an approxmaton of the actual antenna pattern, where the array gans wthn the half-power beamwdth HPBW) ] are assumed to be the maxmum power gan and the array gans correspondng to the remanng AoDs are approxmated to be the frst mnor maxmum gan of the actual pattern, as shown n Fg. 2a) 5]. However, ths approxmaton wll ntroduce huge dscrepances when we evaluate the network coverage probablty. A smulaton result n Fg. 2b) shows that there s a large gap between the SINR coverage probabltes usng two dfferent antenna patterns. The parameter settng s the same as that of Fg. 4 n Secton V. 6) More mportantly, gven the operatng freuency and the antenna spacng, the antenna pattern s crtcally determned by the array sze. In the flat-top pattern, however, t s very dffcult to uanttatvely and accurately depct the varaton of the HPBW and the frst mnor maxmum for dfferent array szes and AoDs. In other words, the smplfed antenna patterns oblterate the possblty of analyzng the mpact of drectonal antenna arrays, whch s a crtcal and unue ssue n mmwave systems. Hence, t s necessary to retan the actual antenna pattern n the analyss of mmwave cellular networks, n order to nvestgate the role of the antenna array n mmwave networks and to ensure the consstency wth practcal systems. IV. COVERAGE ANALYSIS In ths secton, we wll derve a tractable expresson for the coverage probablty consderng the actual antenna pattern. The bottleneck of the analyss s the dstrbuton of the nterference power due to the complcated form of the normalzed array gan Gθ, ϕ ). Based on some approxmatons, we wll provde an analytcal result whch can be easly evaluated. A. Sgnal-to-nterference-plus-nose Rato We assume that each BS has full nformaton about the AoD of the channel between tself and ts servng user, and can algn the beam to the AoD drecton usng analog beamformng. From )-6), the receve SINR s gven by SINR = = P t N t α 2 a H t θ )w 2 βr α σ 2 + Φ\ P tn t α 2 a H t θ )w 2 βr α g r α σ 2 n + Φ\ g Gθ, ϕ )R α = g r α σn 2 + I, where g s a Gamma dstrbuted random varable accordng to the Nakagam fadng assumpton, and σn 2 = σ2 βp t N t s the normalzed nose. In ths paper, we wll evaluate the coverage probablty, whch s defned as the probablty that the receved SINR s greater than a gven threshold γ,.e., where SINR s gven n 7). B. Analyss of Coverage Probablty 7) p c γ) = P SINR > γ), 8) Snce we assume that each user s served by the nearest BS and the PPP Φ s n a fnte area wth radus R, the condtonal dstrbuton of the dstance r between the servng BS and the typcal user s fr) = 2πλr e πλr 2, 9) e λπr2 and for dense networks, we have e λπr2. Based on 7) and 8), the coverage probablty s gven by p c γ) = R fr)p g > γr α σ 2 n + I) ] dr, ) where g s a normalzed gamma random varable wth the Nakagam fadng parameter N.

4 Ir) = r exp r exp ξnγσ 2 nr α πλr 2 + 4π2 λe a kdan t p= ) p = ξnγσnr 2 2 πλr 2 + 2π2 λξnγlnr lnr) kdan t r 2 + 4π2 λe a kdan t ] p R 2 α r α r 2) Ip, ) ] p R 2 2 r 2 r 2 )Ip, ) p=2 ) p =2 α 2 α = 2 2) Lemma. From 2]) For a normalzed gamma random varable g wth parameter N, the probablty Pg < γ) can be tghtly upper bounded by Pg < γ) < e ξγ) N, ) where ξ = NN!) N. where a) follows the Laplace functonal of the PPP Φ, and b) follows the Jensen s neualty. One ntutve way to manpulate the moment-generatng functon wth respect to the array gan n 3) s to drectly derve the dstrbuton of the array gan. However, ths s hghly ntractable due to the complcated form of 6). We wll next gve an analytcal result based on some reasonable approxmatons. Approxmaton : We approxmate the denomnator of the array gan functon as 2 2 kdcos θ cos ϕ ) ] 2. In 6), the antenna spacng should be less than half-wavelength. Therefore, the term 2 kdcos θ cos ϕ ) should be wthn a small range near zero, and the approxmaton s establshed due to the fact that sn x x when x s small. Approxmaton 2: The dstrbuton of the random varable cos θ cos ϕ cannot be expressed n a closed-form, whch s the man obstacle for further analyss. Here, we propose to use a unform dstrbuton n A, A] to approxmate the orgnal dstrbuton. Although ths approxmaton s heurstc, to some extent, t creates the possblty to analyze the coverage probablty wth the actual antenna pattern, and the accuracy wll be tested through smulaton n Secton V. 2 Lemma also gves a tght lower bound of the coverage probablty n ). 3 The array gan Gθ, ϕ ) s abbrevated as G for notatonal smplcty. Based on Approxmatons and 2, the array gan 6) s approxmated as the suare of a snc functon wth respect to a unformly dstrbuted varable φ as follows Gφ ) sn2 N t 2 kdφ ) 2 kdφ ) 2. 4) Lemma 2. The defnte ntegral from to of the -th power Based on Lemma, a tght lower bound of the probablty P g > γr α σn 2 + I) ] of the suared snc functon s gven by can be derved as 2 P g > γr α σn 2 + I) ] sn 2 φ ) k k) 2 > E I e ξγrα σ 2 +I)) ] φ 2 dφ = π. 5) k!2 k)! k= N n 2) Proof: Through Euler s formula and the bnomal theorem, the orgnal ntegral can be manpulated as N N ) = ) n+ e ξnγrα σ 2 n LI ξnγr α ). n 2 ) n= 2 lm ϵ It can be seen from 2) that the man task to obtan the coverage probablty p c γ) s to derve the Laplace transformaton Accordng to the resdue theorem and Cauchy dfferentaton + 2 2j) 2 ) k e jφ2 k) dφ. 6) k k= x jϵ) of the nterference L I s), whch can be further derved as 3 formula, we fnd the orgnal ntegral euals { R L I s) a) = exp 2πλ Eg,G exp sg G x α )] ) } ) 2 2πj xdx r 2 2j) 2 ) k d 2 k 2 )! dφ 2 ejφ2 k), 7) { b) R exp 2πλ EG exp sg x α )] ) } k= whch can be smplfed as 5). xdx, r Based on Lemma 2 and 4), we can tackle the momentgeneratng functon n 3), whch s the bottleneck of the 3) analyss. Lemma 3. The moment-generatng functon wth respect to the array gan E G exp sg x α )] can be expressed as E G exp sg x α )] + 2πea ) p kdan t p= x α a p Proof: See Appendx A. k= p = ) p s ) k k) 2. k!2 k)! 8) Now, wth Lemma 3, we present the man result on the SINR coverage probablty p c γ) as follows. Theorem. The SINR coverage probablty p c γ) can be computed as 2πλ N ) p c γ) ) n+ N R Ir)dr, 9) e λπr2 n n= where Ir) s gven by 2) and ) p ξnγ) a p Ip, ) = 2 α k= ) k k) 2. 2) k!2 k)!

5 Emprcal Cumulatve Dstrbuton Functon SINR Coverage Probablty Proof: The proof s establshed by applyng Lemma 3 to 3) wth some basc manpulatons on ntegral operatons and then substtutng nto 9), ), and 2). Remark : The analytcal result of the coverage probablty gven n 9) can be easly evaluated by numercal ntegraton. Although 2) nvolves a summaton of nfnte terms, t wll be shown n Secton V that the hgh-order terms contrbute lttle to the whole summaton, and usng fnte terms s accurate enough for numercal computaton. Furthermore, a proper choce of the parameter a at whch the seres expands wll save the number of terms to be calculated. V. NUMERICAL RESULTS In ths secton, we wll frst examne the approxmaton made n Secton IV. We wll then demonstrate our result through smulatons. We assume that the bandwdth s 5 MHz, and the transmt power of all the BSs s set to Watt. The separaton between the antenna elements s uarterwavelength to avod gratng lobes. From the recent measurements of LOS mmwave sgnal propagatons 3], the path loss exponent α s eual to or slghtly larger than two and β = 6.4 db. All smulaton results are averaged over 5 thousand realzatons. A. Justfcaton of Approxmaton 2 In Approxmaton 2, we proposed to approxmate the dstrbuton of cos θ cos ϕ as a unform dstrbuton. Here we wll justfy ths approxmaton through evaluatng the cumulatve dstrbuton functons of Gθ, ϕ ) n 6) and Gφ ) n 4). We approxmate the parameter n unform dstrbuton A =.8, and ths value wll be used n all the followng numercal evaluatons N t =64, 28, 256 Actual Dstrbuton 6) Unform Approxmaton 4) G Fg. 3. The cumulatve dstrbuton functons of Gθ, ϕ ) n 6) and Gφ ) n 4). From Fg. 3, we can see that, based on Approxmaton 2, two cumulatve dstrbuton functons almost concde wth each other. It ndcates that the dstrbutons of two array gans are almost the same. Ths justfes that Approxmaton 2 s reasonable n the sense of the dstrbuton of array gans, whch s the crtcal part n the analyss. B. Coverage Probablty Evaluaton We shall now provde the smulaton results to verfy the analytcal result n Secton IV. The Nakagam fadng parameter N s set to 2, and the path loss exponent s α = 2. We compare the smulaton results and the analytcal results from Theorem. To verfy the assumpton that mmwave networks are LOS nterference lmted, we also nclude the NLOS BSs n the smulaton, where we assume that N =,.e., Raylegh fadng, β n = 72 db, and the path loss exponent α = 4 for the NLOS nterferng sgnals =2-3, R=5, N t =28 = -3, R=2, N t =64 Smulaton LOS Smulaton LOS+NLOS Analytcal Result SINR Threshold db) Fg. 4. SINR coverage probablty n dense mmwave cellular networks wth dfferent parameter settngs. Fg. 4 verfes that dense mmwave cellular networks are ndeed LOS nterference lmted, whch means that the NLOS nterference s small compared to the total nterference due to the more severe path loss and small scale fadng. When we numercally calculate 9) n Theorem, a fnte number of terms are ncluded n the summaton, as llustrated n Remark, untl the numercal values converge. We propose to set a = ξnγ/2 to save the number of the terms nvolved n the summaton. Fg. 4 also shows that the approxmatons and the dervatons n Secton IV are reasonable and accurate for dfferent settngs of the network, so the coverage probablty p c γ) provded n Theorem can be regarded as a relable analytcal result when consderng the drectonal antenna pattern n mmwave cellular networks. C. Impact of Array Sze In ths subsecton, we wll nvestgate the mpact of drectonal antenna arrays and compare mmwave cellular systems wth conventonal sub-6 GHz ones. In mmwave networks, we assume that the LOS radus s 2 meters, the Nakagam fadng parameter N = 3, and the path loss exponent α = 2.. In conventonal networks, we assume that small scale fadng s..d. Raylegh fadng and the path loss exponent euals 4. All BSs use maxmum rato transmsson beamformng to transmt a sngle data stream wth MHz avalable bandwdth. For both systems, we assume a dense BS deployment wth densty of 3 BSs per unt area, and evaluate the rate coverage probablty wth the antenna array sze from 64 to 256. The rate coverage probablty s defned as PW log 2 + SINR) > ˆγ] = p c 2ˆγ/W ), where W s the bandwdth

6 Rate Coverage Probablty Conventonal Networks MmWave Networks Antenna Array Sze Fg. 5. Rate coverage probablty n dense mmwave cellular networks wth dfferent parameter settngs. and ˆγ s the rate threshold. The results n Fg. 5 show that, for a gven rate reurement, large-scale antenna arrays are needed n mmwave cellular networks to guarantee an acceptable coverage probablty. It also shows that the coverage probablty of conventonal systems s much lower than that of mmwave networks. Ths phenomenon shows that the large bandwdth and drectonal antenna arrays beneft mmwave systems. Moreover, as t s much easer to mplement largescale antenna arrays n mmwave systems than conventonal ones, thanks to the small wavelength, mmwave networks stand out as an excellent canddate for future 5G systems to provde mult-ggabts per second data rate. VI. CONCLUSIONS One man contrbuton of ths paper was dentfyng the mportance of usng the actual antenna pattern n the coverage analyss for mmwave networks. In partcular, t was demonstrated that the wdely used flat-top antenna pattern wll result n a large performance gap compared to practcal systems. On the contrary, wth the actual antenna pattern, we were able to nvestgate the mpact of the array sze on the coverage probablty. Numercal results showed that largescale drectonal antenna arrays are needed n mmwave cellular systems to guarantee an acceptable coverage probablty. MmWave cellular networks, takng advantage of the wde bandwdth and explotng drectonal transmsson, were shown to acheve a much hgher rate coverage than conventonal sub- 6 GHz networks. It wll be nterestng to extend the coverage analyss to mmwave cellular networks wth more advanced precodng technues, e.g., hybrd precodng 3], 4], whch can support spatal multplexng. APPENDIX A PROOF OF LEMMA 2 Based on the approxmated array gan 4), we can derve the moment-generatng functon as E G exp sg x α )] { A sx α sn2 N t c) = ea 2A A p= 2 kdφ ) 2 kdφ ) 2 a ] p } dφ, 22) where c) follows the seres expanson of the exponental functon at pont a. Then swtchng the order of the ntegral and summaton, we can fnd that 22) s eual to e a ) p p ) p Aa p + s A p= = A x α a p sn 2 N t 2 kdφ ) ] ) 2 dφ d) < ea A ) p Aa p + p= 2 kdφ p ) p s = x α p 2π a kdn t k= ) k k) 2 k!2 k)! ], 23) whch gves the expresson n 8), and d) follows Lemma 2 gven that, for the the tals of the 2-th power of the snc functon, the ntegrals are neglgble when N t s not too small. REFERENCES ] E. Peraha, C. Cordero, M. Park, and L. Yang, IEEE 82.ad: Defnng the next generaton mult-gbps W-F, n Proc. 2 7th IEEE Consumer Commun. and Netw. Conf. CCNC), Jan. 2, pp. 5. 2] G. R. MacCartney, M. K. Samm, and T. S. Rappaport, Explotng drectonalty for mllmeter-wave wreless system mprovement, n Proc. 25 IEEE Int. Conf. Commun. ICC), London, UK, June 25, pp ] M. Akdenz, Y. Lu, M. Samm, S. Sun, S. Rangan, T. Rappaport, and E. Erkp, Mllmeter wave channel modelng and cellular capacty evaluaton, IEEE J. Sel. Areas Commun., vol. 32, no. 6, pp , June 24. 4] C. L, J. Zhang, and K. B. Letaef, Throughput and energy effcency analyss of small cell networks wth mult-antenna base statons, IEEE Trans. Wreless Commun., vol. 3, no. 5, pp , May 24. 5] T. Ba and R. W. Heath, Coverage and rate analyss for mllmeterwave cellular networks, IEEE Trans. Wreless Commun., vol. 4, no. 2, pp. 4, Feb ] M. Kulkarn, S. Sngh, and J. Andrews, Coverage and rate trends n dense urban mmwave cellular networks, n Proc. 24 IEEE Global Commun. Conf. GLOBECOM), Austn, TX, Dec. 24, pp ] A. Thornburg, T. Ba, and R. W. Heath, Mmwave ad hoc network coverage and capacty, n Proc. 25 IEEE Int. Conf. Commun. ICC), London, UK, June 25, pp ] K. Venugopal, M. C. Valent, and R. W. Heath, Interference n fnteszed hghly dense mllmeter wave networks, n Proc. 25 Inf. Theory and Appl. ITA), San Dego, CA, Feb. 25, pp ] F. Babch and M. Comsso, A relable approach for modelng the actual antenna pattern n mllmeter-wave communcaton, IEEE Commun. Lett., vol. 9, no. 8, pp , Aug. 25. ] M. Haengg, Stochastc Geometry for Wreless Networks. Cambrdge Unversty Press, 22. ] C. A. Balans, Antenna Theory: Analyss and Desgn. John Wley & Sons, 25. 2] H. Alzer, On some neualtes for the ncomplete Gamma functon, Math. Comput., vol. 66, no. 28, pp , Apr ] O. El Ayach, S. Rajagopal, S. Abu-Surra, Z. P, and R. W. Heath, Spatally sparse precodng n mllmeter wave MIMO systems, IEEE Trans. Wreless Commun., vol. 3, no. 3, pp , Mar ] X. Yu, J.-C. Shen, J. Zhang, and K. B. Letaef, Alternatng mnmzaton algorthms for hybrd precodng n mllmeter wave MIMO systems, IEEE J. Sel. Topcs Sgnal Process., to appear.