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Optimizing the Energy Efficiency for Future 5G Networks

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Optimizing the Energy Efficiency for Future 5G Networks
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  Optimizing the Energy Efficiency for Future 5GNetworks Betim Maloku 1 , Bujar Krasniqi 1 , Fjolla Ademaj 21 Faculty of Electrical and Computer Engineering, University of Prishtina, Prishtina 10000, Kosovo 2 Vienna University of Technology, Institute of TelecommunicationsGusshausstrasse 25/389, A-1040 Vienna, Austria { betim.maloku, bujar.krasniqi } @uni-pr.edu, fademaj@nt.tuwien.ac.at  Abstract —One of the challenges for the 5th generation of mobile cellular networks is the energy consumption efficiency.In this paper we propose an optimization algorithm thatmaximizes the energy efficiency. We apply the proposedoptimization algorithm for both indoor- and outdoor users.The energy efficiency optimization is carried under variousnetwork conditions such as maximizing the bit rate, minimizingthe transmit power and minimizing the interference. We usethe dual decomposition technique to find the optimal solutionin terms of energy efficiency, under the condition for optimalpower allocation. Our results reveal an increase in terms of energy efficiency by applying our algorithm. Furthermore theyshow a higher energy efficiency achieved under the dynamicpower allocation compared to a static power allocation.  Keywords — 5G, energy efficiency, convex optimization, dualdecomposition, dynamic power allocation I. I NTRODUCTION With the rapid growth of mobile internet and internet of things, the demands for high-speed data applications have beenincreased rapidly. According to [1], in the Western Europeancountries, the daily mobile traffic will be 67 times higher inyears from 2010 to 2020. The 5G mobile network, whichis expected to be deployed for the first time by 2020, willprovide 1000 times higher data rates and save up to  90% of energy expenditure per service compared with the LongTerm Evolution (LTE). Furthermore, it is expected that thebattery lifetime of connected devices on a 5G netowork, willbe extended by 10 times, and that a spectral efficiency of morethan 1000  Gbit 󰀯 s 󰀯 km 2 can be achieved in dense urban areaswith 5 times less latency in end-to-end (E2E) connections [2].The architecture and radio access technologies of all-IP-based networks have to be extended from the legacy 4G systemin order to support the requirements for 5G. There in anongoing discussion regarding new technologies that 5G shouldsupport. According to [2], ultra-densification and millimeterwave are two of the future 5G technologies. More active nodesper unit area and Hz will improve the spectral efficiency. Withthe densification of access nodes, the total energy expense willalso increase. It is known that a considerable part of totalenergy consumption goes for the radiated antennas energy.Hence, in order to reduce the energy expenditure, it is neces-sary to optimize the amount of energy radiated by antennas.In this paper we present an optimization algorithm thatmaximizes the energy efficiency in terms of transmit data rateand assigned power. We consider a frequency range of 28GHz and assume a network consisting of outdoor micro sitesand indoor access points. We optimize the power allocation forboth indoor and outdoor users. The results indicate an increasein terms of energy efficiency, by setting an optimal transmitpower to both micro site and indoor access points. Furthermoreby applying a dynamic power allocation scheme, the energyefficiency is increased.This contribution outlines as follows. In Section II ispresented the system model and the path loss attenuationmodels used to describe the channel properties. In Section III,we formulate the energy efficiency optimization problem andsolve it using Water-Filling-Like method for power allocation.In Section IV we present the simulation results for optimalpower allocation in order to increase the energy efficiency.Section V concludes the work.II. S YSTEM  M ODEL In our system model, we consider seven micro sites eachequipped with three sector antennas, yielding three cells persite. In each cell, indoor access points (IAPs) are randomlydistributed. The system model is illustrated in Fig.1.In order to classify the users, we use the indoor path lossmodel (IPLM) threshold criteria. If the users’ IPLM is smallerthan the IPLM threshold then the user is classified as an indooruser, otherwise is classified as an outdoor user. In the systemmodel, indoor users denoted as  M  IU are served by IAPs, andoutdoor users denoted as  M  OU , are served by micro sites. Inour hexagonal grid, the center micro site is indicated as  BS  0 with its three sectors denoted by S  01 , S  02  and S  03 , where index 0  denotes  BS  0  and indexes  1 ,  2  and  3  denote first, secondand third sector, respectively. Outdoor users that belong to thesector  S  01  of micro site  BS  0  receive power from sector  S  01 ,and interference from two other sectors ( S  02  and  S  03 ) of theirserving micro site, and from all sectors ( S  k 1 ,  S  k 2  and  S  k 3 )of other micro sites,  BS  k , where  k  =  1 , 2 , 3 , 4 , 5 , 6 . Assumingthat transmit power of IAP is low, the interference from IAPis neglected. The energy efficiency  η OU m  of an outdoor user  m                      Figure 1 . System model comprising micro sites and IAPs.Outdoor and indoor users are denotes as  M  OU and  M  IU ,respectively.located in sector  S  01  is defined as [3] η OU m  =  R OU m  p OU  0 ,  (1)where  R OU m  denotes the transmission rate, and  p OU  0  is thetransmit power assigned to an outdoor user  m .The transmission rate achieved by an outdoor user m locatedin sector  S  01  is defined as [4],[5] R OU m  =  B OU m  log 2 ⎛⎜⎜⎜⎝ 1 +  G OU  0 m  p OU  0 N  0 B OU m  +  6 ∑ k = 1 G OU km  p OU k ⎞⎟⎟⎟⎠ ,  (2)where  B OU m  represents the bandwidth assigned to the outdooruser  m ,  N  0  is the noise spectral density, and  p OU k  is theinterference power. The path loss model of the desired channelis denoted with  G OU  0 m , while  G OU km  represents the path lossmodel of the interference channels. The path loss model forboth desired and interfering channels is defined in (5), exceptthat for the interfering channels, the small-scale fading  F   isnot taken into account. Indoor users which belong to sector S  01  of base station  BS  0  receive power from IAP located insector  S  01  of base station  BS  0 , and interference from allsectors ( S  k 1 ,  S  k 2  and  S  k 3 ) of base stations,  BS  k , where k  =  0 , 1 , 2 , 3 , 4 , 5 , 6 . Assuming the effect of wall penetrationlosses and that transmit power of IAP is low, we neglect theinterference from other IAPs. The energy efficiency  η IU m  of anindoor user  m  located in sector  S  01  is defined as η IU m  =  R IU m  p IU  0 ,  (3)where  R IU m  denotes the transmission rate, and  p IU  0  is thetransmit power assigned to an indoor user  m . The transmissionrate achieved by an indoor user  m  located in sector  S  01  is A LGORITHM  1  A LGORITHM FOR  U SER  C LASSIFICATION Require:  IPLM  threshold if   IPLM   <  IPLM  threshold  thenIUser  →  IAPelseOUser  →  OBSend if  Calculate:  p IU  0  ,p IU k  ,p OU  0  ,p OU k using Equation (9) for power allocation. defined as R IU m  =  B IU m  log 2 ⎛⎜⎜⎜⎝ 1 +  G IU  0 m  p IU  0 N  0 B IU m  +  6 ∑ k = 0 G IU km  p IU k ⎞⎟⎟⎟⎠ ,  (4)where  B IU m  denotes the bandwidth assigned to the indooruser  m ,  N  0  represents the noise spectral density, and  p IU k is the interference power from micro site. The terms  G IU  0 m and  G IU km  denote the path loss model of the desired channeland interfering channels, respectively. The path loss model fordesired channel is defined by (6), while the path loss model forinterference channels is expressed by (5), except that small-scale fading  F   is not taken in consideration and a penetrationloss factor  L  pen  is added.  A. Path-loss Attenuation Models To calculate the path loss, we consider the path loss modelfrom [6]. In order to account for the small-scale fading, thepath loss  G km  measured in  dB  is expressed as G km  = −[ 79 . 2 + 26log 10 d + X  σ  − A + F  ]  (5)where  d  represents the distance between outdoor (indoor) userand base station in  m ,  X  σ  is the log-normal shadowing in  dB , A  is the sum of user antenna gain and base station antennagain in  dBi , and  F   denotes the small-scale fading in  dB . Theconstants  79 . 2  and  26  are specific for the center frequency of  28GHz . The antenna gain  A  is defined by horizontal antennapattern [7], [8]. The path loss  G IU  0 m  of the desired channel forindoor users is given as [9] G IU  0 m  = −[ 20log 10 f   + N   log 10 d + P  f  ( n )− 28 ] ,  (6)where  f   is the center frequency in  MHz ,  N   is the distancepower loss coefficient,  d  denotes the distance between indooruser and IAP in  m ,  P  f  ( n )  is the floor penetration loss factorin  dB ,  n  is the number of floors between indoor user and IAP.III. O PTIMIZATION  A LGORITHM In the following, we present the algorithm for classifyingthe users as indoor or outdoor, using the IPLM thresholdcriteria. User classification is outlined in Algorithm 1. Basedon this algorithm, if the user is classified to be indoor, thenthe user will be connected to an IAP, otherwise the user will                      be connected to the micro site. At the end of algorithm, thepower is allocated optimally in order to maximize the energyefficiency.Using a vector-matrix form, the power and bandwidthallocation problem to optimize the energy efficiency is givenasmaximize p,b 1 T  η OU  + 1 T  η IU  (7a)subject to α IU  p OU  0  + α OU  p IU  0  ≤  P  max (7b) Λ × 󰁛  pb 󰁝  = Γ ,  (7c) p ≥ 0 ,  (7d) b ≥ 0 ,  (7e)where  R OU  and  R IU  are the vector elements of indoor andoutdoor user rates. The term η OU  denotes the energy efficiency(EE) for outdoor users and  η IU  represents the EE for indoorusers. The terms  p OU  0  and  p IU  0  are the vector elements of indoor and outdoor user powers, respectively. The coefficients α IU   and  α OU  , in constrain (7b), are used to express theportion of power radiated from IAP and outdoor micro sites,respectively. User rates and user powers are defined as R OU  = [ R OU  1  ,R OU  2  ,...,R OU M  OU  ] , R IU  = [ R IU  1  ,R IU  2  ,...,R IU M  IU  ] , p OU  0  = [  p OU  0 1 ,p OU  0 2 ,...,p OU  0 M OU  ] , p IU  0  = [  p IU  0 1 ,p IU  0 2 ,...,p IU  0 M IU  ] , Γ = [ P  max ,B max ] T  , Λ = 􀁛 1 1 0 00 0 1 1 󰁝 . (8)where, the term  P  max denotes the maximum power while theterm  B max denotes the maximum bandwidth in the assignedcell. Vectors of the power and the bandwidth are defined as p = [  p OU  0  ,p IU  0  ] T  , b = [ B OU  ,B IU  ] T  . (9)The constraints (7b), (7c), (7d) and (7e), in (7), are linear, andthus convex as well. As the energy-efficiency maximizationis contained by the energy-efficiency maximization problemin (7), in standard power control as a particular case [4],the energy-efficiency maximization problem is not convex.Assuming that indoor and outdoor users are served by cellswhich use equal powers  p OU  0  =  p OU k  and  α OU   p IU  0  =  p IU k  , theenergy-efficiency maximization problem is solvable in Water-filling Like method as it becomes a convex problem.The optimization problem shown above, can be used forthe case of multiple users on multiple cells. Due to simplicity,we focus on a single-cell case with users located withinthe cell. The second derivative of   η OU m  with respect to  p OU  0 is concave. Consequently, we get that  ˆ η OU  ( B OU m  ,p OU  0  ) = B OU m  η OU m  (  p OU  0  󰀯 B OU m  )  is concave since it is the prospect of aconcave function [10]. Moreover, as  η IU m  is also concave dueto its form as  η OU m  ’s, the optimization problem (7) is concavetoo.  A. Power Allocation with Water-filling-like method  The analytic solution for optimization problem as presentedin (7) is not feasible. In order to achieve a solution, we furtherconsider that the bandwidth allocation is unchangeable. Byfixing the bandwidth, an algorithm for power allocation can bederived from the Karush-Kuhn-Tucker (KKT) conditions foroptimality [10]. For simplicity, we are replacing  G OU  0 m  =  s m , 6 ∑ k = 1 G OU km  =  t m ,  G IU  0 m  =  u m  and 6 ∑ k = 0 G IU km  =  v m  in the writtenequations. We formulate the optimization problem (7) throughthe Lagrangian as given inŁ ( p , µ , λ ) = 1 T  η OU  + 1 T  η IU  − µ T  ( 1 T  p − P  max )+ λ T  p . (10)where the notations  µ  and  λ  = [ λ OU  ,λ IU  ]  denote the La-grange multipliers,  µ  is the sum-power and  λ  is the positivityconstraint. By applying the Karush-Kuhn-Tucker conditions[10] [11], we have: p ≥ 0 ,  (11a) 1 T  p − P  max ≤  0 ,  (11b) λ  ≥ 0 ,  (11c) λ OU   p OU  =  0 ,  (11d) λ IU   p IU  =  0 ,  (11e) ∂L∂p OU  0 = − M  OU  󲈑 m = 1 B OU m  p OU  0  ln21 ( N  0 B OU m  + t m  p OU  0  )×  s m N  0 B OU m [ N  0 B OU m  +( s m  + t m )  p OU  0  ] +  B OU m (  p OU  0  ) 2  log 2 󰀨 1 +  s m t m 󰀩+ µ − λ OU  =  ψ OU  (  p OU  0  )− λ OU  =  0 ,  (11f) ∂L∂p IU  0 = − M  IU  󲈑 m = 1 B IU m  p IU  0  ln21 ( N  0 B IU m  + α OU  v m  p IU  0  )×  u m N  0 B IU m [ N  0 B IU m  +( u m  + α OU  v m )  p IU  0  ]+  B IU m (  p IU  0  ) 2  log 2 󰀨 1 +  u m α OU  v m 󰀩+ µ − λ IU  =  ψ IU  (  p IU  0  )− λ IU  =  0 .  (11g)The equalities (11f) and (11g), derived from the KKT con-ditions, are the first derivatives with respect to  p OU  0  and  p IU  0  , respectively, of the Lagrangian function expressed inEquation (10). We show that the optimal power allocationcan be calculated efficiently when variable  µ  is constant.We get the optimal  p OU  0  and optimal  p IU  0  as function of  µ , considering the constraints (11c), (11d), (11e), (11f) and(11g), from the roots of functions  ψ OU  (  p OU  0  )  and  ψ IU  (  p IU  0  ) ,respectively. The roots of the above functions can be calcu-lated using Ferrari-Lagrange method [12]. Considering that µ OU  =  M  OU  ∑ m = 1 [( s m 󰀯( N  0 ln 2 )]  and  µ IU  =  M  IU  ∑ m = 1 [( u m 󰀯( N  0 ln 2 )] ,                      we get the optimal power allocation for the outdoor users andthe indoor users given as  p OU  0  = ⎧⎪⎪⎨⎪⎪⎩  p OU  0  ( µ ) ,  if   1 µ  ≥  1 µ OU  , 0 ,  otherwise , (12)  p IU  0  = ⎧⎪⎪⎨⎪⎪⎩  p IU  0  ( µ ) ,  if   1 µ  ≥  1 µ IU  , 0 ,  otherwise , (13)When  M  OU  =  1  and  M  IU  =  1 , the roots shown above can becalculated analytically and give the optimal power assigned tooutdoor user as presented in the following  p OU  0  = ⎧⎪⎪⎨⎪⎪⎩−  k 31 4 k 41 + Z  1  +  12 √ − 4 Z  21  − 2 d 1  −  e 1 Z  1 ,  if   1 µ  ≥  1 µ OU  , 0 ,  otherwise , (14)where parameters  d 1 ,  e 1 ,  Z  1 ,  Q 1 ,  ∆ 01 ,  ∆ 11 ,  k 01 ,  k 11 ,  k 21 , k 31  and  k 41  are given in the following d 1  =  8 k 41 k 21  − 3 k 231 8 k 241 ,  (15a) e 1  =  k 331  − 4 k 41 k 31 k 21  + 8 k 241 k 11 8 k 341 ,  (15b) Z  1  =  12 󲈚 − 23 d 1  +  13 k 41 󰀨 Q 1  +  ∆ 01 Q 1 󰀩 ,  (15c) Q 1  =  3   ∆ 11  +􂈚  ∆ 211  − 4∆ 301 2  ,  (15d) ∆ 01  =  k 221  − 3 k 31 k 11  + 12 k 41 k 01  (15e) ∆ 11  =  2 k 321  − 9 k 31 k 21 k 11  + 27 k 231 k 01  + 27 k 41 k 211 − 72 k 41 k 21 k 01  (15f) k 01  =  B OU  1  ( N  0 B OU  1  ) 2 log 2 󰀨 1 +  s 1 t 1 󰀩  (15g) k 11  =  B OU  1  ( N  0 B OU  1  )󰁛( s 1  + 2 t 1 ) log 2 󰀨 1 +  s 1 t 1 󰀩−  s 1 ln2  (15h) k 21  =  t 1 ( s 1  + t 1 ) B OU  1  log 2 󰀨 1 +  s 1 t 1 󰀩+( N  0 B OU  1  ) 2 µ  (15i) k 31  =  N  0 B OU  1  ( s 1  + 2 t 1 ) µ  (15j) k 41  =  t 1 ( s 1  + t 1 ) µ  (15k)The optimal power assigned to indoor user is  p IU  0  = ⎧⎪⎪⎨⎪⎪⎩−  k 32 4 k 42 + Z  2  +  12 √ − 4 Z  22  − 2 d 2  −  e 2 Z  2 ,  if   1 µ  ≥  1 µ   IU  , 0 ,  otherwise , (16)where parameters  d 2 ,  e 2 ,  Z  2 ,  Q 2 ,  ∆ 02 ,  ∆ 12 ,  k 02 ,  k 12 ,  k 22 , k 32  and  k 42  are given in the following d 2  =  8 k 42 k 22  − 3 k 232 8 k 242 (17a) e 2  =  k 332  − 4 k 42 k 32 k 22  + 8 k 242 k 12 8 k 341 (17b) Z  2  =  12 󲈚 − 23 d 2  +  13 k 42 󰀨 Q 2  +  ∆ 02 Q 2 󰀩  (17c)TABLE I:S IMULATION PARAMETERS Parameters Values Outdoor base station maximum power 5 WIndoor access point maximum power 0.1 WMaximum bandwidth  B max 100MHzCenter carrier frequency  f  c  28GHzOutdoor power coefficient  α OU   50Indoor power coefficient  α IU   0.02Outdoor user position (90 m, 160 ○ )Indoor user position (160 m, 160 ○ )Indoor access point position (165 m, 160 ○ )Inter base station distance  R  300 mShadowing  X  σ  N(0,9.6)dBFast fading  F X  22  dBPenetration loss  L pen  20 dB Q 2  =  3   ∆ 12  +􂈚  ∆ 212  − 4∆ 302 2  (17d) ∆ 02  =  k 222  − 3 k 32 k 12  + 12 k 42 k 02  (17e) ∆ 12  =  2 k 322  − 9 k 32 k 22 k 12  + 27 k 232 k 02  + 27 k 42 k 212 − 72 k 42 k 22 k 02  (17f) k 02  =  B IU  1  ( N  0 B IU  1  ) 2 log 2 󰀨 1 +  u 1 α OU  v 1 󰀩  (17g) k 12  =  B IU  1  ( N  0 B IU  1  )[( u 1  + 2 α OU  v 1 )×  log 2 󰀨 1 +  u 1 α OU  v 1 󰀩−  u 1 ln2   (17h) k 22  =  α OU  v 1 ( u 1  + α OU  v 1 ) B IU  1  log 2 󰀨 1 +  u 1 α OU  v 1 󰀩+( N  0 B IU  1  ) 2 µ  (17i) k 32  =  N  0 B IU  1  ( u 1  + 2 α OU  v 1 ) µ  (17j) k 42  =  α OU  v 1 ( u 1  + α OU  v 1 ) µ  (17k)Using the search via simple bisection, the water-filling-level 1 µ  achieves the optimum.IV. S IMULATIONS In this section, we evaluate our proposed optimizationalgorithm. For simulations, we consider two users, an outdooruser located at position  90m , 160 ○ expresses in polar coor-dinates, and an indoor user located at position  160m , 160 ○ .The IAP is located at coordinates  165m , 160 ○ . The maximumpower of micro site is considered to be 5 W, and maximumpower of IAP is set to 0.1 mW. For simulations, 110 channelrealizations are performed. The simulation parameters arepresented in Table I.Considering Equations (14) and (15) for searching theoptimal water-filling-level  1 µ  via simple bisection searching,we search the optimal power allocation for both outdoorand indoor user. Fig.2 shows the energy efficiency measuredin [Mbit/J] for the outdoor user served by the micro site.The energy efficiency results are expressed as a function of assigned power to user  m  for a fixed and optimal power                      012345012345Outside Base Station Power [W]    E  n  e  r  g  y   E   f   f   i  c   i  e  n  c  y   [   M   b   i   t   /   J   ] 012305001000150020002500  EE for optimal power allocationEE for fixed power allocation x 10 3 Figure 2 . Energy efficiency for the outdoor user   00.020.040.060.080.10123456x 10 6 Inside Access Point Power [W]    E  n  e  r  g  y   E   f   f   i  c   i  e  n  c  y   [   M   b   i   t   /   J   ] 00.010.02051015x 10 5  EE for optimal power allocationEE for fixed power allocation Figure 3 . Energy efficiency for the indoor userallocation. The results indicate an increase in energy efficiencywhen the optimal power allocation algorithm is performed,while for a fixed power allocation scheme, the gain is smaller.The energy efficiency for the indoor user served by anIAP is provided in Fig.3. Results indicate a gain in terms of energy efficiency for optimal power allocation in comparisonto a fixed power allocation scheme. For the users served byIAPs, a gain of 7.5% in terms of energy efficiency is achieved,compared with static power allocation method. Furthermore,comparing Fig.2 and Fig.3, it is observed that the energyefficiency of the indoor user is several times greater than theenergy efficiency of the outdoor user.V. C ONCLUSION This work presented an optimal power allocation scheme interms of energy efficiency, considering macro sites and IAPs.We first classified the users to be either indoor or outdoorbased on indoor path loss threshold criteria, then formulatedthe optimization problem to increase the energy efficiencyconsidering the power allocation and bit rate as constraints.It is known that these problems are almost non-convex andhence, it is difficult to derive the solution. Assuming that anequal power is allocated to the outdoor and indoor users, thenon-convex optimization problem for maximizing the energyefficiency is transformed into a convex optimization problem.By using techniques of dual decomposition, we derived theoptimal power assignment for the indoor and the outdoor usersfor a fixed bandwidth. The optimal assigned power increasesthe energy efficiency and the bit rate simultaneously. Thesimulation results show that by using the algorithm for optimalpower allocation we achieve a gain of 76% in terms of energyefficiency when the users are served by micro sites. For theusers served by IAPs, a gain of 7.5% in terms of energyefficiency is achieved, compared with a static power allocationmethod.A CKNOWLEDGEMENTS This work has been funded by the Christian Doppler Lab-oratory for Wireless Technologies for Sustainable Mobility,the A1 Telekom Austria AG, and the KATHREIN-Werke KG.The financial support by the Federal Ministry of Economy,Family and Youth and the National Foundation for Research,Technology and Development is gratefully acknowledged.R EFERENCES [1] UMTS Forum Report 44, “Mobile traffic forecasts 2010-2020report.” UMTS Forum, Jan 2011.[2] J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano,A. C. K. Soong, and J. C. Zhang, “What will 5g be?”  IEEE  Journal on Selected Areas in Communications , vol. 32, no. 6,pp. 1065–1082, June 2014.[3] G. Y. Li, Z. Xu, C. Xiong, C. Yang, S. Zhang, Y. Chen, andS. Xu, “Energy-efficient wireless communications: tutorial, sur-vey, and open issues,”  IEEE Wireless Communications , vol. 18,no. 6, pp. 28–35, December 2011.[4] B. Krasniqi, M. Wolkerstorfer, C. Mehlf ¨uhrer, and C. F. Meck-lenbr¨auker, “Sum-rate maximization for multiple users in partialfrequency reuse cellular networks,” in  2010 IEEE GlobecomWorkshops , Dec 2010, pp. 814–818.[5] B. Krasniqi, “Partial frequency reuse for long term evolution,”Ph.D. dissertation, E389, Vienna University of Technology,2011.[6] T. S. Rappaport, “5G Channel Measurements and Models forMillimeter-Wave Wireless Communications,” NYU PolytechnicSchool of Engineering, Brooklyn, New York. North American5G Workshop, November 2014.[7]  Physical Layer Aspects for Evolved Universal Terrestial Radio Access (UTRA) (release 7) , 3GPP Technical Report TR 25.814v7.1.0, September 2006.[8]  Further Advancements for Evolved Universal Terrestial Radio Access (E-UTRA) (release 9) , 3GPP Technical Report TR36.814 v0.4.1, February 2009.[9] ITU-R,  Propagation data and prediction methods for the plan-ning of indoor radiocommunication systems and radio localarea networks in the frequency range 900 MHz to 100 GHz ,ITU, Electronic Publication Geneva, 2012.[10] S. Boyd and L. Vandenberghe, “Convex optimization.” Cam-bridge University Press, 2004.[11] L. X. S. Boyd and A. Mutapcic, “Notes on decompositionmethods-Lecture Notes for Course EE392o.” Stanford Uni-versity, October 2003.[12] H. Turnbull, “Theory of Equations.” fourth ed., Oliver andBoyd, London, 1947.                    
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