Analysis of massive MIMO networks using stochastic geometry Tianyang Bai and Robert W. Heath Jr. Wireless Networking and Communications Group Department of Electrical and Computer Engineering The University of Texas at Austin http://www.profheath.org Funded by the NSF under Grant No. NSF-CCF-1218338 and a gift from Huawei
Cellular communication Base station Uplink Downlink User To network Distributions of base stations in a major UK city* (1 mile by 0.5 mile area) Illustration of a cell in cellular networks Irregular base station locations motivate the applications of stochastic geometry * Data taken from sitefinder.ofcom.org.uk 2
5G cellular networks achieving 1000x better Other work Robert W. Heath Jr. (2015) More spectrum More base stations More spectrum efficiency Millimeter wave spectrum Network densification Multiple antennas (MIMO) This talk 3
Massive MIMO concept Conventional cell 1 to 8 antennas MIMO (multiple-input multiple-output) a type of wireless system with multiple antennas at transmitter and receiver Massive MIMO cell > 64 antennas 1 or 2 uses sharing same resources 10 to 30 users sharing same resources Potential for better area spectral efficiency with massive MIMO * T. Marzetta, Noncooperative cellular wireless with unlimited numbers of base station antennas, IEEE Trans. Wireless Commun., Nov. 2011 **X. Gao, O. Edfors, F. Rusek, and F. Tufvesson, Massive MIMO in real propagation environments, To appear in IEEE Trans. Wireless Commun., 2015 7
Three-stage TDD mode (1): uplink training Assume full pilot reuse Assume perfect synchronizat Pilot contamination Uplink training Users: Send pilots to the base stations Base stations: Estimate channels based on training Channel estimation polluted by pilot contamination * T. Marzetta, Noncooperative cellular wireless with unlimited numbers of base station antennas, IEEE Trans. Wireless Commun., Nov. 2011 7
Three-stage TDD mode (2): uplink data Users: Send data to base stations Uplink data Base stations: Matched filtering combining based on channel estimates Simple matched filter receive combining based on channel estimate * T. Marzetta, Noncooperative cellular wireless with unlimited numbers of base station antennas, IEEE Trans. Wireless Commun., Nov. 2011 7
Three-stage TDD mode (3): downlink data Users: Decode received signals Downlink data Base stations: Beamforming based on channel estimates Simple matched filter transmit beamforming based on channel estimates * T. Marzetta, Noncooperative cellular wireless with unlimited numbers of base station antennas, IEEE Trans. Wireless Commun., Nov. 2011 7
Advantages of massive MIMO & implications Robert W. Heath Jr. (2015) TDD (time-division multiplexing) avoids downlink training overhead [Include pilot contamination] Fading and noise become minor with large arrays [Ignore noise in analysis] Out-of-cell interference reduced due to asymptotic orthogonality of channels [Show SIR convergence] Large antenna arrays serve more users to increase cell throughput [Compare sum rate w/ small cells] Simple signal processing becomes near-optimal, with large arrays [Assume simple beamforming] 8
Modeling cellular system performance using stochastic geometry Robert W. Heath Jr. (2015)
Stochastic geometry in cellular systems Serving BS Desired signal power Desired signal Interference link Typical user Interference from PPP interferers Thermal noise (often ignored) Modeling base stations locations as Poisson point process Stochastic geometry allows for simple characterizations of SINR distributions T. X Brown, ``Practical Cellular Performance Bounds via Shotgun Cellular System,'' IEEE JSAC, Nov. 2000. M. Haenggi, J. G. Andrews, F. Baccelli, O. Dousse, and M. Franceschetti, Stochastic geometry and random graph for the analysis and design of wireless networks, IEEEJSAC 09 J. G. Andrews, F. Baccelli, and R. K. Ganti, A tractable approach to coverage and rate in cellular networks, IEEE TCOM 2011. H. S. Dhillon, R. K. Ganti, F. Baccelli, and J. G. Andrews, Modeling and analysis of K-tier downlink heterogeneous cellular networks, IEEE JSAC, 2012 & many more 10
Who* cares about antennas anyway? Diversity Changes fading distribution Interference cancelation Changes received interference Multiplexing Multivariate performance measures Beamforming Changes caused interference * why should non-engineers care at all about antennas 11
Challenges of analyzing massive MIMO Does not directly extend to massive MIMO X Single user per cell Single base station antenna Rayleigh fading No channel estimation Mainly focus on downlink Multiple user per cell Massive base station antennas Correlated fading MIMO channel Pilot contamination Analyze both uplink and downlink 13
Related work on massive MIMO w/ SG Asymptotic analysis using stochastic geometry [1] Derived distribution for asymptotic SIR with infinite BS antennas Considered IID fading channel, not include correlations Assumed BSs distributed as PPP marked with fixed-circles as cells Nearby cells in the model may heavily overlap (not allowed in reality) Concluded same SIR distributions in UL/DL (not matched to simulations) Scaling law between user and BS antennas [2] BS antennas linearly scale with users to maintain mean interference The distribution of SIR is a more relevant performance metric Need advanced system model for massive MIMO analysis [1] P. Madhusudhanan, X. Li, Y. Liu, and T. Brown, Stochastic geometric modeling and interference analysis for massive MIMO systems, Proc.of WiOpt, 2013 [2] N. Liang, W. Zhang, and C. Shen, An uplink interference analysis for massive MIMO systems with MRC and ZF receivers, Proc. of WCNC, 2015. 14
Massive MIMO system model Robert W. Heath Jr. (2015)
Proposed system model Each BS has M antennas serving K users : n-th base station : k-th scheduled user in n-th cell Scheduled user Unscheduled user Base stations distributed as a PPP Users uniformly distributed w/ high density (each BS has at least K associated users) Need to characterize scheduled users distributions 16
of UL SIR distribution Robert W. Heath Jr. (2015) Scheduled users distribution Base station 1st scheduled user 2nd scheduled user Presence of a red user in one cell prevents those of the other red Locations of scheduled users are correlated and do not form a PPP Non-PPP users distributions make exact analysis difficult [1] H. El Sawy and E. Hossain, On stochastic geometry modeling of cellular uplink transmission with truncated channel inversion power control IEEE TCOM, 2014 [2] S. Singh, X. Zhang, and J. Andrews, Joint rate and SINR coverage analysis for decoupled uplink-downlink biased cell association in HetNet, Arxiv, 2014 17
Approximating the scheduled process Distance to associated user is a Rayleigh random variable Base station 1st scheduled user 2nd scheduled user Other-cell scheduled user Exclusion ball with fixed radius r Other-cell scheduled users as PPP outside exclusion ball Use hardcore model for scheduled users locations [1] H. El Sawy and E. Hossain, On stochastic geometry modeling of cellular uplink transmission with truncated channel inversion power control IEEE TCOM, 2014 [2] S. Singh, X. Zhang, and J. Andrews, Joint rate and SINR coverage analysis for decoupled uplink-downlink biased cell association in HetNet, Arxiv, 2014 18
Channel model Channel vector from IID Gaussian vector for fading Bounded path loss model Covariance matrix for correlated fading Path loss of a link with length R mean square of eigenvalues uniformly bounded Address near-field effects in path loss Reasonable for rich scattering channel 19
Uplink channel estimation Assume perfect time synchronization & full pilot reuse in the network Channel estimate of -th BS to its k-th user Error from pilot contamination Need to incorporate pilot contamination in system analysis
SIR in uplink transmission BSs perform maximum ratio combining based on channel estimates As M grows large Disappears from expression 21
SIR in downlink transmission BSs perform match-filtering beamforming based on channel estimates Match-filtering precoder: As M grows large Disappears from expression 22
Asymptotic performance analysis when # of BS antennas goes to infinity Robert W. Heath Jr. (2015)
Toy example with IID fading & finite BSs UL received signal desired signal pilot contamination interference By LLN for IID variables swap limit and sum in finite sum What about spatial correlation and infinite number of BSs?? * T. Marzetta, Noncooperative cellular wireless with unlimited numbers of base station antennas, IEEE Trans. Wireless Commun., Nov. 2011 24
Dealing with correlations in fading Lemma 1: [LLN for non-iid fading ] If the eigenvalues of the fading covariance matrices satisfies then for any two different channels, the asymptotic orthogonality of channel vectors still holds as and for any channel vector, Stronger convergence may holds, but convergence in probability is sufficient for our purposes Law of large numbers holds for certain non IID cases Asymptotic orthogonality holds with certain correlations in fading 25
Dealing with infinite interferers. Fixed ball with radius R0 goes to 0 as in the finite BS example # of nodes in the ball is almost surely finite BS X0 Vanishes by choosing sufficiently large R0 as shown in next page Infinite sum outside ball contributes little to total sum Separate infinite sum into a dominating finite sum and an arbitrarily small infinite sum 26
Dealing with infinite sum outside the ball Show the infinite sum can be made arbitrarily small as Use Markov s inequality Since norm and path loss are positive Use Campbell s formula to compute the mean Hardcore user model: user density Choose R0 as Can be made smaller than any positive constant 27
Asymptotic SIR results in uplink Robert W. Heath Jr. (2015) Theorem 1 [UL SIR convergence] As the number of base station goes to infinity, the uplink SIR converges in probability as Same form as in finite BS case Small-scale fading vanishes Path loss exponent doubles in massive MIMO 28
Asymptotic uplink SIR plots Asymptotic better than SISO Require >10,000 antennas to approach asymptotic curves Convergence to asymptotic SIR (IID fading, K=10, α=4) 29
Asymptotic UL distributions Corollary 1.1 [Distribution of UL asymptotic SIR] Decrease with path loss exponent in low SIR regime Increase with path loss exponent in high SIR regime Analysis match simulations well above 0 db Asymptotic SIR CCDF 30
Asymptotic SIR results in downlink Robert W. Heath Jr. (2015) Theorem 2 [DL SIR convergence] As the number of base station goes to infinity, the downlink SIR converges in probability as where a dual form of UL SIR except for the normalization constan Normalization constant for equal TX power in DL to all users Corollary 2.1 [Distribution of DL asymptotic SIR] Increase with α when T> 0 db Proof of convergence in DL similar to UL 31
Comparing UL and DL distribution Much different SIR distribution observed in DL and UL Indicate decoupled system design for DL and UL 32
Uplink analysis with finite antennas: How should antennas scales with users?
Exact uplink SIR difficult to analyze Terms in exact SIR expression coupled due to pilot contamination Need decoupling approximation to simplify SIR analysis 34
Approximation for uplink SIR Take average over small-scale fading Dropping certain terms not scale with M but causing coupling Approximate the underlying points as an independent PPP Approximate SIR expression easier to analyze using SG 35
Uplink SIR distribution with finite antennas Robert W. Heath Jr. (2015) Theorem 3 [UL non-asymptotic SIR CCDF ] Given the number of base station antennas M, and the number of simultaneously served user K, the CCDF of uplink SIR can be computed as where N is the number of terms used, Moreover, the scaling constant is defined as, where is the average square of eigenvalues of the covariance matrices for fading. Expression is found to be accurate with N>5 terms μ defines the scaling law between K and M 36
Scaling law to maintain uplink SIR Corollary 3.1 [Scaling law for UL SIR] To maintain the uplink SIR distribution Robert W. Heath Jr. (2015) γ is the average square of eigenvalues of the covariance matrices for fading Exponent in the scaling determined by path loss exponent α Need 2γ 2 more antennas than IID fading to maintain SIR distribution Superlinear scaling law when α>2 Correlations in fading reduce SIR coverage 37
Verification of proposed scaling law Superlinear scaling law b/w M and K K: Scheduled User per cell M: # of BS antennas : correlation coefficient of fading Correlations reduce SIR coverage Simulations using 19 cell hexagonal model (α = 4) Scaling law from PPP model applies to hexagonal model 38
Rate comparison w/ small cell 39
Rate comparison setup Massive MIMO Small cell # served user/ cell Varies 1 # BS antenna 8x8 2 # BS antenna 1 2 1. Small cell serves its user by 2x2 spatial multiplexing or SISO 2. Assume perfect channel knowledge for small cell case 3. Compute training overhead of massive MIMO as in [1] 4. Assume user 60x macro massive MIMO BSs 5. UL/DL each takes 50% time/ bandwidth Compare throughput per unit area b/w massive MIMO and small cell [1] T. Marzetta, Noncooperative cellular wireless with unlimited numbers of base station antennas, IEEE Trans. Wireless Commun., Nov. 2011 40
# number of user/ cell for massive Robert W. Heath Jr. (2015) Rate comparison results 8 db Gain for massive MIMO Higher area throughput in massive MIMO by serving multiple users 0 db Ratio of small cell density to massive MIMO Small cell using SISO 8 db Gain for small cell Higher area throughput in small cell due to higher BS density Small cell using 2x2 SM Massive MIMO achieves comparable area throughput w/ sparser BS deployment 41
Conclusions Robert W. Heath Jr. (2015)
Concluding remarks SIR converges to asymptotic equivalence in PPP networks DL and UL asymptotic SIR distributions are different Asymptotic SIR coverage superior to SISO # of antennas scales superlinearly with # of users to keep SIR Correlation reduces SIR coverage Path loss exponent determines the scaling exponent Future directions Application to millimeter wave massive MIMO More sophisticated forms of beamforming, including BS coordination 43
Questions 44