massive Analysis of MIMO networks using stochastic geometry

© Robert W. Heath Jr. (2015)

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

User

Base station

Uplink

Downlink

To network

Illustration of a cell in cellular networks

Distributions of base stations in a major UK city*

(1 mile by 0.5 mile area)

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

More spectrum Millimeter wave spectrum

More base stations Network densification

More spectrum efficiency 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



Massive MIMO: multi-user MIMO with lots of base station antennas*

 Allows more users per cell simultaneously served

 Analyses show large gains in sum cell rate using massive MIMO

 Real measurements w/ prototyping confirm theory**


Three-stage TDD mode (1): uplink training

Assume perfect synchronization

Assume full pilot reuse

Pilot contamination

Users:

Send pilots to the base stations

Uplink training Base stations:

Estimate channels based on training

Channel estimation polluted by pilot contamination


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


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


7

Advantages of massive MIMO & implications


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


Stochastic geometry in cellular systems

Desired signal power

Serving BS

Desired signal

Typical user

Interference link

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.

& many more…

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

10


Who* cares about antennas anyway?

Diversity

Changes fading distribution

Interference cancelation

Changes received interference

Multiplexing

Multivariate performance measures

* why should non-engineers care at all about antennas

Beamforming

Changes caused interference

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


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

 Locations of scheduled users are correlated and do not form a PPP

[1,2]

 Correlations prevent the exact analysis of UL SIR distributions


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

NonPPP 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

Bounded path loss model

Path loss of a link with length R


IID Gaussian vector for fading

Covariance matrix for correlated fading 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

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??


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

goes to 0 as in the finite BS example

Fixed ball with radius R

0

# of nodes in the ball is almost surely finite

BS X

0

Vanishes by choosing sufficiently large

R

0 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

 The interference field can be divided into two parts

 For any ball with fixed radius, there is a.s. finite nodes inside the ball

 Infinite users outside the ball contribute “little” to the sum interference

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 R

0

as

Can be made smaller than any positive constant

27


Asymptotic SIR results in uplink

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

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 constant

Normalization constant for equal TX power in DL to all users

Corollary 2.1 [Distribution of DL asymptotic SIR]

Proof of convergence in DL similar to UL

Increase with

α when T> 0 dB

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

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 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

# served user/ cell

# BS antenna

# BS antenna

Massive MIMO

Varies

8x8

1

Small cell

1

2

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


Rate comparison results

8 dB

Gain for massive MIMO

Higher area throughput in massive MIMO by serving multiple users

0 dB

Higher area throughput in small cell due to higher BS density

Ratio of small cell density to massive MIMO

Small cell using SISO

8 dB

Gain for small cell Small cell using 2x2 SM

Massive MIMO achieves comparable area throughput w/ sparser BS deployment

41

Conclusions



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

massive Analysis of MIMO networks using stochastic geometry

Analysis of

massive

MIMO networks using stochastic geometry

Modeling cellular system performance using stochastic geometry

Massive MIMO system model

Asymptotic performance analysis when # of BS antennas goes to infinity

Uplink analysis with finite antennas:

How should antennas scales with users?

Rate comparison w/ small cell

Conclusions

Questions

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massive Analysis of MIMO networks using stochastic geometry

Analysis of

massive

MIMO networks using stochastic geometry

Modeling cellular system performance using stochastic geometry

Massive MIMO system model

Asymptotic performance analysis when # of BS antennas goes to infinity

Uplink analysis with finite antennas:

How should antennas scales with users?

Rate comparison w/ small cell

Conclusions

Questions

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