massive MIMO networks using stochastic geometry · PDF fileAnalysis of massive MIMO networks using stochastic geometry ... (multiple-input multiple-output) ... Massive MIMO:

© 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


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Cellular communication

Distributions of base stations in a major UK city*

(1 mile by 0.5 mile area)

* Data taken from sitefinder.ofcom.org.uk

Base station

Illustration of a cell in cellular networks

User

Uplink Downlink

To network

Irregular base station locations motivate the applications of stochastic geometry


More spectrum Millimeter wave spectrum

More base stations Network densification

More spectrum efficiency Multiple antennas (MIMO)

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5G cellular networks – achieving 1000x better

This talk

Other work


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Massive MIMO concept

* 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

Potential for better area spectral efficiency with massive MIMO

> 64 antennas 1 to 8 antennas

1 or 2 uses sharing same resources 10 to 30 users sharing same resources

Conventional cell Massive MIMO cell

MIMO (multiple-input multiple-output) a type of wireless

system with multiple antennas at transmitter and receiver


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**

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Three-stage TDD mode (1): uplink training

Users:

Send pilots to the base stations


Base stations:

Estimate channels based on

training

Uplink training

Pilot contamination

Channel estimation polluted by pilot contamination

Assume perfect synchronization Assume full pilot reuse






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Three-stage TDD mode (2): uplink data

Users:

Send data to base stations


Base stations:

Matched filtering combining

based on

channel estimates

Uplink data

Simple matched filter receive combining based on channel estimate






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Three-stage TDD mode (3): downlink data

Users:

Decode received signals


Base stations:

Beamforming based on channel

estimates

Downlink data

Simple matched filter transmit beamforming based on channel

estimates


Fading and noise become minor with large arrays [Ignore noise in analysis]

TDD (time-division multiplexing) avoids downlink training overhead [Include pilot contamination]

Simple signal processing becomes near-optimal, with large arrays [Assume simple beamforming]

Large antenna arrays serve more users to increase cell throughput [Compare sum rate w/ small cells]

Advantages of massive MIMO & implications

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Out-of-cell interference reduced due to asymptotic orthogonality of channels [Show SIR convergence]


Modeling cellular system performance

using stochastic geometry


Stochastic geometry in cellular systems

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Desired signal

Serving BS

Typical user Interference link

Stochastic geometry allows for simple characterizations of SINR distributions

Desired signal power

Interference from PPP interferers

Modeling base stations locations as Poisson point process

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

Thermal noise

(often ignored)

& many more…


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Who* cares about antennas anyway?

Diversity

Changes fading distribution

Multiplexing

Multivariate performance measures

Interference cancelation

Changes received interference

Beamforming

Changes caused interference

* why should non-engineers care at all about antennas


Challenges of analyzing massive MIMO

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Does not directly extend to massive MIMO

X

Single user per cell Multiple user per cell

Single base station antenna Massive base station antennas

Rayleigh fading Correlated fading MIMO channel

No channel estimation Pilot contamination

Mainly focus on downlink Analyze both uplink and downlink


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

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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.


Massive MIMO system model


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Proposed system model

Each BS has M antennas serving K users

Base stations distributed as a PPP

: n-th base station

: k-th scheduled user in n-th cell

Scheduled user

Unscheduled user

Need to characterize scheduled users’ distributions

Users uniformly distributed w/ high density

(each BS has at least K associated users)


Scheduled users’ distribution

Locations of scheduled users are

correlated and do not form a PPP [1,2]

Correlations prevent the exact analysis

of UL SIR distributions

17 [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

1st scheduled user

2nd scheduled user

Base station

Locations of scheduled users are

correlated and do not form a PPP

Non-PPP users’ distributions make exact analysis difficult

Presence of a “red” user in one cell

prevents those of the other red


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Approximating the scheduled process

[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

Distance to associated user

is a Rayleigh random variable

Exclusion ball with fixed radius r

Other-cell scheduled users

as PPP outside exclusion ball

1st scheduled user

2nd scheduled user

Base station

Other-cell scheduled user

Use hardcore model for scheduled users’ locations


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Channel model

Channel vector from

Bounded path loss model

IID Gaussian vector for fading

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


Channel estimate of -th BS to its k-th user

Assume perfect time synchronization & full pilot reuse in the network

Uplink channel estimation

Error from pilot contamination

Need to incorporate pilot contamination in system analysis


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SIR in uplink transmission

BSs perform maximum ratio combining based on channel estimates

Disappears from expression As M grows large


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SIR in downlink transmission

BSs perform match-filtering beamforming based on channel estimates

Match-filtering precoder:

Disappears from expression As M grows large


Asymptotic performance analysis

when # of BS antennas goes to infinity


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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??


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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,

Dealing with correlations in fading

Asymptotic orthogonality holds with certain correlations in fading

Stronger convergence may holds,

but convergence in probability is sufficient for our purposes

Law of large numbers holds for certain non IID cases


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Dealing with infinite interferers

BS X0

Fixed ball with radius R0

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

# of nodes in the ball

is almost surely finite

.

Infinite sum outside ball contributes

little to total sum

goes to 0 as in the finite BS example Vanishes by choosing sufficiently large

R0

as shown in next page

Separate infinite sum into a dominating finite sum

and an arbitrarily small infinite sum


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Show the infinite sum can be made arbitrarily small as

Dealing with infinite sum outside the ball

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


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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

Small-scale fading vanishes Path loss exponent doubles in

massive MIMO

Same form as in finite BS case


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Asymptotic uplink SIR plots

Convergence to asymptotic SIR

(IID fading, K=10, α=4)

Require >10,000 antennas to

approach asymptotic curves

Asymptotic better than SISO


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Asymptotic UL distributions

Corollary 1.1 [Distribution of UL asymptotic SIR]

Asymptotic SIR CCDF

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


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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

Corollary 2.1 [Distribution of DL asymptotic SIR]

a dual form of UL SIR

except for the normalization constant

Proof of convergence in DL similar to UL

Normalization constant for equal TX power in DL to all users

Increase with α when T> 0 dB


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Comparing UL and DL distribution

Indicate decoupled system design for DL and UL

Much different SIR distribution

observed in DL and UL


Uplink analysis with finite antennas:

How should antennas scales with users?


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Exact uplink SIR difficult to analyze

Terms in exact SIR expression coupled

due to pilot contamination

Need decoupling approximation to simplify SIR analysis


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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


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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


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Scaling law to maintain uplink SIR

Corollary 3.1 [Scaling law for UL SIR] To maintain the uplink SIR

distribution

Exponent in the scaling determined by

path loss exponent α

Superlinear scaling law when α>2

Need 2𝛾 − 2 more antennas than IID fading

to maintain SIR distribution

Correlations in fading reduce SIR coverage

𝛾 is the average square of eigenvalues of

the covariance matrices for fading


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Verification of proposed scaling law

K: Scheduled User per cell M: # of BS antennas : correlation coefficient of fading

Superlinear scaling law b/w M and K

Simulations using 19 cell hexagonal model

(𝛼 = 4)

Correlations reduce SIR coverage

Scaling law from PPP model applies to hexagonal model


Rate comparison w/ small cell

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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


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Rate comparison results 8 dB

Gain for massive MIMO

8 dB

Gain for small cell

0 dB

# n

um

be

r o

f u

se

r/ c

ell

for

ma

ssiv

e

Higher area throughput

in small cell due to

higher BS density

Higher area throughput in

massive MIMO by serving

multiple users

Massive MIMO achieves comparable area throughput w/ sparser BS deployment

Small cell using SISO Small cell using 2x2 SM Ratio of small cell density to massive MIMO


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

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Questions

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Documents

massive MIMO networks using stochastic geometry · PDF fileAnalysis of massive MIMO networks using stochastic geometry ... (multiple-input multiple-output) ... Massive MIMO: