massive MIMO networks using stochastic geometry · Potential for better area spectral efficiency...
Transcript of massive MIMO networks using stochastic geometry · Potential for better area spectral efficiency...
© 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
© Robert W. Heath Jr. (2015)
2
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
© Robert W. Heath Jr. (2015)
More spectrum Millimeter wave spectrum
More base stations Network densification
More spectrum efficiency Multiple antennas (MIMO)
3
5G cellular networks – achieving 1000x better
This talk
Other work
© Robert W. Heath Jr. (2015)
7
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
© Robert W. Heath Jr. (2015)
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**
7
Three-stage TDD mode (1): uplink training
Users:
Send pilots to the base stations
* T. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,” IEEE Trans. Wireless Commun., Nov. 2011
Base stations:
Estimate channels based on
training
Uplink training
Pilot contamination
Channel estimation polluted by pilot contamination
Assume perfect synchronization Assume full pilot reuse
© Robert W. Heath Jr. (2015)
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**
7
Three-stage TDD mode (2): uplink data
Users:
Send data to base stations
* T. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,” IEEE Trans. Wireless Commun., Nov. 2011
Base stations:
Matched filtering combining
based on
channel estimates
Uplink data
Simple matched filter receive combining based on channel estimate
© Robert W. Heath Jr. (2015)
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**
7
Three-stage TDD mode (3): downlink data
Users:
Decode received signals
* T. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,” IEEE Trans. Wireless Commun., Nov. 2011
Base stations:
Beamforming based on channel
estimates
Downlink data
Simple matched filter transmit beamforming based on channel
estimates
© Robert W. Heath Jr. (2015)
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
8
Out-of-cell interference reduced due to asymptotic orthogonality of channels [Show SIR convergence]
© Robert W. Heath Jr. (2015)
Modeling cellular system performance
using stochastic geometry
© Robert W. Heath Jr. (2015)
Stochastic geometry in cellular systems
10
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…
© Robert W. Heath Jr. (2015)
11
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
© Robert W. Heath Jr. (2015)
Challenges of analyzing massive MIMO
13
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
© Robert W. Heath Jr. (2015)
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
14
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.
© Robert W. Heath Jr. (2015)
Massive MIMO system model
© Robert W. Heath Jr. (2015)
16
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)
© Robert W. Heath Jr. (2015)
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
© Robert W. Heath Jr. (2015)
18
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
© Robert W. Heath Jr. (2015)
19
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
© Robert W. Heath Jr. (2015)
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
© Robert W. Heath Jr. (2015)
21
SIR in uplink transmission
BSs perform maximum ratio combining based on channel estimates
Disappears from expression As M grows large
© Robert W. Heath Jr. (2015)
22
SIR in downlink transmission
BSs perform match-filtering beamforming based on channel estimates
Match-filtering precoder:
Disappears from expression As M grows large
© Robert W. Heath Jr. (2015)
Asymptotic performance analysis
when # of BS antennas goes to infinity
© Robert W. Heath Jr. (2015)
24
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
* T. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,” IEEE Trans. Wireless Commun., Nov. 2011
What about spatial correlation and infinite number of BSs??
© Robert W. Heath Jr. (2015)
25
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
© Robert W. Heath Jr. (2015)
26
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
© Robert W. Heath Jr. (2015)
27
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
© Robert W. Heath Jr. (2015)
28
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
© Robert W. Heath Jr. (2015)
29
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
© Robert W. Heath Jr. (2015)
30
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
© Robert W. Heath Jr. (2015)
31
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
© Robert W. Heath Jr. (2015)
32
Comparing UL and DL distribution
Indicate decoupled system design for DL and UL
Much different SIR distribution
observed in DL and UL
© Robert W. Heath Jr. (2015)
Uplink analysis with finite antennas:
How should antennas scales with users?
© Robert W. Heath Jr. (2015)
34
Exact uplink SIR difficult to analyze
Terms in exact SIR expression coupled
due to pilot contamination
Need decoupling approximation to simplify SIR analysis
© Robert W. Heath Jr. (2015)
35
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
© Robert W. Heath Jr. (2015)
36
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
© Robert W. Heath Jr. (2015)
37
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
© Robert W. Heath Jr. (2015)
38
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
© Robert W. Heath Jr. (2015)
Rate comparison w/ small cell
39
© Robert W. Heath Jr. (2015)
40
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
© Robert W. Heath Jr. (2015)
41
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
© Robert W. Heath Jr. (2015)
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
© Robert W. Heath Jr. (2015)
Questions
44