Optimization of Cooperative WirelessMulticell Networks
Wei Yu
Electrical and Computer Engineering DepartmentUniversity of Toronto
MIIS 2012
Wei Yu July 2012 1 / 43
Wireless Multicell Network
• Inter-cell interference is the fundamental limiting factor
• Coordination at the signal level: Network MIMO
• Base-stations form an giant antenna array• Joint signal processing
• Coordination of signalling strategies:
• Coordinating power spectrum
• Coordinating beamforming
• Coordinating scheduling
Control Info
• Network optimization plays a key role.
Wei Yu July 2012 2 / 43
Outline
• Coordinated beamforming, power allocation and scheduling
• Nonconvex because of interference• Mixed discrete and continue due to scheduling• Many local optima because of the structure of beamforming
• Network MIMO with limited backhaul capacity
• Heuristic algorithms, but with realistic system simulations
• Goal of this talk:
• Design insights.• Performance projection.
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Part I: Coordinated Beamforming, Scheduling, Power Allocation
Wei Yu July 2012 4 / 43
Multicell Coordination at Signaling Strategy Level
• Consider a cooperative system at signaling-strategy level.
• System setup:
• Multi-cell, sectorized, MIMO-OFDM
• Multiple antennas at both BS/MS
• Users occupy orthogonal dimensions
• Optimization objective:
• Network utility maximization
• Optimization variables:• Scheduling: Which user in each dimension: k = f (l , s, b, n).• Beamforming: What are the transmit/receive BF: (unlsb, v
nlsk).
• Power control: What are the power levels for each beam: Pnlsb.
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Mathematical Formulation
• For lth cell, sth sector, bth beam, kth user, nth frequency
SINRnlsbk =
Pnlsb|(unlsk)THn
ls,lskvnlsb|2
Γ(σ2 +∑
(j ,t,c) 6=(l ,s,b) Pnjtc |(unlsk)THn
jt,lskvnjtc |2)
.
• Optimization problem:
max∑l ,s,k
log(R̄lsk
)s.t. Rlsk =
∑(b,n):k=f (l ,s,b,n)
log (1 + SINRnlsbk)
• This is a challenging (non-convex) problem
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Divide and Conquer
• Three key steps, thus can iterate among them:• Power spectrum optimization• Coordinated beamforming• Proportionally fair scheduling
• Fixing two, the other step is a well-formulated problem.
Scheduling Beamforming Power Allocation
• Heuristic
• How to best do each step?• How well does it work?
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Power Spectrum Optimization
• Fixing scheduling and beamformers, the problem becomes:
max∑lsb
wlsb
∑n
log
(1 +
Pnlsb|hnlsb,lsk |2
Γ(σ2 +∑
(j ,t,c)6=(l ,s,b) Pnjtc |hnjtc,lsk |2)
)
• Many different approaches proposed in the literature:• Game-theory based approach (Ji-Huang ’95 and many others)• Geometric programming (Chiang ’02, ’07)• SCALE algorithm (Papandriopoulos-Evans ’06)• Pricing based approach (Huang-Berry-Honig ’06, Yu ’07)• Load-spillage (Hande-Rangan-Chiang ’08)• Binary power control (Gjendemsjo-Gesbert-Oien-Kiani ’08)• MAPEL/Polyblock optimization (Qian-Zhang-Huang ’09, ’10)• Iterative function evaluation (Dahrouj-Yu ’10)
• No known efficient way to circumvent nonconvexity:
• Fundamentally a difficult problem (Luo-Zhang ’08).
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Pricing-Based Approach
• Each user sets its power to make the derivative of theobjective zero, assuming that all other users powers are fixed.
• Leads to a water-filling-like power update:
Pnlsb =
wlsk∑j 6=l
tnjtc,lsb + λlsb−
Γ(σ2 +∑
(jtc)6=(lsb)
Pnjtc |hnjtc,lsk |2)
|hnlsb,lsk |2
Smax
0
• where tnjtc,lsb is the interference price
tnjtc,lsb = wjtk ′∂rnjtk ′
∂Pnlsb
= wjtk ′Γ|hnlsb,jtk ′ |2
Pnjtc |hnjtc,jtk ′ |2
(SINRnjtc)2
1 + SINRnjtc
Wei Yu July 2012 9 / 43
Coordinated Beamforming
• Fundamental tool: Uplink-downlink duality
Uplink-Downlink Duality
X
X ′1
X ′2
Y1
Y2
Y ′
Z1
Z2
Z
P P
H1
H2
HT1
HT2
Theorem 1. Under the same power constraint, the Gaussian vectormultiple-access channel and the Gaussian vector broadcast channel havethe same capacity.
Wei Yu, University of Toronto 19
• Single-cell: (Schubert-Boche ’04, Bengtsson-Ottersten ’02,Visotsky-Madhow ’99, Wiesel et al ’06, Song et al ’07)
• Multi-cell: (Rashid-Farrokhi et al ’98, Dahrouj-Yu, ’10)
• Use MMSE receive BF of the dual channel for transmit BF.
• Only works for power minimization for fixed SINR target.• Iterate between rate maximization and power minimization.
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Uplink-Downlink Duality
• SINR Duality is Equivalent to Lagrangian Duality
minωi
ωH1 ω1 + ωH
2 ω2 minλi
λ1σ2 + λ2σ
2
s.t.1
γ1(hH1 ω1)2 ≥ (hH1 ω2)2 + σ2 s.t.
λ1
γ1(ωH
1 h1)2 ≥ λ2(ωH1 h2)2 + ωH
1 ω1
1
γ2(hH2 ω2)2 ≥ (hH2 ω1)2 + σ2 λ2
γ2(ωH
2 h2)2 ≥ λ1(ωH2 h1)2 + ωH
2 ω2
• Downlink and uplink problems have the same Lagrangian:
L = ωH1 ω1 + ω
H2 ω2 + λ1
((hH1 ω2)2 + σ
2 −1
γ1
(hH1 ω1)2)
+ λ2
((hH2 ω1)2 + σ
2 −1
γ2
(hH2 ω2)2)
= λ1σ2 + λ2σ
2 +
(ωH1 ω −
λ1
γ1
(ωH1 h1)2 + λ2(ωH
2 h2)2)
+
(ωH2 ω2 −
λ2
γ2
(ωH2 h2)2 + λ1(ωH
2 h1)2)
Wei Yu July 2012 11 / 43
Scheduling
• Choose the best set of users to serve across multiple cells.• Full spatial multiplex: Schedule as many users as BS antennas
• Considerations:• Load balancing• Traffic shaping• Interference avoidance
• Downlink:• Interference is independent of scheduling• Venturino-Prasad-Wang ’09, Stolyar-Viswanathan ’09
• Uplink:• Discrete combinatoric optimization problem• Difficult problem, no known optimal solution
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Proportional Fair Scheduling
• Fairness is crucial as many users share the same resources
• Proportional fairness scheduling:
• Originally proposed for single-cell Qualcom Downlink HDR.• Main idea (Tse 99):
k∗(l , s, b, n) = argmaxrnlskR̄lsk
where the average rate is exponentially updated.
• Over the long run, this is equivalent to:
max∑lsk
log(R̄lsk)
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Coordinated Beamforming, Scheduling, Power SpectrumAdaptation
• Iterate among the three steps:• Fixing two, the other step is a well-formulated problem.
Scheduling Beamforming Power Allocation
• Remarks:
• Not completely rigorous: The beamforming step minimizespower rather than maximize utility.
• The overall algorithm represents a complex negotiation amongthe cells.
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How Well does It Work?
• 7-cell, 3-sector/cell, 4-antenna at BS, 2 at MS, full reuse
0 2 4 6 8 10 12 14 16 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Downlink User Rates (Mbps)
Cu
mu
lative
Dis
trib
utio
n
CP−ZF
CP−CBF
DP−ZF
DP−CBF
• BS-to-BS d=2.8km (Joint work with T. Kwon, C. Shin ’11)• 100% rate improvement for the 25th percentile user• 50% rate improvement for the 40th percentile user
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Heterogeneous Topology
• 3-cell macro, 3-sector/cell, 3-femto/sector, 4 tx antenna
DP 0 −5 −10 −15 −20 −25 −30 −35 −40 −45 −50−100
−50
0
50
100
150
200
250
Lo
g U
tilit
y
Femtostation Power Backoff (dB)
Downlink
Uplink
• Coordinated BF and power control outperform constant powerbackoff
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Can We do Better?
• What about interference alignment? (Cadambe-Jafar ’08)
• For a K -user SISO interference channelcoded across time or frequencydimensions.
• Degree-of-Freedom (DoF) per user is K2
• For each user, the signal vector must notlie in the subspace spanned byinterference.
• Alignment for cellular network:Suh-Ho-Tse ’11,
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Interference Alignment Through Linear Beamforming
M ant. N ant.
• What about without symbol extension?
• Consider K-user MIMO (M × N) case
• Goal: deliver one data stream per user.
• Hij : channel between i th tx. and j th rx.vj : tx. beamformer at the j th tx.uj : rx. beamformer at the i th rx.
• We need:
uTj Hijvi = 0 if i 6= j
uTj Hijvi 6= 0 if i = j
• When is this possible?
• Bezout’s Theorem (Yetis, et al ’10)• Counting # of eqs. vs # of unknowns
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Interference Alignment for Cellular Networks
• Consider a 3-sector intersection:• M antennas/BS, N antennas/MS;• K users per sector.
• Assuming no symbol extensions, to aligninterference, we need:
uTpqHipqvij = 0 if i 6= p or j 6= q
uTpqHipqvij 6= 0 if i = p and j = q
• Alignment is feasible only if: (Zhuang-Berry-Honig ’12)
N + M ≥3∑
i=1
Ki + 1
• For a 3-cell system with 4 ant. at the BS and 3 ant. at theuser, only 2 users/cell can be scheduled (with no extension).
• Detailed analysis: Razaviyayn-Lyubeznik-Luo ’11,Wang-Gou-Jafar ’11, Bresler-Cartwright-Tse ’11
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What is the Right Number of Users to be Scheduled?
• Need system-level optimization to find out.• Fewer users open up more dimensions to ‘hide’ interference in;• More users can better utilize the available spatial dimensions.
• System setting:• Downlink, 3 cell sectors• 45 users/sector• One stream per scheduled user• 64 frequency tones• M tx antennas• N rx antennas
• (Joint work with G. Sridharan)
• Simplifying the setup:• Round-robin scheduling• Maximizing the sum rate• Iterate between duality-based BF and power optimizations
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Flowchart for Optimization: Full Spatial Multiplexing
assume BFs are fixed
Power optimization
SIN
Rs
Normalized BF
for given SINR constraints
tx−rx beamformer design
Round robin scheduling
Final power, BFs
Compute SINRs
Allocate equal power
Initialize to random BFs • Schedule full set of users.
• Initialize with random BF, equalpower.
• Use duality-based algorithm to findthe BF iteratively
• Use interior-point method for accountfor per-BS sum power constraint
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Flowchart for Optimization: What about Alignment?
assume BFs are fixed
Power optimization
SIN
Rs
Normalized BF
for given SINR constraints
tx−rx beamformer design
Round robin scheduling
Final power, BFs
Compute SINRs
Initialize to aligned BFs
Allocate equal power• Only schedule as many users as
alignment allows.
• Many sets of aligned BFs exist.
• Aligned BFs neglect direct channel.
• We further use duality-based tx-rx BFdesign to refine the BF design.
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Computing Aligned Beamformers
• Use algorithm of Gomadam-Cadambe-Jafar ’08 (see alsoPeters-Heath ’09)
• Interference at the j th user in the i th cell is given by:
Iij =∑
(p,q)6=(i ,j)
uHij Hpijvpq
• Covariance of Iij is given by:
Cov(Iij) = uHij
∑(p,q)6=(i ,j)
HpijvpqvHpqH
Hpij
uij , uHij Qijuij
• To minimize interference Iij , set uij to νL(Qij), where νL(Qij)is the eigenvector of Qij with the smallest eigenvalue.
• Update all uij ; use reciprocity to update vij similarly.
• Repeat until convergence.
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What is the Optimal Number of Users to Schedule?
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2150
160
170
180
190
200
210
220
230
B2B distance in km
Sum
rate
per
secto
r in
Mbps
randI−dBF−NPC−2u
randI−dBF−NPC−4u
randI−dBF−PB−2u
randI−dBF−PB−4u
randI−dBF−SP−2u
randI−dBF−SP−4u
• M=4, N=3
• 2 or 4 users/cell/tone
• power opt:no power controlper beam const.sum power const.
• Initialization:Random BFUniform power init.
• Observations:fewer users better atsmaller dist.Crossover point shiftsleft
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At Convergence, are BFs Aligned?Are Aligned BFs Easy to Find?
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2150
200
250
300
350
400
B2B distance in km
Sum
rate
per
secto
r in
Mbps
randI−dBF−NPC−2u
randI−dBF−NPC−4u
randI−dBF−PB−2u
randI−dBF−PB−4u
randI−dBF−SP−2u
randI−dBF−SP−4u
alignI−dBF−SP−2u
alignI−dBF−NPC−2u
• M=4, N=3
• 2 or 4 users/cell/tone
• power opt:no power controlper beam const.sum power const.
• Initialization:Random/aligned BFUniform power init.
• Observations:– aligned initialization
significantly better– aligned BF hard to
find w/ random init.– # users scheduled
plays important role
• Similar conclusion byZhuang-Berry-Honig’12
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Optimization of the ‘# of Users to Schedule’
0 0.5 1 1.5 2 2.5 3 3.5100
150
200
250
300
350
400
450
B2B distance in km
Sum
rate
per
secto
r in
Mbps
randI−dBF−SP−2u
randI−dBF−SP−4u
alignI−dBF−SP−2u
partialalignI−dBF−SP−4u
• M=4, N=3
• 2 or 4 users/cell/tone
• Initialization:Random/aligned BFsPartially aligned BFsUniform power init.
• power opt:sum power const.
• BF design:duality tx-rx BF
• Observations:– partial align init.circumvents sch. issue
– some performanceloss at higher distances
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Summary of Part I
• There are substantial benefits for system-level optimization• for both cellular and (especially) heterogeneous networks.
• Optimization is also quite difficult:
• Nonconvexity of power optimization is difficult to circumvent;• Beamforming is intricately connected to power control;• Discrete nature of user scheduling is hard.
• Interference alignment opens up a new dimension
• What is the optimal number of users to schedule?• Aligned solutions are not unique, how to identify the best one?
(Schmidt-Utschick-Honig ’10)
• Alignment can significantly outperform local optimum• Iterative optimization cannot shut off users to enable alignment• The optimization landscape is extremely non-smooth.
• Many practical issues:• Channel estimation and feedback.
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Part II: Network MIMO Systems with Limited Cooperation
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Network MIMO Systems
• Multi-cell processing can improve performance in wirelesscellular networks
• Multiple base-stations (BS) cooperate by jointly encoding andtransmitting data to Mobile Users (MU)
� helps mitigate interference� requires high amount of data sharing on the backhaul links
• How to achieve the gain from multi-cell processing whenbackhaul capacity is limited?
� Assume perfect and instantaneous CSI� Ignore delay in CSI and data sharing
• (Joint work with A. Chowdhery, S. Mehryar ’12)
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Network Model
• Downlink of a wireless multi-antenna, multi-cell, OFDMAnetwork
� n frequency tones, L cells, K MU’s per cell, Q antennas perBS, single-antenna MU’s
• Cooperation among neighbouring BS’s
� Partial zero-forcing (ZF) to pre-subtract shared data streams� Cooperation links are directional: beam to cell
• Backhaul capacity is limited for each BS
� A sub-set of frequency tones used to share the data stream
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Network Model
BS 2
BS 7BS 5
BS 4
BS 1
BS 6
BS 3
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Problem Formulation
max∑m,k
log(R̄mk)
s.t (A1): Rmk =∑
n∈Dmklog2(1 + SINRn
mk) ∀m, k(A2): 0 ≤
∑l ′,q |wn
mb,l ′q|2pnl ′q ≤ Smax ∀m, b, n(A3): C̄m ≤ Cmax
m ∀m
• Objective function: maximize overall log network utility
� ensures proportional fairness among all users in all cells
• Optimization variables:
� Cooperation-link selection� User-scheduling� Precoder-coefficient and power-spectrum adaptation
• Mixed discrete & continuous optimization problem
� impractical to find the network-wide optimal solution jointlyacross all BSs & tones
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Decoupling & Heuristic Approach
• Dualize the problem w.r.t the backhaul-capacity constraint:
max∑m,k
log(R̄mk)−L∑
m=1
λm(C̄m − Cmax
m
)s.t (A1): Rmk =
∑n∈Dmk
log2(1 + SINRnmk) ∀m, k
(A2): 0 ≤∑
l ′,q |wnmb,l ′q|2pnl ′q ≤ Smax ∀m, b, n
• We propose a heuristic approach that iteratively:
� chooses cooperation links� jointly schedules users� adapts precoding coefficients and power spectra
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Cooperation Link Selection
• Add/Remove a cooperation link from a beam to another BSusing the approximation:
Q∑b=1
1
R̄mk(˜̃rnmk,l ′q′ − r̃nmk) > λl r̃
nl ′k ′′
• All Q beams of receiving BS pre-subtract the interference ofcooperating BS’s beam
• Assume adding the link completely eliminates the interferenceof cooperating BS’s beam
• Only the intra-cell beamformers are used
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User Scheduling
• Select the active user on each beam in each cell and in eachfrequency tone using the approximation:
f (m, b, n) = argmaxk
(1
R̄mk− λmNn
mb
)r̃nmk
• Proportionally fair scheduler
• Accounts for an additional penalty for the backhaul-capacityconstraint
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Precoder Coefficient Optimization & Power SpectraAdaptation
• ZF Precoder Coefficients:
� Invert the channel gain matrix restricted to the cooperatingBSs on a beam-to-beam basis
• Power Spectra Adaptation:
� Solve a weighted rate-sum maximization problem on atone-by-tone basis using interior-point method, withαnmk =
((1/R̄mk)− λmNn
mb
)max
∑m,b
αnmk r
nmk
s.t (A4) : 0 ≤∑
l ,q |wnmb,lq|2pnlq ≤ Smax ∀m, b
Wei Yu July 2012 36 / 43
Summary of Algorithm:
Initialize Powers
Select a set of Cooperation Linksbased on approximate gains
Schedule Users(proportional fairness)
Optimize ZF beamformers,
Optimize Powers
InstantaneousRates Converged?
Update Average Rates and
Proportional Fairness Weights
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Simulation Parameters
Cellular Layout Hexagonal, 7-cell Wrap-around
BS-to-BS Distance 800m
Frequency Reuse 1
Number of Users per Cell 40
Channel Bandwidth 10 MHz (with 64 frequency tones)
BS Max Tx Power 49 dBm
BS Max PSD -27 dBm/Hz
Antenna Gain 15 dBi
SNR Gap (with coding) 6
Background Noise -162 dBm/Hz
Path Loss L = 128.1 + 37.1 log10 (d0)
Multipath Time Delay Profile ITU-R M.1225 PedA
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Performance Gain: Q = 1
0 100 200 300 400 500 600 700 80030
40
50
60
70
80
90
100
110
120
Average Backhaul Capacity per BS (Mbps)
Do
wn
link S
um
Ra
te p
er
Ce
ll (M
bp
s)
Fixed Link Selection
Dynamic Link Selection
Figure: Downlink sum rate per cell vs. backhaul capacity for fixedcooperation link selection and dynamic cooperation link selection forQ = 1 antenna per BS with BS-to-BS distance d = 800m.
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Performance Gain: Q = 4
0 500 1000 1500 2000 2500 3000100
150
200
250
300
350
400
450
500
Average Backhaul Capacity per BS (Mbps)
Do
wn
link S
um
Ra
te p
er
Ce
ll (M
bp
s)
Dynamic Link Selection
Fixed Link Selection
Figure: Downlink sum rate per cell vs. backhaul capacity for fixedcooperation link selection and dynamic cooperation link selection forQ = 4 antennas per BS with BS-to-BS distance d = 800m.
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Proportional Fairness
0 5 10 15 20 25 30 35 40 450
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Downlink User Rates (Mbps)
Cu
mu
lativ
e D
istr
ibu
tion
No Coop.
825 Mbps
1658 Mbps
2558 Mbps (Full Coop.)
Increasing BackhaulCapacity per BS
Figure: Cumulative distribution functions of downlink user rates fordifferent backhaul capacities for the case of BS-to-BS distance of 800mwith Q = 4 antennas per BS.
Wei Yu July 2012 41 / 43
Summary of Part II
• Proposed a numerical algorithm that maximizes thenetwork-wide utility subject to backhaul-capacity constraints
� Selects a subset of tones at each BS for cooperation� Uses a proportional fairness objective to ensure a fair
distribution of gains among the users
• Numerical results suggest that the gain of network MIMOscales linearly with the backhaul capacity.
• Algorithm is iterative and heuristic in nature
� Future work is needed to assess these heuristics in light ofmany local minima present in the optimization landscape
Wei Yu July 2012 42 / 43
Concluding Remarks
• System-level optimization is very useful in the design ofwireless cellular networks and heterogeneous networks
� Coordination is crucial� Potential benefit is significant
• Global optimum is very hard to obtain in aninterference-limited environment
� Many local optima exist in the optimization landscape
• Optimality vs. robustness?
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