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.

Wei Yu July 2012 3 / 43

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.

Wei Yu July 2012 5 / 43

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

Wei Yu July 2012 6 / 43

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?

Wei Yu July 2012 7 / 43

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).

Wei Yu July 2012 8 / 43

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.

Wei Yu July 2012 10 / 43

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

Wei Yu July 2012 12 / 43

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)

Wei Yu July 2012 13 / 43

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

Wei Yu July 2012 16 / 43

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

Wei Yu July 2012 18 / 43

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

Wei Yu July 2012 19 / 43

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

Wei Yu July 2012 21 / 43

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.

Wei Yu July 2012 22 / 43

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.

Wei Yu July 2012 23 / 43

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

Wei Yu July 2012 24 / 43

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

Wei Yu July 2012 25 / 43

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

Wei Yu July 2012 26 / 43

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.

Wei Yu July 2012 27 / 43

Part II: Network MIMO Systems with Limited Cooperation

Wei Yu July 2012 28 / 43

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)

Wei Yu July 2012 29 / 43

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

Wei Yu July 2012 30 / 43

Network Model

BS 2

BS 7BS 5

BS 4

BS 1

BS 6

BS 3

Wei Yu July 2012 31 / 43

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

Wei Yu July 2012 32 / 43

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

Wei Yu July 2012 33 / 43

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

Wei Yu July 2012 34 / 43

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

Wei Yu July 2012 35 / 43

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

Wei Yu July 2012 37 / 43

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

Wei Yu July 2012 38 / 43

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?

Wei Yu July 2012 43 / 43