Spatial Interference Management for Dense Wireless Networks

Spatial Interference Management for Dense WirelessNetworks: Alignment and Other Techniques

Maxime GuillaudVienna University of Technology

[email protected]

http://www.nt.tuwien.ac.at/about-us/staff/maxime-guillaud

Joint work with Roland Tresch (FTW)

Communication Theory WorkshopCancun, May 10, 2010

M. Guillaud (Vienna UT) Cancun, 10.05.2010 1 / 23

mailto:[email protected]

http://www.nt.tuwien.ac.at/about-us/staff/maxime-guillaud/

Current Systems are Interference-Limited

traffic growth + expensive spectrum = frequency reuse + interference(the equation faced by wireless communication engineers...)

b a

b

b


Goals of this PresentationFocus on the Gaussian MIMO Interference Channel (MIMO-IC)

Analyze interference alignment (IA) performance forI non-asymptotic SNRI many users, many antennas

Compare to distributed Tx covariance optimization

NOT Covered HereNumerous derivative works sparked by IA:

I asymmetric complex signalingI lattice alignmentI rational dimension alignment...

Robustness w.r.t. imperfect CSII some clues available from [Tresch,Guillaud ICC’09 ]

Stochastic geometry-based analysis of IA in dense networksI Per-Cluster IA addressed in [Tresch,Guillaud ISIT’10 ]


Recent Focus on the K-User MIMO-IC

K-User MIMO Gaussian Interference Channel

y i =K∑

j=1Hĳx j + ni ∀i = 1 . . .K .

Rx i is interested in the message of Tx i only

Interference Alignment[Gomadam,Cadambe,Jafar IT’08 , Cadambe,Jafar IT’08 ] proposes

I low rank (less than channel rank) linear precodingI create a subspace without interference at the receiver


Interference Alignment in Pictures

s1 H11

H21

H33

H23

n1

n2

n3

s2

s3

NT

NT

NT

NR

NR

NR

x i = Vis i ∀i , where Vi is tall, and s i contains the symbols to transmit

interference∑

j 6=i HĳVjs j does not occupy all receive dimensions


Why is IA Attractive ?Simple formulation: find matrices Vi and Ui (resp. NTi × di and NRi × di)

s.t.{

UHi HĳVj = 0, ∀j 6= i

rank(UH

i HiiVi)

= di .

No complex multiuser codes, interference treated as noiseAchievable rate per user does not saturate at high SNR

limSNR→+∞

Ci (SNR)

log SNR = di

(a.s. under reasonable channel assumptions)

I di interference-free signaling dimensions, or degree of freedom (DoF)available to user i .

“each user gets half of the cake”I In symmetric systems with square channels (NR = NT = N, di = d ∀i),

IA is feasible for d up to N2 .

I Point-to-Point MIMO: limSNR→+∞C(SNR)log SNR = N a.s.


Limitations of IA

Requires extensive CSI

Can only be achieved for limited number of users K[Yetis,Jafar,Kayran GC’09 ]

I For symmetric systems with K users, IA is achievable a.s. iff

NT + NR − (K + 1)d ≥ 0.

I Example with NT = 4,NR = 2, d = 1: solutions for K ≤ 5.

IA requires multi-dimensional fading channels:I multiple antennas (space)I OFDM (frequency)I channel extension (time)


Solutions to IA over the MIMO-IC

Iterative (slow !) scheme based on the minimization of the interferenceleakage metric [Gomadam,Cadambe,Jafar IT’08 ]:

Iw =K∑

i=1

K∑j=1,j 6=i

∣∣UHi HĳVj

∣∣2Frob.

Closed-form solutions only for certain K or in particular cases(NT = NR = K − 1, [Tresch,Guillaud,Riegler SSP’09 ])


Cellular Network Simulation

Compare IA (among K cells out of L) to frequency reuse 1/3

Flat-fading channels (no attempt at scheduling)

Compare the ergodic rate of IA Ri,IA(pi ) to that of frequency reuse +single-link waterfilling Ri,FR(pi ) at each position pi .

K L NT NRScenario A 3 27 2 2Scenario B 7 37 6 2

Parameter ValueTransmit Power per Subcarrier ρi = 17 dBm ∀iPath Loss Model γĳ = 128.1 + 37.6 · log10(rĳ [km]) dBPath Loss Exponent α = 3.76Fast Fading RayleighCell Radius r = 1 kmAntenna Radiation Pattern omnidirectionalCell Shape HexagonalFrequency Reuse Factor κ = 1/3


Cellular Network Simulation Results

(Ri,IA/Ri,FR) in % for Scenario A (K=3) and B (K=7).

−2000 −1000 0 1000 2000−2000

−1000

0

1000

2000

0

20

40

60

80

100

120

−3000 −2000 −1000 0 1000 2000 3000−3000

−2000

−1000

0

1000

2000

3000

20

40

60

80

100

120

from [Tresch,Guillaud PIMRC’09 ]

Some localized gains (hotspots ?)

Average gain over space little or negative


IA at low SNR: Improve Diversity (play with DMT)Let the codimension of the interference-free subspace (at Rx) be greaterthan the user’s transmit DoF: d ′i ≥ di .

I Letting Ui NRi × d ′i and Vi NTi × di ,

IA⇔{

UHi Hĳ Vj = 0d′i ×dj , ∀j 6= i

rank(

UHi Hii Vi

)= di

}I Equivalent channel after interference suppression UH

i HiiVi is tall(diversity).

Updated feasibility criterionI In the symmetric case (NR × NT, di = d , d ′i = d ′), solution exists a.s.

iffd(NT − d) + d ′(NR − d ′)− dd ′(K − 1) ≥ 0.

I Not symmetric in NT,NR: e.g. d = 1, d ′ = 2 is feasible iff2K ≤ NT + 2NR − 3.

The iterative interference leakage minimization generalizes to this case


IA Maximum Achievable Rate

With optimum receiver, assuming that interference is not decodable

Maximum rate achievable by user i :

I(y i ; s i |H) = log det(

Idi + VHi HH

ii(QI

i)−1 HiiViQi

),

whereI Qi = E

[s isH

i]

is the covariance at Tx i ,I QI

i =∑

k 6=i HikVkQkVHk HH

ik + σ2I is the covariance ofinterference+noise at Rx i ,

I H = {Hĳ} i = 1 . . .Kj = 1 . . .K


IA Achievable Rate with Projection ReceiverProjection receiver

y i = UHi y i =UH

i HiiVis i +∑j 6=i

UHi HĳVjs j + UH

i ni

= UHi HiiVi︸︷︷︸

Hii

s i + ni .

Suboptimal but simpleI interference cancellation in the analog front-end can ease the digital

front-end requirements (EC FP7 project MIMAX)

Ergodic MI is trivially (using truncated unitary Ui and Vi)

Ri =EH

[I(y i ; s i |H)

]= EHii

[log det

(ID + HiiQiHH

ii)].

I if Hii is Gaussian i.i.d. fading and channels are independent betweenusers, then Hii is d ′i × di Gaussian i.i.d.


Ergodic Achievable Rates under IA

Symmetric case with Tx power P spread equally over d signaling dimensionsI Qi = Pi

diIdi , Vi ’s truncated unitary matrices (power normalization)

We want

Ri =EH

[I(y i ; s i |H)

]= EH

[log det

(ID + VH

i HHii(QI

i)−1 HiiViQi

)]I QI

i occupies a random (dependent on the channel realization) subspaceof the NR i -dimensional receiver subspace.

I We obtain a tight lower bound

EH

[I(y i ; s i |H)

]? EH

[log det

(Idi + AΨAH)]

where A is a di × NRi matrix with complex Gaussian i.i.d. coefficients

of unit variance, and Ψ = diag(

Pidiσ2 Id′i ,

Pidi (σ2+

∑k 6=i

Pk )INRi−d′i

).

I Evaluated using a recent RMT result [Chiani,Win,Shin IT’10 ]


Validation of the Ergodic MI Formulas

−15 −10 −5 0 5 10 15 20 25 300

1

2

3

4

5

6

7

SNR (dB)

Erg

odic

MI p

er u

ser

(nat

s)

K=7 users, 5x7 (Nr x Nt) channels

IA (d=1,d’=2), projection receiver, MC sim.

IA (d=1,d’=2), projection receiver, analytical

IA (d=1,d’=2) with optimum receiver, MC sim.

IA (d=1,d’=2) with optimum receiver, analytical


Influence of d ′ > d , Per-user Rate

−15 −10 −5 0 5 10 15 20 25 300

1

2

3

4

5

6

7

SNR (dB)

Erg

odic

rat

e pe

r us

er (

nats

)

5x7 (Nr x Nt) channels

IA (K=7,d=1,d’=2), projection receiver

IA (K=7,d=1,d’=2), optimum receiver



Optimum receiver yieldssmall gain, only at low SNR

11 users,d’=1

7 users,d’=2


Influence of d ′ > d , Sum-Rate

−15 −10 −5 0 5 10 15 20 25 300

10

20

30

40

50

60

70

SNR (dB)

Erg

odic

Sum

−ra

te (

nats

)






11 users,d’=1

7 users,d’=2


Comparison with Non-Cooperative Methods

Comparison with the method of [Scutari,Palomar,Barbarossa IT’09 ], basedon a game-theoretic optimization of the transmit covariances.

I Each Tx does waterfilling based on the interference covariance at hisintended receiver resulting from the previous iterations

Properties:I Inherently distributed method, requires only local channel knowledgeI Faster convergence than the leakage-based IA algorithmI Does not always reach a Nash equilibrium (depending on H). If not,

draw another channel realization.


Cooperative (IA) vs. Non-Cooperative (Waterf. Game),Per-User Rate

−15 −10 −5 0 5 10 15 20 25 300

1

2

3

4

5

6

7

SNR (dB)

Erg

odic

rat

e pe

r us

er (

nats

)



Non−cooperative (waterfilling game), K=7



11 users

7 users

IA beats waterfilling gamefor SNR >5dB


Cooperative (IA) vs. Non-Cooperative (Waterf. Game),Sum-Rate

−15 −10 −5 0 5 10 15 20 25 300

10

20

30

40

50

60

70

SNR (dB)

Erg

odic

Sum

−ra

te (

nats

)






waterfilling game showslittle dependency on K

slope proportionalto Kd for IA


Summary

Analyzed the ergodic performance of IA under Rayleigh fadingI Closed-form formulas enable asymptotic (in users, antennas) and

low-SNR analysisI Facilitates optimization of system parameters (d , d ′,NT . . .)

Comparison with distributed transmit covariance optimization (game theory)

Introduced “diversity receivers” in IA

Outlook:

Performance of “pure” IA in the low SNR regime is far from spectacularI arguably due to the subspace-based definitionI answer in approximate alignment, properly weigthing the interference

power ?


Thank you for your attention

Questions ?


Bibliography[Cadambe,Jafar IT’08 ] V. R. Cadambe and S. A. Jafar. Interference alignment and degrees of freedom of the K-user

interference channel. IEEE Transactions on Information Theory, 54(8):3425–3441, August 2008.

[Chiani,Win,Shin IT’10 ] M. Chiani, M. Z. Win, and Hyundong Shin. MIMO Networks: The Effects of Interference. IEEETransactions on Information Theory, 56(1):336–349, January 2010.

[Gomadam,Cadambe,Jafar IT’08 ] K. Gomadam, V. R. Cadambe, and S.A. Jafar. Approaching the Capacity of WirelessNetworks through Distributed Interference Alignment. IEEE Transactions on Information Theory. submitted 2008.

[Guillaud,Tresch Newcom’09 ] M. Guillaud and R. Tresch. Alignment-based interference mitigation technique for cellularnetworks. In Proc. Joint Newcom++ - ACoRN Workshop, Barcelona, Spain, March 2009.

[Scutari,Palomar,Barbarossa IT’09 ] G. Scutari, D. P. Palomar and S. Barbarossa. The MIMO Iterative Waterfilling Algorithm.IEEE Transactions on Information Theory, 57(5):1917–1935, May 2009.

[Tresch,Guillaud ICC’09 ] R. Tresch and M. Guillaud. Cellular interference alignment with imperfect channel knowledge. InInternational Workshop on LTE Evolution, Proc. IEEE International Conference on Communications (ICC), Dresden,Germany, June 2009.

[Tresch,Guillaud ISIT’10 ] R. Tresch and M. Guillaud. Performance of interference alignment in clustered wireless ad hocnetworks. In Proc. International Symposium on Information Theory (ISIT), June 2010.

[Tresch,Guillaud PIMRC’09 ] R. Tresch and M. Guillaud. Clustered interference alignment in large cellular networks. In Proc.IEEE International Symposium On Personal, Indoor and Mobile Radio Communications (PIMRC), Tokyo, Japan, September2009.

[Tresch,Guillaud,Riegler SSP’09 ] R. Tresch, M. Guillaud and E. Riegler. On the achievability of interference alignment in theK-user constant MIMO interference channel. In Proc. IEEE Workshop on Statistical Signal Processing (SSP), Cardiff, U.K.,September 2009.

[Yetis,Jafar,Kayran GC’09 ] C. M. Yetis, S. A. Jafar, and A. H. Kayran. Feasibility conditions for interference alignment. InProc. IEEE Globecom, Honolulu, HI, USA, November 2009.


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Spatial Interference Management for Dense Wireless Networks

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