Spatial Interference Management for Dense WirelessNetworks: Alignment and Other Techniques
Maxime GuillaudVienna University of Technology
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
Current Systems are Interference-Limited
traffic growth + expensive spectrum = frequency reuse + interference(the equation faced by wireless communication engineers...)
b a
b
b
M. Guillaud (Vienna UT) Cancun, 10.05.2010 2 / 23
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 ]
M. Guillaud (Vienna UT) Cancun, 10.05.2010 3 / 23
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
M. Guillaud (Vienna UT) Cancun, 10.05.2010 4 / 23
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
M. Guillaud (Vienna UT) Cancun, 10.05.2010 5 / 23
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.
M. Guillaud (Vienna UT) Cancun, 10.05.2010 6 / 23
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)
M. Guillaud (Vienna UT) Cancun, 10.05.2010 7 / 23
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 ])
M. Guillaud (Vienna UT) Cancun, 10.05.2010 8 / 23
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
M. Guillaud (Vienna UT) Cancun, 10.05.2010 9 / 23
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
M. Guillaud (Vienna UT) Cancun, 10.05.2010 10 / 23
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
M. Guillaud (Vienna UT) Cancun, 10.05.2010 11 / 23
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
M. Guillaud (Vienna UT) Cancun, 10.05.2010 12 / 23
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.
M. Guillaud (Vienna UT) Cancun, 10.05.2010 13 / 23
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 ]
M. Guillaud (Vienna UT) Cancun, 10.05.2010 14 / 23
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
M. Guillaud (Vienna UT) Cancun, 10.05.2010 15 / 23
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
IA (K=11,d=1,d’=1), projection receiver
IA (K=11,d=1,d’=1), optimum receiver
Optimum receiver yieldssmall gain, only at low SNR
11 users,d’=1
7 users,d’=2
M. Guillaud (Vienna UT) Cancun, 10.05.2010 16 / 23
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
)
5x7 (Nr x Nt) channels
IA (K=7,d=1,d’=2), projection receiver
IA (K=7,d=1,d’=2), optimum receiver
IA (K=11,d=1,d’=1), projection receiver
IA (K=11,d=1,d’=1), optimum receiver
11 users,d’=1
7 users,d’=2
M. Guillaud (Vienna UT) Cancun, 10.05.2010 17 / 23
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.
M. Guillaud (Vienna UT) Cancun, 10.05.2010 18 / 23
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
)
5x7 (Nr x Nt) channels
IA (K=7,d=1,d’=2), optimum receiver
Non−cooperative (waterfilling game), K=7
IA (K=11,d=1,d’=1), optimum receiver
Non−cooperative (waterfilling game), K=11
11 users
7 users
IA beats waterfilling gamefor SNR >5dB
M. Guillaud (Vienna UT) Cancun, 10.05.2010 19 / 23
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
)
5x7 (Nr x Nt) channels
IA (K=7,d=1,d’=2), optimum receiver
Non−cooperative (waterfilling game), K=7
IA (K=11,d=1,d’=1), optimum receiver
Non−cooperative (waterfilling game), K=11
waterfilling game showslittle dependency on K
slope proportionalto Kd for IA
M. Guillaud (Vienna UT) Cancun, 10.05.2010 20 / 23
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 ?
M. Guillaud (Vienna UT) Cancun, 10.05.2010 21 / 23
Thank you for your attention
Questions ?
M. Guillaud (Vienna UT) Cancun, 10.05.2010 22 / 23
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.
M. Guillaud (Vienna UT) Cancun, 10.05.2010 23 / 23