Interference Management in Wireless Networks · 2015-08-12 · IS-95 standard) Interference from...

Interference Management in Wireless Networks

Venu Veeravalli

Coordinated Science LabDepartment of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign

Aly El Gamal

Department of Electrical and Computer EngineeringPurdue University

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Part 1: Introduction

ECE Illinois & Purdue

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Interference in Wireless Networks


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Interference Management

Wireless World

Interference management is critical


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Interference Management in Cellular Networks:Historical Perspective


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Single Isolated Cell

• Manhattan mobile phone system (1946)• Interference managed by simply orthogonalizing users in

time-frequency


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Multiple Access Alternatives


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Cellular Concept (Bell Labs)

Reuse spectrum geographically through cell splitting

BASE STATION Mobile

Telephone Switching Office


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Narrowband Cellular (AMPS,GSM)

• Bandwidth divided into narrowbandchannels (200 Khz in GSM), andusers are assigned time slots (8 perchannel in GSM)

• No in-cell interference (orthogonalusers in cell)

• Interference across cells on samechannel is minimized by reusing samechannels only in cells far apart


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Frequency Reuse

• Spectral efficiency of narrowband cellular is reduced by reusefactor

• Interference localized to narrow band


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Wideband Cellular (CDMA)

• Universal frequency reuse – all users in allcells share same bandwidth (1.25 MHz inIS-95 standard)

• Interference from in-cell users can becontrolled by using orthogonal codes(downlink) or successive interferencecancellation (uplink)

• Interference from other cells is averaged andsimply raises the noise floor

• Advantages:• No reduction in capacity due to reuse (no frequency planning

needed)• Graceful degradation in performance with users (soft capacity)• Any technique that reduces power of interferers (soft handoff, voice

activity detection) increases capacity


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Disadvantages of CDMA

• In-cell interference reduces capacity (cannot be eliminatedcompletely)

• Tight power control is needed to manage interference, and maybe too expensive for data applications (with low duty cycle)

Can we simultaneously haveuniversal reuse and keep in-cellusers orthogonal?


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Wideband Cellular OFDM (LTE)

• Split up bandwidth into narrowband sub-channels, with every userhaving access to all sub-channels

• Basic unit of resource is virtual channel (hopping sequence) ?virtual channels are orthogonal in TF

From Tse & Viswanath, Fundamentals of Wireless Comm


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Wideband Cellular OFDM

• In-cell Orthogonalization• Hopping sequences of users within cell are

designed to be orthogonal• Users are assigned one or more virtual

channels

• Out-of-Cell Interference Averaging• Hopping patterns in adjacent cells are

chosen so that there is minimal overlapbetween any pairs

• Interference from out-of-cell is averagedover band rather than being localized


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Additional Resources for Interference Management


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Multiple Antennas (MIMO)

• Multiple antennas providediversity against fading andmultiplexing gain forpoint-to-point links

• Can also be used for spatialseparation of users –beamforming

• Additional degrees of freedomfor interference management!


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Basestation Cooperation

• Out-of-cell users become in-cell users

• Beamforming (downlink), joint decoding (uplink)


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Relaying and User Cooperation

• Potential interferers become helpers (relays of information)

• Particularly useful in distributed interference management (adhoc, mesh networks)


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Dynamic Spectrum Access (Cognitive Radio)

Primary Tx

range

guard band

secondary users

• Primary and secondary usersof spectrum

• Secondary users sensechannels to determinepresence/absence of primaryusers

• Probability of interfering withprimary is constrained


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Explosion in Wireless Data Traffic

How to accommodate exponential growth without new useful spectrum?


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Through Improved PHY?

• Point-to-Point wireless technology mature• Modulation/demodulation• Synchronization• Coding/decoding (near Shannon limits)• MIMO

• Centralized (in-cell) multiuser wireless technology also mature• Orthogonalize users when possible• Otherwise use successive interference cancellation

Spectral efficiency gains from further improvements in PHY are limited!


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By Adding More Basestations?


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Through Improved Interference Management

Several useful engineering solutions for managing interference

But...

What are fundamental limits?


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References

1 T.S. Rappaport. Wireless communications: Principles andpractice. New Jersey: Prentice Hall 1996.

2 A. J. Viterbi. CDMA: Principles of spread spectrumcommunication. Addison Wesley, 1995.

3 D. Tse and P. Viswanath. Fundamentals of wirelesscommunication. Cambridge University Press, 2005.

4 E. Biglieri et al. Principles of Cognitive Radio. CambridgeUniversity Press, 2012.


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Part 2: Information Theory for Interference Channels


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Claude Shannon’s Information Theory

Point-to-point Communication on AWGN channel

Capacity = log(1 + SNR) bits/sec/Hz


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Two User Gaussian Interference Channel

hX1

X2

Y1

Y2

Z2

Z1

• Gaussian noise with unit variance; transmit power constraint P

• Capacity region?

• Known when h ≥ 1


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Strong Interference Setting: h ≥ 1

hX1

X2

Y1

Y2

Z2

Z1

• Both receivers decode both messages (compound MAC)• Capacity region

R1 ≤1

2log(1 + P )

R2 ≤1

2log(1 + P )

R1 +R2 ≤1

2log(1 + P + h2P )


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Very Strong Interference Setting: h ≥√1 + P

hX1

X2

Y1

Y2

Z2

Z1

• Each user first decodes interference treating intended input asnoise; then subtracts interference to decode intended input

• Capacity region

R1 ≤1

2log(1 + P )

R2 ≤1

2log(1 + P )

same as when interference is absent


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Two User Gaussian Interference Channel

hX1

X2

Y1

Y2

Z2

Z1

• Gaussian noise with unit variance; transmit power constraint P

• Capacity region?

Open problem when h < 1


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What is Known?

• Simple schemes• Treat interference as noise• Orthogonalize users• Single user coding/decoding

• Sophisticated schemes• Exploit structure in interference• Joint coding/decoding

• Han-Kobayshi achievable scheme• Power splitting (common/private) and time-sharing• Best known inner bound to capacity region for two-user GIC

• Special case of H-K scheme achieves capacity to within 1 bit!• Etkin, Tse, Wang 2007


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Capacity in Low Interference Regime [Annapureddy&VVV ’09]

Treating interference as noise is optimal in low interference regime

0 5 10 15 20 25 30 35 40−10

−5

0

5

10

15

SNR

INR

Th

resh

old


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Main Result

Theorem

For the two user symmetric Gaussian interference channel satisfying

h(1 + h2P ) ≤ 0.5

treating interference as noise achieves the sum capacity, which is givenby

Csum = log

[1 +

P

1 + h2P

]

Requires new outer bound!


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Genie-aided Outer Bound

• Genie gives side-information to receivers

hX1

X2

Z1

Z2

Y1

Y2

S1

S2

• Sum capacity of genie-aided channel is obvious outer bound tooriginal channel


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Properties of Genie: Usefulness

• Useful: if sum capacity of genie-aided channel (easily) derivable

• Gaussian inputs are optimal=⇒ treating interference as noise is optimal

• Csum(genie-aided) = I(X1G;Y1G, S1G) + I(X2G;Y2G, S2G)

• Eg: S1 = X2 and S2 = X1

Y1

Z2

Z1

Y2

X1

X2


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Properties of Genie: Smartness

• Smart: if when Gaussian inputs are used, genie does not improvesum capacity

⇒ I(X1G;Y1G, S1G) = I(X1G;Y1G) andI(X2G;Y2G, S2G) = I(X2G;Y2G)

• Example: Genie that does not interact with receivers

not useful!

• We want a genie that is both useful and smart

⇒ treating interference as noise is optimal for original channel

Csum = I(X1G;Y1G) + I(X2G;Y2G)


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Quest for Divine Genie

• Restrict class of genies considered

S2 = X2 + ηW2

X1

X2

Z1

Z2

Y1

h

S1 = X1 + ηW1

Y2

• If Wi independent of Zi, i = 1, 2• genie is useful in proving one bit result of Etkin, Tse, and Wang

• Useful and asymptotically smart!


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Geometric Representation of Genie

Y

(√1 + h2P, 0)

(η, θ)

ηW

I + ZX

S

• cos θ: Correlation coefficient between I + Z and ηW

(equivalently between Z and W )


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Useful Genie

Y

(√1 + h2P, 0)

(η, θ)

ηW

I + ZX

S

(0, 1h )

Inside green curve: useful


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Condition: Useful Genie

• Mutual Information for the genie-aided channel

I(Xni ;Y n

i , Sni ) = I(Xn

i ;Sni ) + I(Xni ;Y n

i |Sni )

≤ h(Sni )− nh(SiG|XiG) + nh(YiG|SiG)− h(Y ni |Xn

i , Sni )

• Terms that are not maximized by Gaussian inputs:

h(Sn1 ) = h(hXn1 + hηWn

1 )

h(Y n2 |Xn

2 , Sn2 ) = h(hXn

1 + Zn2 |Wn2 )

and similar terms obtained by swapping “1” and “2”

• Difference between terms is maximized by X1G (worst case noiseresult) if:

|hη| ≤√

1− cos2 θ


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Smart Genie

YX

S

I + Z

ηW

Smart

On the blue line: smart


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Divine Genie

YX

S

I + Z

ηW

Smart

Useful

Green curve intersects with blue line:Treating interference as noise is optimal


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Divine Genie

Smart

Useful

√1 + h2P

0.5h(

√1+h2P)

Intersection if h(1 + h2P ) ≤ 0.5

• Also [Shang, Kramer, Chen 08] &[Motahari, Khandani 08]


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Capacity in Low Interference Regime [Annapureddy&VVV ’09]

INR = h2P and SNR = P

0 5 10 15 20 25 30 35 40−10

−5

0

5

10

15

SNR

INR

Th

resh

old


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Extensions – Outer Bound on Capacity Region

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.5

1

1.5

2

R1

R2

HK Inner BoundETW Outer BoundBroadcast Outer BoundNew Outer Bound


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Interference Network

X3

YM

Y3

Y2

Y1

XM

X2

X1

• Is treating interference as noise still optimal in low interferenceregime?

• Yes, using same “scalar” genie we can find threshold below whichtreating interference as noise is optimal


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Many-One and One-Many Channels

X3

YM

Y3

Y2

Y1

XM

X2

X1

M∑i=2

h21i ≤ 1

X3

YM

Y3

Y2

Y1

XM

X2

X1

M∑i=2

h2i1P1 + h2

i1

h2i1P1 + 1

≤ 1

Treating interference as noise is optimal in low interference regime


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Symmetric M -User Interference Channel

X3

YM

Y3

Y2

Y1

XM

X2

X1

• Interference threshold is characterized by

h(1 + h2P ) ≤ 0.5, where h2 = (M − 1)h2

• Threshold on total interference at receiver below which it isoptimal to treat interference as noise remains constant as Mincreases!


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Case Study – Three User Symmetric Channel

X1

X3

X2

Y1

Y2

Y3

• Requires new genie construction

• No explicit equation for threshold on interference parameter h

• But can compute admissible values of h for given P numerically


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Interference Threshold

2 4 6 8 10 12 14 16 18 20−6

−5

−4

−3

−2

−1

0

1

2

3

4

SNR

INR

Th

resh

old

Two users

Three users


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Extension to MIMO Interference Channels

H21

X2

Z1

Z2

Y 1

Y 2

X1

H22

H11

H12

Input covariance constraints: Qi � 0,Tr (Qi) ≤ P


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Genie-Aided MIMO Interference Channel

Y 2

Hc

Hd

Hd

X2

Z1

Z2

X1

S1

Y 1

S2

Si = Hc Xi +W i

[ZiW i

]∼ N

(0,

[Σ A

A> Σ

])

Genie parameters: Ψ = {Σ, A}ECE Illinois & Purdue

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Genie-aided outer bound

H21

X2

Z1

Z2

Y 1

Y 2

X1

H22

H11

H12

Y 2

Hc

Hd

Hd

X2

Z1

Z2

X1

S1

Y 1

S2

Csum ≤ CGA-ICsum (Ψ)

≥ Useful Genie=

maxQ

∑i

I(XiG;Y iG)Smart Genie

? maxQ

∑i

I(XiG;Y iG, SiG)


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Useful and Smart Genie Conditions

Useful Genie: If Σ � Σ−AΣ−1A> then:

CGA-ICsum (Ψ) = maxQ

RGA-ICTIN (Q,Ψ)

Smart Genie: If(A>(HcQH

>c + Σ)−1Hd −Hc

)Q = 0, then

RICTIN(Q) = RGA-IC

TIN (Q,Ψ)

i.e., the genie is smart w.r.t. input covariance Q.


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Genie-aided outer bound: Fix Q

H21

X2

Z1

Z2

Y 1

Y 2

X1

H22

H11

H12

Y 2

Hc

Hd

Hd

X2

Z1

Z2

X1

S1

Y 1

S2

Csum(Q) ≤ CGA-ICsum (Q,Ψ)

≥ Useful Genie=

RICTIN(Q)

Smart Genie w.r.t. Q= RGA-IC

TIN (Q,Ψ)

Sufficient Condition: Treating interference as noise is sum rate optimalif for every Q satisfying power constraint, there exists a genie that isuseful and smart w.r.t. Q [Shang, Chen, Kramer & Poor, Allerton 08]


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Low Interference Regime – Simpler Condition

Theorem

Let Q∗ be a global maximum of RICTIN(Q). If

• there exists a genie that is useful and smart w.r.t. Q∗

• Q∗ is full rank

then the channel is in low interference regime, i.e.,

Csum = RICTIN(Q∗).


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Proof Ilustration

Q∗

RGA−IC

RIC

Q


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MISO Channel

hc

Z1

Y1X1

Y2

Z2

X2 d

d

θ: angle between d and c.


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Application of General MIMO Result?

• Is Q∗ = arg maxRICTIN(Q) full rank?

No! Q∗ is unit rank

• Beamforming along direction b is optimal, i.e.,

Q∗ = Pbb>

where b is generalized eigenvector of matrix pair(Pdd>, I + h2Pcc>

)corresponding to largest generalized eigenvalue λmax


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Genie-aided outer bound

Csum ≤ CGA-ICsum (Ψ)

≥ Useful Genie=

maxQRICTIN(Q) maxQR

GA-ICTIN (Q,Ψ)

= ≤

maxQRICTIN(Q) maxQRrelaxed(Q,Ψ)

=Additional Constraints

=

RICTIN(Q∗)

Smart Genie w.r.t.Q∗= Rrelaxed(Q∗,Ψ)


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Outer Bound

0 20 40 60 80 100 120 140 160 1800

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Angle between Beamforming vector and cross channel vector

Su

m R

ate

Achievable Rate

Genie−aided outer bound

Relaxed outer bound


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Low Interference Regime

Theorem

The sum capacity of the MISO interference channel is achieved byusing Gaussian inputs, transmit beamforming, and treating interferenceas noise at the receivers, and is given by

CICsum =1

2log

(1 +

P cos2 θ

1 + h2P+ P sin2 θ

)if the channel gain parameter h satisfies:

|h| < h0(θ, P ) (1)

with h0(θ, P ) being the positive solution to the implicit equation

h2 − sin2 θ =

(cos θ

1 + h2P− h)2

+

. (2)


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0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

θ

Thr

esho

ld o

n h sin(θ)

P = 10

P = 1

For θ = 0, h(1 + h2P ) ≤ 0.5 (SISO result)


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Low Interference Regime - Simplified

Theorem

For any P , ifh ≤ sin(θ)

then sum capacity of MISO interference channel is achieved by usingGaussian inputs, transmit beamforming, and treating interference asnoise at receivers.


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SIMO Channel

Z2

d

d

hc

Z1

X1

X2 Y 2

Y 1

θ: angle between d and c.

No covariance optimization


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Improved usefulness condition

• h(hcX +W 1)− h(hcX +W 2) maximized by Gaussiandistribution?

• Σ1 � Σ2 is a sufficient condition, but not necessary.

• We can show that (c>Σ−11 c)−1 ≤ (c>Σ−1

2 c)−1 is sufficient.


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Genie Construction

• Symmetric genie signal

Si = hcXi +W i

[ZiW i

]∼ N

(0,

[I A

A> Σ

])• Useful Genie:(

c>Σ−1c)−1≤(c>(I −AΣ−1A>

)−1c

)−1

• Smart Genie:A>(h2Pcc> + I)−1d− hc = 0

• Goal: Find A and Σ that result in largest threshold on h for lowinterference regime


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Theorem

The sum capacity of the SIMO interference channel is achieved byusing Gaussian inputs, receive beamforming, and treating interferenceas noise at the receivers, and is given by

CICsum =1

2log

(1 +

P cos2 θ

1 + h2P+ P sin2 θ

)if the channel gain parameter h satisfies:

|h| < h0(θ, P ) (3)

with h0(θ, P ) being the positive solution to the implicit equation

h2 − sin2 θ =

(cos θ

1 + h2P− h)2

+

. (4)

Same threshold as the MISO channel


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0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

θ

Thr

esho

ld o

n h sin(θ)

P = 10

P = 1

For θ = 0, h(1 + h2P ) ≤ 0.5 (SISO result)


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Low Interference Regime - Simplified

Theorem

For any P , ifh ≤ sin(θ)

then sum capacity of MISO interference channel is achieved by usingGaussian inputs, transmit beamforming, and treating interference asnoise at receivers.


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Interference Management – Lessons from IT

• Treating interference as noise is optimal as long as it is veryweak (low) – counterpart of very strong interference regime

• Low interference regime can be significant with multipleantennas

• What if interference is not low (but still weak)?

• SISO: Han-Kobayashi scheme with joint decoding in two-user case(one bit optimal [Etkin,Tse,Wang 07])

• Simple schemes that exploit structure in interferencewithout requiring joint decoding?


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References

1 T.S. Han and K. Kobayashi. “A new achievable rate region forthe interference channel.” IEEE Transactions on InformationTheory Jan 1981.

2 R.H. Etkin, D. NC Tse, and H. Wang. “Gaussian interferencechannel capacity to within one bit.” IEEE TransactionsInformation on Theory, Dec 2008.

3 X. Shang, G. Kramer, and B. Chen, “A new outer bound andnoisy-interference sum-rate capacity for the Gaussian interferencechannels, IEEE Trans. on Inform. Theory, Feb. 2009.

4 A. S. Motahari and A. K. Khandani, “Capacity bounds for theGaussian interference channel, IEEE Trans. on Inform. Theory,Feb. 2009.

5 V.S. Annapureddy and V.V. Veeravalli. “Gaussian interferencenetworks: Sum Capacity in the Low Interference Regime and NewOuter Bounds on the Capacity Region,” IEEE Trans. on Inform.Theory, July 2009.

6 V.S. Annapureddy and V.V. Veeravalli. “On the Sum Capacity ofMIMO Interference Channel in the Low Interference Regime.”IEEE Trans. on Inform. Theory, Mar. 2010.


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Part 3: Degrees of Freedom Characterization ofInterference Channels


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Information Theory for IC: State-of-the-art

• Exact characterization• Very hard problem, still open even after > 30 years

• Approximate characterization• Within constant number of bits/sec• Provides some architectural insights

• Degrees of freedom (or multiplexing gain)

DoF = limSNR→∞

sum capacity

log SNR

• Pre-log factor of sum-capacity in high SNR regime• Number of interference free sessions per channel use• Simplest of the three, but can provide useful insight


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K-user (SISO) Interference Channel

W1 W1Tx1 Rx1

W2 W2Tx2 Rx2

W3 W3Tx3 Rx3

How many Degrees of Freedom (DoF)?


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Degrees of Freedom with Orthogonalization

• One active user per channel use• Every user gets an interference free channel once every K channel

uses• DoF per user is 1/K; total DoF equals 1

• Special Case: K = 2• Can easily show that outer bound on DOF equals 1

=⇒ TDMA optimal from DoF viewpoint for K = 2


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Degrees of Freedom for general K

• Outer Bound on DoF [Host-Madsen, Nosratinia ’05]

• There are K(K− 1)/2 pairs and each user appears in (K− 1) pairs• Thus DoF ≤ K/2 or per user DoF ≤ 1/2

• Amazingly, this outer bound is achievable via linear interferencesuppression!

Interference Alignment [Cadambe & Jafar ’08]


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Linear Transmit/Receive Strategies

Interference Channel with Tx/Rx Linear CodingU1 0 00 U2 00 0 U3

†︸︷︷︸

Receive Beams

H1,1 H1,2 H1,3

H2,1 H2,2 H2,3

H3,1 H3,2 H3,3

︸︷︷︸

Channel

V 1 0 00 V 2 00 0 V 3

︸︷︷︸

Transmit Beams

End-to-End matrix is Diagonal =⇒ No Interference!

# streams = Size of the Diagonal matrix


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DoF of Linear Strategies

U1 0 00 U2 00 0 U3

† H1,1 H1,2 H1,3

H2,1 H2,2 H2,3

H3,1 H3,2 H3,3

V 1 0 00 V 2 00 0 V 3

H i,j : NT ×NT block-diagonal matrix

• (Symmetric) MIMO:N = # antennas

• Symbol Extensions (Time or Frequency)T = # symbol extensions

DoF (T ) = (#streams)/T


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Interference Alignment with Symbol Extensions [Cadambe& Jafar ’08]

Tx1

Tx2

Tx3

Rx1

Rx2

Rx3

Hi, j =× 0 00 × 00 0 ×

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

3 Symbol Extensions

4 interference free streams =⇒ PUDoF = 4/9


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Interference Alignment with Symbol Extensions

Tx1

Tx2

Tx3

Rx1

Rx2

Rx3

Hi, j =× 0 00 × 00 0 ×

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

• Set v1a to some fixed direction

• Align v2 with v11 at Rx 3:

H3,2v2 = H3,1v1a

v2 = H−13,2H3,1v1a

• Align v3 with v2 at Rx 1:

H1,3v3 = H1,2v2

v3 = H−11,3H1,2v2

• Align v1b with v3 at Rx 2:

H2,1v1b = H2,3v3

v1b = H−12,1H2,3v3


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Interference Alignment with Symbol Extensions

• Signaling matrix for transmitter `: V ` = [v`1 v`2 · · ·v`m]

• Fix Ik as the interference space at receiver k

• Alignment conditions are:Receiver 1

H1,1V 1 ∩ I1 = ∅H1,2V 2 ⊆ I1

...

H1,KV K ⊆ I1

Receiver 2

H2,1V 1 ⊆ I2

H2,2V 2 ∩ I2 = ∅...

H2,KV K ⊆ I2

. . .

Receiver K

HK,1V 1 ⊆ IKHK,2V 2 ⊆ IK

...

HK,KV K ∩ IK = ∅• For alignment we require m/2 dimensions each for V k and Ik• May not be possible in general


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Asymptotic Interference Alignment

• Enumerate all S∆= K(K − 1) cross-channels with single index:

T = {T i} = {Hk,` : k 6= `}

• Use same signal space V(m) at all Tx’s, defined recursively as:

V(0) = {1}V(m) =

{v,T 1v, . . . ,T Sv : v ∈ V(m−1)

}={T α1

1 T α22 · · ·T αS

S 1 : α1 + α2 + · · ·+ αS ≤ m}

• Dimension of signal space (# symbol extensions):∣∣∣V(m)∣∣∣ =

(m+ S

m

)


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Asymptotic Interference Alignment

• Interference space at Rx k is Ik = ∪ 6=kHk,`V(m) ⊂ V(m+1)

• Desired signal space at Rx k is Hk,kV(m)

• Almost surely, no overlap between desired signal space andinterference space as long as channel coefficients are changingover symbol extensions and are generic

• Also ∣∣V(m)∣∣∣∣V(m+1)∣∣ =

(m+Sm

)(m+1+Sm+1

) =m+ 1

m+ 1 + S

m→∞−→ 1

• Desired signal space gets half the dimensions asymptotically:∣∣Hk,kV(m)∣∣∣∣Hk,kV(m)

∣∣+ |Ik|m→∞−→ 1

2


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Complexity of asymptotic Interference Alignment

# symbol extensions 0 20 40 60 80 100

0.44

0.45

0.46

0.47

0.48

0.49

0.5

PUDoF

PUDoF of 0.5 is achieved asymptotically


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Complexity of asymptotic Interference Alignment

# symbol extensions

PUDoF

0 200 400 600 800 1000 1200 14000.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.53 User

4 User

[Choi, Jafar, and Chung, ’09]


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Interference Alignment Summary

+ Achieves optimal PUDoF for fully connected channel

- Requires global channel state information (CSI)

- Requires large number of symbol extensions


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References

1 V.R. Cadambe and S.A. Jafar. “Interference Alignment andDegrees of Freedom of the K-User Interference Channel.” IEEETrans. Inform. Theory, August 2008.

2 S. A. Jafar. “Interference Alignment – A New Look at SignalDimensions in a Communication Network.” In Foundations andTrends in Communications and Information Theory, NOWPublications, 2010.

3 A. Host-Madsen and A. Nosratinia, “The multiplexing gain ofwireless networks,” in Proceedings of ISIT, 2005.

4 S.W. Choi, S.A. Jafar, and S.-Y. Chung. “On the beamformingdesign for efficient interference alignment.” IEEE CommunicationsLetters, 2009.


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Part 4: Finite Diversity and Iterative Algorithms


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Finite Diversity and Interference alignment

• Asymptotic interference alignment schemes provide degree offreedom gains by letting number of symbol extensions go toinfinity

• In practice, we only have finite number of symbol extensions fromdiversity in time, frequency, or multiple antennas

• How do we apply idea of interference alignment to finite diversitycase?


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Outline

• Look at two cases of IA with finite diversity

1 Spatial diversity through multiple antennas2 Time and frequency diversity

• Provide theoretical guarantees for DoF using IA schemes

• Iterative algorithms inspired by IA for MIMO IC


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Constant MIMO Channel

• Fully connected K-user MIMO interference channel

• N (k)t transmit antennas and N

(k)r receive antennas for user k

• Usual channel model for receiver k given by

Y k =

K∑j=1

HkjXj + Zk

• Y k - signal at receiver k (N(k)r × 1)

• Hkj - channel from transmitter j to receiver k (N(k)r ×N (j)

t )

• Xj - signal sent by transmitter j (N(j)t × 1)

• Zk - noise at receiver k (N(k)r × 1)

• Channels {Hkj} drawn once from continuous joint distributionhence name constant MIMO channel


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Constant MIMO Channel

Tx1 Rx1

Tx2 Rx2

Tx3 Rx3


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Interference Alignment for Constant MIMO Channel

• Consider linear transmit and receive strategies:

U†kY k = U†k

K∑j=1

Hkj Vjxj︸︷︷︸Xj

+U†kZk

• Transmitter constructs channel input Xj = Vjxj with Vj size

N(j)t × dj

• Receiver processes channel output U†kY k with Uk size N

(k)r × dk

• IA for Constant MIMO IC: Construct {Vk} and {Uk} such thatreceivers can zero force interference yielding interference freechannels between each Tx/Rx pair k


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• Goal: to create dk interference free streams between Tx/Rx pair k

• To zero force interference, we need

U†kHkjVj = 0 ∀ k 6= j

• Zero forcing all interference at Rx k yields effective channel

U†kY k = U†kHkkVkxk + U†kZk

• This is MIMO channel with channel matrix U†kHkkVk

• To create dk interference free channels, we need

rank(U†kHkkVk

)= dk


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• For IA in constant MIMO channel with dk interference freechannels between Tx/Rx pair k, we need

rank(U†kHkkVk

)= dk ∀k

U†kHkjVj = 0 ∀k 6= j

• Choose transmit and receive vectors {Vk} and {Uk} using onlyinterfering channels {Hkj}k 6=j

• Combined with our assumption that {Hkj} are drawn from jointdistribution, (almost surely) it holds that

rank(U†kHkkVk

)= dk

Rank condition is automatic


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Solving IA Equations

• We only need to solve IA equations


When is this possible?

• Feasibility of solving IA equations will depend on K, {N (k)t },

{N (k)r }, and {dk}

• Easy to derive necessary conditions for solving IA equations usingvariable and equation counting [Yetis10,Razaviyayn12,Gonzalez12]

• For some specific system configurations, possible to derivesufficient conditions using algebraic geometry (branch of mathfocused on answering when polynomial equations have solutions)


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Symmetric Constant MIMO IC - Necessary

• To illustrate necessary condition, consider symmetric constantMIMO IC:

1 N(k)t = N

(k)r = N

2 dk = d

• In order to solve IA equations, number of variables should be noless than number of equations

• This intuitive idea can be proved rigorously using algebraicgeometry or algebraic field theory

• The no interference equation for Tx j at Rx k

U†kHkjVj = 0

consists of d2 equations

• There are total of K(K − 1)d2 equations to ensure no interference


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• Counting number of variables is little bit trickier

• Satisfying IA equations only depends on subspaces {span(Uk)}and {span(Vk)} and not particular choice of {Uk} and {Vk}

• To illustrate this point, suppose that U†kHkjVj = 0 ∀k 6= j

• For any other basis {Vj} and {Uj} of {span(Uk)} and{span(Vk)}, there exist d× d full rank matrices {Qk} and {Pk}such that

Vj = VjQj

Uk = UkPk

• Then we have U†kHkjVj = P†kU†kHkjVj︸︷︷︸

=0

Qj = 0


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• Since satisfying IA equations only depends on subspaces spannedby {Vk} and {Uk}, number of variables is less than 2K ×Nd

• By preceding argument, due to freedom in choosing Pk and Qk infact we have

dim (span (Vk)) = dim (span (Uk)) = Nd− d2 = (N − d)d

• This implies that number of variables is 2K × d(N − d)

• Necessary condition for IA in symmetric constant MIMO ICbecomes

(K + 1)d ≤ 2N

• This implies that

DoF ≤ K⌊

2N

K + 1

⌋≤ 2N

K

K + 1≤ 2N


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Solving IA Equations - Feasability

• This necessary condition is not always sufficient [Yetis10]

• N(k)t = N

(k)r = 3, dk = 2, and K = 2 system satisfies necessary

condition but is not achievable

• For symmetric constant MIMO channel with K ≥ 3, necessarycondition (K + 1)d ≤ 2N is sufficient [Bresler14]

• Necessary and sufficient conditions for feasibility of IA for otherconfigurations too using similar ideas[Razaviyayn12,Ruan13,Gonzalez12]

• Conditions for feasibility of solving IA equations for constantMIMO IC not known in general


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Finite Time and Frequency Diversity

• To model finite diversity, suppose that we have L symbolextensions, equivalently a diagonal channel matrix

Hkj =

Hkj(1) 0. . .

0 Hkj(L)

• For IA in K-user IC, L ≥ 2K

2sufficient [OzgurTse09]

• Three user case has been exactly characterized exactly usingvector IA achievable scheme [BreslerTse09]

DoF =3

2

(1− 1

4L− 2bL/2c − 1

)= O

(3

2

(1− 1

L

))• As L→∞, recover usual 3

2 DoF


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Finite Diversity and IA

• For K ≥ 4 case with finite diversity, there are sum DoF upperbounds [LiOzgur14]

DoF ≤ K

2

(1− 1

11√L

)• Achievable scheme yields

DoF ≥ K

2

(1− C1

C2√L/2

)

with C1 a constant and C2 = (K − 1)(K − 2)− 1

• Substantial gap between upper bound and achievable scheme

• As L→∞, recover usual K2 DoF


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Time/Frequency vs. Spatial Diversity

• For time and frequency diversity, when amount of diversity (L) islarge, DoF ≈ K

2

• DoF gains for time/frequency diversity scale with K but forK ≥ 4 may need large L to get close to K

2• Coherence time of channel can be issue for large L

• For spatial diversity in symmetric constant MIMO IC, DoF ≈ 2N

• DoF gains for spatial diversity do not scale with K but no need tocode over large number of time/frequency slots

• Finally, for time-varying symmetric MIMO channel with Nantennas at all Tx/Rx [CadambeJafar08]

DoF =KN

2


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IA Inspired Algorithms for Constant MIMO IC

• Examined when we can solve IA equations

rank(U†kHkkVk

)= dk ∀k


• Analysis does not tell us how to actually solve IA equations

• Iterative algorithms based on idea of IA for constant MIMO IC

• Channel State Information: Assume each transmitter and receiverknows all connected channels


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IA Inspired Algorithms for Constant MIMO IC

• Look at two algorithms inspired by IA [Gomadam08]

• Designing transmit vectors to minimize interference is difficultbecause each transmit vector affects interference at all receivers

• Similarly, designing transmit vectors to maximize sum rate isnon-convex problem and difficult to solve

• In contrast, each receive vector is affected only by interferenceseen at receiver and is easy to design

• Idea of both algorithms is to only design receive vectors butalternate the direction of communication

• Role of transmit and receive vectors alternates• Appropriate for TDD systems with natural reversal of directions of

communication


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Constant MIMO IC Algorithm Overview

Tx1 Rx1

Tx2 Rx2

Tx3 Rx3

−→V1(n)−→V1(n)

−→U1(n)←−V1(n)

←−U1(n)

←−V1(n)

−→V1(n+ 1)

−→V2(n)−→V2(n)

−→U2(n)←−V2(n)

←−U2(n)

←−V2(n)

−→V2(n+ 1)

−→V3(n)−→V3(n)

−→U3(n)←−V3(n)

←−U3(n)

←−V3(n)

−→V3(n+ 1)

Forward Direction - Design Receive Vectors Reverse Direction ofCommunication: ←−Vk(n) =

−→Uk(n) Reverse Direction - Design Receive

Vectors Reverse Direction of Communication and Repeat:−→Vk(n+ 1) =

←−Uk(n)

−→←−ECE Illinois & Purdue

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Min Leakage Algorithm

• For fixed transmit vectors {Vk}, interference leakage power at

receiver k given by Ik = trace(U†kQkUk) with

Qk , P

K∑j=1j 6=k

HkjVjV†jH†kj

• If Ik = 0, then we achieve IA

• To minimize interference leakage power, choose dk least dominanteigenvectors of Qk, i.e, those eigenvectors corresponding to dksmallest eigenvalues of Qk

• Alternate directions reversing role of transmit and receive vectors

• Sum interference leakage power converges


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Min Leakage Algorithm

1 Start with arbitrary {−→Vk(0)}2 Set

−→Uk(n) to be dk least dominant eigenvectors of

−→Qk(n)

3 Reverse direction of communication

←−Vk(n) =

−→Uk(n)

4 Set←−Uk(n) to be dk least dominant eigenvectors of

←−Qk(n)

5 Reverse direction of communication and repeat

−→Vk(n+ 1) =

←−Uk(n)


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Max SINR Algorithm

• At high SNR, Min Leakage provides good performance -interference limited regime

• At lower SNR, Min Leakage does not provide good performance -noise becomes more important

• Max SINR same form as Min Leakage algorithm but with MMSEreceive vectors

Uk =

I + P

K∑j=1j 6=k

HkjVjV†jH†kj

−1

HkkVk


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Max SINR Algorithm

• At high SNR, MMSE filter becomes zero forcing filter - focuses oninterference

• At low SNR, MMSE filter becomes matched filter - focuses onnoise

• At intermediate SNR, MMSE filter trades off between interferenceand noise

• Max SINR has better performance at low SNR than Min Leakageor other purely IA oriented algorithms

• No proof of convergence but appears to converge in practice


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Performance Comparison

SNR Per User [dB]0 5 10 15 20 25 30 35 40

Ave

rage

Sum

Rat

e [b

its/u

se]

0

10

20

30

40

50

60

70Max SINRMin Leakage

Similar performance at high SNR Max SINR much better at low SNR


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Other Algorithms

• Min Leakage and Max SINR make sense in TDD system due tonatural alternation of transmitting and receiving

• IA Algorithms for FDD channels have been developed[PetersHeath09]

• Similar in spirit to Min Leakage algorithm but allows for FDD

• Convergent version of Max SINR [WilsonVVV11]• Power control step to produce non-decreasing sum rate• Same performance as Max SINR in simulations

• Several other similar iterative algorithms inspired by IA inliterature


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References

1 C. Yetis, T. Gou, S. Jafar, and A. Kayran, “On feasibility ofinterference alignment in MIMO interference networks,” IEEETrans. Signal Process., 2010.

2 M. Razaviyayn, G. Lyubeznik, and Z.Q. Luo, “On the Degrees ofFreedom Achievable Through Interference Alignment in a MIMOInterference Channel,” IEEE Trans. Signal Process., 2012

3 O. Gonzalez, C. Beltran, and I. Santamaria, “A feasibility test forlinear interference alignment in MIMO channels with constantcoefficients,” IEEE Trans. on Info. Theory, 2014

4 G. Bresler, D. Cartwright, and D. Tse, “Feasibility of InterferenceAlignment for the MIMO Interference Channel,” in IEEE Trans.Info. Theory, 2014

5 L. Ruan, V.K.N. Lau, M.Z. Win, “The feasibility conditions forinterference alignment in MIMO networks,” IEEE Trans. SignalProcess., 2013

6 A. Ozgur and D. Tse, “Achieving Linear Scaling with InterferenceAlignment”, ISIT, 2009

7 G. Bresler and D. Tse, “Degrees-of-freedom for the 3-userGaussian interference channel as a function of channel diversity”,Allerton Conf., 2009


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References

8 V. Cadambe and S. Jafar, “Interference alignment and degrees offreedom of the-user interference channel,” IEEE Trans. Info.Theory, 2008

9 C.T. Li, and A. Ozgur, “Channel Diversity needed for VectorInterference Alignment”, on arXiV, 2014

10 K. Gomadam, V. R. Cadambe, and S. A. Jafar, “Approaching thecapacity of wireless networks through distributed interferencealignment,” in GLOBECOM, 2008

11 S. W. Peters and R. W. Heath, “Interference alignment viaalternating minimization, ICASSP, 2009

12 C. Wilson and V. Veeravalli, “A Convergent Version of the MaxSINR Algorithm for the MIMO Interference Channel,” IEEETrans. Wireless Comm., 2011

13 C. Wilson and V. Veeravalli, “Degrees of Freedom for theConstant MIMO Interference Channel with CoMP Transmission,”IEEE Trans. Comm., 2014


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Part 5: Coordinated Multi-Point Transmission


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K-User Interference Channel

Channel State Information known at all nodes.

W1 W1Tx1 Rx1

W2 W2Tx2 Rx2

W3 W3Tx3 Rx3


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Coordinated Multi-Point (CoMP) Transmission

Messages are jointly transmitted using multiple transmitters.

W1 W1Tx1 Rx1

W2 W2Tx2 Rx2

W3 W3Tx3 Rx3


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CoMP Transmission

• Each message is jointly transmitted using M transmitters

• Message i is transmitted jointly using the transmitters in Ti• For all i ∈ [K], |Ti| ≤M

• We consider all message assignments that satisfy the cooperationconstraint


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Degrees of Freedom (DoF)

DoF = limSNR→∞

sum capacity

log SNR

Objective: Determine the DoF as a function of K and M

PUDoF(M) = limK→∞

DoF(K,M)

K

Is PUDoF(M)>PUDoF(1) for M > 1?


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Example: Two-user Interference Channel

W1 W1Tx1 Rx1

W2 W2Tx2 Rx2

No Cooperation, DoF=1, Time Sharing

Full Cooperation, DoF=2, ZF Transmit Beamforming


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No Cooperation (M = 1)

• For M = 1, outer bound = K/2

• The outer bound can be achieved by jointly coding across multipleparallel channels [Cadambe & Jafar ’08]:

DoF(K,M = 1) = limL→∞

DoF(K,M = 1, L)

L= K/2

where L is the number of parallel channels

Corollary

Without cooperation, the Per User DoF number is given by

PUDoF(M = 1) =1

2


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Full Cooperation (M = K)

• In this case, the channel is a MISO Broadcast channel.

• Each message is available at K antennas, and hence, can becanceled at K − 1 receivers.

• Each user achieves 1 DoF,

DoF(K,M = K) = K.

What happens with partial cooperation (1 < M < K)?


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Clustering

W1 W1Tx1 Rx1

W2 W2Tx2 Rx2

W3 W3Tx3 Rx3

W4 W4Tx4 Rx4

No Degrees of Freedom Gain


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Spiral Message Assignment

Ti = {i, i+ 1, . . . , i+M − 1}

W1 W1Tx1 Rx1

W2 W2Tx2 Rx2

W3 W3Tx3 Rx3


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Spiral Assignment: Matrix Interpretation


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Spiral Assignment: Matrix Interpretation

M : # of non-zero blocks in the columns of V


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Example: K = 3,M = 2

PUDoF= 23


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Spiral Message Assignment: Results

Theorem

The DoF of interference channel with a spiral message assignmentsatisfies

K +M − 1

2≤ DoF(K,M) ≤

⌈K +M − 1

2

⌉

Proof of Achievability: First M − 1 users are interference-free, andinterference occupies half the signal space at each other receiver

Generalizes the Asymptotic Interference Alignment scheme


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Vector Space Interference Alignment [Cadambe-Jafar ’08]

• Asymptotic interference alignment works only if the (Signal +Interference) matrix is full rank

• The generic channel coefficients assumption (they have a jointpdf) is crucial

• With CoMP, the same point to point channel can carry bothdesired and interfering signals

• It is not clear whether the generic channel coefficients assumptionsuffices to prove full rankness of the (Signal + Interference) matrix


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Key Tool: Algebraic Independence

Consider the system of polynomial equations

s1 = f1(t1, t2, · · · , tn)

s2 = f2(t1, t2, · · · , tn)

...

sm = fm(t1, t2, · · · , tn).

Definition

The polynomials f1, f2, · · · , fm are said to be algebraically dependentif and only if there exists an annihilating polynomial F (s1, s2, · · · , sm)such that F (f1, f2, · · · , fm) = 0.


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Jacobian Criterion

The polynomials f1, f2, · · · , fm are algebraically independent if andonly if the Jacobian matrix(

∂fi∂tj

)1≤i≤m,1≤j≤n

has full structural row rank equal to m.


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Algebraic Independence: Applications

Consider the system of polynomial equations

s1 = f1(t1, t2, · · · , tn)

s2 = f2(t1, t2, · · · , tn)

...

sm = fm(t1, t2, · · · , tn).

Theorem

If the variables {t1, . . . , tn} have a continuous joint pdf and thepolynomials f1, f2, . . . , fm are algebraically independent, then thevariables {s1, . . . , sm} have a continuous joint pdf.


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Outline of the Achievable Scheme

OriginalChannel

ZFEncoder

IAEncoder

IADecoder

Derived Channel

Approach:

1 ZF Step: Exploit cooperation to transform the interferencechannel into a derived channel (with single-point transmission)

2 IA Step: Use the known IA techniques to design beams forderived channel

3 Prove that the asymptotic IA step works for generic channelcoefficients


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Example: M = 2

12 DoF

12 DoF

W(1)1

W(2)1

X1 Y1 1 DoF

12 DoF W

(1)2

X2 Y212 DoF

12 DoF W

(1)3

X3 Y312 DoF

h1,1

h2,2

h3,3


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Derived Channel Example: M = 2

X(1)1

X(2)1

Y1 1 DoF

X(1)2

Y212 DoF

X(1)3

Y312 DoF

g(1)1,1

g(2)1,1

g(1)2,2

g(1)3,3

Asymptotic Interference Alignment is used to pack the interference inhalf the signal space at Rx 2 and 3.


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Asymptotic IA for Derived Channel

• (Signal + Interference) matrix has full column rank if the

variables {g(m)i,j } have a continuous joint pdf

• {g(m)i,j } have a continuous joint pdf because {hi,j} have a

continuous joint pdf and the polynomials defining thetransformations from original to derived channel coefficients arealgebraically independent

Prove algebraic independence using Jacobian Criterion


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DoF Outer Bound

W1 W1Tx1 Rx1

W2 W2Tx2 Rx2

W3 W3Tx3 Rx3

W4 W4Tx4 Rx4

W5 W5Tx5 Rx5


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DoF Outer Bound

W1 W1

W2 W2

W3 W3

W4 W4

W5 W5

Tx1

Tx2

Tx3

Tx4

Tx5

Rx1

Rx2

Rx3

Generic channel coefficients → DoF ≤ 3


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DoF Outer Bound: Bipartite Graph Representation

W1

W2

W3

W4

W5

Tx1

Tx2

Tx3

Tx4

Tx5

|Neighborhood({Tx1,Tx2})| = 3 ≥ K − |{Tx1,Tx2}| =⇒ DoF ≤ 3


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Theorem

DoF ≤ minS⊆{1,...,K}

max (|Neighborhood(S)|,K − |S|)


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Definition

DoFout(K,M) = max{Ti}

minS⊆{1,...,K}

max(|Neighborhood(S)|,K − |S|)

PUDoFout(M) = limK→∞

DoFout(K,M)

K

DoF(K,M) ≤ DoFout(K,M)

PUDoF(M) ≤ PUDoFout(M)


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DoF Outer Bound: Results

Definition

We say that a message assignment satisfies a local cooperationconstraint if and only if ∃r(K) = o(K), and for all K−user channels,

Ti ⊆ {i− r(K), i− r(K) + 1, . . . , i+ r(K)},∀i ∈ [K]

Theorem

With the restriction to local cooperation,

PUDoFloc(M) =1

2

Local cooperation cannot achieve a scalable Dof gain


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DoF Outer Bound: Results

Theorem

For M ≥ 2,

PUDoF(M) ≤ PUDoFout(M) ≤ M − 1

M

Corollary

PUDoF(2) =1

2

Assigning each message to two transmitters cannot achieve a scalableDoF gain


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Fully Connected IC with CoMP: Summary

• Considered general cooperation constraint that limits the numberof transmitters at which each message can be available

• Asymptotic Interference Alignment can be used to achieve DoFgains

• Dual results can be obtained with receiver cooperation butrequires sharing of analog received signals

• Iterative Max SINR type algorithms algorithms can be designed inconstant MIMO channel setting [WilsonVVV-14]

• Scalable DoF gains cannot be achieved through:

• Local Cooperation

• Arbitrary assignment of each message to two transmitters


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References

1 P. Marsch and G. P. Fettweis “Coordinated Multi-Point in MobileCommunications: from theory to practice,” First Edition,Cambridge, 2011.

2 A. Host-Madsen and A. Nosratinia, “The multiplexing gain ofwireless networks,” in Proc. IEEE International Symp. Inf. Theory(ISIT), 2007.

3 V. Cadambe and S. A. Jafar, “Interference alignment and degreesof freedom of the K-user interference channel,” IEEE Trans. Inf.Theory, 2008.

4 V. S. Annapureddy, A. El Gamal, and V. V. Veeravalli, “Degreesof Freedom of Interference Channels with CoMP Transmission andReception,” IEEE Trans. Inf. Theory, 2012.

5 A. El Gamal, V. S. Annapureddy, and V. V. Veeravalli, “OnOptimal Message Assignments for Interference Channels withCoMP Transmission,” in Proc. CISS, 2012

6 C. Wilson and V. Veeravalli, “Degrees of Freedom for theConstant MIMO Interference Channel with CoMP Transmission,”IEEE Trans. Comm., 2014


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Part 6: Locally Connected Networks


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Locally Connected Model

Tx i is connected to receivers {i, i+ 1, . . . , i+ L}.

Wyner Model: L =1

Rx1 Tx1

Rx2 Tx2

Rx3 Tx3

Rx4 Tx4

Tx5 Rx5

L = 2

Rx1 Tx1

Rx2 Tx2

Rx3 Tx3

Rx4 Tx4

Tx5 Rx5


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Results for Wyner Model [Lapidoth-Shamai-Wigger ’07]

Rx2 Tx2

Rx3 Tx3

Rx4 Tx4

Tx1 Rx1

Rx5 Tx5

Rx6 Tx6

W2

W3

W4

W1

W5

W6

Backhaul load factor =1

PUDoF (L=1,M=2) = 2/3 > 1/2

W1

W2

W4

W5


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Results for Wyner Model [Lapidoth-Shamai-Wigger ’07]

• Spiral transmit sets: Ti = {i, i+ 1, . . . , i+M − 1}

• PUDoF(L = 1,M) = MM+1

Backhaul load factor = M2

• Local cooperation can achieve PUDoF gains for locally connectedchannels

• Achievable scheme relies on only:• Zero-forcing transmit beamforming• Local CSI• Fractional reuse

Is spiral message assignment optimal?


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Example: No Cooperation

PUDoF(L =1,M =1) = 1

2 PUDoF(L =1,M =1) = 23

Rx1 Tx1

Rx2 Tx2

Rx3 Tx3

W1

W2

W3

Rx1 Tx1

Rx2 Tx2

Rx3 Tx3

W1

W2

W3

Interference-aware message assignment + Fractional reuse


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Locally Connected IC with CoMP: Main Result

Theorem

Under the general cooperation constraint |Ti| ≤M, ∀i ∈ {1, 2, . . . ,K},

2M

2M + L≤ PUDoF(L,M) ≤ 2M + L− 1

2M + L

and the optimal message assignment satisfies a local cooperationconstraint.

Corollary

PUDoF(L = 1,M) =2M

2M + 1


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DoF Achieving Scheme

Rx2 Tx2

Rx3 Tx3

Rx4 Tx4

Tx1 Rx1

Rx5 X5

W2

W4

W1

W2

W4

W1

W5

W3

W5

Backhaul load factor =6/5 PUDoF (L=1,M=2) = 4/5 > 2/3


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DoF Outer Bound

Have to consider all possible message assignments satisfying|Ti| ≤M, ∀i ∈ [K]

1 First simplify the combinatorial aspect of the problem byidentifying useful message assignments

2 Then derive an equivalent model with fewer receivers and sameDoF


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DoF Outer Bound: Useful Message Assignments

An assignment of a message Wx to a transmitter Ty is useful only ifone of the following conditions holds:

1 Signal delivery: Ty is connected to the designated receiver Rx

2 Interference mitigation: Ty is interfering with anothertransmitter Tz, both carrying the message Wx


156 / 177


Tx1 Rx1

Tx2 Rx2

Tx3 Rx3

Tx4 Rx4

W3 W3

Assigning W3 to Tx1 is not useful.


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Corollary

An assignment of a message Wx to a transmitter Ty is useful only ifthere exists a chain of interfering transmitters carrying Wx thatincludes Ty and another transmitter Tz that is connected to Rx

Proves optimality of local cooperation


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Extensions: Multiple-Antenna Transmitters

Theorem

For the locally connected interference channel with N−antennatransmitters, if MN ≥ L+M then PUDoF(L,M,N) = 1, otherwise,

2MN

M(N + 1) + L≤ PUDoF(L,M,N) ≤ M(N + 1) + L− 1

M(N + 1) + L

and the optimal message assignment satisfies a local cooperationconstraint.


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CoMP Transmission for IC: Summary

• Local Cooperation• no PUDoF gain for fully connected channel• is optimal for locally connected channel

• Interference aware message assignments allow for higherthroughput

• Fractional reuse and zero-forcing transmit beam-forming aresufficient to achieve PUDoF gains, without need for symbolextensions and interference alignment


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References

1 V. S. Annapureddy, A. El Gamal, and V. V. Veeravalli, “Degreesof Freedom of Interference Channels with CoMP Transmission andReception,” IEEE Trans. Inf. Theory, 2012.

2 A. Lapidoth, N. Levy, S. Shamai (Shitz) and M. A. Wigger“Cognitive Wyner networks with clustered decoding,” IEEE Trans.Inf. Theory 2014

3 A. Wyner, “Shannon-theoretic approach to a Gaussian cellularmultiple-access channel,” IEEE Trans. Inf. Theory, 1994.

4 S. Shamai and M. A. Wigger, “Rate-limited TransmitterCooperation in Wyner’s Asymmetric Interference Network,” inProc. IEEE Int. Symp. Inf. Theory, 2014


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Part 7: Cellular Networks and Backhaul Load Constraint


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Backhaul Load Constraint

More natural cooperation constraint that takes into account overallbackhaul load: ∑

i∈[K] |Ti|K

≤ B

Solution under transmit set size constraint |Ti| ≤M, ∀i ∈ [M ], can beused to provide solutions under backhaul load constraint


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Wyner’s Model with Backhaul Load Constraint

Rx1 Tx1

Rx2 Tx2

Rx3 Tx3

Theorem

Under cooperation constraint∑i∈[K] |Ti| ≤ BK,

PUDoF(B) =4B − 1

4B

Recall that |Ti| ≤M,∀i ∈ [K]⇒ PUDoF(M) = 2M2M+1


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Coding Scheme: B = 1

Rx1 Tx1

Rx2 Tx2

Rx3 Tx3

W1

W2

W3

B = 2

3 PUDoF = 2

3

B = 6

5 PUDoF = 4

5

3K

8users

5K

8users

PUDoF (B =1) = 3

4

Rx2 Tx2

Rx3 Tx3

Rx4 Tx4

Tx1 Rx1

Rx5 X5

W2

W4

W1

W3

W5


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Application in Denser Networks

Tx i is connected to receivers {i, i+ 1, . . . , i+ L}.

Rx1 Tx1

Rx2 Tx2

Rx3 Tx3

Rx4 Tx4

Tx5 Rx5

L = 2

Result: Using only zero-forcing transmitbeamforming and fractional reuse:

PUDoF(L,B = 1) ≥ 1

2,∀L ≤ 6.

without need for interference alignmentand symbol extensions


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Application in Denser Networks

PUDoF(M = 1) = 12 PUDoF(B = 1) ≥ 5

9


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Interference in Cellular Networks

Locally (partially) connected interference channel!


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Interference Graph for Single Tier

Tx,Rx pair

Each node represents a Tx-Rx pair


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No Intra-sector Interference


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Partition into Noninterfering Tx-Rx Pairs


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Full Cooperation (M = 6)

Rx2 Tx2

Rx3 Tx3

Rx4 Tx4

Tx1 Rx1

Rx5 Tx5

Rx6 Tx6

W2

W3

W4

W1

W5

W6

2

3 4 5

6

1

B = 6 × 6

9= 4; PUDoF = 6

9= 2

3


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Under-utlizing Backhaul Resources (M = 2)

Rx2 Tx2

Rx3 Tx3

Rx4

Tx1

Rx5

Tx6 Rx6

Tx5

W3

Tx4

W5

W2

W4

W6

Rx1 W1

1

2

3 4 5

6

B = 6

9= 2

3; PUDoF = 4

9


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No Extra Backhaul Load

Backhaul Load B = 1, PUDoF= 715


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Discussion: Cooperation through Backhaul

• Similar gains in DoF for other cellular interference models, withonly zero-forcing and fractional reuse

• Gains improve with asymmetric cooperation and interferenceaware message assignment

• Gains in DoF can also be obtained for uplink with decodedmessages being exchanged through backhaul [Ntranos et al ’14]

• Requires multiple antennas at both mobiles and basestations• For same backhaul load factor, gain is smaller than on downlink

with Tx cooperation


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Summary

• Infrastructure enhancements in backhaul can be exploited throughcooperative transmission to lead to significant rate gains

• Minimal or no increase in backhaul load• Fractional reuse and zero-forcing transmit beam-forming are

sufficient to achieve rate gains• No need for symbol extensions and interference alignment

• Open Questions:• Partial/unknown CSI• Network dynamics and robustness to link erasures• Joint design with message passing schemes for uplink


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References

1 A. El Gamal and V. V. Veeravalli, “Flexible Backhaul Design andDegrees of Freedom for Linear Interference Channels,” in Proc.IEEE Int. Symp. Inf. Theory, 2014.

2 M. Bande, A. El Gamal, and V. V. Veeravalli, “Flexible BackhaulDesign with Cooperative Transmission in Cellular InterferenceNetworks,” in Proc. IEEE Int. Symp. Inf. Theory, 2015

3 V. Ntranos, M. A. Maddah-Ali, and G. Caire, “CellularInterference Alignment,” arXiv, 2014.

4 V. Ntranos, M. A. Maddah-Ali, and G. Caire, “OnUplink-Downlink Duality for Cellular IA,” arXiv, 2014.


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We are writing a book!

A. El Gamal, V. V. Veeravalli “Cellular Interference Management:Flexible Backhaul Design and Cooperation”, Artech House, 2016


Interference Management in Wireless Networks · 2015-08-12 · IS-95 standard) Interference from...

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