“Crystallization” in Large Fading Networks

Veniamin Morgenshternjoint work with Helmut Bolcskei

1

An Interference Network with Relays

K Relay Terminals

Source 2

Source 1

Source M

Destination M Destination 1

Destination 2

• Relays have no traffic requirements

• No direct links between sourcesand destinations

• Single-antenna transceivers

• No cooperation between sourcesand between destinations

• Zone free of relays around eachsource and destination

• Bounded area

2

Cut-Set Upper Bound on Network Capacity

Source 2

Source 1

Source M

Destination M

Destination 1

Destination 2

Sc

S• Max-flow min-cut theorem∑i∈S,j ∈Sc

R(i,j) ≤ I(X(S);Y(Sc)|X(Sc)

)

yields (for largeK)

C ≤ M

2log(K) +O(1)

• This bound is achieved withcooperation in a MIMO system

3

Motivation

Question: Can we achieve the cut set bound without cooperation?

Yes: [Bolcskei, Nabar 2004] show a protocol that

• ForM fixed andK →∞ realizes distributed orthogonalization

• C = (M/2) log(K) +O(1)

4

System and Channel Model

.

.

.

...

...

...

Destination terminals

Source terminals Relay terminals

First hop Second hop

S1

S2

SM

R1

R2

RK

D1

D2

DM

hk,m fm,k

Pm,kEk,m

s1

s2

sM

r1

r2

rK

t1

t2

tK

y1

y2

yM

5

System and Channel Model Cont’d

• rk =∑Mm=1Ek,mhk,msm + zk

• ym =∑Kk=1Pm,kfm,ktk + wm

• E ≤ Ek,m ≤ E, P ≤ Pm,k ≤ P ∀k,m

• hk,m, fm,k ∼ CN (0, 1), i.i.d.

• zk, wm ∼ CN (0, σ2), i.i.d.

• Gaussian codebooks are used

• E[|sm|2] ≤ 1/M, E[|tm|2] ≤ Prel/K

6

Protocol 1 (P1) from [Bolcskei, Nabar ’04]

...

S1

S2

SM

Source

terminals

Destination

terminals

.

.

.

D1

D2

DM

Relay terminals

First hop Second hop

...

...

...

...

...

...

...

...

• K relay terminals arepartitioned intoMgroups of equal size

⇒ K/M relays in eachgroup

• Each group is assignedto one S-D pair

• Each relay knowsphases of its assignedbackward and forwardchannels

7

Smart Scattering

...

S1

S2

SM

.

.

.

D1

D2

DM

h1

h2

hM fM

f2

f1

MF MF

Scalingh!

1 = e"j arg(h1) f!

1 = e"j arg(f1)

Distributed multi-stream separation through smart scatterersperforming matched-filtering

8

Capacity Scaling for largeK and fixedM• For fixedM andK → ∞, lower bound approaches upper bound and

the network capacity converges (w.p.1) to

C =M

2log(K) +O(1)

• Asymptotically inK cooperation between destination terminals isnot needed to achieve network capacity

Questions:

• Is distributed orthogonalization possible if bothM,K →∞?

• If so, howK should scale withM?

9

[VM et al., 2005]: K should grow asM3

ForM,K →∞, the per S-D pair capacity scales as

CP1 =1

2log

(1 + Θ

(K

M3

))

Question: Is there protocol which requires less relays to realizedistributed orthogonalization?

10

First...

Proof Techniques

11

I-O relation of Sm→ Dm link

Independent decoding. I-O relation of Sm → Dm link is written as

ym = sm

K∑k=1

am,mk︸ ︷︷ ︸effective channel gain (gm)

+∑m6=m

sm

K∑k=1

am,mk︸ ︷︷ ︸interference (im)

+K∑k=1

bmk zk + wm√K︸ ︷︷ ︸

noise (nm)

where

am,mk ∼ f∗p(k),k fm,k h∗k,p(k) hk,m

p(k) = m iff relay k servesmth S-D pair

12

Lower Bound

• I-O relation of Sm → Dm link

ym = E[gm]sm + (gm − E[gm])︸ ︷︷ ︸gm

sm + im + nm︸ ︷︷ ︸wm

– effective channel gain has non-zero mean, i.e., E[gm] > 0

– zero-mean wm is not Gaussian– wm and gm are not statistically independent

• Slight modification of a technique from [Medard, 2000] yields

I(ym; sm) ≥ log

(1 +

(E[gm])2

Var[gm] + Var[wm]

)

13

Outage Analysis: “Crystallization”• Each destination terminal knows fading coefficients in entire network

• I-O relation of Sm → Dm link given by

ym = gmsm + im + nm

• Conditioned on {hk,m, fm,k}∀m,k interference im and noise nm areGaussian

⇒ Im =1

2log(

1 + SINRm|{hk,m,fm,k})

whereSINRm|{hk,m,fm,k} =

|gm|2

σ2i + σ2

n

Goal: Analyze behavior of the random variable SINR whenM,K →∞

14

Proof Techniques for Concentration Results

SINR of Sm → Dm link given by

SINR =

∣∣∣∑k:p(k)=m am,mk +

∑k:p(k)6=m a

m,mk

∣∣∣2∑m6=m

∣∣∣∑Kk=1 a

m,mk

∣∣∣2 + σ2M∑Kk=1 |bmk |

2+KMσ2

• Consider each term in the numerator and denominator separately

• Use Chernoff bound to estimate large deviations from mean

– gives asymptotically tight results– independence of summands is required, which is not the case fora’s and b’s

15

Main Tool: Truncation Lemma (thanks to O. Zeitouni)Have to deal with sums of the form SN =

∑Ni=1AiXiφi, where

• {Xi}∞i=1 (not necessarily independent) with common cdf FX

• i.i.d. {φi}∞i=1 (−1 ≤ φi ≤ 1)

• positive and uniformly bounded coefficients {Ai}∞i=1 (0 ≤ Ai ≤ A∗)

• for all x ≥ x0 > 0, we have 1− FX(x) + FX(−x) ≤ Ae−αxβ

Then, for allN and t such that δ2 ≥ x0

P{|SN − E{SN}| ≥

√Nδ}≤ 2 exp

{−2δ2β/(β+2)

(A∗)2γ

}+NA exp

{−αδ2β/(β+2)

}

16

SINR is in Narrow Interval Around Mean with High Prob.Theorem 1. There exist constants C1, C2, C3, C4, C5, C6,M0 andK0 suchthat for anyM ≥ M0 andK ≥ K0 for any x > 1, the probability Pout,P1

of the event SINRP1 /∈ [LP1, UP1], where

LP1 =π2

16

P E2

P E2

(max

[0,K − C1M

√Kx])2

M2(M − 1)K + C2M5/2Kx+ C3M3

UP1 =π2

16

P E2

P E2

(K + C4M

√Kx)2

max[0,M2(M − 1)K − C5M5/2Kx] + C6M3

satisfies the following inequality

Pout,P1 ≤ Poly1(M,K)e−∆1x2/7

17

Outage Interpretation

UC

Pout PDF of I

L R

• FixK = M3

• C = 12 log

(1 + π2

16P E2

P E2

)• ChooseR < C

• Choose Pout (gives x)

• ChooseK such thatL = R

18

Proof of Ergodic Capacity Upper Bound

• Upper bound on per S-D pair capacity

C ≤ 1

2log(1 + E {SINR(H,F)})

• Use thatE{X} =

∫ ∞0

x pX(x)dx ≤∞∑n=0

nP {X > n}

• Using the tail behavior result for SINR, we get

P

{SINR > (K + o(K)x)

2

M3K − o(M3K)x

}≤ Poly(M,K) e−∆xβ

19

Network “Crystallization”

• SINRs of effective channels Sm → Dm (m = 1, 2, . . . ,M ) converge todeterministic limit asM,K →∞

• Per-stream diversity order→∞ asM,K →∞

• Individual SISO fading links in the network converge to independentAWGN links (network “crystallizes”)

• The exponent 2/7, which characterizes the speed of convergence, isunlikely to be fundamental

• Theorems 1 can be reformulated to provide bounds on outageprobability

20

Can we do better thanK = M 3?

21

Protocol 2 (P2) from [Dana and Hassibi, 2003]

Source

terminals

Destination

terminalsRelay terminals

First hop Second hop

...

S1

S2

SM

...

D1

D2

DM

......

• No relay partitioning

• Each relay knowsphases of all Mbackward and all Mforward channels

22

P2: Each Relay Assists All S-D Pairs

...

S1

S2

SM

.

.

.

D1

D2

DM

Phase matching with respect toall backward and all forward channels,

normalization to ensure power constraint

kth relay terminal

hk,M

hk,1

hk,2

rk

f1,k

f2,k

fM,k

tk

tk = !k

!

M"

m=1

f!

m,kh!

k,m

#

rk

ForM,K →∞, the per S-D pair capacity scales as

CP2 =1

2log

(1 + Θ

(K

M2

))

23

Conclusions

• Network decouples if rate of growth ofK as function ofM issufficiently fast

• P1 and P2 trade amount of CSI at relays for required rate of growth ofrelays

• The individual Sm → Dm fading links converge to independent AWGNlinks asM,K →∞⇒ Network crystallizes

• Back from infinity: Characterizing “crystallization rate” could serve asa general tool to study large wireless networks

• Network capacity scaling for P2 is√T , where T = 2M +K

24

Cooperation

25

Interference Relay Network with Cooperation at Relays

...

S1

S2

SM

.

.

.

Destination

terminals

Source

terminals

D1

D2

DM

...

...

R1

Relay terminals

...

...

RK

R2

...

...

...

...

First hop Second hop

Ek,l Pl,k

fl,khk,l

1

N

N

N

1

1

N

N

N

1

1

1

26

P1 with Cooperation

...

Phasematching

.

.

.

Phasematching

!k

...

S1

S2

SM

.

.

.

D1

D2

DM

rk tk

tk = !k f!

1,khHk,1rk

kth relay terminal

(assigned to S1 ! D1)

hk,1

hk,2

hk,M

f1,k

f2,k

fM,k

ForM,K →∞, the per S-D pair capacity scales as

CP1 =1

2log

(1 + Θ

(KN

M3

))

27

P2 with Cooperation

...

S1

S2

SM

.

.

.

D1

D2

DM

hk,1

hk,2

hk,M

f1,k

f2,k

fM,k

rk tk

...

.

.

.

Phase matching with respect to

all backward and all forward channels,

normalization to ensure power constraint

tk = !k

!

M"

l=1

f!

l,khH

k,l

#

rk

kth relay terminal

ForM,K →∞, the per S-D pair capacity scales as

CP2 =1

2log

(1 + Θ

(KN

M2

))

28

Cooperation Increases Per-Stream Array Gain

• Cooperation at the relay level increases the per-stream array gain

• Per-stream array gain can be decomposed asA = AdAc, where

– Distributed array gain

Ad,P1 = KN/M3 Ad,P2 = KN/M2

– Array gain due to cooperation at relay level

Ac,P1 = Ac,P2 = N

29

Cooperation vs. No Cooperation

• Consider a network with a total of T relay antenna elements

• No cooperation at the relay level

C(nc)P1 =

1

2log

(1 + Θ

(T

M3

))

• Cooperation at the relay level in groups ofN antenna elements

C(c)P1 =

1

2log

(1 + Θ

(TN

M3

))Cooperation leads to N -fold reduction in total number of relayantenna elements needed to achieve given per S-D pair capacity

30

Thank You!

31

Backup

32

Convergence of SINR CDF to Step-Function

0 2 4 6 8 10 12 14 16 18 20 22 240

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SINR

M = 20

M = 35

M = 5

N = 4

N = 1 limiting CDF

M = 35

M = 5

M = 20

P1 for K = M3

0 2 4 6 8 10 12 14 16 18 20 22 240

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SINR

M = 20

M = 35

M = 5

N = 4

N = 1 limiting CDF

M = 35

M = 5

M = 20

P2 for K = M2

33