# Jacobi matrices and wireless

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21 Νοε 2013 (πριν από 4 χρόνια και 5 μήνες)

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Jacobimatricesandwireless
communication
NathanLevy
Dept.ofEE,Technion-IIT,TechnionCity,Haifa32000,ISRAEL
DMA,EcoleNormalesup
´
erieure,Paris75012,FRANCE
e-mail:nlevy@tx.technion.ac.il
RMTandWirelessCommunications
July16,2008
Jointworkwith:ShlomoShamai(Shitz)andOferZeitouni
Jacobimatricesandwirelesscommunication
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Chautauqua
Jacobimatricesandwirelesscommunication
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Chautauqua
•Forcellularmodels,jointprocessingofsignalsrelatedtodifferentusersisa
Jacobimatricesandwirelesscommunication
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GeneralSystemModel
Cell 3
Cell 4
Cell 5
Cell 6
Ant. 2
Ant. 3
Ant 4
Ant 5
d=2 & K=2
N
C
(0,1)
N
C
(0,1)
N
C
(0,1)
N
C
(0,1)
•AWyner-typemulti-cellmodelwithm+dcellsorderedonaline
•m+dcellswithKsingleantennauserspercellarearrangedonaline,m
singleantennaBStsarelocatedinthecells
•Eachuserseesonlythed+1nearestcell-sites
•Weareinterestedintheasymptoticm→∞
Jacobimatricesandwirelesscommunication
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GeneralSystemModel(Cont'd)
Cell 3
Cell 4
Cell 5
Cell 6
Ant. 2
Ant. 3
Ant 4
Ant 5
d=2 & K=2
ζ
2,4
ζ
3,4
ζ
4,4
ζ
3,5
ζ
4,5
ζ
5,5

5,6
)
1

5,6
)
2
N
C
(0,1)
N
C
(0,1)
N
C
(0,1)
N
C
(0,1)
i,j
(1×Kvector)
•Channeltransfermatrix:
H
m
:=

ζ
1,1
ζ
1,2
∙∙∙ζ
1,d+1
0∙∙∙0

2,2
∙∙∙ζ
2,d+1
ζ
2,d+2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0
0∙∙∙0ζ
m,m
ζ
m,m+1
∙∙∙ζ
m,d+m

Jacobimatricesandwirelesscommunication
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AveragePer-CellSum-RateCapacity
•Hypothesis:
only
-Userscannotcooperatetheirtransmissionsinanyway
-Afullysynchronous,optimallycodedsystemisassumed
-ThetransmitpowerperuserisdenotedbyP
-Thetotalintra-celltransmitpowerisdenotedby
¯
P￿KP
hypothesis:ergodicity,stationarity,integrability,ζ
i−d,i
ζ

i,i
non-zeroa.s
givenby
C
ul
(
¯
P)=
1
m
E
H
m

logdet
I
m
+
¯
P
K
H
m
H

m

Jacobimatricesandwirelesscommunication
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High-SNRapproximation
Thehigh-SNRregimeischaracterizedthroughtheafnecapacityapproximation:
C
ul
(
¯
P)≈
¯
P≫1
S


¯
P|
dB
3|
dB
−L


=
S

3|
dB

¯
P|
dB
−3|
dB
L


•S

(multiplexinggain)￿lim
¯
P→∞
¯
P
˙
C
ul
(
¯
P)
•L

(poweroffset)￿lim
¯
P→∞
(log
2
¯
P−C
ul
(
¯
P)/S

)
Jacobimatricesandwirelesscommunication
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Previouswork
•[Wyner'94]Derivationoftheper-cellsum-ratecapacityinabsenceof
•[Narula'97]Derivationoftheper-cellsum-ratecapacityinpresenceofi.i.d
•[Somekh-Shamai'00]Derivationoftheper-cellsum-ratecapacityforK
large
•[Somekh-Shamai'00][Liang-Goldsmith'06]Boundsontheper-cell
sum-ratecapacity
•[L-Somekh-Shamai-Zeitouni'07]Derivationofthehigh-SNRmultiplexing
d=1andTDMAscheduling
•[Somekh-Zaidel-Shamai'07]andreferencesthereinforresultsonthe
Jacobimatricesandwirelesscommunication
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Fromnowon...
Fromnowond=1
•Containstherelevantideas
•Technicallyeasier
•d6=1attheend
Jacobimatricesandwirelesscommunication
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TheThoulessformula
•Thegoalistorelatetheper-cellsumratecapacitywiththegrowthrateofa
randomsequence
•Wegiveaproofsimilarto[Narula'97].
-DeneG
m
=I
m
+PH
m
H

m
.
-[Narula'97]:tocomputedetG
m
,writeG
m
=L
m
U
m
,whereL
m
is
lowertriangularandU
m
isuppertriangular.
-Weusethefollowingdecomposition:G
m
U
m
=L
m
.
-Denote
G
m
=

d
1
c

2
00
c
2
d
2
c

3
0
0
.
.
.
.
.
.
c

m
00c
m
d
m

-c
i
=Pζ
i,i
ζ

i−1,i
,d
i
=1+P


i,i
|
2
+|ζ
i,i+1
|
2

.
Jacobimatricesandwirelesscommunication
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TheThoulessformula(Cont'd)
ThedecompositionG
m
U
m
=L
m
isasfollows:

d
1
c

2
00
c
2
d
2
c

3
0
0
.
.
.
.
.
.
c

m
00c
m
d
m

x
1
x
1
∙∙∙x
1
0x
2
∙∙∙x
2
.
.
.
.
.
.
.
.
.
.
.
.
0∙∙∙0x
m

=

−c

2
x
2
0∙∙∙0
.
.
.
−c

3
x
3
.
.
.
.
.
.
0
.
.
.
.
.
.
0
00
.
.
.
−c

m+1
x
m+1

.
•x
0
=0,x
1
=1
•ci
x
i−1
+d
i
x
i
+c†
i+1
x
i+1
=0.
•x
i
isgovernedbyarandomsecond-orderlinearrecurrence.

1
m
log|detG
m
|=
1
m
P
m+1
i=2
log|ci
|+
1
m
log|x
m+1
|.
•Itisenoughtostudythesequencex
i
.
Jacobimatricesandwirelesscommunication
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Matrixformulation
•DeneM
i
=

01
−c
i
/c

i+1
−di
/c

i+1

.

x
m
x
m+1

isobtainedbyapplyingtheM
m
on

0
1

.

x
m
x
m+1

=M
m

x
m−1
x
m

=M
m
M
1

0
1

.
Jacobimatricesandwirelesscommunication
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Lyapunovexponents
Let(M
i
)
i∈N
beasequenceofrandommatriceswithgoodhypothesis:ergodicity,
stationarity,integrability
FurstenbergH.,KestenH.(1960):a.s,thefollowingquantityconvergestowarda
constant,thetopLyapunovexponent
γ(M)￿lim
n→∞
1
n
logkM
n
M
1
k.
Example:(a
1
i
)
i∈N
,...,(a
k
i
)
i∈N
,ksequencesofi.i.drandomvariables.
M
i
=diag(a
1
i
,...,a
k
i
).γ(M)=
max

Elog

a
1
1

,...,Elog

a
k
1

.
Jacobimatricesandwirelesscommunication
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Lyapunovexponents(Cont'd)
•Problem:det(G
m
)growslike

01

M
m
M
1

0
1

.
-Thegrowthratemaydependonthestartingvector
-Differentcoordinatesmayhavedifferentgrowthrate
•Notaproblemford=1butwewantarobustsolution
•Twodirections
-BythestudyofthespectralpropertiesofH
m
H

m
,whichwillgivethe
high-SNRbehaviour
-Bythestudyoftherandomsequence

x
m
x
m+1

,whichwillgivethe
uctuationsoftheper-cellsum-ratecapacity
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Convergenceoftheper-cellsum-ratecapacity
Proposition
1Undergoodhypothesis:ergodicity,stationarity,integrability,ζ
i−1,i
ζ

i,i
non-zeroa.s
I.
1
m
logdet

I
m
+
¯
P
K
H
m
H

m

convergesa.sandinexpectation
II.Thelimitis
logP+Elog

ζ
i−1,i
ζ

i,i

+γ(M)
III.Wegetasequenceofupperbounds,asymptoticallytight:forasub-multiplicative
norm(kABk≤kAkkBk)
lim
m→∞
C
ul
(
¯
P)≤logP+Elog

ζ
i−1,i
ζ

i,i

+
1
k
ElogkM
k
...M
1
k
IV.Thehigh-SNRregimeischaracterizedby
S

=1;L

=logK−

Elog

ζ
i−1,i
ζ

i,i

+γ(M[P=∞])

Jacobimatricesandwirelesscommunication
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CorollaryforK=1and(d=1ord=2)
Proposition
2AssumeK=1.Undergoodhypothesis:ergodicity,stationarity,
integrability,ζ
i−1,i
ζ

i,i
non-zeroa.s
I.Ford=1,
S

=1;L

=−max

Elog|ζ
i−1,i
|
2
;Elog|ζ
i,i
|
2

II.Ford=2,intheasymmetricWynermodel
(forexampleζ
i−2,i
∼α×Rayleigh,ζ
i−1,i
∼β×Rayleigh,ζ
i,i
∼Rayleigh,i.i.d
withα,β≤0.4)
S

=1;L

=−Elog|ζ
i,i
|
2
Ford=1,itextendstheresultof[L-Somekh-Shamai-Zeitouni'07]tonon
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ProofofProposition1
•[Craig-Simon'83]If(thereexistsabasissuchthat)onecoefcientofthe
matrixM
m
M
1
growsatleastasfastaalltheothercoefcients,its
growthrateisthetopLyapunovexponent.
•AcoefcientofthematrixM
m
M
1
growslike

α
1
β
1

M
m
M
1

α
2
β
2

HAND-WAIVING
Changingα
1

2

1
andβ
2
isequivalenttoslightlymodifyingG
m
into
e
G
m
inthetop-leftandbottom-rightcornersandthissmallperturbationdoesnot
Jacobimatricesandwirelesscommunication
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ProofofProposition1(Cont'd)
•Upperbound
-Forasub-multiplicativenormandk∈N:
kM
nk
M
1
k≤

M
nk
M
(n−1)k

kM
k
M
1
k.
-Therefore
γ(M)≤
1
k
ElogkM
k
M
1
k.
•Continuityinthehigh-SNRregime:weusethepropertiesoftheintegrated
densityofstatesofH
m
H

m
(thelimitofitseigenvaluedistribution)
Jacobimatricesandwirelesscommunication
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Fluctuationsoftheper-cellsum-rate
beginningofalltimes
•DeneC
m
(
¯
P)￿
1
m
logdet

I
m
+
¯
P
K
H
m
H

m

Proposition
cellsarei.i.dandthatthesupportoftheirdistributionisC
K(d+1)
orR
K(d+1)
.We
moreoverassumeintegrability,ζ
i−1,i
ζ

i,i
non-zeroa.s.
I.C
m
(ρ)convergesa.sandinexpectation.WedenotebyC(ρ)thelimit
II.

m(C
m
(ρ)−C(ρ))convergesinlawtoanon-degeneratedcenteredGaussian
randomvariable
III.Forallε>0,thereexistsα>0suchthat
limsup
m→∞
1
m
logP(|C
m
(ρ)−C(ρ)|>ε)<−α
Jacobimatricesandwirelesscommunication
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ProofofProposition3
Backtod=1
M
i
=

01

ζ
i,i
ζ

i−1,i
ζ

i+1,i+1
ζ
i,i+1

i,i
|
2
+|ζ
i,i+1
|
2
+
1
P
ζ

i+1,i+1
ζ
i,i+1

.
•Wewanttousethetheoryofproductsofi.i.drandommatrices.
Unfortunately,theM
i
arenoti.i.d.
•M
i
dependsonζ
i−1,i

i,i

i,i+1
andζ
i+1,i+1
.OurgoalistondΔ
i
that
dependsonlyonζ
i−1,i
andζ
i,i
suchthatγ(Δ)=γ(M)
•Theideaistoseparatethedependencyon(ζ
i−1,i

i,i
)andon

i,i+1

i+1,i+1
)
Jacobimatricesandwirelesscommunication
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ProofofProposition3(Cont'd)
P
1
(i)￿

−ζ
i,i
ζ

i−1,i
−|ζ
i,i
|
2

1
P
01

andP
2
(i)=

01
1
ζ

i,i
ζ
i−1,i

ζ

i−1,i
ζ

i,i

•M
i
=P
2
(i+1)P
1
(i).
•WesetΔ
i
=P
1
(i)P
2
(i)andget
M
n
M
1
=P
2
(n+1)P
1
(n)P
2
(n)P
2
(2)P
1
(1)
=P
2
(n+1)Δ
n
Δ
2
P
1
(1)
•γ(Δ)=γ(M)
•det(G
m
)growslike

01

Δ
m
Δ
1

1
0

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ProofofProposition3(Cont'd)
em
￿
Δ
m
Δ
1

1
0

Δ
m
Δ
1

1
0

•Thesequenceem
isaMarkovChainontheprojectivespaceofdimension1
•Itiscontractiveandstronglyirreducible(uptorestrictiontoasubspace)
•Thelawofem
convergesveryquicklytoanon-degeneratedistribution,
whichgivesProposition3
•Thestrongirreducibilityisprovedbycomputer-basedproof(ford=1and
d=2)
•Toolsin[Bougerol-Lacroix'85]
Jacobimatricesandwirelesscommunication
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Atasteofd6=1
•H
m
istwo-diagonalbyblocks(blocksofsized).Exampleford=2:
H
m
=

ζ
1,1
ζ
1,2
ζ
1,3
0
0

2,2
ζ
2,3
ζ
2,4
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0
0

m,m
ζ
m,m+1
ζ
m,m+2

.

1
m
log|detG
m
|=
1
m
P
m+1
i=2
log|detc
i
|+
1
m
log|detx
m+1
|,wherec
i
and
x
i
ared×dmatrices.
•Inordertodealwithdetx
m+1
,onneedstouseexteriorproducts
•Westudytheproductofrandommatricesofsize

2d
d

•Fortheuctuations,therelevantMarkovChainisontheprojectivespaceof
dimension

2d
d

−1
Jacobimatricesandwirelesscommunication
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ComparaisonwithclassicalRMT
Numberof
linear
nonzerocoefcients
inthesizeofthematrix
inthesizeofthematrix
Limitingper-cell
Functionofthedistribution
Functionofthesecond
sum-rate
moment
Fluctuationsofthe
1

m
1
m
per-cellsum-rate
LargeDeviations
P(|C
m
−C|>ε)≤e−mα
P(|C
m
−C|>ε)≤e−m2
α
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Jacobimatricesandwirelesscommunication
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