Code

Optim

izationin

CD

MA

Systems

SennurU

lukus

AT

&T

Labs-R

esearch

ulukus@research.att.com

Roy

D.Y

ates

WIN

LA

B,R

utgersU

niversity

ryates@w

inlab.rutgers.edu

1

Introduction�

Each

userhas

apow

er(p

i ),asignature

sequence(si )

anda

receiverfilter

(ci )

ps

k ,k

ps

i ,i

ps

j ,j

mo

bile

k

mo

bile

j

cccc

1ijN

mo

bile

i

......

BS

�

Whatare

thejointly

optimum

pi ,si ,c

i foralli?

2

From

Continuous

Signalsto

Vectors

+1

-1+1

+1

-1-1+1

-1+1

-1

i (t)

si (t)

c

Pro

cessing

gain

= L

L =

T

b

Tc

si=

T

T

c

b

t t

�

Chip

sampled

receivedsignal:

r �

∑Ni �

1�p

i ai si �

n

�

Receiver

filtering:y

i �

r �

ci

�

This

work

isgeneralizable

toany

comm

unicationsystem

where

transmit

waveform

sare

representablein

afinite

dimensionalvector

space.3

ConventionalC

DM

ASystem

�

Generate

si randomly

(nosignature

optimization)

�

Use

matched

filtersc

i �

si (nofilter

optimization)

�

Controltransm

itterpow

er(SIR

-basedpow

ercontrol)

pi� n�

1� �

γ �

i

γi� n� p

i� n�

�

Power

controlconvergesto

componentw

isesm

allestpower

vectorw

here

alluserssatisfy

theirSIR

requirements

�

Furtherim

provementcan

beachieved

ifthe

receiverfilters

aredesigned

tosuppress

interference

4

JointP

ower

andR

eceiverF

ilterO

ptimization

�

S.Ulukus

andR

.D.Y

ates,“Adaptive

Power

Controland

MM

SE

InterferenceSuppression,”

AC

MJ.on

Wireless

Netw

orks.

�

Fixedsi (no

signatureoptim

ization)

�

Choose

powers

(pi )

andlinear

receiverfilters

(ci )

jointlyoptim

ally

�

Iterativealgorithm

–For

fixedpow

ers,updatefilters

tom

inimize

theM

SE(equivalently

maxim

izethe

SIR)

–M

MSE

multiuser

detectors

–For

fixedreceiver

filtersupdate

powers

inthe

usualway

pi� n�

1� �γ �

i

γi� n� p

i� n�

�

Results

dependonly

onthe

signaturesequences;further

improvem

entcan

beachieved

ifthe

signaturesare

designedto

avoidinterference

5

Review

:Inform

ationT

heoreticC

DM

AC

apacity�

User

ihassignature

sequencesi ;letS �

s1���� sN .

�

Sumcapacity

ofa

generalmultiaccess

channel

Csum

�

max

� R1 ������

RN��� �

N∑i �

1 Ri

�

Csum

:m

aximum

totalnumber

ofbits

Nusers

cantransm

itonthe

uplink.

�

Forasingle

cellsynchronousC

DM

Asystem

,with

pi �

pforalli,(V

erdu)

Csum

�

12log� det� IL �

pσ2 SS �

��6

Maxim

izationof

theSum

Capacity

�

Form

atricesA

K �M

andB

M �

K

det� IK �

AB� �

det� IM �

BA�

�

CD

MA

sumcapacity

becomes

Csum

�

12log� det� IL �

pσ2 SS �

���

12log� det� IN �

pσ2 S �

S��

�

Tom

aximize

thesum

capacity(R

upf,Massey)

–If

N

�

L,S �

S �

IN(N

orthonormalsequences)

–If

N

�

L,SS �

�

NLIL

(NW

elchB

oundE

qualitysequences)

7

Review

:N

etwork

Capacity

�

Netw

orkcapacity:

Maxim

umnum

berof

admissible

usersgiven

processinggain

Land

SIRtargetβ

�

Nusers

areadm

issibleif

thereare

positivepow

ersp

i andsignature

sequencessi such

thatSIRi �

β

�

Netw

orkcapacity

with

MM

SEreceivers

(Visw

anath,Anantharam

,Tse)

N�

L1�

1β

�

The

maxim

umis

achievedw

ith

Equalreceived

powers:

pi �

pfor

alli

WB

Esignature

sequences:SS �

�� N� L� IL

8

Netw

orkC

apacityII

�

MM

SEreceiver

forthe

ithuser

is

ci �

�

pi B

1si

�

When

pi �

pfor

alli,B

�

pSS ��

σ2IL

�

With

WB

Esequences

SS ��

NLIL

ci �

αi si

scaledm

atchedfilters!

�

Netw

orkcapacity

with

matched

filters(V

iswanath,A

nantharam,Tse)

N

�

L1�

1β

�

Max

isachieved

with

equalrec’dpow

ersand

WB

Esequences

�

“Power

/signaturesequence

/receiverfilter

optimization”

problem

actuallyhas

two

degreesoffreedom

:pow

ersand

signatures.

9

Simple

example,L

!

2,N

!

1

10

Simple

example,L

!

2,N

!

2

11

Simple

example,L

!

2,N

!

3

12

Welch’s

Bound

�

Originally

alow

erbound

form

axi�

j� s �is

j� 2kin

asetof

unitenergyvectors

�

The

derivationuses

thelow

erbound:

N∑i �1

N∑j �

1 � s �

is

j� 2k�

N2

" L#

k

1k

$

�

Fork �

1,alow

erbound

forTotalSquared

Correlation:

TSC

�N∑i �

1

N∑j �1 � s �

is

j� 2�

N2

L

�

IfN

�

L,thebound

isloose.

ForN

orthonormalvectors,T

SC

�

N

�

IfN

�

L,thebound

isachieved

iffSS �

�

NLIL

(Massey,M

ittelholzer)

13

WB

ESequences,M

inimum

TSC

,andO

ptimality

�

Minim

umT

SCsequences:

–O

rthonormalsequences

forN

�

L

–W

BE

sequencesfor

N

�L

�

Fora

singlecellC

DM

Asystem

,minim

umT

SCsequences

maxim

ize

–Inform

ationtheoretic

sumcapacity

–N

etwork

capacity

�

Goal:A

simple

algorithmw

hichconverges

toa

setofm

inimum

TSC

sequences.

14

WB

ESequences

andM

inimum

TSC

�

Optim

alsignaturesare

minim

umT

SCsignatures

�

Startingpointfor

TSC

reduction:

TSC

�� s �

ksk� 2

%&'(

1

�2s �

k∑j) �

k sj s �

jsk �

∑i) �

k ∑j) �

k � s �

is

j� 2

�

Many

ways

toreduce

TSC

:

–e.g.

choosesk

tobe

eigenvectorof∑

j) �k s

j s �j

with

min

eigenvalue

15

An

IterativeT

SCR

eductionA

lgorithm�

Method:R

eplacesk

with

ck �

A

1k

sk

� s �kA

2k

sk� 1* 2

where

Ak �

∑j) �

k sj s �

j �a

2IL .

�

ck

isa

generalizednorm

alizedM

MSE

filterfor

userk.

�

PracticalImplem

entation:

–U

sea

blindadaptive

MM

SEdetector

foreach

user.

–W

henreceiver

filterfor

userk

converges,transmitfilter

coefficients

backto

thetransm

itter

16

Algorithm

Properties

�

Algorithm

:Replace

skw

ithM

MSE

filterck

Old

Signatures:S �

s1���� sk

1 sk sk#

1���� sN

New

Signatures:S +

� s1 ��� sk

1 ck sk#

1���� sN

�

Theorem

:TSC� S +

� �

TSC� S�

andT

SC� S +� �

TSC� S�

iffc

k �

sk .

17

The

IterativeA

lgorithm

sN (n-1)

s s(n-1)

1(n-1)2

sN (n-1)

s s1(n-1)2

sN (n-1)

s s(n)

12

sN (n-1)

s s(n)

1(n)2

sN (n)

s s(n)

1(n)2

sN (n)

s s1(n)2

(n)

(n)

(n+1)

iteration n

step 1step 2

step Nstep (N

-1)

iteration (n+1)

SS

SS

SS

(n-1)(n)

1(n)

2(n)

N-1

N (n)(n+1)

1

(n)S =

iteration (n-1)end of. . .

. . .

. . .

. . .

. . .

. . .

. . .

�

TSC� n ,

1� �

TSC

1� n� �����

TSC

N

1� n� �

TSC

N� n� �

TSC� n�

18

Algorithm

Convergence

�

TSC� n�

isdecreasing

andlow

erbounded �-

TSC� n�

converges

�

TSC� n�

converges �-

S� n�/.S

�

Does

TSC

reachglobalm

inimum

?19

Properties

ofthe

Fixed

Point

�

LetS �

limn0

∞S� n�

�

N

�

L:S

convergesto

S �

S �

IN

–Sufficientcondition: initialsignature

sequencesS� 0�

arelinearly

independent

�

N

�

L:S

convergesto

SS ��

NLIL

–Sufficientcondition: L

ofN

initialsignaturesequences

S� 0�

are

linearlyindependent

–N

ecessarycondition:S� 0�

doesnothave

orthogonalsubsets

�

Forboth

casessufficientconditions

canbe

relaxed

20

Min/M

axE

igenvaluesofS 1

2 n3 S2 n3

andT

SC

2 n3

02

46

810

0

0.5 1

1.5 2

2.5 3

iteration (n)

min. and max. eigenvalues of STS

min. and m

ax. e.v. of S TS for N=5m

in. and max. e.v. of S TS for N=10

1

02

46

810

4 6 8 10 12 14 16 18 20

iteration (n)

TSC = trace(SSTSST)N=5

N=10

TSCN

N

�

Lcase:N

�

5 10and

L �

10

21

Min/M

axE

igenvaluesofS2 n3 S 1

2 n3

andT

SC

2 n3

02

46

810

0 1 2 3 4 5 6 7 8 9 10

iteration (n)

min. and max. eigenvalues of SST

N=20

N=30

N=40

N=50

min. and m

ax. e.v. of SS T

N/L

02

46

810

0 50

100

150

200

250

300

350

iteration (n)

TSC = trace(SSTSST)

N=20

N=30

N=40

N=50

TSCN

2/L

N

�

Lcase:N

�

20 30 40 50and

L �10

22

Conclusions

�

This

work

developeda

distributedalgorithm

foradapting

users’

signaturesthatcan

beused

forinterference

avoidance.

�

The

algorithmis

shown

toconverge

toa

setofusers’

signaturesthatare

optimalboth

interm

sof

information

theoreticcapacity

andnetw

ork

capacity.

�

The

algorithmcan

beim

plemented

usingfeedback

tothe

transmitter

from

anadaptive

MM

SEreceiver.

23

Work

inP

rogress

�

Extensions

toasynchronous

systems.

�

Analysis

ofm

ultipathchannels

�

Multiple

receivers(m

ulticellsystems)

�

Implem

entationbased

onblind

adaptivedetectors.

�

Effectiveness

inunlicensed

environments.

24