Universal Linked Multiple Access Source Codes

Sidharth JaggiProf. Michelle Effros

Source Codes (SCs)

Multiple Access SCs

Models for Source Coding

)ˆ,ˆ( nn YX

Y

X

Z

RX(Q)

RY(Q)

nX

nY

Slepian-Wolf

Slepian-WolfRate Region

),( YXHQ

)(YHQ

)|( YXHQ

)|( XYHQ

)(XHQ

RX(Q)

RY(Q)

Source Codes (SCs)

Multiple Access SCs

Universal MASCs?

Models for Source Coding

?))(),((?,

Rate Wolf-Slepian0

QRQRPQ

YX

e

} memoryless distributions on{

Universal MASCs?Let

),(),()()(),()(:,

YXHYXHyQyQxQxQQQ

QQ 21 212121

),( YXHQ1

)(YHQ

)(XHQ

),( YXHQ2

1Q2Q

Universal MASCs?

nY

?),(),( yxQyxQ 21 or

or or

22

11

?)(?)()()(

QRQRQRQR

YX

YXnX

1Q2Q

Source Codes (SCs)

Universal SCs Multiple Access SCs Missing Link

Linked MASCs

Models for Source Coding

Linked MASC (LMASC) Model

Y

X

Z

Xavier

Yvonne

Zorba

XR

YR

nX

nY

YXr XYr )ˆ,ˆ( nn YX

(0,0)-LMASCs

Y

X

Z

Xavier

Yvonne

Zorba

nX

nY

),(, 00 XYYX rr

YXr XYr

XR

YR

)ˆ,ˆ( nn YX

),( YXHQ

)(YHQ

)|( YXHQ

)|( XYHQ

)(XHQ

RX

RY

(0,0)-LMASC Rate Region

(0,0)-LMASC Rate Region =Slepian-WolfRate Region

Source Codes (SCs)

Universal SCs Multiple Access SCs

Linked MASCs

Universal LMASCs?

Universal LMASCs?

?),(?,

Region RateSW on point Any 0

YX

e

RRPQ

Universal (0,0)-LMASCsCode

nC

Y

X

Z

Xavier

Yvonne

Zorba

nX

nY

)(QRX

)(QRY

Universal (0,0)-LMASCsCode

0))(( QCP ne

nC

Y

X

Z

Xavier

Yvonne

Zorba

nX

nY

YXr XYr

XR

YR

)ˆ,ˆ( nn YX

),())(( 00QCr n

),())(( 00 QCR n

))(),(()())(( , QRQRRREQCR YXYXQn

Results for (0,0)-LMASCsIf ,)( r

nenCr

,)(

eP

nn

e CP 2

,)( Rn

enCR

Example:

,)( /31nCr n

.)(/ 31

2 ne nCP

,)( /61 nCR n

then

Tradeoffs

12 PRr eee

Y

X

Z

Xavier

Yvonne

Zorba

nX

nY

),(),( 00 XYYX rr

XR

YR

LMASCs ),( XYYX rr

YXrYHQ )(

YXrYXHQ )|(

),( YXHQ

YXr

YXr

XR

YR

Y

X

ZYXr 0

YXrYXHQ )|(

YXrYHQ )(

Achievable Region

Universal Coding possible

LMASCs ),( XYYX rr

Y

X

Z

Yvonne

Zorba

nX

nY

XR

YR

)ˆ,ˆ,...,ˆ( nnn YXA

Xavier

AlgernonA

AR

nA -encoder LMASCs=

l -encoder MASCl

-encoder LMASCsl

Universal Coding possible

Y

X

Z

Xavier

Yvonne

Zorba

XR

YR

nX

nY

XZr

YZr

(0,0)-FMASCs =(0,0)-LMASCs( , )-FMASCs =(0,0)-LMASCsUniversal Coding possible

XZr YZr

Feedback MASCs

Y

X

Z

Xavier

Yvonne

Zorba)ˆ( nPRY

)(nmX )(nmY

nY

nX

)ˆ,ˆ( nn YX

)ˆ( nPRX

)ˆ( nPCn

Proof Sketch - Universal LMASCs

nP̂ ),,( )()( nmnm YXLet be the type of

nP

Q

Proof Sketch - Universal LMASCs

What could possibly go wrong?

||ˆ|| QPn)(n

nP̂

Q

•Estimate “far off”

Probability of Error Rate Redundancy

),( nn YX 11

nP̂

Q

What could possibly go wrong?

),( nn YX 22

Probability of Error Redundancy• Atypicalit

y• Code failsnC

• Source

mismatch

ConclusionsX

ZMASC

(0,0)-LMASC

Universality

Universality

Y

XZ

Y

-

LMASCUniversality

XZ

Y

),( XYYX rr

ConclusionsX

Z(0,0)-FMASCY

l -encoder - LMASC Universality

Universality

XZ

Y

Universality -

FMASC

),( YZXZ rr

Complicated diagrams

The bottom line is…

It WORKS!