12/06/2016

TuesdaySuperposition SCL

HTML,F¥g±IfBw]±stB if R

,> Iluisl

KDlmhm)

Rtthz > IN ,v ;D

G- P WTC

Pathetic¥I÷ExFthn¥t*ti÷IfIa.

÷n

Idealamt*D⇒÷¥¥YII*÷÷EEE±"

sn

Difference: Psnmimjlm # Qsnmimilm

Notice : t.sn/m=PI by problem definition

Pmimilsnm is anything

we want ( will be LE)

By superposition softCovering

Lemma :

If Ru'

> Ilu ;s )

Ritri > In ,v;s )then Qsqna PI

see the firstpicture

today'

Define Pmimilsnu =

Qmimrysnm × Psnlunvnlsnlunlmibv"lMiTmi,€

G has

T3 inputs !

T boyesrule

Thisis LE.

Thereforegiven

rate constraints PoQ ( the entire systems ! )

↳ This is due to scl .

According10 Q :

Secrecy : R 'u > IN ;Z )

Rita'

> Ilan ;z ,}⇒QzynaPI

OIfig;IyY"

"¥sµ>[(v;zµ)⇒Q±µ=EMmFPuunHmm#÷mil,Vlmi

,mi ,m )

,YYETEMReliability "

pp; TRYR

-

T 7

|Mi|mim_|dribution=tksanD,

wrong right QWQYRilIHNi# doesn't matter

right | wayQuvqyiu|RitR< INMIU )fbeagth.si/wrgwngQuvQ#utRitRsICu.v&

Wyner- Ziv :

rorcausal

hossg compressionwith

Tideinformation at decoder

.

Setting : yn

Given dlx,xY

xn_T#eEnD#µg D is achievable it

H,Y)~F* , EFKEIHK,x?) ]

Equivalently ,

Pxnynmxnn = Fxnyn Pmxn Pxymyn ⇒ M - X

'- yn

X"

- ( M,

yn) - In

Theorem :( Wyner- Ziv 1976)

RID) - min ICU ;x ) - I( a ;Y )

Puix-

E- fly ,u )Ilu ;X1y )

Efdlx,xT]sD

( distribution used in theorem

Pxxuxntpxypulxppxjynau.µp

⇒ xujxuj,"g( Similarto distribution constraints (on sequence) in problem statement

Main Idea of achievability :

Codebook of U sequences to cover X.

Y boosts the effect in two ways

:

1.) Random binning : The codeword can be only partially transmitted. ( - IN ;D )

2.) Decoder uses U and Y to produce X.

RIn

binary symmetric :

↳D

ii•\eodysiibfmatin

Prof of achievability : ( specific)FixF sit.

R > Iplu ; XIY ) = Ilu ;x ) - IN ;D

D > Eptdlxixl)

Choose R'

C- ( I ( Ujx ) - R,

Ilu ; Y ))Let n' EEIR ]

Codebook C={ u" (m.mil }~Fu

Encoder : Observe X"

and produce M ( somehow)

Decoder : Receive M. Dec.de UMM

,m

') (somehow)

Compute ,for each i

,I ;=f( Yi

,Ui )

↳given by F lie

. Fxyya )

Aslong as sequences are J.T. ~F

,distortion satisfied

.

Jointly typical