¢ Continuity property ( see TV properties, properly

1 in

page59 )

Ee[ Etpldlsn, FD]{Eefeadcsisttdmaxllfolku ]

= E.dk ,54 tdmt Eellfflkr

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'° % Hmmmm

. !=*.us ,s ):here Ogn|µ is a look

of(determinist ' 4

choosea pjys st

.

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11/1512016

Gelfand - Pinker ( 1980)Tuesday

m-tffyapyf.sn#nkdei-

f S" iid B

This is the moot general bk you may

replace Pylx ,s← Py

,s1×,sthis

gives

t.es#*EFtfEatmD*tarinxfgsEfFBttEpIBnp;xIixin

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It Pxls

S" iid B

m_yfzxnfch.me#dji~PWps → the answer is not the same as above !www.gniidPSExample 1 : Write twice to optical disc

.

in send time you use you can change

the Os to 1 's but not ow.

around

in here encoder gets non . causal info (you can read the whole disk) but decoder has no info .

Example 2 : Gaussian channel w/ known noise (eg . stray interference)

→ EX 1 and 2 are equiv . in [f_p and I Versions.

That :

c= max Ilu ;y ) - I( uis )Puns

Again uses synthetic channel at encoder

↳block dig.

of Thm : (Not the problem )

¥t÷x#±¥Opentionptmi

µrministic actually

,

doesn'tMhaapfaty

n→HoIIuI÷thExfEt"sn~Ps

Let's see Thm in action first

In Example 1 : Channel : Y=XvS s~Ber(p)

Let U=×={ 1 if st

2- if so where

ZnBerlydZISiePuasatt@oYoYsyY.Xvs )Pu=Hs=o=Y2 =Uvs

Therefore,

Y=U ( see' KUVS when so # st

,at aid

' H 4

Il Yiukhlu )

I ( y ;u ) - I( U;sl= Hall - Ilu ;s )

=H( uls )

=lP[ so ] .= I→ This actolly maximises ICUIY ) - Il Uis )

, just consider that

decoder also has 59,

then it is tp becauseyou can ignore ended part

Example 2 :

"

Writing on Dirty Paper"

( cost ) :(channel : Y - XTSTN,

SIN )both Gaussian

Choose U,

X,

S to be Jointly Gaussian. and XIS and HIUIX ,s)=O

Optimize Ui WLOG U=s+XX+No¥%

c=tlgkPN#

noneiowwithout state

Nforhen)

Gelled I Pink

G- P Achievability proof

Start by choosingFux

,sst

. RLIPTU ; Y) - Iplu ; s )

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digto make transmission

undated with state

choose Rgt ( Ilu ,s ),

Iluiy - R )

RTRG( IHIY ) ( so that we can decode it by channel ady)

-

total rate

Random Codebook ~PI : E{ unlm.my) }

ProofusingJointtypicolilyenadefwearesuppressiyXintkegljne.mbd.wEncoderipun1p@GimnmandSYfindmgs.t

. @" (m.my , g) e Jdn'

bwrt F

Decode: Find unique m sit .

Fmgsit

.( Mm

, mgl,

Y" ) € To

"

whey,

.

→E=ranalysis :

EdigerSmall because Rg

> Iluis )

DecodigerroriForlmmg@ynnFnmttnoeFormttruthluMm.m

g) IY"

small era because Rtihg( IIU

:

PIthL6lhdead.En@GinenmandSnPpy1msnxFgn1unGnlu4m.myl)Punlmsn

a- look -

up

Un =un( m.mg)

e-

Induced distribution

PmPsnP@PunlmmgPxnlunsnPynlxnsnPmyynuniFapPKwkupYtnh.H;Idaho

,

deaden(Payne"°%areµ, .go.m.oh.ge that part)f¥tainmLet

Qsnmgµ=m=Qmg*%mmg

unit. induced ↳look

up a" (Ming)

of M

pjnµ(snlunlmmgl)

Efllpsnmghnmtssnaylmnttr ]→0 htanase

as Icu ;D

( soft

coveyKuen

.)now Ute property of TV same channel

different input

output of channel is

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