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On the inner and outer bounds for 2-receiver discrete memoryless broadcast channels

On the inner and outer bounds for 2-receiver discrete memoryless broadcast channels
On the inner and outer bounds for 2-receiver discrete memoryless broadcast channels

a r X i v :0804.3825v 1 [c s .I T ] 24 A p r 2008

1

On the inner and outer bounds for 2-receiver discrete memoryless broadcast channels

Chandra Nair,CUHK and Vincent Wang Zizhou,CUHK

Abstract —We study the best known general inner bound[1]and outer bound[2]for the capacity region of the two user discrete memory less channel.We prove that a seemingly stronger outer bound is identical to a weaker form of the outer bound that was also presented in [2].We are able to further express the best outer bound in a form that is computable,i.e.there are bounds on the cardinalities of the auxiliary random variables.The inner and outer bounds coincide for all channels for which the capacity region is known and it is not known whether the regions described by these bounds are same or different.We present a channel,where assuming a certain conjecture backed by simulations and partial theoretical results,one can show that the bounds are different.

I.I NTRODUCTION

In [3],Cover introduced the notion of a broadcast channel through which one sender transmits information to two or more receivers.For the purpose of this paper we focus our attention on broadcast channels with precisely two receivers.De?nition:A broadcast channel (BC)consists of an input alphabet X and output alphabets Y 1and Y 2and a probability transition function p (y 1,y 2|x ).A ((2nR 1,2nR 2),n )code for a broadcast channel consists of an encoder

x n :2nR 1×2nR 2→X n ,

and two decoders

?W 1:Y n 1→2nR 1?W 2:Y n 2

→2nR 2.The probability of error P (n )

e is de?ned to be the probability

that the decoded message is not equal to the transmitted message,i.e.,

P (n )e

=P

{?W 1(Y n 1

)=W 1}∪

{?W 2(Y n 2

)=W 2}

where the message is assumed to be uniformly distributed over 2nR 1×2nR 2.

A rate pair (R 1,R 2)is said to be achievable for the broad-cast channel if there exists a sequence of ((2nR 1,2nR 2),n )

codes with P (n )

e →0.The capacity region o

f the broadcast channel with is the closure of the set of achievable rates.The capacity region of the two user discrete memoryless channel is unknown.

The capacity region is known for lots of special cases such as degraded,less noisy,more capable,deterministic,semi-deterministic,etc.-see [4]and the references therein.

General inner and outer bounds for the two-user discrete memoryless broadcast channel have also been known in liter-ature.Here we state the best known inner and outer bounds for the region from the literature.Bound 1:[M¨a rton ’79]The following rate pairs are achiev-able:

R 1≤I (U,W ;Y 1)R 2≤I (V,W ;Y 2)

R 1+R 2≤min {I (W ;Y 1),I (W ;Y 2)}+I (U ;Y 1|W )

+I (V ;Y 2|W )?I (U ;V |W )for any p (u,v,w,x )such that (U,V,W )→X →(Y 1,Y 2)form a Markov chain.

Bound 2:[Nair-El Gamal ’07]The region R de?ned by the union over the rate pairs satisfying

R 1≤I (U,W ;Y 1)R 2≤I (V,W ;Y 2)

R 1+R 2≤min {I (U,W ;Y 1)+I (V ;Y 2|U,W ),

I (V,W ;Y 2)+I (U ;Y 1|V,W )}

over all p (u )p (v )p (w,x |u,v )such that (U,V,W )→X →(Y 1,Y 2)form a Markov chain forms an outer bound to the capacity region.

Remark 1:Both the bounds are tight for all the special classes of two-user broadcast channels for which the capacity region is known.However,since the bounds are dif?cult to evaluate in general it is not known whether the tightness of these bounds is speci?c to the scenarios or whether they coincide yielding the capacity region.

A possibly weaker form of the outer bound was also presented in [2]by removing the independence between U and V .Under this relaxation we have the following:

Bound 3:[Nair-El Gamal ’07]The region R 1de?ned by the union over the rate pairs satisfying

R 1≤I (U ;Y 1)

R 2≤I (V ;Y 2)

R 1+R 2≤min {I (U ;Y 1)+I (V ;Y 2|U ),

I (V ;Y 2)+I (U ;Y 1|V )}

over all p (u,v,x )such that (U,V )→X →(Y 1,Y 2)form a Markov chain constitutes an outer bound to the capacity region.

One of the main results of the paper is the following:The regions described by Bounds 2and 3are identical.

The organization of the paper is as follows.In Section II we show that the regions described by Bound 2and Bound 3

2 are the same.We also present a different representation of the

the bound which allows us to have bounds on the cardinalities

of the auxiliary random variables.In Section III we study the

binary skew-symmetric channel[5]and conjecture that the

inner and outer bounds are different for this channel.

II.O N EVALUATION OF THE OUTER BOUND

A.Identity of the bounds

Theorem1:The regions R and R1coincide,i.e.R=R1.

Proof:Clearly,by setting U′=(U,W)and V′= (V,W),we have that R?R1.Therefore it suf?ces to show that R1?R.

The idea of the proof1is as follows:Given a(U,V)we will produce a(U?,V?,W?)with U?,V?being independent such that

I(U;Y1)=I(U?,W?;Y1)

I(V;Y2)=I(V?,W?;Y2)

I(U;Y1|V)=I(U?;Y1|V?,W?)(1)

I(V;Y2|U)=I(V?;Y2|U?,W?).

Let(U,V,X)be a triple such that(U,V)→X→(Y1,Y2)form a Markov chain.Let V={0,1,...,m?1}. De?ne new random variables U?,V?,W?and a distribution p(u?,v?,w?,x)according to

P(U?=u,V?=i,W?=j,X=x)

=

1

m

P(U=u)

and hence independent,

P(U?=u,W?=i,X=x)=

1

m

P(V=(i+j)m,X=x). From the above it follows in a straightforward manner that (1)holds and thus completes the proof.

l

P(U=u,V=v,X=(i?j)l),

P(X?=k|U?=u i,V?=v j)

= 1if k=(i?j)l

0otherwise,

(2)

one obtains

I(U;Y1)=I(U?;Y1)

I(V;Y2)=I(V?;Y2)(3) I(X;Y1|V)=I(X;Y1|V?)=I(U?;Y1|V?)

I(X;Y2|U)=I(X;Y2|U?)=I(V?;Y2|U?).

Thus R2?R1.

III.T

HE BINARY SKEW-SYMMETRIC CHANNEL

A.On evaluating M¨a rton inner bound

We consider the following channel[5]called the Binary skew-symmetric channel,BSSC.For ease we restrict ourselves to the case p=1

This implies that for η≤η0=

1

5.

Assuming Conjecture 1is true we can now analyze the sum rate of

the Marton inner bound with the random variable W .Theorem 1implies

R 1+R 2≤min {I (W ;Y 1),I (W ;Y 2)}

+I (U ;Y 1|W )+I (V ;Y 2|W )?I (U ;V |W ).Let W 0={w :P(X =0|W =w )≤0.5}and W 1={w :P(X =0|W =w )>0.5}.Let T be a function of W de?ned by T =

0if w ∈W 0

1if w ∈W 1.We have the following bound on the sum rate

R 1+R 2≤min {I (W,T ;Y 1),I (W,T ;Y 2)}

+I (U ;Y 1|W,T )+I (V ;Y 2|W,T )?I (U ;V |W,T )

(a )

≤min {I (W,T ;Y 1),I (W,T ;Y 2)}+P(T =0)I (X ;Y 1|W,T =0)+P(T =1)I (X ;Y 2|W,T =1)

(b )

≤min {I (T ;Y 1),I (T ;Y 2)}+P(T =0)I (X ;Y 1|T =0)

+P(T =1)I (X ;Y 2|T =1).

Here (a )follows from Conjecture 1and (b )follows from the fact that

P(T =1)I (W ;Y 1|T =1)≤P(T =1)I (W ;Y 1|T =1),P(T =0)I (W ;Y 2|T =0)≤P(T =0)I (W ;Y 1|T =0).In [2]the bound on sum rate,min {I (T ;Y 1),I (T ;Y 2)}+P(T =0)I (X ;Y 1|T =0)+P(T =1)I (X ;Y 2|T =1)has been studied and the maximum was evaluated as ≈0.3616.This could also be inferred from [5]and the evaluation of the Cover-van-der-Meulen region for this channel.

Thus assuming Conjecture 1we have that the sum rate of the M¨a rton inner bound is bounded by 0.3616...(correct to 4decimal places).

B.Evaluating outer bound -BSSC

In [2]the sum rate of the pairs (R 1,R 2)described by Bound 4was evaluated and it was shown that the maximum sum rate was bounded by 0.3711..(correct to 4decimal places).Thus we have that the region described by Bound 2is strictly larger than that described by Bound 1(assuming Conjecture 1)and thus the inner and outer bounds differ for BSSC.

IV.C ONCLUSION

In this paper,we study the inner and outer bounds for the 2-user discrete memoryless broadcast channel.We prove that for the purpose of evaluating the outer bound the region described by a weaker version (which is easier to evaluate)indeed coincides with a stronger version.

The bounds matched for all the special classes of channels for which the capacity was known.It is not known if the bounds were inherently different or not.We then studied the bounds for the particular case of the binary skew symmetric channel (BSSC).We present a conjecture that,if proved,would establish that the inner and the outer bounds are indeed not tight for BSSC.Numerical simulations also indicate that the bounds differ for BSSC.

This de?nitely indicates that one of the bounds or possibly both are weak.We have demonstrated that resolving the capacity region for the BSSC would de?nitely give a strong hint on the capacity region of the broadcast channel for two users.

A CKNOWLEDGMENTS

The authors would like to acknowledge Prof.Bruce Hajek for very stimulating discussions during his visit to CUHK.The authors would also like to acknowledge some valuable sugges-tions and stimulating exchanges on the broadcast channel and on BSSC by Prof.Abbas El Gamal.

R EFERENCES

[1]K.Marton,“A coding theorem for the discrete memoryless broadcast

channel,”IEEE https://www.sodocs.net/doc/c015445165.html,.Theory ,vol.IT-25,pp.306–311,May,1979.[2] C.Nair and A.El Gamal,“An outer bound to the capacity region of the

broadcast channel,”IEEE https://www.sodocs.net/doc/c015445165.html,.Theory ,vol.IT-53,pp.350–355,January,2007.

[3]T.Cover,“Broadcast channels,”IEEE https://www.sodocs.net/doc/c015445165.html,.Theory ,vol.IT-18,pp.

2–14,January,1972.

[4]——,“Comments on broadcast channels,”IEEE https://www.sodocs.net/doc/c015445165.html,.Theory ,vol.

IT-44,pp.2524–2530,October,1998.

[5] B.Hajek and M.Pursley,“Evaluation of an achievable rate region for

the broadcast channel,”IEEE https://www.sodocs.net/doc/c015445165.html,.Theory ,vol.IT-25,pp.36–46,January,1979.[6]I.Csiz′a r and J.K¨o rner,“Broadcast channels with con?dential messages,”

IEEE https://www.sodocs.net/doc/c015445165.html,.Theory ,vol.IT-24,pp.339–348,May,1978.

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