Three-Receiver Broadcast Channels with Side Information

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In this paper, we derive achievable rate regions for two classes of ... Cover and Chiang [7] ... omitted when the choice of n is clear, thus we only use boldfaceย ...
Three-Receiver Broadcast Channels with Side Information Saeed Hajizadeh (Undergraduate Student) Department of Electrical Engineering Ferdowsi University of Mashhad Mashhad, Iran [email protected] Abstractโ€”Three-receiver broadcast channel (BC) is of interest due to its information theoretical differences with two receiver one. In this paper, we derive achievable rate regions for two classes of 3-receiver BC with side information available at the transmitter, Multilevel BC and 3-receiver less noisy BC, by using superposition coding, Gelโ€™fand-Pinsker binning scheme and Nair-El Gamal indirect decoding. Our rate region for multilevel BC subsumes the Steinberg rate region for 2-receiver degraded BC with side information as its special case. We also find the capacity region of 3-receiver less noisy BC when side information is available both at the transmitter and at the receivers. Keywords: 3-receiver broadcast channel, less noisy, Multilevel broadcast channel

I.

INTRODUCTION

The k-receiver, ๐‘˜๐‘˜ โ‰ฅ 3, broadcast channel (BC) was first studied by Borade et al. in [1] where they simply surmised that straightforward extension of Kรถrner-Martonโ€™s capacity region for two-receiver BCs with degraded message sets [2] to k-receiver multilevel broadcast networks is optimal. Nair-El Gamal [3] showed that the capacity region of a special class of 3-receiver BCs with two degraded message sets when one of the receivers is a degraded version of the other, is a superset of [1], thus proving that direct extension of [2] is not in general optimal. Nair and Wang later in [4] established the capacity region of the 3-receiver less noisy BC. Channels with Side information (SI), were first studied by Shannon [5], where he found the capacity region of the Single-Input-Single-Output channel when SI is causally available at the encoder. Gelfโ€™and and Pinsker [6] found the capacity region of a single-user channel when SI is non-causally available at the transmitter while the receiver is kept ignorant of it. Cover and Chiang [7] extended the results of [6] to the case where SI is available at both the encoder and the decoder. Multiple user channels with side information were studied in [8] where inner and outer bounds for degraded BC with non-causal SI and capacity region of degraded BC with causal SI were found. Moreover, in [9] inner and outer bounds were given to general two-user BCs with SI available at the transmitter and other special cases both for BCs and MACs were also found. In this paper, we find the achievable rate region of Multilevel BC and 3-receiver less noisy BC both with SI non-

Ghosheh Abed Hodtani Department of Electrical Engineering Ferdowsi University of Mashhad Mashhad, Iran [email protected]

causally available at the encoder. Our achievable rate regions reduce to that of [3] and [4] when there is no side information. We also find the capacity region of the latter when side information is also available at the receivers. The rest of the paper is organized as follows. In section II, basic definitions and notations are presented. In sections III and IV, new achievable rate regions are given for the Multilevel BC and 3receiver less noisy BC, respectively. In section V, conclusion is given. II.

DEFINITIONS

Random variables and their realizations are denoted by uppercase and lowercase letters, respectively, e.g. x is a realization of X. Let ๐’ณ๐’ณ, ๐’ด๐’ด1 , ๐’ด๐’ด2 , ๐’ด๐’ด3 , ๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž ๐’ฎ๐’ฎ be finite sets showing alphabets of random variables. The n-sequence of a random variable is given by ๐‘‹๐‘‹ ๐‘›๐‘› where the superscript is omitted when the choice of n is clear, thus we only use boldface letters for the random variable itself, i.e. ๐’™๐’™ = ๐‘ฅ๐‘ฅ ๐‘›๐‘› . is the Throughout, we assume that ๐‘‹๐‘‹๐‘–๐‘–๐‘›๐‘› sequence (๐‘‹๐‘‹๐‘–๐‘– , ๐‘‹๐‘‹๐‘–๐‘–+1 , โ€ฆ , ๐‘‹๐‘‹๐‘›๐‘› ). Definition 1: A channel ๐‘‹๐‘‹ โ†’ ๐‘๐‘ is said to be a degraded version of the channel ๐‘‹๐‘‹ โ†’ ๐‘Œ๐‘Œ with SI if ๐‘‹๐‘‹ โ†’ ๐‘Œ๐‘Œ โ†’ ๐‘๐‘ be a Markov chain conditioned on every ๐‘ ๐‘  โˆˆ ๐’ฎ๐’ฎ for all ๐‘๐‘(๐‘ข๐‘ข, ๐‘ฅ๐‘ฅ|๐‘ ๐‘ ). Multilevel BC with side information, denoted by ๏ฟฝ๐’ณ๐’ณ, ๐’ฎ๐’ฎ, ๐’ด๐’ด1 , ๐’ด๐’ด2 , ๐’ด๐’ด3 , ๐‘๐‘(๐‘ฆ๐‘ฆ1 , ๐‘ฆ๐‘ฆ3 |๐‘ฅ๐‘ฅ, ๐‘ ๐‘ ), ๐‘๐‘(๐‘ฆ๐‘ฆ2 |๐‘ฆ๐‘ฆ1 )๏ฟฝ , is a 3-receiver BC with 2-degraded message sets with input alphabet ๐’ณ๐’ณ and output alphabets ๐’ด๐’ด1 , ๐’ด๐’ด2 , and ๐’ด๐’ด3 . The side information is the random variable S distributed over the set ๐’ฎ๐’ฎ according to ๐‘๐‘(๐‘ ๐‘ ). The transition probability function ๐‘๐‘(๐‘ฆ๐‘ฆ1 , ๐‘ฆ๐‘ฆ3 |๐‘ฅ๐‘ฅ, ๐‘ ๐‘ ) describes the relationship between channel input X, side information S, and channel outputs ๐‘Œ๐‘Œ1 and ๐‘Œ๐‘Œ3 while the probability function ๐‘๐‘(๐‘ฆ๐‘ฆ2 |๐‘ฆ๐‘ฆ1 ) shows the virtual channel modeling the output ๐‘Œ๐‘Œ2 as the degraded version of ๐‘Œ๐‘Œ1 . Independent message sets ๐‘š๐‘š0 โˆˆ โ„ณ0 and ๐‘š๐‘š1 โˆˆ โ„ณ1 are to be reliably sent, m0 being the common message for all the receivers and m1 the private message only for Y1 . Channel model is depicted in Fig. 1.

Definition 2: A (๐‘›๐‘›, 2๐‘›๐‘›๐‘…๐‘…0 , 2๐‘›๐‘›๐‘…๐‘…1 , ๐œ–๐œ–) two-degraded message set code for the Multilevel BC with side information ๏ฟฝ๐‘๐‘(๐‘ฆ๐‘ฆ1 , ๐‘ฆ๐‘ฆ3 |๐‘ฅ๐‘ฅ, ๐‘ ๐‘ ), ๐‘๐‘(๐‘ฆ๐‘ฆ2 |๐‘ฆ๐‘ฆ1 )๏ฟฝ consists of an encoder map

The messages ๐‘š๐‘š1 โˆˆ โ„ณ1 , ๐‘š๐‘š2 โˆˆ โ„ณ2 , ๐‘š๐‘š3 โˆˆ โ„ณ3 are to be reliably sent to receivers ๐‘Œ๐‘Œ1 , ๐‘Œ๐‘Œ2 , ๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž ๐‘Œ๐‘Œ3 , respectively. The code and rate tuple definitions are as follows (๐‘›๐‘›, 2๐‘›๐‘›๐‘…๐‘…1 , 2๐‘›๐‘›๐‘…๐‘…2 , 2๐‘›๐‘›๐‘…๐‘…3 , ๐œ–๐œ–) 1

(๐‘…๐‘…1 , ๐‘…๐‘…2 , ๐‘…๐‘…3 ) = (๐‘™๐‘™๐‘™๐‘™๐‘™๐‘™๐‘€๐‘€1 , ๐‘™๐‘™๐‘™๐‘™๐‘™๐‘™๐‘€๐‘€2 , ๐‘™๐‘™๐‘™๐‘™๐‘™๐‘™๐‘€๐‘€3 ) ๐‘›๐‘›

Figure 1. Multilevel broadcast channel with side information.

๐‘“๐‘“ โˆถ {1,2, โ€ฆ , ๐‘€๐‘€0 } ร— {1,2, โ€ฆ , ๐‘€๐‘€1 } ร— ๐’ฎ๐’ฎ ๐‘›๐‘› โŸถ ๐’ณ๐’ณ ๐‘›๐‘›

and a tuple of decoding maps

๐‘”๐‘”๐‘ฆ๐‘ฆ1 โˆถ ๐’ด๐’ด1๐‘›๐‘› โŸถ {1,2, โ€ฆ , ๐‘€๐‘€0 } ร— {1,2, โ€ฆ , ๐‘€๐‘€1 } ๐‘”๐‘”๐‘ฆ๐‘ฆ2 โˆถ ๐’ด๐’ด2๐‘›๐‘› โŸถ {1,2, โ€ฆ , ๐‘€๐‘€0 } ๐‘”๐‘”๐‘ฆ๐‘ฆ3 โˆถ ๐’ด๐’ด3๐‘›๐‘› โŸถ {1,2, โ€ฆ , ๐‘€๐‘€0 } (๐‘›๐‘›)

Such that ๐‘ƒ๐‘ƒ๐‘’๐‘’ ๐‘€๐‘€0

โ‰ค ๐œ–๐œ–, i.e.

๐‘€๐‘€1

1 ๏ฟฝ ๏ฟฝ ๏ฟฝ ๐‘๐‘(๐’”๐’”)๐‘๐‘{๐‘”๐‘”๐‘ฆ๐‘ฆ1 (๐’š๐’š1 ) โ‰  (๐‘š๐‘š0 , ๐‘š๐‘š1 ) ๐‘œ๐‘œ๐‘œ๐‘œ ๐‘€๐‘€0 ๐‘€๐‘€1 ๐‘›๐‘› ๐‘›๐‘› ๐‘š๐‘š 0=1 ๐‘š๐‘š 1 =1 ๐‘ ๐‘  โˆˆ๐’ฎ๐’ฎ

๐‘”๐‘”๐‘ฆ๐‘ฆ2 (๐’š๐’š2 ) โ‰  ๐‘š๐‘š0 ๐‘œ๐‘œ๐‘œ๐‘œ ๐‘”๐‘”๐‘ฆ๐‘ฆ3 (๐’š๐’š3 ) โ‰  ๐‘š๐‘š0 |๐’”๐’”, ๐’™๐’™(๐‘š๐‘š0 , ๐‘š๐‘š1 , ๐’”๐’”)} โ‰ค ๐๐

The rate pair of the code is defined as (๐‘…๐‘…0 , ๐‘…๐‘…1 ) =

1 (log ๐‘€๐‘€0 , ๐‘™๐‘™๐‘™๐‘™๐‘™๐‘™๐‘€๐‘€1 ) ๐‘›๐‘›

A rate pair (๐‘…๐‘…0 , ๐‘…๐‘…1 ) is said to be ๐œ–๐œ–-achievable if for any ๐œ‚๐œ‚ > 0 there is an integer ๐‘›๐‘›0 such that for all ๐‘›๐‘› โ‰ฅ ๐‘›๐‘›0 we have a code for (๐‘›๐‘›, 2๐‘›๐‘›(๐‘…๐‘…0 โˆ’๐œ‚๐œ‚ ) , 2๐‘›๐‘›(๐‘…๐‘…1 โˆ’๐œ‚๐œ‚ ) , ๐œ–๐œ–) ๏ฟฝ๐‘๐‘(๐‘ฆ๐‘ฆ1 , ๐‘ฆ๐‘ฆ3 |๐‘ฅ๐‘ฅ, ๐‘ ๐‘ ), ๐‘๐‘(๐‘ฆ๐‘ฆ2 |๐‘ฆ๐‘ฆ1 )๏ฟฝ. The union of the closure of all ๐œ–๐œ–achievable rate pairs is called the capacity region ๐’ž๐’ž๐‘€๐‘€๐‘€๐‘€๐‘€๐‘€ . Definition 3: A channel ๐‘‹๐‘‹ โ†’ ๐‘Œ๐‘Œ is said to be less noisy than the channel ๐‘‹๐‘‹ โ†’ ๐‘๐‘ in the presence of side information if ๐ผ๐ผ(๐‘ˆ๐‘ˆ; ๐‘Œ๐‘Œ|๐‘†๐‘† = ๐‘ ๐‘ ) โ‰ฅ ๐ผ๐ผ(๐‘ˆ๐‘ˆ; ๐‘๐‘|๐‘†๐‘† = ๐‘ ๐‘ ) โˆ€๐‘๐‘(๐‘ข๐‘ข, ๐‘ฅ๐‘ฅ, ๐‘ฆ๐‘ฆ, ๐‘ง๐‘ง|๐‘ ๐‘ ) = ๐‘๐‘(๐‘ข๐‘ข|๐‘ ๐‘ )๐‘๐‘(๐‘ฅ๐‘ฅ|๐‘ข๐‘ข, ๐‘ ๐‘ )๐‘๐‘(๐‘ฆ๐‘ฆ, ๐‘ง๐‘ง|๐‘ฅ๐‘ฅ, ๐‘ ๐‘ ) ๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž โˆ€๐‘ ๐‘  โˆˆ ๐’ฎ๐’ฎ.

The 3-receiver less noisy BC with side information is depicted in Fig. 2, where ๐‘Œ๐‘Œ1 is less noisy than ๐‘Œ๐‘Œ2 and ๐‘Œ๐‘Œ2 is less noisy than ๐‘Œ๐‘Œ3 , i.e. according to [4], ๐‘Œ๐‘Œ1 โ‰ฝ ๐‘Œ๐‘Œ2 โ‰ฝ ๐‘Œ๐‘Œ3 .

Figure.2. Three-receiver less noisy broadcast channel with side information.

Achievable rate tuples and the achievable rate region and the capacity region ๐’ž๐’ž๐ฟ๐ฟ are defined in just the same way as Multilevel BC. III.

MULTILEVEL BROADCAST CHANNEL WITH SIDE INFORMATION

Define ๐’ซ๐’ซ as the collection of all random variables (๐‘ˆ๐‘ˆ, ๐‘‰๐‘‰, ๐‘†๐‘†, ๐‘‹๐‘‹, ๐‘Œ๐‘Œ1 , ๐‘Œ๐‘Œ2 , ๐‘Œ๐‘Œ3 ) with finite alphabets such that ๐‘๐‘(๐‘ข๐‘ข, ๐‘ฃ๐‘ฃ, ๐‘ ๐‘ , ๐‘ฅ๐‘ฅ, ๐‘ฆ๐‘ฆ1 , ๐‘ฆ๐‘ฆ2 , ๐‘ฆ๐‘ฆ3 ) = ๐‘๐‘(๐‘ ๐‘ )๐‘๐‘(๐‘ข๐‘ข|๐‘ ๐‘ )๐‘๐‘(๐‘ฃ๐‘ฃ|๐‘ข๐‘ข, ๐‘ ๐‘ )๐‘๐‘(๐‘ฅ๐‘ฅ|๐‘ฃ๐‘ฃ, ๐‘ ๐‘ )๐‘๐‘(๐‘ฆ๐‘ฆ1 , ๐‘ฆ๐‘ฆ3 |๐‘ฅ๐‘ฅ, ๐‘ ๐‘ )๐‘๐‘(๐‘ฆ๐‘ฆ2 |๐‘ฆ๐‘ฆ1 )

(1)

By (1), the following Markov chains hold:

(2)

(๐‘ˆ๐‘ˆ, ๐‘‰๐‘‰) โ†’ (๐‘‹๐‘‹, ๐‘†๐‘†) โ†’ (๐‘Œ๐‘Œ1 , ๐‘Œ๐‘Œ3 )

(3)

(๐‘†๐‘†, ๐‘‹๐‘‹, ๐‘Œ๐‘Œ3 ) โ†’ ๐‘Œ๐‘Œ1 โ†’ ๐‘Œ๐‘Œ2

Theorem 1: A pair of nonnegative numbers (๐‘…๐‘…0 , ๐‘…๐‘…1 ) is achievable for Multilevel BC with side information noncausally available at the transmitter provided that ๐‘…๐‘…0 โ‰ค min{๐ผ๐ผ(๐‘ˆ๐‘ˆ; ๐‘Œ๐‘Œ2 ) โˆ’ ๐ผ๐ผ(๐‘ˆ๐‘ˆ; ๐‘†๐‘†), ๐ผ๐ผ(๐‘‰๐‘‰; ๐‘Œ๐‘Œ3 ) โˆ’ ๐ผ๐ผ(๐‘ˆ๐‘ˆ๐‘ˆ๐‘ˆ; ๐‘†๐‘†)} ๐‘…๐‘…1 โ‰ค ๐ผ๐ผ(๐‘‹๐‘‹; ๐‘Œ๐‘Œ1 |๐‘ˆ๐‘ˆ) โˆ’ ๐ผ๐ผ(๐‘‰๐‘‰; ๐‘†๐‘†|๐‘ˆ๐‘ˆ) โˆ’ ๐ผ๐ผ(๐‘‹๐‘‹; ๐‘†๐‘†|๐‘‰๐‘‰) (4) ๐‘…๐‘…0 + ๐‘…๐‘…1 โ‰ค ๐ผ๐ผ(๐‘‰๐‘‰; ๐‘Œ๐‘Œ3 ) + ๐ผ๐ผ(๐‘‹๐‘‹; ๐‘Œ๐‘Œ1 |๐‘‰๐‘‰) โˆ’ ๐ผ๐ผ(๐‘‹๐‘‹; ๐‘†๐‘†|๐‘‰๐‘‰) โˆ’ ๐ผ๐ผ(๐‘ˆ๐‘ˆ๐‘ˆ๐‘ˆ; ๐‘†๐‘†) for some (๐‘ˆ๐‘ˆ, ๐‘‰๐‘‰, ๐‘†๐‘†, ๐‘‹๐‘‹, ๐‘Œ๐‘Œ1 , ๐‘Œ๐‘Œ2 , ๐‘Œ๐‘Œ3 ) โˆˆ ๐’ซ๐’ซ.

Corollary 1.1: By setting ๐‘†๐‘† โ‰ก โˆ… in (4), our achievable rate region in Theorem 1 is reduced to the capacity region of Multilevel BC given in [3].

Corollary 1.2: By setting ๐‘Œ๐‘Œ3 = ๐‘Œ๐‘Œ1 and ๐‘‰๐‘‰ = ๐‘ˆ๐‘ˆ in (4), our achievable rate region reduces to that of [8] for the two-user degraded BC with side information. Proof: Fix n and a joint distribution on ๐’ซ๐’ซ. Note that side information is distributed i.i.d according to ๐‘›๐‘›

๐‘๐‘(๐’”๐’”) = ๏ฟฝ ๐‘๐‘(๐‘ ๐‘ ๐‘–๐‘– ) ๐‘–๐‘–=1

Split the โ„ณ1 message into two independent submessage sets โ„ณ11 , ๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž โ„ณ12 so that ๐‘…๐‘…1 = ๐‘…๐‘…11 + ๐‘…๐‘…12 .

Codebook Generation: First randomly and independently โ€ฒ โ€ฒ generate 2๐‘›๐‘›๏ฟฝ๐‘…๐‘…0 +๐‘…๐‘…0 ๏ฟฝ sequences ๐’–๐’–(๐‘š๐‘š0โ€ฒ , ๐‘š๐‘š0 ), ๐‘š๐‘š0โ€ฒ โˆˆ ๏ฟฝ1,2, โ€ฆ , 2๐‘›๐‘›๐‘…๐‘…0 ๏ฟฝ, ๐‘š๐‘š0 โˆˆ {1,2, โ€ฆ , 2๐‘›๐‘›๐‘…๐‘…0 }, each one i.i.d according to โˆ๐‘›๐‘›๐‘–๐‘–=1 ๐‘๐‘(๐‘ข๐‘ข๐‘–๐‘– ) and then randomly throw them into 2๐‘›๐‘›๐‘…๐‘…0 bins. It is clear that โ€ฒ we have 2๐‘›๐‘›๐‘…๐‘…0 sequences in each bin. Now for each ๐’–๐’–(๐‘š๐‘š0โ€ฒ , ๐‘š๐‘š0 ), randomly and independently โ€ฒ โ€ฒ โ€ฒ , ๐‘š๐‘š11 ), ๐‘š๐‘š11 โˆˆ generate 2๐‘›๐‘›(๐‘…๐‘…11 +๐‘…๐‘…11 ) sequences ๐’—๐’—(๐‘š๐‘š0โ€ฒ , ๐‘š๐‘š0 , ๐‘š๐‘š11

๏ฟฝ1, โ€ฆ , 2๐‘›๐‘›๐‘…๐‘…11 ๏ฟฝ, ๐‘š๐‘š11 โˆˆ {1, โ€ฆ , 2๐‘›๐‘›๐‘…๐‘…11 } each one i.i.d according to โˆ๐‘›๐‘›๐‘–๐‘–=1 ๐‘๐‘๐‘‰๐‘‰|๐‘ˆ๐‘ˆ ๏ฟฝ๐‘ฃ๐‘ฃ๐‘–๐‘– ๏ฟฝ๐‘ข๐‘ข๐‘–๐‘– (๐‘š๐‘š0โ€ฒ , ๐‘š๐‘š0 )๏ฟฝ, and randomly throw them into 2๐‘›๐‘›๐‘…๐‘…11 bins. โ€ฒ Now for each sequence ๐’—๐’—(๐‘š๐‘š0โ€ฒ , ๐‘š๐‘š0 , ๐‘š๐‘š11 , ๐‘š๐‘š11 ), randomly and โ€ฒ independently generate 2๐‘›๐‘›(๐‘…๐‘…12 +๐‘…๐‘…12 ) โ€ฒ โ€ฒ , ๐‘š๐‘š11 , ๐‘š๐‘š12 , ๐‘š๐‘š12 ) each one i.i.d sequences ๐’™๐’™(๐‘š๐‘š0โ€ฒ , ๐‘š๐‘š0 , ๐‘š๐‘š11 ๐‘›๐‘› according to โˆ๐‘–๐‘–=1 ๐‘๐‘๐‘‹๐‘‹|๐‘ˆ๐‘ˆ,๐‘‰๐‘‰ (๐‘ฅ๐‘ฅ๐‘–๐‘– |๐‘ฃ๐‘ฃ๐‘–๐‘– , ๐‘ข๐‘ข๐‘–๐‘– ) = โˆ๐‘›๐‘›๐‘–๐‘–=1 ๐‘๐‘๐‘‹๐‘‹|๐‘‰๐‘‰ (๐‘ฅ๐‘ฅ๐‘–๐‘– |๐‘ฃ๐‘ฃ๐‘–๐‘– ). Then randomly throw them into 2๐‘›๐‘›๐‘…๐‘…12 bins. Then provide the transmitter and all the receivers with bins and their codewords. Encoding: We are given the side information ๐’”๐’” and the message pair (๐‘š๐‘š0 , ๐‘š๐‘š1 ). Indeed, our messages are bin indices. We find ๐‘š๐‘š11 , ๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž ๐‘š๐‘š12 . Now in the bin ๐‘š๐‘š0 of ๐’–๐’– sequences (๐’๐’) look for a ๐‘š๐‘š0โ€ฒ such that (๐’–๐’–(๐‘š๐‘š0โ€ฒ , ๐‘š๐‘š0 ), ๐’”๐’”) โˆˆ ๐ด๐ด๐๐ , i.e. the sequence ๐’–๐’– that is jointly typical with the ๐’”๐’” given where definitions of typical sequences are given in [12]. Then in the โ€ฒ such that bin ๐‘š๐‘š11 of ๐’—๐’— sequences look for some ๐‘š๐‘š11

โ€ฒ , 1), ๐ธ๐ธ11 = {( ๐’–๐’–(๐‘€๐‘€0โ€ฒ , 1), ๐’—๐’—(๐‘€๐‘€0โ€ฒ , 1, ๐‘€๐‘€11 (๐‘›๐‘›) โ€ฒ โ€ฒ โ€ฒ ๐’™๐’™(๐‘€๐‘€0 , 1, ๐‘€๐‘€11 , 1, ๐‘€๐‘€12 , 1), ๐’š๐’š1 ) โˆ‰ ๐ด๐ดโˆˆ }

โ€ฒ Now in the bin ๐‘š๐‘š12 of ๐’™๐’™ sequences look for some ๐‘š๐‘š12 such that

โ€ฒ โ‰ค ๐ผ๐ผ(๐‘‹๐‘‹; ๐‘Œ๐‘Œ1 |๐‘‰๐‘‰) โˆ’ 6๐œ–๐œ– ๐‘…๐‘…12 + ๐‘…๐‘…12 โ€ฒ โ€ฒ ๐‘…๐‘…11 + ๐‘…๐‘…11 + ๐‘…๐‘…12 + ๐‘…๐‘…12 โ‰ค ๐ผ๐ผ(๐‘‹๐‘‹; ๐‘Œ๐‘Œ1 |๐‘ˆ๐‘ˆ) โˆ’ 6๐œ–๐œ– โ€ฒ โ€ฒ โ€ฒ ๐‘…๐‘…0 + ๐‘…๐‘…0 + ๐‘…๐‘…11 + ๐‘…๐‘…11 + ๐‘…๐‘…12 + ๐‘…๐‘…12 โ‰ค ๐ผ๐ผ(๐‘‹๐‘‹; ๐‘Œ๐‘Œ1 ) โˆ’ 5๐œ–๐œ–

โ€ฒ

โ€ฒ (๐’–๐’–(๐‘š๐‘š0โ€ฒ , ๐‘š๐‘š0 ), ๐’—๐’—(๐‘š๐‘š0โ€ฒ , ๐‘š๐‘š0 , ๐‘š๐‘š11 , ๐‘š๐‘š11 ), ๐’”๐’”) โˆˆ

(๐’๐’) ๐‘จ๐‘จ๐๐

โ€ฒ , ๐‘š๐‘š11 ), (๐’–๐’–(๐‘š๐‘š0โ€ฒ , ๐‘š๐‘š0 ), ๐’—๐’—(๐‘š๐‘š0โ€ฒ , ๐‘š๐‘š0 , ๐‘š๐‘š11 (๐’๐’) โ€ฒ โ€ฒ โ€ฒ , ๐‘š๐‘š11 , ๐‘š๐‘š12 , ๐‘š๐‘š12 ), ๐’”๐’”) โˆˆ ๐‘จ๐‘จ๐๐ ๐’™๐’™(๐‘š๐‘š0 , ๐‘š๐‘š0 , ๐‘š๐‘š11

We send the found ๐’™๐’™ sequence. Before bumping into decoding, assume that the correct indices are found through โ€ฒ โ€ฒ = ๐‘€๐‘€11 the encoding procedure, i.e. ๐‘š๐‘š0โ€ฒ = ๐‘€๐‘€0โ€ฒ , ๐‘š๐‘š11 โ€ฒ โ€ฒ and ๐‘š๐‘š12 = ๐‘€๐‘€12 .

Decoding: Since the messages are uniformly distributed over their respective ranges, we can assume, without loss of generality, that the tuple (๐‘š๐‘š0 , ๐‘š๐‘š11 , ๐‘š๐‘š12 ) = (1,1,1) is sent. The second receiver ๐‘Œ๐‘Œ2 receives ๐’š๐’š2 thus having the following error events (๐‘›๐‘›)

๐ธ๐ธ21 = {(๐’–๐’–(๐‘€๐‘€0โ€ฒ , 1), ๐’š๐’š2 ) โˆ‰ ๐ด๐ด๐œ–๐œ– } (๐‘›๐‘›) ๐ธ๐ธ22 = {(๐’–๐’–(๐‘š๐‘š0โ€ฒ , ๐‘š๐‘š0 ), ๐’š๐’š2 ) โˆˆ ๐ด๐ด๐œ–๐œ– ๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘  ๐‘š๐‘š0 โ‰  1 โ€ฒ โ€ฒ ๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž ๐‘š๐‘š0 โ‰  ๐‘€๐‘€0 } (๐‘›๐‘›)

๐ธ๐ธ23 = {(๐’–๐’–(๐‘€๐‘€0โ€ฒ , ๐‘š๐‘š0 ), ๐’š๐’š1 ) โˆˆ ๐ด๐ด๐œ–๐œ– ๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘  ๐‘š๐‘š0 โ‰  1}

leads us to a redundant inequality.

Now by the weak law of large numbers (WLLN) [14], ๐‘๐‘(๐ธ๐ธ21 ) โ‰ค ๐œ–๐œ–, โˆ€๐œ–๐œ– > 0 as ๐‘›๐‘› โ†’ โˆž. For the second error event we have โ€ฒ

๐‘๐‘(๐ธ๐ธ22 ) = ๏ฟฝ ๏ฟฝ ๐‘๐‘(๐’–๐’–)๐‘๐‘(๐’š๐’š2 ) โ‰ค 2๐‘›๐‘›๏ฟฝ๐‘…๐‘…0 +๐‘…๐‘…0 ๏ฟฝ 2๐‘›๐‘›(๐ป๐ป(๐‘ˆ๐‘ˆ,๐‘Œ๐‘Œ2 )+๐œ–๐œ–) 2

โ€ฒ

(๐‘›๐‘› )

2

๐‘…๐‘…0 + ๐‘…๐‘…0โ€ฒ โ‰ค ๐ผ๐ผ(๐‘ˆ๐‘ˆ; ๐‘Œ๐‘Œ2 ) โˆ’ 3๐œ–๐œ–

โ€ฒ ๐ธ๐ธ14 = {( ๐’–๐’–(๐‘š๐‘š0โ€ฒ , ๐‘š๐‘š0 ), ๐’—๐’—(๐‘š๐‘š0โ€ฒ , ๐‘š๐‘š0 , ๐‘š๐‘š11 , ๐‘š๐‘š11 ), (๐‘›๐‘›) โ€ฒ โ€ฒ โ€ฒ ๐’™๐’™(๐‘š๐‘š0 , ๐‘š๐‘š0 , ๐‘š๐‘š11 , ๐‘š๐‘š11 , ๐‘š๐‘š12 , ๐‘š๐‘š12 ), ๐’š๐’š1 ) โˆˆ ๐ด๐ดโˆˆ ๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘  ๐‘š๐‘š0 โ‰  1 ๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž ๐‘š๐‘š0โ€ฒ โ‰  ๐‘€๐‘€0โ€ฒ ๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘  โ€ฒ โ€ฒ ๐‘š๐‘š1๐‘–๐‘– โ‰  1 ๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž ๐‘š๐‘š1๐‘–๐‘– โ‰  ๐‘€๐‘€1๐‘–๐‘– , ๐‘–๐‘– = 1,2}

The first receiverโ€™s probability of error can be arbitrarily made small provided that (6) (7) (8)

The third receiver ๐‘Œ๐‘Œ3 receives ๐’š๐’š3 and needs to decode only the common message indirectly by decoding the message ๐‘š๐‘š11 . The error events are (๐‘›๐‘›)

โ€ฒ , 1), ๐’š๐’š3 ) โˆ‰ ๐ด๐ดโˆˆ } ๐ธ๐ธ31 = {( ๐’–๐’–(๐‘€๐‘€0โ€ฒ , 1), ๐’—๐’—(๐‘€๐‘€0โ€ฒ , 1, ๐‘€๐‘€11 (๐‘›๐‘›) โ€ฒ โ€ฒ โ€ฒ ๐ธ๐ธ32 = {( ๐’–๐’–(๐‘€๐‘€0 , 1), ๐’—๐’—(๐‘€๐‘€0 , 1, ๐‘š๐‘š11 , ๐‘š๐‘š11 ), ๐’š๐’š3 ) โˆˆ ๐ด๐ดโˆˆ ๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“ โ€ฒ โ€ฒ โ€ฒ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘  ๐‘š๐‘š11 โ‰  1 ๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž ๐‘š๐‘š11 โ‰  ๐‘€๐‘€11 } (๐‘›๐‘›) โ€ฒ โ€ฒ โ€ฒ ๐ธ๐ธ33 = {( ๐’–๐’–(๐‘š๐‘š0 , ๐‘š๐‘š0 ), ๐’—๐’—(๐‘š๐‘š0 , ๐‘š๐‘š0 , ๐‘š๐‘š11 , ๐‘š๐‘š11 ), ๐’š๐’š3 ) โˆˆ ๐ด๐ดโˆˆ ๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“ โ€ฒ โ€ฒ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘  ๐‘š๐‘š0 โ‰  1, ๐‘š๐‘š11 โ‰  1, ๐‘š๐‘š0โ€ฒ โ‰  ๐‘€๐‘€0โ€ฒ , ๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž ๐‘š๐‘š11 โ‰  ๐‘€๐‘€11 }

Again by using WLLN and AEP, we see that the third receiverโ€™s error probabilities can be arbitrarily made small as ๐‘›๐‘› โ†’ โˆž provided that (9)

Using Gelโ€™fand-Pinsker coding we see that the encoders can โ€ฒ โ€ฒ choose the proper ๐‘š๐‘š0โ€ฒ , ๐‘š๐‘š11 , ๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž ๐‘š๐‘š12 indices with vanishing probability of error provided that for every ๐œ–๐œ– > 0 and sufficiently large n ๐‘…๐‘…0โ€ฒ โ‰ฅ ๐ผ๐ผ(๐‘ˆ๐‘ˆ; ๐‘†๐‘†) + 2๐œ–๐œ– โ€ฒ ๐‘…๐‘…11 โ‰ฅ ๐ผ๐ผ(๐‘‰๐‘‰; ๐‘†๐‘†|๐‘ˆ๐‘ˆ) + 2๐œ–๐œ– โ€ฒ ๐‘…๐‘…12 โ‰ฅ ๐ผ๐ผ(๐‘‹๐‘‹; ๐‘†๐‘†|๐‘‰๐‘‰) + 2๐œ–๐œ–

(10) (11) (12)

๐ผ๐ผ(๐‘‰๐‘‰; ๐‘†๐‘†|๐‘ˆ๐‘ˆ) + ๐ผ๐ผ(๐‘ˆ๐‘ˆ; ๐‘†๐‘†) = ๐ผ๐ผ(๐‘‰๐‘‰๐‘‰๐‘‰; ๐‘†๐‘†)

(13)

Now combining (5) - (9) and (10) - (12) and noting that

โ€ฒ

= 2โˆ’๐‘›๐‘›๏ฟฝ๐ผ๐ผ(๐‘ˆ๐‘ˆ;๐‘Œ๐‘Œ2 )โˆ’3๐œ–๐œ–โˆ’๐‘…๐‘…0 โˆ’๐‘…๐‘…0 ๏ฟฝ

We see that โˆ€๐œ–๐œ– > 0, ๐‘๐‘(๐ธ๐ธ22 ) โ‰ค ๐œ–๐œ– as ๐‘›๐‘› โ†’ โˆž provided that

โ€ฒ ๐ธ๐ธ13 = {( ๐’–๐’–(๐‘€๐‘€0โ€ฒ , 1), ๐’—๐’—(๐‘€๐‘€0โ€ฒ , 1, ๐‘š๐‘š11 , ๐‘š๐‘š11 ), (๐‘›๐‘›) โ€ฒ โ€ฒ โ€ฒ ๐’™๐’™(๐‘€๐‘€0 , 1, ๐‘š๐‘š11 , ๐‘š๐‘š11 , ๐‘š๐‘š12 , ๐‘š๐‘š12 ), ๐’š๐’š1 ) โˆˆ ๐ด๐ดโˆˆ โ€ฒ โ€ฒ ๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘’๐‘’ ๐‘š๐‘š1๐‘–๐‘– โ‰  1 ๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž ๐‘š๐‘š1๐‘–๐‘– โ‰  ๐‘€๐‘€1๐‘–๐‘– , ๐‘–๐‘– = 1,2}

โ€ฒ ๐‘…๐‘…0 + ๐‘…๐‘…0โ€ฒ + ๐‘…๐‘…11 + ๐‘…๐‘…11 โ‰ค ๐ผ๐ผ(๐‘‰๐‘‰; ๐‘Œ๐‘Œ3 ) โˆ’ 3๐œ–๐œ–

Remark 1: The following error event

๐‘š๐‘š 0 ,๐‘š๐‘š 0 ๐ด๐ด ๐œ–๐œ– โˆ’๐‘›๐‘›(๐ป๐ป(๐‘ˆ๐‘ˆ)โˆ’๐œ–๐œ–) โˆ’๐‘›๐‘›(๐ป๐ป(๐‘Œ๐‘Œ2 )โˆ’๐œ–๐œ–)

โ€ฒ ๐ธ๐ธ12 = {( ๐’–๐’–(๐‘€๐‘€0โ€ฒ , 1), ๐’—๐’—(๐‘€๐‘€0โ€ฒ , 1, ๐‘€๐‘€11 , 1), (๐‘›๐‘›) โ€ฒ โ€ฒ โ€ฒ ๐’™๐’™(๐‘€๐‘€0 , 1, ๐‘€๐‘€11 , 1, ๐‘š๐‘š12 , ๐‘š๐‘š12 ), ๐’š๐’š1 ) โˆˆ ๐ด๐ดโˆˆ โ€ฒ โ€ฒ ๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘  ๐‘š๐‘š12 โ‰  1 ๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž ๐‘š๐‘š12 โ‰  ๐‘€๐‘€12 }

(5)

The first receiver ๐‘Œ๐‘Œ1 receives ๐’š๐’š1 and needs to decode both ๐‘š๐‘š0 and ๐‘š๐‘š1 . Therefore, the error events are

and using Fourier-Motzkin procedure afterwards to eliminate ๐‘…๐‘…11 ๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž ๐‘…๐‘…12 , we obtain (4) as an achievable rate region for Multilevel BC with side information. โˆŽ

IV.

THREE-RECEIVER LESS NOISY BROADCAST CHANNEL WITH SIDE INFORMATION

Define ๐’ซ๐’ซ โˆ— as the collection of all random variables (๐‘ˆ๐‘ˆ, ๐‘‰๐‘‰, ๐‘†๐‘†, ๐‘‹๐‘‹, ๐‘Œ๐‘Œ1 , ๐‘Œ๐‘Œ2 , ๐‘Œ๐‘Œ3 ) with finite alphabets such that ๐‘๐‘(๐‘ข๐‘ข, ๐‘ฃ๐‘ฃ, ๐‘ ๐‘ , ๐‘ฅ๐‘ฅ, ๐‘ฆ๐‘ฆ1 , ๐‘ฆ๐‘ฆ2 , ๐‘ฆ๐‘ฆ3 ) = ๐‘๐‘(๐‘ ๐‘ )๐‘๐‘(๐‘ข๐‘ข|๐‘ ๐‘ )๐‘๐‘(๐‘ฃ๐‘ฃ|๐‘ข๐‘ข, ๐‘ ๐‘ )๐‘๐‘(๐‘ฅ๐‘ฅ|๐‘ฃ๐‘ฃ, ๐‘ ๐‘ )๐‘๐‘(๐‘ฆ๐‘ฆ1 , ๐‘ฆ๐‘ฆ2 , ๐‘ฆ๐‘ฆ3 |๐‘ฅ๐‘ฅ, ๐‘ ๐‘ )

(14)

Theorem 2: A rate triple(๐‘…๐‘…1 , ๐‘…๐‘…2 , ๐‘…๐‘…3 ) is achievable for 3receiver less noisy BC with side information non-causally available at the transmitter provided that

(15)

๐‘…๐‘…1 โ‰ค ๐ผ๐ผ(๐‘‹๐‘‹; ๐‘Œ๐‘Œ1 |๐‘‰๐‘‰๐‘‰๐‘‰) ๐‘…๐‘…2 โ‰ค ๐ผ๐ผ(๐‘‰๐‘‰; ๐‘Œ๐‘Œ2 |๐‘ˆ๐‘ˆ๐‘ˆ๐‘ˆ) ๐‘…๐‘…3 โ‰ค ๐ผ๐ผ(๐‘ˆ๐‘ˆ; ๐‘Œ๐‘Œ3 |๐‘†๐‘†)

(22)

Achievability: The direct part of the proof is achieved if you set ๐‘Œ๐‘Œ๏ฟฝ๐‘˜๐‘˜ = (๐‘Œ๐‘Œ๐‘˜๐‘˜ , ๐‘†๐‘†), ๐‘˜๐‘˜ = 1,2,3 in (15).

Converse: The converse part uses an extension of lemma 1 in [4].

for some joint distribution on ๐’ซ๐’ซ โˆ— .

Corollary 2.1: By setting ๐‘†๐‘† โ‰ก โˆ… in the above rate region, it reduces to the capacity region of 3-receiver less noisy BC given in [4]. Proof: The proof uses Coverโ€™s superposition [15] and Gelโ€™fand-Pinsker random binning coding [6] procedures along with Nairโ€™s indirect decoding and is similar to the last proof provided and thus only an outline is provided. Fix n and a distribution on ๐’ซ๐’ซ โˆ— . Again note that side information is distributed i.i.d according to ๐‘›๐‘›

๐‘๐‘(๐’”๐’”) = ๏ฟฝ ๐‘๐‘(๐‘ ๐‘ ๐‘–๐‘– ) ๐‘–๐‘–=1

Randomly and independently generate โ€ฒ ) sequences ๐’–๐’–(๐‘š๐‘š3 , ๐‘š๐‘š3 , each distributed i.i.d 2 according to โˆ๐‘›๐‘›๐‘–๐‘–=1 ๐‘๐‘(๐‘ข๐‘ข๐‘–๐‘– ) and randomly throw them into 2๐‘›๐‘›๐‘…๐‘…3 bins. For each ๐’–๐’–(๐‘š๐‘š3โ€ฒ , ๐‘š๐‘š3 ), randomly and independently generate ๐‘›๐‘›(๐‘…๐‘…2โ€ฒ +๐‘…๐‘…2 ) sequences ๐’—๐’—(๐‘š๐‘š3โ€ฒ , ๐‘š๐‘š3 , ๐‘š๐‘š2โ€ฒ , ๐‘š๐‘š2 ) each distributed i.i.d 2 according to โˆ๐‘›๐‘›๐‘–๐‘–=1 ๐‘๐‘๐‘‰๐‘‰|๐‘ˆ๐‘ˆ (๐‘ฃ๐‘ฃ๐‘–๐‘– |๐‘ข๐‘ข๐‘–๐‘– ) and randomly throw them into 2๐‘›๐‘›๐‘…๐‘…2 bins. Now for each generated ๐’—๐’—(๐‘š๐‘š3โ€ฒ , ๐‘š๐‘š3 , ๐‘š๐‘š2โ€ฒ , ๐‘š๐‘š2 ), randomly and independently generate โ€ฒ โ€ฒ ๐‘›๐‘›(๐‘…๐‘…1โ€ฒ +๐‘…๐‘…1 ) โ€ฒ sequences ๐’™๐’™(๐‘š๐‘š3 , ๐‘š๐‘š3 , ๐‘š๐‘š2 , ๐‘š๐‘š2 , ๐‘š๐‘š1 , ๐‘š๐‘š1 ) , each one 2 distributed i.i.d according to โˆ๐‘›๐‘›๐‘–๐‘–=1 ๐‘๐‘๐‘‹๐‘‹|๐‘‰๐‘‰ (๐‘ฅ๐‘ฅ๐‘–๐‘– |๐‘ฃ๐‘ฃ๐‘–๐‘– ) and randomly throw them into 2๐‘›๐‘›๐‘…๐‘…1 bins. Encoding is succeeded with small probability of error provided that ๐‘…๐‘…3โ€ฒ โ‰ฅ ๐ผ๐ผ(๐‘ˆ๐‘ˆ; ๐‘†๐‘†) ๐‘…๐‘…2โ€ฒ โ‰ฅ ๐ผ๐ผ(๐‘‰๐‘‰; ๐‘†๐‘†|๐‘ˆ๐‘ˆ) ๐‘…๐‘…1โ€ฒ โ‰ฅ ๐ผ๐ผ(๐‘‹๐‘‹; ๐‘†๐‘†|๐‘‰๐‘‰)

(16) (17) (18)

๐‘…๐‘…3 + ๐‘…๐‘…3โ€ฒ ๐‘…๐‘…2 + ๐‘…๐‘…2โ€ฒ ๐‘…๐‘…1 + ๐‘…๐‘…1โ€ฒ

(19) (20) (21)

and decoding is succeeded if โ‰ค ๐ผ๐ผ(๐‘ˆ๐‘ˆ; ๐‘Œ๐‘Œ3 ) โ‰ค ๐ผ๐ผ(๐‘‰๐‘‰; ๐‘Œ๐‘Œ2 |๐‘ˆ๐‘ˆ) โ‰ค ๐ผ๐ผ(๐‘‹๐‘‹; ๐‘Œ๐‘Œ1 |๐‘‰๐‘‰)

Theorem 3: The capacity region of the 3-receiver less noisy BC with side information, non-causally available at the transmitter and the receivers is the set of all rate triples(๐‘…๐‘…1 , ๐‘…๐‘…2 , ๐‘…๐‘…3 ) such that

Proof:

๐‘…๐‘…1 โ‰ค ๐ผ๐ผ(๐‘‹๐‘‹; ๐‘Œ๐‘Œ1 |๐‘‰๐‘‰) โˆ’ ๐ผ๐ผ(๐‘‹๐‘‹; ๐‘†๐‘†|๐‘‰๐‘‰) ๐‘…๐‘…2 โ‰ค ๐ผ๐ผ(๐‘‰๐‘‰; ๐‘Œ๐‘Œ2 |๐‘ˆ๐‘ˆ) โˆ’ ๐ผ๐ผ(๐‘‰๐‘‰; ๐‘†๐‘†|๐‘ˆ๐‘ˆ) ๐‘…๐‘…3 โ‰ค ๐ผ๐ผ(๐‘ˆ๐‘ˆ; ๐‘Œ๐‘Œ3 ) โˆ’ ๐ผ๐ผ(๐‘ˆ๐‘ˆ; ๐‘†๐‘†)

๐‘›๐‘›(๐‘…๐‘…3โ€ฒ +๐‘…๐‘…3 )

Now combining (16), (17) and (18) with (19), (20) and (21) gives us (15). โˆŽ

Lemma 1: [4] Let the channel ๐‘‹๐‘‹ โ†’ ๐‘Œ๐‘Œ be less noisy than the channel ๐‘‹๐‘‹ โ†’ ๐‘๐‘. Consider (๐‘€๐‘€, ๐‘†๐‘† ๐‘›๐‘› ) to be any random vector such that (๐‘€๐‘€, ๐‘†๐‘† ๐‘›๐‘› ) โ†’ ๐‘‹๐‘‹ ๐‘›๐‘› โ†’ (๐‘Œ๐‘Œ ๐‘›๐‘› , ๐‘๐‘ ๐‘›๐‘› )

forms a Markov chain. Then

1.

๐ผ๐ผ๏ฟฝ๐‘Œ๐‘Œ ๐‘–๐‘–โˆ’1 ; ๐‘๐‘๐‘–๐‘– ๏ฟฝ๐‘€๐‘€, ๐‘†๐‘† ๐‘›๐‘› ๏ฟฝ โ‰ฅ ๐ผ๐ผ๏ฟฝ๐‘๐‘ ๐‘–๐‘–โˆ’1 ; ๐‘๐‘๐‘–๐‘– ๏ฟฝ๐‘€๐‘€, ๐‘†๐‘† ๐‘›๐‘› ๏ฟฝ

2. ๐ผ๐ผ๏ฟฝ๐‘Œ๐‘Œ ๐‘–๐‘–โˆ’1 ; ๐‘Œ๐‘Œ๐‘–๐‘– ๏ฟฝ๐‘€๐‘€, ๐‘†๐‘† ๐‘›๐‘› ๏ฟฝ โ‰ฅ ๐ผ๐ผ๏ฟฝ๐‘๐‘ ๐‘–๐‘–โˆ’1 ; ๐‘Œ๐‘Œ๐‘–๐‘– ๏ฟฝ๐‘€๐‘€, ๐‘†๐‘† ๐‘›๐‘› ๏ฟฝ

Proof: First of all note that since the channel is memoryless we have ๐‘›๐‘› (๐‘€๐‘€1 , ๐‘€๐‘€2 , ๐‘€๐‘€3 , ๐‘Œ๐‘Œ1๐‘–๐‘–โˆ’1 , ๐‘Œ๐‘Œ2๐‘–๐‘–โˆ’1 , ๐‘Œ๐‘Œ3๐‘–๐‘–โˆ’1 , ๐‘†๐‘† ๐‘–๐‘–โˆ’1 , ๐‘†๐‘†๐‘–๐‘–+1 ) โ†’ (๐‘‹๐‘‹๐‘–๐‘– , ๐‘†๐‘†๐‘–๐‘– ) โ†’ (๐‘Œ๐‘Œ1๐‘–๐‘– , ๐‘Œ๐‘Œ2๐‘–๐‘– , ๐‘Œ๐‘Œ3๐‘–๐‘– )

Just like [4], for any 1 โ‰ค ๐‘Ÿ๐‘Ÿ โ‰ค ๐‘–๐‘– โˆ’ 1 ๐ผ๐ผ(๐‘๐‘ ๐‘Ÿ๐‘Ÿโˆ’1 , ๐‘Œ๐‘Œ๐‘Ÿ๐‘Ÿ๐‘–๐‘–โˆ’1 ; ๐‘Œ๐‘Œ๐‘–๐‘– ๏ฟฝ๐‘€๐‘€, ๐‘†๐‘† ๐‘›๐‘› )

๐‘–๐‘–โˆ’1 ; ๐‘Œ๐‘Œ๐‘–๐‘– ๏ฟฝ๐‘€๐‘€, ๐‘†๐‘† ๐‘›๐‘› ๏ฟฝ = ๐ผ๐ผ๏ฟฝ๐‘๐‘ ๐‘Ÿ๐‘Ÿ โˆ’1 , ๐‘Œ๐‘Œ๐‘Ÿ๐‘Ÿ+1

๐‘–๐‘–โˆ’1 ๏ฟฝ +๐ผ๐ผ๏ฟฝ๐‘Œ๐‘Œ๐‘Ÿ๐‘Ÿ ; ๐‘Œ๐‘Œ๐‘–๐‘– ๏ฟฝ๐‘€๐‘€, ๐‘†๐‘† ๐‘›๐‘› , ๐‘๐‘ ๐‘Ÿ๐‘Ÿ โˆ’1 , ๐‘Œ๐‘Œ๐‘Ÿ๐‘Ÿ+1

๐‘›๐‘› โ‰ฅ ๐ผ๐ผ๏ฟฝ๐‘๐‘ ๐‘Ÿ๐‘Ÿ โˆ’1 , ๐‘Œ๐‘Œ๐‘Ÿ๐‘Ÿ๐‘–๐‘–โˆ’1 +1 ; ๐‘Œ๐‘Œ๐‘–๐‘– ๏ฟฝ๐‘€๐‘€, ๐‘†๐‘† ๏ฟฝ

๐‘–๐‘–โˆ’1 ๏ฟฝ +๐ผ๐ผ๏ฟฝ๐‘๐‘๐‘Ÿ๐‘Ÿ ; ๐‘Œ๐‘Œ๐‘–๐‘– ๏ฟฝ๐‘€๐‘€, ๐‘†๐‘† ๐‘›๐‘› , ๐‘๐‘ ๐‘Ÿ๐‘Ÿโˆ’1 , ๐‘Œ๐‘Œ๐‘Ÿ๐‘Ÿ+1

๐‘–๐‘–โˆ’1 ; ๐‘Œ๐‘Œ๐‘–๐‘– ๏ฟฝ๐‘€๐‘€, ๐‘†๐‘† ๐‘›๐‘› ๏ฟฝ = ๐ผ๐ผ๏ฟฝ๐‘๐‘ ๐‘Ÿ๐‘Ÿ , ๐‘Œ๐‘Œ๐‘Ÿ๐‘Ÿ+1

where the inequality follows from the memorylessness of the channel and the fact that ๐‘Œ๐‘Œ is less noisy than ๐‘๐‘, i.e. ๐‘–๐‘–โˆ’1 ๐‘–๐‘–โˆ’1 ๏ฟฝ โ‰ฅ ๐ผ๐ผ๏ฟฝ๐‘๐‘๐‘Ÿ๐‘Ÿ ; ๐‘Œ๐‘Œ๐‘–๐‘– ๏ฟฝ๐‘€๐‘€, ๐‘†๐‘† ๐‘›๐‘› , ๐‘๐‘ ๐‘Ÿ๐‘Ÿโˆ’1 , ๐‘Œ๐‘Œ๐‘Ÿ๐‘Ÿ+1 ๏ฟฝ. ๐ผ๐ผ๏ฟฝ๐‘Œ๐‘Œ๐‘Ÿ๐‘Ÿ ; ๐‘Œ๐‘Œ๐‘–๐‘– ๏ฟฝ๐‘€๐‘€, ๐‘†๐‘† ๐‘›๐‘› , ๐‘๐‘ ๐‘Ÿ๐‘Ÿโˆ’1 , ๐‘Œ๐‘Œ๐‘Ÿ๐‘Ÿ+1

Proof of the second part follows the same as the first part โˆŽ with negligible variations. Now we stick to the proof of the converse

๐‘›๐‘›๐‘…๐‘…3 = ๐ป๐ป(๐‘€๐‘€3 ) = ๐ป๐ป(๐‘€๐‘€3 |๐‘†๐‘† ๐‘›๐‘› ) = ๐ป๐ป(๐‘€๐‘€3 |๐‘†๐‘† ๐‘›๐‘› , ๐‘Œ๐‘Œ3๐‘›๐‘› )

+๐ผ๐ผ(๐‘€๐‘€3 ; ๐‘Œ๐‘Œ3๐‘›๐‘› |๐‘†๐‘† ๐‘›๐‘› ) โ‰ค ๐ป๐ป(๐‘€๐‘€3 |๐‘Œ๐‘Œ3๐‘›๐‘› ) + ๐ผ๐ผ(๐‘€๐‘€3 ; ๐‘Œ๐‘Œ3๐‘›๐‘› |๐‘†๐‘† ๐‘›๐‘› ) ๐‘›๐‘›

โ‰ค ๐‘›๐‘›๐œ–๐œ–3๐‘›๐‘› + ๏ฟฝ ๐ผ๐ผ(๐‘€๐‘€3 ; ๐‘Œ๐‘Œ3๐‘–๐‘– |๐‘†๐‘† ๐‘›๐‘› , ๐‘Œ๐‘Œ3๐‘–๐‘–โˆ’1 ) โ‰ค ๐‘›๐‘›๐œ–๐œ–3๐‘›๐‘› ๐‘–๐‘–=1

๐‘›๐‘›

+ ๏ฟฝ ๐ผ๐ผ(๐‘€๐‘€3 ; ๐‘Œ๐‘Œ3๐‘–๐‘– |๐‘†๐‘† ๐‘–๐‘–=1 ๐‘›๐‘›

๐‘–๐‘–โˆ’1

๐‘›๐‘› , ๐‘†๐‘†๐‘–๐‘– , ๐‘†๐‘†๐‘–๐‘–+1 , ๐‘Œ๐‘Œ3๐‘–๐‘–โˆ’1 )

๐‘›๐‘›

+ ๏ฟฝ ๐ผ๐ผ(๐‘‹๐‘‹๐‘–๐‘– ; ๐‘Œ๐‘Œ1๐‘–๐‘– |๐‘ˆ๐‘ˆ๐‘–๐‘– , ๐‘†๐‘†๐‘–๐‘– ),

โ‰ค ๐‘›๐‘›๐œ–๐œ–3๐‘›๐‘›

๐‘–๐‘–=1

where (a) follows from the memorylessness of the channel and (b) follows from Lemma 1.

๐‘›๐‘› , ๐‘Œ๐‘Œ3๐‘–๐‘–โˆ’1 ; ๐‘Œ๐‘Œ3๐‘–๐‘– ๏ฟฝ๐‘†๐‘†๐‘–๐‘– ๏ฟฝ โ‰ค ๐‘›๐‘›๐œ–๐œ–3๐‘›๐‘› + ๏ฟฝ ๐ผ๐ผ๏ฟฝ๐‘€๐‘€3 , ๐‘†๐‘† ๐‘–๐‘–โˆ’1 , ๐‘†๐‘†๐‘–๐‘–+1 ๐‘–๐‘–=1 ๐‘›๐‘›

๐‘›๐‘›

๐‘–๐‘–=1

๐‘–๐‘–=1

๐‘›๐‘› , ๐‘Œ๐‘Œ2๐‘–๐‘–โˆ’1 ; ๐‘Œ๐‘Œ3๐‘–๐‘– ๏ฟฝ๐‘†๐‘†๐‘–๐‘– ๏ฟฝ = ๐‘›๐‘›๐œ–๐œ–3๐‘›๐‘› + ๏ฟฝ ๐ผ๐ผ(๐‘ˆ๐‘ˆ๐‘–๐‘– ; ๐‘Œ๐‘Œ3๐‘–๐‘– |๐‘†๐‘†๐‘–๐‘– ) + ๏ฟฝ ๐ผ๐ผ๏ฟฝ๐‘€๐‘€3 , ๐‘†๐‘† ๐‘–๐‘–โˆ’1 , ๐‘†๐‘†๐‘–๐‘–+1 ๐‘›๐‘› , ๐‘Œ๐‘Œ2๐‘–๐‘–โˆ’1 ๏ฟฝ and the last inequality where ๐‘ˆ๐‘ˆ๐‘–๐‘– โ‰œ ๏ฟฝ๐‘€๐‘€3 , ๐‘†๐‘† ๐‘–๐‘–โˆ’1 , ๐‘†๐‘†๐‘–๐‘–+1 follows from Lemma 1.

๐‘›๐‘›๐‘…๐‘…2 = ๐ป๐ป(๐‘€๐‘€2 ) = ๐ป๐ป(๐‘€๐‘€2 |๐‘€๐‘€3 , ๐‘†๐‘†

๐‘›๐‘› )

= ๐ป๐ป(๐‘€๐‘€2 |๐‘€๐‘€3 , ๐‘†๐‘†

๐‘›๐‘›

, ๐‘Œ๐‘Œ2๐‘›๐‘› )

+๐ผ๐ผ(๐‘€๐‘€2 ; ๐‘Œ๐‘Œ2๐‘›๐‘› |๐‘€๐‘€3 , ๐‘†๐‘† ๐‘›๐‘› ) โ‰ค ๐ป๐ป(๐‘€๐‘€2 |๐‘Œ๐‘Œ2๐‘›๐‘› ) + ๐ผ๐ผ(๐‘€๐‘€2 ; ๐‘Œ๐‘Œ2๐‘›๐‘› |๐‘€๐‘€3 , ๐‘†๐‘† ๐‘›๐‘› ) ๐‘›๐‘›

๐‘›๐‘› โ‰ค ๐‘›๐‘›๐œ–๐œ–2๐‘›๐‘› + ๏ฟฝ ๐ผ๐ผ๏ฟฝ๐‘€๐‘€2 ; ๐‘Œ๐‘Œ2๐‘–๐‘– ๏ฟฝ๐‘€๐‘€3 , ๐‘†๐‘† ๐‘–๐‘–โˆ’1 , ๐‘†๐‘†๐‘–๐‘– , ๐‘†๐‘†๐‘–๐‘–+1 , ๐‘Œ๐‘Œ2๐‘–๐‘–โˆ’1 ๏ฟฝ = ๐‘›๐‘›๐œ–๐œ–2๐‘›๐‘› ๐‘›๐‘›

๐‘–๐‘–=1

๐‘›๐‘› ๐‘›๐‘› , ๐‘Œ๐‘Œ2๐‘–๐‘–โˆ’1 ; ๐‘Œ๐‘Œ2๐‘–๐‘– ๏ฟฝ๐‘€๐‘€3 , ๐‘†๐‘† ๐‘–๐‘–โˆ’1 , ๐‘†๐‘†๐‘–๐‘–+1 , ๐‘Œ๐‘Œ2๐‘–๐‘–โˆ’1 , ๐‘†๐‘†๐‘–๐‘– ๏ฟฝ + ๏ฟฝ ๐ผ๐ผ๏ฟฝ๐‘€๐‘€2 , ๐‘€๐‘€3 , ๐‘†๐‘† ๐‘–๐‘–โˆ’1 , ๐‘†๐‘†๐‘–๐‘–+1 ๐‘–๐‘–=1

๐‘›๐‘›

๐‘›๐‘› โ‰ค ๐‘›๐‘›๐œ–๐œ–1๐‘›๐‘› + ๏ฟฝ ๐ผ๐ผ๏ฟฝ๐‘€๐‘€1 ; ๐‘Œ๐‘Œ1๐‘–๐‘– ๏ฟฝ๐‘€๐‘€2 , ๐‘€๐‘€3 , ๐‘†๐‘† ๐‘–๐‘–โˆ’1 , ๐‘†๐‘†๐‘–๐‘– , ๐‘†๐‘†๐‘–๐‘–+1 , ๐‘Œ๐‘Œ1๐‘–๐‘–โˆ’1 ๏ฟฝ ๐‘–๐‘–=1

+ ๏ฟฝ ๐ผ๐ผ๏ฟฝ๐‘‹๐‘‹๐‘–๐‘– ; ๐‘Œ๐‘Œ1๐‘–๐‘– ๏ฟฝ๐‘€๐‘€2 , ๐‘€๐‘€3 , ๐‘†๐‘†

๐‘–๐‘–โˆ’1

๐‘›๐‘› , ๐‘†๐‘†๐‘–๐‘– , ๐‘†๐‘†๐‘–๐‘–+1 , ๐‘Œ๐‘Œ1๐‘–๐‘–โˆ’1 ๏ฟฝ

๐‘›๐‘› ) + ๏ฟฝ ๐ผ๐ผ(๐‘‹๐‘‹๐‘–๐‘– ; ๐‘Œ๐‘Œ1๐‘–๐‘– ๏ฟฝ๐‘€๐‘€2 , ๐‘€๐‘€3 , ๐‘†๐‘† ๐‘–๐‘–โˆ’1 , ๐‘†๐‘†๐‘–๐‘– , ๐‘†๐‘†๐‘–๐‘–+1 ๐‘–๐‘–=1 ๐‘›๐‘›

๐‘›๐‘› ๏ฟฝ โˆ’ ๏ฟฝ ๐ผ๐ผ๏ฟฝ๐‘Œ๐‘Œ1๐‘–๐‘–โˆ’1 ; ๐‘Œ๐‘Œ1๐‘–๐‘– ๏ฟฝ๐‘€๐‘€2 , ๐‘€๐‘€3 , ๐‘†๐‘† ๐‘–๐‘–โˆ’1 , ๐‘†๐‘†๐‘–๐‘– , ๐‘†๐‘†๐‘–๐‘–+1 ๐‘–๐‘–=1 ๐‘›๐‘›

[1] [2]

[4]

+๐ผ๐ผ(๐‘€๐‘€1 ; ๐‘Œ๐‘Œ1๐‘›๐‘› |๐‘€๐‘€2 , ๐‘€๐‘€3 , ๐‘†๐‘† ๐‘›๐‘› ) โ‰ค ๐ป๐ป(๐‘€๐‘€1 |๐‘Œ๐‘Œ1๐‘›๐‘› ) + ๐ผ๐ผ(๐‘€๐‘€1 ; ๐‘Œ๐‘Œ1๐‘›๐‘› |๐‘€๐‘€2 , ๐‘€๐‘€3 , ๐‘†๐‘† ๐‘›๐‘› )

๐‘›๐‘›

REFERENCES

๐‘–๐‘–=1

๐‘›๐‘›๐‘…๐‘…1 = ๐ป๐ป(๐‘€๐‘€1 |๐‘†๐‘† ๐‘›๐‘› , ๐‘€๐‘€2 , ๐‘€๐‘€3 ) = ๐ป๐ป(๐‘€๐‘€1 |๐‘€๐‘€2 , ๐‘€๐‘€3 , ๐‘†๐‘† ๐‘›๐‘› , ๐‘Œ๐‘Œ1๐‘›๐‘› )

๐‘›๐‘› ) + ๏ฟฝ ๐ผ๐ผ(๐‘‹๐‘‹๐‘–๐‘– ; ๐‘Œ๐‘Œ1๐‘–๐‘– ๏ฟฝ๐‘€๐‘€2 , ๐‘€๐‘€3 , ๐‘†๐‘† ๐‘–๐‘–โˆ’1 , ๐‘†๐‘†๐‘–๐‘– , ๐‘†๐‘†๐‘–๐‘–+1

(๐‘Ž๐‘Ž) ๐‘›๐‘›๐œ–๐œ– โ‰ค 1๐‘›๐‘›

= ๐‘›๐‘›๐œ–๐œ–1๐‘›๐‘› +

[5] [6] [7] [8]

[9] [10]

(๐‘๐‘) ๐‘›๐‘›๐œ–๐œ– โ‰ค 1๐‘›๐‘›

๐‘–๐‘–=1 ๐‘›๐‘›

[11] [12] [13]

๐‘›๐‘› ๏ฟฝ = ๐‘›๐‘›๐œ–๐œ–1๐‘›๐‘› โˆ’ ๏ฟฝ ๐ผ๐ผ๏ฟฝ๐‘Œ๐‘Œ2๐‘–๐‘–โˆ’1 ; ๐‘Œ๐‘Œ1๐‘–๐‘– ๏ฟฝ๐‘€๐‘€2 , ๐‘€๐‘€3 , ๐‘†๐‘† ๐‘–๐‘–โˆ’1 , ๐‘†๐‘†๐‘–๐‘– , ๐‘†๐‘†๐‘–๐‘–+1

[14]

๐‘›๐‘› , ๐‘Œ๐‘Œ2๐‘–๐‘–โˆ’1 , ๐‘†๐‘†๐‘–๐‘– ๏ฟฝ = ๐‘›๐‘›๐œ–๐œ–1๐‘›๐‘› + ๏ฟฝ ๐ผ๐ผ๏ฟฝ๐‘‹๐‘‹๐‘–๐‘– ; ๐‘Œ๐‘Œ1๐‘–๐‘– ๏ฟฝ๐‘€๐‘€2 , ๐‘€๐‘€3 , ๐‘†๐‘† ๐‘–๐‘–โˆ’1 , ๐‘†๐‘†๐‘–๐‘–+1

[16]

๐‘–๐‘–=1 ๐‘›๐‘›

๐‘–๐‘–=1

CONCLUSION

We established two achievable rate regions for two special classes of 3-receiver BCs with side information. We also found the capacity region of 3-receiver less noisy BC when side information is available both at the transmitter and at the receivers.

[3]

๐‘›๐‘› , ๐‘Œ๐‘Œ2๐‘–๐‘–โˆ’1 ๏ฟฝ. It is clear that for the where ๐‘‰๐‘‰๐‘–๐‘– โ‰œ ๏ฟฝ๐‘€๐‘€2 , ๐‘€๐‘€3 , ๐‘†๐‘† ๐‘–๐‘–โˆ’1 , ๐‘†๐‘†๐‘–๐‘–+1 given choice of ๐‘ˆ๐‘ˆ๐‘–๐‘– and ๐‘‰๐‘‰๐‘–๐‘– , we have the Markov chain (2) satisfied for the channel is assumed to be memoryless.

๐‘–๐‘–=1

V.

๐‘›๐‘›

= ๐‘›๐‘›๐œ–๐œ–2๐‘›๐‘› + ๏ฟฝ ๐ผ๐ผ(๐‘‰๐‘‰๐‘–๐‘– ; ๐‘Œ๐‘Œ2๐‘–๐‘– |๐‘ˆ๐‘ˆ๐‘–๐‘– , ๐‘†๐‘†๐‘–๐‘– ),

๐‘›๐‘›

Now using the standard time sharing scheme, we can easily conclude that any achievable rate triple for the three-receiver less noisy broadcast channel with side information nancausally available at the transmitter and at the receivers, must satisfy (22) and the proof is complete. โˆŽ

[15]

S. Borade, L. Zheng, and M. Trott., โ€œMultilevel Broadcast Networks,โ€ Int. Symp. on Inf. Theory, pp. 1151-1155, June 2007. J. Kรถrner and K. Marton, โ€œGeneral broadcast channels with degraded message sets,โ€ IEEE Trans. On Inf. Theory IT-23, Jan. 1977, pp. 60-64. C. Nair and A. El Gamal, โ€The capacity region of a class of 3-receiver broadcast channels with degraded message sets,โ€ IEEE Trans. On Inf. Theory, vol. 55, no. 10, pp. 4479-4493, Oct. 2009. C. Nair and Z. V. Wang, โ€œThe Capacity Region of the Three-receiver Less Noisy Broadcast Channel,โ€ IEEE Trans. On Inf. Theory, Vol. 57, pp. 4058-4062, July 2011. C. E. Shannon, โ€œChannels with Side Information at the Transmitter,โ€ IBM Journal of Research and Development, Vol. 2, pp. 289-293, Oct. 1958. S. I. Gelโ€™fand and M. S. Pinsker, โ€œCoding for Channel with random parameters,โ€ Probl. Contr. And Inform. Theory, Vol. 9, no. 1, pp. 19-31, 1980. T. M. Cover; M. Chiang, โ€œDuality Between Channel Capacity and Rate Distortion with two-sided State Information,โ€ IEEE Trans. On Inf. Theory, Vol. 48, pp. 1629-1638, June 2002. Y. Steinberg, โ€œCoding for the Degraded Broadcast Channel with Random Parameters, with Causal and Noncausal Side Information,โ€ IEEE Trans. On Inform. Theory, Vol. 51, no. 8, pp. 2867-2877, Aug. 2005. Reza Khosravi-Farsani; Farokh Marvasti, โ€œCapacity Bounds for Multiuser Channels with Non-Causal Channel State Information at the Transmitters,โ€ Inf. Theory. Workshop (ITW), pp. 195-199, Oct. 2011. J. Kรถrner and K. Marton, โ€œComparison of two noisy Channelsโ€, Topics on Information Theory (ed. by I. Csiszar and P.Elias), Keszthely, Hungry, August 1975, 411-423. C. Nair, โ€œCapacity Regions of two new classes of two-receiver broadcast channelsโ€, IEEE Trans. On Inf. Theory, Vol. 56, pp. 4207-4214, Sept. 2010. T. M. Cover and J. A. Thomas, โ€œElements of Information Theory,โ€ John Wiley & Sons, 1991. A. El Gamal and Y. H. Kim, โ€œNetwork Information Theory,โ€ Cambridge University Press, 2011. S. M. Ross, โ€œA First Course in Probability Theory,โ€ Prentice Hall, printed in the United States of America, 5th edition, 1997. T. M. Cover, โ€œAn Achievable Rate Region for the Broadcast Channel,โ€ IEEE Trans. On Inf. Theory, Vol. IT-21, pp. 399-404, July 1975. Y. Stein berg and S. Shamai, โ€œAchievable rates for the broadcast channel with states known at the transmitter,โ€ Int. Symp. on Inf. Theory, 2005, Adelaide, SA, pp. 2184-2188.