On the Achievable Rate Region of the K-Receiver Broadcast Channels via Exhaustive Message Splitting

This paper focuses on K-receiver discrete-time memoryless broadcast channels (DM-BCs) with private messages, where the transmitter wishes to convey K private messages to K receivers. A general inner bound on the capacity region is proposed bas...

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Autores principales:	Rui Tang, Songjie Xie, Youlong Wu
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	broadcast channel capacity region covering lemma Science Q Astrophysics QB460-466 Physics QC1-999
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spelling	oai:doaj.org-article:42428703bd874d18b795406103efb14e2021-11-25T17:29:25ZOn the Achievable Rate Region of the <i>K</i>-Receiver Broadcast Channels via Exhaustive Message Splitting10.3390/e231114081099-4300https://doaj.org/article/42428703bd874d18b795406103efb14e2021-10-01T00:00:00Zhttps://www.mdpi.com/1099-4300/23/11/1408https://doaj.org/toc/1099-4300This paper focuses on <i>K</i>-receiver discrete-time memoryless broadcast channels (DM-BCs) with private messages, where the transmitter wishes to convey <i>K</i> private messages to <i>K</i> receivers. A general inner bound on the capacity region is proposed based on an exhaustive message splitting and a <i>K</i>-level modified Marton’s coding. The key idea is to split every message into <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>K</mi></msubsup><mfenced separators="" open="(" close=")"><mfrac linethickness="0pt"><mi>K</mi><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mfenced></mrow></semantics></math></inline-formula> submessages each corresponding to a set of users who are assigned to recover them, and then send these submessages via codewords chosen from a <i>K</i>-level structure codebooks. To guarantee the joint typicality among all transmitted codewords, a sufficient condition on the subcodebooks’ sizes is derived through a newly establishing hierarchical covering lemma, which extends the 2-level multivariate covering lemma to the <i>K</i>-level case with more intricate dependences. As the number of auxiliary random variables and rate conditions both increase exponentially with <i>K</i>, the standard Fourier–Motzkin elimination procedure becomes infeasible when <i>K</i> is large. To tackle this problem, we obtain a <i>closed form</i> of achievable rate region with a special observation of disjoint unions of sets that constitute the power set of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>K</mi><mo>}</mo></mrow></semantics></math></inline-formula>. The proposed achievable rate region allows arbitrary input probability mass functions and improves over previously known achievable (closed form) rate regions for <i>K</i>-receiver (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>≥</mo><mn>3</mn></mrow></semantics></math></inline-formula>) BCs.Rui TangSongjie XieYoulong WuMDPI AGarticlebroadcast channelcapacity regioncovering lemmaScienceQAstrophysicsQB460-466PhysicsQC1-999ENEntropy, Vol 23, Iss 1408, p 1408 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	broadcast channel capacity region covering lemma Science Q Astrophysics QB460-466 Physics QC1-999
spellingShingle	broadcast channel capacity region covering lemma Science Q Astrophysics QB460-466 Physics QC1-999 Rui Tang Songjie Xie Youlong Wu On the Achievable Rate Region of the <i>K</i>-Receiver Broadcast Channels via Exhaustive Message Splitting
description	This paper focuses on <i>K</i>-receiver discrete-time memoryless broadcast channels (DM-BCs) with private messages, where the transmitter wishes to convey <i>K</i> private messages to <i>K</i> receivers. A general inner bound on the capacity region is proposed based on an exhaustive message splitting and a <i>K</i>-level modified Marton’s coding. The key idea is to split every message into <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>K</mi></msubsup><mfenced separators="" open="(" close=")"><mfrac linethickness="0pt"><mi>K</mi><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mfenced></mrow></semantics></math></inline-formula> submessages each corresponding to a set of users who are assigned to recover them, and then send these submessages via codewords chosen from a <i>K</i>-level structure codebooks. To guarantee the joint typicality among all transmitted codewords, a sufficient condition on the subcodebooks’ sizes is derived through a newly establishing hierarchical covering lemma, which extends the 2-level multivariate covering lemma to the <i>K</i>-level case with more intricate dependences. As the number of auxiliary random variables and rate conditions both increase exponentially with <i>K</i>, the standard Fourier–Motzkin elimination procedure becomes infeasible when <i>K</i> is large. To tackle this problem, we obtain a <i>closed form</i> of achievable rate region with a special observation of disjoint unions of sets that constitute the power set of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>K</mi><mo>}</mo></mrow></semantics></math></inline-formula>. The proposed achievable rate region allows arbitrary input probability mass functions and improves over previously known achievable (closed form) rate regions for <i>K</i>-receiver (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>≥</mo><mn>3</mn></mrow></semantics></math></inline-formula>) BCs.
format	article
author	Rui Tang Songjie Xie Youlong Wu
author_facet	Rui Tang Songjie Xie Youlong Wu
author_sort	Rui Tang
title	On the Achievable Rate Region of the <i>K</i>-Receiver Broadcast Channels via Exhaustive Message Splitting
title_short	On the Achievable Rate Region of the <i>K</i>-Receiver Broadcast Channels via Exhaustive Message Splitting
title_full	On the Achievable Rate Region of the <i>K</i>-Receiver Broadcast Channels via Exhaustive Message Splitting
title_fullStr	On the Achievable Rate Region of the <i>K</i>-Receiver Broadcast Channels via Exhaustive Message Splitting
title_full_unstemmed	On the Achievable Rate Region of the <i>K</i>-Receiver Broadcast Channels via Exhaustive Message Splitting
title_sort	on the achievable rate region of the <i>k</i>-receiver broadcast channels via exhaustive message splitting
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/42428703bd874d18b795406103efb14e
work_keys_str_mv	AT ruitang ontheachievablerateregionoftheikireceiverbroadcastchannelsviaexhaustivemessagesplitting AT songjiexie ontheachievablerateregionoftheikireceiverbroadcastchannelsviaexhaustivemessagesplitting AT youlongwu ontheachievablerateregionoftheikireceiverbroadcastchannelsviaexhaustivemessagesplitting
_version_	1718412280375279616

On the Achievable Rate Region of the <i>K</i>-Receiver Broadcast Channels via Exhaustive Message Splitting

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