Compound channel with feedback: Opportunistic capacity and error - PowerPoint PPT Presentation

Compound channel with feedback: Opportunistic capacity and error exponents Aditya Mahajan and Sekhar Tatikonda Yale University March 17, 2010 CISS

Compound channel Channel model � | � � , � �−� � = � ∘ �� ��� � |� � � � ∘ ∈ � , � known to encoder and decoder

Compound channel Channel model Capacity � | � � , � �−� � = � ∘ �� ��� � |� � � � ∘ ∈ � , � known to encoder and decoder ���� = max �∈𝛦�𝕐� inf �∈ℚ 𝐽��, ��

Compound channel Channel model Capacity Capacity with feedback � | � � , � �−� � = � ∘ �� ��� � |� � � � ∘ ∈ � , � known to encoder and decoder ���� = max �∈𝛦�𝕐� inf �∈ℚ 𝐽��, �� � 𝐺 ��� = inf �∈ℚ max �∈𝛦�𝕐� 𝐽��, ��

Feedback capacity is defined pessimistically

Outline Variable length coding scheme Achievable rate and opportunistic capacity Probability of error and error exponents Literature Overview Variable length communication over DMC Variable length communication over compound channel Main Result Lower bound on error exponent region Achievable coding scheme Example

Variable length coding Assume � = {� � , …, � 𝑀 } . Variable length coding scheme is a tuple �𝐍, �, �, 𝜐� Compound message: 𝐍 = �� � , …, � 𝑀 � . Let 𝕅 = ∏ 𝑀 ℓ=� {�, …, � ℓ } . Encoding strategy: � = �� � , � � , …� 𝑢 = 𝕅 × � 𝑢−� ↦ � � Decoding strategy: � = �� � , � � , …� 𝑀 � 𝑢 : � 𝑢 ↦ ⋃ {�ℓ, ��, �ℓ, ��, …, �ℓ, � ℓ �} ℓ=� Stopping time 𝜐 with respect to the channel output � 𝑢

Operation of the scheme Decoder: Encoder Compound message 𝐗 = �� � , …, � 𝑀 � � ℓ is uniformly distributed in {�, …, � ℓ } � � = � � �𝐗�, � � = � � �𝐗, � � �, ⋯ �ˆ �, ˆ �� = � 𝜐 �� � , …, � 𝜐 �

Performance metrics Probability of error 𝐐 = �� � , …, � 𝑀 � ℓ = � ℓ � ˆ � � ≠ � ˆ 𝑀 � Rate: 𝐒 = �� � , …, � 𝑀 � � ℓ = 𝔽 ℓ [log � ˆ 𝑀 ] 𝔽 ℓ [𝜐]

Operational interpretation Encoder Channel Decoder Higher level application generates an infinite bit-stream Variable length communication using �𝐍, �, �, 𝜐�

Operational interpretation Encoder Channel Decoder Higher level application generates an infinite bit-stream Variable length communication using �𝐍, �, �, 𝜐� Let � max = max{� � , …, � 𝑀 } and � min = min{� � , …, � 𝑀 }

Operational interpretation Encoder Channel Decoder Higher level application generates an infinite bit-stream Encoding Variable length communication using �𝐍, �, �, 𝜐� Let � max = max{� � , …, � 𝑀 } and � min = min{� � , …, � 𝑀 } Transmitter picks log � � max bits from the bit-stream. � ℓ is the decimal expansion of the first log � � ℓ of these bits.

Operational interpretation Encoder Channel Decoder Decoding application. chosen bits and returns the remaining bits to the bit-stream. At stopping time 𝜐 , the receiver passes � ˆ �, ˆ �� to a higher layer The transmitter removes log � � ˆ 𝑀 bits from the log � � max initially

Operational interpretation Encoder Channel Decoder Decoding application. chosen bits and returns the remaining bits to the bit-stream. At stopping time 𝜐 , the receiver passes � ˆ �, ˆ �� to a higher layer The transmitter removes log � � ˆ 𝑀 bits from the log � � max initially Advantage of being opportunistic: log � � ˆ 𝑀 − log � � min

Opportunistic capacity Achievable Rate Rate 𝐒 = �� � , …, � 𝑀 � is achievable if ∃ sequence of coding schemes such that for 𝜁 > � and sufficiently large � , and for all ℓ = �, …, � , 1. lim �→∞ 𝔽 ℓ [𝜐 ��� ] = ∞ 2. � ��� < 𝜁 and � ��� � � ℓ − 𝜁 ℓ ℓ

Opportunistic capacity Achievable Rate The union of all achievable rates is called Rate 𝐒 = �� � , …, � 𝑀 � is achievable if ∃ sequence of coding schemes such that for 𝜁 > � and sufficiently large � , and for all ℓ = �, …, � , 1. lim �→∞ 𝔽 ℓ [𝜐 ��� ] = ∞ 2. � ��� < 𝜁 and � ��� � � ℓ − 𝜁 ℓ ℓ the opportunistic capacity region ℂ 𝐺 ���

Error Exponents Error exponent Given a sequence of coding scheme that achieve a rate vector 𝐒 , the error exponent 𝐅 = �� � , …, � 𝑀 � is given by �→∞ −log � ��� ℓ � ℓ = lim 𝔽 ℓ [𝜐 ��� ]

Error Exponents Error exponent Given a sequence of coding scheme that achieve a rate vector 𝐒 , the error exponent 𝐅 = �� � , …, � 𝑀 � is given by �→∞ −log � ��� ℓ � ℓ = lim 𝔽 ℓ [𝜐 ��� ] For a particular rate 𝐒 , the union of all possible error exponents is called the error exponent region 𝔽�𝐒� .

Outline Variable length coding scheme Achievable rate and opportunistic capacity Probability of error and error exponents Literature Overview Variable length communication over DMC Variable length communication over compound channel Main Result Lower bound on error exponent region Achievable coding scheme Example

Variable length communication over DMC Burnashev-76, “Data transmission over a discrete channel with feedback: Random transmission time” Burnashev exponent Special case of a compound channel when |�| = � . ���, �� = � � �� − �� where � = �/� .

Variable length communication over DMC Achievability scheme Yamamoto-Itoh-79, “Asymptotic performance of a modified Schalkwijk-Barron scheme with noiseless feedback”. Repeat until ACCEPT is received Message mode : Fixed length code at rate � − 𝜁 and length �� Control mode : Send ACCEPT or REJECT for length �� − ���

Advantage of variable length comm Rate Exponent 0.728 0.728 0.322 0.322 2.536 2.536 Burnashev's exponent BSC �.� � = 0.53 bits � = 0.53 bits

Variable length comm over compound channel Tchamkerten-Telatar-06, “Variable length coding over unknown channel” Can we achieve Burnashev exponent even if we do not know the channel?

Variable length comm over compound channel Tchamkerten-Telatar-06, “Variable length coding over unknown channel” Can we achieve Burnashev exponent even if we do not know the channel? Negative result Under some restricted conditions, yes. In general, no. Restricted attention to � ℓ /� ℓ = 𝑑����𝑏��

Counterexample: { BSC 𝑞 , BSC �−𝑞 } 2.5 Burnashev exponent Zero−rate error exponent Combined upper bound 2.0 Exponent of proposed scheme Error Exponent 1.5 1.0 0.5 0.0 0.0 0.1 0.2 0.3 0.4 0.5 Rate

Questions What are the error exponents when conditions of Tchamkerten-Telatar-06 are not met? Which coding schemes achieve the best exponent? What about rates when � ℓ /� ℓ is not a constant?

Outline Variable length coding scheme Achievable rate and opportunistic capacity Probability of error and error exponents Literature Overview Variable length communication over DMC Variable length communication over compound channel Main Result Lower bound on error exponent region Achievable coding scheme Example

Main Result Opportunistic Capacity ℂ 𝐺 ��� = {�� � , …, � 𝑀 � : � � � ℓ < � ℓ , ℓ = �, …, �} where � ℓ is the capacity of DMC � ℓ .

Main Result Opportunistic Capacity Error Exponent Region ℂ 𝐺 ��� = {�� � , …, � 𝑀 � : � � � ℓ < � ℓ , ℓ = �, …, �} where � ℓ is the capacity of DMC � ℓ . Let 𝑈 𝑑 ℓ be the exponent of the channel estimation error when the channel is � ℓ . For any channel estimation scheme, �𝑈 𝑑 � , …, 𝑈 𝑑 𝑀 � ∈ 𝕌 * .

Main Result Opportunistic Capacity Error Exponent Region ℂ 𝐺 ��� = {�� � , …, � 𝑀 � : � � � ℓ < � ℓ , ℓ = �, …, �} where � ℓ is the capacity of DMC � ℓ . Let 𝑈 𝑑 ℓ be the exponent of the channel estimation error when the channel is � ℓ . For any channel estimation scheme, �𝑈 𝑑 � , …, 𝑈 𝑑 𝑀 � ∈ 𝕌 * . At rate 𝐒 = �� � , …, � 𝑀 � , the error exponent is 𝑈 𝑑 � � ℓ (� − � ℓ ℓ � ℓ � ) 𝑈 𝑑 ℓ + � � ℓ � ℓ where � � ℓ = max 𝑦 𝐵 ,𝑦 𝑆 ∈𝕐 ��� ℓ �⋅|𝑦 𝐵 �, � ℓ �⋅|𝑦 � ��