Bounds for the capacity error function for unidirectional channels - PowerPoint PPT Presentation

Bounds for the capacity error function for unidirectional channels with noiseless feedback Christian Deppe 1 , Vladimir Lebedev 2 and Georg Maringer 1 1 Technical University of Munich Institute for Communications Engineering 2 Kharkevich Institute for Information Transmission Problems Russian Academy of Sciences June 5th, 2020

Motivation • Large ratio between the error probability 0 → 1 and 1 → 0 • practically only one type of error is relevant • asymmetric error model • example: fiber optic communication, data storages 1 0 0 1 1 Figure: Asymmetric channel 1 J.H. Weber, “Bounds and Constructions for Binary Block Codes Correcting Asymmetric or Unidirectional Errors“, TU Delft, Dissertation 1989. Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 2

Encoding with feedback • sender chooses message m out of set of possible messages M • input and output alphabet of channel Q := { 0 , . . . , q − 1 } • Encoding algorithm ◮ c i : M × Q i − 1 → Q , i ∈ { 1 , . . . , n } ◮ c ( m , y n − 1 ) = (( c 1 ( m ) , c 2 ( m , y 1 ) , . . . , c n ( m , y n − 1 )) , where y k = ( y 1 , . . . , y k ) • error vector e := ( e 1 , e 2 , . . . , e n ) , where e i = y i − c i ( m ) ✛ feedback noise ❄ ✲ ✲ SENDER CHANNEL � � RECEIVER ❄ Figure: Channel with feedback Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 3

Error model • n blocklength • t maximal number of errors within each block • τ := t n , maximal error fraction • analysis is purely combinatorial • asymptotic case ( n → ∞ ) for fixed τ • maximal amount of errors proportional to blocklength Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 4

Capacity error function and zero error capacity • maximal achievable rate for a channel Γ with at most t = τ n errors in the combinatorial setting with noiseless feedback • denoted as C f q (Γ , τ ) • for τ = 1 this was denoted as the zero error capacity by Shannon 2 , in the following denoted as C f 0 , q (Γ) 2 C.E. Shannon, “The zero error capacity of a noisy channel“, IRE Trans. Inform. Theory, vol. 2, num. 3, page 8-19, 1956. Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 5

Discrete channels • one to one mapping to bipartite graphs • set of input vertices V in • set of output vertices V out • set of edges E ⊂ V in × V out Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 6

Unidirectional channels • composed out of 2 asymmetric channels ( E � = V in × V out ) ◮ one channel for which only positive error vectors are possible ( e i ≥ 0 ∀ i ) ◮ for other channel only negative error vectors are possible ( e i ≤ 0 ∀ i ) • beginning of block: one channel is selected at random • channel remains the same within each block 0 0 0 0 1 1 1 1 Figure: binary Z-channel and inverse Z-channel, together Γ 2 U Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 7

Generalized Z-channel and generalized inverse Z-channel 0 0 0 0 1 1 1 1 2 2 2 2 . . . . . . . . . . . . . . . . . . q − 2 q − 2 q − 2 q − 2 q − 1 q − 1 q − 1 q − 1 Figure: Generalized Z-channel Γ q Z and generalized inverse Z-channel Γ q , together Γ q Z U Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 8

Zero error capacity of Γ q U By applying Theorem 7 of 2 �� q �� 0 , q (Γ q C f Z ) = log q 2 0 , q (Γ q �� q Strategy to show that C f �� U ) = log q : 2 • Define S := { k ∈ { 0 , . . . , q − 1 } : k ≡ 0 (mod 2 ) } • Encode the message into the first n − 1 symbols of the block and use the last symbol to specify to which channel the unidirectional channel corresponds. 0 0 0 0 1 1 1 1 2 2 2 2 Figure: Γ 3 U 2 C.E. Shannon, “The zero error capacity of a noisy channel“, IRE Trans. Inform. Theory, vol. 2, num. 3, page 8-19, 1956. Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 9

q (Γ q Upper bound on the capacity error function C f U , τ ) ∗ ∗ 0 0 1 1 2 2 . . . . . . q − 2 q − 2 q − 1 q − 1 Figure: Γ q ∗ • maximal set of messages for Γ q U is denoted as M U q ( n , t ) • M ∗ q ( n , t ) , where M ∗ q ( n , t ) is a maximal set of messages for Γ q ∗ q ( n , t ) ≥ M U Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 10

q (Γ q Upper bound on the capacity error function C f U , τ ) An improvement is possible by bounding the set of output sequences and using a sphere packing argument: � t � n q n − i � M upper i = 0 i ( n , t ) := q � t � n � i = 0 i An asymptotic analysis of M upper ( n , t ) leads to the result q � � �� � � � � �� 1 1 τ, 1 q (Γ q C f U , τ ) ≤ 1 + h q min − min − h q min τ, τ, q + 1 q + 1 2 where h q denotes the binary entropy function to the base q . Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 11

q (Γ q Lower bound on the capacity error function C f U , τ ) Let ˜ Γ 2 ( q ) denote the set of bipartite graphs with V in = V out = { 0 , . . . , q − 1 } where all vertices have at most degree 2. If there exists an encoding strategy ∆ which achieves a positive rate R > 0 for τ = 1, then for 0 ≤ τ ≤ 1 2 there exists an algorithm that achieves an asymptotic rate 3 R DL ( τ ) = 1 − h ( τ ) log q ( 2 ) and from a previous result it follows that � 1 � � q �� q � �� 0 , q (Γ q = C f R DL = log q ≤ log q U ) 2 2 2 � � � q + 1 � U , 1 Γ q 0 , q (Γ q C f = C f = log q U ) for odd q q 2 2 3 C. Deppe and V. Lebedev, “Algorithms for Q-ary Error-Correcting-Codes with Partial Feedback and Limited Magnitude“, ISIT 2019. Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 12

Results for q = 5 and q = 6 Upper and lower bound on capacity error function of unidirectional channels 1 Lower bound C f 5 (Γ 5 U , τ ) Upper bound C f 5 (Γ 5 U , τ ) 0 . 95 Lower bound C f 6 (Γ 6 U , τ ) Upper bound C f 6 (Γ 6 U , τ ) 0 . 9 Asymptotic rate 0 . 85 0 . 8 0 . 75 0 . 7 0 . 65 0 . 6 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 τ Figure: Lower and upper bound on the capacity error function of the unidirectional channels Γ 5 U and Γ 6 U Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 13

Rubber method • proposed by Ahlswede, Deppe and Lebedev 4 • Idea: Reserve a sequence of symbols of length r (rubber sequence) which does not occur in the sequence of information symbols • use this sequence to tell the receiver that an error occurred • receiver deletes the rubber sequence plus one previous symbol • retransmit the information symbol that was not disturbed by channel • continue transmitting information symbols... • asymptotic rate proportional to ( 1 − ( r + 1 ) τ ) 4 R. Ahlswede, C. Deppe and V. Lebedev, “Non binary error correcting codes with noiseless feedback, localized errors, or both, Annals of the European Academy of Sciences 1, 285-309, 2005. Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 14

Modified rubber method for Γ q Z and Γ q Z • Consider a channel Γ ∈ ˜ Γ 2 ( q ) • Knowing the error positions ⇒ knowing error values as well • use rubber method to make receiver aware of erroneous positions • no retransmissions necessary ⇒ asymptotic rate proportional to ( 1 − r τ ) • choose rubber sequence such that rubber symbols cannot be created by channel disturbance • r consecutive zeros for Γ q Z • r consecutive q − 1 symbols for Γ q Z • length of rubber sequence r can be varied for different τ ⇒ optimization parameter to achieve higher asymptotic rate Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 15

Modified rubber method for Γ q U , encoding strategy • Sender and receiver do not know to which asymmetric channel the unidirectional channel corresponds before the first error. • Both sender and receiver assume that the channel corresponds to Γ q Z . • If they are correct they are successful in transmission because the modified rubber method works. • Otherwise send 0 until r of them are received. • apply function f to all remaining information symbols plus the symbol to be retransmitted • retransmit erroneous symbol • use modified rubber method for Γ q to transmit the remaining information symbols Z • send 0-symbols to fill the block if the channel was Γ q Z and q − 1 symbols otherwise • The asymptotic rate of the modified rubber method for Γ q U and Γ q Z are the same. f : Q → Q k �→ k + 1 mod q . Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 16

Modified rubber method for Γ q U , decoding strategy • receiver checks last symbol to find out which channel was active • if Γ q Z was active ⇒ use decoder for Z-channel • treat symbol before the first rubber sequence as the first erroneous one • use decoder for Γ q to obtain the remaining information symbols Z • use f − 1 on the information symbols after the first rubber sequence to obtain the message Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 17