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 Deppe1, Vladimir Lebedev2 and Georg Maringer1

1Technical University of Munich

Institute for Communications Engineering

2Kharkevich 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 storages1

1 1

Figure: Asymmetric channel

1J.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

◮ ci : M × Qi−1 → Q,

i ∈ {1, . . . , n}

◮ c(m, yn−1) = ((c1(m), c2(m, y1), . . . , cn(m, yn−1)), where yk = (y1, . . . , yk)

• error vector e := (e1, e2, . . . , en), where ei = yi − ci(m)

SENDER CHANNEL RECEIVER

✲ ✲ ✛ ❄

feedback noise

• ❄

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 = τn errors in the combinatorial setting with

noiseless feedback

• denoted as Cf

q(Γ, τ)

• for τ = 1 this was denoted as the zero error capacity by Shannon2, in the following denoted as Cf

0,q(Γ)

2C.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 Vin
• set of output vertices Vout
• set of edges E ⊂ Vin × Vout

Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 6

Unidirectional channels

• composed out of 2 asymmetric channels (E = Vin × Vout)

◮ one channel for which only positive error vectors are possible (ei ≥ 0 ∀i) ◮ for other channel only negative error vectors are possible (ei ≤ 0 ∀i)

• beginning of block: one channel is selected at random
• channel remains the same within each block

1 1 1 1

Figure: binary Z-channel and inverse Z-channel, together Γ2

U

Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 7

Generalized Z-channel and generalized inverse Z-channel

1 1 2 2 . . . . . . . . . q − 2 q − 2 q − 1 q − 1 1 1 2 2 . . . . . . . . . q − 2 q − 2 q − 1 q − 1

Figure: Generalized Z-channel Γq

Z and generalized inverse Z-channel Γq

Z , together Γq

U

Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 8

Zero error capacity of Γq

U By applying Theorem 7 of2 Cf

0,q(Γq Z) = logq

q

2

• Strategy to show that Cf

0,q(Γq U) = logq

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. 1 1 2 2 1 1 2 2

Figure: Γ3

U

2C.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

Upper bound on the capacity error function Cf

q(Γq U, τ)

∗ ∗

1 1 2 2 . . . . . . q − 2 q − 2 q − 1 q − 1

Figure: Γq∗

• maximal set of messages for Γq

U is denoted as MU q (n, t)

• M∗

q(n, t) ≥ MU q (n, t), where M∗ q(n, t) is a maximal set of messages for Γq∗ Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 10

Upper bound on the capacity error function Cf

q(Γq U, τ) An improvement is possible by bounding the set of output sequences and using a sphere packing argument:

Mupper

q

(n, t) := t

i=0

n

i

• qn−i

t

i=0

n

i

• An asymptotic analysis of Mupper

q

(n, t) leads to the result

Cf

q(Γq U, τ) ≤ 1 + hq

• min
• τ,

1 q + 1

• − min
• τ,

1 q + 1

• − hq
• min
• τ, 1

2

• where hq denotes the binary entropy function to the base q.

Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 11

Lower bound on the capacity error function Cf

q(Γq U, τ) Let ˜

Γ2(q) denote the set of bipartite graphs with Vin = Vout = {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 rate3

RDL(τ) = 1 − h(τ) logq(2) and from a previous result it follows that RDL

1

2

• = logq

q

2

• ≤ logq

q

2

• = Cf

0,q(Γq U)

Cf

q

• Γq

U, 1

2

• = logq

q + 1

2

• = Cf

0,q(Γq U) for odd q

• 3C. 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

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

τ

Asymptotic rate Upper and lower bound on capacity error function of unidirectional channels Lower bound Cf

5(Γ5 U, τ)

Upper bound Cf

5(Γ5 U, τ)

Lower bound Cf

6(Γ6 U, τ)

Upper bound Cf

6(Γ6 U, τ)

Figure: Lower and upper bound on the capacity error function of the unidirectional channels Γ5

U and Γ6 U

Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 13

Rubber method

• proposed by Ahlswede, Deppe and Lebedev4
• 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)
• 4R. 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 Γq

Z and Γq

Z

• Consider a channel Γ ∈ ˜

Γ2(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 − rτ)
• choose rubber sequence such that rubber symbols cannot be created by channel disturbance
• r consecutive zeros for Γq

Z

• r consecutive q − 1 symbols for Γq

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 Γq

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 Γq

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 Γq

Z to transmit the remaining information symbols

• send 0-symbols to fill the block if the channel was Γq

Z and q − 1 symbols otherwise

• The asymptotic rate of the modified rubber method for Γq

U and Γq Z are the same.

f : Q → Q k → k + 1

mod q.

Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 16

Modified rubber method for Γq

U, decoding strategy

• receiver checks last symbol to find out which channel was active
• if Γq

Z was active ⇒ use decoder for Z-channel

• treat symbol before the first rubber sequence as the first erroneous one
• use decoder for Γq

Z to obtain the remaining information symbols

• use f −1 on the information symbols after the first rubber sequence to obtain the message

Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 17

Symmetric channel Γ2

• capacity error function has been derived by Berlekamp5 and Zigangirov6
• unidirectional channel is easier than symmetric channel
• modified rubber method for unidirectional channel achieves larger asymptotic rate than Cf

2(Γ2, τ)

1 1

Figure: Symmetric Channel Γ2

5E.R. Berlekamp, “Block coding for the binary symmetric channel with noiseless delayless feedback, Proc. Symposium on Error Correcting Codes, Univ.

Wisconsin, 1968.

6K.Sh. Zigangirov, “On the number of correctable errors for transmission over a binary symmetrical channel with feedback, Problems Information

Transmission 12, No. 2, 85-97, 1976. Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 18

Results for the binary case

0 5 · 10−20.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

τ

Achieved rate Comparison: Unidirectional channel, symmetric channel Lower bound Cf

2(Γ2 U, τ)

Cf

2(Γ2, τ)

Figure: Comparison of the lower bound on Cf

2(Γ2 U, τ) obtained by using the modified rubber method and the capacity error

function of the symmetric channel Cf

2(Γ2, τ)

Christian Deppe, Vladimir Lebedev, Georg Maringer (TUM) 19