[PPT] - Time-Varying Variable-Length Error-Correcting Codes Victor PowerPoint Presentation

SLIDE 1

Time-Varying Variable-Length Error-Correcting Codes

Victor Buttigieg Johann A. Briffa ISITA 2018, 31 October 2018, Singapore

Department of Communications and Computer Engineering University of Malta Msida MSD 2080, Malta

SLIDE 2

Outline

Introduction and motivation Time-Varying Variable-Length Error-Correcting (TV-VLEC) codes Sequence-level MAP decoder Symbol-Constrained Free Distance (SCFD) Results Conclusions

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SLIDE 3

Introduction and Motivation

Variable-Length Codes

Used mainly for source coding
Suffer from loss of synchronization under error conditions

(Takishima et al., 1994)

Symbols may be inserted and/or deleted at the decoder

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SLIDE 4

Introduction and Motivation

Variable-Length Codes

Used mainly for source coding
Suffer from loss of synchronization under error conditions

(Takishima et al., 1994)

Symbols may be inserted and/or deleted at the decoder

a b 10 c 11 10 11 b c 00 11 aa c Message: Encoded Message: Received: Decoded Message: Simple VL Code

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SLIDE 5

Introduction and Motivation

Motivation

Introduce mechanisms that aid in the detection and avoidance of

inserted/deleted symbols

Redundancy

– Variable-Length Error-Correcting (VLEC) codes

Decoder keeping track of number of symbols

– Sequence MAP Decoder

Time-dependency

– Time-Varying Variable-Length Error-Correcting (TV-VLEC) codes

Improves
symbol-level synchronization
error performance

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SLIDE 6

TV-VLEC Codes

Definitions

Define a TV-VLEC code of order m, as the set of m distinct

sub-codes C = {C✵, C✶, . . . , Cm−✶}, where each sub-code Ci consists of s codewords.

Two sub-codes are distinct if they differ in at least one codeword.
The length of codeword ❝ is denoted by |❝|.
In regular TV-VLEC codes, all Ci have the same length distribution

Encoding

Let X = {x✵, x✶, . . . , xs−✶} be an s-symbol memoryless source.
Let P(xj) ≥ P(xj+✶), where P(xj) is the probability of occurrence
f symbol xj
The more probable symbols are encoded using the shorter
codewords. For sub-code Ci, symbol xj is mapped to codeword

❈ i(xj), where |❈ i(xj)| ≤ |❈ i(xj+✶)|

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SLIDE 7

TV-VLEC Codes

TV-VLEC Sequence-Use

Consider the message ❛ = (a✵, a✶, . . . , aK−✶) consisting of K

symbols from source X

Message ❛ is encoded as the sequence

❜ = ❈ u✵(a✵) ❈ u✶(a✶) . . . ❈ uK−✶(aK−✶)

The TV-VLEC sequence-use is defined by ✉ = (u✵, u✶, . . . , uK−✶),

uk ∈ {✵, ✶, . . . , m − ✶}, k = ✵, ✶, . . . , K − ✶

The sequence-use could be set to be
Sequential (each sub-code used in turn)
Random (sub-code selected randomly)
Fixed (sub-codes used in fixed repeating pattern)

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SLIDE 8

TV-VLEC Codes

Encoding Example

TV-VLEC Code B C0 C1 00 111 10011 01001 11 000 01100 10110 TV-VLEC Code B C0 C1 00 111 10011 01001 11 000 01100 10110 x0 x1 x2 x3 Source Symbol Sequence-Use: 0 1 0 1 Message: x0 x0 x2 x1 Encoded massage: 001110011000 Sequence-Use: 0 1 0 1 Message: x0 x0 x2 x1 Encoded massage: 001110011000

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SLIDE 9

Sequence-Level MAP Decoder

Decoders for VLEC Codes

In the literature, several decoders for VLEC codes are given
Maximum-likelihood (Buttigieg & Farrell, 1993)
Symbol-level and sequence-level MAP decoders – both exact and

approximate (Park & Miller, 2000)

Different channel models were also considered - BSC, Gaussian and

Additive-Markov (Subbalakshmi & Vaisey, 2003)

The changes required to accommodate TV-VLEC codes are

minimal

Decoder complexity remains the same

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SLIDE 10

Sequence-Level MAP Decoder

1 2 4 8 19 18 17 15 11 16 13 10 5 3 9 7 6 14 12 1 2 3

Symbol Index, k Bit Index, n TV-VLEC Code B C0 C1 00 111 10011 01001 11 000 01100 10110 TV-VLEC Code B C0 C1 00 111 10011 01001 11 000 01100 10110 x0 x1 x2 x3 Source Symbol Sequence-Use: 0 1 0 1 Message: x0 x0 x2 x1 Encoded massage: 001110011000 Sequence-Use: 0 1 0 1 Message: x0 x0 x2 x1 Encoded massage: 001110011000 9/21

SLIDE 11

Sequence-Level MAP Decoder

Path Metric

The path metric to be maximized is given by

M(ˆ ❛)

K−✶

k=✵

(log p − log (✶ − p)) d❤

❈ uk(ˆ

ak), ②(k) + log P(ˆ ak)

p is the BSC cross-over probability
d❤
❈ uk (ˆ

ak), ②(k) is the Hamming distance between the codeword from sub-code uk corresponding to the kth decoded symbol ˆ ak and the corresponding received sub-sequence ②(k)

P(ˆ

ak) is the a priori probability of the kth decoded symbol

Maximization can be achieved using dynamic programming on a

trellis

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SLIDE 12

Symbol-Constrained Free Distance

Distance Measures

In decoders that ignore the number of decoded symbols, paths

merge when a sequence of codewords have an equal number of bits

The appropriate distance measure to use in this case is the free

distance

In decoders that are constrained on the number of decoded

symbols, paths merge when a sequence of codewords have an equal number of bits and an equal number of codewords

In this work we introduce the Symbol-Constrained Free Distance

(SCFD) to take this into account

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SLIDE 13

Symbol-Constrained Free Distance

1 2 4 8 19 18 17 15 11 16 13 10 5 3 9 7 6 14 12

Bit Index, n TV-VLEC Code B C0 C1 00 111 10011 01001 11 000 01100 10110 TV-VLEC Code B C0 C1 00 111 10011 01001 11 000 01100 10110 x0 x1 x2 x3 Source Symbol Sequence-Use: 0 1 0 Message 1: x0 x1 x2 Encoded massage 1: 0000010011 Sequence-Use: 0 1 0 Message 1: x0 x1 x2 Encoded massage 1: 0000010011 Encoded massage 2: 1001111111 Message 2: x2 x0 x1 12/21

SLIDE 14

Symbol-Constrained Free Distance

Free Distance

The free distance of a TV-VLEC code C = {C✵, C✶, . . . , Cm−✶} of
rder m and with sequence-use ✉ = (u✵, u✶, . . . , uK−✶) is the

minimum Hamming distance between any two codeword concatenations ❈ ui′ (a′

✵) ❈ ui′+✶(a′ ✶) . . . ❈ ui′+K′−✶

a′

K ′−✶

and

❈ ui′′ (a′′

✵) ❈ ui′′+✶(a′′ ✶) . . . ❈ ui′′+K′′−✶

a′′

K ′′−✶

,

where a′

j, a′′ j are two source symbols such that a′ ✵ = a′′ ✵, K ′−✶

j=✵
❈ ui′+j
a′

j

=

K ′′−✶

j=✵
❈ ui′′+j
a′′

j

and

k′−✶

j=✵
❈ ui′+j
a′

j

=

k′′−✶

j=✵
❈ ui′′+j
a′′

j

for any k′ < K ′ and k′′ < K ′′, K ′, K ′′ = ✶, ✷, . . . , K,

i′ = ✵, ✶, . . . , K − K ′, and i′′ = ✵, ✶, . . . , K − K ′′.

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SLIDE 15

Symbol-Constrained Free Distance

1 2 4 8 19 18 17 15 11 16 13 10 5 3 9 7 6 14 12 1 2 3

Symbol Index, k Bit Index, n TV-VLEC Code B C0 C1 00 111 10011 01001 11 000 01100 10110 TV-VLEC Code B C0 C1 00 111 10011 01001 11 000 01100 10110 x0 x1 x2 x3 Source Symbol Sequence-Use: 0 1 0 Message 1: x0 x1 x2 Encoded massage 1: 0000010011 Sequence-Use: 0 1 0 Message 1: x0 x1 x2 Encoded massage 1: 0000010011 Encoded massage 2: 1001111111 Message 2: x2 x0 x1 14/21

SLIDE 16

Symbol-Constrained Free Distance

The symbol-constrained free distance of a TV-VLEC code

C = {C✵, C✶, . . . , Cm−✶} of order m and with sub-code sequence-use ✉ = (u✵, u✶, . . . , uK−✶) is the minimum Hamming distance between any two codeword concatenations ❈ ui(a′

✵) ❈ ui+✶(a′ ✶) . . . ❈ ui+K′−✶

a′

K ′−✶

and

❈ ui(a′′

✵) ❈ ui+✶(a′′ ✶) . . . ❈ ui+K′−✶

a′′

K ′−✶

,

where a′

j, a′′ j are two source symbols such that a′ ✵ = a′′ ✵, K ′−✶

j=✵
❈ ui+j
a′

j

=

K ′−✶

j=✵
❈ ui+j
a′′

j

and

k−✶

j=✵
❈ ui+j
a′

j

=

k−✶

j=✵
❈ ui+j
a′′

j

for k < K ′, K ′ = ✶, ✷, . . . , K and i = ✵, ✶, . . . , K − K ′.

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SLIDE 17

Symbol-Constrained Free Distance

Problem with Order One TV-VLEC Codes

An order one TV-VLEC code is a VLEC code
A symbol-constrained MAP decoder can recover symbol sync by

a "reverse" error event

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SLIDE 18

Symbol-Constrained Free Distance

Problem with Order One TV-VLEC Codes

An order one TV-VLEC code is a VLEC code
A symbol-constrained MAP decoder can recover symbol sync by

a "reverse" error event

VLEC Code A 1011 11010 x0 x1 x2 Source Symbol

0101111010 1101001011

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SLIDE 19

Symbol-Constrained Free Distance

Problem with Order One TV-VLEC Codes

An order one TV-VLEC code is a VLEC code
A symbol-constrained MAP decoder can recover symbol sync by

a "reverse" error event

VLEC Code A 1011 11010 x0 x1 x2 Source Symbol

0101111010 1101001011 Constrained Symbol Distance = 4

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SLIDE 20

Symbol-Constrained Free Distance

Problem with Order One TV-VLEC Codes

An order one TV-VLEC code is a VLEC code
A symbol-constrained MAP decoder can recover symbol sync by

a "reverse" error event

VLEC Code A 1011 11010 x0 x1 x2 Source Symbol

Constrained Symbol Distance = 4 0101110111101011010 1101010111101001011 0101110111101011010 1101010111101001011

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SLIDE 21

Symbol-Constrained Free Distance

Problem with Order One TV-VLEC Codes

An order one TV-VLEC code is a VLEC code
A symbol-constrained MAP decoder can recover symbol sync by

a "reverse" error event

VLEC Code A 1011 11010 x0 x1 x2 Source Symbol

Constrained Symbol Distance = 4 0101110111101011010 1101010111101001011 0101110111101011010 1101010111101001011

The symbol-constrained free distance is not relevant in this case
Problem is solved by using TV-VLEC codes with order 2 or higher

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SLIDE 22

Results – Levenshtein Symbol Error Rate

2 4 6 8 10 12 10-6 10-5 10-4 10-3 10-2 10-1 100

Code B: SCFD = ✸, L = ✷.✾ bits Code C: dfree = ✷, L = ✷.✾ bits Code D: dfree = ✸, L = ✸.✸ bits As expected, symbol-constrained decoder offers better performance

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Results – Levenshtein Symbol Error Rate

2 4 6 8 10 12 10-6 10-5 10-4 10-3 10-2 10-1 100

Code B: SCFD = ✸, L = ✷.✾ bits Code C: dfree = ✷, L = ✷.✾ bits Code D: dfree = ✸, L = ✸.✸ bits TV-VLEC code B performs better compared to same average codeword length or same dfree VLEC codes

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SLIDE 24

Results – Hamming Symbol Error Rate

2 4 6 8 10 12 10-6 10-5 10-4 10-3 10-2 10-1 100

Code B: SCFD = ✸, L = ✷.✾ bits Code C: dfree = ✷, L = ✷.✾ bits Code D: dfree = ✸, L = ✸.✸ bits Performance gains more pronounced with Hamming SER – greater capability to maintain symbol-level synchronization with code B

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SLIDE 25

Results – Frame Error Rate

2 4 6 8 10 12 10-4 10-3 10-2 10-1 100

Code B: SCFD = ✸, L = ✷.✾ bits Code C: dfree = ✷, L = ✷.✾ bits Code D: dfree = ✸, L = ✸.✸ bits TV-VLEC code B gives a coding gain of 2dB when compared to VLEC code C and a gain of 0.5dB when compared to VLEC code D

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SLIDE 26

Conclusions

Contributions

Introduced Time-Varying Variable-Length Error-Correcting Codes
Defined the Symbol-Constrained Free Distance (SCFD) that takes

into account the number of symbols in compared paths Advantages of TV-VLEC codes

Significant improvement in performance can be achieved without

increasing decoding complexity over normal VLEC code decoders

SCFD allows the construction of better performing codes

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SLIDE 27

Conclusions

Future Work

Using concatenated codes
Implementing a symbol-level soft-in soft-out decoder should be

relatively straight forward

Performance with an outer error-correcting code is expected to be

significantly better

Decoder complexity may be reduced using similar approximate

MAP decoding techniques as for VLEC codes (e.g. Huang et al., 2008)

More efficient algorithms to design TV-VLEC codes with a given

SCFD are required

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