1 introduction 2 coding and uncoding siso (soft input soft ...jean-marc.brossier/... · a scheme of...

IntroductionTurbo Principle

SISO (Soft Input Soft Output)Example of a product code

1 Introduction

2 Turbo PrincipleCoding and uncoding

3 SISO (Soft Input Soft Output)Definition of a soft information, how to use it?Convolutional codesBlock codes

4 Example of a product code

J.-M. Brossier Turbo codes.

Achieving channel capacity ...

Shannon says this is possible ... how ?

Make use of the gap between the source rate and the channelcapacity: coding scheme

Claude Shannon, 1953

A scheme of coding and decoding can be found allowing correctionof all transmission errors, if the information rate is inferior or equalto the channel capacity.

Coding and uncoding

Turbo coding

Serial encoding

Code 1

Code 2Interleaver

Data

u

p

q

Two short systematic codes are used to build a large codeData uRedundancy of the first coder pRedundancy of the second coder q

Coding and uncoding

Turbo coding

Serial encoding

Code 1

Code 2Interleaver

Data

u

p

q

Coding and uncoding

Turbo coding

Serial encoding

Code 1

Code 2Interleaver

Data

u

p

q

Coding and uncoding

Turbo coding

Serial encoding

Code 1

Code 2Interleaver

Data

u

p

q

Coding and uncoding

Turbo decoding

Iterative Decoding Scheme

Decoder 1

Decoder 2

u

p

q

E

D

Coding and uncoding

Turbo decoding

Decoder 1

Decoder 2

u

p

q

E

D

First iteration

The two decoders provide afirst estimation of thetransmitted symbols.

Each decoder transmits itsoutput to the input of theother one for the seconditeration.

Coding and uncoding

Turbo decoding

Decoder 1

Decoder 2

u

p

q

E

D

First iteration

The two decoders provide afirst estimation of thetransmitted symbols.

Each decoder transmits itsoutput to the input of theother one for the seconditeration.

Coding and uncoding

Turbo decoding

Decoder 1

Decoder 2

u

p

q

E

D

Second Iteration

Using the outputs computedat the first iteration, the twodecoders provide a secondestimation of the transmittedsymbols.

the same sequence ofoperations is applied for alliterations ...

Coding and uncoding

Turbo decoding

Decoder 1

Decoder 2

u

p

q

E

D

Second Iteration

Using the outputs computedat the first iteration, the twodecoders provide a secondestimation of the transmittedsymbols.

the same sequence ofoperations is applied for alliterations ...

Definition of a soft information, how to use it?Convolutional codesBlock codes

Soft information

What is a soft information?

A log-likelihood ratio

Example: the Additive White Gaussian Noise Channel

Its output is given by r = x + b with x = ±1 and b zero-meanGaussian random variable with variance σ2.

LLR (Log Likelihood Ratio):

logp (r |+ 1)p (r | − 1)

= log

1σ√2π exp(− (r−1)

2

2σ2

)1

σ√

2πexp

(− (r+1)

2

2σ2

) = 2

σ2r

Interpretation:

The sign of the LLR is a hard decisionIts module indicates the reliability of this decision.

Soft information

What is a soft information?

A log-likelihood ratio

Example: the Additive White Gaussian Noise Channel

Its output is given by r = x + b with x = ±1 and b zero-meanGaussian random variable with variance σ2.

LLR (Log Likelihood Ratio):

logp (r |+ 1)p (r | − 1)

= log

1σ√2π exp(− (r−1)

2

2σ2

)1

σ√

2πexp

(− (r+1)

2

2σ2

) = 2

σ2r

Interpretation:

The sign of the LLR is a hard decisionIts module indicates the reliability of this decision.

Modelisation of the decoder input

Emitted codeword

X = (X1, · · · ,Xn)

Soft received sequence of values

R = (LLR1, · · · , LLRn)

LLRi are Log-Likelihood Ratios

Iteration 1: information ishard or soft.

Following iterations: onlysoft information.

Decomposition of the informationgiven by LLRs

Hard decision: the LLR signprovides a hard decision:

Yi = sgn [LLRi ]

Reliability: the LLR moduleprovides its reliability:

αi = |LLRi |

Emitted codeword

X = (X1, · · · ,Xn)

R = (LLR1, · · · , LLRn)

Yi = sgn [LLRi ]

αi = |LLRi |

Emitted codeword

X = (X1, · · · ,Xn)

R = (LLR1, · · · , LLRn)

Yi = sgn [LLRi ]

αi = |LLRi |

Emitted codeword

X = (X1, · · · ,Xn)

R = (LLR1, · · · , LLRn)

Yi = sgn [LLRi ]

αi = |LLRi |

Emitted codeword

X = (X1, · · · ,Xn)

R = (LLR1, · · · , LLRn)

Yi = sgn [LLRi ]

αi = |LLRi |

Emitted codeword

X = (X1, · · · ,Xn)

R = (LLR1, · · · , LLRn)

Yi = sgn [LLRi ]

αi = |LLRi |

Emitted codeword

X = (X1, · · · ,Xn)

R = (LLR1, · · · , LLRn)

Yi = sgn [LLRi ]

αi = |LLRi |

Emitted codeword

X = (X1, · · · ,Xn)

R = (LLR1, · · · , LLRn)

Yi = sgn [LLRi ]

αi = |LLRi |

Several kinds of decoders

Notations

Vector of errors Zm = Y ⊕ Xm

Weights of errors W (Zm) =n∑

i=1Zmi

Analog weight (soft) Wα (Zm) =

∑ni=1 αiZ

mi .

Incomplete Decoder (Hard input)

It only uses the hard information.

It gives the word Xm = (Xm1 , · · · ,Xmn ) whose Hammingdistance to Y = (Y1, · · · ,Yn) is minimum.

1 word is found if W (Zm) ≤⌊

dmin−12

⌋else no word found.

The decision is right if the number of errors is less than⌊dmin−1

2

⌋.

Several kinds of decoders

Notations

Vector of errors Zm = Y ⊕ Xm

Weights of errors W (Zm) =n∑

i=1Zmi

Analog weight (soft) Wα (Zm) =

∑ni=1 αiZ

mi .

Complete Decoder (Soft input)

It uses the whole information.

Complete decoder: minm Wα (Y ⊕ Xm) can provide acodeword even if the number of errors is greater than⌊

dmin−12

⌋Complete Soft Decoder: minm Wα (Y ⊕ Xm) with analogweight Wα (Z

m).

SISO Convolutional

Soft input

The Viterbi algorithm is able to use soft inputs: it only needs touse an Euclidian metric.

Soft output

The Viterbi must be modified:

Soft Output Viterbi Algorithm (SOVA)

Idea: keep more than a single path: the difference betweenmetrics of the two best paths is an indication about thereliability of the decision.

SISO Convolutional

Soft input

The Viterbi algorithm is able to use soft inputs: it only needs touse an Euclidian metric.

Soft output

The Viterbi must be modified:

Soft Output Viterbi Algorithm (SOVA)

Idea: keep more than a single path: the difference betweenmetrics of the two best paths is an indication about thereliability of the decision.

Block turbo codes

Soft input - Chase Algorithm

If only hard decisions are known. Vector R is received. A hardversion Y of R is usable by a usual algebric decoder. Analgebraic decoder provides a codeword XA.

For an incomplete decoder, the procedure stops here.

But, if reliabilities are known, it is possible to improve theestimation:

Find weak positions (weak LLRs)Modify Y for these positions and produce a small set ofdecoded codewords using this set of decisions about R.Select the codeword whose analog distance to R is minimum.

Block turbo codes

Soft outputs - Pyndiah Algorithm

Aim : computation of reliabilities

Λ (dj) = logP (aj = +1|R)P (aj = −1|R)

Reliabilities

S±1j is the set of words with cij = ±1,

P (aj = ±1|R) =∑

C i∈S±1j

P(E = C i |R

)

Block turbo codes

Λ (dj) = logP (aj = +1|R)P (aj = −1|R)

Reliabilities

S±1j is the set of words with cij = ±1,

P (aj = ±1|R) =∑

C i∈S±1j

P(E = C i |R

)

Block turbo codes

Λ (dj) = logP (aj = +1|R)P (aj = −1|R)

Reliabilities

Λ (dj) = log

∑C i∈S+1j

P(R|E = C i

)∑C i∈S−1j

P (R|E = C i )

with P(R|E = C i

)=(

1√2πσ

)nexp

(−|R−C

i |22σ2

)

Block turbo codes

Λ (dj) = logP (aj = +1|R)P (aj = −1|R)

Reliabilities

Λ (dj) = log

∑C i∈S+1j

P(R|E = C i

)∑C i∈S−1j

P (R|E = C i )

with P(R|E = C i

)=(

1√2πσ

)nexp

(−|R−C

i |22σ2

)

Block turbo codes

Λ (dj) = logP (aj = +1|R)P (aj = −1|R)

A good approximation of reliabilities is given by:

Λ (dj) ≈1

2σ2

(∣∣∣R − C−1(j)∣∣∣2 − ∣∣∣R − C+1(j)∣∣∣2)C±1(j) are words in S±1j whose Euclidian distance to R is minimum.

S±1j is the set of words with cij = ±1

Block turbo codes

Λ (dj) = logP (aj = +1|R)P (aj = −1|R)

A good approximation of reliabilities is given by:

Λ (dj) ≈1

2σ2

(∣∣∣R − C−1(j)∣∣∣2 − ∣∣∣R − C+1(j)∣∣∣2)C±1(j) are words in S±1j whose Euclidian distance to R is minimum.

S±1j is the set of words with cij = ±1

1 introduction 2 coding and uncoding siso (soft input soft ...jean-marc.brossier/... · a scheme of...

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