Introduction Turbo Principle SISO (Soft Input Soft Output) Example of a product code Achieving channel capacity ... Shannon says this is possible ... how ? Make use of the gap between the source rate and the channel capacity: coding scheme Claude Shannon, 1953 A scheme of coding and decoding can be found allowing correction of all transmission errors, if the information rate is inferior or equal to the channel capacity. J.-M. Brossier Turbo codes.
Introduction Turbo Principle Coding and uncoding SISO (Soft Input Soft Output) Example of a product code Turbo coding Serial encoding u Data Code 1 p Interleaver Code 2 q Two short systematic codes are used to build a large code Data u Redundancy of the first coder p Redundancy of the second coder q J.-M. Brossier Turbo codes.
Introduction Turbo Principle Coding and uncoding SISO (Soft Input Soft Output) Example of a product code Turbo decoding Iterative Decoding Scheme u First iteration The two decoders provide a p E Decoder 1 first estimation of the transmitted symbols. Each decoder transmits its output to the input of the other one for the second E Decoder 2 D iteration. q J.-M. Brossier Turbo codes.
Introduction Turbo Principle Coding and uncoding SISO (Soft Input Soft Output) Example of a product code Turbo decoding Iterative Decoding Scheme Second Iteration u Using the outputs computed p E at the first iteration, the two Decoder 1 decoders provide a second estimation of the transmitted symbols. the same sequence of E Decoder 2 D operations is applied for all iterations ... q J.-M. Brossier Turbo codes.
Introduction Definition of a soft information, how to use it? Turbo Principle Convolutional codes SISO (Soft Input Soft Output) Block codes Example of a product code 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-mean Gaussian random variable with variance σ 2 . LLR (Log Likelihood Ratio): − ( r − 1) 2 � � 1 2 π exp √ log p ( r | + 1) = 2 2 σ 2 σ p ( r | − 1) = log σ 2 r − ( r +1) 2 � � 1 2 π exp √ 2 σ 2 σ Interpretation: The sign of the LLR is a hard decision Its module indicates the reliability of this decision. J.-M. Brossier Turbo codes.
Introduction Definition of a soft information, how to use it? Turbo Principle Convolutional codes SISO (Soft Input Soft Output) Block codes Example of a product code Modelisation of the decoder input Decomposition of the information Emitted codeword given by LLRs X = ( X 1 , · · · , X n ) Hard decision: the LLR sign provides a hard decision: Soft received sequence of values Y i = sgn [ LLR i ] R = ( LLR 1 , · · · , LLR n ) LLR i are Log-Likelihood Ratios Reliability: the LLR module provides its reliability: Iteration 1: information is hard or soft. α i = | LLR i | Following iterations: only soft information. J.-M. Brossier Turbo codes.
Introduction Definition of a soft information, how to use it? Turbo Principle Convolutional codes SISO (Soft Input Soft Output) Block codes Example of a product code Several kinds of decoders Notations Vector of errors Z m = Y ⊕ X m n Weights of errors W ( Z m ) = Z m � i i =1 Analog weight (soft) W α ( Z m ) = � n i =1 α i Z m i . Incomplete Decoder (Hard input) It only uses the hard information. It gives the word X m = ( X m 1 , · · · , X m n ) whose Hamming distance to Y = ( Y 1 , · · · , Y n ) is minimum. � d min − 1 1 word is found if W ( Z m ) ≤ � else no word found. 2 The decision is right if the number of errors is less than � d min − 1 � . 2 J.-M. Brossier Turbo codes.
Introduction Definition of a soft information, how to use it? Turbo Principle Convolutional codes SISO (Soft Input Soft Output) Block codes Example of a product code Several kinds of decoders Notations Vector of errors Z m = Y ⊕ X m n Weights of errors W ( Z m ) = Z m � i i =1 Analog weight (soft) W α ( Z m ) = � n i =1 α i Z m i . Complete Decoder (Soft input) It uses the whole information. Complete decoder: min m W α ( Y ⊕ X m ) can provide a codeword even if the number of errors is greater than � d min − 1 � 2 Complete Soft Decoder: min m W α ( Y ⊕ X m ) with analog weight W α ( Z m ). J.-M. Brossier Turbo codes.
Introduction Definition of a soft information, how to use it? Turbo Principle Convolutional codes SISO (Soft Input Soft Output) Block codes Example of a product code SISO Convolutional Soft input The Viterbi algorithm is able to use soft inputs: it only needs to use an Euclidian metric. Soft output The Viterbi must be modified: Soft Output Viterbi Algorithm (SOVA) Idea: keep more than a single path: the difference between metrics of the two best paths is an indication about the reliability of the decision. J.-M. Brossier Turbo codes.
Introduction Definition of a soft information, how to use it? Turbo Principle Convolutional codes SISO (Soft Input Soft Output) Block codes Example of a product code Block turbo codes Soft input - Chase Algorithm If only hard decisions are known. Vector R is received. A hard version Y of R is usable by a usual algebric decoder. An algebraic decoder provides a codeword XA. For an incomplete decoder, the procedure stops here. But, if reliabilities are known, it is possible to improve the estimation: Find weak positions (weak LLRs) Modify Y for these positions and produce a small set of decoded codewords using this set of decisions about R. Select the codeword whose analog distance to R is minimum. J.-M. Brossier Turbo codes.
Introduction Definition of a soft information, how to use it? Turbo Principle Convolutional codes SISO (Soft Input Soft Output) Block codes Example of a product code Block turbo codes Soft input - Chase Algorithm If only hard decisions are known. Vector R is received. A hard version Y of R is usable by a usual algebric decoder. An algebraic decoder provides a codeword XA. For an incomplete decoder, the procedure stops here. But, if reliabilities are known, it is possible to improve the estimation: Find weak positions (weak LLRs) Modify Y for these positions and produce a small set of decoded codewords using this set of decisions about R. Select the codeword whose analog distance to R is minimum. J.-M. Brossier Turbo codes.
Introduction Definition of a soft information, how to use it? Turbo Principle Convolutional codes SISO (Soft Input Soft Output) Block codes Example of a product code Block turbo codes Soft outputs - Pyndiah Algorithm Aim : computation of reliabilities Λ ( d j ) = log P ( a j = +1 | R ) P ( a j = − 1 | R ) Reliabilities S ± 1 is the set of words with c i j = ± 1, j � E = C i | R � � P ( a j = ± 1 | R ) = P C i ∈ S ± 1 j J.-M. Brossier Turbo codes.
Introduction Definition of a soft information, how to use it? Turbo Principle Convolutional codes SISO (Soft Input Soft Output) Block codes Example of a product code Block turbo codes Soft outputs - Pyndiah Algorithm Aim : computation of reliabilities Λ ( d j ) = log P ( a j = +1 | R ) P ( a j = − 1 | R ) Reliabilities � R | E = C i � � P C i ∈ S +1 j Λ ( d j ) = log � P ( R | E = C i ) C i ∈ S − 1 j � 2 � � n −| R − C i | � R | E = C i � 1 � with P = exp √ 2 σ 2 2 πσ J.-M. Brossier Turbo codes.
Introduction Definition of a soft information, how to use it? Turbo Principle Convolutional codes SISO (Soft Input Soft Output) Block codes Example of a product code Block turbo codes Soft outputs - Pyndiah Algorithm Aim : computation of reliabilities Λ ( d j ) = log P ( a j = +1 | R ) P ( a j = − 1 | R ) A good approximation of reliabilities is given by: 1 �� 2 2 � � R − C − 1( j ) � � � R − C +1( j ) � Λ ( d j ) ≈ − � � � � 2 σ 2 � � C ± 1( j ) are words in S ± 1 whose Euclidian distance to R is minimum. j S ± 1 is the set of words with c i j = ± 1 j J.-M. Brossier Turbo codes.
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