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Introduction Turbo Principle SISO (Soft Input Soft Output) Example of a product code Introduction 1 Turbo Principle 2 Coding and uncoding SISO (Soft Input Soft Output) 3 Definition of a soft information, how to use it? Convolutional

  1. Introduction Turbo Principle SISO (Soft Input Soft Output) Example of a product code Introduction 1 Turbo Principle 2 Coding and uncoding SISO (Soft Input Soft Output) 3 Definition of a soft information, how to use it? Convolutional codes Block codes Example of a product code 4 J.-M. Brossier Turbo codes.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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.

  7. 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.

  8. 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.

  9. Introduction Turbo Principle Coding and uncoding SISO (Soft Input Soft Output) Example of a product code Turbo decoding Iterative Decoding Scheme u p E Decoder 1 E Decoder 2 D q J.-M. Brossier Turbo codes.

  10. 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.

  11. 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.

  12. 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.

  13. 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.

  14. 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.

  15. 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.

  16. 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.

  17. 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.

  18. 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.

  19. 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.

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