State-of-the-Art Channel Coding

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Institut für State-of-the-Art Channel Coding Prof. Dr.

Engineering University of Rostock, Germany Email:

1 Institut für State-of-the-Art Channel Coding Prof. Dr.-Ing. Volker Kühn Institute of Communications Engineering University of Rostock, Germany September 2 Volker Kühn - State-of-the-Art Channel Coding UNIVERSITÄT ROSTOCK FAKULTÄT INFORMATIK UND ELEKTROTECHNIK

2 Outline of Lectures Lesson : One Lesson of Information Theory Principle structure of communication systems Definitions of entropy, mutual information, Channel coding theorem of Shannon Lesson 2: Introduction to Error Correcting Codes Basics of error correcting codes Linear block codes Convolutional codes (if time permits) Lesson 3: State-of-the-art channel coding Coding strategies to approach the capacity limits Definition of soft-information and turbo decoding principle Examples for state-of-the-art error correcting codes 2

3 Shorty Review of Milestones 948: Shannon defines his information theory Definition of entropy and mutual information Channel coding theorem 963: Robert G. Gallager: Low Density Parity Check Codes 966: G. David Forney: Concatenated Codes Computers to that time not strong enough to demonstrate potential of investigated coding schemes Turbo decoding was implicitly already invented 993: First presentation of Turbo-Codes by Berrou, Glavieux, et al. Approaching Shannon s capacity for half-rate code by.5 db 2: Stephan ten Brink: EXIT Chart Analysis Leads to further understanding of iterative decoding principles Allows design / optimization of powerful concatenated codes Repeat Accumulate Code approaches capacity up to.8 db 3

4 Potential of Turbo Codes P b Comparison conv. codes / turbo codes for R c =/2 Lc=3 Lc=5 - Lc=7 Lc=9 TC db db log E / N b Optimized interleaver of length 256 x 256 = bit For this interleaver gain of nearly 3 db over conv. code with L c = 9 Gap to Shannon s channel capacity only.5 db Tremendous performance loss for smaller interleavers World record:. db gap to Shannon capacity by Stephan ten Brink 4

5 Serial and Parallel Code Concatenation Serial Code Concatenation Example: Repeat Accumulate Codes inner code outer code D 2 D C C 2 D 2 D Parallel Code Concatenation Example: Turbo Codes C P C 2 C q S 5

6 Interleaving Simple block interleaver write x x 3 x 6 x 9 x 2 read interleaving depth: 5 x x 4 x 7 x x 3 x 2 x 5 x 8 x x 4 Input sequence: x, x, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9, x, x, x 2, x 3, x 4 Output sequence: x, x 3, x 6, x 9, x 2, x, x 4, x 7, x, x 3, x 2, x 5, x 8, x, x 4 Convolutional interleaver Random interleaver 6

7 Simple Example of Serial Concatenation Concatenation of (3,2,2)-SPC and (4,3,2)-SPC code C C Total code rate: R c = 2/4 =.5 u c c 2 w H (c 2 ) d min = 2 Minimum Hamming distance is not improved by code concatenation 7

8 Another Example of Serial Concatenation Concatenation of (4,3,2)-SPC and (7,4,3)-Hamming code Total code rate: Rc = 3/7 C C uc c c 2 c 2 w H (c 2 ) w H (c 2 ) c 2 w H (c 2 ) original concatenation: d min = 3 optimized concatenation: d min = 4 Minimum Hamming distance can only be improved by careful selection of subset of inner code 8

9 Serial Code Concatenation: Product Codes k n -k Information bits arranged in (k,k - )-matrix k u p C Row-wise encoding with code C - of rate k - / n - Column-wise encoding with code C of rate k / n n -k p C p + Entire code rate: R c = k k n n = R c R c Minimum Hamming distance: d min = d min d min 9

10 Examples of Product Codes () (2,6,4) product code x x 4 x 8 x x 5 x 9 x 2 x 6 x x 3 x 7 x Horizontal: (3,2,2)-SPC code no error correction possible Vertical: (4,3,2)-SPC code no error correction possible Code rate: /2 d min = 2 2 = 4 Correction of error possible

11 Examples of Product Codes (2) (28,2,6) product code x x 7 x 4 x 2 x x 7 x 4 x 2 x x x x 5 x 8 x 5 x 22 x x 8 x 5 x 22 x 2 x 9 x 6 x 23 x 2 x 9 x 6 x 23 x 3 x x 7 x 24 x 3 x x 7 x 24 x 4 x x 8 x 25 x 4 x x 8 x 25 x 5 x 2 x 9 x 26 x 5 x 2 x 9 x 26 x 6 x 3 x 2 x 27 x 6 x 3 x 2 x 27 Horizontal: (4,3,2)-SPC code no error correction possible Vertical: (7,4,3)-Hamming code single error correction possible d min = 2 3 = 6 2 errors correctable

12 Parallel Code Concatenation n -k k k - n - -k - u p C p - Information bits u row-wise encoded with C - column-wise encoded with C Parity check bits of component codes not encoded (no checks on checks) C - Entire code rate R c k k n n ( n k ) ( n k ) R R / c / c Minimum Hamming distance: d d d min min min 2

13 Example of Turbo Code 2 systematic, recursive convolutional encoders (L c = 3) Constituent code rates R c = 2/3 total code rate R c = /2 u c c u T T C u 2 T T c 2 P C 2 3

14 Turbo Code from Berrou and Glavieux 2 systematic, recursive convolutional encoders (L c = 5) Constituent code rates R c = 2/3 total code rate R c = /2 Pseudo random interleaver of length bits u c c u T T T T C c 2 u 2 C 2 T T T T P 4

15 Repeat Accumulate Code from ten Brink Outer half-rate repetition code Inner convolutional code (scrambler) of rate total code rate R c = /2 Random interleaver of different lengths Code doping: replace a few code bits (%) by information bits for improving the convergence of iterative decoding process u repetition encoder repetition encoder T T T inner convolutional code 5

16 Outline of Lectures Lesson : One Lesson of Information Theory Principle structure of communication systems Definitions of entropy, mutual information, Channel coding theorem of Shannon Lesson 2: Introduction to Error Correcting Codes Basics of error correcting codes Linear block codes Convolutional codes (if time permits) Lesson 3: State-of-the-art channel coding Coding strategies to approach the capacity limits Definition of soft-information and turbo decoding principle Examples for state-of-the-art error correcting codes 6

17 Log-Likelihood Ratios (LLRs) Definition log-likelihood ratio: Sign determines hard decision Magnitude represents reliability of hard decision 8 Probability of correct decision: P correct = e L(x) + e L(x) Expectation of LLR (soft bit) E{x} = Pr{x = +} Pr{x = } = el(x) + e L(x) + e µ L(x) L(x) = tanh 2 L(x) L(x) = log Pr{x = +} Pr{x = } Pr{x = +}

18 Log-Likelihood Ratios at AWGN Channel Output Scaled matched filter output equals LLR p(y x = +) L(y x) = log p(y x = ) = log exp[ (y )2 /2/σN 2 ] exp[ (y + ) 2 /2/σN 2 ] = 2 6 σ 2 N y 4 L(y x) 2-2 db 2 db -4 4 db 6 db 8dB y 8

19 Example for Soft-Output Decoding Single parity check code: u u 2 p p = u u 2 Question: What is the LLR of u given the LLRs of u 2 and p? Resolving parity check equation w.r.t. u : Extrinsic LLR does not depend on u itself: u = u 2 p L e (u ) = log Pr{u 2 p = } Pr{u 2 p = } = log Pr{u 2 =, p = } + Pr{u 2 =, p = } Pr{u 2 =, p = } + Pr{u 2 =, p = } = log Pr{u 2 = } Pr{p = } + Pr{u 2 = } Pr{p = } Pr{u 2 = } Pr{p = } + Pr{u 2 = } Pr{p = }. = 2 atanh tanh µ L(u2 ) 2 µ L(p) tanh = 2 atanh E{u 2 } E{p} 2 9

20 L-Algebra mod-2-sum of 2 statistical h independent random variables: L(x x 2 ) = 2 atanh sgn tanh L(x ) L(x sgn )/2 L(x 2 ) tanh ³L(x min 2 )/2 i L(x ), L(x 2 ) ª L(x ) L(x 2 ) tanh(x/2) + - tanh(x/2) mod-2-sum of n variables: 2 artanh(x) - + " Y n µ # L(xi ) L(x x n ) = 2artanh tanh 2 i= ny sgn L(x i ) min L(xi ) ª i 2 i= L(x x 2 )

21 General Approach for Soft-Output Decoding For systematic encoders, soft-output of decoder can be split into 3 statistically independent parts: L(û i ) = log p(u i =, y) p(u i =, y) = log P P c Γ () i c Γ () i L ch y i L a (u i ) p(y x) Pr{c} p(y x) Pr{c} = log p(y i x i = +) p(y i x i = ) + log Pr{u i = } Pr{u i = } + log P c Γ () i P c Γ () i nq j= j6=i nq j= j6=i p(y j x j ) p(y j x j ) kq j= j6=i kq j= j6=i Pr{c j } Pr{c j } Intrinsic LLR (systematic part) A-priori LLR L e (û i ) Extrinsic LLR 2

22 Outline of Lectures Lesson : One Lesson of Information Theory Principle structure of communication systems Definitions of entropy, mutual information, Channel coding theorem of Shannon Lesson 2: Introduction to Error Correcting Codes Basics of error correcting codes Linear block codes Convolutional codes (if time permits) Lesson 3: State-of-the-art channel coding Coding strategies to approach the capacity limits Definition of soft-information and turbo decoding principle Examples for state-of-the-art error correcting codes 22

23 Soft-Output Decoding for (4,3,2)-SPC-Code E s /N = 2dB u L ch y encoding + approximation c L e (û) Pr{û correct} BPSK x = L ch y+ L e (û) L(û) HD AWGN error corrected y HD error detected. but not corrected 23

24 Turbo Decoding of (24,6,3)-Produktcode () u x LLR encoding AWGN BPSK SNR=2 db L a, (û) = L e, (û). vertical extrinsic decoding information L (û) L e, (û) L ch y L e, (û)

25 Turbo Decoding of (24,6,3)-Produktcode (2) L 2.5 e, (û) L (û) L ch y + L a, (û) horizontal L y + ch decoding L e, (û) + L a, (û) L ch y + L a,2 (u) L e, (û) = L a,2 (û) û

26 Turbo Decoding of (24,6,3)-Produktcode (3) L ch y + L a,2 (u) û 2 L 2 (û) x x vertical decoding L ch y + L a,2 (u) L ch y + L e,2 (û) + L a,2 (û) e,2 (û) L ch y + L ch y + L 2 (û) horizontal L e,2 (û) L a,2 (û) L e,2 (û) + L a,2 (û) decoding

27 Turbo Decoding of (24,6,3)-Produktcode (4). 3.4 L ch y + L a,3 (u) û 3 L 2 (û) vertical decoding L ch y + L a,3 (u) L ch y + L e,3 (û) + L a,3 (û) e,3 (û) L 3 (û) L e,3 (û) L ch y + L ch y horizontal L a,3 (û) L e,3 (û) + L a,3 (û) decoding

28 General Concept of Turbo Decoding L 2 ˆ a u Le ( ) ( u) Lch y D D 2 L ( u ˆ) L L ch e L 2 e y ( u ˆ) ( uˆ ) s L ( u) L y L ( uˆ ) 2 a ch s e L 2 ( uˆ ) L L ch e L 2 e y ( uˆ ) ( uˆ ) s Each decoder supplies extrinsic information as a priori information to other decoder L e (û) is incorporated in L(û) for systematic encoders Improvement by additional decoding iteration with a-priori knowledge if L e (û), L a (û) and L ch y s are statistically independent 28

29 Simulation Results for Product Code (7,4,3)-Hamming Codes, parallel concatenation It. It.2 It.3 analyt. -2 P b log E / N b 29

30 Simulation Results for Product Codes (5,,3)-Hamming-Codes, parallel concatenation It. It.2 It.3 analyt. -2 P b log E / N 3 b

31 Simulation Results for Product Codes (3,26,3)-Hamming-Codes, parallel concatenation It. It.2 It.3 analyt. -2 P b log E / N b 3

32 Simulation Results for Turbo Codes (L c = 3) Simple Block Interleaver No significant improvements after third decoding iteration P b x Block-Interleaver - -2 It. It. 2 It. 3 It. 4 It. 5 It. 6 P b 3x3 Block-Interleaver - -2 It. It. 2 It. 3 It. 4 It. 5 It log E / N b log E / N b 32

33 Simulation Results for Turbo Codes (L c = 3) P b Block and Random Interleavers Iterative process gains significantly even after sixth iteration Increasing interleaver size improves performance remarkably 9-Random-Interleaver, Rc=/ It. It. 2 It. 3 It. 4 It. 6 It. P b Comparison of different interleavers - -2 FC, Lc=9 BIL- BIL-4 BIL-9 RIL-9 RIL-9,Rc=/ E b / N in db Eb / N in db 33

34 Repeat Accumulate Code by ten Brink Half-rate outer repetition encoder Rate-one inner recursive convolutional encoder Approximately decoding iterations are needed - -2 BER E b /N in db 34

35 Repeat Accumulate Code by ten Brink 35

36 Application Areas of Turbo Detection Application of turbo processing not restricted to concatenated codes Applicable for any concatenated system Concatenation of source and channel coding (exploitation of residual redundancy from source coding) Concatenation of coding and modulation (bit-interleaved coded modulation) Channel equalization and decoding can be performed iteratively Multi-user detection and decoding can be performed iteratively 36

37 Institut für Thanks for your attention! September 2 Volker Kühn - State-of-the-Art Channel Coding UNIVERSITÄT ROSTOCK FAKULTÄT INFORMATIK UND ELEKTROTECHNIK

One Lesson of Information Theory

Institut für One Lesson of Information Theory Prof. Dr.-Ing. Volker Kühn Institute of Communications Engineering University of Rostock, Germany Email: volker.kuehn@uni-rostock.de http://www.int.uni-rostock.de/