Analysis of Iterative Decoding

1. Analysis of Iterative Decoding Alexei Ashikhmin Research Department of Mathematics of Communications Bell Laboratories

2. Mutual Information and Channel Capacity • LDPC Codes • Density Evolution Analysis of LDPC Codes • EXIT Functions Analysis of LDPC Codes • Binary Erasure Channel • Gaussian Channel • MIMO Channel • Expander Codes

3. Shannon’s Channel Coding Theorem • In 1948, Claude Shannon, generally regarded as the father of the Information Age, published the paper: “A Mathematical Theory of Communications” which laid the foundations of Information Theory.

4. In this remarkable paper, he formulated the notion of channel capacity, defining the maximal rate by which information can be transmitted reliably over the channel. Channel

5. Encoder • Shannon proved that for any channel, there exists a family of codes (including linear block codes) that achieve arbitrary small probability of error at any communication rate up to the channel capacity Channel Decoder

6. Linear binary codes • A binary linear [n,k] code C is a k-dimensional subspace of • R=k/n is the code rate • Example of an [6,2] code:

7. Repetition code of length 3 • Single parity check code of length 3 sum of code bits of any codeword equals zero by mod 2, i.e. the number of ones in any codeword is even

8. Shannon’s Channel Coding Theorem (Cont.) • Shannon proved that if R<C then a typical (random) code has the probability of error decreasing exponentially fast with the code length (SNR is the Signal to Noise Ratio)

9. The complexity of decoding of a random code is • We need codes with a nice structure • Algebraic codes (BCH, Reed-Solomon, Algebraic Geometry) have nice structure, but do not allow one to achieve capacity

10. Mutual Information and Channel Capacity • LDPC Codes • Density Evolution Analysis of LDPC Codes • EXIT Functions Analysis of LDPC Codes • Binary Erasure Channel • AWGN and Other Channels • MIMO Channel • Expander Codes

11. 1 1 1 Low Density Parity Check (LDPC) Codes • LDPC codes can be defined with the help of bipartite graphs 0 1 Variable nodes Check nodes

12. LDPC codes – Definition (Cont.) Sparse graph Average variable node degree dv Average check node degree dc n is the code length m is the number of parity checks n-m is the number of information symbols

13. Belief Propagation Decoding • We receive from the channel a vector of corrupted symbols • For each symbol we compute log-likelyhood ratio

14. Belief PropagationDecoding(Cont.) Sparse graph

15. Mutual Information and Channel Capacity • LDPC Codes • Density Evolution Analysis of LDPC Codes • EXIT Functions Analysis of LDPC Codes • Binary Erasure Channel • AWGN and Other Channels • MIMO Channel • Expander Codes

16. Density Evolution Analysis T.Richardson and R. Urbanke • Assume that we transmit +1, -1 through Gaussian channel. Then • received symbols are Gaussian random variables • their log-likelihood ratios (LLR) are also Gaussian random variables

17. Density Evolution Analysis Sparse graph

18. Mutual Information and Channel Capacity • LDPC Codes • Density Evolution Analysis of LDPC Codes • EXIT Functions Analysis of LDPC Codes • Binary Erasure Channel • AWGN and Other Channels • MIMO Channel • Expander Codes

19. Extrinsic Information Transfer (EXIT) Functions • Stephen ten Brink in 1999 came up with EXIT functions for analysis of iterative decoding of TURBO codes • Ashikhmin, Kramer, ten Brink 2002: EXIT functions analysis of LDPC codes and properties of EXIT functions in the binary erasure channel • E.Sharon, A.Ashikhmin, S.Litsyn 2003: EXIT functions for continues channels • I.Sutskover, S.Shamai, J.Ziv 2003: bounds on EXIT functions • I.Land, S.Huettinger, P. Hoeher, J.Huber 2003: bounds on EXIT functions • Others

20. Extrinsic APP Decoder Source Extrinsic Channel Encoder EXIT Functions (cont) • Average a priori information: • Average extrinsic information: • EXIT function:

21. R=4/15 Simplex [15,4] Code and a Good Code of Infinite Length with R=4/15

22. Encoder 1 Source Extrinsic APP Decoder Extrinsic Channel Communication Channel Encoder 2 • Average a priori information: • Average communication information: • Average extrinsic information: • EXIT function:

23. Mutual Information and Channel Capacity • LDPC Codes • Density Evolution Analysis of LDPC Codes • EXIT Functions Analysis of LDPC Codes • Binary Erasure Channel • AWGN and Other Channels • MIMO Channel • Expander Codes

24. EXIT Function in Binary Erasure ChannelA.Ashikhmin, G.Kramer, S. ten Brink • are split support weights (or generalized Hamming weights) of a code, i.e. the number of subspaces of the code that have dimension r and support weight i on the first n positions and support weight j on the second m positions. • Let • Then

25. Extrinsic Chan. Comm. Chan. Decoder Decoder

26. Examples for BEC with erasure probability q • Let dv=2 and dc=4, the code rate R=1-dv/dc =1/2 • and • This code does not achieve capacity

27. In BEC with q=0.3

28. Extrinsic APP Decoder Source Extrinsic Channel Encoder Area Theorems for Binary Erasure Channel • Theorem:

29. Code with large minimum distance Code with small minimum distance

30. Encoder 1 Source Extrinsic APP Decoder Extrinsic Channel Communication Channel Encoder 2 • Theorem: where C is the capacity of the communication channel

32. For successful decoding we must guarantee that EXIT functions do not intersect with each other • This is possible only if the area under the variable nodes function is larger than the area under the check nodes function

33. To construct an LDPC code that achieves capacity in BEC we must match the EXIT functions of variable and check nodes. • Tornado LDPC codes (A. Shokrollahi) • Right-regular LDPC codes (A. Shokrollahi), obtained with the help of the Taylor series expansion of the EXIT functions:

34. Mutual Information and Channel Capacity • LDPC Codes • Density Evolution Analysis of LDPC Codes • EXIT Functions Analysis of LDPC Codes • Binary Erasure Channel • AWGN and Other Channels • MIMO Channel • Expander Codes

35. AWGN and other Communication ChannelsE.Sharon, A. Ashikhmin, S. Litsyn • To analyse LDPC codes we need EXIT functions of the repetition and single parity check codes in other (not BEC) channels • EXIT function of repetition codes for AWGN channel where

36. Let be “soft bits” • Let be the conditional probability density of T given 1 was transmitted. • If the channel is T-consistent, i.e. if then the EXIT function of single repetition code of length n is

37. How accurate can we be with EXIT functions • Let us take the following LDPC code: • the variable nodes degree distribution • the check node degree distribution • According to Density Evolution analysis this code can work in AWGN channel with Eb/No=0.3dB • According to the EXIT function analyses the code can work at Eb/No=0.30046dB, the difference is only 0.00046dB

38. Mutual Information and Channel Capacity • LDPC Codes • Density Evolution Analysis of LDPC Codes • EXIT Functions Analysis of LDPC Codes • Binary Erasure Channel • Gaussian Channel • MIMO Channel • Expander Codes

39. Application to Multiple Antenna Channel • Capacity of the Multiple Input Multiple Output channel grows linearly with the number of antennas • We assume that detector knows coefficients

40. Design of LDPC Code for MIMO ChannelS. ten Brink, G. Kramer, A. Ashikhmin • We construct combined EXIT function of detector and variable nodes and match it with the EXIT function of the check nodes • The resulting node degree distribution is different from the AWGN channel

42. Probability of Error and Decoding Complexity • Let A be an LDPC code with rate R=(1-)C • Conjecture 1: the probability of decoding error of A decreases only polynomially with the code length • Conjecture 2: the complexity of decoding behaves like

43. Mutual Information and Channel Capacity • LDPC Codes • Density Evolution Analysis of LDPC Codes • EXIT Functions Analysis of LDPC Codes • Binary Erasure Channel • Gaussian Channel • MIMO Channel • Expander Codes

44. Exapnder CodesM.Sipser and D.A.Spielman (1996) • Let us take a bipartite expander graph • Assign to edges code bits such that bits on edges conneted to a left (right) node form a codeword of code C1 (C2) Bits form a code word of C2 Bits form a code word of C1 Bits form a code word of C2 Bits form a code word of C1

45. An (V,E) graph is called (,)-expander if every subset of at most |V| has at least |V| neighbors

46. Decoding of Exapnder Codes Maximum Likelihood Decoding of C2 Maximum Likelihood Decoding of C1 Maximum Likelihood Decoding of C2 Maximum Likelihood Decoding of C1

47. M. Sipser and D. Spielman (1996) showed that an Expander code can decode d/48 errors and that d grows linearly with N • G. Zemor (2001) proved that an Expander code can decode d/4 errors • A. Barg and G. Zemor proved that Expander codes have positive error exponent if R<C • R. Roth and V. Skachek (2003) proved that an Expander codes can decode d/2 errors • What is the complexity of decoding of Expander codes?

48. Complexity of Decoding of Expander Codes • Let N be the entire code length, let n be the length of codes C1 and C2 • We choose n=log2N and allow N tends to infinity • The complexity of ML decoding of C1 and C2 is O(2n)=O(N) • The overall complexity of decoding is linear in N • At the same time if R=(1-)C then the complexity of decoding is • Can we replace ML decoding with decoding up to half min.dist?

49. R1C Threshold of Decoding of Expander Codes • Barg and Zemor: Choose C2 to be a good code with R2 1 and C1 to be a good code with R1C (capacity). The rate of the expander code is R=R1+R2-1 C (capacity). C2Expander Code:

50. Codes with Polynomial Decoding Complexity and Positive Error ExponentA.Ashikhmin and V.Skachek (preliminary results) • We assume that in there exist LDPC codes such that • Conjecture 1. • Conjecture 2. The complexity of decoding • Let us use such kind of LDPC codes as constituent codes C1 and C2 in a Expander code Cexp with rate R=(1-)C.