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

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

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.

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

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 Encoder Channel Decoder

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:

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

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)

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

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

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

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

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

Belief Propagation Decoding (Cont.) Sparse graph

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

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

Density Evolution Analysis Sparse graph

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

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

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

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

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

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

EXIT Function in Binary Erasure Channel A. Ashikhmin, G. Kramer, S EXIT Function in Binary Erasure Channel A.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

Extrinsic Chan. Comm. Chan. Decoder Decoder

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

In BEC with q=0.3

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

Code with large minimum distance Code with small minimum distance

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

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

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:

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

AWGN and other Communication Channels E. Sharon, A. Ashikhmin, S AWGN and other Communication Channels E.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

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

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

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

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

Design of LDPC Code for MIMO Channel S. ten Brink, G. Kramer, A Design of LDPC Code for MIMO Channel S. 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

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

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

Exapnder Codes M.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

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

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

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?

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=log2 N 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?

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 R1 C (capacity). The rate of the expander code is R=R1+R2-1 C (capacity). C2 Expander Code: R1 C

Codes with Polynomial Decoding Complexity and Positive Error Exponent A.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.

Theorem. The complexity of decoding of Cexp is The error exponent is where i maximizes the expression: E is small, but positive. Hence Perror=2-NE is decreasing exponentially fast with the code length N.

Concatenation of LDPC and Expander codes Random codes

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EXIT Function source