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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
Channel
Decoder
Encoder
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 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)
Shannon’s Channel Coding Theorem (Cont.)
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 ChannelsMIMO Channel
Expander Codes
Low Density Parity Check (LDPC) Codes
LDPC codes can be defined with the help of bipartite graphs
Var
iabl
e no
des
Che
ck n
odes
0
1
1
1
1
LDPC codes – Definition (Cont.)Sparse graph
Average variablenode degree dv
Average checknode degree dc
n is the code length m is the number of parity checksn-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 ChannelsMIMO Channel
Expander Codes
Density Evolution Analysis
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
T.Richardson and R. Urbanke
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 ChannelsMIMO 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)
Average a priori information:
Average extrinsic information:
EXIT function:
EncoderExtrinsi
c
Channel
Extrinsic
APP
Decoder
Source
Simplex [15,4] Code and a Good Code of Infinite Length with R=4/15
R=4/15
Average a priori information: Average communication information: Average extrinsic information: EXIT function:
Encoder
1Source
Extrinsic
Channel Extrinsic
APP
Decoder
Encoder
2
Communication
Channel
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 ChannelsMIMO Channel
Expander Codes
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
Dec
oder
Dec
oder
Comm. Chan.
Extrinsic Chan.
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
Theorem:
EncoderExtrinsic
Channel
Extrinsic
APP
Decoder
Source
Code with large
minimum distance
Code with small
minimum distance
Theorem:
where C is the capacity of the communication channel
Encoder
1Source
Extrinsic
Channel Extrinsic
APP
Decoder
Encoder
2
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 ChannelsMIMO Channel
Expander Codes
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
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 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
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 C1
Bits form a code word of C1
Bits form a code word of C2
Bits form a code word of C2
An (V,E) graph is called (,)-expander if every subset of at most |V| has at least |V| neighbors
Maximum Likelihood Decoding of C1
Decoding of Exapnder Codes
Maximum Likelihood Decoding of C1
Maximum Likelihood Decoding of C2
Maximum Likelihood Decoding of C2
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.
Random codes
Concatenation of LDPC and Expander codes
Thank you for your attention
YOURTHANK
ATTENTION!YOUFOR
EXIT Function
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