Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels

A method is proposed, called channel polarization, to construct code sequences that achieve the symmetric capacity I(W) of any given binary-input discrete memoryless channel (B-DMC) W . The symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal...

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Published in:IEEE transactions on information theory Vol. 55; no. 7; pp. 3051 - 3073
Main Author: Arikan, Erdal
Format: Journal Article
Language:English
Published: New York, NY IEEE 01.07.2009
Institute of Electrical and Electronics Engineers
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
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ISSN:0018-9448, 1557-9654
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Abstract A method is proposed, called channel polarization, to construct code sequences that achieve the symmetric capacity I(W) of any given binary-input discrete memoryless channel (B-DMC) W . The symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability. Channel polarization refers to the fact that it is possible to synthesize, out of N independent copies of a given B-DMC W , a second set of N binary-input channels \{W_N^{(i)}:1\le i\le N\} such that, as N becomes large, the fraction of indices i for which I(W_N^{(i)}) is near 1 approaches I(W) and the fraction for which I(W_N^{(i)}) is near 0 approaches 1-I(W) . The polarized channels \{W_N^{(i)}\} are well-conditioned for channel coding: one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining. Codes constructed on the basis of this idea are called polar codes. The paper proves that, given any B-DMC W with I(W)> 0 and any target rate R ≪ I(W) , there exists a sequence of polar codes \{{\Fraktur {C}}_n;n\ge 1\} such that {\Fraktur {C}}_n has block-length N=2^n , rate \ge R , and probability of block error under successive cancellation decoding bounded as P_{e}(N,R) \le O(N^{-{1\over 4}}) independently of the code rate. This performance is achievable by encoders and decoders with complexity O(N\log N) for each.
AbstractList A method is proposed, called channel polarization, to construct code sequences that achieve the symmetric capacity I(W) of any given binary-input discrete memoryless channel (B-DMC) W. The symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability. Channel polarization refers to the fact that it is possible to synthesize, out of N independent copies of a given B-DMC W, a second set of N binary-input channels {WN(i)1 les i les N} such that, as N becomes large, the fraction of indices i for which I(WN(i)) is near 1 approaches I(W) and the fraction for which I(WN(i)) is near 0 approaches 1-I(W). The polarized channels {WN(i)} are well-conditioned for channel coding: one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining. Codes constructed on the basis of this idea are called polar codes. The paper proves that, given any B-DMC W with I(W) > 0 and any target rate R
A method is proposed, called channel polarization, to construct code sequences that achieve the symmetric capacity I(W) of any given binary-input discrete memoryless channel (B-DMC) W . The symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability. Channel polarization refers to the fact that it is possible to synthesize, out of N independent copies of a given B-DMC W , a second set of N binary-input channels \{W_N^{(i)}:1\le i\le N\} such that, as N becomes large, the fraction of indices i for which I(W_N^{(i)}) is near 1 approaches I(W) and the fraction for which I(W_N^{(i)}) is near 0 approaches 1-I(W) . The polarized channels \{W_N^{(i)}\} are well-conditioned for channel coding: one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining. Codes constructed on the basis of this idea are called polar codes. The paper proves that, given any B-DMC W with I(W)> 0 and any target rate R ≪ I(W) , there exists a sequence of polar codes \{{\Fraktur {C}}_n;n\ge 1\} such that {\Fraktur {C}}_n has block-length N=2^n , rate \ge R , and probability of block error under successive cancellation decoding bounded as P_{e}(N,R) \le O(N^{-{1\over 4}}) independently of the code rate. This performance is achievable by encoders and decoders with complexity O(N\log N) for each.
A method is proposed, called channel polarization, to construct code sequences that achieve the symmetric capacity $I(W)$ of any given binary-input discrete memoryless channel (B-DMC) $W$. The symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability. Channel polarization refers to the fact that it is possible to synthesize, out of $N$ independent copies of a given B-DMC $W$, a second set of $N$ binary-input channels $\{W_N^{(i)}:1\le i\le N\}$ such that, as $N$ becomes large, the fraction of indices $i$ for which $I(W_N^{(i)})$ is near 1 approaches $I(W)$ and the fraction for which $I(W_N^{(i)})$ is near 0 approaches $1-I(W)$. The polarized channels $\{W_N^{(i)}\}$ are well-conditioned for channel coding: one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining. Codes constructed on the basis of this idea are called polar codes. The paper proves that, given any B-DMC $W$ with $I(W)>0$ and any target rate $R < I(W)$, there exists a sequence of polar codes $\{{\mathscr C}_n;n\ge 1\}$ such that ${\mathscr C}_n$ has block-length $N=2^n$, rate $\ge R$, and probability of block error under successive cancellation decoding bounded as $P_{e}(N,R) \le \bigoh(N^{-\frac14})$ independently of the code rate. This performance is achievable by encoders and decoders with complexity $O(N\log N)$ for each. [PUBLICATION ABSTRACT]
Author Arikan, Erdal
Author_xml – sequence: 1
  givenname: Erdal
  surname: Arikan
  fullname: Arikan, Erdal
  email: arikan@ee.bilkent.edu.tr
  organization: Dept. of Electr.-Electron. Eng., Bilkent Univ., Ankara
BackLink http://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=21826048$$DView record in Pascal Francis
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Issue 7
Keywords Performance evaluation
Binary channel
Channel capacity
Discrete channel
successive cancellation decoding
Information rate
n channel
Decoding
Channel coding
Information transmission
Memoryless channel
Codec
Capacity-achieving codes
Algorithm complexity
Reed-Muller (RM) codes
Plotkin construction
polar codes
channel polarization
Reed Muller code
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PublicationTitle IEEE transactions on information theory
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References ref15
ref14
ref11
lin (ref6) 2004
gallager (ref10) 1968
ar?kan (ref12) 2008
ref2
sahai (ref13) 2008
ref1
ref17
muller (ref3) 1954; ec 3
ref16
reed (ref4) 1954; it 4
ref5
blahut (ref7) 1983
chung (ref9) 1974
forney (ref8) 2005
References_xml – year: 1974
  ident: ref9
  publication-title: A Course in Probability Theory
– ident: ref2
  doi: 10.1109/TIT.2005.862081
– ident: ref11
  doi: 10.2307/2003354
– ident: ref14
  doi: 10.1016/S0019-9958(58)90239-0
– year: 1968
  ident: ref10
  publication-title: Information Theory and Reliable Communication
– ident: ref15
  doi: 10.1109/TIT.1962.1057683
– year: 2008
  ident: ref12
  publication-title: On the Rate of Channel Polarization
– ident: ref17
  doi: 10.1109/LCOMM.2008.080017
– ident: ref5
  doi: 10.1109/TIT.1960.1057584
– ident: ref1
  doi: 10.1002/j.1538-7305.1948.tb01338.x
– ident: ref16
  doi: 10.1109/18.910573
– year: 2004
  ident: ref6
  publication-title: Error Control Coding
– year: 2005
  ident: ref8
  publication-title: MIT 6 451 Lecture Notes
– year: 1983
  ident: ref7
  publication-title: Theory and Practice of Error Control Codes
– volume: it 4
  start-page: 39
  year: 1954
  ident: ref4
  article-title: A class of multiple-error-correcting codes and the decoding scheme
  publication-title: IRE Trans Inf Theory
– year: 2008
  ident: ref13
  publication-title: private communication
– volume: ec 3
  start-page: 6
  year: 1954
  ident: ref3
  article-title: Application of Boolean algebra to switching circuit design and to error correction
  publication-title: IRE Trans Electron Comput
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Snippet A method is proposed, called channel polarization, to construct code sequences that achieve the symmetric capacity I(W) of any given binary-input discrete...
A method is proposed, called channel polarization, to construct code sequences that achieve the symmetric capacity $I(W)$ of any given binary-input discrete...
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SubjectTerms Applied sciences
Binary system
Blocking
Cancellation
Capacity planning
Capacity-achieving codes
Channel capacity
Channel coding
channel polarization
Channels
Codes
Coding, codes
Communication channels
Construction
Councils
Decoding
Encoders
Exact sciences and technology
Information theory
Information, signal and communications theory
Memoryless systems
Noise cancellation
Plotkin construction
polar codes
Polarization
Reed- Muller (RM) codes
Reproduction
Signal and communications theory
successive cancellation decoding
Systems, networks and services of telecommunications
Telecommunications
Telecommunications and information theory
Transmission and modulation (techniques and equipments)
Title Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels
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