Channel Polarization and Polar Codes by x5YKdrJ

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									                  By
Fakhruddin Mahmood
           Anlei Rao
Outline
 Introduction
 Channel Polarization
    Channel Combining
    Channel Splitting
 Polar Codes
    Polar coding
    Successive Decoding
 Conclusion
Introduction
 Shannon’s proof of noisy channel coding theorem is
  the random coding method that he used to show the
  existence of capacity achieving code sequences.
 Construction of capacity-achieving code sequences
  has been an elusive goal
 Polar codes [Arikan] were the first provably capacity
  achieving codes for any symmetric B-DMC
 Low encoding and decoding complexity O(NlogN)
 Main idea of polar codes is based on the phenomenon
  of channel polarization
Introduction
 By recursively combining and splitting individual
  channels, some channels become error free while
  others turn into complete noise
 Those fraction of channels that become noiseless are
  given by I(W) which is the symmetric capacity
 I(W) is equal to Shannon capacity C under the
  condition that the B-DMC is symmetric
 Shannon capacity C is the highest rate at which
  reliable communication is possible across W using the
  inputs letters of the channel with equal probability.
Introduction
 Polar coding is the construction of codes that achieve
 I(W) by taking advantage of the polarizing effect.

 Basic idea is to create a coding system where each
 coordinate channel can be accessed individually and
 send data only through those whose capacity is close to
 I(W)
Channel Polarization
 An operation converting N ind. copies of B-DMC W to
 a polarized channel set of {   }
Channel Polarization
 An operation converting N ind. copies of B-DMC into
  a polarized channel set of { }
 The polarized channel becomes either noisy or
  noiseless as block length N goes to infinity.
Channel Polarization
 An operation converting N ind. copies of B-DMC into
  a polarized channel set of { }
 The polarized channel becomes either noisy or
  noiseless as block length N goes to infinity.
 By sending the information bits through these
  noiseless channels, we can achieve the symmetric
  capacity of B-DMC.
Channel Polarization
 An operation converting N ind. copies of B-DMC into
  a polarized channel set of { }
 The polarized channel becomes either noisy or
  noiseless as block length N goes to infinity.
 By sending the information bits through these
  noiseless channels, we can achieve the symmetric
  capacity of B-DMC.
 Channel Polarization consists of two parts: channel
  combining and channel splitting
Channel Polarization
 Channel Combining:


 with the transition prob:
Channel Polarization
 Channel Combining:


  with the transition prob:
    : generating matrix calculated in a recursive way:
Channel Polarization
 Channel Combining:


  with the transition prob:
    : generating matrix calculated in a recursive way:




     : {1, 2, 3……N}         {1, 3……N-1, 2, 4……N}
Channel Polarization
 Structure :
Example with N=8

Channel Combining:
Example with N=8
  With simulation we can calculate the generating matrix
  for N=8:
Channel Polarization
 Channel Splitting:


 with the transition prob:
Example with N=8

After channel combining:
Example with N=8
Example with N=8
Example with N=8
Example with N=8
Example with N=8
Example with N=8
Polar Codes
 Polar Coding
 Based on the process of channel combining
Polar Codes
 Polar Coding
 Based on the process of channel combining
 Using the generating matrix for coding:
Polar Codes
 Polar Coding
 Based on the process of channel combining
 Using the generating matrix for coding:



 Choose the information set S={i:            }
Polar Codes
 Polar Coding
 Based on the process of channel combining
 Using the generating matrix for coding:



 Choose the information set S={i:            }
 Choose the frozen bits at will
Polar Codes
 Successive Decoding
 Based on the process of channel splitting
Polar Codes
 Successive Decoding
 Based on the process of channel splitting
 Use ML rule to make decisions
Polar Codes
 Successive Decoding
 Based on the process of channel splitting
 Use ML rule to make decisions
 Probability of block error bounded as
Polar Codes
 Successive Decoding
 Based on the process of channel splitting
 Use ML rule to make decisions
 Probability of block error bounded as
 Coding and decoding complexity: O(NlogN)
Example of N=8
Example of N=8
Example of N=8
Conclusion
 By combining and splitting the N-ind. copies of B-
  DMCs, we can get error free or pure-noise polarized
  channels.
 Transmitting information bits only through noiseless
  channels while fixing symbols transmitted through the
  pure-noise ones, the Shannon capacity of the
  symmetric B-DMC can be achieved.
 Polar codes, based on the phenomenon of channel
  polarization, are capacity-achieving for any symmetric
  B-DMC with low encoding and decoding complexity
  O(NlogN) and block error

								
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