# 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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