By Fakhruddin Mahmood

Anlei Rao

OutlineIntroductionChannel Polarization

Channel CombiningChannel Splitting

Polar CodesPolar codingSuccessive Decoding

Conclusion

IntroductionShannon’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

IntroductionBy 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.

IntroductionPolar 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 PolarizationAn operation converting N ind. copies of B-

DMC W to a polarized channel set of { }

Channel PolarizationAn 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 PolarizationAn 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 PolarizationAn 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 PolarizationChannel Combining:

with the transition prob:

Channel PolarizationChannel Combining:

with the transition prob: : generating matrix calculated in a

recursive way:

Channel PolarizationChannel Combining:

with the transition prob: : generating matrix calculated in a

recursive way:

: {1, 2, 3……N} {1, 3……N-1, 2, 4……N}

Channel PolarizationStructure :

Example with N=8

Channel Combining:

Example with N=8

With simulation we can calculate the generating matrix for N=8:

Channel PolarizationChannel 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 CodesPolar CodingBased on the process of channel combining

Polar CodesPolar CodingBased on the process of channel combiningUsing the generating matrix for coding:

Polar CodesPolar CodingBased on the process of channel combiningUsing the generating matrix for coding:

Choose the information set S={i: }

Polar CodesPolar CodingBased on the process of channel combiningUsing the generating matrix for coding:

Choose the information set S={i: }

Choose the frozen bits at will

Polar CodesSuccessive DecodingBased on the process of channel splitting

Polar CodesSuccessive DecodingBased on the process of channel splittingUse ML rule to make decisions

Polar CodesSuccessive DecodingBased on the process of channel splittingUse ML rule to make decisionsProbability of block error bounded as

Polar CodesSuccessive DecodingBased on the process of channel splittingUse ML rule to make decisionsProbability of block error bounded asCoding and decoding complexity: O(NlogN)

Example of N=8

Example of N=8

Example of N=8

ConclusionBy 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