1. Channel Polarization and Polar CodesBy
Fakhruddin Mahmood
Anlei Rao

2. Outline Introduction Channel Polarization Polar Codes Conclusion
Channel Combining
Channel Splitting
Polar Codes
Polar coding
Successive Decoding
Conclusion

3. 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

4. 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.

5. 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)

6. Channel Polarization
An operation converting N ind. copies of B-DMC W to a polarized channel set of { }

7. 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.

8. 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.

9. 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

10. Channel Polarization
Channel Combining:
with the transition prob:

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

12. 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}

13. Channel Polarization
Structure :

14. Example with N=8
Channel Combining:

15. Example with N=8
With simulation we can calculate the generating matrix for N=8:

16. Channel Polarization
Channel Splitting:
with the transition prob:

17. Example with N=8
After channel combining:

18. Example with N=8

19. Example with N=8

20. Example with N=8

21. Example with N=8

22. Example with N=8

23. Example with N=8

24. Polar Codes
Polar Coding
Based on the process of channel combining

25. Polar Codes Polar Coding Based on the process of channel combining
Using the generating matrix for coding:

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

27. 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

28. Polar Codes Successive Decoding
Based on the process of channel splitting

29. Polar Codes Successive Decoding
Based on the process of channel splitting
Use ML rule to make decisions

30. Polar Codes Successive Decoding
Based on the process of channel splitting
Use ML rule to make decisions
Probability of block error bounded as

31. 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)

32. Example of N=8

33. Example of N=8

34. Example of N=8

35. 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