Practical Design and Decoding of Polar Codes Vera Miloslavskaya - - PowerPoint PPT Presentation

Practical Design and Decoding of Polar Codes

Vera Miloslavskaya

Saint-Petersburg State Polytechnic University veram@dcn.icc.spbstu.ru

January 2015

Polar Codes

Polar codes can achieve the capacity of an arbitrary binary-input

• utput-symmetric memoryless channel

An (n = lm, k) polar code is generated by k rows of matrix Gn = A⊗m, where A = 1 1 1

• Problems:

The successive cancellation decoder is far from maximum likelihood Minimum distance of polar code ∼ √n The original Arikan construction results in codes of length 2m

                          =

⊗

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

3

A

u0=0 u4=0 u2=0 u6=a0 u1=0 u5=a1 u3=a2 u7=a3 c0 c1 c2 c3 c4 c5 c6 c7 a0 a1 a2 a3

Vera Miloslavskaya (SPbSPU) Practical Design and Decoding of Polar Codes January 2015 2 / 4

Design of Polar Codes

A generalization of polar codes: dynamic frozen symbols A dynamic frozen symbol is equal to some linear combination of information symbols Polar subcodes of extended BCH codes An algorithm for construction of shortened polar codes Error probability under the successive cancellation decoding is minimized Optimal for the code length lower than 64, suboptimal for other lengths

Length=1024, Dimension=512

10-6 10-5 10-4 10-3 10-2 10-1 100 0.5 1 1.5 2 2.5 FER Eb/N0, dB Subcode of e-BCH, d=24, L=32 Classical polar, L=32 Polar-CRC, L=32 WiMAX LDPC (1032,516)

Length=768, Dimension=512

10-6 10-5 10-4 10-3 10-2 10-1 100 1 1.5 2 2.5 3 3.5 4 FER Eb/N0, dB LDPC Random shortening Random puncturing Proposed shortening

Vera Miloslavskaya (SPbSPU) Practical Design and Decoding of Polar Codes January 2015 3 / 4

Decoding of Polar Codes

A sequential decoding algorithm is proposed An instance of the stack successive cancellation (SC) decoding method A new metric for paths ui

0 is used

Significant complexity reduction compared to the list/stack SC decoding The algorithm can be applied to Polar codes with the 2 × 2 kernel Polar codes with an arbitrary binary kernel Reed-Solomon codes represented as polar codes with dynamic frozen symbols

Average decoding complexity for (1024, 512) codes, ×103 real operations

Eb/N0, dB Polar code, proposed approach LDPC, Belief prop. Additions Comparisons Additions log tanh( x

2 )

L = 32 L = 256 L = 2048 L = 32 L = 256 L = 2048 ≤ 200 iter. ≤ 200 iter. 141 833 5231 227 1332 8374 2617 1307 0.5 133 752 4265 218 1224 6968 2333 1112 1 73 286 1232 122 477 2065 1469 722 1.5 32 88 267 54 151 461 394 185 2 18 27 42 31 48 74 140 62

Vera Miloslavskaya (SPbSPU) Practical Design and Decoding of Polar Codes January 2015 4 / 4