A Fast Polar Code List Decoder Architecture Based on Sphere - PowerPoint PPT Presentation

A Fast Polar Code List Decoder Architecture Based on Sphere Decoding Seyyed Ali Hashemi , Carlo Condo, Warren J. Gross Department of Electrical and Computer Engineering McGill University Montr´ eal, Qu´ ebec, Canada May 31, 2017 Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 0/15

What is the problem? ◮ Polar Codes are adopted in 5G ◮ High speed and good error-correction performance ◮ Successive-Cancellation List (SCL) Decoding ◮ Very good error-correction performance but high complexity ◮ Very slow : there are many redundant calculations In this talk: We show how to speed up SCL without losing error-correction performance! Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 1/15

Polar Codes ◮ Can provably achieve channel capacity ◮ Encoding is based on polarizing matrix G ⊗ n ◮ Input bits are divided into Information bits and Frozen bits ◮ Decoding schemes: List-SD Speed SC ◮ Successive- Cancellation (SC) SCL SD ◮ SC List (SCL) ◮ Sphere Decoding (SD) Error-Correction Performance ◮ List-SD E. Arıkan, ”Channel polarization: A method for constructing capacity achieving codes for symmetric binary input memoryless channels,” T-IT , July 2009. Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 2/15

Polar Codes ◮ Can provably achieve channel capacity ◮ Encoding is based on polarizing matrix G ⊗ n ◮ Input bits are divided into Information bits and Frozen bits List-SD � ◮ Decoding schemes: Speed SC ◮ Successive- Cancellation (SC) SCL SD ◮ SC List (SCL) ◮ Sphere Decoding (SD) Error-Correction Performance ◮ List-SD E. Arıkan, ”Channel polarization: A method for constructing capacity achieving codes for symmetric binary input memoryless channels,” T-IT , July 2009. Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 2/15

SC Decoding α β β r α l β l α r ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 3/15

SC Decoding α β α l β r β l α r ˆ ˆ ˆ u 0 u 1 ˆ u 2 ˆ u 3 u 4 ˆ u 5 ˆ u 6 ˆ u 7 Exact formulation � α i + N ν � � α i �� � α l i = 2 arctanh tanh tanh 2 , 2 2 α r 2 + ( 1 − 2 β l i = α i + N ν i ) α i , Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 3/15

SC Decoding α β α l β r β l α r ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 Hardware-friendly formulation α l i = sgn ( α i ) sgn ( α i + N ν 2 ) min ( | α i | , | α i + 2 s − 1 | ) α r 2 + ( 1 − 2 β l i = α i + N ν i ) α i Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 3/15

SCL Decoding ◮ For finite practical code-lengths, SCL estimates each information bit as either 0 or 1 ◮ L codeword candidates survive to limit complexity ◮ CRC-aided SCL can outperform LDPC codes ◮ A path metric helps the selection of the surviving candidates Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 4/15

SCL Decoding ◮ For finite practical code-lengths, SCL estimates each information bit as either 0 or 1 ◮ L codeword candidates survive to limit complexity ◮ CRC-aided SCL can outperform LDPC codes ◮ A path metric helps the selection of the surviving candidates Exact formulation i � � 1 + e − ( 1 − 2 ˆ � u jl ) α jl PM i l = ln j = 0 Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 4/15

SCL Decoding ◮ For finite practical code-lengths, SCL estimates each information bit as either 0 or 1 ◮ L codeword candidates survive to limit complexity ◮ CRC-aided SCL can outperform LDPC codes ◮ A path metric helps the selection of the surviving candidates Exact formulation i � � 1 + e − ( 1 − 2 ˆ � u jl ) α jl PM i l = ln j = 0 Hardware-friendly formulation � u i l � = 1 if ˆ � � �� PM i − 1 l + | α i l | , 1 − sgn α i l , 2 PM i l = PM i − 1 l , otherwise Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 4/15

Simplified SCL (SSCL) ◮ SCL requires traversing the whole decoding tree ◮ Very slow SSCL and SSCL-SPC: ◮ Faster: simplified Rate-1, Rate-0, Rep and SPC nodes ◮ No need to traverse the decoding tree ◮ Guaranteed to preserve error-correction performance for SSCL SSCL SSCL-SPC Rep Rep SPC Rep Rate-1 Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 5/15

Simplified Nodes ◮ Rate-0: LLRs should be positive → Negative LLRs penalize the path ◮ Rep: Based on info. bit → LLRs are either all positive or all negative ◮ Rate-1: Each bit is estimated as either 0 or 1 → N ν bits ◮ SPC: Even Parity Constraint → 1) Least reliable bit is estimated first (1). 2) All other bits are estimated ( N ν − 1). 3) Parity constraint is imposed (1) [ α 0 , α 1 , α 2 , α 3 ] ⇔ [ β 0 , β 1 , β 2 , β 3 ] Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 6/15

SSCL (SSCL-SPC) Issues ◮ SSCL (SSCL-SPC) requires estimating all the bits in Rate-1 (and SPC) nodes ◮ N ν ( N ν + 1) time-steps Fast-SSCL (Fast-SSCL-SPC): ◮ Very fast: Rate-1 (and SPC) nodes can be further simplified ◮ A specific number of bit-estimations is required for every list size ◮ Guaranteed to preserve error-correction performance Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 7/15

Fast-SSCL Theorem In SCL decoding with list size L, the maximum number of bit estimations in a Rate-1 node of length N ν required to get the exact same results as the conventional SCL decoder is min ( L − 1 , N ν ) . In practical polar codes: ◮ There are many instances where L − 1 < N ν . ◮ Savings in number of time-steps can be achieved. Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 8/15

Fast-SSCL Example L = 2, N ν = 4, | α 0 l | ≤ | α 1 l | ≤ | α 2 l | ≤ | α 3 l | , PM 0 0 < PM 0 1 PM 00 PM 01 Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 9/15

Fast-SSCL Example L = 2, N ν = 4, | α 0 l | ≤ | α 1 l | ≤ | α 2 l | ≤ | α 3 l | , PM 0 0 < PM 0 1 PM 00 PM 01 PM 00 PM 00 + | α 00 | PM 01 PM 01 + | α 01 | Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 9/15

Fast-SSCL Example L = 2, N ν = 4, | α 0 l | ≤ | α 1 l | ≤ | α 2 l | ≤ | α 3 l | , PM 0 0 < PM 0 1 PM 00 PM 01 PM 00 PM 00 + | α 00 | PM 01 PM 01 + | α 01 | PM 00 PM 00 + | α 10 | PM 01 PM 01 + | α 11 | Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 9/15

Fast-SSCL Example L = 2, N ν = 4, | α 0 l | ≤ | α 1 l | ≤ | α 2 l | ≤ | α 3 l | , PM 0 0 < PM 0 1 PM 00 PM 01 PM 00 PM 00 + | α 00 | PM 01 PM 01 + | α 01 | PM 00 PM 00 + | α 10 | PM 01 PM 01 + | α 11 | Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 9/15

Fast-SSCL Example L = 2, N ν = 4, | α 0 l | ≤ | α 1 l | ≤ | α 2 l | ≤ | α 3 l | , PM 0 0 < PM 0 1 PM 00 PM 01 PM 00 PM 00 + | α 00 | PM 01 PM 01 + | α 01 | PM 00 PM 00 + | α 10 | PM 00 + | α 00 | PM 00 + | α 00 | + | α 10 | Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 9/15

Fast-SSCL Example L = 2, N ν = 4, | α 0 l | ≤ | α 1 l | ≤ | α 2 l | ≤ | α 3 l | , PM 0 0 < PM 0 1 PM 00 PM 01 PM 00 PM 00 + | α 00 | PM 01 PM 01 + | α 01 | PM 00 PM 00 + | α 10 | PM 00 + | α 00 | PM 00 + | α 00 | + | α 10 | α 1 l is always discarded! Result: We prove that there is no need to estimate the bits after min ( L − 1 , N ν ) bits are estimated! Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 9/15

Fast-SSCL Implementation Issue The LLR values at the top of the tree have to be sorted Solution At every time-step i , the i -th least reliable bit is found Example: L = 4, N ν = 4, | α 0 l | ≤ | α 1 l | ≤ | α 2 l | ≤ | α 3 l | Time-step 1: Find | α 0 l | + bit estimation Time-step 2: Find | α 1 l | + bit estimation Time-step 3: Find | α 2 l | + bit estimation Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 10/15

Time-Step Reduction P ( 1024 , 256 ) P ( 1024 , 512 ) P ( 1024 , 768 ) 1200 1200 1200 1000 1000 1000 800 800 800 600 600 600 400 400 400 200 200 200 0 0 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 L L L SSCL SSCL-SPC Fast-SSCL Fast-SSCL-SPC Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 11/15

Decoder Architecture Node Sequence Controller CRC Unit PM Computation and Sorting SC Decoders PE PE 1 . . . . . . · · · Memories . . . PE PE P · · · 1 L Channel LLRs Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 12/15