.inue.uni-stuttgart.de CRC-Aided Belief Propagation List Decoding of Polar Codes M. Geiselhart, A. Elkelesh, M. Ebada S. Cammerer, S. ten Brink www University of Stuttgart ISIT 2020 21. - 26. June 2020 Institute of Telecommunications Prof. Dr. Ing. Stephan ten Brink
Institute of Telecommunications Can Iterative Decoding Approach CA-SCL Performance? 10 − 1 10 − 2 BLER 10 − 3 10 − 4 BP 10 − 5 CRC-aided-SCL ( L = 32 ) ML estimate via OSD-4 10 − 6 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5 E b / N 0 [dB] (128,64) 5G Polar+CRC Code, g CRC ( x ) = x 6 + x 5 + 1 1/26 Marvin Geiselhart CA-BPL Decoding of Polar Codes 21. - 26. June 2020
Institute of Telecommunications Can Iterative Decoding Approach CA-SCL Performance? 10 − 1 10 − 2 BLER 10 − 3 10 − 4 BP CRC-aided-BPL 10 − 5 CRC-aided-SCL ( L = 32 ) ML estimate via OSD-4 10 − 6 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5 E b / N 0 [dB] (128,64) 5G Polar+CRC Code, g CRC ( x ) = x 6 + x 5 + 1 1/26 Marvin Geiselhart CA-BPL Decoding of Polar Codes 21. - 26. June 2020
Institute of Telecommunications Outline 1 Motivation 2 Belief Propagation Decoding of Polar Codes 3 CRC-Aided BPL Decoding 4 Results 5 Conclusion 2/26 Marvin Geiselhart CA-BPL Decoding of Polar Codes 21. - 26. June 2020
Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion Institute of Telecommunications Agenda 1 Motivation 2 Belief Propagation Decoding of Polar Codes 3 CRC-Aided BPL Decoding 4 Results 5 Conclusion 3/26 Marvin Geiselhart CA-BPL Decoding of Polar Codes 21. - 26. June 2020
Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion Institute of Telecommunications Polar Codes u 0 x 0 y 0 W u 1 x 1 y 1 Hadamard matrix W y 2 u 2 x 2 � � ⊗ n W 1 0 G N = 1 1 u 3 x 3 y 3 W u 4 x 4 y 4 x = uG N W u 5 x 5 y 5 u A ∈ { 0 , 1 } W u A C = 0 u 6 x 6 y 6 W u 7 x 7 y 7 W • Introduced by Arıkan [Arıkan, 2009] • Shown to be capacity achieving for B-DMC for N → ∞ • Low complexity encoding of O ( N log N ) • State-of-the-art CA-SCL decoder using outer CRC code 4/26 Marvin Geiselhart CA-BPL Decoding of Polar Codes 21. - 26. June 2020
Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion Institute of Telecommunications Why Iterative Decoding? SCL has some limitations in practice • SCL is hard output • SCL is inherently sequential Potential benefits of iterative decoding • Soft-in / Soft-out → Enables iterative detection and decoding → Allows Turbo style decoding of concatenated codes • Easily parallelizable → Preferable for hardware implementations → Can reduce overall decoding latency 5/26 Marvin Geiselhart CA-BPL Decoding of Polar Codes 21. - 26. June 2020
Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion Institute of Telecommunications Brief History of Iterative Polar Decoding Polar Codes as Factor Graph Codes on Graphs Permutations → BP decoding [Schwartz and Vardy, 2006] [Forney, 2001] [Arıkan, 2010] [Hussami et al., 2009] [Dimnik and Be’ery, 2009] Permutation Selection Ensemble Decoding Methods (e.g. BPL decoding) [Tosun, 2019] [Hehn et al., 2007] [Raviv et al., 2020] [Elkelesh et al., 2018] 6/26 Marvin Geiselhart CA-BPL Decoding of Polar Codes 21. - 26. June 2020
Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion Institute of Telecommunications Agenda 1 Motivation 2 Belief Propagation Decoding of Polar Codes 3 CRC-Aided BPL Decoding 4 Results 5 Conclusion 7/26 Marvin Geiselhart CA-BPL Decoding of Polar Codes 21. - 26. June 2020
Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion Institute of Telecommunications Belief Propagation Decoding L -messages propagation PE R in , 1 R out , 1 L out , 1 L in , 1 L frozen L ch R in , 2 R out , 2 L out , 2 L in , 2 R out , 1 = R in , 1 ⊞ ( L in , 2 + R in , 2 ) R out , 2 = ( R in , 1 ⊞ L in , 1 )+ R in , 2 R -messages propagation L out , 1 = L in , 1 ⊞ ( L in , 2 + R in , 2 ) • R 0 , i = ∞ for i ∈ A c L out , 2 = ( R in , 1 ⊞ L in , 1 )+ L in , 2 • Pass messages up to N it , max times • Hard Decision on L 0 , i + R 0 , i and L n , i + R n , i yields ˆ u i and ˆ x i • Early stopping if ˆ uG N = ˆ x 8/26 Marvin Geiselhart CA-BPL Decoding of Polar Codes 21. - 26. June 2020
Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion Institute of Telecommunications Permuted Factor Graphs Stage 0 Stage 1 Stage 2 u 0 x 0 u 1 x 1 u 2 x 2 u 3 x 3 u 4 x 4 u 5 x 5 u 6 x 6 u 7 x 7 • Permutation of the stages have the same encoding function • ( log 2 N ) ! valid factor graph permutations [Hussami et al., 2009] • Contain different cycles ⇒ different convergence behavior 9/26 Marvin Geiselhart CA-BPL Decoding of Polar Codes 21. - 26. June 2020
Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion Institute of Telecommunications Permuted Factor Graphs Stage 0 Stage 1 Stage 2 Stage 2 Stage 0 Stage 1 u 0 x 0 u 0 x 0 u 1 x 1 u 1 x 1 u 2 x 2 u 2 x 2 u 3 x 3 u 3 x 3 u 4 x 4 u 4 x 4 u 5 x 5 u 5 x 5 u 6 x 6 u 6 x 6 u 7 x 7 u 7 x 7 • Permutation of the stages have the same encoding function • ( log 2 N ) ! valid factor graph permutations [Hussami et al., 2009] • Contain different cycles ⇒ different convergence behavior 9/26 Marvin Geiselhart CA-BPL Decoding of Polar Codes 21. - 26. June 2020
Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion Institute of Telecommunications Belief Propagation List Decoding x 1 ˆ yes x 1 ˆ ˆ x == ˆ u · G ? BP ( Π Π Π 1 ) ˆ u 1 no y , A • Decode received sequence y with L different permutations → e.g., cyclic shifts, random permutations • Of all converged decoders, take the ML-in-the-list decision x BPL = argmin ˆ ˆ x i � y − ˆ x i � • For correct decoding, at least one decoder has to converge correctly 10/26 Marvin Geiselhart CA-BPL Decoding of Polar Codes 21. - 26. June 2020
Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion Institute of Telecommunications Belief Propagation List Decoding x 1 ˆ yes ˆ x 1 x == ˆ ˆ u · G ? BP ( Π Π 1 ) Π ˆ u 1 no x 2 ˆ yes ˆ x 2 BP ( Π Π 2 ) ˆ x == ˆ u · G ? Π u 2 ˆ no y , A • Decode received sequence y with L different permutations → e.g., cyclic shifts, random permutations • Of all converged decoders, take the ML-in-the-list decision x BPL = argmin ˆ ˆ x i � y − ˆ x i � • For correct decoding, at least one decoder has to converge correctly 10/26 Marvin Geiselhart CA-BPL Decoding of Polar Codes 21. - 26. June 2020
Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion Institute of Telecommunications Belief Propagation List Decoding y x 1 ˆ yes ˆ x 1 ˆ x == ˆ u · G ? BP ( Π Π 1 ) Π ˆ u 1 no x i � � y − ˆ x 2 ˆ yes ˆ x 2 BP ( Π Π 2 ) x i , i ∈{ 1 ,..., L } x == ˆ ˆ u · G ? Π x BPL ˆ argmin u 2 ˆ no y , A ˆ x BPL = ˆ ˆ ˆ x L yes x L ˆ x == ˆ u · G ? BP ( Π Π L ) Π ˆ u L no • Decode received sequence y with L different permutations → e.g., cyclic shifts, random permutations • Of all converged decoders, take the ML-in-the-list decision x BPL = argmin ˆ ˆ x i � y − ˆ x i � • For correct decoding, at least one decoder has to converge correctly 10/26 Marvin Geiselhart CA-BPL Decoding of Polar Codes 21. - 26. June 2020
Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion Institute of Telecommunications (2048,1024)-Polar Code [Elkelesh et al., 2018] 10 − 2 BP 200 iterations BPL ( L = 32 , N it , max = 200 ) SCL L = 32 10 − 3 BLER 10 − 4 2 2 . 2 2 . 4 2 . 6 2 . 8 3 3 . 2 E b / N 0 [dB] • BPL approaches SCL already for L = 32 • What about CRC? 11/26 Marvin Geiselhart CA-BPL Decoding of Polar Codes 21. - 26. June 2020
Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion Institute of Telecommunications Agenda 1 Motivation 2 Belief Propagation Decoding of Polar Codes 3 CRC-Aided BPL Decoding 4 Results 5 Conclusion 12/26 Marvin Geiselhart CA-BPL Decoding of Polar Codes 21. - 26. June 2020
Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion Institute of Telecommunications CRC Code g ( x ) = x 2 + x + 1 y i D D • Definition: C = { m ( x ) g ( x ) : deg ( m ( x )) ≤ k } • Specified by CRC polynomial g ( x ) with degree r • Hard decision error detection: y ( x ) ≡ 0 mod g ( x ) • Usage in decoding CRC-Polar concatenated codes: Candidate selection in SCL decoding → CA-SCL Stopping condition for iterative decoding • Can we do better? 13/26 Marvin Geiselhart CA-BPL Decoding of Polar Codes 21. - 26. June 2020
Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion Institute of Telecommunications Soft-In/Soft-Out Decoding of CRC Code L in , 0 L in , 1 L in , 2 L in , 3 L in , 4 00 g ( x ) = x 2 + x + 1 State s 10 01 11 • Restriction to 2 r states, independent of N CRC • MAP decoding using BCJR algorithm [Bahl et al., 1974] 14/26 Marvin Geiselhart CA-BPL Decoding of Polar Codes 21. - 26. June 2020
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