Simplified Successive-Cancellation List Decoding of Polar Codes Seyyed Ali Hashemi , Carlo Condo, Warren J. Gross Department of Electrical and Computer Engineering McGill University Montr´ eal, Qu´ ebec, Canada July 12, 2016 Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 1 / 14
Motivation What is the problem? 5G requirements are stringent Polar Codes are a good match 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) SSCL Decoding of Polar Codes ISIT 2016 2 / 14
Background 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 Frozen bits help decoding Decoding schemes: List-SD Successive-Cancellation (SC) Speed SC SC List (SCL) Sphere Decoding (SD) SCL List-SD SD Error-Correction Performance E. Arıkan, ”Channel polarization: A method for constructing capacity achieving codes for symmetric binary input memoryless channels,” IEEE Transactions on Information Theory , vol. 55, no. 7, pp. 3051-3073, July 2009. Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 3 / 14
Background 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 Frozen bits help decoding List-SD � Decoding schemes: Successive-Cancellation (SC) Speed SC SC List (SCL) Sphere Decoding (SD) SCL List-SD SD Error-Correction Performance E. Arıkan, ”Channel polarization: A method for constructing capacity achieving codes for symmetric binary input memoryless channels,” IEEE Transactions on Information Theory , vol. 55, no. 7, pp. 3051-3073, July 2009. Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 3 / 14
Background Encoding and SC Decoding s = 0 s = 1 s = 2 s = 3 s = 3 α ���� ���� � �� � � �� � x 0 0 β s = 2 α l β r x 1 0 β l α r x 2 0 s = 1 u 3 x 3 s = 0 x 4 0 u 5 x 5 u 6 x 6 u 7 x 7 Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 4 / 14
Background Encoding and SC Decoding s = 0 s = 1 s = 2 s = 3 s = 3 α ���� ���� � �� � � �� � x 0 0 β s = 2 α l β r x 1 0 β l α r x 2 0 s = 1 u 3 x 3 s = 0 x 4 0 u 5 x 5 Exact formulation u 6 x 6 � 1 + e α i + α i +2 s − 1 � u 7 x 7 α l i = ln e α i + e α i +2 s − 1 α r i = α i +2 s − 1 + (1 − 2 β l i ) α i Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 4 / 14
Background Encoding and SC Decoding s = 0 s = 1 s = 2 s = 3 s = 3 α ���� ���� � �� � � �� � x 0 0 β s = 2 α l β r x 1 0 α r β l x 2 0 s = 1 u 3 x 3 s = 0 x 4 0 u 5 x 5 Exact formulation u 6 x 6 � 1 + e α i + α i +2 s − 1 � u 7 x 7 α l i = ln e α i + e α i +2 s − 1 α r i = α i +2 s − 1 + (1 − 2 β l i ) α i Hardware-friendly formulation α l i = sgn( α i ) sgn( α i +2 s − 1 ) min( | α i | , | α i +2 s − 1 | ) α r i = α i +2 s − 1 + (1 − 2 β l i ) α i Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 4 / 14
Background SSC and Fast-SSC SC s = 3 s = 2 s = 1 s = 0 T = 14 time-steps Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 5 / 14
Background SSC and Fast-SSC Simplified SC (SSC) s = 3 s = 2 s = 1 Rate-0 Rate-1 s = 0 T = 10 time-steps Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 5 / 14
Background SSC and Fast-SSC Fast-SSC s = 3 s = 2 Rep SPC s = 1 s = 0 T = 2 time-steps Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 5 / 14
Background 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) SSCL Decoding of Polar Codes ISIT 2016 6 / 14
Background 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 − 2ˆ u jl ) α jl PM i l = ln j =0 Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 6 / 14
Background 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 − 2ˆ u jl ) α jl PM i l = ln j =0 Hardware-friendly formulation � u i l � = 1 PM i − 1 l + | α i l | , if ˆ 2 (1 − sgn ( α i l )) , PM i l = PM i − 1 l , otherwise Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 6 / 14
Proposed Algorithm SCL Issues SCL requires sorting the path metrics Adds N more time-steps to the decoding process SSC and Fast-SSC are equivalent to SC They are guaranteed to preserve the error-correction performance Simplified SCL: Faster: simplified Rate-1, Rate-0 and Rep nodes Guaranteed to preserve error-correction performance Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 7 / 14
Proposed Algorithm Simplified SCL Theorem The path metric for a Rate-1 node of length 2 s can be calculated as 2 s − 1 � � � 1 + e − (1 − 2 β i ) α i PM 2 s − 1 = ln , i =0 where α i and β i are relative to the top of the Rate-1 node tree. Based on List-SD idea [ α 0 , α 1 , α 2 , α 3 ] ⇔ [ β 0 , β 1 , β 2 , β 3 ] s = 2 s = 1 s = 0 Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 8 / 14
Proposed Algorithm Simplified SCL Proof. Induction ( η i = 1 − 2 β i ) For Rate-1 node with N = 2: � � 1+ e α 0+ α 1 α l 0 = ln e α 0 + e α 1 α r 0 = α 1 + η l 0 α 0 SC η l 0 = η 0 η 1 η r 0 = η 1 � � � � 1 + e − η l 0 α l 1 + e − η r 0 α r PM 1 = ln +ln 0 0 Substituting for α l 0 , α r 0 , η l 0 and η r 0 : � 1 + e − η 0 α 0 � � 1 + e − η 1 α 1 � PM 1 = ln +ln The theorem holds for N = 2. Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 9 / 14
Proposed Algorithm Simplified SCL Proof. Proof. For Rate-1 node with N = 2 s : Induction ( η i = 1 − 2 β i ) For Rate-1 node with N = 2: PM 2 s − 1 = PM l 2 s − 1 − 1 + PM r 2 s − 1 − 1 � � 1+ e α 0+ α 1 α l 0 = ln = Σ 2 s − 1 − 1 ln � 1 + e − η i α i � e α 0 + e α 1 i =0 α r 0 = α 1 + η l 0 α 0 � 1 + e − η i +2 s − 1 α i +2 s − 1 � SC + ln η l 0 = η 0 η 1 = Σ 2 s − 1 � 1 + e − η i α i � i =0 ln , η r 0 = η 1 The theorem holds for all N . � � � � 1 + e − η l 0 α l 1 + e − η r 0 α r PM 1 = ln +ln 0 0 β 0 u 0 x 0 Substituting for α l 0 , α r 0 , η l 0 and η r 0 : β 1 u 1 x 1 β 2 � 1 + e − η 0 α 0 � � 1 + e − η 1 α 1 � PM 1 = ln +ln u 2 x 2 β 3 u 3 x 3 The theorem holds for N = 2. Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 9 / 14
Proposed Algorithm Simplified SCL Theorem In the hardware-friendly formulation of SCL algorithm, the path metric for a Rate-1 node of length 2 s can be calculated as 2 s − 1 PM 2 s − 1 = 1 � sgn ( α i ) α i − (1 − 2 β i ) α i , 2 i =0 where α i and β i are relative to the top of the Rate-1 node tree. Simplified Rate-1 node can be decoded in N time-steps About a factor of 3 faster than conventional SCL Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 10 / 14
Proposed Algorithm Simplified SCL Theorem The path metric for a Rate-0 node of length 2 s can be calculated as 2 s − 1 � 1 + e − α i � � PM 2 s − 1 = ln , i =0 where α i is the LLR value at the top of the Rate-0 node tree. [ α 0 , α 1 , α 2 , α 3 ] ⇔ [0 , 0 , 0 , 0] s = 2 s = 1 s = 0 Simplified Rate-0 nodes can be decoded in at most log 2 N time-steps The time complexity is reduced from O ( N ) to O (log 2 N ) Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 11 / 14
Proposed Algorithm Simplified SCL Theorem The path metric for a Rep node of length 2 s can be calculated as 2 s − 1 � � � 1 + e − (1 − 2 β 2 s − 1 ) α i PM 2 s − 1 = ln , i =0 where α i is relative to the top of the Rep node tree and β 2 s − 1 is relative to the information bit in the Rep node tree. [ α 0 , α 1 , α 2 , α 3 ] ⇔ [ β 3 , β 3 , β 3 , β 3 ] s = 2 s = 1 s = 0 Simplified Rep nodes can be decoded in at most log 2 N time-steps The time complexity is reduced from O ( N ) to O (log 2 N ) Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 12 / 14
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