Lower Bounds on the Probability of Error of Polar Codes Boaz Shuval and Ido Tal Andrew and Erna Viterbi Department of Electrical Engineering Technion - Israel Institute of Technology Haifa 32000, Israel June 2017
Introduction ◮ Arıkan’s Polar Codes asymptotically achieve capacity ◮ Analysis based on upper bounds on P e ◮ How tight is the upper bound? ◮ Existing lower bounds on P e are trivial ◮ In this work: Improved lower bounds Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 2 / 20
Preliminaries ◮ BMS Channel W ( y | u ) ◮ Polar Construction → N = 2 n synthetic channels 1 , u i − 1 W i ( y N | u i ) 1 ◮ Polarize to “good” ( A ) and “bad” channels ◮ Transmit frozen bits on “bad” channels ◮ Successive Cancellation Decoding � i ∈ A c u i ˆ 1 , ˆ u i − 1 U i ( y N ) = 1 , ˆ u i − 1 1 arg max u i W i ( y N | u i ) i ∈ A 1 Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 3 / 20
SC Probability of Error Let E i = event that W i errs SC probability of error: �� � = P E i P SC e i ∈ A Bounds: � i ∈ A P { E i } ≤ P SC ≤ P { E i } max e i ∈ A Question How do we improve the lower bound? Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 4 / 20
Improving the Lower Bound Two ingredients: ◮ If A ′ ⊆ A then �� � � � � P E i ≥ P E i i ∈ A i ∈ A ′ ◮ Bonferroni bound: �� � � � P E i ≥ P { E i } − P { E i ∩ E j } i ∈ A i ∈ A i , j ∈ A , i < j Recall P { E i ∩ E j } = P { E i } + P { E j } − P { E i ∪ E j } Approach Lower bounds on P { E i ∪ E j } = ⇒ better lower bounds on P SC e Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 5 / 20
Previous Work ◮ Mori & Tanaka [2009] ◮ Density evolution to approximate joint distribution ⇒ P { E i ∪ E j } ◮ Exact for BEC ◮ Parizi & Telatar [2013] ◮ Only for BEC ◮ Track correlation between erasure events ◮ Showed: union bound asymptotically tight for BEC Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 6 / 20
New Lower Bound ◮ Works for any initial BMS channel W ◮ Provable lower bound on P SC e ◮ Approximates joint distribution of two synthetic channels W a , b ◮ Controls output alphabet sizes ◮ Coincides with lower bounds for BEC ◮ Better than existing lower bound for general BMS channels Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 7 / 20
Numerical Results 10 − 3 Upper Bound Previous Lower Bound Probability of Error 10 − 5 10 − 7 10 − 9 10 − 11 0 . 12 0 . 14 0 . 16 0 . 18 0 . 2 0 . 22 Crossover Probability Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 8 / 20
Numerical Results 10 − 3 Upper Bound Previous Lower Bound New Lower Bound Probability of Error 10 − 5 10 − 7 10 − 9 10 − 11 0 . 12 0 . 14 0 . 16 0 . 18 0 . 2 0 . 22 Crossover Probability Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 8 / 20
Conceptual Algorithm Input: ◮ BMS channel W ◮ a-channel transform list α 1 , α 2 , . . . , α n α, β ∈ {− , + } ◮ b-channel transform list β 1 , β 2 , . . . , β n e ( W a n , b n ) Output: Lower bound on P SC Steps: 1. Initialize: W 0 , 0 = W 2. For i = 1 , . . . , n , do: ◮ W a i , b i ← � JointlyPolarize α i ,β i ( W a i − 1 , b i − 1 ) LowerBound ( P SC e ( W a n , b n )) 3. Compute: Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 9 / 20
Conceptual Algorithm Input: ◮ BMS channel W ◮ a-channel transform list α 1 , α 2 , . . . , α n α, β ∈ {− , + } ◮ b-channel transform list β 1 , β 2 , . . . , β n e ( W a n , b n ) Output: Lower bound on P SC alphabet Steps: size grows 1. Initialize: W 0 , 0 = W 2. For i = 1 , . . . , n , do: control ◮ W a i , b i ← � JointlyPolarize α i ,β i ( W a i − 1 , b i − 1 ) alphabet size ◮ W a i , b i ← � JointlyUpgrade ( W a i , b i ) LowerBound ( P SC e ( W a n , b n )) 3. Compute: Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 9 / 20
SC Decoding – suboptimal ( y a , y b ) ( u a , u b ) ( 0 , 0 ) ( 0 , 1 ) ( 1 , 0 ) ( 1 , 1 ) ( 0 , 0 ) 0 . 30 0 . 04 0 . 04 0 . 62 ( 0 , 1 ) 0 . 44 0 . 46 0 . 01 0 . 09 ( 1 , 0 ) 0 . 22 0 . 49 0 . 24 0 . 05 ( 1 , 1 ) 0 . 05 0 . 54 0 . 32 0 . 09 Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 10 / 20
SC Decoding – suboptimal ( y a , y b ) ( u a , u b ) ( 0 , 0 ) ( 0 , 1 ) ( 1 , 0 ) ( 1 , 1 ) ( 0 , 0 ) 0 . 30 0 . 04 0 . 04 0 . 62 ( 0 , 1 ) 0 . 44 0 . 46 0 . 01 0 . 09 ( 1 , 0 ) 0 . 22 0 . 49 0 . 24 0 . 05 ( 1 , 1 ) 0 . 05 0 . 54 0 . 32 0 . 09 ◮ Optimal decoder: P e = 0 . 52 Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 10 / 20
SC Decoding – suboptimal ( y a , y b ) ( u a , u b ) ( 0 , 0 ) ( 0 , 1 ) ( 1 , 0 ) ( 1 , 1 ) ( 0 , 0 ) 0 . 30 0 . 04 0 . 04 0 . 62 ( 0 , 1 ) 0 . 44 0 . 46 0 . 01 0 . 09 ( 1 , 0 ) 0 . 22 0 . 49 0 . 24 0 . 05 ( 1 , 1 ) 0 . 05 0 . 54 0 . 32 0 . 09 ◮ Optimal decoder: P e = 0 . 52 ◮ SC decoder: P SC = 0 . 7075 e Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 10 / 20
SC Decoding – suboptimal Degrade: ( 0 , 0 ) , ( 1 , 1 ) → ( 0 ′ , 0 ′ ) ( 0 , 1 ) , ( 1 , 0 ) → ( 1 ′ , 1 ′ ) ( y a , y b ) ( y a , y b ) ( u a , u b ) ( u a , u b ) ( 0 ′ , 0 ′ ) ( 1 ′ , 1 ′ ) ( 0 , 0 ) ( 0 , 1 ) ( 1 , 0 ) ( 1 , 1 ) ( 0 , 0 ) ( 0 , 0 ) 0 . 30 0 . 04 0 . 04 0 . 62 0 . 92 0 . 08 ( 0 , 1 ) ( 0 , 1 ) 0 . 44 0 . 46 0 . 01 0 . 09 0 . 53 0 . 47 ( 1 , 0 ) ( 1 , 0 ) 0 . 22 0 . 49 0 . 24 0 . 05 0 . 27 0 . 73 ( 1 , 1 ) ( 1 , 1 ) 0 . 05 0 . 54 0 . 32 0 . 09 0 . 14 0 . 86 ◮ Optimal decoder: P e = 0 . 52 ◮ SC decoder: P SC = 0 . 7075 e Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 10 / 20
SC Decoding – suboptimal Degrade: ( 0 , 0 ) , ( 1 , 1 ) → ( 0 ′ , 0 ′ ) ( 0 , 1 ) , ( 1 , 0 ) → ( 1 ′ , 1 ′ ) ( y a , y b ) ( y a , y b ) ( u a , u b ) ( u a , u b ) ( 0 ′ , 0 ′ ) ( 1 ′ , 1 ′ ) ( 0 , 0 ) ( 0 , 1 ) ( 1 , 0 ) ( 1 , 1 ) ( 0 , 0 ) ( 0 , 0 ) 0 . 30 0 . 04 0 . 04 0 . 62 0 . 92 0 . 08 ( 0 , 1 ) ( 0 , 1 ) 0 . 44 0 . 46 0 . 01 0 . 09 0 . 53 0 . 47 ( 1 , 0 ) ( 1 , 0 ) 0 . 22 0 . 49 0 . 24 0 . 05 0 . 27 0 . 73 ( 1 , 1 ) ( 1 , 1 ) 0 . 05 0 . 54 0 . 32 0 . 09 0 . 14 0 . 86 ◮ Optimal decoder: P e = 0 . 52 ◮ Optimal decoder: P e = 0 . 555 ◮ SC decoder: P SC = 0 . 7075 ◮ SC decoder: P SC = 0 . 555 e e Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 10 / 20
SC Decoding – suboptimal Degrade: ( 0 , 0 ) , ( 1 , 1 ) → ( 0 ′ , 0 ′ ) ( 0 , 1 ) , ( 1 , 0 ) → ( 1 ′ , 1 ′ ) ( y a , y b ) ( y a , y b ) ( u a , u b ) ( u a , u b ) ( 0 ′ , 0 ′ ) ( 1 ′ , 1 ′ ) ( 0 , 0 ) ( 0 , 1 ) ( 1 , 0 ) ( 1 , 1 ) ( 0 , 0 ) ( 0 , 0 ) 0 . 30 0 . 04 0 . 04 0 . 62 0 . 92 0 . 08 ( 0 , 1 ) ( 0 , 1 ) 0 . 44 0 . 46 0 . 01 0 . 09 0 . 53 0 . 47 ( 1 , 0 ) ( 1 , 0 ) 0 . 22 0 . 49 0 . 24 0 . 05 0 . 27 0 . 73 ( 1 , 1 ) ( 1 , 1 ) 0 . 05 0 . 54 0 . 32 0 . 09 0 . 14 0 . 86 ◮ Optimal decoder: P e = 0 . 52 ◮ Optimal decoder: P e = 0 . 555 ◮ SC decoder: P SC = 0 . 7075 ◮ SC decoder: P SC = 0 . 555 e e Conclusion SC decoder is not ordered by degradation Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 10 / 20
New Decoder Joint channel: W a , b ( y a , y b | u a , u b ) ◮ New Decoder: minimize P { E a ∪ E b } using ˆ u a = φ a ( y a ) ˆ u b = φ b ( y b ) ◮ Notation: P ∗ e ◮ Generally requires exhaustive search Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 11 / 20
New Decoder Joint channel: W a , b ( y a , y b | u a , u b ) ◮ New Decoder: minimize P { E a ∪ E b } using ˆ u a = φ a ( y a ) ˆ u b = φ b ( y b ) ◮ Notation: P ∗ e ( W a , b ) ≥ P ∗ e ( W a , b ) P SC e ◮ Generally requires exhaustive search Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 11 / 20
New Decoder Joint channel: W a , b ( y a , y b | u a , u b ) ◮ New Decoder: minimize P { E a ∪ E b } using ˆ u a = φ a ( y a ) ˆ u b = φ b ( y b ) ◮ Notation: P ∗ e ( W a , b ) ≥ P ∗ e ( W a , b ) P SC e ◮ Generally requires exhaustive search ◮ For polar codes: ◮ easily found ◮ ordered by (proper) joint degradation: p � W a , b ⇒ P ∗ e ( W a , b ) ≥ P ∗ e ( Q a , b ) Q a , b Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 11 / 20
Upgrading Procedures Overview Goal: p � W a , b ◮ Find Q a , b ◮ Reduce output alphabet of one marginal ◮ Leave other marginal unchanged New joint channel upgrading procedures: ◮ A-channel upgrade ◮ B-channel upgrade Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 12 / 20
Joint Synthetic Channels – D -value Representation General form of Joint channel: W a , b ( y a , u a , y r | u a , u b ) � �� � y b D -values for BMS channel: d ( y ) = W ( y | 0 ) − W ( y | 1 ) W ( y | 0 ) + W ( y | 1 ) May switch to D -value representation: W a , b ( y a , u a , d b | u a , u b ) Lemma W a , b ( y a , u a , y r | u a , u b ) ≡ W a , b ( y a , u a , d b | u a , u b ) Boaz Shuval and Ido Tal (Technion) Lower bounds on P e of Polar Codes June 2017 13 / 20
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