[PPT] - CRC-Aided Belief Propagation List Decoding of Polar Codes M. PowerPoint Presentation

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www .inue.uni-stuttgart.de

CRC-Aided Belief Propagation List Decoding of Polar Codes

M. Geiselhart, A. Elkelesh, M. Ebada
S. Cammerer, S. ten Brink

University of Stuttgart ISIT 2020

21. - 26. June 2020

Institute of Telecommunications

Prof. Dr. Ing. Stephan ten Brink

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Institute of Telecommunications

Can Iterative Decoding Approach CA-SCL Performance?

1.5 2 2.5 3 3.5 4 4.5 5 10−6 10−5 10−4 10−3 10−2 10−1 Eb/N0 [dB] BLER

BP CRC-aided-SCL (L = 32) ML estimate via OSD-4

(128,64) 5G Polar+CRC Code, gCRC(x) = x6 +x5 +1

Marvin Geiselhart CA-BPL Decoding of Polar Codes

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Institute of Telecommunications

Can Iterative Decoding Approach CA-SCL Performance?

1.5 2 2.5 3 3.5 4 4.5 5 10−6 10−5 10−4 10−3 10−2 10−1 Eb/N0 [dB] BLER

BP CRC-aided-BPL CRC-aided-SCL (L = 32) ML estimate via OSD-4

(128,64) 5G Polar+CRC Code, gCRC(x) = x6 +x5 +1

Marvin Geiselhart CA-BPL Decoding of Polar Codes

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Institute of Telecommunications

Outline

1 Motivation 2 Belief Propagation Decoding of Polar Codes 3 CRC-Aided BPL Decoding 4 Results 5 Conclusion

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Agenda

1 Motivation 2 Belief Propagation Decoding of Polar Codes 3 CRC-Aided BPL Decoding 4 Results 5 Conclusion

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Polar Codes

u0 x0 W y0 u1 x1 W y1 u2 x2 W y2 u3 x3 W y3 u4 x4 W y4 u5 x5 W y5 u6 x6 W y6 u7 x7 W y7

Hadamard matrix GN =

1

1 1 ⊗n x = uGN uA ∈ {0,1} uAC = 0

Introduced by Arıkan [Arıkan, 2009]
Shown to be capacity achieving for B-DMC for N → ∞
Low complexity encoding of O(N logN)
State-of-the-art CA-SCL decoder using outer CRC code

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

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

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Brief History of Iterative Polar Decoding Polar Codes as Codes on Graphs → BP decoding

[Forney, 2001] [Arıkan, 2010]

Factor Graph Permutations

[Schwartz and Vardy, 2006] [Hussami et al., 2009] [Dimnik and Be’ery, 2009]

Permutation Selection Methods

[Tosun, 2019] [Raviv et al., 2020]

Ensemble Decoding (e.g. BPL decoding)

[Hehn et al., 2007] [Elkelesh et al., 2018]

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Agenda

1 Motivation 2 Belief Propagation Decoding of Polar Codes 3 CRC-Aided BPL Decoding 4 Results 5 Conclusion

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Belief Propagation Decoding

Lch Lfrozen L-messages propagation R-messages propagation

PE

Rin,1 Lout,1 Rout,1 Lin,1 Rout,2 Lin,2 Rin,2 Lout,2

Rout,1 = Rin,1 ⊞(Lin,2 +Rin,2) Rout,2 = (Rin,1 ⊞Lin,1)+Rin,2 Lout,1 = Lin,1 ⊞(Lin,2 +Rin,2) Lout,2 = (Rin,1 ⊞Lin,1)+Lin,2

R0,i = ∞ for i ∈ Ac
Pass messages up to Nit,max times
Hard Decision on L0,i +R0,i and Ln,i +Rn,i yields ˆ

ui and ˆ xi

Early stopping if ˆ

uGN = ˆ x

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Permuted Factor Graphs

u0 u1 u2 u3 u4 u5 u6 u7

Stage 0 Stage 1 Stage 2

x0 x1 x2 x3 x4 x5 x6 x7

Permutation of the stages have the same encoding function
(log2 N)! valid factor graph permutations [Hussami et al., 2009]
Contain different cycles ⇒ different convergence behavior

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Permuted Factor Graphs

u0 u1 u2 u3 u4 u5 u6 u7

Stage 0 Stage 1 Stage 2

x0 x1 x2 x3 x4 x5 x6 x7 u0 u1 u2 u3 u4 u5 u6 u7

Stage 2 Stage 0 Stage 1

x0 x1 x2 x3 x4 x5 x6 x7

Permutation of the stages have the same encoding function
(log2 N)! valid factor graph permutations [Hussami et al., 2009]
Contain different cycles ⇒ different convergence behavior

Marvin Geiselhart CA-BPL Decoding of Polar Codes

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Belief Propagation List Decoding

y,A BP(Π Π Π1) ˆ x == ˆ u·G? ˆ x1 ˆ u1 no yes ˆ x1

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

ˆ xBPL = argminˆ

xiy− ˆ

xi

For correct decoding, at least one decoder has to converge correctly

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Belief Propagation List Decoding

y,A BP(Π Π Π1) ˆ x == ˆ u·G? ˆ x1 ˆ u1 no yes ˆ x1 BP(Π Π Π2) ˆ x == ˆ u·G? ˆ x2 ˆ u2 no yes ˆ x2

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

ˆ xBPL = argminˆ

xiy− ˆ

xi

For correct decoding, at least one decoder has to converge correctly

Marvin Geiselhart CA-BPL Decoding of Polar Codes

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Belief Propagation List Decoding

y,A BP(Π Π Π1) ˆ x == ˆ u·G? ˆ x1 ˆ u1 no yes ˆ x1 BP(Π Π Π2) ˆ x == ˆ u·G? ˆ x2 ˆ u2 no yes ˆ x2 BP(Π Π ΠL) ˆ x == ˆ u·G? ˆ xL ˆ uL no yes ˆ xL ˆ xBPL = argmin

ˆ xi,i∈{1,...,L}

y− ˆ xi y ˆ xBPL

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

ˆ xBPL = argminˆ

xiy− ˆ

xi

For correct decoding, at least one decoder has to converge correctly

Marvin Geiselhart CA-BPL Decoding of Polar Codes

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

(2048,1024)-Polar Code [Elkelesh et al., 2018]

2 2.2 2.4 2.6 2.8 3 3.2 10−4 10−3 10−2 Eb/N0 [dB] BLER

BP 200 iterations BPL (L = 32,Nit,max = 200) SCL L = 32

BPL approaches SCL already for L = 32
What about CRC?

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Agenda

1 Motivation 2 Belief Propagation Decoding of Polar Codes 3 CRC-Aided BPL Decoding 4 Results 5 Conclusion

Marvin Geiselhart CA-BPL Decoding of Polar Codes

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

CRC Code yi D D

g(x) = x2 +x+1

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?

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Soft-In/Soft-Out Decoding of CRC Code

00 10 01 11 State s Lin,0 Lin,1 Lin,2 Lin,3 Lin,4 g(x) = x2 +x+1

Restriction to 2r states, independent of NCRC
MAP decoding using BCJR algorithm [Bahl et al., 1974]

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

CRC-Aided BP Decoding (BCJR)

Lch Lf L-messages propagation R-messages propagation Lin Lout CRC Trellis Polar FG

Turbo style decoding:

Left pass BP → CRC-BCJR → Right pass BP → ···

Stopping if ˆ

uGN = ˆ x and ˆ uA has valid CRC

Problem: Complexity of BCJR dominates O(NCRC2r)

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

CRC-Aided BP Decoding (SPA)

Lch Lf L-messages propagation R-messages propagation CRC FG Polar FG

HCRC =

1

1 1 1 1 1

Sum-Product Algorithm (SPA, well known from LDPC codes)
Surprisingly, the exact form of HCRC has only minor influence

→ naïve construction of HCRC from systematic GCRC

Extension with learned weights (“Neural BP” [Doan et al., 2019] )

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

CRC-Aided BPL Decoding

. . . . . . Lch Marvin Geiselhart CA-BPL Decoding of Polar Codes

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Agenda

1 Motivation 2 Belief Propagation Decoding of Polar Codes 3 CRC-Aided BPL Decoding 4 Results 5 Conclusion

Marvin Geiselhart CA-BPL Decoding of Polar Codes

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

(128,64)-CRC-Polar Code

1.5 2 2.5 3 3.5 4 4.5 5 10−6 10−5 10−4 10−3 10−2 10−1

BP

Eb/N0 [dB] BLER

Iterative Decoding CRC-based stopping Non-Iterative Decoding CA-SCL (L = 32) ML estimate via OSD-4

5G information set construction (70 non-frozen bits)
CRC-6 g(x) = x6 +x5 +1 from 5G standard
Iterative dec. with Nit,max = 200 (average @4dB: 5.93, @5dB: 3.03)

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

(128,64)-CRC-Polar Code

1.5 2 2.5 3 3.5 4 4.5 5 10−6 10−5 10−4 10−3 10−2 10−1

BP

Eb/N0 [dB] BLER

Iterative Decoding CRC-based stopping CRC-aided BCJR Non-Iterative Decoding CA-SCL (L = 32) ML estimate via OSD-4

5G information set construction (70 non-frozen bits)
CRC-6 g(x) = x6 +x5 +1 from 5G standard
Iterative dec. with Nit,max = 200 (average @4dB: 5.93, @5dB: 3.03)

Marvin Geiselhart CA-BPL Decoding of Polar Codes

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

(128,64)-CRC-Polar Code

1.5 2 2.5 3 3.5 4 4.5 5 10−6 10−5 10−4 10−3 10−2 10−1

BP

Eb/N0 [dB] BLER

Iterative Decoding CRC-based stopping CRC-aided SPA CRC-aided BCJR Non-Iterative Decoding CA-SCL (L = 32) ML estimate via OSD-4

5G information set construction (70 non-frozen bits)
CRC-6 g(x) = x6 +x5 +1 from 5G standard
Iterative dec. with Nit,max = 200 (average @4dB: 5.93, @5dB: 3.03)

Marvin Geiselhart CA-BPL Decoding of Polar Codes

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

(128,64)-CRC-Polar Code

1.5 2 2.5 3 3.5 4 4.5 5 10−6 10−5 10−4 10−3 10−2 10−1

BP BPL (L = 32)

Eb/N0 [dB] BLER

Iterative Decoding CRC-based stopping CRC-aided SPA CRC-aided BCJR Non-Iterative Decoding CA-SCL (L = 32) ML estimate via OSD-4

5G information set construction (70 non-frozen bits)
CRC-6 g(x) = x6 +x5 +1 from 5G standard
Iterative dec. with Nit,max = 200 (average @4dB: 5.93, @5dB: 3.03)

Marvin Geiselhart CA-BPL Decoding of Polar Codes

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

(128,64)-CRC-Polar Code

1.5 2 2.5 3 3.5 4 4.5 5 10−6 10−5 10−4 10−3 10−2 10−1

BP BPL (L = 32)

Eb/N0 [dB] BLER

Iterative Decoding CRC-based stopping CRC-aided SPA CRC-aided BCJR Non-Iterative Decoding CA-SCL (L = 32) ML estimate via OSD-4

5G information set construction (70 non-frozen bits)
CRC-6 g(x) = x6 +x5 +1 from 5G standard
Iterative dec. with Nit,max = 200 (average @4dB: 5.93, @5dB: 3.03)

Marvin Geiselhart CA-BPL Decoding of Polar Codes

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Which Graph Permutations to use for BPL?

For N = 128, there are 7! = 5040 stage permutations in total
Targeting a list size of L = 32: ∼ 1083 options
How to find a good set S of L permutations?

Similar permutations will likely output the same estimate ˆ xi Goal: diverse permutations rather than multiple good ones

We will formulate this as an optimization problem
Notation:

EML event of a (hypothetic) ML decoder failing Ei event of CA-BP decoder with permutation i failing D default permutation, i.e. stage order (0,1,...,n−1)

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

List Selection Optimization Problem

Regard only ML decodable y
Optimize total BLER performance of permutation set S

minimize

S

P

EBPL(S)
EML
= P
i∈S

Ei

EML
subject to

|S| = L.

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

List Selection Optimization Problem

Regard only ML decodable y
Optimize total BLER performance of permutation set S

minimize

S

P

EBPL(S)
EML
= P
i∈S

Ei

EML
subject to

|S| = L.

Factor by assuming one member D from S (the default FG):

P

i∈S

Ei

EML
D∈S

= P

ED
EML
·P

 

i∈S\{D}

Ei

ED,EML

 

List Gain
Exploit correlation of BP decoders

→ Conditioning on ED increases error probability → Fewer samples in Monte Carlo simulation required

Numeric optimization of the list gain (e.g. genetic algorithm)

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Comparison of Selection Methods

2.5 3 3.5 4 4.5 5 10−5 10−4 10−3 10−2

List Gain FG Selection Gain

Eb/N0 [dB] BLER

L = 1, default permutation D Factor Graph (FG) Selection L = 7, random permutations L = 7, cyclic shifts L = 7, genetic optimized

(128,64)-CRC-Polar code with CRC-6
L = n = 7 for fair comparison

→ Random permutations (drawn new for each sample) → Cyclic shifts → Offline genetic optimized S outperforms both methods

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Agenda

1 Motivation 2 Belief Propagation Decoding of Polar Codes 3 CRC-Aided BPL Decoding 4 Results 5 Conclusion

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Conclusion

We can decode CRC-aided polar codes iteratively
We can make use of the outer CRC code

... for early stopping ... for SISO decoding using BCJR or SPA

Iterative decoding can approach CA-SCL performance
Optimized permutation selection reduces complexity at equal

performance

Open Problems

From offline to smart online permutation selection CRC/outer code design for iterative decoding Soft CRC decoding

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Conclusion

We can decode CRC-aided polar codes iteratively
We can make use of the outer CRC code

... for early stopping ... for SISO decoding using BCJR or SPA

Iterative decoding can approach CA-SCL performance
Optimized permutation selection reduces complexity at equal

performance

Open Problems

From offline to smart online permutation selection CRC/outer code design for iterative decoding Soft CRC decoding

Marvin Geiselhart CA-BPL Decoding of Polar Codes

21. - 26. June 2020

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Conclusion

We can decode CRC-aided polar codes iteratively
We can make use of the outer CRC code

... for early stopping ... for SISO decoding using BCJR or SPA

Iterative decoding can approach CA-SCL performance
Optimized permutation selection reduces complexity at equal

performance

Open Problems

From offline to smart online permutation selection CRC/outer code design for iterative decoding Soft CRC decoding

Marvin Geiselhart CA-BPL Decoding of Polar Codes

21. - 26. June 2020

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Conclusion

We can decode CRC-aided polar codes iteratively
We can make use of the outer CRC code

... for early stopping ... for SISO decoding using BCJR or SPA

Iterative decoding can approach CA-SCL performance
Optimized permutation selection reduces complexity at equal

performance

Open Problems

From offline to smart online permutation selection CRC/outer code design for iterative decoding Soft CRC decoding

Marvin Geiselhart CA-BPL Decoding of Polar Codes

21. - 26. June 2020

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Conclusion

We can decode CRC-aided polar codes iteratively
We can make use of the outer CRC code

... for early stopping ... for SISO decoding using BCJR or SPA

Iterative decoding can approach CA-SCL performance
Optimized permutation selection reduces complexity at equal

performance

Open Problems

From offline to smart online permutation selection CRC/outer code design for iterative decoding Soft CRC decoding

Marvin Geiselhart CA-BPL Decoding of Polar Codes

21. - 26. June 2020

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Conclusion

We can decode CRC-aided polar codes iteratively
We can make use of the outer CRC code

... for early stopping ... for SISO decoding using BCJR or SPA

Iterative decoding can approach CA-SCL performance
Optimized permutation selection reduces complexity at equal

performance

Open Problems

From offline to smart online permutation selection CRC/outer code design for iterative decoding Soft CRC decoding

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Institute of Telecommunications Motivation BP Decoding CRC-Aided BPL Decoding Results Conclusion

Thank you for your attention!

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References

[Arıkan, 2009] Arıkan, E. (2009). Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels. IEEE Trans. Inf. Theory, 55(7):3051–3073. [Arıkan, 2010] Arıkan, E. (2010). Polar Codes: A Pipelined Implementation.

Proc. 4th ISBC, pages 11–14.

[Bahl et al., 1974] Bahl, L., Cocke, J., Jelinek, F ., and Raviv, J. (1974). Optimal Decoding of Linear Codes for Minimizing Symbol Error Rate. IEEE Trans. Inf. Theory, 20(2):284–287. [Dimnik and Be’ery, 2009] Dimnik, I. and Be’ery, Y. (2009). Improved Random Redundant Iterative HDPC Decoding. IEEE Transactions on Communications, 57(7):1982–1985. [Doan et al., 2019] Doan, N., Hashemi, S. A., Mambou, E. N., Tonnellier, T., and Gross, W. J. (2019). Neural Belief Propagation Decoding of CRC-Polar Concatenated Codes. In IEEE Inter. Conf. on Commun. (ICC). [Elkelesh et al., 2018] Elkelesh, A., Ebada, M., Cammerer, S., and ten Brink, S. (2018). Belief Propagation List Decoding of Polar Codes. IEEE Commun. Lett., 22(8):1536–1539. [Forney, 2001] Forney, G. D. (2001). Codes on graphs: normal realizations. IEEE Transactions on Information Theory, 47(2):520–548. [Hehn et al., 2007] Hehn, T., Huber, J. B., Laendner, S., and Milenkovic, O. (2007). Multiple-Bases Belief-Propagation for Decoding of Short Block Codes. In IEEE Inter. Symp. Inf. Theory (ISIT), pages 311–315. [Hussami et al., 2009] Hussami, N., Korada, S. B., and Urbanke, R. (2009). Performance of Polar Codes for Channel and Source Coding. In IEEE Inter. Symp. Inf. Theory (ISIT), pages 1488–1492. [Raviv et al., 2020] Raviv, N., Caciularu, A., Raviv, T., Goldberger, J., and Be’ery, Y. (2020). perm2vec: Graph permutation selection for decoding of error correction codes using self-attention. [Schwartz and Vardy, 2006] Schwartz, M. and Vardy, A. (2006). On the Stopping Distance and the Stopping Redundancy of Codes. IEEE Trans. Inf. Theory, 52(3):922–932. [Tosun, 2019] Tosun, B. (2019). Belief Propagation Decoding Using Factor Graph Permutations. Master thesis, Middle East Technical University.

Marvin Geiselhart CA-BPL Decoding of Polar Codes

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