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On the Decoding of Polar Codeson Permuted Factor Graphs
Nghia Doan1, Seyyed Ali Hashemi1, Marco Mondelli2, andWarren Gross1
1McGill University, Quebec, Canada2Standford University, California, USA
GLOBECOM 2018Abu Dhabi, UAEDec 10, 2018
Doan et. al On the Decoding of Polar Codes on Permuted Factor Graphs 0/17
Problem Statement
I Polar codes: selected for the eMBB control channel in 5G
I Successive Cancellation (SC) List (SCL): gooderror-correction performance, serial nature
I Belief Propagation (BP): reasonable error-correctionperformance, highly parallel→ enable high decodingthroughput !
Doan et. al On the Decoding of Polar Codes on Permuted Factor Graphs 1/17
Problem Statement
This paper
I Improve the error-correction performance of polar-codedecoding by exploiting factor-graph permutations
I Provide a hardware friendly representation of thefactor-graph permutations
Doan et. al On the Decoding of Polar Codes on Permuted Factor Graphs 2/17
Polar codesI Introduced by Arıkan1 in 2009I P(N,K ), N: code length, K : message lengthI Code construction: based on polarization phenomenon
I K most reliable channels: information bitsI (N − K ) least reliable channels: frozen bits
P(8, 5) with u0, u1, and u2 are frozen bits
1E. Arıkan. “Channel Polarization: A Method for Constructing Capacity-Achieving Codes for SymmetricBinary-Input Memoryless Channels”. In: 55.7 (2009), pp. 3051–3073. ISSN: 0018-9448. DOI:10.1109/TIT.2009.2021379.Doan et. al On the Decoding of Polar Codes on Permuted Factor Graphs 3/17
Belief Propagation (BP) Decoding
I An iterative message passing algorithmI Belief messages are updated through Processing
Elements (PEs)I Termination conditions: CRC-based condition2, maximum
number of iterations
BP decoding for P(8, 5) A polar PE
2Y. Ren et al. “Efficient early termination schemes for belief-propagation decoding of polar codes”. In: IEEE 11thInt. Conf. on ASIC. 2015, pp. 1–4. DOI: 10.1109/ASICON.2015.7517046.
Doan et. al On the Decoding of Polar Codes on Permuted Factor Graphs 4/17
BP Decoding on Permuted Factor Graphs
I Permuting the factor-graph layers preserves the code34
I Running BP decoding on multiple factor-graphpermutations improves the error-correction performance56
Various factor-graph permutations of P(8, 5)
3S. B. Korada. “Polar Codes for Channel and Source Coding”. PhD thesis. Lausanne, Switzerland: EPFL, 2009.4Nadine Hussami, Satish Babu Korada, and Rudiger Urbanke. “Performance of polar codes for channel and
source coding”. In: IEEE Int. Symp. on Inf. Theory. 2009, pp. 1488–1492.5A. Elkelesh et al. “Belief propagation decoding of polar codes on permuted factor graphs”. In: IEEE Wireless
Commun. and Net. Conf. 2018, pp. 1–6. DOI: 10.1109/WCNC.2018.8377158.6Ahmed Elkelesh et al. “Belief Propagation List Decoding of Polar Codes”. In: IEEE Communications Letters 22
(2018), pp. 1536–1539.
Doan et. al On the Decoding of Polar Codes on Permuted Factor Graphs 5/17
BP Decoding on Permuted Factor Graphs
Permuted factor graph representations for P(8, 5)
I Problem: each factor-graph permutation requires adifferent decoding schedule→ require different decodersin hardware
I Solution: transform factor-graph layer permutation tocodeword-position permutation
Doan et. al On the Decoding of Polar Codes on Permuted Factor Graphs 6/17
Factor-graph Permutation to Codeword-positionPermutation
I Definitions:
I Permutation π : {0,1, . . . ,n − 1} → {0,1, . . . ,n − 1}where n = log2(N)
I Apply a permutation π to the original graph layerL = {ln−1, . . . , l1, l0}:
L = {ln−1, . . . , l1, l0}π−→ Lπ = {lπ(n−1), . . . , lπ(1), lπ(0)}
I Binary expansion of the integer i
{b(i)n−1, . . . ,b
(i)1 ,b(i)
0 }
where b(i)j ∈ {0,1} (0 ≤ i ≤ N − 1;0 ≤ j ≤ n − 1)
Doan et. al On the Decoding of Polar Codes on Permuted Factor Graphs 7/17
Factor-graph Permutation to Codeword-positionPermutation
Theorem 1Apply permutation π to the original factor graph of a polar code
L = {ln−1, . . . , l1, l0}π−→ Lπ = {lπ(n−1), . . . , lπ(1), lπ(0)}.
Then, the synthetic channel associated with the binaryexpansion
{b(i)n−1, . . . ,b
(i)1 ,b(i)
0 }
of the original factor graph L is the same as the syntheticchannel associated with the binary expansion
{b(i)π(n−1), . . . ,b
(i)π(1),b
(i)π(0)}
of the permuted factor graph Lπ.
Doan et. al On the Decoding of Polar Codes on Permuted Factor Graphs 8/17
Factor-graph Permutation to Codeword-positionPermutation
Proof.
I On L, the synthetic channel associated with the binaryexpansion {b(i)
n−1, . . . ,b(i)1 ,b(i)
0 } is
W (i)L = ((((W b(i)
n−1)...)b(i)1 )b(i)
0 )
I On Lπ, the synthetic channel associated with the binaryexpansion {b(i)
π(n−1), . . . ,b(i)π(1),b
(i)π(0)} is
W (i)Lπ = ((((W b(i)
π(π(n−1)))...)b(i)π(π(1)))
b(i)π(π(0))
= ((((W b(i)n−1)...)b(i)
1 )b(i)0 ) = W (i)
L
Doan et. al On the Decoding of Polar Codes on Permuted Factor Graphs 9/17
Factor-graph Permutation to Codeword-positionPermutation
Apply Theorem 1 to a permuted factor graph Lπ:
Lπ = {lπ(n−1), . . . , lπ(1), lπ(0)}π−→ L = {ln−1, . . . , l1, l0}.
Then, the synthetic channel associated with the binaryexpansion
{b(i)n−1, . . . ,b
(i)1 ,b(i)
0 }
of the permuted factor graph Lπ is the same as the syntheticchannel associated with the binary expansion
{b(i)π(n−1), . . . ,b
(i)π(1),b
(i)π(0)}
of the original factor graph L.
Doan et. al On the Decoding of Polar Codes on Permuted Factor Graphs 10/17
Factor-graph Permutation to Codeword-positionPermutation
An example:
Doan et. al On the Decoding of Polar Codes on Permuted Factor Graphs 11/17
Selection of Good Permutations
I Construct a set of permutations P
I Fix (n − k) left-most layers of the original graph
I Construct k ! permutations of the k right-most layers
I |P| = k !
I If the decoding algorithm fails on the original graph
I Run the decoding algorithm on each permutation p of PI Numerically evaluate the probability of successful decoding
for each permutation p
I Select the M best permutations of P
Doan et. al On the Decoding of Polar Codes on Permuted Factor Graphs 12/17
Selection of Good Permutations
2 2.5 3 3.5
10−4
10−3
10−2
Eb/N0 [dB]
FER
k = 4
k = 5
k = 6
k = 7
k = 8
k = 9
FER performance of BP decoding on 16 best permuted factor graphsof P(1024,512) with 24-bit CRC, and |P| = k !.
I Good permutations: found by permuting the layers on theright-most side of the original factor graph
Doan et. al On the Decoding of Polar Codes on Permuted Factor Graphs 13/17
5G P(1024,512), no CRC, AWGN channel
2 2.5 3 3.5
10−4
10−3
10−2
10−1
Eb/N0 [dB]
FER
PBP-CS78 PBP-R109
PBP-B10 PSC-B10
SC BP
SCL32
7Korada, “Polar Codes for Channel and Source Coding”.8Hussami, Korada, and Urbanke, “Performance of polar codes for channel and source coding”.9Elkelesh et al., “Belief propagation decoding of polar codes on permuted factor graphs”.
Doan et. al On the Decoding of Polar Codes on Permuted Factor Graphs 14/17
5G P(1024,512), 24-bit CRC, AWGN channel
2 2.5 3 3.510−6
10−5
10−4
10−3
10−2
10−1
Eb/N0 [dB]
FER
SCL32
SCL32-CRC24
PBP-CS-CRC241011
PBP-R10-CRC2412
PBP-B10-CRC24
PSC-B10-CRC24
PBP-R32-CRC2412
PBP-B32-CRC24
PSC-B32-CRC24
PBP-R128-CRC2412
PBP-B128-CRC24
PSC-B128-CRC24
10Korada, “Polar Codes for Channel and Source Coding”.11Hussami, Korada, and Urbanke, “Performance of polar codes for channel and source coding”.12Elkelesh et al., “Belief propagation decoding of polar codes on permuted factor graphs”.
Doan et. al On the Decoding of Polar Codes on Permuted Factor Graphs 15/17
Conclusion
I Factor-graph permutations can be mapped tocodeword-position permutations→ require a single decoder for hardware implementation
I Propose a method to construct good permutations fordifferent polar-code decoders
I 5G P(1024,512), no CRC, at FER = 10−4: PSC-B10 is 0.4dB better than PSC-R10, and almost equivalent to SCL32
I 5G P(1024,512), 24-bit CRC, at FER = 10−4: PBP-B128is 0.25 dB better than PBP-R128, but 0.3 dB worse thanSCL32
Doan et. al On the Decoding of Polar Codes on Permuted Factor Graphs 16/17
Thank you !
Doan et. al On the Decoding of Polar Codes on Permuted Factor Graphs 17/17