[PPT] - A Fast Polar Code List Decoder Architecture Based on Sphere PowerPoint Presentation

SLIDE 1

A Fast Polar Code List Decoder Architecture Based on Sphere Decoding

Seyyed Ali Hashemi, Carlo Condo, Warren J. Gross

Department of Electrical and Computer Engineering McGill University Montr´ eal, Qu´ ebec, Canada May 31, 2017

Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 0/15

SLIDE 2

What is the problem?

◮ Polar Codes are adopted in 5G

◮ High speed and good error-correction performance

◮ 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) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 1/15

SLIDE 3

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

◮ Decoding schemes:

◮ Successive-

Cancellation (SC)

◮ SC List (SCL) ◮ Sphere Decoding (SD) ◮ List-SD

Speed Error-Correction Performance SCL SD SC List-SD

E. Arıkan, ”Channel polarization: A method for constructing capacity achieving codes for symmetric binary

input memoryless channels,” T-IT, July 2009. Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 2/15

SLIDE 4

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

◮ Decoding schemes:

◮ Successive-

Cancellation (SC)

◮ SC List (SCL) ◮ Sphere Decoding (SD) ◮ List-SD

Speed Error-Correction Performance SCL SD SC List-SD

E. Arıkan, ”Channel polarization: A method for constructing capacity achieving codes for symmetric binary

input memoryless channels,” T-IT, July 2009. Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 2/15

SLIDE 5

SC Decoding

ˆ u0 ˆ u1 ˆ u2 ˆ u3 ˆ u4 ˆ u5 ˆ u6 ˆ u7 α β αl βl βr αr

Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 3/15

SLIDE 6

SC Decoding

ˆ u0 ˆ u1 ˆ u2 ˆ u3 ˆ u4 ˆ u5 ˆ u6 ˆ u7 α β αl βl βr αr

Exact formulation

αl

i =2 arctanh

tanh

αi 2

tanh

αi+ Nν

2

,

αr

i =αi+ Nν

2 + (1 − 2βl

i)αi,

Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 3/15

SLIDE 7

SC Decoding

ˆ u0 ˆ u1 ˆ u2 ˆ u3 ˆ u4 ˆ u5 ˆ u6 ˆ u7 α β αl βl βr αr

Hardware-friendly formulation

αl

i = sgn(αi) sgn(αi+ Nν

2 ) min(|αi|, |αi+2s−1|)

αr

i =αi+ Nν

2 + (1 − 2βl

i)αi

Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 3/15

SLIDE 8

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) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 4/15

SLIDE 9

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

PMil =

i

j=0

ln

1 + e−(1−2ˆ

ujl )αjl

Seyyed Ali Hashemi (McGill)

A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 4/15

SLIDE 10

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

PMil =

i

j=0

ln

1 + e−(1−2ˆ

ujl )αjl

Hardware-friendly formulation

PMil =

PMi−1l +|αil|,

if ˆ uil = 1

2

1 − sgn
αil
,

PMi−1l,

therwise

Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 4/15

SLIDE 11

Simplified SCL (SSCL)

◮ SCL requires traversing the whole decoding tree

◮ Very slow

SSCL and SSCL-SPC:

◮ Faster: simplified Rate-1, Rate-0, Rep and SPC nodes

◮ No need to traverse the decoding tree

◮ Guaranteed to preserve error-correction performance for

SSCL SSCL Rep Rep Rate-1 SSCL-SPC Rep SPC

Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 5/15

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Simplified Nodes

◮ Rate-0: LLRs should be positive → Negative LLRs

penalize the path

◮ Rep: Based on info. bit → LLRs are either all positive or all

negative

◮ Rate-1: Each bit is estimated as either 0 or 1 → Nν bits ◮ SPC: Even Parity Constraint → 1) Least reliable bit is

estimated first (1). 2) All other bits are estimated (Nν − 1). 3) Parity constraint is imposed (1) [α0, α1, α2, α3] ⇔ [β0, β1, β2, β3]

Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 6/15

SLIDE 13

SSCL (SSCL-SPC) Issues

◮ SSCL (SSCL-SPC) requires estimating all the bits in

Rate-1 (and SPC) nodes

◮ Nν (Nν + 1) time-steps

Fast-SSCL (Fast-SSCL-SPC):

◮ Very fast: Rate-1 (and SPC) nodes can be further

simplified

◮ A specific number of bit-estimations is required for every list

size

◮ Guaranteed to preserve error-correction performance

Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 7/15

SLIDE 14

Fast-SSCL

Theorem

In SCL decoding with list size L, the maximum number of bit estimations in a Rate-1 node of length Nν required to get the exact same results as the conventional SCL decoder is min (L − 1, Nν) .

In practical polar codes:

◮ There are many instances where L − 1 < Nν. ◮ Savings in number of time-steps can be achieved.

Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 8/15

SLIDE 15

Fast-SSCL Example

L = 2, Nν = 4, |α0l| ≤ |α1l| ≤ |α2l| ≤ |α3l|, PM00 < PM01

PM00 PM01 Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 9/15

SLIDE 16

Fast-SSCL Example

L = 2, Nν = 4, |α0l| ≤ |α1l| ≤ |α2l| ≤ |α3l|, PM00 < PM01

PM00 PM01 PM00 PM01 +|α01 | PM00 +|α00 | PM01 Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 9/15

SLIDE 17

Fast-SSCL Example

L = 2, Nν = 4, |α0l| ≤ |α1l| ≤ |α2l| ≤ |α3l|, PM00 < PM01

PM00 PM01 PM00 PM01 +|α01 | PM00 PM01 +|α11 | PM00 +|α00 | PM01 PM00 +|α10 | PM01 Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 9/15

SLIDE 18

Fast-SSCL Example

L = 2, Nν = 4, |α0l| ≤ |α1l| ≤ |α2l| ≤ |α3l|, PM00 < PM01

PM00 PM01 PM00 PM01 +|α01 | PM00 PM01 +|α11 | PM00 +|α00 | PM01 PM00 +|α10 | PM01 Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 9/15

SLIDE 19

Fast-SSCL Example

L = 2, Nν = 4, |α0l| ≤ |α1l| ≤ |α2l| ≤ |α3l|, PM00 < PM01

PM00 PM01 PM00 PM01 +|α01 | PM00 PM00 +|α00 | + |α10 | PM00 +|α00 | PM01 PM00 +|α10 | PM00 +|α00 | Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 9/15

SLIDE 20

Fast-SSCL Example

L = 2, Nν = 4, |α0l| ≤ |α1l| ≤ |α2l| ≤ |α3l|, PM00 < PM01

PM00 PM01 PM00 PM01 +|α01 | PM00 PM00 +|α00 | + |α10 | PM00 +|α00 | PM01 PM00 +|α10 | PM00 +|α00 |

α1l is always discarded!

Result:

We prove that there is no need to estimate the bits after min (L − 1, Nν) bits are estimated!

Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 9/15

SLIDE 21

Fast-SSCL Implementation

Issue

The LLR values at the top of the tree have to be sorted

Solution

At every time-step i, the i-th least reliable bit is found

Example: L = 4, Nν = 4, |α0l| ≤ |α1l| ≤ |α2l| ≤ |α3l|

Time-step 1: Find |α0l| + bit estimation Time-step 2: Find |α1l| + bit estimation Time-step 3: Find |α2l| + bit estimation

Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 10/15

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Time-Step Reduction

P(1024, 256)

21 22 23 24 25 26 27 28 29 200 400 600 800 1000 1200 L

P(1024, 512)

21 22 23 24 25 26 27 28 29 200 400 600 800 1000 1200 L

P(1024, 768)

21 22 23 24 25 26 27 28 29 200 400 600 800 1000 1200 L

SSCL SSCL-SPC Fast-SSCL Fast-SSCL-SPC

Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 11/15

SLIDE 23

Decoder Architecture

PM Computation and Sorting Controller CRC Unit SC Decoders · · · . . . PE PE . . . PE PE 1 · · · L 1 . . . P Memories Node Sequence Channel LLRs

Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 12/15

SLIDE 24

ASIC Synthesis Comparisons

10−0.3 10−0.2 10−0.1 100 100.1 100.2 100.3 10−0.2 100 100.2 100.4 100.6

Fast-SSCL [Lin’16] [Hashemi’16] Fast-SSCL [Yuan’17] [Xiong’16] [Lin’16] [XiongMM’16] [Hashemi’16] Fast-SSCL [Hashemi’16]

Latency [µs] Area [mm2]

L = 2 L = 4 L = 8

Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 13/15

SLIDE 25

Conclusion

◮ We proposed SSCL, SSCL-SPC, Fast-SSCL, and

Fast-SSCL-SPC

◮ Fast decoding of Rate-1 and SPC nodes ◮ SSCL: Nν bits need to be estimated for Rate-1 nodes ◮ SSCL-SPC: Nν + 1 bits need to be estimated for SPC nodes ◮ Fast-SSCL: min(L − 1, Nν) bits need to be estimated for

Rate-1 nodes

◮ Fast-SSCL-SPC: min(L, Nν) bits need to be estimated for

SPC nodes

◮ Many instances in practice where L < Nν ◮ Non-approximated formulation for SSCL and Fast-SSCL ◮ Guaranteed equivalence with conventional SCL Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 14/15

SLIDE 26

Thank you!

Seyyed Ali Hashemi (McGill) A Fast Polar Code List Decoder Architecture Based on Sphere Decoding 15/15