Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR...
Transcript of Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR...
![Page 1: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/1.jpg)
Are Polar Codes Practical?
Prof. Warren J. GrossMcGill University, Montreal, QC.
Coding: from Practice to TheoryBerkeley, CA.Feb. 10th, 2015.
![Page 2: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/2.jpg)
Achieving the Channel Capacity
Polar Codes20080 dB
using SCas n Ñ 8
Successive cancellation Ñ low throughput.
Mediocre performance at moderate code-length.Becomes excellent for N ° 220.
2/36
![Page 3: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/3.jpg)
Channel Polarization
u0 W y0
u1 W y1
I0 “ IpW q “ I1
u0 ` W y0
u1 W y1
I0 ă IpW q ă I1
u0 ` ` ` y0
u1 ` ` y1
u2 ` ` y2
u3 ` y3
u4 ` ` y4
u5 ` y5
u6 ` y6
u7 y7
`
`
`
`
`
`
`
`
`
`
`
`
Proportion of reliable bits Ñ IpW q.
1/26
![Page 4: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/4.jpg)
Channel Polarization
u0 W y0
u1 W y1
I0 “ IpW q “ I1
u0 ` W y0
u1 W y1
I0 ă IpW q ă I1
u0 ` ` ` y0
u1 ` ` y1
u2 ` ` y2
u3 ` y3
u4 ` ` y4
u5 ` y5
u6 ` y6
u7 y7
`
`
`
`
`
`
`
`
`
`
`
`
Proportion of reliable bits Ñ IpW q.
1/26
![Page 5: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/5.jpg)
Channel Polarization
u0 W y0
u1 W y1
I0 “ IpW q “ I1
u0 ` W y0
u1 W y1
I0 ă IpW q ă I1
u0 ` ` ` y0
u1 ` ` y1
u2 ` ` y2
u3 ` y3
u4 ` ` y4
u5 ` y5
u6 ` y6
u7 y7
`
`
`
`
`
`
`
`
`
`
`
`
Proportion of reliable bits Ñ IpW q.
1/26
![Page 6: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/6.jpg)
Successive Cancellation Decoding
u0 ` ` ` y0
u1 ` ` y1
u2 ` ` y2
u3 ` y3
u4 ` ` y4
u5 ` y5
u6 ` y6
u7 y7
S0 S1 S2
u0
u0 ‘ u1
u2 u1
u0 ‘ u1 ‘ u2 ‘ u3
u4 u1 ‘ u3
u4 ‘ u5 u2 ‘ u3
u6 u5 u3
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
F
G
2/26
![Page 7: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/7.jpg)
Successive Cancellation Decoding
u0 ` ` ` y0
u1 ` ` y1
u2 ` ` y2
u3 ` y3
u4 ` ` y4
u5 ` y5
u6 ` y6
u7 y7
S0 S1 S2
u0
u0 ‘ u1
u2 u1
u0 ‘ u1 ‘ u2 ‘ u3
u4 u1 ‘ u3
u4 ‘ u5 u2 ‘ u3
u6 u5 u3
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
F
G
2/26
![Page 8: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/8.jpg)
Successive Cancellation Decoding
u0 ` ` ` y0
u1 ` ` y1
u2 ` ` y2
u3 ` y3
u4 ` ` y4
u5 ` y5
u6 ` y6
u7 y7
S0 S1 S2
u0
u0 ‘ u1
u2 u1
u0 ‘ u1 ‘ u2 ‘ u3
u4 u1 ‘ u3
u4 ‘ u5 u2 ‘ u3
u6 u5 u3
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
F
G
2/26
![Page 9: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/9.jpg)
Semi-Parallel SC Decoder Implementation
0 50 1000
0.2
0.4
0.6
0.8
1
# PEs
Speed, N “ 210
Utilization, N “ 210
ą 95% speed with 64 PEs.
First polar decoder ASIC.
Code (1024, 512)
Technology 180 nmArea 1.72 mm2
Frequency 150 MHzInfo. T/P 49 Mbps
Power 67 mW
Mishra et al., “A Successive Cancellation Decoder ASIC for a 1024-bit Polar Code in 180nm CMOS,” A-SSCC 2012. 3/26
![Page 10: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/10.jpg)
FPGA Implementation
N LUT FF RAM (bits) f (MHz) T/P (Mbps)
215 3,263 1,304 411,648 167 62R216 3,414 1,316 821,248 157 57R218 3,548 1,349 3,278,848 140 51R220 5,956 1,366 13,109,248 102 38R
4/26
![Page 11: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/11.jpg)
Throughput of SC Decoders
0 20 40 60 80 100 120
SP-SC
SC-ASIC
TP-SC
56
88
106
Info. T/P for R “ 0.9 (Mbps)
Pamuk, ISIT 2013
Mishra, A-SSCC 2012
Raymond, TSP 2013
5/26
![Page 12: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/12.jpg)
Improving Throughput
100 MbpsSC
100 GbpsUnrolled
1000ˆ
![Page 13: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/13.jpg)
Successive Cancellation Tree
§ View the SC decoder graph as a tree.
u0 ` ` ` y0
u1 ` ` y1
u2 ` ` y2
u3 ` y3
u4 ` ` y4
u5 ` y5
u6 ` y6
u7 y7
6/26
![Page 14: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/14.jpg)
Successive Cancellation Tree
§ View the SC decoder graph as a tree.
u0 ` ` ` y0
u1 ` ` y1
u2 ` ` y2
u3 ` y3
u4 ` ` y4
u5 ` y5
u6 ` y6
u7 y7
6/26
![Page 15: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/15.jpg)
Successive Cancellation Tree
§ View the SC decoder graph as a tree.
u0 ` ` ` y0
u1 ` ` y1
u2 ` ` y2
u3 ` y3
u4 ` ` y4
u5 ` y5
u6 ` y6
u7 y7
6/26
![Page 16: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/16.jpg)
Successive Cancellation Tree
§ View the SC decoder graph as a tree.
u0 ` ` ` y0
u1 ` ` y1
u2 ` ` y2
u3 ` y3
u4 ` ` y4
u5 ` y5
u6 ` y6
u7 y7
6/26
![Page 17: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/17.jpg)
Successive Cancellation Tree
§ View the SC decoder graph as a tree.
u0 ` ` ` y0
u1 ` ` y1
u2 ` ` y2
u3 ` y3
u4 ` ` y4
u5 ` y5
u6 ` y6
u7 y7
6/26
![Page 18: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/18.jpg)
Successive Cancellation Tree
§ View the SC decoder graph as a tree.
u0 ` ` ` y0
u1 ` ` y1
u2 ` ` y2
u3 ` y3
u4 ` ` y4
u5 ` y5
u6 ` y6
u7 y7
6/26
![Page 19: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/19.jpg)
Simplified Successive-Cancellation Decoding
Rate-0 Rate-1
Alamdar-Yazdi and Kschischang., “A Simplified Successive-Cancellation Decoder for Polar Codes,” Comm. Lett., 2011. 7/26
![Page 20: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/20.jpg)
Simplified Successive-Cancellation Decoding
Rate-0
Rate-1
Alamdar-Yazdi and Kschischang., “A Simplified Successive-Cancellation Decoder for Polar Codes,” Comm. Lett., 2011. 7/26
![Page 21: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/21.jpg)
Simplified Successive-Cancellation Decoding
Rate-0 Rate-1
Alamdar-Yazdi and Kschischang., “A Simplified Successive-Cancellation Decoder for Polar Codes,” Comm. Lett., 2011. 7/26
![Page 22: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/22.jpg)
Simplified Successive-Cancellation Decoding
Rate-0
Rate-1
Alamdar-Yazdi and Kschischang., “A Simplified Successive-Cancellation Decoder for Polar Codes,” Comm. Lett., 2011. 7/26
![Page 23: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/23.jpg)
Simplified Successive-Cancellation Decoding
Rate-0 Rate-1
Alamdar-Yazdi and Kschischang., “A Simplified Successive-Cancellation Decoder for Polar Codes,” Comm. Lett., 2011. 7/26
![Page 24: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/24.jpg)
Fast-SSC Decoding
RepetitionSingleParityCheck
Sarkis et al., “Fast Polar Decoders: Algorithm and Implementation,” IEEE JSAC, 2014. 8/26
![Page 25: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/25.jpg)
Fast-SSC Decoding
Repetition
SingleParityCheck
Sarkis et al., “Fast Polar Decoders: Algorithm and Implementation,” IEEE JSAC, 2014. 8/26
![Page 26: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/26.jpg)
Fast-SSC Decoding
RepetitionSingleParityCheck
Sarkis et al., “Fast Polar Decoders: Algorithm and Implementation,” IEEE JSAC, 2014. 8/26
![Page 27: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/27.jpg)
Fast-SSC Decoding
Repetition
SingleParityCheck
Sarkis et al., “Fast Polar Decoders: Algorithm and Implementation,” IEEE JSAC, 2014. 8/26
![Page 28: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/28.jpg)
Fast-SSC Decoding
RepetitionSingleParityCheck
Sarkis et al., “Fast Polar Decoders: Algorithm and Implementation,” IEEE JSAC, 2014. 8/26
![Page 29: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/29.jpg)
Fast-SSC Decoding—New Nodes
Repetition
SPC
N “ 8
Rep-SPC
N “ 4
ML-0011
9/26
![Page 30: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/30.jpg)
SC Tree—(256, 230)
10/26
![Page 31: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/31.jpg)
SSC and Fast-SSC Trees—(256, 230)
Developed a compiler to convert the graph to instructions.
11/26
![Page 32: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/32.jpg)
SSC and Fast-SSC Trees—(256, 230)
Developed a compiler to convert the graph to instructions.
11/26
![Page 33: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/33.jpg)
Fast-SSC Decoder Implementation
R LUT FF RAM (bits) f (MHz) T/P (Mbps)
Fast-SSC
0.83 25,866 7,209 536,126 108 7910.90 1,081
SSC
0.90 276
For any code of length = 32,768.
Sarkis et al. “Fast polar decoders: algorithm and implementation,” IEEE JSAC, 2014. 12/26
![Page 34: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/34.jpg)
Throughput of Hardware Decoders
0 500 1,000 1,500 2,000 2,500 3,000
TP-SC
Fast-SSC
BP
106
1,081
2,338
Info. T/P (Mbps)
Park, Sym. VLSI 2014ASIC, 300 MHz
(1024, 512)
Sarkis, JSAC 2014FPGA, 108 MHz(32678, 29492)
Pamuk, ISIT 2013FPGA, 230 MHz(16384, 14746)
13/26
![Page 35: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/35.jpg)
The Unrolled Decoder
§ (8, 4) polar code:
C0C1
C0
C2
C1C0
C3
C2C1 C0
C4
C3C2 C1
C0
C5
C4C3 C2
C1
C0
§ Unroll and pipeline all operations.
F Rep G SPC ‘
C0C1 C0C2 C1 C0C3 C2 C1 C0C4 C3 C2 C1 C0C5 C4 C3 C2 C1 C0
14/26
![Page 36: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/36.jpg)
The Unrolled Decoder
§ (8, 4) polar code:
C0C1
C0
C2
C1C0
C3
C2C1 C0
C4
C3C2 C1
C0
C5
C4C3 C2
C1
C0
§ Unroll and pipeline all operations.
F Rep G SPC ‘
C0C1 C0C2 C1 C0C3 C2 C1 C0C4 C3 C2 C1 C0C5 C4 C3 C2 C1 C0
14/26
![Page 37: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/37.jpg)
The Unrolled Decoder
§ (8, 4) polar code:
C0
C1
C0
C2
C1C0
C3
C2C1 C0
C4
C3C2 C1
C0
C5
C4C3 C2
C1
C0
§ Unroll and pipeline all operations.
F Rep G SPC ‘
C0
C1 C0C2 C1 C0C3 C2 C1 C0C4 C3 C2 C1 C0C5 C4 C3 C2 C1 C0
14/26
![Page 38: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/38.jpg)
The Unrolled Decoder
§ (8, 4) polar code:
C0
C1
C0
C2
C1C0
C3
C2C1 C0
C4
C3C2 C1
C0
C5
C4C3 C2
C1
C0
§ Unroll and pipeline all operations.
F Rep G SPC ‘
C0
C1 C0
C2 C1 C0C3 C2 C1 C0C4 C3 C2 C1 C0C5 C4 C3 C2 C1 C0
14/26
![Page 39: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/39.jpg)
The Unrolled Decoder
§ (8, 4) polar code:
C0C1
C0
C2
C1C0
C3
C2C1 C0
C4
C3C2 C1
C0
C5
C4C3 C2
C1
C0
§ Unroll and pipeline all operations.
F Rep G SPC ‘
C0C1 C0
C2 C1 C0
C3 C2 C1 C0C4 C3 C2 C1 C0C5 C4 C3 C2 C1 C0
14/26
![Page 40: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/40.jpg)
The Unrolled Decoder
§ (8, 4) polar code:
C0C1
C0
C2
C1C0
C3
C2C1 C0
C4
C3C2 C1
C0
C5
C4C3 C2
C1
C0
§ Unroll and pipeline all operations.
F Rep G SPC ‘
C0C1 C0C2 C1 C0
C3 C2 C1 C0
C4 C3 C2 C1 C0C5 C4 C3 C2 C1 C0
14/26
![Page 41: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/41.jpg)
The Unrolled Decoder
§ (8, 4) polar code:
C0C1
C0
C2
C1C0
C3
C2C1 C0
C4
C3C2 C1
C0
C5
C4C3 C2
C1
C0
§ Unroll and pipeline all operations.
F Rep G SPC ‘
C0C1 C0C2 C1 C0C3 C2 C1 C0
C4 C3 C2 C1 C0
C5 C4 C3 C2 C1 C0
14/26
![Page 42: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/42.jpg)
The Unrolled Decoder
§ (8, 4) polar code:
C0C1
C0
C2
C1C0
C3
C2C1 C0
C4
C3C2 C1
C0
C5
C4C3 C2
C1
C0
§ Unroll and pipeline all operations.
F Rep G SPC ‘
C0C1 C0C2 C1 C0C3 C2 C1 C0C4 C3 C2 C1 C0
C5 C4 C3 C2 C1 C0
14/26
![Page 43: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/43.jpg)
The Unrolled Decoder Implementation
LUT FF RAM (bits) f (MHz) Info. T/P (Gbps)
SP-SC4,130 1,388 11,904 196 0.04
Unrolled155,858 158,185 285,120 231 118.5
BP (ASIC)300 2.3
For a (1024, 512) polar code.
Giard et al. “A 237 Gbps unrolled hardware polar decoder,” submitted to IET Letters and arXiv, Dec. 2014 15/26
![Page 44: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/44.jpg)
SDR and Modern Software ECC
107 108 109 1010
Demodulation
ECC
Instructions
Viterbi decoder
OFDM
Demel et al. “A LTE Receiver Framework Using GNU Radio,” JSPS, 2015.
2 4 6 8 10 12
LDPC
Turbo
Info. T/P (Mbps) per Core
Wu et al.Asilomar 2013
6 iterations
Sarkis et al.GlobalSIP 2014
30 iterations
16/26
![Page 45: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/45.jpg)
SDR and Modern Software ECC
107 108 109 1010
Demodulation
ECC
Instructions
Viterbi decoder
OFDM
Demel et al. “A LTE Receiver Framework Using GNU Radio,” JSPS, 2015.
2 4 6 8 10 12
LDPC
Turbo
Info. T/P (Mbps) per Core
Wu et al.Asilomar 2013
6 iterations
Sarkis et al.GlobalSIP 2014
30 iterations
16/26
![Page 46: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/46.jpg)
SDR and Modern Software ECC
107 108 109 1010
Demodulation
ECC
Instructions
Viterbi decoder
OFDM
Demel et al. “A LTE Receiver Framework Using GNU Radio,” JSPS, 2015.
2 4 6 8 10 12
LDPC
Turbo
Info. T/P (Mbps) per Core
Wu et al.Asilomar 2013
6 iterations
Sarkis et al.GlobalSIP 2014
30 iterations
Goal: 50Mbps!16/26
![Page 47: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/47.jpg)
Floating-Point Throughput
(2048, 1707) (16384, 14746) (32768, 29492)
150
200
250
300
350In
fo.
T/P
(Mbp
s)ICASSP 2014
GlobalSIP 2014
§ SIMD.
§ Branchless.
§ Single core.
Giard et al. “Fast Software Polar Decoders,” ICASSP 2014.Sarkis et al. “Autogenerating Software Polar Decoders,” GlobalSIP 2014. 17/26
![Page 48: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/48.jpg)
Fixed-Point Software Polar Decoders
Decoder f pGHzq Info. T/P (Mbps) Latency µs
Le Gal et al. 3.6+ 1,557 605Fast-SSC-int8 3.1+ 1,412 21
(32768, 29492) polar code.
Le Gal et al. “Multi-Gb/s Software decoding of Polar Codes,” TSP 2015.Giard et al. “Low-latency Software Polar Decoders,” in preparation. 18/26
![Page 49: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/49.jpg)
Improving Finite-Length Error-Correction Performance
Eb{N0
BE
R
LDPC
PC
PC List-CRC
19/26
![Page 50: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/50.jpg)
List Decoding
§ For each information bit, continue decoding assuming both0 and 1.
u0 u1 u2 u3
ˆ
ˆ
00000001
01000101
000
01
000001
010011
u0
u1
u2
u3
0
00 01
000 001 010 011
0000 0001 0010 0011 0100 0101 0110 0111
§ Select paths according to reliability values.
§ Using a CRC to select the output yields a significantimprovement.
20/26
![Page 51: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/51.jpg)
List Decoding
§ For each information bit, continue decoding assuming both0 and 1.
u0 u1 u2 u3
ˆ
ˆ
00000001
01000101
000
01
000001
010011
u0
u1
u2
u3
0
00 01
000 001 010 011
0000 0001 0010 0011 0100 0101 0110 0111
§ Select paths according to reliability values.
§ Using a CRC to select the output yields a significantimprovement.
20/26
![Page 52: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/52.jpg)
List Decoding
§ For each information bit, continue decoding assuming both0 and 1.
u0 u1 u2 u3
ˆ
ˆ
00000001
01000101
000
01
000001
010011
u0
u1
u2
u3
0
00 01
000
001
010
011
0000 0001
0010 0011
0100 0101
0110 0111
§ Select paths according to reliability values.
§ Using a CRC to select the output yields a significantimprovement.
20/26
![Page 53: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/53.jpg)
List Decoding
§ For each information bit, continue decoding assuming both0 and 1.
u0 u1 u2 u3
ˆ
ˆ
00000001
01000101
000
01
000001
010011
u0
u1
u2
u3
0
00 01
000
001
010
011
0000 0001
0010 0011
0100 0101
0110 0111
§ Select paths according to reliability values.
§ Using a CRC to select the output yields a significantimprovement.
20/26
![Page 54: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/54.jpg)
List Decoding
§ For each information bit, continue decoding assuming both0 and 1.
u0 u1 u2 u3
ˆ
ˆ
00000001
01000101
000
01
000001
010011
u0
u1
u2
u3
0
00
01
000 001
010
011
0000 0001 0010 0011
0100
0101 0110 0111
§ Select paths according to reliability values.
§ Using a CRC to select the output yields a significantimprovement.
20/26
![Page 55: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/55.jpg)
List Decoding
§ For each information bit, continue decoding assuming both0 and 1.
u0 u1 u2 u3
ˆ
ˆ
00000001
01000101
000
01
000001
010011
u0
u1
u2
u3
0
00
01
000 001
010
011
0000 0001 0010 0011
0100
0101 0110 0111
§ Select paths according to reliability values.
§ Using a CRC to select the output yields a significantimprovement.
20/26
![Page 56: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/56.jpg)
ASIC Implementation
§ Synthesis results for N “ 1024 using UMC 90nm:
L Area (mm2) f (MHz) T/P (Mbps)
2 1.60 459 181R4 3.53 314 89R
Balatsoukas-Stimming et al. “Hardware Architecture for List Successive Cancellation Decoding of Polar Codes,” TCAS II, 2014. 21/26
![Page 57: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/57.jpg)
Two-Step SW List Decoder
§ List decoders are slower than Fast-SSC decoders.
§ List have better error-correction performance thanFast-SSC decoders.
§ Combine both:
Fast-SSC
ValidCRC?
Done
List-CRC
yes
no
22/26
![Page 58: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/58.jpg)
Two-Step SW List Decoder
§ List decoders are slower than Fast-SSC decoders.
§ List have better error-correction performance thanFast-SSC decoders.
§ Combine both:
Fast-SSC
ValidCRC?
Done
List-CRC
yes
no
Sarkis et al. “Increasing the Speed of Polar List Decoders,” SiPS, 2014. 22/26
![Page 59: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/59.jpg)
Two-Step SW List Decoder
Eb{N0
FER
(2048, 1723)
Fast-SSC
SSC-List-CRC
23/26
![Page 60: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/60.jpg)
Two-Step SW List Decoder
Eb{N0
FER
(2048, 1723)
Fast-SSC
SSC-List-CRC
T “ kp1´FERFqLpFast-SSCq`FERFLpListq bit/s
23/26
![Page 61: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/61.jpg)
Two-Step SW List Decoder
3.5 4 4.5
10´8
10´7
10´6
10´5
10´4
10´3
10´2
10´1
Eb{N0 (dB)
FER
LDPC, Imax “ 30 SSC-List-CRC, L “ 32 SSC-List-CRC, L “ 64
(2048, 1723)
24/26
![Page 62: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/62.jpg)
Two-Step SW List Decoder
3.5 4 4.50
10
20
30
40
50
60
Eb{N0 (dB)
Info
rmat
ion
T/P
(Mbp
s)
LDPC, Imax “ 30 SSC-List-CRC, L “ 32 SSC-List-CRC, L “ 64
(2048, 1723)
25/26
![Page 63: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/63.jpg)
Conclusion
§ Polar codes can have excellent error-correctionperformance.
§ Polar codes can have throughput in the hundred-Gbpsrange.
§ Once throughput and performance improvements arecombined, we believe that polar codes will be a viableoption for multiple applications.
Thank you!
26/26
![Page 64: Are Polar Codes Practical? Polar Codes Practical? Prof. Warren J. Gross McGill University, ... SDR and Modern Software ECC 107 108 109 1010 Demodulation …](https://reader034.fdocument.pub/reader034/viewer/2022051507/5a77c68a7f8b9a93088e166f/html5/thumbnails/64.jpg)
Conclusion
§ Polar codes can have excellent error-correctionperformance.
§ Polar codes can have throughput in the hundred-Gbpsrange.
§ Once throughput and performance improvements arecombined, we believe that polar codes will be a viableoption for multiple applications.
Thank you!
26/26