concatenated polar codes - dept. of ie, cuhk staff web …sjaggi/files/slides.pdf · mayank bakshi...
TRANSCRIPT
![Page 1: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/1.jpg)
Mayank Bakshi Caltech
Concatenated Polar Codes
Sidharth JaggiCUHK
Michelle EffrosCaltech
![Page 2: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/2.jpg)
Channel
Typical multiuser system
Sources:
- Capacity bounds : known in many cases
- Practical coding schemes not known for most channels
- encoding/decoding complexity
Key challenges:
- good error probability decay
![Page 3: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/3.jpg)
Channel
Typical multiuser system
Sources:
- Capacity bounds : known in many cases
- Practical coding schemes not known for most channels
- encoding/decoding complexity
Key challenges:
- good error probability decay
Capacity achieving:
Multi-user channels
- MAC, degraded broadcast, Gelfand-Pinsker
Network Source Coding- Slepian-Wolf, coded side information...
Polar Codes:
Many point-to-point channels
[Arikan’09, Korada et al’09, Hussami et al ’09]
![Page 4: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/4.jpg)
Channel
Typical multiuser system
Sources:
- Capacity bounds : known in many cases
- Practical coding schemes not known for most channels
- encoding/decoding complexity
Key challenges:
- good error probability decay
Encoding complexity: O(n log n)
Decoding complexity: O(n log n)
Capacity achieving:
Error probability: 2−√
n
Multi-user channels
- MAC, degraded broadcast, Gelfand-Pinsker
Network Source Coding- Slepian-Wolf, coded side information...
Many point-to-point channels
Polar Codes:[Arikan’09, Korada et al’09, Hussami et al ’09]
![Page 5: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/5.jpg)
Encoding complexity
Decoding complexity
Random Codes
Spielman Codes
LP Decoding
Polar Codes
Modified Polar Codes
Desirable
Pe
O(N log N) O(N log N)
exp(−θ(N))
exp(−θ(N))
exp(−θ(N))
O(N log N) O(N log N)exp(−θ(N))
exp(−o(√
N))
Point-to-point channel: other codes
θ(N)
exp(−o(Nβ(l))) O(2l N log N)O(lN log N)
exp(O(N))exp(O(N))
θ(N · 21/)
O(N log N) O(N log N)
![Page 6: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/6.jpg)
Improve error probability?
Desirable O(N log N) O(N log N)exp(−θ(N))
Polar Codes O(N log N) O(N log N)exp(−o(√
N))
![Page 7: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/7.jpg)
Improve error probability?
u1
u2
uk
R-S
P
P
P
x1
x2
xn
Py1
y2
yn
−1
P−1
P−1
R-S−1
u1
u2
uk
Encoding Decoding
Concatenation [Forney ’66]
Desirable O(N log N) O(N log N)exp(−θ(N))
Polar Codes O(N log N) O(N log N)exp(−o(√
N))
![Page 8: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/8.jpg)
Encoding
. . .
mRO
nRI
N = m · noverall
blocklength
![Page 9: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/9.jpg)
Encoding
. . .
mRO
nRI
N = m · n
R = RO · RI
K = (mRO) · (nRI)
overall blocklength
message blocklength
outercode
innercode
![Page 10: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/10.jpg)
Encoding
. . .
. . .
mRO
nRI
nRI
m
R-S encoder
N = m · n
R = RO · RI
. . .
K = (mRO) · (nRI)
overall blocklength
message blocklength
outercode
innercode
![Page 11: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/11.jpg)
Encoding
. . .
. . .
. . .
. . .
mRO
nRI
nRI
m
m
n
R-S encoder
N = m · n
R = RO · RI
. . .
K = (mRO) · (nRI)
overall blocklength
message blocklength
outercode
innercode
Polar encoder
![Page 12: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/12.jpg)
Decoding
. . .
. . .
. . .
. . .
mRO
nRI
nRI
m
m
n
R-S decoder
Polar decoder
. . .
![Page 13: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/13.jpg)
Does it work well?
. . .
. . .
. . .
. . .
mRO
nRI
nRI
m
m
n
R-S encoder
Polar encoder
. . .
![Page 14: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/14.jpg)
Does it work well?
. . .
. . .
. . .
. . .
mRO
nRI
nRI
m
m
n
R-S encoder
Polar encoder
First attempt:
m = N/log N
n = log N
. . .
![Page 15: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/15.jpg)
Does it work well?
. . .
. . .
. . .
. . .
mRO
nRI
nRI
m
m
n
R-S encoder
Polar encoder
First attempt:
m = N/log N
n = log N
Error Probability:
Error if more than inner codes failm(1− 2RO). . .
![Page 16: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/16.jpg)
Does it work well?
. . .
. . .
. . .
. . .
mRO
nRI
nRI
m
m
n
R-S encoder
Polar encoder
First attempt:
m = N/log N
n = log N
Error Probability:
Error if more than inner codes failm(1− 2RO). . .
Pr(error) ∼ exp(−O(N
log N))
![Page 17: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/17.jpg)
Does it work well?
. . .
. . .
. . .
. . .
mRO
nRI
nRI
m
m
n
R-S encoder
Polar encoder
First attempt:
m = N/log N
n = log N
Error Probability:
Error if more than inner codes failm(1− 2RO)
Encoding complexity: O(N log N)
. . .Pr(error) ∼ exp(−O(
Nlog N
))
![Page 18: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/18.jpg)
Does it work well?
. . .
. . .
. . .
. . .
. . .
mRO
nRI
nRI
m
m
n
R-S encoder
Polar encoder
First attempt:
m = N/log N
n = log N
Error Probability:
Error if more than inner codes failm(1− 2RO)
Encoding complexity: O(N log N)
Decoding complexity: O(N2 log N)
Pr(error) ∼ exp(−O(N
log N))
![Page 19: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/19.jpg)
Can decoding be performed using encoding?
. . .
. . .
. . .
. . .
. . .
mRO
nRI
nRI
m
m
n
R-S encoder
Polar encoder
![Page 20: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/20.jpg)
Can decoding be performed using encoding?
. . .
. . .
. . .
. . .
. . .
mRO
nRI
nRI
m
m
n
Systematic R-S encoder
Polar encoder
![Page 21: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/21.jpg)
Decoding
. . . . . .
m
n
![Page 22: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/22.jpg)
Decoding
. . .
. . .
. . .
. . .
nRI
m
m
n
Polar decoder
![Page 23: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/23.jpg)
Decoding
. . .
. . .
. . .
. . .
nRI
m
n
mRO m(1− RO)
Polar decoder
![Page 24: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/24.jpg)
Decoding
. . .
. . . . . .
mRO
m
n
. . . nRI
m(1− RO)
. . . . . . nRI
mRO m(1− RO)
Polar decoder
Systematic R-S encoder
![Page 25: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/25.jpg)
Decoding
. . .
. . . . . .
mRO
m
n
. . . nRI
m(1− RO)
. . . . . . nRI
mRO m(1− RO)
Polar decoder
Systematic R-S encoder . . . . . .=?
![Page 26: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/26.jpg)
Decoding
. . .
. . . . . .
mRO
m
n
. . . nRI
m(1− RO)
. . . . . . nRI
mRO m(1− RO)
. . . . . .=?
. . .Yes
Polar decoder
Systematic R-S encoder
mRO
nRI
![Page 27: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/27.jpg)
Decoding
. . .
. . . . . .
mRO
m
n
. . . nRI
m(1− RO)
. . . . . . nRI
mRO m(1− RO)
. . . . . .=?
. . .
No
Yes
. . .R-S decoder
Polar decoder
Systematic R-S encoder
mRO
mRO
nRI
nRI
![Page 28: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/28.jpg)
Decoding
. . .
. . . . . .
mRO
m
n
. . . nRI
m(1− RO)
. . . . . . nRI
mRO m(1− RO)
. . . . . .=?
. . .
No
Yes
. . .R-S decoder
Polar decoder
Systematic R-S encoder
mRO
mRO
nRI
nRI
Ps
1− Ps
Decoding complexity:
(1− Ps)
x encoding complexityPs
+ x R-S decoder complexity
![Page 29: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/29.jpg)
. . .
. . .
. . .
. . .
. . .
mRO
nRI
nRI
m
m
n
Systematic R-S encoder
Polar encoder
Does it help?
First attempt:
m = N/log N
n = log N
Error Probability:
Error if more than inner codes failm(1− 2RO)
Encoding complexity: O(N log N)
Pr(error) ∼ exp(−O(N
log N))
![Page 30: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/30.jpg)
. . .
. . .
. . .
. . .
. . .
mRO
nRI
nRI
m
m
n
Systematic R-S encoder
Polar encoder
Does it help?
First attempt:
m = N/log N
n = log N
Error Probability:
Error if more than inner codes failm(1− 2RO)
Encoding complexity: O(N log N)
Decoding complexity:
(1− Ps)
x encoding complexityPs
+ x R-S decoder complexity
Pr(error) ∼ exp(−O(N
log N))
![Page 31: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/31.jpg)
. . .
. . .
. . .
. . .
. . .
mRO
nRI
nRI
m
m
n
Systematic R-S encoder
Polar encoder
Does it help?
First attempt:
m = N/log N
n = log N
Error Probability:
Error if more than inner codes failm(1− 2RO)
Encoding complexity: O(N log N)
Decoding complexity:
(1− Ps)
x encoding complexityPs
+ x R-S decoder complexity
= O(N2 log N)
Pr(error) ∼ exp(−O(N
log N))
![Page 32: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/32.jpg)
. . .
. . .
. . .
. . .
. . .
mRO
nRI
nRI
m
m
n
Systematic R-S encoder
Polar encoder
Can it ever help?
smaller ? m
![Page 33: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/33.jpg)
. . .
. . .
. . .
. . .
. . .
mRO
nRI
nRI
m
m
n
Systematic R-S encoder
Polar encoder
Can it ever help?
smaller ? m
m = Nα (α < 1)n = N1−α
![Page 34: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/34.jpg)
. . .
. . .
. . .
. . .
. . .
mRO
nRI
nRI
m
m
n
Systematic R-S encoder
Polar encoder
Can it ever help?
smaller ?
Decoding complexity:
(1− Ps)
x encoding complexityPs
+ x R-S decoder complexity
m
m = Nα (α < 1)n = N1−α
= O(N log N)
![Page 35: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/35.jpg)
. . .
. . .
. . .
. . .
. . .
mRO
nRI
nRI
m
m
n
Systematic R-S encoder
Polar encoder
Can it ever help?
smaller ?
Encoding complexity: O(N log N)
Decoding complexity:
(1− Ps)
x encoding complexityPs
+ x R-S decoder complexity
m
m = Nα (α < 1)n = N1−α
= O(N log N)
![Page 36: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/36.jpg)
. . .
. . .
. . .
. . .
. . .
mRO
nRI
nRI
m
m
n
Systematic R-S encoder
Polar encoder
Can it ever help?
smaller ?
Error Probability:
Error if more than inner codes failm(1− 2RO)
Encoding complexity: O(N log N)
Decoding complexity:
(1− Ps)
x encoding complexityPs
+ x R-S decoder complexity
m
m = Nα (α < 1)n = N1−α
= O(N log N)
Pr(error) ∼ exp(−O(Nα/2))
![Page 37: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/37.jpg)
Encoding complexity
Decoding complexityPe
m n Pem n
desirable: O(N log N) O(N log N)exp(−θ(N))
exp(−o(Nα/2)) O(N log N)
O(N log N) O(N2 log N)
O(N log N)
exp(−O(N
log N))
θ(Nα) θ(N1−α)
θ(log N)θ(N
log N)
![Page 38: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/38.jpg)
Encoding complexity
Decoding complexityPe
m n
θ(log N)θ(N
log N)
Pem n
desirable: O(N log N) O(N log N)exp(−θ(N))
θ(N)θ(1)
θ(Nα) exp(−o(Nα/2))θ(N1−α) O(N log N)
O(N log N) O(N2 log N)
O(N log N)
θ(log2 N)θ(N
log2 N)
θ(log3 N)
θ(log4 N)θ(N
log4 N)
θ(N
log3 N)
exp(−O(N
log N))
![Page 39: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/39.jpg)
Encoding complexity
Decoding complexityPe
m n
θ(log N)θ(N
log N)
Pem n
desirable: O(N log N) O(N log N)exp(−θ(N))
θ(N)θ(1) exp(−o(√
N) O(N log N) O(N log N)
θ(Nα) exp(−o(Nα/2))θ(N1−α) O(N log N)
O(N log N)
O(N log N)
O(N log N)
O(N log N) O(N2 log N)
O(N log N)
O(N log N)
θ(log2 N)θ(N
log2 N)
θ(log3 N)
θ(log4 N)θ(N
log4 N)
θ(N
log3 N)
O(N2 log N)
O(N log N)exp(−Ω(N
log27/8 N))
exp(−O(N
log4 N))
exp(−O(N
log2 N))
exp(−O(N
log N))
![Page 40: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/40.jpg)
Encoding complexity
Decoding complexityPe
m n Pem n
desirable: O(N log N) O(N log N)exp(−θ(N))
exp(−o(√
N) O(N log N) O(N log N)
exp(−o(Nα/2)) O(N log N)
O(N log N)
O(N log N)
O(N log N)
O(N log N) O(N2 log N)
O(N log N)
O(N log N)
O(N2 log N)
O(N log N)exp(−Ω(N
log27/8 N))
exp(−O(N
log4 N))
exp(−O(N
log2 N))
exp(−O(N
log N))θ(log N)θ(
Nlog N
)
θ(N)θ(1)
θ(Nα) θ(N1−α)
θ(log2 N)θ(N
log2 N)
θ(log3 N)
θ(log4 N)θ(N
log4 N)
θ(N
log3 N)
![Page 41: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/41.jpg)
Concatenated polar codes
Encoding Complexity:
Decoding Complexity:
Achieve capacity for arbitrary point-to-point channels
Error probability:
. . .
. . .
. . .
. . .
. . .
Systematic R-S encoder
Polar encoder
2−N/(log N)3.375
O(N log N)
O(N log N)
![Page 42: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/42.jpg)
e.g. Multiple access channel
Concatenation in multi-user channels
XY
Z
p(y|x, z)
![Page 43: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/43.jpg)
e.g. Multiple access channel
Concatenation in multi-user channels
XY
Z
p(y|x, z)
- Apply concatenation to each message
. . .
. . .
. . .
. . .
. . .
Systematic R-S encoder
Polar encoder
- Polar inner code
![Page 44: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/44.jpg)
e.g. Multiple access channel
Concatenation in multi-user channels
XY
Z
p(y|x, z)
- Apply concatenation to each message
. . .
. . .
. . .
. . .
. . .
Systematic R-S encoder
Polar encoder
- Polar inner code
Achieve capacity
Encoding Complexity:
Decoding Complexity:
Error probability: 2−N/(log N)3.375
O(N log N)
O(N log N)
![Page 45: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/45.jpg)
Concatenation in network source coding
e.g. Coded side information
X
Y
X
(X, Y) ∼ p(x, y)
![Page 46: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/46.jpg)
Concatenation in network source coding
X
Y
X
e.g. Coded side information. . .
. . . . . .
Systematic R-S encoder
... ......
... ......
(X, Y) ∼ p(x, y)
![Page 47: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/47.jpg)
Concatenation in network source coding
X
Y
X
e.g. Coded side information. . .
. . . . . .
Systematic R-S encoder
... ......
... ......
(X, Y) ∼ p(x, y)
- Systematic bits ∼ p(x, y)
![Page 48: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/48.jpg)
Concatenation in network source coding
X
Y
X
e.g. Coded side information. . .
. . . . . .
Systematic R-S encoder
... ......
... ......
(X, Y) ∼ p(x, y)
- Systematic bits ∼ p(x, y)
- Parity bits: not even i.i.d.
- Polar code may not work
![Page 49: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/49.jpg)
Concatenation in network source coding
. . .
. . .
. . .
. . .
Systematic R-S encoder
Polar encoder
. . .
e.g. Coded side information
X
Y
X
... ......
... ......
(X, Y) ∼ p(x, y)
- Systematic bits ∼ p(x, y)
- Parity bits: not even i.i.d.
- Polar code may not work
- Transmit parity bits without coding!
![Page 50: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/50.jpg)
Concatenation in network source coding
Systematic R-S encoder
Polar encoder
. . .
. . .
. . .
. . .
. . .
e.g. Coded side information
X
Y
X
(X, Y) ∼ p(x, y)
Achieve optimal rates
.. .. .... ..
.. .. .... ..
Encoding Complexity:
Decoding Complexity:
Error probability: 2−N/(log N)3.375
O(N log N)
O(N log N)
![Page 51: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/51.jpg)
Ke
y
ide
as • Concatenate: Systematic R-S outer, Polar inner
• Suitable inner/outer codelength => reduced average decoding complexity
• Modification for source coding: apply source code only to message bits, not parity
Concluding remarks
![Page 52: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/52.jpg)
Ke
y
ide
asR
esu
lts
• Concatenate: Systematic R-S outer, Polar inner
• Suitable inner/outer codelength => reduced average decoding complexity
• Modification for source coding: apply source code only to message bits, not parity
• Efficient, capacity achieving codes • Arbitrary point-to-point• Several multi-user channels: Degraded broadcast channel, multiple-access channel• Network Source coding problems: e.g. Slepian-Wolf, Coded Side Information
Concluding remarks
![Page 53: Concatenated Polar Codes - Dept. of IE, CUHK Staff Web …sjaggi/files/Slides.pdf · Mayank Bakshi Caltech Concatenated Polar Codes Sidharth Jaggi CUHK Michelle Effros Caltech](https://reader031.vdocuments.us/reader031/viewer/2022030816/5b29fdb97f8b9ae1288b5583/html5/thumbnails/53.jpg)
Concluding remarks
Desirable O(N log N) O(N log N)exp(−θ(N))
Polar Codes O(N log N) O(N log N)exp(−o(√
N))
Concatenated Codes
exp(−Ω(N
log27/8 N)) O(N log N) O(N log N)
• Use insights from concatenated code to design a better single stage code• Joint decoding improves the performance in some cases
• Other parameters? •Dependence of error probability on coding rate ?N
ex
t S
tep
s