Efficient Codes using Channel Polarizationmedard/pimit/EffrosMayank.pdf · 2009-09-18 · Find...
Transcript of Efficient Codes using Channel Polarizationmedard/pimit/EffrosMayank.pdf · 2009-09-18 · Find...
![Page 1: Efficient Codes using Channel Polarizationmedard/pimit/EffrosMayank.pdf · 2009-09-18 · Find Polar Codes or a modification to achieve capacity for other types of channel.! Characterize](https://reader031.fdocuments.us/reader031/viewer/2022030816/5b29f86e7f8b9a251e8b6b39/html5/thumbnails/1.jpg)
Concatenating Polar and R-S codes gives the best properties of both! - Use Polar codes as Code 2 as they achieve capacity!- Use R-S codes as Code 1 to reduce error probability!- Complexity !
At each encoder:
How it works:
- Divide input of blocklength N into N/f(N) sub-blocksof length f(N) each
- Apply high rate R-S code on the entire input followed by a polar code on each sub-block
- Decode the two stages one by one
- When the polar code fails on few of the sub-blocks, the R-S code can correct the error
- P(error) decays as exp(-o(N)); Complexity is O(N poly log N); excess rate goes to 0 asymptotically
Assumptions and limitations:
• Works for channels where capacity-achieving codes are known (e.g. point-to-point channels, degraded broadcast channels, multiple access channels)
• Dependence of error probability on excess rate unknown
- Joint decoding of the two stages may lead to a better error performance – we know this in special cases - Use insight from concatenated coding scheme to design a better single stage coding scheme
Efficient Codes using Channel Polarization !Bakshi, Jaggi, and Effros!
- Practical capacity achieving schemes are not known for general multi-input multi-output channels!
- Codes based on channel polarization that achieve capacity for point-to-point, degraded broadcast and MAC have poor error performance!
Find Polar Codes or a modification to achieve capacity for other types of channel.!
Characterize the dependence on other parameters e.g., excess rate.!
Concatenating Polar and R-S codes leads to more efficient codes for several different channels E
ND
-OF-
PH
AS
E G
OA
L C
OM
MU
NIT
Y C
HA
LLE
NG
E
ACHIEVEMENT DESCRIPTION
ST
AT
US
QU
O N
EW
IN
SIG
HT
S
Code 1 Code 2
High rate R-S code Polar Code
![Page 2: Efficient Codes using Channel Polarizationmedard/pimit/EffrosMayank.pdf · 2009-09-18 · Find Polar Codes or a modification to achieve capacity for other types of channel.! Characterize](https://reader031.fdocuments.us/reader031/viewer/2022030816/5b29f86e7f8b9a251e8b6b39/html5/thumbnails/2.jpg)
Mayank Bakshi Department of Electrical Engineering,California Institute of Technology
Efficient Codes based on Channel Polarization
(joint work with Sidharth Jaggi, CUHK and Michelle Effros, Caltech)
![Page 3: Efficient Codes using Channel Polarizationmedard/pimit/EffrosMayank.pdf · 2009-09-18 · Find Polar Codes or a modification to achieve capacity for other types of channel.! Characterize](https://reader031.fdocuments.us/reader031/viewer/2022030816/5b29f86e7f8b9a251e8b6b39/html5/thumbnails/3.jpg)
Channel
Motivation
Typical multiuser system
- Capacity bounds known in many cases
Sources:
- Practical coding schemes unknown for most channels
- Encoding/Decoding Complexity
- Blocklength required to achieve desired error probability
Key Challenges:
![Page 4: Efficient Codes using Channel Polarizationmedard/pimit/EffrosMayank.pdf · 2009-09-18 · Find Polar Codes or a modification to achieve capacity for other types of channel.! Characterize](https://reader031.fdocuments.us/reader031/viewer/2022030816/5b29f86e7f8b9a251e8b6b39/html5/thumbnails/4.jpg)
Channel Polarization
p(y1|x1)
p(y2|x2)
p(yn|xn)
x1
x2
xn
y1
y2
yn
x1
x2
xn
y1
y2
yn
p(y1|x1)
p(y2|x2)
p(yn|xn)un
u2
u1
P
Channel seen by each is same (statistically)
Different see different channels xi ui
Choose matrix s.t. each either sees a channel of capacity either close to 1 or close to 0 (depending on the value of i)
P ui
e.g. Point-to-point channel
Channel polarization:
xi −→ yi ui −→ (yn, ui−1)
![Page 5: Efficient Codes using Channel Polarizationmedard/pimit/EffrosMayank.pdf · 2009-09-18 · Find Polar Codes or a modification to achieve capacity for other types of channel.! Characterize](https://reader031.fdocuments.us/reader031/viewer/2022030816/5b29f86e7f8b9a251e8b6b39/html5/thumbnails/5.jpg)
Polar Codes
- Systematic procedure to construct P
- Successive cancellation based decoding rule
Main features:
Encoding Complexity: O(n log n)
Decoding Complexity: O(n log n)
Achieve capacity for arbitrary point-to-point channels
Error probability: 2−√
n
Can be applied to several multi-user channels as well
- Multiple access channel, degraded broadcast channel, Gelfand-Pinsker channel
Channel Polarization
(Close to linear)
(Close to linear)
(long block length required to get a desired error probability)
![Page 6: Efficient Codes using Channel Polarizationmedard/pimit/EffrosMayank.pdf · 2009-09-18 · Find Polar Codes or a modification to achieve capacity for other types of channel.! Characterize](https://reader031.fdocuments.us/reader031/viewer/2022030816/5b29f86e7f8b9a251e8b6b39/html5/thumbnails/6.jpg)
Reed-Solomon Codes
(u1, u2, . . . , uk) f (x) = u1 + u2x + . . . + ukxk−1�
f (x1), f (x2), . . . , f (xn)�
Data packets Codeword
Main features:
Encoding Complexity:
Decoding Complexity:
Not capacity achieving in general
Error probability:
O(n(log n)2)
O(n(log n)2)
2−αn
Easily scale to large field sizes
(Close to linear)
(Close to linear)
(short block lengths suffice to get a desired error probability)
![Page 7: Efficient Codes using Channel Polarizationmedard/pimit/EffrosMayank.pdf · 2009-09-18 · Find Polar Codes or a modification to achieve capacity for other types of channel.! Characterize](https://reader031.fdocuments.us/reader031/viewer/2022030816/5b29f86e7f8b9a251e8b6b39/html5/thumbnails/7.jpg)
Q: Can we get the best of both worlds?
+ = ?
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
A: Yes, almost
Concatenation
![Page 8: Efficient Codes using Channel Polarizationmedard/pimit/EffrosMayank.pdf · 2009-09-18 · Find Polar Codes or a modification to achieve capacity for other types of channel.! Characterize](https://reader031.fdocuments.us/reader031/viewer/2022030816/5b29f86e7f8b9a251e8b6b39/html5/thumbnails/8.jpg)
Concatenation
- Encode and decode in two steps
- Polarization based codes help correct channel errors at rate close to capacity
Main features:
Encoding Complexity:
Decoding Complexity:
Achieve capacity for arbitrary point-to-point channels
Error probability:
- R-S code encodes across blocks of Polar code to correct block errors when Polar codes fail
O(n(log n)2)
O(n(log n)2)
2−n/ log n
(Close to linear)
(Close to linear)
(block length required to get a desired error probability is almost of the same order as R-S)
![Page 9: Efficient Codes using Channel Polarizationmedard/pimit/EffrosMayank.pdf · 2009-09-18 · Find Polar Codes or a modification to achieve capacity for other types of channel.! Characterize](https://reader031.fdocuments.us/reader031/viewer/2022030816/5b29f86e7f8b9a251e8b6b39/html5/thumbnails/9.jpg)
e.g. Multiple access channel
Concatenation in multi-user channels
XY
Z
p(y|x, z)
- Perform separate concatenation at each encoder
- R-S code adds redundancy to each message set
- Polarization based codes achieve the capacity
- By a careful choice of parameters:
Encoding Complexity:
Decoding Complexity:
Error probability:
O(n(log n)2)
O(n(log n)2)
2−n/ log n
Achieve capacity
![Page 10: Efficient Codes using Channel Polarizationmedard/pimit/EffrosMayank.pdf · 2009-09-18 · Find Polar Codes or a modification to achieve capacity for other types of channel.! Characterize](https://reader031.fdocuments.us/reader031/viewer/2022030816/5b29f86e7f8b9a251e8b6b39/html5/thumbnails/10.jpg)
Concatenation in network source coding
- Encode the message symbols by an optimal code
Encoding Complexity:
Decoding Complexity:
Error probability:
O(n(log n)2)
O(n(log n)2)
2−n/ log n
Achieve optimal rates
- Use systematic R-S codes to compute redundancy packets at each encoder
- Transmit the redundancy packets without coding
General idea:
- At each decoder, use redundancy packets to correct block errors
- Similar performance boost as in channel coding
- e.g., when combined with Polar codes for Coded Side Information problem,
![Page 11: Efficient Codes using Channel Polarizationmedard/pimit/EffrosMayank.pdf · 2009-09-18 · Find Polar Codes or a modification to achieve capacity for other types of channel.! Characterize](https://reader031.fdocuments.us/reader031/viewer/2022030816/5b29f86e7f8b9a251e8b6b39/html5/thumbnails/11.jpg)
Ke
y
ide
asR
esu
lts
• Concatenation helps reduce the error probability of coding schemes even in networked scenario
• Complexity is largely determined by outer code - R-S code
• Rate is determined by inner code - Polar Code
• Efficient codes for • Several multi-user channels: Degraded broadcast channel, multiple-access channel• Network Source coding problems: e.g. Slepian-Wolf, Coded Side Information
Summary