Polar Coding - Status and Prospectskedart/ee376a_winter1617/...(uniform) Wvec Vector channel Combine WN WN−1 b b b W1 Split New channels (polarized) Polarization Polar coding Performance

Polarization Polar coding Performance

Polar CodingStatus and Prospects

Erdal Arıkan

Electrical-Electronics Engineering DepartmentBilkent UniversityAnkara, Turkey

1 August 2011The IEEE International Symposium on Information Theory

ISIT’2011Saint Petersburg, Russia


The channel

Let W : X → Y be a binary-input discrete memoryless channel

WX Y

I input alphabet: X = {0, 1},

I output alphabet: Y,

I transition probabilities:

W (y |x), x ∈ X , y ∈ Y


The channel


WX Y




W (y |x), x ∈ X , y ∈ Y


The channel


WX Y




W (y |x), x ∈ X , y ∈ Y


The channel


WX Y




W (y |x), x ∈ X , y ∈ Y


Symmetry assumption

Assume that the channel has “input-output symmetry.”


Symmetry assumption


Examples:

1− ε

1− ε

ε

ε

1

0

1

0

BSC(ε)


Symmetry assumption


Examples:

1− ε

1− ε

ε

ε

1

0

1

0

BSC(ε)

1− ε

1− ε

ε

ε

1

0

1

0

?

BEC(ε)


Capacity

For channels with input-output symmetry, the capacity is given by

C (W )∆= I (X ;Y ), with X ∼ unif. {0, 1}


Capacity

For channels with input-output symmetry, the capacity is given by

C (W )∆= I (X ;Y ), with X ∼ unif. {0, 1}

Use base-2 logarithms:

0 ≤ C (W ) ≤ 1


The main idea

I Channel coding problem trivial for two types of channelsI Perfect: C (W ) = 1I Useless: C (W ) = 0

I Transform ordinary W into such extreme channels


The main idea




The main idea




The main idea




The method: aggregate and redistribute capacity

W

W

b

b

b

W

Original channels(uniform)



W

W

b

b

b

W


Wvec

Vectorchannel

Combine



W

W

b

b

b

W


Wvec

Vectorchannel

Combine

WN

WN−1

b

b

b

W1

Split

New channels(polarized)


Combining

I Begin with N copies of W ,

I use a 1-1 mapping

GN : {0, 1}N → {0, 1}N

I to create a vector channel

Wvec : UN → Y N

W

W

WXN

X2

X1

YN

Y2

Y1


Combining


I use a 1-1 mapping

GN : {0, 1}N → {0, 1}N


Wvec : UN → Y N

W

W

WXN

X2

X1

YN

Y2

Y1

GN

UN

U2

U1


Combining


I use a 1-1 mapping

GN : {0, 1}N → {0, 1}N


Wvec : UN → Y N

W

W

WXN

X2

X1

YN

Y2

Y1

GN

UN

U2

U1

Wvec


Conservation of capacity

Combining operation is lossless:

I Take U1, . . . ,UN i.i.d. unif. {0, 1}

I then, X1, . . . ,XN i.i.d. unif. {0, 1}

I and

C (Wvec) = I (UN ;Y N)

= I (XN ;Y N)

= NC (W )

W

W

W

GN

XN

X2

X1

YN

Y2

Y1

UN

U2

U1

Wvec






I and


= I (XN ;Y N)

= NC (W )

W

W

W

GN

XN

X2

X1

YN

Y2

Y1

UN

U2

U1

Wvec






I and


= I (XN ;Y N)

= NC (W )

W

W

W

GN

XN

X2

X1

YN

Y2

Y1

UN

U2

U1

Wvec


Splitting


Wvec

UN

Ui+1

Ui

Ui−1

U1

YN

Yi

Y1


Splitting


=N∑

i=1

I (Ui ;YN ,U i−1)

Wvec

UN

Ui+1

Ui

Ui−1

U1

YN

Yi

Y1


Splitting


=N∑

i=1

I (Ui ;YN ,U i−1)

Define bit-channels

Wi : Ui → (Y N ,U i−1)

Wvec

UN

Ui+1

Ui

Ui−1

U1

U1

Ui−1

YN

Yi

Y1

Wi


Splitting


=N∑

i=1

I (Ui ;YN ,U i−1)

=N∑

i=1

C (Wi )

Define bit-channels

Wi : Ui → (Y N ,U i−1)

Wvec

UN

Ui+1

Ui

Ui−1

U1

U1

Ui−1

YN

Yi

Y1

Wi


Polarization is commonplace

I Polarization is the rule not theexception

I A random permutation

GN : {0, 1}N → {0, 1}N

is a good polarizer with highprobability

I Equivalent to Shannon’s randomcoding approach

W

W

W

GN

XN

X2

X1

YN

Y2

Y1

UN

U2

U1





GN : {0, 1}N → {0, 1}N



W

W

W

GN

XN

X2

X1

YN

Y2

Y1

UN

U2

U1





GN : {0, 1}N → {0, 1}N



W

W

W

GN

XN

X2

X1

YN

Y2

Y1

UN

U2

U1


Random polarizers: stepwise, isotropic

5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bit channel index

Cap

acity


Random polarizers: stepwise, isotropic

5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bit channel index

Cap

acity

Isotropy: any redistribution order is as good as any other.


The complexity issue

I Random polarizers lack structure, too complex to implement

I Need a low-complexity polarizer

I May sacrifice stepwise, isotropic properties of randompolarizers in return for less complexity












Basic module for a low-complexity scheme

Combine two copies of W

W

W

Y2

Y1

X2

X1




+

U2

U1

G2

W

W

Y2

Y1

X2

X1




+

U2

U1

G2

W

W

Y2

Y1

X2

X1

and split to create two bit-channels

W1 : U1 → (Y1,Y2)

W2 : U2 → (Y1,Y2,U1)


The first bit-channel W1

W1 : U1 → (Y1,Y2)

+

random U2

U1

W

W

Y2

Y1


The first bit-channel W1

W1 : U1 → (Y1,Y2)

+

random U2

U1

W

W

Y2

Y1

C (W1) = I (U1;Y1,Y2)


The second bit-channel W2

W2 : U2 → (Y1,Y2,U1)

+

U2

U1

W

W

Y2

Y1


The second bit-channel W2

W2 : U2 → (Y1,Y2,U1)

+

U2

U1

W

W

Y2

Y1

C (W2) = I (U2;Y1,Y2,U1)


Capacity conserved but redistributed unevenly

+

U2

U1

W

W

Y2

Y1

X2

X1

I Conservation:

C (W1) + C (W2) = 2C (W )

I Extremization:

C (W1) ≤ C (W ) ≤ C (W2)

with equality iff C (W ) equals 0 or 1.


Capacity conserved but redistributed unevenly

+

U2

U1

W

W

Y2

Y1

X2

X1

I Conservation:

C (W1) + C (W2) = 2C (W )

I Extremization:

C (W1) ≤ C (W ) ≤ C (W2)

with equality iff C (W ) equals 0 or 1.


Notation

The two channels created by the basic transform

(W ,W ) → (W1,W2)

will be denoted also as

W− = W1 and W+ = W2


Notation

The two channels created by the basic transform

(W ,W ) → (W1,W2)

will be denoted also as

W− = W1 and W+ = W2

Likewise, we write W−−, W−+ for descendants of W−; and W+−,W++ for descendants of W+.


For the size-4 construction

+

W

W


... duplicate the basic transform

+

+

W

W

W

W


... obtain a pair of W − and W+ each

W+

W+

W−

W−


... apply basic transform on each pair

+

+

W+

W+

W−

W−


... decode in the indicated order

+

+

W+

W+

W−

W−

U4

U2

U3

U1


... obtain the four new bit-channels

W++

W−+

W+−

W−−

U4

U2

U3

U1


Overall size-4 construction

+

+

+

+

W

W

W

W

U4

U2

U3

U1

Y4

Y2

Y3

Y1

X4

X2

X3

X1


“Rewire” for standard-form size-4 construction

+

+

+

+

W

W

W

W

U4

U3

U2

U1

Y4

Y3

Y2

Y1

X4

X3

X2

X1


Size 8 construction

+

+

+

+

+

+

+

+

+

+

+

+

W

W

W

W

W

W

W

W

Y8

Y7

Y6

Y5

Y4

Y3

Y2

Y1

U8

U7

U6

U5

U4

U3

U2

U1

X8

X7

X6

X5

X4

X3

X2

X1


Demonstration of polarization

Polarization is easy to analyze when W is a BEC.

If W is a BEC(ε), then so are W−

and W+, with erasure probabili-ties

ε−∆= 2ε− ε2

andε+

∆= ε2

respectively.1− ε

1− ε

ε

ε

1

0

1

0

?

W






ε−∆= 2ε− ε2

andε+

∆= ε2

respectively.1− ε−

1− ε−

ε−

ε−

1

0

1

0

?

W−






ε−∆= 2ε− ε2

andε+

∆= ε2

respectively.1− ε+

1− ε+

ε+

ε+

1

0

1

0

?

W+


Polarization for BEC(12): N = 16

2 4 6 8 10 12 14 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bit channel index

Cap

acity

Capacity of bit channels

N=16



5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bit channel index

Cap

acity


N=32



10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bit channel index

Cap

acity


N=64



20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bit channel index

Cap

acity


N=128



50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bit channel index

Cap

acity


N=256



50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bit channel index

Cap

acity


N=512



100 200 300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bit channel index

Cap

acity


N=1024


Polarization martingale

0

1

C(W )



0

1

1

C(W )

C(W2)

C(W1)



0

1

1 22

C(W )

C(W2)

C(W1)

C(W++)

C(W−+)

C(W+−)

C(W−−)



0

1

1 22

C(W )

C(W2)

C(W1)

C(W++)

C(W−+)

C(W+−)

C(W−−)



0

1

1 22 3333

C(W )

C(W2)

C(W1)

C(W++)

C(W−+)

C(W+−)

C(W−−)



0

1

1 22 3333 44444444

C(W )

C(W2)

C(W1)

C(W++)

C(W−+)

C(W+−)

C(W−−)



0

1

1 22 3333 44444444 5555555555555555

C(W )

C(W2)

C(W1)

C(W++)

C(W−+)

C(W+−)

C(W−−)



0

1

1 22 3333 44444444 5555555555555555 66666666666666666666666666666666

C(W )

C(W2)

C(W1)

C(W++)

C(W−+)

C(W+−)

C(W−−)



0

1

1 22 3333 44444444 5555555555555555 66666666666666666666666666666666 7777777777777777777777777777777777777777777777777777777777777777

C(W )

C(W2)

C(W1)

C(W++)

C(W−+)

C(W+−)

C(W−−)



0

1

1 22 3333 44444444 5555555555555555 66666666666666666666666666666666 7777777777777777777777777777777777777777777777777777777777777777 88888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888

C(W )

C(W2)

C(W1)

C(W++)

C(W−+)

C(W+−)

C(W−−)


Theorem (Polarization, A. 2007)

The bit-channel capacities {C (Wi )} polarize: for any

δ ∈ (0, 1), as the construction size N grows

[

no. channels with C (Wi ) > 1− δ

N

]

−→ C (W )

and[

no. channels with C (Wi ) < δ

N

]

−→ 1− C (W )

Theorem (Rate of polarization, A. and Telatar (2008))

Above theorem holds with δ ≈ 2−√

N . 0

δ

1− δ

1


Theorem (Polarization, A. 2007)

The bit-channel capacities {C (Wi )} polarize: for any

δ ∈ (0, 1), as the construction size N grows

[

no. channels with C (Wi ) > 1− δ

N

]

−→ C (W )

and[

no. channels with C (Wi ) < δ

N

]

−→ 1− C (W )

Theorem (Rate of polarization, A. and Telatar (2008))

Above theorem holds with δ ≈ 2−√

N . 0

δ

1− δ

1


Polar code example: W = BEC(12), N = 8, rate 1/2

C(Wi )

0.0039

0.1211

0.1914

0.6836

0.3164

0.8086

0.8789

0.9961

+

+

+

+

+

+

+

+

+

+

+

+

W

W

W

W

W

W

W

W

Y8

Y7

Y6

Y5

Y4

Y3

Y2

Y1

U8

U7

U6

U5

U4

U3

U2

U1



C(Wi )

0.0039

0.1211

0.1914

0.6836

0.3164

0.8086

0.8789

0.9961

Rank

8

7

6

4

5

3

2

1

+

+

+

+

+

+

+

+

+

+

+

+

W

W

W

W

W

W

W

W

Y8

Y7

Y6

Y5

Y4

Y3

Y2

Y1

U8

U7

U6

U5

U4

U3

U2

U1



C(Wi )

0.0039

0.1211

0.1914

0.6836

0.3164

0.8086

0.8789

0.9961

Rank

8

7

6

4

5

3

2

1

+

+

+

+

+

+

+

+

+

+

+

+

W

W

W

W

W

W

W

W

Y8

Y7

Y6

Y5

Y4

Y3

Y2

Y1

U8

U7

U6

U5

U4

U3

U2

U1

data



C(Wi )

0.0039

0.1211

0.1914

0.6836

0.3164

0.8086

0.8789

0.9961

Rank

8

7

6

4

5

3

2

1

+

+

+

+

+

+

+

+

+

+

+

+

W

W

W

W

W

W

W

W

Y8

Y7

Y6

Y5

Y4

Y3

Y2

Y1

U8

U7

U6

U5

U4

U3

U2

U1

data

data



C(Wi )

0.0039

0.1211

0.1914

0.6836

0.3164

0.8086

0.8789

0.9961

Rank

8

7

6

4

5

3

2

1

+

+

+

+

+

+

+

+

+

+

+

+

W

W

W

W

W

W

W

W

Y8

Y7

Y6

Y5

Y4

Y3

Y2

Y1

U8

U7

U6

U5

U4

U3

U2

U1

data

data

data



C(Wi )

0.0039

0.1211

0.1914

0.6836

0.3164

0.8086

0.8789

0.9961

Rank

8

7

6

4

5

3

2

1

+

+

+

+

+

+

+

+

+

+

+

+

W

W

W

W

W

W

W

W

Y8

Y7

Y6

Y5

Y4

Y3

Y2

Y1

U8

U7

U6

U5

U4

U3

U2

U1

data

data

data

data



C(Wi )

0.0039

0.1211

0.1914

0.6836

0.3164

0.8086

0.8789

0.9961

Rank

8

7

6

4

5

3

2

1

+

+

+

+

+

+

+

+

+

+

+

+

W

W

W

W

W

W

W

W

Y8

Y7

Y6

Y5

Y4

Y3

Y2

Y1

U8

U7

U6

U5

U4

U3

U2

U1

data

data

data

frozen

data

frozen

frozen

frozen



C(Wi )

0.0039

0.1211

0.1914

0.6836

0.3164

0.8086

0.8789

0.9961

Rank

8

7

6

4

5

3

2

1

+

+

+

+

+

+

+

+

+

+

+

+

W

W

W

W

W

W

W

W

Y8

Y7

Y6

Y5

Y4

Y3

Y2

Y1

U8

U7

U6

0

U4

0

0

0

data

data

data

frozen

data

frozen

frozen

frozen


Construction complexity

I An O(N) construction algorithm exists that usesdensity-evolution

I First proposed by Mori and Tanaka, without finite-precisionimplementation details

I Tal and Vardy introduced smart quantization methods for apractical implementation

I The algorithm works well in practice but a precise proof ofO(N) complexity still lacking

I Recent work: Pedarsani, Hassani, Tal, and Telatar (ISIT’2011)






























Encoding complexity

Encoding complexity for polar coding is O(N logN).


Encoding: an example

+

+

+

+

+

+

+

+

+

+

+

+

W

W

W

W

W

W

W

W

Y8

Y7

Y6

Y5

Y4

Y3

Y2

Y1

1

0

1

0

1

0

0

0

free

free

free

frozen

free

frozen

frozen

frozen



+

+

+

+

+

+

+

+

+

+

+

+

W

W

W

W

W

W

W

W

Y8

Y7

Y6

Y5

Y4

Y3

Y2

Y1

1

0

1

0

1

0

0

0

1

1

1

1

1

1

0

0

free

free

free

frozen

free

frozen

frozen

frozen



+

+

+

+

+

+

+

+

+

+

+

+

W

W

W

W

W

W

W

W

Y8

Y7

Y6

Y5

Y4

Y3

Y2

Y1

1

0

1

0

1

0

0

0

1

1

1

1

1

1

0

0

1

1

0

0

1

1

1

1

free

free

free

frozen

free

frozen

frozen

frozen



+

+

+

+

+

+

+

+

+

+

+

+

W

W

W

W

W

W

W

W

Y8

Y7

Y6

Y5

Y4

Y3

Y2

Y1

1

0

1

0

1

0

0

0

1

1

1

1

1

1

0

0

1

1

0

0

1

1

1

1

1

1

0

0

0

0

1

1

free

free

free

frozen

free

frozen

frozen

frozen


Successive cancellation decoding complexity

(A. 2007)

Complexity of successive cancellation decoding for polar codes isO(N logN).


Successive cancellation decoding complexity

(A. 2007)

Complexity of successive cancellation decoding for polar codes isO(N logN).

Earlier work on similar decoders:

I Kabatiansky (1990)

I Schnabl and Bossert (1996)

I Dumer and co-authors (from 1990s)

I Burnashev and Dumer (2006-2009)


Performance of SC decoder

(A. and Telatar, 2008)

For any rate R < C (W ) and block-length N, the probability offrame error for polar codes under SC decoding is bounded roughlyas

Pe(N,R) = o(

2−√

N

)

I Prior result (A. 2007): Pe(N,R) = o(

N−1/4)

.

I Latest result: A rate-dependent refinement of the“square-root” bound has been given by Tanaka & Mori(2010) and Hassani & Urbanke (2010)





Pe(N,R) = o(

2−√

N

)


N−1/4)

.






Pe(N,R) = o(

2−√

N

)


N−1/4)

.



Polar coding summary

Given W , N = 2n, and R < C (W ), a polar code with these param-eters has

I construction complexity O(N) (conjecture),

I encoding complexity ≈ N logN,

I decoding complexity ≈ N logN,

I and frame error probability Pe(N,R) ≈ o(2−√

N)








N)








N)








N)


Polar coding in other contexts

I Source coding (lossless)

I Source coding in the presence of memory

I Lossy source coding

I Slepian-Wolf problem

I Wyner-Ziv problem

I Gelfand-Pinsker problem

I MAC

I Degraded-broadcast channel

I Wyner wiretap channel

I Randomness extraction

I ...


Channel-coding scenarios

Topic Ref.

Single-user q-ary channels Sasoglu, Telatar, A. (2009)Multi-access channels Sasoglu, Telatar, Yeh (2010)

m-user MAC Abbe and Telatar (2010)Wyner wiretap channel Mahdavifar and Vardy (2009)

′′ Hof and Shamai (2010)′′ Koyluoglu and El Gamal (2010)′′ Andersson et al. (2010)

Relay channel Andersson et al. (2010)′′ Blasco-Serrano et al. (2010)′′ Karzand (2011)

Compund channel coding Hassani, Korada, Urbanke (2009)


Source-coding scenarios

Topic Ref.

Lossless source coding Hussami, Korada, Urbanke (2009)Rate-distortion coding Korada and Urbanke (2009)

q-ary lossless source coding Karzand and Telatar (2010)Direct source polarization A. (2010)Universal polar coding Abbe (2010)

Sparse recovery Abbe (2010)Randomness extraction Abbe (2011)

Ergodic source polarization Sasoglu (2011)


Scenarios with side-information

Topic Ref.

Wyner-Ziv coding Korada and Urbanke (2009)Gelfand-Pinsker coding Korada and Urbanke (2009)Slepian-Wolf coding Hussami, Korada, Urbanke (2009)


Generalized polarization schemesq: alphabet size`: dimension of basic transform (kernel)E : rate of polarization exponent

q ` Exponent E Kernel Ref.

2 2 to 15 ≤ 1/2 Any linear KSU (2009)2 16 0.51828 BCH KSU (2009)2 31 0.52643 BCH KSU (2009)4 4 0.573120 Reed-Solomon MT (2010)2 14 0.50193 Nonlinear PSL (2011)2 15 0.50773 Nonlinear PSL (2011)2 16 0.52742 Nonlinear PSL (2011)

KSU: Korada, Sasoglu, UrbankeMT: Mori and TanakaPSL: Presman, Shapira, Litsyn


Performance comparison: Polar vs. Turbo

Turbo code

I WiMAX CTC

I Duobinary, memory 3

I QAM over AWGN channel

I Gray mapping

I BICM

I Simulator: “CodedModulation Library”

Polar code

I Standard construction

I Successive cancellationdecoding

I QAM over AWGN channel

I Natural mapping

I Multi-level PAM

I PAM over AWGN channel


Example: 8-PAM as 3 bit channels

I PAM signals selected by three bits (b1, b2, b3)

I Three layers of binary channels created

I Each layer encoded independently

I Layers decoded in the order b3, b2, b1

Bit b1 0 1

-4 42-PAM







Bit b2 0 1 10

-6 -2 2 64-PAM

Bit b1 0 1







Bit b2 0 1 10

Bit b1 0 1

-7 -5 -3 -1 1 3 5 7

0 1 0 1 0 1 0 1Bit b3

8-PAM







Bit b2 0 1 10

Bit b1 0 1

-7 -5 -3 -1 1 3 5 7

0 1 0 1 0 1 0 1Bit b3

8-PAM







Bit b2 0 1 10

Bit b1 0 1

-7 -5 -3 -1 1 3 5 7

0 1 0 1 0 1 0 1Bit b3

8-PAM







Bit b2 0 1 10

Bit b1 0 1

-7 -5 -3 -1 1 3 5 7

0 1 0 1 0 1 0 1Bit b3

8-PAM


Multi-layering jump-starts polarization

0 5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

SNR (dB)

Cap

acity

(bi

ts)

Layer 1 capacityLayer 2 capacityLayer 3 capacitySum of three layersShannon limit


4-QAM, Rate 1/2

−1 0 1 2 3 4 5 610

−6

10−5

10−4

10−3

10−2

10−1

100

EbNo (dB)

FE

R

Polar(512,256) 4−QAMPolar(1024,512) 4−QAMCTC(480,240) 4−QAMCTC(960,480) 4−QAM

Turbo

Polar


16-QAM, Rate 3/4

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.510

−6

10−5

10−4

10−3

10−2

10−1

100

EbNo (dB)

FE

R

Polar(512,384) 16−QAMCTC(192,144) 16−QAMCTC(384,288) 16−QAMCTC(576,432) 16−QAM


64-QAM, Rate 5/6

7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 1310

−5

10−4

10−3

10−2

10−1

100

EbNo (dB)

FE

R

Polar(768,640) 64−QAMPolar(384,320) 64−QAMCTC(576,480) 64−QAM


Complexity comparison: 64-QAM, Rate 5/6

Average decoding time in milliseconds per codeword (ms/cw)

Eb/N0 CTC(576,432) Polar(768,640) Polar(384,320)

10 dB 6.23 0.92 0.4811 dB 1.83 1.01 0.53

Both decoders implemented as MATLAB mex functions. Polar decoder is a successive

cancellation decoder. CTC decoder is a public domain decoder (CML). Profiling done

by MATLAB Profiler. Iteration limit for CTC decoder was 10; average no of iterations

was 10 at 10 dB and 3.3 at 11 dB. CTC decoder used a linear approximation to

log-MAP while polar decoder used exact log-MAP.


Complexity comparison: 64-QAM, Rate 5/6

Average decoding time in milliseconds per codeword (ms/cw)

Eb/N0 CTC(576,432) Polar(768,640) Polar(384,320)

10 dB 6.23 0.92 0.4811 dB 1.83 1.01 0.53

Polar codes show a complexity advantage against CTC codes.

Both decoders implemented as MATLAB mex functions. Polar decoder is a successive

cancellation decoder. CTC decoder is a public domain decoder (CML). Profiling done

by MATLAB Profiler. Iteration limit for CTC decoder was 10; average no of iterations

was 10 at 10 dB and 3.3 at 11 dB. CTC decoder used a linear approximation to

log-MAP while polar decoder used exact log-MAP.


Performance improvement for polar codes

I Concatenation to improve minimum distance

I List decoding to improve SC decoder performance


Performance improvement for polar codes

I Concatenation to improve minimum distance

I List decoding to improve SC decoder performance


Concatenation

Method Ref

Block turbo coding with polar constituents AKMOP (2009)Generalized concatenated coding with polar inner AM (2009)Reed-Solomon outer, polar inner BJE (2010)Polar outer, block inner SH (2010)Polar outer, LDPC inner EP (ISIT’2011)

AKMOP: A., Kim, Markarian, Ozgur, PoyrazGCC: A., MarkarianBJE: Bakshi, Jaggi, and EffrosSH: Seidl and HuberEP: Eslami and Pishro-Nik


Tal-Vardy list decoder for polar codes

I First produce L candidate decisions

I Pick the most likely word from the list

I Complexity O(LN logN)












Tal-Vardy list decoder performanceLength n = 2048, rate R = 0.5, BPSK-AWGN channel, list-size L.








Tal-Vardy list decoder performance

Length n = 2048, rate R = 0.5, BPSK-AWGN channel, list-size L.







List-of-L performance quickly approaches ML performance!


Tal-Vardy list decoder with CRC

I Same decoder as before but data contains a built-in CRC

I Selection made by CRC and relative likelihood



I Same decoder as before but data contains a built-in CRC

I Selection made by CRC and relative likelihood










Polar codes (+CRC) achieve state-of-the-art performance!


Hardware implementation of polar codes

I AdvantagesI Regular structure simplifies resource reuseI Lack of randomness helps avoid memory conflicts

I DisadvantagesI High latency: O(N)I Throughput bottleneck: 1/2 bits per clock-period

References: A. (2007, 2010), Leroux, Tal, Vardy, Gross (2010),Leroux, Sarkis, and Gross (2011), Pamuk and A. (2011).



























Summary

I Polarization is a commonplace phenomenon – almostunavoidable

I Polar codes are low-complexity methods designed to exploitpolarization for achieving Shannon limits

I Polar codes with some help from other methods performcompetitively with the state-of-the-art codes in terms ofcomplexity and performance


Summary





Summary





Acknowledgements

Research sponsors

Special thanks to Emre Telatar, Alex Vardy, Ido Tal, and WarrenGross for sharing ideas, software and slides, and to MatthewValenti for Coded Modulation Library.


Thank you!

Polar Coding - Status and Prospectskedart/ee376a_winter1617/...(uniform) Wvec Vector channel Combine WN WN−1 b b b W1 Split New channels (polarized) Polarization Polar coding Performance

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