Polar codes for q-ary channels, q=2r - A. James Clark ...abarg/talks/MultilevelPolarTalk.pdfPolar...

Polar codes for q-ary channels, q = 2r

Alexander BargJoint work with Woo-Myoung Park

Dept of ECE/Inst. for Systems ResearchUniversity of Maryland, College Park, MD 20742

February 6, 2012

Paper arXiv:1107.4965v3(Dept of ECE/Inst. for Systems Research University of Maryland, College Park, MD 20742)Polar codes for q-ary channels, q = 2r February 6, 2012 1 / 19

1 Introduction: Polar codesBinary Polar Codes

2 q-ary polar codes

3 Polar codes for q-ary alphabets, q = 2r

Extremal configurationsConvergence theoremsRate of polarization

4 Transmission scheme

(Dept of ECE/Inst. for Systems Research University of Maryland, College Park, MD 20742)Polar codes for q-ary channels, q = 2r February 6, 2012 2 / 19

Introduction: Polar codes Binary Polar Codes

Binary polar codes (Arikan ’09)

Discrete memoryless channel W : X → Y with capacity I(W )

Channel transformationW−(y1, y2|u1) =

∑u2∈X

12W (y1|u1 + u2)W (y2|u2)

W+(y1, y2,u1|u2) = 12W (y1|u1 + u2)W (y2|u2)

W−−,W−+,W+−,W++, . . .

After n steps we obtain N = 2n channels W b,b ∈ +,−n

The virtual channels polarize: For any ε > 0

limn→∞

|b ∈ +,−n : I(W b) ∈ (ε,1− ε)|2n = 0.

Transmission schemeRate of polarization (Arikan-Telatar ’09)






∑u2∈X

12W (y1|u1 + u2)W (y2|u2)

W+(y1, y2,u1|u2) = 12W (y1|u1 + u2)W (y2|u2)

W−−,W−+,W+−,W++, . . .



limn→∞

|b ∈ +,−n : I(W b) ∈ (ε,1− ε)|2n = 0.







∑u2∈X

12W (y1|u1 + u2)W (y2|u2)

W+(y1, y2,u1|u2) = 12W (y1|u1 + u2)W (y2|u2)

W−−,W−+,W+−,W++, . . .



limn→∞

|b ∈ +,−n : I(W b) ∈ (ε,1− ε)|2n = 0.







∑u2∈X

12W (y1|u1 + u2)W (y2|u2)

W+(y1, y2,u1|u2) = 12W (y1|u1 + u2)W (y2|u2)

W−−,W−+,W+−,W++, . . .



limn→∞

|b ∈ +,−n : I(W b) ∈ (ε,1− ε)|2n = 0.







∑u2∈X

12W (y1|u1 + u2)W (y2|u2)

W+(y1, y2,u1|u2) = 12W (y1|u1 + u2)W (y2|u2)

W−−,W−+,W+−,W++, . . .



limn→∞

|b ∈ +,−n : I(W b) ∈ (ε,1− ε)|2n = 0.





A way to prove polarization:I(W−) + I(W+) = 2I(W )

Given a DMC V : X → Y define the Bhattacharyya parameterZ (V ) =

∑y∈Y

√V (y |0)V (y |1). We have

Z (W+) = Z (W )2

Z (W−) ≤ 2Z (W )(1− Z (W ))

I(V )2 + Z (V )2 ≤ 1I(V ) + Z (V ) ≥ 1


q-ary polar codes

q-ary polar codes

Arikan’s kernel: (u1,u2)H2 = (x1, x2)t , where H2 =(1 0

1 1

).

Z (Wx ,x ′) =∑y∈Y

√W (y |x)W (y |x ′)

Zv (W ) ,12r

∑x∈X

Z (Wx ,x+v)

We haveZv (W+) = Zv (W )2

Zv (W−) ≤ 2Zv (W ) +∑

δ∈X\0,−v

Zδ(W )Zv+δ(W ).

(E. Sasoglu, E. Telatar, and E. Arıkan (’09))

Mori-Tanaka (2010(a), 2010(b)); Abbe-Telatar (2010)


q-ary polar codes

q-ary polar codes


1 1

).

Z (Wx ,x ′) =∑y∈Y

√W (y |x)W (y |x ′)

Zv (W ) ,12r

∑x∈X

Z (Wx ,x+v)


Zv (W−) ≤ 2Zv (W ) +∑

δ∈X\0,−v

Zδ(W )Zv+δ(W ).


Mori-Tanaka (2010(a), 2010(b)); Abbe-Telatar (2010)


q-ary polar codes

q-ary polar codes


1 1

).

Z (Wx ,x ′) =∑y∈Y

√W (y |x)W (y |x ′)

Zv (W ) ,12r

∑x∈X

Z (Wx ,x+v)


Zv (W−) ≤ 2Zv (W ) +∑

δ∈X\0,−v

Zδ(W )Zv+δ(W ).


Mori-Tanaka (2010(a), 2010(b)); Abbe-Telatar (2010)(Dept of ECE/Inst. for Systems Research University of Maryland, College Park, MD 20742)Polar codes for q-ary channels, q = 2r February 6, 2012 5 / 19

Polar codes for q-ary alphabets, q = 2r

Polar codes for q-ary alphabets, q = 2r

Let Wn be the random channel at step n,

Pr(Wn = W B,B ∈ +,−n) = 2−n

In = I(Wn) – symmetric capacity

TheoremIn → I∞ a.e., where I∞ is supported on the set 0,1, . . . , r andEI∞ = I(W ).


Polar codes for q-ary alphabets, q = 2r Extremal configurations

Extremal configurations

The virtual channels converge to one of r + 1 possibilities:

1 1 1 . . . 1 10 1 1 . . . 1 10 0 1 . . . 1 1...

......

0 0 0 . . . 0 10 0 0 . . . 0 0



Extremal configurations

Define the channel “for the last k bits”:

W [k ](y |u) =1

2r−k

∑x∈X :x r

r−k+1=u

W (y |x), u ∈ 0,1k

Theorem

For any DMC W : X → Y the channels W (i)N polarize to one of the

r + 1 extremal configurations. Namely, let Vi = W (i)N and

πk ,N =|i ∈ [N] : |I(Vi)− k | < δ ∧ |I(V [k ]

i )− k | < δ|N

,

where δ > 0, then limN→∞ πk ,N = P(I∞ = k) for all k = 0,1, . . . , r .Consequently

r∑k=1

kπk → I(W ).



Extremal configurations: Example

E. Sasoglu, E. Telatar, and E. Arıkan (’09): the following channel is astable point, and does not polarize:

001

1 01

2 11

1

3

I(W ) = 1, capacity achieved by transmitting one bit

This is an extremal configuration, so it is already polarized!



Extremal configurations: Example

E. Sasoglu, E. Telatar, and E. Arıkan (’09): the following channel is astable point, and does not polarize:

001

1 01

2 11

1

3

I(W ) = 1, capacity achieved by transmitting one bitThis is an extremal configuration, so it is already polarized!



Extremal configurations: Example시트5

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Channel Index

Cap

acity



Convergence proofs

Define Zi(W ) = 12i−1

∑v∈Xi

Zv (W ), i = 1,2, . . . , r

TheoremFor all i = 1,2, . . . , r

limn→∞

Zi,n = Zi,∞ a.e.,

where the variables Zi,∞ take values 0 and 1. With probability one thevector (Zi,∞, i = 1, . . . , r) takes one of the following values:

(Z1,∞ = 0, Z2,∞ = 0, . . . , Zr−1,∞ = 0, Zr ,∞ = 0)(Z1,∞ = 1, Z2,∞ = 0, . . . , Zr−1,∞ = 0, Zr ,∞ = 0)(Z1,∞ = 1, Z2,∞ = 1, . . . , Zr−1,∞ = 0, Zr ,∞ = 0)

......

...(Z1,∞ = 1, Z2,∞ = 1, . . . , Zr−1,∞ = 1, Zr ,∞ = 0)(Z1,∞ = 1, Z2,∞ = 1, . . . , Zr−1,∞ = 1, Zr ,∞ = 1).



Lemma

For a DMC with q-ary input, I(W ) and Z (W ) are related by

I(W ) ≥ log2r

1 +∑r

i=1 2i−1Zi(W )

I(W ) ≤r∑

i=1

√1− Zi(W )2.


Polar codes for q-ary alphabets, q = 2r Convergence theorems

Convergence of Zv

Zv (W+) = Zv (W )2

Zv (W−) ≤ 2Zv (W ) +∑

δ∈X\0,−v

Zδ(W )Zv+δ(W ).

P

∣∣∣∣ Zmax,n+1(W+) = Zmax,n(W )2

Zmax,n(W−) ≤ min(qZmax,n(W ),1)



Convergence of Zv

Zv (W+) = Zv (W )2

Zv (W−) ≤ 2Zv (W ) +∑

δ∈X\0,−v

Zδ(W )Zv+δ(W ).

P

∣∣∣∣ Zmax,n+1(W+) = Zmax,n(W )2

Zmax,n(W−) ≤ min(qZmax,n(W ),1)



Convergence of Zv

10 iterations of P

0 200 400 600 800 1000

0.0

0.2

0.4

0.6

0.8

1.0

1.2



Convergence of Zv

30 iterations of P

0 200 400 600 800 1000

0.0

0.2

0.4

0.6

0.8

1.0

1.2



Convergence of Zv

Starting point of the argument:Process P converges to 0,1 a.e.

LemmaLet Un,n ≥ 0 be a sequence of random variables adapted to a filtrationFn with the following properties:(i) Un ∈ [0,1](ii) P(Un+1 = U2

n |Fn) ≥ 1/2(iii) Un+1 ≤ qUn for some q ∈ Z+.Then there are events Ω0,Ω1 such that P(Ω0 ∪ Ω1) = 1 and Un(ω)→ ifor ω ∈ Ωi , i = 0,1.



Convergence of Zv

Starting point of the argument:Process P converges to 0,1 a.e.

LemmaLet Un,n ≥ 0 be a sequence of random variables adapted to a filtrationFn with the following properties:(i) Un ∈ [0,1](ii) P(Un+1 = U2

n |Fn) ≥ 1/2(iii) Un+1 ≤ qUn for some q ∈ Z+.Then there are events Ω0,Ω1 such that P(Ω0 ∪ Ω1) = 1 and Un(ω)→ ifor ω ∈ Ωi , i = 0,1.


Polar codes for q-ary alphabets, q = 2r Rate of polarization

Rate of polarization

TheoremLet 0 < α < 1/2. For any DMC W : X → Y with I(W ) > 0 and anyR < I(W ) there exists a sequence of r -tuples of disjoint subsetsA0,N , . . . ,Ar−1,N of [N] such that

∑k |Ak ,N |(r − k) ≥ NR and

Zv (W (i)N ) < 2−Nα

for all i ∈ Ak ,N , all v ∈⋃r

l=k+1Xl , and allk = 0,1, . . . , r − 1.

TheoremLet 0 < α < 1/2 and let 0 < R < I(W ), where W : X → Y is a DMC.The best achievable error probability of block error under successivecancellation decoding at block length N = 2n and rate R satisfies

Pe = O(2−Nα).


Polar codes for q-ary alphabets, q = 2r Rate of polarization

Rate of polarization

TheoremLet 0 < α < 1/2. For any DMC W : X → Y with I(W ) > 0 and anyR < I(W ) there exists a sequence of r -tuples of disjoint subsetsA0,N , . . . ,Ar−1,N of [N] such that

∑k |Ak ,N |(r − k) ≥ NR and

Zv (W (i)N ) < 2−Nα

for all i ∈ Ak ,N , all v ∈⋃r

l=k+1Xl , and allk = 0,1, . . . , r − 1.

TheoremLet 0 < α < 1/2 and let 0 < R < I(W ), where W : X → Y is a DMC.The best achievable error probability of block error under successivecancellation decoding at block length N = 2n and rate R satisfies

Pe = O(2−Nα).


Transmission scheme

Transmission scheme

The channels that carry k bits have∑

i Zi ≈ r − k . Use this informationto locate the subsets of coordinates Ak ,n.

Encoding:Message uN

1 = (u1, . . . ,uN) : uj contains r − k ”frozen” bits if j ∈ Ak ,n

“Successive cancellation” decoding: for j = 1, . . . ,N put

uj =

uj , j ∈ Ar ,n

arg maxx W (j)N (yN

1 , uj−11 |x), j ∈ ∪k≤r−1Ak ,n

where if j ∈ Ak ,n, k = 0,1, . . . , r − 1, then the maximum is computedover the symbols x ∈ X with the fixed (known) values of the first r − kbits.


Transmission scheme

Transmission scheme



Encoding:Message uN



uj =

uj , j ∈ Ar ,n

arg maxx W (j)N (yN

1 , uj−11 |x), j ∈ ∪k≤r−1Ak ,n



Transmission scheme

Transmission scheme



Encoding:Message uN



uj =

uj , j ∈ Ar ,n

arg maxx W (j)N (yN

1 , uj−11 |x), j ∈ ∪k≤r−1Ak ,n



Transmission scheme

q = 29시트7

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Index

Capacity

Paper: arXiv:1107.4965v3


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