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Polar Coding Tutorial · 2017-07-19 · Polarization Encoding Decoding Construction Performance Polar Coding Tutorial Erdal Arıkan Electrical-Electronics Engineering Department Bilkent

Jun 20, 2018

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  • Polarization Encoding Decoding Construction Performance

    Polar Coding Tutorial

    Erdal Arkan

    Electrical-Electronics Engineering DepartmentBilkent UniversityAnkara, Turkey

    Jan. 15, 2015Simons InstituteUC Berkeley

  • Polarization Encoding Decoding Construction Performance

    The channel

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

    WX Y

    input alphabet: X = {0, 1},

    output alphabet: Y,

    transition probabilities:

    W (y |x), x X , y Y

  • Polarization Encoding Decoding Construction Performance

    The channel

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

    WX Y

    input alphabet: X = {0, 1},

    output alphabet: Y,

    transition probabilities:

    W (y |x), x X , y Y

  • Polarization Encoding Decoding Construction Performance

    The channel

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

    WX Y

    input alphabet: X = {0, 1},

    output alphabet: Y,

    transition probabilities:

    W (y |x), x X , y Y

  • Polarization Encoding Decoding Construction Performance

    The channel

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

    WX Y

    input alphabet: X = {0, 1},

    output alphabet: Y,

    transition probabilities:

    W (y |x), x X , y Y

  • Polarization Encoding Decoding Construction Performance

    Symmetry assumption

    Assume that the channel has input-output symmetry.

  • Polarization Encoding Decoding Construction Performance

    Symmetry assumption

    Assume that the channel has input-output symmetry.

    Examples:

    1

    1

    1

    0

    1

    0

    BSC()

  • Polarization Encoding Decoding Construction Performance

    Symmetry assumption

    Assume that the channel has input-output symmetry.

    Examples:

    1

    1

    1

    0

    1

    0

    BSC()

    1

    1

    1

    0

    1

    0

    ?

    BEC()

  • Polarization Encoding Decoding Construction Performance

    Capacity

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

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

  • Polarization Encoding Decoding Construction Performance

    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

  • Polarization Encoding Decoding Construction Performance

    The main idea

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

    Transform ordinary W into such extreme channels

  • Polarization Encoding Decoding Construction Performance

    The main idea

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

    Transform ordinary W into such extreme channels

  • Polarization Encoding Decoding Construction Performance

    The main idea

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

    Transform ordinary W into such extreme channels

  • Polarization Encoding Decoding Construction Performance

    The main idea

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

    Transform ordinary W into such extreme channels

  • Polarization Encoding Decoding Construction Performance

    The method: aggregate and redistribute capacity

    W

    W

    b

    b

    b

    W

    Original channels(uniform)

  • Polarization Encoding Decoding Construction Performance

    The method: aggregate and redistribute capacity

    W

    W

    b

    b

    b

    W

    Original channels(uniform)

    Wvec

    Vectorchannel

    Combine

  • Polarization Encoding Decoding Construction Performance

    The method: aggregate and redistribute capacity

    W

    W

    b

    b

    b

    W

    Original channels(uniform)

    Wvec

    Vectorchannel

    Combine

    WN

    WN1

    b

    b

    b

    W1

    Split

    New channels(polarized)

  • Polarization Encoding Decoding Construction Performance

    Combining

    Begin with N copies of W ,

    use a 1-1 mapping

    GN : {0, 1}N {0, 1}N

    to create a vector channel

    Wvec : UN Y N

    W

    W

    WXN

    X2

    X1

    YN

    Y2

    Y1

  • Polarization Encoding Decoding Construction Performance

    Combining

    Begin with N copies of W ,

    use a 1-1 mapping

    GN : {0, 1}N {0, 1}N

    to create a vector channel

    Wvec : UN Y N

    W

    W

    WXN

    X2

    X1

    YN

    Y2

    Y1

    GN

    UN

    U2

    U1

  • Polarization Encoding Decoding Construction Performance

    Combining

    Begin with N copies of W ,

    use a 1-1 mapping

    GN : {0, 1}N {0, 1}N

    to create a vector channel

    Wvec : UN Y N

    W

    W

    WXN

    X2

    X1

    YN

    Y2

    Y1

    GN

    UN

    U2

    U1

    Wvec

  • Polarization Encoding Decoding Construction Performance

    Conservation of capacity

    Combining operation is lossless:

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

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

    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

  • Polarization Encoding Decoding Construction Performance

    Conservation of capacity

    Combining operation is lossless:

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

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

    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

  • Polarization Encoding Decoding Construction Performance

    Conservation of capacity

    Combining operation is lossless:

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

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

    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

  • Polarization Encoding Decoding Construction Performance

    Splitting

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

    Wvec

    UN

    Ui+1

    Ui

    Ui1

    U1

    YN

    Yi

    Y1

  • Polarization Encoding Decoding Construction Performance

    Splitting

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

    =N

    i=1

    I (Ui ;YN ,U i1)

    Wvec

    UN

    Ui+1

    Ui

    Ui1

    U1

    YN

    Yi

    Y1

  • Polarization Encoding Decoding Construction Performance

    Splitting

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

    =N

    i=1

    I (Ui ;YN ,U i1)

    Define bit-channels

    Wi : Ui (YN ,U i1)

    Wvec

    UN

    Ui+1

    Ui

    Ui1

    U1

    U1

    Ui1

    YN

    Yi

    Y1

    Wi

  • Polarization Encoding Decoding Construction Performance

    Splitting

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

    =N

    i=1

    I (Ui ;YN ,U i1)

    =N

    i=1

    C (Wi )

    Define bit-channels

    Wi : Ui (YN ,U i1)

    Wvec

    UN

    Ui+1

    Ui

    Ui1

    U1

    U1

    Ui1

    YN

    Yi

    Y1

    Wi

  • Polarization Encoding Decoding Construction Performance

    Polarization is commonplace

    Polarization is the rule not theexception

    A random permutation

    GN : {0, 1}N {0, 1}N

    is a good polarizer with highprobability

    Equivalent to Shannons randomcoding approach

    W

    W

    W

    GN

    XN

    X2

    X1

    YN

    Y2

    Y1

    UN

    U2

    U1

  • Polarization Encoding Decoding Construction Performance

    Polarization is commonplace

    Polarization is the rule not theexception

    A random permutation

    GN : {0, 1}N {0, 1}N

    is a good polarizer with highprobability

    Equivalent to Shannons randomcoding approach

    W

    W

    W

    GN

    XN

    X2

    X1

    YN

    Y2

    Y1

    UN

    U2

    U1

  • Polarization Encoding Decoding Construction Performance

    Polarization is commonplace

    Polarization is the rule not theexception

    A random permutation

    GN : {0, 1}N {0, 1}N

    is a good polarizer with highprobability

    Equivalent to Shannons randomcoding approach

    W

    W

    W

    GN

    XN

    X2

    X1

    YN

    Y2

    Y1

    UN

    U2

    U1

  • Polarization Encoding Decoding Construction Performance

    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

  • Polarization Encoding Decoding Construction Performance

    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.

  • Polarization Encoding Decoding Construction Performance

    The complexity issue

    Random polarizers lack structure, too complex to implement

    Need a low-complexity polarizer

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

  • Polarization Encoding Decoding Construction Performance

    The complexity issue

    Random polarizers lack structure, too complex to implement

    Need a low-complexity polarizer

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

  • Polarization Encoding Decoding Construction Performance

    The complexity issue

    Random polarizers lack structure, too complex to implement

    Need a low-complexity polarizer

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

  • Polarization Encoding Decoding Construction Performance

    Basic module for a low-complexity scheme

    Combine two copies of W

    W

    W

    Y2

    Y1

    X2

    X1

  • Polarization Encoding Decoding Construction Performance

    Basic module for a low-complexity scheme

    Combine two copies of W

    +

    U2

    U1

    G2

    W

    W

    Y2

    Y1

    X2

    X1

  • Polarization Encoding Decoding Construction Performance

    Basic module for a low-complexity scheme

    Combine two copies of W

    +

    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)

  • Polarization Encoding Decoding Construction Performance

    The first bit-channel W1

    W1 : U1 (Y1,Y2)

    +

    random U2

    U1

    W

    W

    Y2

    Y1

  • Polarization Encoding Decoding Construction Performance

    The first bit-channel W1

    W1 : U1 (Y1,Y2)

    +

    random U2

    U1

    W

    W

    Y2

    Y1

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

  • Polarization Encoding Decoding Construction Performance

    The second bit-channel W2

    W2 : U2 (Y1,Y2,U1)

    +

    U2

    U1

    W

    W

    Y2

    Y1

  • Polarization Encoding Decoding Construction Performance

    The second bit-channel W2

    W2 : U2 (Y1,Y2,U1)

    +

    U2

    U1

    W

    W

    Y2

    Y1

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

  • Polarization Encoding Decoding Construction Performance

    Capacity conserved but redistributed unevenly

    +

    U2

    U1

    W

    W

    Y2

    Y1

    X2

    X1

    Conservation:

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

    Extremization:

    C (W1) C (W ) C (W2)

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

  • Polarization Encoding Decoding Construction Performance

    Capacity conserved but redistributed unevenly

    +

    U2

    U1

    W

    W

    Y2

    Y1

    X2

    X1

    Conservation:

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

    Extremization:

    C (W1) C (W ) C (W2)

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

  • Polarization Encoding Decoding Construction Performance

    Notation

    The two channels created by the basic transform

    (W ,W ) (W1,W2)

    will be denoted also as

    W = W1 and W+ = W2

  • Polarization Encoding Decoding Construction Performance

    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+.

  • Polarization Encoding Decoding Construction Performance

    For the size-4 construction

    +

    W

    W

  • Polarization Encoding Decoding Construction Performance

    ... duplicate the basic transform

    +

    +

    W

    W

    W

    W

  • Polarization Encoding Decoding Construction Performance

    ... obtain a pair of W and W + each

    W+

    W+

    W

    W

  • Polarization Encoding Decoding Construction Performance

    ... apply basic transform on each pair

    +

    +

    W+

    W+

    W

    W

  • Polarization Encoding Decoding Construction Performance

    ... decode in the indicated order

    +

    +

    W+

    W+

    W

    W

    U4

    U2

    U3

    U1

  • Polarization Encoding Decoding Construction Performance

    ... obtain the four new bit-channels

    W++

    W+

    W+

    W

    U4

    U2

    U3

    U1

  • Polarization Encoding Decoding Construction Performance

    Overall size-4 construction

    +

    +

    +

    +

    W

    W

    W

    W

    U4

    U2

    U3

    U1

    Y4

    Y2

    Y3

    Y1

    X4

    X2

    X3

    X1

  • Polarization Encoding Decoding Construction Performance

    Rewire for standard-form size-4 construction

    +

    +

    +

    +

    W

    W

    W

    W

    U4

    U3

    U2

    U1

    Y4

    Y3

    Y2

    Y1

    X4

    X3

    X2

    X1

  • Polarization Encoding Decoding Construction Performance

    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

  • Polarization Encoding Decoding Construction Performance

    Demonstration of polarization

    Polarization is easy to analyze when W is a BEC.

    If W is a BEC(), then so are W

    and W+, with erasure probabili-ties

    = 2 2

    and+

    = 2

    respectively.1

    1

    1

    0

    1

    0

    ?

    W

  • Polarization Encoding Decoding Construction Performance

    Demonstration of polarization

    Polarization is easy to analyze when W is a BEC.

    If W is a BEC(), then so are W

    and W+, with erasure probabili-ties

    = 2 2

    and+

    = 2

    respectively.1

    1

    1

    0

    1

    0

    ?

    W

  • Polarization Encoding Decoding Construction Performance

    Demonstration of polarization

    Polarization is easy to analyze when W is a BEC.

    If W is a BEC(), then so are W

    and W+, with erasure probabili-ties

    = 2 2

    and+

    = 2

    respectively.1 +

    1 +

    +

    +

    1

    0

    1

    0

    ?

    W+

  • Polarization Encoding Decoding Construction Performance

    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

  • Polarization Encoding Decoding Construction Performance

    Polarization for BEC(12): N = 32

    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

    Capacity of bit channels

    N=32

  • Polarization Encoding Decoding Construction Performance

    Polarization for BEC(12): N = 64

    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

    Capacity of bit channels

    N=64

  • Polarization Encoding Decoding Construction Performance

    Polarization for BEC(12): N = 128

    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

    Capacity of bit channels

    N=128

  • Polarization Encoding Decoding Construction Performance

    Polarization for BEC(12): N = 256

    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

    Capacity of bit channels

    N=256

  • Polarization Encoding Decoding Construction Performance

    Polarization for BEC(12): N = 512

    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

    Capacity of bit channels

    N=512

  • Polarization Encoding Decoding Construction Performance

    Polarization for BEC(12): N = 1024

    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

    Capacity of bit channels

    N=1024

  • Polarization Encoding Decoding Construction Performance

    Polarization martingale

    0

    1

    C(W )

  • Polarization Encoding Decoding Construction Performance

    Polarization martingale

    0

    1

    1

    C(W )

    C(W2)

    C(W1)

  • Polarization Encoding Decoding Construction Performance

    Polarization martingale

    0

    1

    1 22

    C(W )

    C(W2)

    C(W1)

    C(W++)

    C(W+)

    C(W+)

    C(W)

  • Polarization Encoding Decoding Construction Performance

    Polarization martingale

    0

    1

    1 22

    C(W )

    C(W2)

    C(W1)

    C(W++)

    C(W+)

    C(W+)

    C(W)

  • Polarization Encoding Decoding Construction Performance

    Polarization martingale

    0

    1

    1 22 3333

    C(W )

    C(W2)

    C(W1)

    C(W++)

    C(W+)

    C(W+)

    C(W)

  • Polarization Encoding Decoding Construction Performance

    Polarization martingale

    0

    1

    1 22 3333 44444444

    C(W )

    C(W2)

    C(W1)

    C(W++)

    C(W+)

    C(W+)

    C(W)

  • Polarization Encoding Decoding Construction Performance

    Polarization martingale

    0

    1

    1 22 3333 44444444 5555555555555555

    C(W )

    C(W2)

    C(W1)

    C(W++)

    C(W+)

    C(W+)

    C(W)

  • Polarization Encoding Decoding Construction Performance

    Polarization martingale

    0

    1

    1 22 3333 44444444 5555555555555555 66666666666666666666666666666666

    C(W )

    C(W2)

    C(W1)

    C(W++)

    C(W+)

    C(W+)

    C(W)

  • Polarization Encoding Decoding Construction Performance

    Polarization martingale

    0

    1

    1 22 3333 44444444 5555555555555555 66666666666666666666666666666666 7777777777777777777777777777777777777777777777777777777777777777

    C(W )

    C(W2)

    C(W1)

    C(W++)

    C(W+)

    C(W+)

    C(W)

  • Polarization Encoding Decoding Construction Performance

    Polarization martingale

    0

    1

    1 22 3333 44444444 5555555555555555 66666666666666666666666666666666 7777777777777777777777777777777777777777777777777777777777777777 88888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888

    C(W )

    C(W2)

    C(W1)

    C(W++)

    C(W+)

    C(W+)

    C(W)

  • Polarization Encoding Decoding Construction Performance

    Theorem (Polarization, A. 2007)

    The bit-channel capacities {C (Wi )} polarize: for any (0, 1), as the construction size N grows

    [

    no. channels with C (Wi ) > 1

    N

    ]

    C (W )

    and[

    no. channels with C (Wi ) <

    N

    ]

    1 C (W )

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

    Above theorem holds with 2

    N . 0

    1

    1

  • Polarization Encoding Decoding Construction Performance

    Theorem (Polarization, A. 2007)

    The bit-channel capacities {C (Wi )} polarize: for any (0, 1), as the construction size N grows

    [

    no. channels with C (Wi ) > 1

    N

    ]

    C (W )

    and[

    no. channels with C (Wi ) <

    N

    ]

    1 C (W )

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

    Above theorem holds with 2

    N . 0

    1

    1

  • Polarization Encoding Decoding Construction Performance

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

    I (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

  • Polarization Encoding Decoding Construction Performance

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

    I (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

  • Polarization Encoding Decoding Construction Performance

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

    I (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

  • Polarization Encoding Decoding Construction Performance

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

    I (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

  • Polarization Encoding Decoding Construction Performance

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

    I (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

  • Polarization Encoding Decoding Construction Performance

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

    I (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

  • Polarization Encoding Decoding Construction Performance

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

    I (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

  • Polarization Encoding Decoding Construction Performance

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

    I (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

  • Polarization Encoding Decoding Construction Performance

    Encoding complexity

    Theorem

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

    Proof:

    Polar coding transform can be represented as a graph withN[1 + log(N)] variables.

    The graph has (1 + log(N)) levels with N variables at eachlevel.

    Computation begins at the source level and can be carried outlevel by level.

    Space complexity O(N), time complexity O(N logN).

  • Polarization Encoding Decoding Construction Performance

    Encoding complexity

    Theorem

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

    Proof:

    Polar coding transform can be represented as a graph withN[1 + log(N)] variables.

    The graph has (1 + log(N)) levels with N variables at eachlevel.

    Computation begins at the source level and can be carried outlevel by level.

    Space complexity O(N), time complexity O(N logN).

  • Polarization Encoding Decoding Construction Performance

    Encoding complexity

    Theorem

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

    Proof:

    Polar coding transform can be represented as a graph withN[1 + log(N)] variables.

    The graph has (1 + log(N)) levels with N variables at eachlevel.

    Computation begins at the source level and can be carried outlevel by level.

    Space complexity O(N), time complexity O(N logN).

  • Polarization Encoding Decoding Construction Performance

    Encoding complexity

    Theorem

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

    Proof:

    Polar coding transform can be represented as a graph withN[1 + log(N)] variables.

    The graph has (1 + log(N)) levels with N variables at eachlevel.

    Computation begins at the source level and can be carried outlevel by level.

    Space complexity O(N), time complexity O(N logN).

  • Polarization Encoding Decoding Construction Performance

    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

  • Polarization Encoding Decoding Construction Performance

    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

    1

    1

    1

    1

    1

    1

    0

    0

    free

    free

    free

    frozen

    free

    frozen

    frozen

    frozen

  • Polarization Encoding Decoding Construction Performance

    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

    1

    1

    1

    1

    1

    1

    0

    0

    1

    1

    0

    0

    1

    1

    1

    1

    free

    free

    free

    frozen

    free

    frozen

    frozen

    frozen

  • Polarization Encoding Decoding Construction Performance

    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

    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

  • Polarization Encoding Decoding Construction Performance

    Successive Cancellation Decoding (SCD)

    Theorem

    The complexity of successive cancellation decoding for polar codesis O(N logN).

    Proof: Given below.

  • Polarization Encoding Decoding Construction Performance

    SCD: Exploit the x = |a|a+ b| structure

    +

    +

    +

    +

    +

    +

    +

    +

    +

    +

    +

    +

    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

    a4

    a3

    a2

    a1

    b4

    b3

    b2

    b1

  • Polarization Encoding Decoding Construction Performance

    First phase: treat a as noise, decode (u1, u2, u3, u4)

    +

    +

    +

    +

    +

    +

    +

    +

    W

    W

    W

    W

    W

    W

    W

    W

    u4

    u3

    u2

    u1

    x8

    x7

    x6

    x5

    x4

    x3

    x2

    x1

    y8

    y7

    y6

    y5

    y4

    y3

    y2

    y1

    noise a4

    noise a3

    noise a2

    noise a1

    b4

    b3

    b2

    b1

  • Polarization Encoding Decoding Construction Performance

    End of first phase

    +

    +

    +

    +

    +

    +

    +

    +

    +

    +

    +

    +

    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

    a4

    a3

    a2

    a1

    b4

    b3

    b2

    b1

  • Polarization Encoding Decoding Construction Performance

    Second phase: Treat b as known, decode (u5, u6, u7, u8)

    +

    +

    +

    +

    +

    +

    +

    +

    W

    W

    W

    W

    W

    W

    W

    W

    u8

    u7

    u6

    u5

    y8

    y7

    y6

    y5

    y4

    y3

    y2

    y1

    a4

    a3

    a2

    a1

    known b4

    known b3

    known b2

    known b1

  • Polarization Encoding Decoding Construction Performance

    First phase in detail

    +

    +

    +

    +

    +

    +

    +

    +

    W

    W

    W

    W

    W

    W

    W

    W

    u4

    u3

    u2

    u1

    x8

    x7

    x6

    x5

    x4

    x3

    x2

    x1

    y8

    y7

    y6

    y5

    y4

    y3

    y2

    y1

    noise a4

    noise a3

    noise a2

    noise a1

    b4

    b3

    b2

    b1

  • Polarization Encoding Decoding Construction Performance

    Equivalent channel model

    +

    +

    +

    +

    W

    W

    W

    W

    W

    W

    W

    W

    x8

    x7

    x6

    x5

    x4

    x3

    x2

    x1

    y8

    y7

    y6

    y5

    y4

    y3

    y2

    y1

    noise a4

    noise a3

    noise a2

    noise a1

    b4

    b3

    b2

    b1

  • Polarization Encoding Decoding Construction Performance

    First copy of W

    +

    +

    +

    +

    W

    W

    W

    W

    W

    W

    W

    W

    W

    W

    x8

    x7

    x6

    x5

    x4

    x3

    x2

    x1

    y8

    y7

    y6

    y5

    y4

    y3

    y2

    y1

    noise a4

    noise a3

    noise a2

    noise a1

    b4

    b3

    b2

    b1

  • Polarization Encoding Decoding Construction Performance

    Second copy of W

    +

    +

    +

    +

    W

    W

    W

    W

    W

    W

    W

    W

    W

    W

    x8

    x7

    x6

    x5

    x4

    x3

    x2

    x1

    y8

    y7

    y6

    y5

    y4

    y3

    y2

    y1

    noise a4

    noise a3

    noise a2

    noise a1

    b4

    b3

    b2

    b1

  • Polarization Encoding Decoding Construction Performance

    Third copy of W

    +

    +

    +

    +

    W

    W

    W

    W

    W

    W

    W

    W

    W

    W

    x8

    x7

    x6

    x5

    x4

    x3

    x2

    x1

    y8

    y7

    y6

    y5

    y4

    y3

    y2

    y1

    noise a4

    noise a3

    noise a2

    noise a1

    b4

    b3

    b2

    b1

  • Polarization Encoding Decoding Construction Performance

    Fourth copy of W

    +

    +

    +

    +

    W

    W

    W

    W

    W

    W

    W

    W

    W

    W

    x8

    x7

    x6

    x5

    x4

    x3

    x2

    x1

    y8

    y7

    y6

    y5

    y4

    y3

    y2

    y1

    noise a4

    noise a3

    noise a2

    noise a1

    b4

    b3

    b2

    b1

  • Polarization Encoding Decoding Construction Performance

    Decoding on W

    +

    +

    +

    +

    W

    W

    W

    W

    u4

    u3

    u2

    u1

    (y4, y8)

    (y3, y7)

    (y2, y6)

    (y1, y5)

    b4

    b3

    b2

    b1

  • Polarization Encoding Decoding Construction Performance

    b = |t|t+w|

    +

    +

    +

    +

    W

    W

    W

    W

    u4

    u3

    u2

    u1

    (y4, y8)

    (y3, y7)

    (y2, y6)

    (y1, y5)

    b4

    b3

    b2

    b1

    t2

    t1

    w2

    w1

  • Polarization Encoding Decoding Construction Performance

    Decoding on W

    +

    W

    W

    u2

    u1

    (y2, y4, y6, y8)

    (y1, y3, y5, y7)

    w2

    w1

  • Polarization Encoding Decoding Construction Performance

    Decoding on W

    Wu1 (y1, y2, . . . , y8)

  • Polarization Encoding Decoding Construction Performance

    Decoding on W

    Wu1 (y1, y2, . . . , y8)

    Compute

    L=

    W(y1, . . . , y8 | u1 = 0)

    W(y1, . . . , y8 | u1 = 1).

  • Polarization Encoding Decoding Construction Performance

    Decoding on W

    Wu1 (y1, y2, . . . , y8)

    Compute

    L=

    W(y1, . . . , y8 | u1 = 0)

    W(y1, . . . , y8 | u1 = 1).

    Set

    u1 =

    u1 if u1 is frozen

    0 else if L > 0

    1 else

  • Polarization Encoding Decoding Construction Performance

    Decoding on W

    Wu1 (y1, y2, . . . , y8)

    Compute

    L=

    W(y1, . . . , y8 | u1 = 0)

    W(y1, . . . , y8 | u1 = 1).

    Set

    u1 =

    u1 if u1 is frozen

    0 else if L > 0

    1 else

  • Polarization Encoding Decoding Construction Performance

    Decoding on W

    Wu1 (y1, y2, . . . , y8)

    Compute

    L=

    W(y1, . . . , y8 | u1 = 0)

    W(y1, . . . , y8 | u1 = 1).

    Set

    u1 =

    u1 if u1 is frozen

    0 else if L > 0

    1 else

  • Polarization Encoding Decoding Construction Performance

    Decoding on W +

    +

    W

    W

    u2

    known u1

    (y2, y4, y6, y8)

    (y1, y3, y5, y7)

  • Polarization Encoding Decoding Construction Performance

    Decoding on W +

    W+u2 (y1, . . . , y8, u1)

  • Polarization Encoding Decoding Construction Performance

    Decoding on W +

    W+u2 (y1, . . . , y8, u1)

    Compute

    L+=

    W+(y1, . . . , y8, u1 | u2 = 0)

    W+(y1, . . . , y8, u1 | u2 = 1).

  • Polarization Encoding Decoding Construction Performance

    Decoding on W +

    W+u2 (y1, . . . , y8, u1)

    Compute

    L+=

    W+(y1, . . . , y8, u1 | u2 = 0)

    W+(y1, . . . , y8, u1 | u2 = 1).

    Set

    u2 =

    u2 if u2 is frozen

    0 else if L+ > 0

    1 else

  • Polarization Encoding Decoding Construction Performance

    Decoding on W +

    W+u2 (y1, . . . , y8, u1)

    Compute

    L+=

    W+(y1, . . . , y8, u1 | u2 = 0)

    W+(y1, . . . , y8, u1 | u2 = 1).

    Set

    u2 =

    u2 if u2 is frozen

    0 else if L+ > 0

    1 else

  • Polarization Encoding Decoding Construction Performance

    Decoding on W +

    W+u2 (y1, . . . , y8, u1)

    Compute

    L+=

    W+(y1, . . . , y8, u1 | u2 = 0)

    W+(y1, . . . , y8, u1 | u2 = 1).

    Set

    u2 =

    u2 if u2 is frozen

    0 else if L+ > 0

    1 else

  • Polarization Encoding Decoding Construction Performance

    Complexity for successive cancelation decoding

    Let CN be the complexity of decoding a code of length N

    Decoding problem of size N for W reduced to two decodingproblems of size N/2 for W and W+

    SoCN = 2CN/2 + kN

    for some constant k

    This gives CN = O(N logN)

  • Polarization Encoding Decoding Construction Performance

    Complexity for successive cancelation decoding

    Let CN be the complexity of decoding a code of length N

    Decoding problem of size N for W reduced to two decodingproblems of size N/2 for W and W+

    SoCN = 2CN/2 + kN

    for some constant k

    This gives CN = O(N logN)

  • Polarization Encoding Decoding Construction Performance

    Complexity for successive cancelation decoding

    Let CN be the complexity of decoding a code of length N

    Decoding problem of size N for W reduced to two decodingproblems of size N/2 for W and W+

    SoCN = 2CN/2 + kN

    for some constant k

    This gives CN = O(N logN)

  • Polarization Encoding Decoding Construction Performance

    Complexity for successive cancelation decoding

    Let CN be the complexity of decoding a code of length N

    Decoding problem of size N for W reduced to two decodingproblems of size N/2 for W and W+

    SoCN = 2CN/2 + kN

    for some constant k

    This gives CN = O(N logN)

  • Polarization Encoding Decoding Construction Performance

    Performance of polar codes

    Theorem

    For any rate R < I (W ) and block-length N, the probability offrame error for polar codes under successive cancelation decoding isbounded as

    Pe(N,R) = o(

    2

    N+o(

    N))

    Proof: Given in the next presentation.

  • Polarization Encoding Decoding Construction Performance

    Construction complexity

    Theorem

    Given W and a rate R < I (W ), a polar code can be constructed inO(Npoly(log(N))) time that achieves under SCD the performance

    Pe = o(

    2

    N+o(

    N))

    Proof: Given in the next presentation.

  • Polarization Encoding Decoding Construction Performance

    Polar coding summary

    Summary

    Given W , N = 2n, and R < I (W ), a polar code can be constructedsuch that it has

    construction complexity O(Npoly(log(N))),

    encoding complexity N logN,

    successive-cancellation decoding complexity N logN,

    frame error probability Pe(N,R) = o(

    2

    N+o(

    N))

    .

  • Polarization Encoding Decoding Construction Performance

    Polar coding summary

    Summary

    Given W , N = 2n, and R < I (W ), a polar code can be constructedsuch that it has

    construction complexity O(Npoly(log(N))),

    encoding complexity N logN,

    successive-cancellation decoding complexity N logN,

    frame error probability Pe(N,R) = o(

    2

    N+o(

    N))

    .

  • Polarization Encoding Decoding Construction Performance

    Polar coding summary

    Summary

    Given W , N = 2n, and R < I (W ), a polar code can be constructedsuch that it has

    construction complexity O(Npoly(log(N))),

    encoding complexity N logN,

    successive-cancellation decoding complexity N logN,

    frame error probability Pe(N,R) = o(

    2

    N+o(

    N))

    .

  • Polarization Encoding Decoding Construction Performance

    Polar coding summary

    Summary

    Given W , N = 2n, and R < I (W ), a polar code can be constructedsuch that it has

    construction complexity O(Npoly(log(N))),

    encoding complexity N logN,

    successive-cancellation decoding complexity N logN,

    frame error probability Pe(N,R) = o(

    2

    N+o(

    N))

    .

  • Polarization Encoding Decoding Construction Performance

    List decoder for polar codes

    Developed by Tal and Vardy (2011); similar to Dumers listdecoder for Reed-Muller codes.

    First produce L candidate decisions

    Pick the most likely word from the list

    Complexity O(LN logN)

  • Polarization Encoding Decoding Construction Performance

    List decoder for polar codes

    Developed by Tal and Vardy (2011); similar to Dumers listdecoder for Reed-Muller codes.

    First produce L candidate decisions

    Pick the most likely word from the list

    Complexity O(LN logN)

  • Polarization Encoding Decoding Construction Performance

    List decoder for polar codes

    Developed by Tal and Vardy (2011); similar to Dumers listdecoder for Reed-Muller codes.

    First produce L candidate decisions

    Pick the most likely word from the list

    Complexity O(LN logN)

  • Polarization Encoding Decoding Construction Performance

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

  • Polarization Encoding Decoding Construction Performance

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

  • Polarization Encoding Decoding Construction Performance

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

  • Polarization Encoding Decoding Construction Performance

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

  • Polarization Encoding Decoding Construction Performance

    Tal-Vardy list decoder performance

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

  • Polarization Encoding Decoding Construction Performance

    Tal-Vardy list decoder performance

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

  • Polarization Encoding Decoding Construction Performance

    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!

  • Polarization Encoding Decoding Construction Performance

    List decoder with CRC

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

    Selection made by CRC and relative likelihood

  • Polarization Encoding Decoding Construction Performance

    List decoder with CRC

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

    Selection made by CRC and relative likelihood

  • Polarization Encoding Decoding Construction Performance

    Tal-Vardy list decoder with CRC

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

  • Polarization Encoding Decoding Construction Performance

    Tal-Vardy list decoder with CRC

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

  • Polarization Encoding Decoding Construction Performance

    Tal-Vardy list decoder with CRC

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

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

  • Polarization Encoding Decoding Construction Performance

    Summary

    Polarization is a commonplace phenomenon almostunavoidable

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

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

  • Polarization Encoding Decoding Construction Performance

    Summary

    Polarization is a commonplace phenomenon almostunavoidable

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

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

  • Polarization Encoding Decoding Construction Performance

    Summary

    Polarization is a commonplace phenomenon almostunavoidable

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

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

    PolarizationEncodingDecodingConstructionPolar coding performance

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