Polar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics - - PowerPoint PPT Presentation

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Polar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics - - PowerPoint PPT Presentation

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Polar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September 2nd, 2009 Thanks to Emre Telatar and Satish Korada. (for many borrowed figures) http://panorama.epfl.ch Sunday, September 13, 2009 Sunday,

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SLIDE 1

Polar Codes -- A New Paradigm for Coding

  • R. Urbanke, EPFL

Physics of Algorithms, Santa Fe, September 2nd, 2009 Thanks to Emre Telatar and Satish Korada. (for many borrowed figures)

http://panorama.epfl.ch

Sunday, September 13, 2009

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SLIDE 2

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SLIDE 3

Coding

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SLIDE 4

Coding

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SLIDE 5

Coding

code C={000, 010, 101, 111} n ... blocklength

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SLIDE 6

Important Parameters

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SLIDE 7

Important Parameters

rate, error probability, encoding complexity, decoding complexity, blocklength

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SLIDE 8

Linear Codes

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SLIDE 9

Linear Codes

generator matrix

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SLIDE 10

Linear Codes

generator matrix parity-check matrix

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SLIDE 11

Linear Codes

generator matrix parity-check matrix

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SLIDE 12

Bitwise MAP Decoding

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SLIDE 13

Bitwise MAP Decoding

[LDPC -- Gallager ‘60]

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SLIDE 14

Polar Codes: Summary

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SLIDE 15

Polar Codes: Summary

Erdal Arikan, ISIT 2007

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SLIDE 16

Polar Codes: Summary

Erdal Arikan, ISIT 2007 very general phenomenon

Sunday, September 13, 2009

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SLIDE 17

Polar Codes: Summary

Erdal Arikan, ISIT 2007 very general phenomenon information theoretic view why codes work

Sunday, September 13, 2009

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SLIDE 18

Polar Codes: Summary

Erdal Arikan, ISIT 2007 very general phenomenon information theoretic view why codes work first “low complexity” scheme which provably achieves the capacity for a fairly wide array of channels

Sunday, September 13, 2009

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SLIDE 19

Polar Codes: Summary

Erdal Arikan, ISIT 2007 very general phenomenon information theoretic view why codes work first “low complexity” scheme which provably achieves the capacity for a fairly wide array of channels many possible variations on the theme

Sunday, September 13, 2009

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SLIDE 20

Polar Codes: Summary

Erdal Arikan, ISIT 2007 very general phenomenon information theoretic view why codes work first “low complexity” scheme which provably achieves the capacity for a fairly wide array of channels codes not only good for channel coding; work equally well for source coding and more complicated scenarios many possible variations on the theme

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SLIDE 21

References

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SLIDE 22

References

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SLIDE 23

Codes from Kronecker Product of G2

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SLIDE 24

Codes from Kronecker Product of G2

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SLIDE 25

Codes from Kronecker Product of G2

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SLIDE 26

Codes from Kronecker Product of G2

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SLIDE 27

Codes from Kronecker Product of G2

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SLIDE 28

Codes from Kronecker Product of G2

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SLIDE 29

Codes from Kronecker Product of G2

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SLIDE 30

Codes from Kronecker Product of G2

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SLIDE 31

Codes from Kronecker Product of G2

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SLIDE 32

Codes from Kronecker Product of G2

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SLIDE 33

Codes from Kronecker Product of G2

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SLIDE 34

Reed-Muller Codes

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SLIDE 35

Reed-Muller Codes

choose rows of largest weight

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SLIDE 36

Polar Codes

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SLIDE 37

Polar Codes

W -- BMS channel

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SLIDE 38

Polar Codes

W -- BMS channel

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SLIDE 39

Polar Codes

W -- BMS channel

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SLIDE 40

Polar Codes

W -- BMS channel

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SLIDE 41

Polar Codes

W -- BMS channel

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SLIDE 42

Polar Codes

W -- BMS channel

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SLIDE 43

Polar Codes

W -- BMS channel

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SLIDE 44

Polar Codes

W -- BMS channel

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SLIDE 45

Polar Codes

W -- BMS channel

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SLIDE 46

Channel Polarization

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SLIDE 47

Channel Polarization

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SLIDE 48

Channel Polarization

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SLIDE 49

Channel Polarization

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SLIDE 50

Channel Polarization

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SLIDE 51

Channel Polarization

N

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SLIDE 52

Channel Polarization

N

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SLIDE 53

Channel Polarization

N

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SLIDE 54

Channel Polarization

N

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SLIDE 55

Channel Polarization

N

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SLIDE 56

Channel Polarization

N

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SLIDE 57

Channel Polarization

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SLIDE 58

Successive Decoding

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SLIDE 59

Successive Decoding

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SLIDE 60

Successive Decoding

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SLIDE 61

Successive Decoding

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SLIDE 62

Successive Decoding

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SLIDE 63

Successive Decoding

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SLIDE 64

Successive Decoding

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SLIDE 65

Successive Decoding

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SLIDE 66

Successive Decoding

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SLIDE 67

More on Polarization

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SLIDE 68

More on Polarization

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SLIDE 69

More on Polarization

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SLIDE 70

More on Polarization

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SLIDE 71

More on Polarization

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SLIDE 72

More on Polarization

N

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SLIDE 73

More on Polarization

N

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SLIDE 74

More on Polarization

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SLIDE 75

N

More on Polarization

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SLIDE 76

N

More on Polarization

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SLIDE 77

N

More on Polarization

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SLIDE 78

N

More on Polarization

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SLIDE 79

Equivalent “Random” Channel

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SLIDE 80

Equivalent “Random” Channel

Set B1, B2, ... to be i.i.d. {+, -} valued, uniformly distributed random variables

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SLIDE 81

Equivalent “Random” Channel

Define In=I(WB1, B2, ..., Bn) Set B1, B2, ... to be i.i.d. {+, -} valued, uniformly distributed random variables

Sunday, September 13, 2009

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SLIDE 82

Equivalent “Random” Channel

Define In=I(WB1, B2, ..., Bn) Set B1, B2, ... to be i.i.d. {+, -} valued, uniformly distributed random variables Study the distribution of In

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SLIDE 83

Properties of In

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SLIDE 84

Properties of In

I0=I(W) is a constant

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SLIDE 85

Properties of In

In ∈ [0, 1]; so In is bounded I0=I(W) is a constant

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SLIDE 86

Properties of In

In ∈ [0, 1]; so In is bounded I0=I(W) is a constant Conditional on B1, B2, ..., Bn, and with P= WB1, B2, ..., Bn, In+1 can only take on the two values I(P+) and I(P-)

Sunday, September 13, 2009

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SLIDE 87

Properties of In

In ∈ [0, 1]; so In is bounded I0=I(W) is a constant Conditional on B1, B2, ..., Bn, and with P= WB1, B2, ..., Bn, In+1 can only take on the two values I(P+) and I(P-) Further, E[In+1 | B1, B2, ..., Bn]=(I(P+)+ I(P-))/2=I(P), so {In} is a (bounded) martingale

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SLIDE 88

Properties of In

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SLIDE 89

a bounded martingale converges almost surely

Properties of In

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SLIDE 90

I∞ = limn→∞ In exists almost surely; E[I∞]=I0=I(W) a bounded martingale converges almost surely

Properties of In

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SLIDE 91

I∞ = limn→∞ In exists almost surely; E[I∞]=I0=I(W) a bounded martingale converges almost surely Pr{|In+1-In|≤ε}→1; but |In+1-In|=(I(P+)- I(P-))/2

Properties of In

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SLIDE 92

I∞ = limn→∞ In exists almost surely; E[I∞]=I0=I(W) a bounded martingale converges almost surely Pr{|In+1-In|≤ε}→1; but |In+1-In|=(I(P+)- I(P-))/2 from extremes of information combining we know that (I(P+)-I(P-))/2 ≤ε implies that I(P)∉(δ, 1-δ)

Properties of In

(I(P+)- I(P-))/2 I(P)

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SLIDE 93

I∞ = limn→∞ In exists almost surely; E[I∞]=I0=I(W) a bounded martingale converges almost surely Pr{|In+1-In|≤ε}→1; but |In+1-In|=(I(P+)- I(P-))/2 from extremes of information combining we know that (I(P+)-I(P-))/2 ≤ε implies that I(P)∉(δ, 1-δ)

Properties of In

we conclude that I∞ takes values only in {0, 1} (I(P+)- I(P-))/2 I(P)

Sunday, September 13, 2009

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SLIDE 94

Summary of Known Results

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SLIDE 95

Summary of Known Results

achieve capacity on memoryless channels Arikan 2007

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SLIDE 96

Summary of Known Results

Arikan and Telatar 2008 achieve capacity on memoryless channels Arikan 2007

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SLIDE 97

Summary of Known Results

Arikan and Telatar 2008 Korada, Sasoglu, and U. 2009 achieve capacity on memoryless channels Arikan 2007

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SLIDE 98

Polar Codes Based on Larger Matrices

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SLIDE 99

Polar Codes Based on Larger Matrices

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SLIDE 100

Characterization of Exponent

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SLIDE 101

Exponent: Example

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SLIDE 102

Exponent: Example

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SLIDE 103

Exponent: Example

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SLIDE 104

Exponent: Example

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SLIDE 105

Exponent: Example

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SLIDE 106

Exponent: Example

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SLIDE 107

Results

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SLIDE 108

Results

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SLIDE 109

Results

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SLIDE 110

Results

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SLIDE 111

Results

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SLIDE 112

Summary of Known Results

Arikan and Telatar 2008 Korada, Sasoglu, and U. 2009 achieve capacity on memoryless channels Arikan 2007

Sunday, September 13, 2009

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SLIDE 113

Summary of Known Results

Arikan and Telatar 2008 Korada, Sasoglu, and U. 2009

  • ptimal for lossy source

coding, Wyner-Ziv, Gelfand-Pinsker, ... Korada and U. 2009 achieve capacity on memoryless channels Arikan 2007

Sunday, September 13, 2009

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SLIDE 114

Source Coding

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SLIDE 115

Source Coding

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SLIDE 116

Source Coding

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SLIDE 117

Summary of Known Results

Arikan and Telatar 2008 Korada, Sasoglu, and U. 2009

  • ptimal for lossy source

coding, Wyner-Ziv, Gelfand-Pinsker, ... Korada and U. 2009 achieve capacity on memoryless channels Arikan 2007

Sunday, September 13, 2009

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SLIDE 118

Summary of Known Results

Arikan and Telatar 2008 Korada, Sasoglu, and U. 2009

  • ptimal for lossy source

coding, Wyner-Ziv, Gelfand-Pinsker, ... Korada and U. 2009 Mori and Tanaka 2009 efficient construction achieve capacity on memoryless channels Arikan 2007

Sunday, September 13, 2009

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SLIDE 119

Summary of Known Results

Arikan and Telatar 2008 Korada, Sasoglu, and U. 2009

  • ptimal for lossy source

coding, Wyner-Ziv, Gelfand-Pinsker, ... Korada and U. 2009 suboptimal for compound coding Hassani, Korada and U. 2009 Mori and Tanaka 2009 efficient construction achieve capacity on memoryless channels Arikan 2007

Sunday, September 13, 2009

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SLIDE 120

Summary of Known Results

Arikan and Telatar 2008 Korada, Sasoglu, and U. 2009

  • ptimal for lossy source

coding, Wyner-Ziv, Gelfand-Pinsker, ... Korada and U. 2009 suboptimal for compound coding Hassani, Korada and U. 2009 Mori and Tanaka 2009 efficient construction achieve capacity on memoryless channels Arikan 2007 non-binary version and asym. channels Arikan, Sasoglu, and Telatar 2009

Sunday, September 13, 2009

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SLIDE 121

Summary of Known Results

Arikan and Telatar 2008 Korada, Sasoglu, and U. 2009

  • ptimal for lossy source

coding, Wyner-Ziv, Gelfand-Pinsker, ... Korada and U. 2009 suboptimal for compound coding Hassani, Korada and U. 2009 Mori and Tanaka 2009 efficient construction achieve capacity on memoryless channels Arikan 2007 non-binary version and asym. channels Arikan, Sasoglu, and Telatar 2009 scaling

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SLIDE 122

Summary

+ many applications + completely new paradigm of coding + provably achieves capacity + low complexity

  • currently only competitive for VERY large N

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SLIDE 123

Wyner-Ziv and Gelfand-Pinsker

Sunday, September 13, 2009