Polar Codes -- A New Paradigm for Codingjasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September 2nd, 2009

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.chSunday, September 13, 2009

http://panorama.epfl.ch

http://panorama.epfl.ch

Sunday, September 13, 2009

Coding


Coding


Coding

codeC={000, 010, 101, 111}

n ... blocklength


Important Parameters


Important Parameters

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


Linear Codes


Linear Codes

generator matrix


Linear Codes

generator matrix parity-check matrix


Linear Codes

generator matrix parity-check matrix


Bitwise MAP Decoding


Bitwise MAP Decoding

[LDPC -- Gallager ‘60]


Polar Codes: Summary



Erdal Arikan, ISIT 2007




very general phenomenon





information theoretic view whycodes work






first “low complexity” scheme which provablyachieves the capacity for a fairly wide array of

channels







channels

many possible variations on the theme







channels

codes not only good for channel coding;work equally well for source coding and more

complicated scenarios

many possible variations on the theme


References


References


Codes from Kronecker Product of G2






















Reed-Muller Codes


Reed-Muller Codes

choose rows of largest weight


Definition of Channels

Polar Codes



Polar Codes

W -- BMS channel



Polar Codes

W -- BMS channel



Polar Codes

W -- BMS channel



Polar Codes

W -- BMS channel



Polar Codes

W -- BMS channel



Polar Codes

W -- BMS channel



Polar Codes

W -- BMS channel



Polar Codes

W -- BMS channel



Polar Codes

W -- BMS channel



Channel Polarization
















N




N




N




N




N




N






Successive Decoding



Successive Decoding



Successive Decoding



Successive Decoding



Successive Decoding



Successive Decoding



Successive Decoding



Successive Decoding



Successive Decoding



More on Polarization
















N




N






N




N




N




N



Equivalent “Random” Channel



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



Define In=I(WB1, B2, ..., Bn)




Define In=I(WB1, B2, ..., Bn)


Study the distribution of In


Properties of In


Properties of In

I0=I(W) is a constant


Properties of In

In ∈ [0, 1]; so In is bounded



Properties of In



Conditional on B1, B2, ..., Bn, and withP= WB1, B2, ..., Bn, In+1 can only take on thetwo values I(P+) and I(P-)


Properties of In



Conditional on B1, B2, ..., Bn, and withP= WB1, B2, ..., Bn, In+1 can only take on thetwo 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


Properties of In


a bounded martingale converges almost surely

Properties of In


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


Properties of In




Pr{|In+1-In|≤ε}→1; but |In+1-In|=(I(P+)- I(P-))/2

Properties of In





from extremes of information combining we knowthat (I(P+)-I(P-))/2 ≤ε implies that I(P)∉(δ, 1-δ)

Properties of In

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

I(P)





from extremes of information combining we knowthat (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)


Summary of Known Results



achieve capacity on memoryless channels Arikan 2007



Arikan and Telatar 2008





Korada, Sasoglu, and U. 2009




Polar Codes Based on Larger Matrices



Polar Codes Based on Larger Matrices



Characterization of Exponent



Exponent: Example



Exponent: Example



Exponent: Example



Exponent: Example



Exponent: Example



Exponent: Example



Results



Results



Results



Results



Results










optimal for lossy source coding, Wyner-Ziv, Gelfand-Pinsker, ...

Korada and U. 2009



Source Coding



Source Coding



Source Coding







Korada and U. 2009







Korada and U. 2009

Mori and Tanaka 2009efficient construction







Korada and U. 2009

suboptimal for compound coding Hassani, Korada and U. 2009








Korada and U. 2009




non-binary version and asym. channels Arikan, Sasoglu, and Telatar 2009






Korada and U. 2009




non-binary version and asym. channels Arikan, Sasoglu, and Telatar 2009

scaling


Summary

+ many applications

+ completely new paradigm of coding

+ provably achieves capacity

+ low complexity

- currently only competitive for VERY large N


Wyner-Ziv and Gelfand-Pinsker


Polar Codes -- A New Paradigm for Codingjasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September 2nd, 2009

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