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Transcript of Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A...
![Page 1: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/1.jpg)
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
![Page 2: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/2.jpg)
Sunday, September 13, 2009
![Page 3: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/3.jpg)
Coding
Sunday, September 13, 2009
![Page 4: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/4.jpg)
Coding
Sunday, September 13, 2009
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Coding
codeC={000, 010, 101, 111}
n ... blocklength
Sunday, September 13, 2009
![Page 6: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/6.jpg)
Important Parameters
Sunday, September 13, 2009
![Page 7: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/7.jpg)
Important Parameters
rate, error probability, encoding complexity,decoding complexity, blocklength
Sunday, September 13, 2009
![Page 8: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/8.jpg)
Linear Codes
Sunday, September 13, 2009
![Page 9: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/9.jpg)
Linear Codes
generator matrix
Sunday, September 13, 2009
![Page 10: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/10.jpg)
Linear Codes
generator matrix parity-check matrix
Sunday, September 13, 2009
![Page 11: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/11.jpg)
Linear Codes
generator matrix parity-check matrix
Sunday, September 13, 2009
![Page 12: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/12.jpg)
Bitwise MAP Decoding
Sunday, September 13, 2009
![Page 13: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/13.jpg)
Bitwise MAP Decoding
[LDPC -- Gallager ‘60]
Sunday, September 13, 2009
![Page 14: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/14.jpg)
Polar Codes: Summary
Sunday, September 13, 2009
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Polar Codes: Summary
Erdal Arikan, ISIT 2007
Sunday, September 13, 2009
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Polar Codes: Summary
Erdal Arikan, ISIT 2007
very general phenomenon
Sunday, September 13, 2009
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Polar Codes: Summary
Erdal Arikan, ISIT 2007
very general phenomenon
information theoretic view whycodes work
Sunday, September 13, 2009
![Page 18: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/18.jpg)
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
Sunday, September 13, 2009
![Page 19: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/19.jpg)
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
many possible variations on the theme
Sunday, September 13, 2009
![Page 20: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/20.jpg)
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
codes not only good for channel coding;work equally well for source coding and more
complicated scenarios
many possible variations on the theme
Sunday, September 13, 2009
![Page 21: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/21.jpg)
References
Sunday, September 13, 2009
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References
Sunday, September 13, 2009
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Codes from Kronecker Product of G2
Sunday, September 13, 2009
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Codes from Kronecker Product of G2
Sunday, September 13, 2009
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Codes from Kronecker Product of G2
Sunday, September 13, 2009
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Codes from Kronecker Product of G2
Sunday, September 13, 2009
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Codes from Kronecker Product of G2
Sunday, September 13, 2009
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Codes from Kronecker Product of G2
Sunday, September 13, 2009
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Codes from Kronecker Product of G2
Sunday, September 13, 2009
![Page 30: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/30.jpg)
Codes from Kronecker Product of G2
Sunday, September 13, 2009
![Page 31: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/31.jpg)
Codes from Kronecker Product of G2
Sunday, September 13, 2009
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Codes from Kronecker Product of G2
Sunday, September 13, 2009
![Page 33: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/33.jpg)
Codes from Kronecker Product of G2
Sunday, September 13, 2009
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Reed-Muller Codes
Sunday, September 13, 2009
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Reed-Muller Codes
choose rows of largest weight
Sunday, September 13, 2009
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Definition of Channels
Polar Codes
Sunday, September 13, 2009
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Definition of Channels
Polar Codes
W -- BMS channel
Sunday, September 13, 2009
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Definition of Channels
Polar Codes
W -- BMS channel
Sunday, September 13, 2009
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Definition of Channels
Polar Codes
W -- BMS channel
Sunday, September 13, 2009
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Definition of Channels
Polar Codes
W -- BMS channel
Sunday, September 13, 2009
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Definition of Channels
Polar Codes
W -- BMS channel
Sunday, September 13, 2009
![Page 42: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/42.jpg)
Definition of Channels
Polar Codes
W -- BMS channel
Sunday, September 13, 2009
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Definition of Channels
Polar Codes
W -- BMS channel
Sunday, September 13, 2009
![Page 44: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/44.jpg)
Definition of Channels
Polar Codes
W -- BMS channel
Sunday, September 13, 2009
![Page 45: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/45.jpg)
Definition of Channels
Polar Codes
W -- BMS channel
Sunday, September 13, 2009
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Definition of Channels
Channel Polarization
Sunday, September 13, 2009
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Definition of Channels
Channel Polarization
Sunday, September 13, 2009
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Definition of Channels
Channel Polarization
Sunday, September 13, 2009
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Definition of Channels
Channel Polarization
Sunday, September 13, 2009
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Channel Polarization
Definition of Channels
Sunday, September 13, 2009
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Channel Polarization
Definition of Channels
N
Sunday, September 13, 2009
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Channel Polarization
Definition of Channels
N
Sunday, September 13, 2009
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Channel Polarization
Definition of Channels
N
Sunday, September 13, 2009
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Channel Polarization
Definition of Channels
N
Sunday, September 13, 2009
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Channel Polarization
Definition of Channels
N
Sunday, September 13, 2009
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Channel Polarization
Definition of Channels
N
Sunday, September 13, 2009
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Channel Polarization
Definition of Channels
Sunday, September 13, 2009
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Definition of Channels
Successive Decoding
Sunday, September 13, 2009
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Definition of Channels
Successive Decoding
Sunday, September 13, 2009
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Definition of Channels
Successive Decoding
Sunday, September 13, 2009
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Definition of Channels
Successive Decoding
Sunday, September 13, 2009
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Definition of Channels
Successive Decoding
Sunday, September 13, 2009
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Definition of Channels
Successive Decoding
Sunday, September 13, 2009
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Definition of Channels
Successive Decoding
Sunday, September 13, 2009
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Definition of Channels
Successive Decoding
Sunday, September 13, 2009
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Definition of Channels
Successive Decoding
Sunday, September 13, 2009
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Definition of Channels
More on Polarization
Sunday, September 13, 2009
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Definition of Channels
More on Polarization
Sunday, September 13, 2009
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Definition of Channels
More on Polarization
Sunday, September 13, 2009
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Definition of Channels
More on Polarization
Sunday, September 13, 2009
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Definition of Channels
More on Polarization
Sunday, September 13, 2009
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Definition of Channels
More on Polarization
N
Sunday, September 13, 2009
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Definition of Channels
More on Polarization
N
Sunday, September 13, 2009
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Definition of Channels
More on Polarization
Sunday, September 13, 2009
![Page 75: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/75.jpg)
Definition of Channels
N
More on Polarization
Sunday, September 13, 2009
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Definition of Channels
N
More on Polarization
Sunday, September 13, 2009
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Definition of Channels
N
More on Polarization
Sunday, September 13, 2009
![Page 78: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/78.jpg)
Definition of Channels
N
More on Polarization
Sunday, September 13, 2009
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Equivalent “Random” Channel
Sunday, September 13, 2009
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Equivalent “Random” Channel
Set B1, B2, ... to be i.i.d. {+, -} valued, uniformly distributed random variables
Sunday, September 13, 2009
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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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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
Sunday, September 13, 2009
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Properties of In
Sunday, September 13, 2009
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Properties of In
I0=I(W) is a constant
Sunday, September 13, 2009
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Properties of In
In ∈ [0, 1]; so In is bounded
I0=I(W) is a constant
Sunday, September 13, 2009
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Properties of In
In ∈ [0, 1]; so In is bounded
I0=I(W) is a constant
Conditional on B1, B2, ..., Bn, and withP= WB1, B2, ..., Bn, In+1 can only take on thetwo values I(P+) and I(P-)
Sunday, September 13, 2009
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Properties of In
In ∈ [0, 1]; so In is bounded
I0=I(W) is a constant
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
Sunday, September 13, 2009
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Properties of In
Sunday, September 13, 2009
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a bounded martingale converges almost surely
Properties of In
Sunday, September 13, 2009
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I∞ = limn→∞ In exists almost surely; E[I∞]=I0=I(W)
a bounded martingale converges almost surely
Properties of In
Sunday, September 13, 2009
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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
Sunday, September 13, 2009
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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 knowthat (I(P+)-I(P-))/2 ≤ε implies that I(P)∉(δ, 1-δ)
Properties of In
(I(P+)- I(P-))/2
I(P)
Sunday, September 13, 2009
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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 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)
Sunday, September 13, 2009
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Summary of Known Results
Sunday, September 13, 2009
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Summary of Known Results
achieve capacity on memoryless channels Arikan 2007
Sunday, September 13, 2009
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Summary of Known Results
Arikan and Telatar 2008
achieve capacity on memoryless channels Arikan 2007
Sunday, September 13, 2009
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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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Definition of Channels
Polar Codes Based on Larger Matrices
Sunday, September 13, 2009
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Definition of Channels
Polar Codes Based on Larger Matrices
Sunday, September 13, 2009
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Definition of Channels
Characterization of Exponent
Sunday, September 13, 2009
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Definition of Channels
Exponent: Example
Sunday, September 13, 2009
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Definition of Channels
Exponent: Example
Sunday, September 13, 2009
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Definition of Channels
Exponent: Example
Sunday, September 13, 2009
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Definition of Channels
Exponent: Example
Sunday, September 13, 2009
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Definition of Channels
Exponent: Example
Sunday, September 13, 2009
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Definition of Channels
Exponent: Example
Sunday, September 13, 2009
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Definition of Channels
Results
Sunday, September 13, 2009
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Definition of Channels
Results
Sunday, September 13, 2009
![Page 109: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/109.jpg)
Definition of Channels
Results
Sunday, September 13, 2009
![Page 110: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/110.jpg)
Definition of Channels
Results
Sunday, September 13, 2009
![Page 111: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/111.jpg)
Definition of Channels
Results
Sunday, September 13, 2009
![Page 112: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/112.jpg)
Summary of Known Results
Arikan and Telatar 2008
Korada, Sasoglu, and U. 2009
achieve capacity on memoryless channels Arikan 2007
Sunday, September 13, 2009
![Page 113: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/113.jpg)
Summary of Known Results
Arikan and Telatar 2008
Korada, Sasoglu, and U. 2009
optimal for lossy source coding, Wyner-Ziv, Gelfand-Pinsker, ...
Korada and U. 2009
achieve capacity on memoryless channels Arikan 2007
Sunday, September 13, 2009
![Page 114: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/114.jpg)
Source Coding
Definition of Channels
Sunday, September 13, 2009
![Page 115: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/115.jpg)
Source Coding
Definition of Channels
Sunday, September 13, 2009
![Page 116: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/116.jpg)
Source Coding
Definition of Channels
Sunday, September 13, 2009
![Page 117: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/117.jpg)
Summary of Known Results
Arikan and Telatar 2008
Korada, Sasoglu, and U. 2009
optimal for lossy source coding, Wyner-Ziv, Gelfand-Pinsker, ...
Korada and U. 2009
achieve capacity on memoryless channels Arikan 2007
Sunday, September 13, 2009
![Page 118: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/118.jpg)
Summary of Known Results
Arikan and Telatar 2008
Korada, Sasoglu, and U. 2009
optimal for lossy source coding, Wyner-Ziv, Gelfand-Pinsker, ...
Korada and U. 2009
Mori and Tanaka 2009efficient construction
achieve capacity on memoryless channels Arikan 2007
Sunday, September 13, 2009
![Page 119: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/119.jpg)
Summary of Known Results
Arikan and Telatar 2008
Korada, Sasoglu, and U. 2009
optimal for lossy source coding, Wyner-Ziv, Gelfand-Pinsker, ...
Korada and U. 2009
suboptimal for compound coding Hassani, Korada and U. 2009
Mori and Tanaka 2009efficient construction
achieve capacity on memoryless channels Arikan 2007
Sunday, September 13, 2009
![Page 120: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/120.jpg)
Summary of Known Results
Arikan and Telatar 2008
Korada, Sasoglu, and U. 2009
optimal for lossy source coding, Wyner-Ziv, Gelfand-Pinsker, ...
Korada and U. 2009
suboptimal for compound coding Hassani, Korada and U. 2009
Mori and Tanaka 2009efficient construction
achieve capacity on memoryless channels Arikan 2007
non-binary version and asym. channels Arikan, Sasoglu, and Telatar 2009
Sunday, September 13, 2009
![Page 121: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/121.jpg)
Summary of Known Results
Arikan and Telatar 2008
Korada, Sasoglu, and U. 2009
optimal for lossy source coding, Wyner-Ziv, Gelfand-Pinsker, ...
Korada and U. 2009
suboptimal for compound coding Hassani, Korada and U. 2009
Mori and Tanaka 2009efficient construction
achieve capacity on memoryless channels Arikan 2007
non-binary version and asym. channels Arikan, Sasoglu, and Telatar 2009
scaling
Sunday, September 13, 2009
![Page 122: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/122.jpg)
Summary
+ many applications
+ completely new paradigm of coding
+ provably achieves capacity
+ low complexity
- currently only competitive for VERY large N
Sunday, September 13, 2009
![Page 123: Polar Codes -- A New Paradigm for Codingcnls.lanl.gov/~jasonj/poa/slides/urbanke.pdfPolar Codes -- A New Paradigm for Coding R. Urbanke, EPFL Physics of Algorithms, Santa Fe, September](https://reader035.fdocuments.in/reader035/viewer/2022062600/5b032b2e7f8b9a0a548bbd4d/html5/thumbnails/123.jpg)
Wyner-Ziv and Gelfand-Pinsker
Sunday, September 13, 2009