Polar Codes

174 Pages·2019·5.142 MB·English

by  Orhan Gazi

Springer Topics in Signal Processing
Orhan Gazi
Polar Codes
A Non-Trivial Approach
to Channel Coding
Springer Topics in Signal Processing
Volume 15
Series editors
Jacob Benesty, Montreal, Canada
Walter Kellermann, Erlangen, Germany
More information about this series at http://www.springer.com/series/8109
Orhan Gazi
Polar Codes
A Non-Trivial Approach to Channel Coding
123
Orhan Gazi
Department ofElectronic and
Communication Engineering
Çankaya University
Etimesgut/Ankara
Turkey
ISSN 1866-2609 ISSN 1866-2617 (electronic)
SpringerTopics inSignal Processing
ISBN978-981-13-0736-2 ISBN978-981-13-0737-9 (eBook)
https://doi.org/10.1007/978-981-13-0737-9
LibraryofCongressControlNumber:2018943249
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Contents
1 Information Theory Perspective of Polar Codes and Polar
Encoding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Information Theory Perspective of Polar Codes . . . . . . . . . . . . . . 1
1.1.1 The Philosophy of Polar Codes . . . . . . . . . . . . . . . . . . . . 5
1.1.2 Fundamental Idea of Polar Codes . . . . . . . . . . . . . . . . . . . 7
1.2 Polar Encoding Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Recursive Construction of Polar Encoder Structures . . . . . . . . . . . 11
1.4 Generator Matrix and Encoding Formula for Polar Codes. . . . . . . 14
2 Decoding of Polar Codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1 Kernel Encoder and Decoder Units of the Polar Codes. . . . . . . . . 27
2.2 Decoding Tree for the Successive Cancelation Decoding
of Polar Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 Level Indices and Determination of Levels for Bit Distribution. . . 46
2.4 Decoding Algorithm for Polar Codes. . . . . . . . . . . . . . . . . . . . . . 47
2.5 Binary Erasure Channel (BEC) . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.6 Determination of Frozen Bit Locations for BEC Channels . . . . . . 58
2.7 Decoding Operation with Frozen Bits . . . . . . . . . . . . . . . . . . . . . 64
3 Polarization of Binary Erasure Channels . . . . . . . . . . . . . . . . . . . . . 75
3.1 Polar Codes in Binary Erasure Channels . . . . . . . . . . . . . . . . . . . 75
3.1.1 Split Channels and Capacity of Split Channels
When BECs are Employed. . . . . . . . . . . . . . . . . . . . . . . . 76
3.2 Capacity of Split Channels for N = 4. . . . . . . . . . . . . . . . . . . . . . 78
3.3 Bhattacharyya Parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.4 Capacity Bound Using Bhattacharyya Parameter . . . . . . . . . . . . . 84
3.5 Inequalities for the Mutual Information and Bhattacharyya
Parameters of the Split Channels. . . . . . . . . . . . . . . . . . . . . . . . . 86
3.6 Split Binary Erasure Channels. . . . . . . . . . . . . . . . . . . . . . . . . . . 87
v
vi Contents
4 Mathematical Modelling of Polar Codes, Channel Combining
and Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.1 Channel Combining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.1.1 Probability Background Information . . . . . . . . . . . . . . . . . 92
4.1.2 Recursive Construction of Polar Encoder Structures. . . . . . 97
4.2 Channel Splitting and Decoding of Polar Codes. . . . . . . . . . . . . . 103
4.3 Mathematical Description of Successive Cancellation
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.4 Recursive Computation of the Successive Cancellation
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.4.1 Recursive Channel Transformation . . . . . . . . . . . . . . . . . . 115
4.4.2 Butterfly Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.5 Recursive Calculation of Conditional Channel Input
Probabilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.6 Recursive Calculation of Likelihood Ratio. . . . . . . . . . . . . . . . . . 130
4.6.1 Bit Value Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5 Polarization Rate and Performance of Polar Codes . . . . . . . . . . . . . 135
5.1 r-Field or r-Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.1.1 Borel Field or Borel r-Field. . . . . . . . . . . . . . . . . . . . . . . 136
5.1.2 Measure Function and Measure Space . . . . . . . . . . . . . . . 137
5.1.3 Probability Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.2 Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.3 Random or Stochastic Process. . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.3.1 Cylinder Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.3.2 Measurable Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.3.3 Adopted Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.3.4 Martingale Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.3.5 Super-Martingale Process. . . . . . . . . . . . . . . . . . . . . . . . . 146
5.3.6 Sub-Martingale Process . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.3.7 Martingale Process Convergence Theorem . . . . . . . . . . . . 146
5.4 Channel Polarization Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.5 Performance Analysis of Polar Codes . . . . . . . . . . . . . . . . . . . . . 158
References.... .... .... .... ..... .... .... .... .... .... ..... .... 167
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 169
Chapter 1
Information Theory Perspective of Polar
Codes and Polar Encoding
Polar codes are one of the recently discovered capacity achieving channel codes.
Whatmakesthepolarcodesdifferentfromotherchannelcodesisthatpolarcodes
aredesignedmathematicallyandtheirperformancearemathematicallyproven.On
the other hand, the previous channel codes are constructed in a trivial manner. In
thischapterwefirstprovidethefundamentalconceptsfromtheinformationtheory,
suchasentropy,mutualinformation,andchannelcapacity,thendiscussthephiloso-
phybehindtheideaofpolarencodingconsideringthefundamentalssubjectsofthe
informationtheory.
Afterexplainingthephilosophyofthepolarencoding,weinspectthepolarencod-
ingtechniqueprosedbyArıkan[1].Polarencodingoperationcanbeperformedina
recursivemanner.Forthispurpose,therearetwotypesofencoderstructuresdefined
in[1].Afterexplainingtherecursiveencodingmethods,weprovidethemathematical
formulasforthepolarencodingoperation.
1.1 InformationTheoryPerspectiveofPolarCodes
(cid:2) (cid:3)
˜ ˜
ForthediscreterandomvariableU,theentropy H U isdefinedas
(cid:2) (cid:3) (cid:4)
H U˜ (cid:2)− p(u)logp(u) (1.1)
u
(cid:2) (cid:3)
where p(u)istheprobabilitymassfunctionof H U˜ ,andlog(·)isthelogarithmic
˜ ˜
functionwithbase2.ThejointentropyoftwodiscreterandomvariablesU andY is
definedas
(cid:2) (cid:3) (cid:4)
H U˜,Y˜ (cid:2)− p(u,y)logp(u,y) (1.2)
u,y
©SpringerNatureSingaporePteLtd.2019 1
O.Gazi,PolarCodes,SpringerTopicsinSignalProcessing15,
https://doi.org/10.1007/978-981-13-0737-9_1
2 1 InformationTheoryPerspectiveofPolarCodesandPolarEncoding
where p(u,y)isthejointprobabilitymassfunctionofU˜ andY˜.
˜ ˜
ThemutualinformationbetweentwodiscreterandomvariableU andY isdefined
as
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:5) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:5) (cid:3)
(cid:5) (cid:5)
I U˜;Y˜ (cid:2) H U˜ −H U˜(cid:5)Y˜ oras I U˜;Y˜ (cid:2) H Y˜ −H Y˜(cid:5)U˜ (1.3)
whichcanbeexpressedintermsofthejointandmarginalprobabilitymassfunctions
as
I(cid:2)U˜;Y˜(cid:3)(cid:2)(cid:4)p(u,y)log p(u,y) . (1.4)
p(u)p(y)
u,y
Theexpressionin(1.4)isnothingbutaprobabilisticaveragequantityandcanbe
writteninshortas
⎧ (cid:2) (cid:3) ⎫
(cid:2) (cid:3) ⎨ p U˜,Y˜ ⎬
I U˜;Y˜ (cid:2) E log (cid:2) (cid:3) (cid:2) (cid:3) . (1.5)
⎩ ˜ ˜ ⎭
p U p Y
Note:
(cid:2) (cid:2) (cid:3)(cid:3) (cid:4)
E g U˜,Y˜ (cid:2) p(u,y)g(u,y).
u,y
ThemutualinformationbetweenU˜ and(Y˜,Z˜)isdefinedas
I(cid:2)U˜;Y˜,Z˜(cid:3)(cid:2) (cid:4) p(u,y,z)log p(u,y,z) (1.6)
p(u)p(y,z)
u,y,z
whichcanbewrittenas
⎧ (cid:2) (cid:3) ⎫
(cid:2) (cid:3) ⎨ p U˜,Y˜,Z˜ ⎬
I U˜;Y˜,Z˜ (cid:2) E log (cid:2) (cid:3) (cid:2) (cid:3) . (1.7)
⎩ p U˜ p Y˜,Z˜ ⎭
(cid:2) (cid:3)
Inasimilarmanner,wecandefine I U˜ ,U˜ Y˜ ,Y˜ as
1 1 1 2
I(cid:2)U˜ ,U˜ ;Y˜ ,Y˜ (cid:3)(cid:2) (cid:4) p(u ,u ,y ,y )log p(u1,u2,y1,y2) (1.8)
1 2 1 2 1 2 1 2 p(u ,u )p(y ,y )
u1,u2,y1,y2 1 2 1 2
Ingeneral,fortworandomvariablevectors
(cid:12) (cid:13) (cid:12) (cid:13)
U¯ (cid:2) U˜ ,U˜ ,...U˜ Y (cid:2) Y˜ ,Y˜ ,...Y˜ (1.9)
1 2 N 1 2 N
1.1 InformationTheoryPerspectiveofPolarCodes 3
(cid:14) (cid:15)
¯ ¯
themutualinformation I U;Y isdefinedas
I(cid:14)U¯;Y¯(cid:15)(cid:2)(cid:4)p(u¯,y¯)log p(u¯,y¯) . (1.10)
p(u¯)p(y¯)
u¯,y¯
Theorem1.1 Themutualinformation
(cid:2) (cid:3)
I U˜;Y˜,Z˜
canbewrittenas
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:5) (cid:3)
(cid:5)
I U˜;Y˜,Z˜ (cid:2) I U˜;Y˜ +I U˜;Z˜(cid:5)Y˜ . (1.11)
(cid:2) (cid:3)
Proof1.1 Fromdefinition(1.5),wecanwrite I U˜;Y˜,Z˜ as
I(cid:2)U˜;Y˜,Z˜(cid:3)(cid:2) (cid:4) p(u,y,z)log p(u,y,z) (1.12)
p(u)p(y,z)
u,y,z
where
p(u,y,z)
p(u)p(y,z)
canbewrittenas
p(u,y,z) p(u,z|y)p(y)
(cid:2)
p(u)p(y,z) p(u)p(z|y)p(y)
inwhichmultiplyingthenumeratoranddenominatoroftherightsideby p(u,y)we
get
p(u,y,z) p(u,y) p(u,z|y)p(y)
(cid:2)
p(u)p(y,z) p(u,y) p(u)p(z|y)p(y)
wheretherighthandsidecanberearrangedas
p(u,y) p(u,z|y)p(y)
p(u)p(y) p(z|y)p(u,y)
inwhichusing p(u,y)(cid:2) p(u|y)p(y),weget
p(u,y) p(u,z|y)p(y)
p(u)p(y) p(z|y)p(u|y)p(y)
whichissimplifiedas