Lecture2 - Face Research · Lecture2.ppt Author: Ben Jones Created Date: 10/18/2011 2:51:33 PM ...
2.lecture2
-
Upload
mailstonaik -
Category
Documents
-
view
219 -
download
0
Transcript of 2.lecture2
-
8/13/2019 2.lecture2
1/25
CODING THEORYCODING THEORY
A Birds Eye View : ContinuedA Birds Eye View : Continued
Block Codes: BasicsBlock Codes: Basics
-
8/13/2019 2.lecture2
2/25
-
8/13/2019 2.lecture2
3/25
-
8/13/2019 2.lecture2
4/254
Types of Channel CodesTypes of Channel Codes
lock Codeslock Codes ( Codes with( Codes with strong algebraic flavorstrong algebraic flavor))~1950~1950------ Hamming CodeHamming Code ((Single error correctionSingle error correction))All codes in 50s were too weak compared to theAll codes in 50s were too weak compared to thecodes promised by Shannoncodes promised by Shannon
Major Breakthrough..1960Major Breakthrough..1960
BCH CodesBCH Codes
ReedReed--Solomon CodesSolomon Codes
Capable of correctingCapable of correctingMultiple ErrorsMultiple Errors
-
8/13/2019 2.lecture2
5/25
5
-
8/13/2019 2.lecture2
6/25
6
-
8/13/2019 2.lecture2
7/25
7
-
8/13/2019 2.lecture2
8/25
8
ConvolutionalConvolutional CodesCodes
Codes withCodes with Probabilistic flavor.Probabilistic flavor.
Late 1950s but gained popularity after theLate 1950s but gained popularity after theintroduction ofintroduction of Viterbi algorithmViterbi algorithm in 1967.in 1967.
Developed from the idea of sequential decodingDeveloped from the idea of sequential decoding NonNon--block codesblock codes
Codes are generated by a convolution operation onCodes are generated by a convolution operation onthe information sequencethe information sequence
-
8/13/2019 2.lecture2
9/25
-
8/13/2019 2.lecture2
10/25
10
-
8/13/2019 2.lecture2
11/25
11
Coding Schemes: TrendCoding Schemes: Trend
Since 1970s the two avenues of research startedSince 1970s the two avenues of research started
working togetherworking together This resulted in the development towards theThis resulted in the development towards the
codes promised by Shannoncodes promised by Shannon TodayToday Turbo Codes,Turbo Codes, are capable of achievingare capable of achieving
an improvement close to Shannon Limitan improvement close to Shannon Limit
-
8/13/2019 2.lecture2
12/25
12
Coding SchemesCoding Schemes
Applications demand for wide range of dataApplications demand for wide range of data
rates, block sizes, error rates.rates, block sizes, error rates. No single error protection scheme works for allNo single error protection scheme works for all
applications.applications. Some requires the use of multipleSome requires the use of multiple
coding techniques.coding techniques.
A common combination uses anA common combination uses an innerinner
convolutional codeconvolutional codeand anand an outerouterReedReed--Solomon code.Solomon code.
-
8/13/2019 2.lecture2
13/25
13
-
8/13/2019 2.lecture2
14/25
14
-
8/13/2019 2.lecture2
15/25
15
-
8/13/2019 2.lecture2
16/25
16Slide 3
-
8/13/2019 2.lecture2
17/25
17
-
8/13/2019 2.lecture2
18/25
18
-
8/13/2019 2.lecture2
19/25
19
-
8/13/2019 2.lecture2
20/25
-
8/13/2019 2.lecture2
21/25
21
-
8/13/2019 2.lecture2
22/25
22
Block codes: basic deBlock codes: basic defifinitionsnitions AnAn alphabetalphabet is a discrete (usuallyis a discrete (usually fifinite) set ofnite) set of
symbolssymbols..
example:example: BB == {{ 00;; 11}} is the binary alphabetis the binary alphabet
DeDefifinitionnition: A: A block codeblock code ofof blocklengthblocklength nn over anover an
alphabetalphabet XX is a nonemptyis a nonempty set ofset of nn--tuplestuples ofofsymbols fromsymbols from XX..
TheThe nn--tuplestuples of the code are calledof the code are called code wordscode words..
Code words areCode words are vectorsvectors whose components arewhose components aresymbols insymbols in XX..
-
8/13/2019 2.lecture2
23/25
23
Block codes: basic deBlock codes: basic defifinitionsnitions Code words of lengthCode words of length nn are typically generatedare typically generated
by encoding messages ofby encoding messages of kk informationinformation bitsbitsusing an invertible encoding function.using an invertible encoding function.
Number ofNumber of codewordscodewords isis MM == 22kk ,, RateRate RR == k/nk/n
The rate is a dimensionless fractionThe rate is a dimensionless fraction;; the fractionthe fractionof transmitted symbols thatof transmitted symbols that carry information.carry information.
A code withA code with blocklengthblocklength nn and rateand rate kk//nn is calledis calledanan ((n; kn; k)) cocodede
-
8/13/2019 2.lecture2
24/25
24
Systematic encoderSystematic encoder The error protection ability of a block codeThe error protection ability of a block code
depends only on thedepends only on the setset ofof codewordscodewords, not on, not onthe mapping from source messages tothe mapping from source messages tocodewordscodewords..
An encoder isAn encoder is systematicsystematic when it copies thewhen it copies the kkmessage symbols unchanged into themessage symbols unchanged into thecodeword.codeword. CodewordsCodewords are of the formare of the form
cc = [= [mm pp ]] oror cc = [= [ p mp m]]wherewhere mm is the vector ofis the vector of kk message symbolsmessage symbolsandand pp is the vector ofis the vector of nn--kk redundantredundant oror checkcheck
symbols.symbols.
-
8/13/2019 2.lecture2
25/25
25
Linear Block CodesLinear Block Codes
matrixGeneratorG
(vector)wordmessagem
(vector)wordcode
,
=
c
where
Gmc