`
`Irregular Turbo-Like Codes
`
`Brendan J. Frey
`David J. C. MacKay
`
`1
`
`Hughes, Exh. 1034, p. 1
`
`
`
`Suppose you are told the values of
`some bits" in a transmitted codeword
`(or s~~!S.)
`This effectively decreases the rate of
`the remaining code, making decoding
`eas1er
`
`•
`
`Can we get a free lunch
`out of this?
`
`Hughes, Exh. 1034, p. 2
`
`
`
`Irregular codes:
`A free bite of lunch
`
`Bit or
`State
`Variable
`
`Pinned down SLOWLY
`
`Hughes, Exh. 1034, p. 3
`
`
`
`''Irregularizing'' a turbocode
`
`Regular turbocode R=1 0/20
`
`.. l rregularization'~
`
`Parity bits
`
`0
`
`States. trellis 1
`
`Systematic bits
`
`States, trellis 2
`
`Parity bits 0
`
`Irregular turbocode R=8/1 8
`
`Hughes, Exh. 1034, p. 4
`
`
`
`Rate-degree relations
`
`Trellis representing constituent convolutional codes, average rate R'
`
`. . .
`
`. ..
`
`I
`
`Permuter
`
`I
`I ...
`...
`~
`[,..-L-Re___.__,p 21 I Rep 2 ! ~ I,........._R-ep ,__3 .......__,!
`6 ... 6 ~-·· 6
`
`!
`
`I
`
`I
`...
`... I HO
`I •••
`I Rep D
`I I Rep D
`... 6
`~~~·6
`
`l
`
`l
`
`j
`
`-d = Avg degree of codeword bits
`-R = Avg rate of convolutiona l codes
`R = Rate of irregular turbocode
`
`Avg # constraints per bit: d(l- R)
`
`Hughes, Exh. 1034, p. 5
`
`
`
`Simplified degree profiles
`
`Degree 1
`
`Degree 2
`
`Degree de: Fraction fe "elite" bits
`have degree de
`
`d = ( 1 - R) + 2 ( R -
`
`f e) + def e
`
`1- R-
`-
`
`1-R
`(1-R)+2(R- fe)+defe
`
`Hughes, Exh. 1034, p. 6
`
`
`
`K=65536, R=l/2:
`Optimizing fe with de= 10
`
`0. 02 ....-------,-, -
`
`-,.-------,-,--...,.---
`
`----,
`
`0.015 ~
`
`+
`
`;
`
`a:
`w
`OJ
`
`0.01
`
`0.005
`
`+
`+
`+
`
`...
`
`+
`
`* +
`
`+
`
`+
`+
`+
`
`-
`
`+
`+
`
`*
`
`0 L-----1------1...--~----L....-L - - - - -1
`0.02
`0.04
`0.06
`0.08
`0.1
`0
`Fraction fe of degree 1 0 bits
`
`Hughes, Exh. 1034, p. 7
`
`
`
`K==65536, R==l/2:
`Optimizing de with fe == .05
`
`0.06
`
`0.05 r-
`
`0.04
`
`0.03
`0.02 ..
`
`0.01 ~
`
`a:
`w
`co
`
`+
`
`+
`
`+ ...
`
`I
`
`+
`+
`t
`
`-t·
`.....
`+
`+
`
`-
`
`0
`
`"
`20
`15
`10
`5
`0
`Degree of elite bits making up 5°/o of the codeword bits
`
`0
`
`Hughes, Exh. 1034, p. 8
`
`
`
`K=65536, R=l/2:
`Measured bit error rate
`
`------------!!(
`
`\
`\
`
`\ ' ' \
`\ ' .
`' '
`
`\
`
`' '.
`\
`' \
`\ ' '
`
`1 e-1
`
`1e-2
`
`a:
`w 1e-3
`m
`
`'
`
`'
`
`1e-4
`
`\ ' 'I
`1
`\
`
`I
`I
`
`I
`I
`I
`I
`I
`I
`I
`I
`
`i
`I
`I
`I
`
`I
`
`' I
`
`' ' \
`' I . I
`· ~ \I I,
`
`1e-5
`0.1
`
`0.2
`
`0.3
`
`0.5
`/No (dB)
`
`0.6
`
`0.7
`
`0.8
`
`n
`
`Hughes, Exh. 1034, p. 9
`
`
`
`K==8920, R==l/3, CCSDS:
`Optimizing de and fe
`
`1e-1 I
`
`co
`"0
`'roo
`0
`II
`
`0 z ::c w
`
`+J
`ctS
`a: 1e-2
`w
`m
`
`de=12
`
`1 e-3 L - - -L - -.L.-.-- " - - - - " - - - - - ' - - - - - . . 1 . - - - - - J
`0
`0.01
`0.02
`0.03
`0.04
`0.05
`0.06
`0.07
`
`fe
`
`"If'\
`
`Hughes, Exh. 1034, p. 10
`
`
`
`K==8920, R==l/3, CCSDS:
`Measured bit error rate
`
`1e00 ~ ,
`I
`i
`1e-1 r
`
`a:
`w
`ca
`
`1e-2
`
`1e-3
`
`1e-4
`
`1e-5
`
`Irregular
`
`1 e-6 !.,___1 __ _.__ _
`-0.2
`-0.4
`
`_____.~ _
`___.__ __ __.__ _
`0.2
`0
`0.4
`Eb/No (dB)
`
`_____.
`0.6
`
`, ,
`
`Hughes, Exh. 1034, p. 11
`
`
`
`K==8920, R==l/3, CCSDS :
`Measured word error rate
`
`a:
`w
`~
`
`1e-1
`
`1e-2
`
`1e-3 ~
`I
`
`1e-4
`
`Irregular
`
`.l.
`
`1e-5
`-0.4
`
`-0.2
`
`0.2
`0
`Eb/No (dB}
`
`0.4
`
`0.6
`
`11"j
`
`Hughes, Exh. 1034, p. 12
`
`
`
`Summary
`
`Irregular turbocodes are a good idea!
`
`Gain of 0.23 dB for irregular K=65536,
`R=l/2 Berrou et al turbocode
`
`. . . But, ''i rreg u larization" introduces
`low-weight codewords at rate 1/2
`
`Gain of 0.2 dB for irregular K=8920 ,
`R=l/3 CCSDS turbocode- no weight
`problem
`
`For long block lengths: Use density
`evolution
`
`For short block lengths: Search
`probably better
`
`is
`
`Hughes, Exh. 1034, p. 13
`
`