`
`Brendan J. Frey
`
`David J. C. MacKay
`
`Apple 1013
`
`1
`
`
`
`Suppose you are told the values of
`some bits"in a transmitted codeword
`(or s-in-I-es)
`
`This effectively decreases the rate of
`the remaining code, making decoding
`easier
`
`Can we get a free lunch
`
`out of this?
`
`2
`
`
`
`Irregular codes:
`
`A free bite of lunch
`
`Variable
`
`Pinned down SLOWLY
`
`L
`
`i Bit or
`State
`Variable
`
`Pinned down QU|CKLYi
`
`3
`
`
`
`“IrreguIarizing" a turbocode
`
`Regular turbocode R=10/20
`
`”'lrregularization"
`
`4
`
`
`
`Rate-deg ree relations
`
`Trellis representing constituent convolutional codes, average rate R’
`
`1
`
`I
`
`I
`
`E
`
`H
`
`1". I
`§~
`3
`E
`1
`A’!
`W
`
`3
`
`;
`
`I
`
`9i
`
`E
`
`i
`
`I
`
`E
`
`*1
`
`g
`
`1l
`
`. a
`.
`
`J)
`
`<5
`
`t Rep2|
`
`C5
`
`.
`
`Rep3
`
`E
`
`’
`
`H9133
`
`(5
`
`i
`
`I
`
`f3
`
`'
`
`‘s
`
`‘ Rep D
`
`-90
`
`3..
`
`i
`
`fa
`
`sun.
`
`I!
`*
`
`3 Rep .0
`
`g
`
`(5
`
`‘
`2
`«
`
`9
`
`= Avg degree of codeword bits
`
`R = Avg rate of convolutional codes
`
`R 2 Rate of irregular turbocode
`
`Avg # constraints per bit: J(1 -- R)
`
`'1 —R=J(1—R)!
`
`5
`
`
`
`Simplified degree profiles
`
`Degree 1
`
`Degree 2
`
`Degree de: Fraction fe “elite” bts
`have degree de
`
`1-R
`_
`— (1"R)+2(R"fe)+defe
`
`6
`
`
`
`K265:-336, R=1/2:
`
`Optimizing fe with de = 10
`
`0.08
`0.06
`0.04
`0.02
`Fraction fe of degree 10 bits
`
`7
`
`
`
`K=65536, R=1/2:
`
`Optimizing de with f6 = .
`
`0.06
`
`0.05
`
`0.04
`
`0.03
`
`0.02
`
`0.01
`
`0
`
`20
`15
`10
`5
`O
`Degree of elite bits making up 5% of the codeword bits
`
`8
`
`
`
`K=65536, R=1/2:
`
`Measured bit error rate
`
`1e-1
`
`9
`
`
`
`K:-8920, R:-1/3, CCSDS:
`
`Optimizing ale and fa
`
`ii
`
`on
`‘U
`*1
`c:
`u
`_o
`ED
`-’
`(U
`
`M1
`
`0:
`u:
`an
`
`10
`
`
`
`K=8920, R=1/3, CCSDS:
`
`Measured bit error rate
`
`1e00
`
`o
`
`0.2
`
`0.4
`
`Eb/No (dB)
`
`63,000 ‘Hooks s‘uMu\au-I-ac‘
`
`11
`
`
`
`K:-8920, R21/3, CCSDS:
`
`Measured word error rate
`
`1e0Og
`
`S9
`
`1e1
`
`r
`"
`1e-2 I
`
`1e-3
`
`1e-4 I
`i
`‘ge_5E
`
`-DA»
`
`412
`
`0
`
`0
`
`3%»
`
`{L6
`
`Eb/No (us)
`
`12
`
`
`
`Summary
`
`Irregular turbocodes are a good idea!
`
`Gain of 0.23 dB for irregular K=65536,
`R=1/2 Berrou et al turbocode
`
`But, “irregularization” introduces
`low-weight codewords at rate 1/2
`
`Gain of 0.2 dB for irregular K:8920,
`R=1/3 CCSDS turbocode - no weight
`problem
`
`For long block lengths: Use density
`evolution
`
`For short block lengths: Search is
`probably better
`
`13