Data communication line codes and constrained sequences

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Data communicationline codes and constrained
sequences

• A.J. Han Vinck
• Revised December 09, 2010

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A-synchronous arrivals Line codes

• Binary data transmitted using special signal form
• Important aspects
• Enough transitions in signal to gain timing
  information
• Constraints to match sequence to the physical
  channel
• Applications
• Optical transmission
• Magnetic and Optical recording
• Classical line transmission

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Run Length Limited sequences (RLL)

• maximum run of same symbols
• changes in sequence used to synchronize
• minimum run of same symbols
• used to improve transmision efficiency
• Example ( minimum 2, maximum 5 )
• 00111000001111001100111

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Line codes for clock recovery Manchester code
1 0 1 1 0 0 0 0
1 0 1
5V 0
Basic frequency
1 0 1 1
0 0
5V 0 -5V

• Longest run 2, but time to transmit twice as
  large! Efficiency ½

Detection is simple Transition 1 ? 0 gt 1
0 ? 1 gt 0
A.J. Han Vinck
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Line codes for clock recovery Manchester code
1 0 1 1 0 0 0 0
1 0 1
5V 0
Basic frequency
5V 0 -5V

• Longest run 2

Same time to transmit, but basic frequency
factor of 2 higher!
Detection is simple Transition 1 ? 0 gt 1
0 ? 1 gt 0
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4B/5B mapping codes
4B/5B guarantees at least one transition per
block to allow the clock signal to be recovered.
Efficiency decreased by 25. 4B5B is used in
the following standard 100BASE-TX standard
defined by IEEE 802.3u
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(d,k) sequences

• Definition between 2 ones maximum of k
  zeros minimum of d zeros
• ex 0100010010001101
• Conversion from (d,k) into RLL
  0100010010001101
• ?1000011100001001
• or
• ?0111100011110110
• the ones indicate the position of a transition
  in the RLL sequence
• Note, from a (d,k) sequence we can derive 2 RLL
  sequences depending on the starting state (0 or
  1)

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Conclusion from (d,k) to RLL

• RLL sequences have minimum run of d1 and maximum
  of k1
• ex the Manchester code has (d,k) (0,1)
• but efficiency ½, i.e. sequence is twice as
  long
• efficiency can be improved to 0.69, i.e. about
  1.4 times as long!
• Problems generation of (d,k) sequences
• how efficient can this be done

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Shannons way out

• Calculate N(n),
• the maximum number of sequences with constraint
  and length n,
• The capacity is then defined as
• C maximum of information bits/channel use

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Calculations of maximum rate C(d,k)

• the following Markov source generates all (d,k)
  sequences
• Very important for practical applications how
  far is a construction away from optimal?

0 0 0
0 0
0 1 2 d
d1 k
1 1 1
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Ex Modified Frequency Modulation MFM (d,k)
(1,3)

• Use markov state diagram to describe sequence
  generation

1/01 0/10 1/01
A 0/00 B
Decoding rule 0 x0 1 01
Efficiency ½ Maximum C(1,3) 0.55
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Example (MFM)
0 0 0
S1
S2
S3
S4
1 1 1
sequences in state S1 at time n sequences
in state S1 at time n-2 sequences in
state S1 at time n-3 sequences in state
S1 at time n-4 A solution can be of the form
sequences in state S1 at time n ? cZn Then, we
obtain the relation Zn Zn-2 Zn-3 Zn-4
and For a solution ? we calculate C(d,k) gt 1/n
log2 c ?n gt log2? 0.55
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Example (Manchester (0,1)
0
S1
S2
1
1
sequences in state S1 at time n
sequences in state S1 at time n-2
sequences in state S1 at time n-1
sequences in state S1 at time n ? cZn.
Then, Zn Zn-1 Zn-2 For a solution ? we
calculate C(d,k) gt 1/n log2 c ?n gt log2?
0.69
A.J. Han Vinck
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Eight to Fourteen Modulation (EFM)
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Ex CD (EFM) uses d2, k10
Note DvD uses EFM
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in recording and transmissionkeep basic basic
frequency or minimum symbol length!

• no constraint L bits take time ? L/1

1 0
1 0
d 1
d 2
1 0
A.J. Han Vinck
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Why is it used in recording?

• no constraint L bits take time ? L/1

1 0

• with constraint L bits take time ? L/C(d,k),
  where C(d,k) lt 1
• i.e. it takes more symbols with the contraint
  included. The best performance is indicated by
  C(d,k)

1 0

• same spacing (d1) units of time ? ?/(d1)

SURPRISE ?L/C(d,k) ? L/(d1) C(d,k) lt ? L
for (d1) C(d,k) gt 1
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Example d 1

• 00 00 or 11 2 d 1
• 01 000 or 111 3
• 10 0000 or 1111 4
• 11 00000 or 11111 5 R 2/3.5 1/1.75 (
  less than C(d,k) )

• With constraint L bits take
• ?L/R ? L/(d1) R 0.875 ? L lt ? L

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Example (contd)
Example 4 symbols 00,01,10,11 encode
sequence 00, 01, 10, 00, 00... as
00, 111, 0000, 11, 00,...,
where after every symbol we change polarity.
The code is uniquely decodable!
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What happened? the symbol duration changed !!
Thus, we can pack more sequences in the same
time.

• no constraint (11,01,10,00)

1 0

• with constraint

1 0

• same minimum spacing, but different run lengths

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Main idea (like in coded modulation, Ungerböck)
uncoded
RLL- coded d 1
?
0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 1
0 0 0 0 0 0 1 1
0 0 1 0
0 0 0 0 1 1 0 0
2k
???
???
gt 2k
1 1 1 1
1 1 1 1 1 1 1 1
0 1 1 1 0 0 1 1
T
0 0 0 0 1 1 1 0
0 0 0 1 1 1 0 0
T
- Same minimum transition time ?, same
information rate k/T 1/?
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Table for the efficiency of (d,k) codes

• d0 d1 d2 d3
• k
• 1 .69
• 2 .88 .41
• 3 .95 .55 .29
• 4 .98 .62 .41 .22
• 5 .99 .65 .46 .32

A.J. Han Vinck
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IBM disk code (d,k) (2,7)
(d,k) RLL

• 10 ? 1000 0000 or 1111
• 11 ? 0100 0111 or 1000
• 011 ? 000100 000111 or 111000
• 010 ? 001000 001111 or 110000
• 000 ? 100100 111000 or 000111
• 0011 ? 00100100 00111000 or 11000111
• 0010 ? 00001000 00001111 or 11110000

Homework show that this code does not violate
the constraint!
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Contd
Remark - for every L information bits, we
produce exactly 2L channel digits Thus, the
efficiency ½ (Maximum 0.51 ) - Sequence
uniquely decodable!
Instead of using L positions to store L bits of
information we need only L/ R(d1) L/1.5
positions or 1/3 less!
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Efficient encoding example d1, k?

• 000 00000 R 3/5 0.6 (Maximum 0.69)
• 001 00001
• 010 00010
• 011 00100 R(d1) 1.2
• 100 00101
• 101 01000
• 110 01001
• 111 01010

• Problem
• What is maximum rate?
• How to encode?

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problem

• For large d
• exact timing
• detection of transition
• erroneous shift of transition is called peak
  shift
• (in recording)

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Other constrained codes
Equal weight codes ones is the same for every
code word example 0011, 0101, 0110, 1001,
1010, 1100 DC-balanced try to balance the
number of ones and zeros example transmit 000
or 111, 010 or 101 001 or 110, 100 or 011
only 1-bit redundancy, for every length Pulse
Position modulation weight 1 code
words example 001, 010, 100