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Cell 1 Cell 2. t i+1. t i. 0,7. 7,7. 13. (5,4). 41. 0,0. 7,0. Decoder Implementation. Transition during any one time interval can be grouped into disjoint cells Branch Metrics  aa’  ab’  ca’  cb’  bc’  bd’  dc’  dd’ Hamming distance for HD

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decoder implementation

Cell 1 Cell 2

ti+1

ti

0,7

7,7

13

(5,4)

41

0,0

7,0

Decoder Implementation
  • Transition during any one time interval can be grouped into disjoint cells
  • Branch Metrics
    • aa’ab’ ca’ cb’
    • bc’ bd’ dc’dd’
    • Hamming distance for HD
    • Euclidean distance/ ED2 for SD
  • State Metrics
    • a b c d
  • Add-Compare-Select (ACS)
add compare select acs cell 1

c

a

ca’

cb’

aa’

ab’

+

+

+

+

Compare

Compare

ˆ

ˆ

ˆ

ˆ

mc

ma

mc

ma

Select

1-of-2

Select

1-of-2

Select

1-of-2

Select

1-of-2

a’

b’

mb’

ma’

ˆ

ˆ

To another logic unit

To another logic unit

Add Compare Select (ACS)Cell 1
free distance
Free Distance
  • Minimum free distance  minimum distance in the set of all arbitrary long paths that diverge and remerge from the all-zero path.
  • Systematic code free distance < Non-Systematic code free distance
  • Error correcting capability  t = (df -1)/2
coding gain
Coding Gain
  • Soft Decision ML decoding
    • Pe  Q ( df/2 ) ; for moderate to high SNR
  • Asymptotic Coding gain G
    • G (dB) = 20 log10 (d f/ d ref)

or

G (dB) = 10 log10 (d f2/ d ref2)

  • Alternately for high SNR and given error probability
    • G (dB) = (Eb/No)U (dB) - (Eb/No)C (dB)
  • TCM goal  to achieve d f > d ref at same info rate , BW & power
set partitioning example of 8 psk

2

3

A0

1

d0= 2 sin(/8) = 0.765

4

0

1

5

2

7

B0

d1=  2

6

3

B1

1

4

d1=  2

0

5

7

6

2

C0

C1

C2

1

3

C3

4

0

5

7

d2= 2

d2= 2

6

Set Partitioning – Example of 8-PSK
4 state trellis

States

0

0

0

C0 C1

4

4

4

2

2

2

0 4 2 6

6

6

6

C2 C3

2

2

2

6

6

6

0

0

0

4

4

4

1 5 3 7

1

1

1

C1 C0

5

5

5

2 6 0 4

3

3

3

3

3

3

7

7

7

7

7

7

1

1

1

C3 C2

5

5

5

3 7 1 5

4-State Trellis

0,1,2,3,4,5,6,7  Waveform Numbers

coding gain for 8 psk 4 state trellis
Coding gain for 8-PSK 4-State trellis
  • Candidate error event paths
    • 2, 1, 2
        • d2 = d12 + d02 + d12 = 2 + 0.585 + 2 = 4.585
        • d = sqrt( 4.585) = 2.2
    • 4
        • d = 2
  • df min ( 2.2, 2 ) = 2, dref 2
  • G (dB) = 10 log10 (d f2/ d ref2) = 3 dB
coded 8 psk 8 state trellis

State

0426

1537

4062

5173

2604

3715

6240

7351

0 0 0

7 6

Coded 8- PSK 8 - State trellis
  • Coded 8-state 8-PSK
  • d2 = d12 + d02 + d12

= 2 + 0.585 + 2

= 4.585

df = 4.585

dref = 2

G (dB)

= 10 log10 (d f2/ d ref2)

= 3.6 dB

6

16-state trellis  4.1 dB coding gain over uncoded 4-PSK

mapping waveforms to trellis transitions
Mapping Waveforms to Trellis Transitions
  • If k bits are to be encoded per modulation interval, trellis must allow for 2k possible transitions from each state to successor state,
  • More than one transition may occur between pairs of states,
  • All waveforms should occur with equal frequency and with a fair amount of regularity and symmetry,
  • Transitions originating from the same state are assigned waveforms either from subset B0 or B1 – never a mixture between them ,
  • Transitions joining into the same state are assigned waveforms either from subset B0 or B1 – never a mixture between them ,
  • Parallel transitions are assigned waveforms either from subset C0 or C1 or C2 or C3 – never a mixture between them.
slide13

d0 = 2 / 10 = 0.632

d1 = 2 d0

d2 = 2 d1 = 2 d0

d3 = 2 d2

slide14

State

0 4 2 6

1 5 3 7

4 0 6 2

5 1 7 3

2 6 0 4

3 7 1 5

6 2 4 0

7 3 5 1

8-state trellis for 16-QAM