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Set Partitioning Example of 8-PSK

6

A0

1 0

1

2

3

4

5 7

d0

= 2 sin(/8) = 0.765

B1

13

5 7

d1= 2B0 0

2

6

4

d1= 2

C2

1

5d2= 2

C3

3

7

C0 04 C1

2

6

d2= 2

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4-State Trellis

0

42

6

2 6 04

1

5

3

7

37 1

5

0

4

2

6

2 6 04

1

5

37

37 1

5

0

4

2

6

2 6 04

1

5

37

37 1

5

StatesC0 C1

C2 C3

C1 C0

C3 C2

0 4 2 6

1 5 3 7

2 6 0 4

3 7 1 5

0,1,2,3,4,5,6,7 Waveform Numbers

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Free Distance

Minimum free distance

minimum distance in the set ofall 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

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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 ref

2)

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

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Coding gain for 8-PSK 4-State trellis

Candidate error event paths 2, 1, 2

d2 = d12 + d0

2 + d12 = 2 + 0.585 + 2 = 4.585

d = sqrt( 4.585) = 2.2

4 d = 2

df min ( 2.2, 2 ) = 2, d

ref 2

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

2) = 3 dB

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Coded 8- PSK 8 - State trellisState

0426

1537

4062

5173

2604

3715

6240

7351

0 0 0

7 6

6Coded 8-state 8-PSK

d2 = d12 + d0

2 + d12

= 2 + 0.585 + 2

= 4.585

df = 4.585

dref = 2

G (dB)= 10 log10 (d f

2/ d ref2)

= 3.6 dB

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

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Waveform encoding

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Decoder Implementation

Transition during any one time interval can be grouped into

disjoint cells

Branch Metrics

aaab cacb bcbd dcdd Hamming distance for HD

Euclidean distance/ ED2 for SD

State Metrics a b c d

Add-Compare-Select (ACS)

ti ti+1

Cell 1 Cell 2

0,0 7,0

0,7 7,7

41

13

(5,4)

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Add Compare Select (ACS)Cell 1

Select1-of-2

a

Select1-of-2

ma

ma mc

Compare

+aa

a

+ca

c

Select1-of-2

b

Select1-of-2

mb

ma mc

Compare

+ab +cb

To another logic unit To another logic unit

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d0 = 2 / 10 = 0 .632d1 = 2 d0

d2 = 2 d1 = 2 d0d3 = 2 d2

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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

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32 QAM Partioning

32point

16point

16point

8

point

8

point

8

point

8

point

4point

4point

4point

4point

4point

4point

4point

4point

2 inputs define oneof 8 states8 state binary coderidentifies one of the

8 subsets

2 other inputsidentify one of4 points in the

subset

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