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Trellis Coded Modulation

© Tallal Elshabrawy

Trellis Coded Modulation: Introduction Increases the constellation size compared to uncoded

communication Increases in throughput (b/s/Hz) Decline in BER performance due to decrease of dmin

Trellis Coded Modulation (TCM) is used to offset loss resulting from constellation size increase

TCM achieves this higher gain by jointly using the distance properties of the code and the distance properties of the constellation, by carefully mapping coded and uncoded bits to the constellation points.

TCM uses “set partitioning” to map the bits to the constellation points

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Ungerboek Set Partitioning Ungerboeck Set partitioning:

Partition a constellation such that in each partition the minimum distance increases.

For binary data, in each stage we partition the constellation (signal set) into two subsets.

The number of stages depends on the particular TCM scheme.

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Ungerboek Partitioning of 8-PSK Constellation

A0

B0 B1

C0 C1 C2C3

( )8πsinE2=d SA

0123

45 6 7

0

2

4

6

13

5 7

04

2

6

3

7

1

5

SB E2=d

SE

SC E2=d

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Convolutional Codes(Reminder)

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IntroductionBlock Codes: Code words are produced on a block by block basis.

In Block Codes, the encoder must buffer an entire block before generating the associated codeword.

Some applications have bits arrive serially rather than in large blocks

Convolutional codes operate on the incoming message sequence continuously in a serial manner

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Convolutional Codes SpecificationA convolutional code is specified by three parameters

(n, k, K), where k/n is the coding rate and determines the number of

data bits per coded bit K is called the constraint length of the encoder where

the encoder has K-1 memory elements

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Convolutional Encoder: Example

0 0

+

+

Input Output

Rate ½ Convolutional Encoder

1 0 1

1

1

c1

c2

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Convolutional Encoder: Example

1 0

+

+

Input Output

Rate ½ Convolutional Encoder

1 0 1

1

0

c1

c2

1 1

© Tallal Elshabrawy

Convolutional Encoder: Example

0 1

+

+

Input Output

Rate ½ Convolutional Encoder

1 0 1

0

0

c1

c2

1 0 1 1

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Convolutional Encoder: Example

1 0

+

+

Input Output

Rate ½ Convolutional Encoder

1 0 1c1

c2

0 0 1 0 1 1

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+

+

Input Output

b0 b1

States (b0b1)s0 00s1 10s2 01s3 11

S0

S3

S1S2

0/00

1/11

1/00

0/011/10

1/01

0/10

0/11

Input 0Input 1

State Diagram Representation

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s0 (0 0)

s1 (1 0)

s2 (0 1)

s3 (1 1)

00 00 00

11 11 11

01

10 10

10

01 01

1100

00 00 00

11 11 11

01 01 01

10 10 10

10 10 10

01 01 01

11 11 1100 00 00

S0

S3

S1S2

0/00

1/11

1/00

0/011/10

1/01

0/10

0/11Trellis Representation

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s0 (0 0)

s1 (1 0)

s2 (0 1)

s3 (1 1)

00 00 00 00 00 00

11 11 11 11 11 11

01 01 01 01

10 10 10 10 10

10 10 10 10

01 01 01 01 01

11 11 11 11

Input: 101 Output:

S0

S3

S1S2

0/00

1/11

1/00

0/011/10

1/01

0/10

0/11

00 00 00 00

001011

Trellis Representation

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Trellis Representation of QPSK

01

2 3SQPSK E2=d

SE

Trellis Representation01

23

01

23

01

23

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Trellis Representation of QPSKDmin for 3 Consecutive Symbols 0 0 0

01

23

01

23

01

23

01

23

01

23

01

23

SE

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Trellis Representation of QPSKDmin for 3 Consecutive Symbols 0 0 0

01

23

01

23

01

23

01

23

01

23

01

23

SE2

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Trellis Representation of QPSKDmin for 3 Consecutive Symbols 0 0 0

01

23

01

23

01

23

01

23

01

23

01

23

0 0 0

SE3

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Trellis Representation of QPSKDmin for 3 Consecutive Symbols 0 0 0

01

23

01

23

01

23

01

23

01

23

01

23

SE

0 0 0

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Trellis Representation of QPSKDmin for 3 Consecutive Symbols 0 0 0

01

23

01

23

01

23

01

23

01

23

01

23

SE2

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Trellis Representation of QPSKDmin for 3 Consecutive Symbols 0 0 0

01

23

01

23

01

23

01

23

01

23

01

23

SE2

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Trellis Representation of QPSKDmin for 3 Consecutive Symbols 0 0 0

01

23

01

23

01

23

01

23

01

23

01

23

0 0 00 0 1SE2

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Trellis Representation of QPSK Summary)dmin for 3 Consecutive Symbols 0 0 0

01

23

01

23

01

23

01

23

01

23

01

23

01

2 3

=QPSK Sd 2E

SE

= + + =2 2min QPSK S Sd 0 0 2E 2E

S0

S0

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Four State Trellis with Parallel Paths0 0 04 4 4

26

26

26

2 6 2 6 2 604

04

04

15

15

15

15

15

15

3 73 7

3 7

3 7 3 7 3 7

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Four State Trellis with Parallel Paths

0 0 04 4 4

26

26

26

2 6 2 6 2 604

04

04

15

15

15

15

15

15

3 73 7

3 7

3 7 3 7 3 7

Dmin for 3 Consecutive Symbols 0 0 0

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Four State Trellis with Parallel Paths

0 0 04 4 4

26

26

26

2 6 2 6 2 604

04

04

15

15

15

15

15

15

3 73 7

3 7

3 7 3 7 3 7

Dmin for 3 Consecutive Symbols 0 0 0

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Four State Trellis with Parallel Paths

0 0 04 4 4

26

26

26

2 6 2 6 2 604

04

04

15

15

15

15

15

15

3 73 7

3 7

3 7 3 7 3 7

Dmin for 3 Consecutive Symbols 0 0 0

Distance between 0 0 0 and 2 1 2

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Four State Trellis with Parallel PathsDmin for 3 Consecutive Symbols 0 0 0

Distance between 0 0 0 and 2 1 2

Is this Dmin?[ ] [ ] ( ) ( )( ) ( )= + + =

2 2 2

S S S S0,0,0 , 2,1,2d 2E 2 E sin π 8 2E 2.14 E

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Four State Trellis with Parallel Paths

0 0 04 4 4

26

26

26

2 6 2 6 2 604

04

04

15

15

15

15

15

15

3 73 7

3 7

3 7 3 7 3 7

Dmin for 3 Consecutive Symbols 0 0 0

Distance between 0 0 0 and 0 0 4

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Four State Trellis with Parallel PathsDmin for 3 Consecutive Symbols 0 0 0

Distance between 0 0 0 and 0 0 4

[ ] [ ] ( ) S2

S4,0,0,0,0,0 E2=E2+0+0=d

Is this Dmin? YES & it is better than that of uncoded QPSK

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Coding Gain Four State Trellis TCM

31

Union Bound

mine

0

d1P M 1 erfc2 2 N

Coding Gain Four State Trellis TCM

2

min 8PSK TCM min 8PSK TCM8PSK TCM 10 10 2

min QPSK min QPSK

d dG dB 20 log 10 log

d d

S8PSK TCM 10

S

4EG dB 10 log 3 dB

2E

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Eight State Trellis without Parallel Paths

32

S0

S1

S2

S3

S4

S5

S6

S7

0 0 0

6

7

6

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

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Eight State Trellis without Parallel Pathsdmin for 3 Consecutive Symbols 0 0 0

[ ] [ ] ( ) ( )( ) ( )-= = + + =2 2 2

min 8PSK TCM S S S S0,0,0 , 6,7,6d d 2E 2 E sin π 8 2E 2.14 E

Distance between 0 0 0 and 6 7 6

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Coding Gain Eight State Trellis TCM

34

Union Bound

mine

0

d1P M 1 erfc2 2 N

Coding Gain Four State Trellis TCM

2

min 8PSK TCM min 8PSK TCM8PSK TCM 10 10 2

min QPSK min QPSK

d dG dB 20 log 10 log

d d

S8PSK TCM 10

S

4.585EG dB 10 log 3.6 dB

2E

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Encoder for Four State Trellis TCM

35

+

+

m1

m2

u1

u2

u3

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Encoder for Four State Trellis TCM

36

0

+

+

m1

m2

u1u2

u3

0

S0 (00)

S1 (10)

S2 (01)

S3 (11)

0 0 0

1 0 0 0 1 1 1 1 1

© Tallal Elshabrawy

Encoder for Four State Trellis TCM

37

0

+

+

m1

m2

u1u2

u3

1

S0 (00)

S1 (10)

S2 (01)

S3 (11)

0 0 0

1 0 0

0 1 0 1 1 0

0 1 1 1 1 1

0 0 1 1 0 1

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Encoder for Four State Trellis TCM

38

1

+

+

m1

m2

u1u2

u3

0

S0 (00)

S1 (10)

S2 (01)

S3 (11)

0 0 0

1 0 0

0 1 1

1 1 1

0 1 0 1 1 0

0 1 1 1 1 1

0 0 0

1 0 0

0 0 1 1 0 1

© Tallal Elshabrawy

Encoder for Four State Trellis TCM

39

1

+

+

m1

m2

u1u2

u3

S0 (00)

S1 (10)

S2 (01)

S3 (11) 1

0 0 0

1 0 0

0 1 1

1 1 1

0 1 0 1 1 0

0 0 1

1 0 1

0 1 1 1 1 1

0 0 0

1 0 0

0 0 1 1 0 1 0 1 0

11 0

© Tallal Elshabrawy

Encoder for Four State Trellis TCM

40

1

+

+

m1

m2

u1u2

u3

S0 (00)

S1 (10)

S2 (01)

S3 (11) 1

0 0 0

1 0 0

0 1 1

1 1 1

0 1 0 1 1 0

0 0 1

1 0 1

0 1 1 1 1 1

0 0 0

1 0 0

0 0 1 1 0 1 0 1 0

11 0

0 0 0 1 0 0

0 1 1

1 1 1

0 1 0

1 1 0

0 0 1

1 0 1