1. © Tallal Elshabrawy Trellis Coded Modulation

2. © 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 d min 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

3. © Tallal Elshabrawy 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.

4. © Tallal Elshabrawy Ungerboek Partitioning of 8-PSK Constellation A0A0 B0B0 B1B1 C0C0 C1C1 C2C2 C3C3 0 1 2 3 4 5 6 7 0 2 4 6 1 3 5 7 0 4 2 6 3 7 1 5

5. © Tallal Elshabrawy Convolutional Codes (Reminder)

6. © Tallal Elshabrawy Introduction Block 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

7. © Tallal Elshabrawy Convolutional Codes Specification A 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

8. © Tallal Elshabrawy Convolutional Encoder: Example 00 + + Input Output Rate ½ Convolutional Encoder 1 0 1 1 1 c1c1 c2c2

9. © Tallal Elshabrawy Convolutional Encoder: Example 10 + + Input Output Rate ½ Convolutional Encoder 1 0 1 1 0 c1c1 c2c2 1

10. © Tallal Elshabrawy Convolutional Encoder: Example 01 + + Input Output Rate ½ Convolutional Encoder 1 0 11 0 1 0 0 c1c1 c2c2 1 0 1 1

11. © Tallal Elshabrawy Convolutional Encoder: Example 10 + + Input Output Rate ½ Convolutional Encoder 1 0 11 0 1 c1c1 c2c2 0 0 1 0 1 1

12. © Tallal Elshabrawy b0b0 b1b1 States (b 0 b 1 ) s 0 00 s 1 10 s 2 01 s 3 11 S0S0 S3S3 S1S1 S2S2 0/00 1/11 1/00 0/01 1/10 1/01 0/10 0/11 Input 0 Input 1 State Diagram Representation

13. © Tallal Elshabrawy s 0 (0 0) s 1 (1 0) s 2 (0 1) s 3 (1 1) 00 11 01 10 01 11 00 11 01 10 01 11 00 S 0 S 3 S 1 S 2 0/00 1/11 1/00 0/01 1/10 1/01 0/10 0/11 Trellis Representation

14. © Tallal Elshabrawy s 0 (0 0) s 1 (1 0) s 2 (0 1) s 3 (1 1) 00 11 01 10 01 11 Input: 101  Output: S 0 S 3 S 1 S 2 0/00 1/11 1/00 0/01 1/10 1/01 0/10 0/11 00 001011 Trellis Representation

15. © Tallal Elshabrawy Trellis Representation of QPSK 0 1 2 3 Trellis Representation 0 1 2 3 0 1 2 3 0 1 2 3

16. © Tallal Elshabrawy Trellis Representation of QPSK D min for 3 Consecutive Symbols 0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3

17. © Tallal Elshabrawy Trellis Representation of QPSK D min for 3 Consecutive Symbols 0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3

18. © Tallal Elshabrawy Trellis Representation of QPSK D min for 3 Consecutive Symbols 0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 0 0

19. © Tallal Elshabrawy Trellis Representation of QPSK D min for 3 Consecutive Symbols 0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 0 0

20. © Tallal Elshabrawy Trellis Representation of QPSK D min for 3 Consecutive Symbols 0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3

21. © Tallal Elshabrawy Trellis Representation of QPSK D min for 3 Consecutive Symbols 0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3

22. © Tallal Elshabrawy Trellis Representation of QPSK D min for 3 Consecutive Symbols 0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 0 0 0 0 1

23. © Tallal Elshabrawy Trellis Representation of QPSK Summary) d min for 3 Consecutive Symbols 0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 01 23 S0S0 S0S0

24. © Tallal Elshabrawy Four State Trellis with Parallel Paths 0 0 0 4 4 4 2 6 2 6 2 6 2 62 6 2 6 0 4 0 4 0 4 1 5 1 5 1 5 1 5 1 5 1 5 3 7 3 7 3 7 3 7 3 7 3 7

25. © Tallal Elshabrawy Four State Trellis with Parallel Paths 0 0 0 4 4 4 2 6 2 6 2 6 2 62 6 2 6 0 4 0 4 0 4 1 5 1 5 1 5 1 5 1 5 1 5 3 7 3 7 3 7 3 7 3 7 3 7 D min for 3 Consecutive Symbols 0 0 0

26. © Tallal Elshabrawy Four State Trellis with Parallel Paths 0 0 0 4 4 4 2 6 2 6 2 6 2 62 6 2 6 0 4 0 4 0 4 1 5 1 5 1 5 1 5 1 5 1 5 3 7 3 7 3 7 3 7 3 7 3 7 D min for 3 Consecutive Symbols 0 0 0

27. © Tallal Elshabrawy Four State Trellis with Parallel Paths 0 0 0 4 4 4 2 6 2 6 2 6 2 62 6 2 6 0 4 0 4 0 4 1 5 1 5 1 5 1 5 1 5 1 5 3 7 3 7 3 7 3 7 3 7 3 7 D min for 3 Consecutive Symbols 0 0 0 Distance between 0 0 0 and 2 1 2

28. © Tallal Elshabrawy Four State Trellis with Parallel Paths D min for 3 Consecutive Symbols 0 0 0 Distance between 0 0 0 and 2 1 2 Is this D min ?

29. © Tallal Elshabrawy Four State Trellis with Parallel Paths 0 0 0 4 4 4 2 6 2 6 2 6 2 62 6 2 6 0 4 0 4 0 4 1 5 1 5 1 5 1 5 1 5 1 5 3 7 3 7 3 7 3 7 3 7 3 7 D min for 3 Consecutive Symbols 0 0 0 Distance between 0 0 0 and 0 0 4

30. © Tallal Elshabrawy Four State Trellis with Parallel Paths D min for 3 Consecutive Symbols 0 0 0 Distance between 0 0 0 and 0 0 4 Is this D min ?  YES & it is better than that of uncoded QPSK

31. © Tallal Elshabrawy Coding Gain Four State Trellis TCM 31 Union Bound Coding Gain Four State Trellis TCM

32. © Tallal Elshabrawy Eight State Trellis without Parallel Paths 32 S0S0 S1S1 S2S2 S3S3 S4S4 S5S5 S6S6 S7S7 0 00 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

33. © Tallal Elshabrawy Eight State Trellis without Parallel Paths d min for 3 Consecutive Symbols 0 0 0 Distance between 0 0 0 and 6 7 6

34. © Tallal Elshabrawy Coding Gain Eight State Trellis TCM 34 Union Bound Coding Gain Four State Trellis TCM

35. © Tallal Elshabrawy Encoder for Four State Trellis TCM 35 + + m1m1 m2m2 u1u1 u2u2 u3u3

36. © Tallal Elshabrawy Encoder for Four State Trellis TCM 36 0 + + m1m1 m2m2 u1u1 u2u2 u3u3 0 S 0 (00) S 1 (10) S 2 (01) S 3 (11) 0 0 0 1 0 0 0 1 1 1 1 1

37. © Tallal Elshabrawy Encoder for Four State Trellis TCM 37 0 + + m1m1 m2m2 u1u1 u2u2 u3u3 1 S 0 (00) S 1 (10) S 2 (01) S 3 (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

38. © Tallal Elshabrawy Encoder for Four State Trellis TCM 38 1 + + m1m1 m2m2 u1u1 u2u2 u3u3 0 S 0 (00) S 1 (10) S 2 (01) S 3 (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

39. © Tallal Elshabrawy Encoder for Four State Trellis TCM 39 1 + + m1m1 m2m2 u1u1 u2u2 u3u3 S 0 (00) S 1 (10) S 2 (01) S 3 (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

40. © Tallal Elshabrawy Encoder for Four State Trellis TCM 40 1 + + m1m1 m2m2 u1u1 u2u2 u3u3 S 0 (00) S 1 (10) S 2 (01) S 3 (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