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Trellis Coded Modulation

Trellis Coded Modulation

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Trellis Coded Modulation

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  1. Trellis Coded Modulation

  2. 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

  3. 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. 2 1 3 0 4 A0 7 5 6 2 1 3 4 0 B0 B1 7 5 6 2 3 1 C3 4 C0 0 C1 C2 7 5 6 Ungerboek Partitioning of 8-PSK Constellation

  5. Convolutional Codes(Reminder)

  6. 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. 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. Convolutional Encoder: Example Rate ½ Convolutional Encoder 1 + c2 Output Input 1 0 1 0 0 c1 + 1

  9. Convolutional Encoder: Example Rate ½ Convolutional Encoder 1 + c2 Output Input 1 01 1 0 1 1 c1 + 0

  10. Convolutional Encoder: Example Rate ½ Convolutional Encoder 0 + c2 Output Input 101 0 1 1 0 1 1 c1 + 0

  11. Convolutional Encoder: Example Rate ½ Convolutional Encoder + c2 Output Input 101 1 0 0 0 1 0 1 1 c1 +

  12. State Diagram Representation 0/00 S0 b1 b0 1/11 0/11 1/00 S2 S1 0/01 States (b0b1) s0 00 s1 10 s2 01 s3 11 1/10 0/10 S3 1/01 Input 0 Input 1

  13. 00 00 00 11 11 11 11 11 11 00 00 00 01 01 01 10 10 10 10 10 10 01 01 01 0/00 Trellis Representation S 0 1/11 0/11 1/00 S S 2 1 0/01 1/10 0/10 S 3 1/01 00 00 00 s0 (0 0) 11 11 11 11 s1 (1 0) 00 01 01 s2 (0 1) 10 10 10 s3 (1 1) 01

  14. 0/00 Trellis Representation S 0 1/11 0/11 1/00 S S 2 1 0/01 Input: 101 Output: 001011 1/10 0/10 S 3 1/01 00 00 00 00 00 00 s0 (0 0) 11 11 11 11 11 11 11 11 11 11 s1 (1 0) 00 00 00 00 01 01 01 01 01 s2 (0 1) 10 10 10 10 10 10 10 10 10 s3 (1 1) 01 01 01 01

  15. Trellis Representation of QPSK 0 1 3 2 Trellis Representation 0 0 0 1 1 1 2 2 2 3 3 3

  16. Trellis Representation of QPSK Dmin for 3 Consecutive Symbols 0 0 0 0 0 0 1 1 1 2 2 2 3 3 3 0 0 0 1 1 1 2 2 2 3 3 3

  17. Trellis Representation of QPSK Dmin for 3 Consecutive Symbols 0 0 0 0 0 0 1 1 1 2 2 2 3 3 3 0 0 0 1 1 1 2 2 2 3 3 3

  18. Trellis Representation of QPSK Dmin for 3 Consecutive Symbols 0 0 0 0 0 0 1 1 1 0 0 0 2 2 2 3 3 3 0 0 0 1 1 1 2 2 2 3 3 3

  19. Trellis Representation of QPSK Dmin for 3 Consecutive Symbols 0 0 0 0 0 0 1 1 1 0 0 0 2 2 2 3 3 3 0 0 0 1 1 1 2 2 2 3 3 3

  20. Trellis Representation of QPSK Dmin for 3 Consecutive Symbols 0 0 0 0 0 0 1 1 1 2 2 2 3 3 3 0 0 0 1 1 1 2 2 2 3 3 3

  21. Trellis Representation of QPSK Dmin for 3 Consecutive Symbols 0 0 0 0 0 0 1 1 1 2 2 2 3 3 3 0 0 0 1 1 1 2 2 2 3 3 3

  22. Trellis Representation of QPSK Dmin for 3 Consecutive Symbols 0 0 0 0 0 0 1 1 1 2 2 2 0 0 0 0 0 1 3 3 3 0 0 0 1 1 1 2 2 2 3 3 3

  23. Trellis Representation of QPSK Summary) dmin for 3 Consecutive Symbols 0 0 0 0 0 0 1 1 1 1 0 S0 2 2 2 3 3 3 0 0 0 1 1 1 S0 2 3 2 2 2 3 3 3

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

  25. Four State Trellis with Parallel Paths Dmin for 3 Consecutive Symbols 0 0 0 0 0 0 4 4 4 2 2 2 6 6 6 2 6 2 6 2 6 0 0 0 4 4 4 1 1 1 5 5 5 7 3 7 3 7 3 3 3 7 3 7 7 1 1 1 5 5 5

  26. Four State Trellis with Parallel Paths Dmin for 3 Consecutive Symbols 0 0 0 0 0 0 4 4 4 2 2 2 6 6 6 2 6 2 6 2 6 0 0 0 4 4 4 1 1 1 5 5 5 7 3 7 3 7 3 3 3 7 3 7 7 1 1 1 5 5 5

  27. Four State Trellis with Parallel Paths Dmin for 3 Consecutive Symbols 0 0 0 0 0 0 4 4 4 2 2 2 6 6 6 2 6 2 6 2 6 0 0 0 4 4 4 1 1 1 5 5 5 7 3 7 3 7 3 3 3 7 3 7 7 1 1 1 5 5 5 Distance between 0 0 0 and 2 1 2

  28. Four State Trellis with Parallel Paths Dmin for 3 Consecutive Symbols 0 0 0 Distance between 0 0 0 and 2 1 2 Is this Dmin?

  29. Four State Trellis with Parallel Paths Dmin for 3 Consecutive Symbols 0 0 0 0 0 0 4 4 4 2 2 2 6 6 6 2 6 2 6 2 6 0 0 0 4 4 4 1 1 1 5 5 5 7 3 7 3 7 3 3 3 7 3 7 7 1 1 1 5 5 5 Distance between 0 0 0 and 0 0 4

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

  31. Coding Gain Four State Trellis TCM Union Bound Coding Gain Four State Trellis TCM

  32. Eight State Trellis without Parallel Paths 0 0 0 S0 0 4 2 6 6 S1 1 5 3 7 S2 4 0 6 2 6 S3 5 1 7 3 S4 2 6 0 4 7 S5 3 7 1 5 S6 6 2 4 0 S7 7 3 5 1

  33. Eight State Trellis without Parallel Paths dmin for 3 Consecutive Symbols 0 0 0 Distance between 0 0 0 and 6 7 6

  34. Coding Gain Eight State Trellis TCM Union Bound Coding Gain Four State Trellis TCM

  35. Encoder for Four State Trellis TCM u1 m1 u2 + m2 + u3

  36. Encoder for Four State Trellis TCM 0 0 0 S0 (00) 1 0 0 0 1 1 1 1 1 S1 (10) m1 u1 S2 (01) u2 + m2 0 0 S3 (11) + u3

  37. Encoder for Four State Trellis TCM 0 0 0 S0 (00) 1 0 0 0 1 1 1 1 1 S1 (10) 0 1 0 1 1 0 m1 u1 S2 (01) u2 + 0 0 1 1 0 1 m2 1 0 S3 (11) + u3

  38. Encoder for Four State Trellis TCM 0 0 0 S0 (00) 1 0 0 0 1 1 0 1 1 1 1 1 1 1 1 S1 (10) 0 0 0 1 0 0 0 1 0 1 1 0 m1 u1 S2 (01) u2 + 0 0 1 1 0 1 m2 0 1 S3 (11) + u3

  39. Encoder for Four State Trellis TCM 0 0 0 S0 (00) 1 0 0 0 1 1 0 1 1 1 1 1 1 1 1 S1 (10) 0 0 0 1 0 0 0 1 0 1 1 0 m1 u1 S2 (01) u2 0 0 1 + 0 0 1 1 0 1 1 0 1 0 1 0 m2 1 1 S3 (11) 11 0 + u3

  40. Encoder for Four State Trellis TCM 0 1 1 0 0 0 S0 (00) 0 1 0 0 0 1 1 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 S1 (10) 0 0 0 1 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 m1 u1 S2 (01) u2 0 0 1 + 0 0 1 1 0 1 1 0 1 0 1 0 m2 1 1 S3 (11) 11 0 + u3