Hybrid concatenated codes and iterative decoding

• US 6,023,783 A
• Filed: 05/15/1997
• Issued: 02/08/2000
• Est. Priority Date: 05/15/1996
• Status: Expired due to Term

• Current Assignee: California Institute of Technology
• Original Assignee: California Institute of Technology
• Inventors: Divsalar, Dariush, Pollara, Fabrizio
• Primary Examiner(s): Baker, Stephen M.
• Application Number:
  US08/857,021
• Time in Patent Office: 999 Days
• Field of Search: 371/37.4, 371/37.6, 371/37.06, 371/43.2, 371/43.4, 371/43.6, 371/43.7, 375/264, 375/265, 375/262, 375/341
• US Class Current: 714/792
• CPC Class Codes: H03M 13/256 with trellis coding, e.g. with convolutional codes and TCM
• CPC Class Codes: H03M 13/258 with turbo codes, e.g. Turbo Trellis Coded Modulation [TTCM]
• CPC Class Codes: H03M 13/27 using interleaving techniques
• CPC Class Codes: H03M 13/2746 S-random interleaver
• CPC Class Codes: H03M 13/2903 Methods and arrangements specifically for encoding, e.g. parallel encoding of a plurality of constituent codes
• CPC Class Codes: H03M 13/296 Particular turbo code structure
• CPC Class Codes: H03M 13/2972 Serial concatenation using convolutional component codes
• CPC Class Codes: H03M 13/2978 Particular arrangement of the component decoders
• CPC Class Codes: H03M 13/3988 for rate k/n convolutional codes, with k>1, obtained by convolutional encoders with k inputs and n outputs
• CPC Class Codes: H04L 1/0041 Arrangements at the transmitter end
• CPC Class Codes: H04L 1/005 Iterative decoding, including iteration between signal detection and decoding operation
• CPC Class Codes: H04L 1/0054 Maximum-likelihood or sequential decoding, e.g. Viterbi, Fano, ZJ algorithms
• CPC Class Codes: H04L 1/0066 Parallel concatenated codes
• CPC Class Codes: H04L 1/0071 Use of interleaving interleaving per se H03M13/27

Abstract

Several improved turbo code apparatuses and methods. The invention encompasses several classes: (1) A data source is applied to two or more encoders with an interleaver between the source and each of the second and subsequent encoders. Each encoder outputs a code element which may be transmitted or stored. A parallel decoder provides the ability to decode the code elements to derive the original source information d without use of a received data signal corresponding to d. The output may be coupled to a multilevel trellis-coded modulator (TCM). (2) A data source d is applied to two or more encoders with an interleaver between the source and each of the second and subsequent encoders. Each of the encoders outputs a code element. In addition, the original data source d is output from the encoder. All of the output elements are coupled to a TCM. (3) At least two data sources are applied to two or more encoders with an interleaver between each source and each of the second and subsequent encoders. The output may be coupled to a TCM. (4) At least two data sources are applied to two or more encoders with at least two interleavers between each source and each of the second and subsequent encoders. (5) At least one data source is applied to one or more serially linked encoders through at least one interleaver. The output may be coupled to a TCM. The invention includes a novel way of terminating a turbo coder.

Claims

1. A system for error-correction coding of a source of original digital data elements, comprising:

(a) a first systematic convolutional encoder, coupled to the source of original digital data elements, for generating a first series of coded output elements derived from the original digital data elements;

(b) at least one interleaver, each coupled to the source of original digital data elements, for modifying the order of the original digital data elements to generate respective sets of interleaved elements; and

(c) at least one next systematic convolutional encoder, each coupled to respective interleaved elements, each for generating a corresponding next series of coded output elements derived from a respective set of interleaved elements, each next series of coded output elements being in parallel with the first series of coded output elements;

wherein the system for error-correction coding outputs only the first series of coded output elements and each next series of coded output elements.

2. The system of claim 1, further including a decoder for receiving signals representative of at least some of the first series of coded output elements and of at least some of each next series of coded output elements, and for generating the original digital data elements from such received signals.

3. The system of claim 1, further including a multilevel modulator, coupled to the coded output elements of each systematic convolutional encoder, for generating an output modulated signal representative of at least some of such coded output elements.

4. The system of claim 3, wherein the multilevel modulator generates a trellis code modulation.

5. The system of claim 3, further including a demodulator for demodulating the output signal of the multilevel modulator into a data signal representative of at least some of the first series of coded output elements and of at least some of each next series of coded output elements, and a decoder, coupled to the demodulator, for generating the original digital data elements from the data signal.

6. A system for error-correction coding of a plurality of sources of original digital data elements, comprising:

(a) a first systematic convolutional encoder, coupled to each source of original digital data elements, for generating a first set of series coded output elements derived from the original digital data elements;

(b) at least one set of interleavers, each set coupled to respective sources of original digital data elements, for modifying the order of the original digital data elements from the respective coupled sources to generate a respective set of interleaved elements; and

(c) at least one next systematic convolutional encoder, each coupled to at least one set of interleaved elements, each for generating a corresponding next set of series coded output elements derived from the coupled sets of interleaved elements, each next set of series coded output elements being in parallel with the first set of series coded output elements.

7. The system of claim 6, wherein the system for error-correction coding further outputs the original digital data elements.

8. The system of claim 6, wherein the system for error-correction coding outputs only the first set of series coded output elements and each next set of series coded output elements.

9. The stem of claim 6, further including a decoder for receiving signals representative of at least some of the first set of series coded output elements and of at least some of each next set of series coded output elements, and for generating the original digital data elements from such received signals.

10. The system of claim 6, further including a multilevel modulator, coupled to the coded output elements, for generating an output modulated signal representative of at least some of the coded output elements.

11. The system of claim 10, wherein the multilevel modulator generates a trellis code modulation.

12. The system of claim 10, further including a demodulator for demodulating the output signal of the multilevel modulator into a data signal representative of at least some of the first set of series coded output elements and of at least some of each next set of series coded output elements, and a decoder, coupled to the demodulator, for generating the original digital data elements from the data signal.

13. A system for error-correction coding of a source of original digital data elements comprising:

(a) at least one interleaver, each coupled to the source of original digital data elements, for modifying the order of the original digital data elements to generate respective interleaved elements; and

(b) a single systematic recursive convolutional encoder module, coupled to the source of original digital data elements and to interleaved elements from at least one interleaver, for generating a set of coded output elements derived from the original digital data elements;

wherein the system for error-correction coding outputs the set of coded output elements and the original digital data elements.

14. The system of claim 13, further including a decoder for receiving signals representative of at least some of the set of coded output elements, and for generating the original digital data elements from such received signals.

15. The system of claim 13, further including a multilevel modulator, coupled to the set of coded output elements of the systematic convolutional encoder, for generating an output modulated signal representative of at least some of the set of coded output elements.

16. The system of claim 15, wherein the multilevel modulator generates a trellis code modulation.

17. The system of claim 15, further including a demodulator for demodulating the output signal of the multilevel modulator into a data signal representative of at least some of the set of coded output elements, and a decoder, coupled to the demodulator, for generating the original digital data elements from the data signal.

18. A system for error-correction coding of a source of original digital data elements, comprising:

(a) a first encoder, coupled to the source of original digital data elements, for generating a plurality of coded intermediate output elements derived from the original digital data elements;

(b) at least one interleaver, each coupled to at least one of the plurality of coded intermediate output elements, for modifying the order of the coded intermediate output elements to generate respective interleaved output elements; and

(c) at least one systematic recursive convolutional encoder, each coupled to at least one interleaver, for generating a set of coded output elements derived from the interleaved output elements from each coupled interleaver.

19. The system of claim 18, further including a decoder for receiving signals representative of at least some of the set of coded output elements, and for generating the original digital data elements from such received signals.

20. The system of claim 18, further including a multilevel modulator, coupled to each set of coded output elements, for generating an output modulated signal representative of at least some of the set of coded output elements.

21. The system of claim 20, wherein the multilevel modulator generates a trellis code modulation.

22. The system of claim 20, further including a demodulator for demodulating the output signal of the multilevel modulator into a data signal representative of at least some of the coded output elements, and a decoder, coupled to the demodulator, for generating the original digital data elements from the data signal.

23. A system for error-correction coding of a source of original digital data elements, comprising:

(a) a first systematic convolutional encoder, coupled to the source of original digital data elements, for generating a first series of coded output elements derived from the original digital data elements;

(b) at least one interleaver, each coupled to the source of original digital data elements, for modifying the order of the original digital data elements to generate respective interleaved elements;

(c) at least one next systematic convolutional encoder, each coupled to respective interleaved elements, each for generating a corresponding next series of coded output elements derived from the respective interleaved elements, each next series of coded output elements being in parallel with the first series of coded output elements; and

(d) a multilevel modulator, directly coupled to the original digital data elements and to the coded output elements of each systematic convolutional encoder, for generating an output modulated signal representative of at least some of such original digital data elements and coded output elements.

24. The system of claim 23, wherein the multilevel modulator generates a trellis code modulation.

25. A system for error-correction coding and multilevel modulation of a plurality of sources of original digital data elements, comprising:

(a) a first systematic convolutional encoder, coupled to each source of original digital data elements, for systematically selecting a first subset of the original digital data elements and generating a first series of coded output elements derived from the first selected subset, and for outputting at least one source of original digital data elements unchanged;

(b) at least two interleavers, each coupled to a respective one of the plurality of sources of original digital data elements, for modifying the order of the original digital data elements to generate respective sets of interleaved elements;

(c) at least one next systematic convolutional encoder, each coupled to at least two interleavers, each for systematically selecting a next subset from the sets of interleaved elements different from each other selected subset, for generating a corresponding next series of coded output elements derived from the corresponding next subset, each next series of coded output elements being in parallel with the first series of coded output elements, and for outputting at least one set of interleaved elements unchanged; and

(d) a multilevel modulator, coupled to the original digital data elements, the unchanged interleaved elements, and the coded output elements, for generating an output modulated signal representative of at least some of such original digital data elements, unchanged interleaved elements, and coded output elements.

26. The system of claim 25, wherein the multilevel modulator generates a trellis code modulation.

27. A system for terminating a turbo encoder comprising:

(a) a plurality of serially connected delay elements D having a tap after each delay element;

(b) a plurality of first selective combinatorial devices, at least one before the first serially connected delay element D, at least one after the last serially connected delay element D, and at least one between each intermediate pair of serially connected delay elements D;

(c) at least one data source line u_b, where b is the number of data source lines, each coupled to each first selective combinatorial device as input lines;

(d) at least one set of next selective combinatorial devices, each set comprising a plurality of selective combinatorial devices each coupled to a corresponding tap and serially coupled together, with an end selective combinatorial device of each set selectively coupled to a corresponding data source line;

wherein, to terminate input to the delay elements D, the sets of next selective combinatorial devices are coupled to the corresponding data source line and selectively actuated to select tap coefficients a_i0, . . . a_i,m-1 for i=1, 2, . . . b, to apply to a corresponding data source line, wherein the tap coefficients are obtained by repeated use of the following equation, and by solving the resulting equations;

##EQU19## where S^k (D) is the state of the turbo encoder at time k with coefficients S^k₀, S^k₁, . . . S^k_m-1 for input u^k₁, . . . u^k_b, and termination in state zero is achieved in at most m clock cycles.

28. A decoder system for decoding a plurality of sequences of received signals y_i, representative of code elements x_i generated by a turbo encoder from a source of original digital data elements u_i, into decoded elements corresponding to the original digital data elements u_i, the decoder system comprising:

(a) at least three decoder modules, each having a received signal input i, a feedback input, and an output, the output of each decoder module being coupled to the feedback input of each other decoder module; and

(b) a summing module, coupled to each output of each decoder module, for generating final decoded elements from the outputs of the decoder modules;

wherein each sequence of received signals y_i is coupled to the received signal input i of a corresponding decoder module.

29. The decoder system of claim 28, wherein each decoder module includes:

(a) a feedback input comprising a combinatorial element;

(b) a permuter, coupled to the combinatorial element;

(c) a probability-based decoder, coupled to the permuter and including a received signal input;

(d) an inverse permuter, coupled to the probability-based decoder;

(e) a differential combinatorial element, coupled to the inverse permuter; and

(f) a delay element, coupled between the combinatorial element and the differential combinatorial element.

30. The decoder system of claim 29, wherein the probability-based decoder uses the maximum a posteriori probability algorithm.

31. The decoder system of claim 29, wherein the probability-based decoder uses the soft output Viterbi algorithm.

32. An iterative decoder system for decoding at least one sequence of received signals y_i, representative of code elements x_i generated by a self-concatenated encoder from a source of original digital data elements u_i, into decoded elements corresponding to the original digital data elements u_i, the decoder system comprising:

(a) a plurality of feedback inputs each comprising a combinatorial element;

(b) a plurality of permuters, each coupled to a corresponding combinatorial element;

(c) a probability-based decoder, coupled to each permuter and at least one combinatorial element, and including a received signal input and an output;

(d) a plurality of inverse permuters, each coupled to the probability-based decoder so as to receive a signal associated with a corresponding permuter;

(e) a plurality of differential combinatorial elements, one coupled to the output of the probability-based decoder and each other coupled to a corresponding inverse permuter, and each coupled to every non-corresponding feedback input; and

(f) a plurality of delay elements, each coupled between a corresponding combinatorial element and a corresponding differential combinatorial element;

wherein each sequence of received signals y_i is coupled to the received signal input of the probability-based decoder.

33. The decoder system of claim 32, wherein the probability-based decoder uses the maximum a posteriori probability algorithm.

34. The decoder system of claim 32, wherein the probability-based decoder uses the soft output Viterbi algorithm.

35. An iterative decoder system for decoding at least one sequence of received signals y_i, representative of code elements x_i generated by a serial encoder from a source of original digital data elements u_i, into decoded elements corresponding to the original digital data elements u_i, the decoder system comprising:

(a) a plurality of permuters each having an input;

(b) at least one first probability-based decoder, each coupled to a corresponding permuter, and including a received signal input;

(c) a plurality of inverse permuters, each coupled to a corresponding probability-based decoder so as to receive a signal associated with a corresponding permuter;

(d) a plurality of first differential combinatorial elements, each coupled to a corresponding inverse permuter;

(e) a plurality of first delay elements, each coupled between the input of a corresponding permuter and a corresponding first differential combinatorial element;

(f) a second probability-based decoder, coupled to each first differential combinatorial element, and including inputs corresponding to each first differential combinatorial element, and an output;

(g) a plurality of second differential combinatorial elements, each coupled to the second probability-based decoder and to the input of a corresponding permuter;

(h) a plurality of second delay elements, each coupled between corresponding inputs of the second probability-based decoder and a corresponding second differential combinatorial element;

wherein the sequence of received signals y_i is coupled to corresponding received signal inputs of the first probability-based decoders.

36. The decoder system of claim 35, wherein at least one probability-based decoder uses the maximum a posteriori probability algorithm.

37. The decoder system of claim 35, wherein at least one probability-based decoder uses the soft output Viterbi algorithm.

38. A method for error-correction coding of a source of original digital data elements, comprising the steps of:

(a) generating a first series of systematic convolutional encoded output elements derived from a source of original digital data elements;

(b) modifying the order of the original digital data elements to generate at least one set of respective interleaved elements; and

(c) generating at least one corresponding next series of systematic convolutional encoded output elements derived from a corresponding set of respective interleaved elements, each next series of coded output elements being in parallel with the first series of coded output elements;

(d) outputting only the first series of systematic convolutional encoded output elements and each next series of systematic convolutional encoded output elements.

39. The method of claim 38, further including the steps of:

(a) receiving signals representative of at least some of the first series of systematic convolutional encoded output elements and of at least some of each next series of systematic convolutional encoded output elements; and

(b) generating the original digital data elements from such received signals.

40. The method of claim 38, further including the step of generating an output multilevel modulated signal representative of at least some of such coded output elements.

41. The method of claim 40, wherein the multilevel modulation is trellis code modulation.

42. The method of claim 40, further including the steps of:

(a) demodulating the output multilevel modulated signal into a data signal representative of at least some of the first series of systematic convolutional encoded output elements and of at least some of each next series of systematic convolutional encoded output elements;

(b) generating the original digital data elements from the data signal.

43. A method for error-correction coding of a plurality of sources of original digital data elements, comprising the steps of:

(a) generating a first set of series systematic convolutional encoded output elements derived from a plurality of sources of original digital data elements;

(b) modifying the order of the original digital data elements from the respective sources to generate at least one set of respective interleaved elements; and

(c) generating at least one corresponding next set of series systematic convolutional encoded output elements derived from a corresponding set of respective interleaved elements, each next set of series systematic convolutional encoded output elements being in parallel with the first set of series systematic convolutional encoded output elements.

44. The method of claim 43, including the further the step of outputting the original digital data elements.

45. The method of claim 43, including the further the step of outputting only the first set of series systematic convolutional encoded output elements and each next set of series systematic convolutional encoded output elements.

46. The method of claim 43, further including the steps of:

(a) receiving signals representative of at least some of the first set of series systematic convolutional encoded output elements and of at least some of each next set of series systematic convolutional encoded output elements; and

(b) generating the original digital data elements from such received signals.

47. The method of claim 43, further including the step of generating an output multilevel modulated signal representative of at least some of such systematic convolutional encoded output elements.

48. The method of claim 47, wherein the multilevel modulation is trellis code modulation.

49. The method of claim 47, further including the steps of:

(a) demodulating the output multilevel modulated signal into a data signal representative of at least some of the first set of series systematic convolutional encoded output elements and of at least some of each next set of series systematic convolutional encoded output elements; and

(b) generating the original digital data elements from the data signal.

50. A method for error-correction coding of a source of original digital data elements, comprising the steps of:

(a) modifying the order of a source of original digital data elements to generate at least one set of interleaved elements; and

(b) generating a set of systematic convolutional encoded output elements derived from the original digital data elements and at least one set of interleaved elements;

(c) outputting only the set of systematic convolutional encoded output elements.

51. The method of claim 50, further including the steps of:

(a) receiving signals representative of at least some of the set of systematic convolutional encoded output elements; and

(b) generating the original digital data elements from such received signals.

52. The method of claim 50, further including the step of generating an output multilevel modulated signal representative of at least some of such systematic convolutional encoded output elements.

53. The method of claim 52, wherein the multilevel modulation is trellis code modulation.

54. The method of claim 52, further including the steps of:

(a) demodulating the output multilevel modulated signal into a data signal representative of at least some of the set of systematic convolutional encoded output elements; and

(b) generating the original digital data elements from the data signal.

55. A method for error-correction coding of a source of original digital data elements, comprising the steps of:

(a) generating a plurality of systematic convolutional encoded intermediate output elements derived from a source of original digital data elements;

(b) modifying the order of the systematic convolutional encoded intermediate output elements to generate at least one set of respective interleaved output elements; and

(c) generating a set of systematic convolutional encoded output elements derived from at least one set of interleaved output elements.

56. The method of claim 55, further including the steps of:

(a) receiving signals representative of at least some of the set of systematic convolutional encoded output elements; and

(b) generating the original digital data elements from such received signals.

57. The method of claim 55, further including the step of generating an output multilevel modulated signal representative of at least some of such systematic convolutional encoded output elements.

58. The method of claim 57, wherein the multilevel modulation is trellis code modulation.

59. The method of claim 57, further including the steps of:

(a) demodulating the output multilevel modulated signal into a data signal representative of at least some of the set of systematic convolutional encoded output elements; and

(b) generating the original digital data elements from the data signal.

60. A method for error-correction coding of a source of original digital data elements, comprising the steps of:

(a) generating a first series of systematic convolutional encoded output elements derived from a source of original digital data elements;

(b) modifying the order of the original digital data elements to generate at least one set of interleaved elements;

(c) generating at least one next series of systematic convolutional encoded output elements derived from at least one set of interleaved elements, each next series of systematic convolutional encoded output elements being in parallel with the first series of systematic convolutional encoded output elements; and

(d) generating an output multilevel modulated signal directly from and representative of at least some of such original digital data elements and systematic convolutional encoded output elements.

61. The method of claim 60, wherein the multilevel modulation is trellis code modulation.

62. A method for error-correction coding and multilevel modulation of a plurality of sources of original digital data elements, comprising the steps of:

(a) systematically selecting a first subset of original digital data elements from a plurality of sources of original digital data elements;

(b) generating a first series of systematic convolutional encoded output elements derived from the first selected subset;

(c) outputting at least one source of original digital data elements unchanged;

(d) modifying the order of the original digital data elements to generate at least two sets of interleaved elements;

(e) systematically selecting a next subset from the sets of interleaved elements different from each other selected subset;

(f) generating at least one next series of systematic convolutional encoded output elements derived from a corresponding next subset, each next series of systematic convolutional encoded output elements being in parallel with the first series of systematic convolutional encoded output elements;

(g) outputting at least one set of interleaved elements unchanged; and

(h) generating an output multilevel modulated signal representative of at least some of such original digital data elements, unchanged interleaved elements, and systematic convolutional encoded output elements.

63. The method of claim 62, wherein the multilevel modulation is trellis code modulation.

64. A method for terminating input to a turbo encoder comprising a plurality of serially connected delay elements D having a tap after each delay element;

a plurality of first selective combinatorial devices, at least one before the first serially connected delay element D, at least one after the last serially connected delay element D, and at least one between each intermediate pair of serially connected delay elements D;

at least one data source line u_b, where b is the number of data source lines, each coupled to each first selective combinatorial device as input lines; and

at least one set of next selective combinatorial devices, each set comprising a plurality of selective combinatorial devices each coupled to a corresponding tap and serially coupled together, with an end selective combinatorial device of each set selectively coupled to a corresponding data source line;

the method comprising the steps of;

(a) coupling the sets of next selective combinatorial devices to the corresponding data source line;

(b) selectively actuating the sets of next selective combinatorial devices to select tap coefficients a_i0, . . . a_i,m-1 for i=1, 2, . . . b, to apply to a corresponding data source line, wherein the tap coefficients are obtained by repeated use of the following equation, and by solving the resulting equations;

##EQU20## where S^k (D) is the state of the turbo encoder at time k with coefficients S^k₀, S^k₁, . . . S^k_m-1 for input u^k₁, . . . u^k_b, and termination in state zero is achieved in at most m clock cycles.

65. A method for decoding a plurality of sequences of received signals y_i, representative of systematic convolutional encoded elements x_i generated from a source of original digital data elements u_i, into decoded elements corresponding to the original digital data elements u_i, the method comprising the steps of:

(a) coupling at least three decoder modules, each having a received signal input i, a feedback input, and an output, such that the output of each decoder module is coupled to the feedback input of each other decoder module;

(b) applying each sequence of received signals y_i to the received signal input i of a corresponding decoder module; and

(c) summing the output of each decoder module to generate final decoded elements.

66. The method of claim 65, wherein each decoder module includes:

(a) a feedback input comprising a combinatorial element;

(b) a permuter, coupled to the combinatorial element;

(c) a probability-based decoder, coupled to the permuter and including a received signal input;

(d) an inverse permuter, coupled to the probability-based decoder;

(e) a differential combinatorial element, coupled to the inverse permuter; and

(f) a delay element, coupled between the combinatorial element and the differential combinatorial element.

67. The method of claim 66, wherein the probability-based decoder uses the maximum a posteriori probability algorithm.

68. The decoder method of claim 66, wherein the probability-based decoder uses the soft output Viterbi algorithm.

69. An iterative method for decoding at least one sequence of received signals y_i, representative of systematic convolutional encoded elements x_i generated by a self-concatenated encoder from a source of original digital data elements u_i, into decoded elements corresponding to the original digital data elements u_i, the method comprising the steps of:

(a) applying feedback signals to inputs of a plurality of combinatorial elements to generate first output signals;

(b) applying the first output signals to a plurality of permuters to generate second output signals;

(c) applying selected ones of the first and second output signals, and each sequence of received signals y_i, to a probability-based decoder to generate third output signals and a decoded output for decoded elements;

(d) applying the third output signals to a plurality of inverse permuters to generate fourth output signals;

(e) applying the fourth output signals to a plurality of differential combinatorial elements to generate feedback signals;

(f) applying the feedback signals to non-corresponding inputs of the plurality of combinatorial elements; and

(g) coupling a plurality of delay elements between a corresponding combinatorial element and a corresponding differential combinatorial element.

70. The decoder method of claim 69, wherein the probability-based decoder uses the maximum a posteriori probability algorithm.

71. The decoder method of claim 69, wherein the probability-based decoder uses the soft output Viterbi algorithm.

72. An iterative method for decoding at least one sequence of received signals y_i, representative of code elements x_i generated by a serial encoder from a source of original digital data elements u_i, into decoded elements corresponding to the original digital data elements u_i, the method comprising the steps of:

(a) applying feedback signals to respective inputs of a plurality of permuters to generate first output signals;

(b) applying the first output signals, and the sequence of received signals y_i, to at least one first probability-based decoder to generate second output signals;

(c) applying the second output signals to a plurality of inverse permuters to generate third output signals;

(d) applying the third output signals to a plurality of first differential combinatorial elements to generate fourth output signals;

(e) coupling a plurality of first delay elements between respective inputs of corresponding permuters and corresponding first differential combinatorial elements;

(f) applying the fourth output signals to a second probability-based decoder to generate fifth output signals and a decoded output for decoded elements;

(g) applying the fifth output signals to a plurality of second differential combinatorial elements, to generate feedback signals coupled to the second probability-based decoder and to the inputs of corresponding permuters;

(h) coupling a plurality of second delay elements between corresponding inputs of the second probability-based decoder and corresponding second differential combinatorial elements.

73. The method of claim 72, wherein at least one probability-based decoder uses the maximum a posteriori probability algorithm.

74. The method of claim 72, wherein at least one probability-based decoder uses the soft output Viterbi algorithm.