Methods of Receiving Data Transmitted Using Non-Uniform Constellations with Overlapping Constellation Point Locations

Abstract

Communication systems are described that use unequally spaced constellations that have increased capacity compared to conventional constellations operating within a similar SNR band. One embodiment is a digital communications system including a transmitter transmitting signals via a communication channel, the transmitter including a coder capable of receiving user bits and outputting encoded bits at a rate, a mapper capable of mapping encoded bits to symbols in a constellation, and a modulator capable of generating a modulated signal for transmission via the communication channel using symbols generated by the mapper, wherein the constellation is unequally spaced and characterizable by assignment of locations and labels of constellation points to maximize parallel decode capacity of the constellation at a given signal-to-noise ratio so that the constellation provides a given capacity at a reduced signal-to-noise ratio compared to a uniform constellation that maximizes the minimum distance between constellation points of the uniform constellation.

Claims

1. A method of receiving data, comprising: receiving a signal at a receiver via a communication channel having a channel signal-to-noise ratio (SNR), where the receiver comprises a demodulator, a demapper, and a decoder; demodulating the received signal into a demodulated signal using the demodulator and providing the demodulated signal to the demapper via a coupling between the demodulator and the demapper; determining likelihoods using the demapper based upon the demodulated signal and a symbol constellation that includes constellation points at a plurality of unique point locations, where: the plurality of unique point locations are unequally spaced; the constellation points each have a location and a different label; and the locations of at least two of the constellation points are the same; providing the determined likelihoods to the decoder via a coupling between the demapper and the decoder; and providing a sequence of received bits from the decoder based upon the determined likelihoods and a low density parity check (LDPC) code having an associated LDPC code rate.

2. The method of claim 1, where the location of at least one of the constellation points of the symbol constellation is different from the location of any other of the constellation points.

3. The method of claim 1, further comprising: selecting the symbol constellation from a plurality of symbol constellations each having a same number of constellation points using the receiver; wherein a first of the plurality of symbol constellations includes constellation points at a first set of unique point locations that are unequally spaced; wherein a second of the plurality of symbol constellations includes constellation points at a second set of unique point locations that are unequally spaced; and wherein the number of unique point locations in the first set of unique point locations is different from the number of unique point locations in the second set of unique point locations.

4. The method of claim 1, wherein the receiver successfully decodes the signal received over the communication channel when the SNR of the communication channel is equal to or less than 5.27 dB and the associated LDPC code rate is below 1/3 and the symbol constellation includes sixty-four constellation points.

5. The method of claim 1, wherein the receiver successfully decodes the signal received over the communication channel when the SNR of the communication channel is equal to or less than 15.42 dB and the associated LDPC code rate is below 5/8 and the symbol constellation includes two-hundred-fifty-six constellation points.

6. The method of claim 1, wherein the receiver successfully decodes the signal received over the communication channel when the SNR of the communication channel is equal to or less than 21.52 dB and the associated LDPC code rate is below 7/10 and the symbol constellation includes one-thousand-twenty-four constellation points.

7. The method of claim 1, further comprising: selecting an LDPC code rate and unequally spaced symbol constellation pair from a plurality of predetermined LDPC code rate and unequally spaced symbol constellation pairs, where each of the plurality of predetermined LDPC code rate and unequally spaced symbol constellation pairs comprises: an LDPC code rate from a plurality of LDPC code rates; and an unequally spaced symbol constellation from a plurality of unequally spaced symbol constellations; wherein the selected LDPC code rate and the unequally spaced symbol constellation pair includes the associated LDPC code rate of the LDPC code and the symbol constellation.

8. The method of claim 7, wherein each of the plurality of unequally spaced symbol constellations is only included in one of the plurality of predetermined LDPC code rate and unequally spaced symbol constellation pairs.

9. The method of claim 7, wherein the plurality of predetermined LDPC code rate and unequally spaced symbol constellation pairs includes a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of a first type characterized by at least two constellation points having locations that are the same and different labels.

10. The method of claim 9, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of a first type comprises: a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixty-four-point symbol constellations of the first type, and a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different two-hundred-fifty-six-point symbol constellations of the first type.

11. The method of claim 10, wherein: the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixty-four-point symbol constellations of the first type comprises multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate below 1/3 and a sixty-four-point unequally spaced symbol constellation of the first type; and the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different two-hundred-fifty-six-point symbol constellations of the first type and multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate below 5/8 and a two-hundred-and-fifty-six-point unequally spaced symbol constellation of the first type.

12. The method of claim 9, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of a first type comprises a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different one-thousand-twenty-four-point symbol constellations of the first type.

13. The method of claim 12, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different one-thousand-twenty-four-point symbol constellations of the first type and multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate below 7/10 and a one-thousand-twenty-four-point unequally spaced symbol constellation of the first type.

14. The method of claim 9, wherein the SNR of the communication channel is: equal to or less than 5.27 dB and the selected LDPC code rate and unequally spaced symbol constellation pair includes an LDPC code rate that is below 1/3 and a sixty-four-point unequally spaced symbol constellation of the first type; and equal to or less than 15.42 dB and the selected LDPC code rate and unequally spaced symbol constellation pair includes an LDPC code rate below 5/8 and a two-hundred-fifty-six-point unequally spaced symbol constellation of the first type.

15. The method of claim 14, wherein the SNR of the communication channel is equal to or less than 21.52 dB and the selected LDPC code rate and unequally spaced symbol constellation pair includes an LDPC code rate below 7/10 and a one-thousand-twenty-four-point unequally spaced symbol constellations of the first type.

16. The method of claim 9, wherein: the plurality of predetermined LDPC code rate and unequally spaced symbol constellation pairs includes a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of a second type characterized in that no two constellation points within an unequally spaced symbol constellation of the second type have locations that are the same; and the method further comprises determining likelihoods using the demapper based upon an unequally spaced symbol constellation of the second type.

17. The method of claim 16, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of the second type comprises: a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixteen-point symbol constellations of the second type; a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixty-four-point symbol constellations of the second type; and a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different two-hundred-fifty-six-point symbol constellations of the second type.

18. The method of claim 17, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of a first type comprises: a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixty-four-point symbol constellations of the first type, a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different two-hundred-fifty-six-point symbol constellations of the first type, and a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different one-thousand-twenty-four-point symbol constellations of the first type.

19. The method of claim 17, wherein: the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixteen-point symbol constellations of the first type comprises multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 3/8 and less than or equal to 6/8 and a sixteen-point unequally spaced symbol constellation of the second type; the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixty-four-point symbol constellations of the first type comprises multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 3/6 and less than or equal to 5/6 and a sixty-four-point unequally spaced symbol constellation of the second type; and the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different two-hundred-and-fifty-six-point symbol constellations of the first type comprises multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 6/8 and less than or equal to 7/8 and a two-hundred-and-fifty-six-point unequally spaced symbol constellation of the second type.

20. The method of claim 17, wherein the SNR of the communication channel is: between 3.11 dB and 9.25 dB while using at least one of the multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 3/8 and less than or equal to 6/8 and a sixteen-point unequally spaced symbol constellation of the second type; between 9.00 dB and 15.93 dB while using at least one of the multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 3/6 and less than or equal to 5/6 and a sixty-four-point unequally spaced symbol constellation of the second type; and between 18.72 dB and 22.13 dB while using at least one of the multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 6/8 and less than or equal to 7/8 and a two-hundred-and-fifty-six-point unequally spaced symbol constellation of the second type.

21. The method of claim 16, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of the second type comprises a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different one-thousand-twenty-four-point symbol constellations.

22. The method of claim 21, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different one-thousand-twenty-four-point symbol constellations of the first type comprises multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 8/10 and less than or equal to 9/10 and a one-thousand-twenty-four-point unequally spaced symbol constellation of the second type.

23. The method of claim 21, wherein the receiver successfully decodes the signal received over the communication channel when the SNR of the communication channel is between 24.79 dB and 28.20 dB while using at least one of the multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 8/10 and less than or equal to 9/10 and a one-thousand-twenty-four-point unequally spaced symbol constellation of the second type.

24. The method of claim 1, wherein the symbol constellations is characterized by the assignment of labels and locations of the constellation points such that the symbol constellation is capable of providing greater parallel decoding capacity when operated at a symbol constellation operating SNR than an equally spaced symbol constellation when operated at the same SNR.

25. The method of claim 1, further comprising: selecting the symbol constellation from a plurality of unequally spaced symbol constellations using the receiver; wherein each of the plurality of unequally spaced symbol constellations is characterized by the assignment of labels and locations of the constellation points such that each of the plurality of unequally spaced symbol constellations is capable of providing greater parallel decoding capacity when operated at a symbol constellation operating SNR than the other plurality of unequally spaced symbol constellations when operated at the same SNR.

26. The method of claim 25, wherein the symbol constellation operating SNR is a channel SNR where the receiver is capable of using the respective unequally spaced symbol constellation to receive data at a frame error rate (FER) of 10.sup.−2 or lower.

27. The method of claim 1, wherein the symbol constellation is characterized by the assignment of labels and spacing of constellation points so as to maximize parallel decoding capacity at a symbol constellation operating SNR.

28. The method of claim 1, further comprising: selecting the symbol constellation from a plurality of unequally spaced symbol constellations using the receiver; wherein each of the plurality of unequally spaced symbol constellations is characterized by the assignment of labels and spacing of constellation points so as to maximize parallel decoding capacity at a symbol constellation operating SNR.

29. The method of claim 1, further comprising: selecting the symbol constellation from a plurality of unequally spaced symbol constellations using the receiver; wherein each of the plurality of unequally spaced symbol constellations is characterized by the assignment of labels and spacing of constellation points so as to maximize parallel decoding capacity at a symbol constellation operating SNR subject to at least one constraint.

30. The method of claim 1, wherein the symbol constellation is an orthogonalized unequally spaced Pulse Amplitude Modulated (PAM) constellation, wherein the PAM constellation comprises at least two constellation points having locations that are the same and different labels.

31. The method of claim 1, further comprising enabling the symbol constellation by an upgrade to at least one of software and firmware of the receiver.

32. The method of claim 1, further comprising: receiving data bits at a transmitter comprising a coder, a mapper, and a modulator; encoding the data bits using the coder and the LDPC code; mapping the encoded bits to symbols using the mapper based upon the symbol constellation; and transmitting a transmission signal via the communication channel based upon the mapped symbols using the modulator.

33. A method of receiving data, comprising: receiving a signal at a receiver via a communication channel having a channel signal-to-noise ratio (SNR); demodulating the received signal into a demodulated signal using the receiver; determining likelihoods using the receiver based upon the demodulated signal and a symbol constellation that includes constellation points at a plurality of unique point locations, where: the plurality of unique point locations are unequally spaced; the constellation points each have a location and a different label; and the locations of at least two of the constellation points are the same; and providing a sequence of received bits from the receiver based upon the determined likelihoods and a low density parity check (LDPC) code having an associated LDPC code rate.

34. The method of claim 33, where the location of at least one of the constellation points of the symbol constellation is different from the location of any other of the constellation points.

35. The method of claim 33, further comprising: selecting the symbol constellation from a plurality of symbol constellations each having a same number of constellation points using the receiver; wherein a first of the plurality of symbol constellations includes constellation points at a first set of unique point locations that are unequally spaced; wherein a second of the plurality of symbol constellations includes constellation points at a second set of unique point locations that are unequally spaced; and wherein the number of unique point locations in the first set of unique point locations is different from the number of unique point locations in the second set of unique point locations.

36. The method of claim 33, wherein the receiver successfully decodes the signal received over the communication channel when the SNR of the communication channel is equal to or less than 5.27 dB and the associated LDPC code rate is below 1/3 and the symbol constellation includes sixty-four constellation points.

37. The method of claim 33, wherein the receiver successfully decodes the signal received over the communication channel when the SNR of the communication channel is equal to or less than 15.42 dB and the associated LDPC code rate is below 5/8 and the symbol constellation includes two-hundred-fifty-six constellation points.

38. The method of claim 33, wherein the receiver successfully decodes the signal received over the communication channel when the SNR of the communication channel is equal to or less than 21.52 dB and the associated LDPC code rate is below 7/10 and the symbol constellation includes one-thousand-twenty-four constellation points.

39. The method of claim 33, further comprising: selecting an LDPC code rate and unequally spaced symbol constellation pair from a plurality of predetermined LDPC code rate and unequally spaced symbol constellation pairs, where each of the plurality of predetermined LDPC code rate and unequally spaced symbol constellation pairs comprises: an LDPC code rate from a plurality of LDPC code rates; and an unequally spaced symbol constellation from a plurality of unequally spaced symbol constellations; wherein the selected LDPC code rate and the unequally spaced symbol constellation pair includes the associated LDPC code rate of the LDPC code and the symbol constellation.

40. The method of claim 39, wherein each of the plurality of unequally spaced symbol constellations is only included in one of the plurality of predetermined LDPC code rate and unequally spaced symbol constellation pairs.

41. The method of claim 39, wherein the plurality of predetermined LDPC code rate and unequally spaced symbol constellation pairs includes a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of a first type characterized by at least two constellation points having locations that are the same and different labels.

42. The method of claim 41, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of a first type comprises: a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixty-four-point symbol constellations of the first type, and a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different two-hundred-fifty-six-point symbol constellations of the first type.

43. The method of claim 42, wherein: the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixty-four-point symbol constellations of the first type comprises multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate below 1/3 and a sixty-four-point unequally spaced symbol constellation of the first type; and the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different two-hundred-fifty-six-point symbol constellations of the first type and multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate below 5/8 and a two-hundred-and-fifty-six-point unequally spaced symbol constellation of the first type.

44. The method of claim 41, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of a first type comprises a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different one-thousand-twenty-four-point symbol constellations of the first type.

45. The method of claim 44, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different one-thousand-twenty-four-point symbol constellations of the first type and multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate below 7/10 and a one-thousand-twenty-four-point unequally spaced symbol constellation of the first type.

46. The method of claim 41, wherein the SNR of the communication channel is: equal to or less than 5.27 dB and the selected LDPC code rate and unequally spaced symbol constellation pair includes an LDPC code rate that is below 1/3 and a sixty-four-point unequally spaced symbol constellation of the first type; and equal to or less than 15.42 dB and the selected LDPC code rate and unequally spaced symbol constellation pair includes an LDPC code rate below 5/8 and a two-hundred-fifty-six-point unequally spaced symbol constellation of the first type.

47. The method of claim 46, wherein the SNR of the communication channel is equal to or less than 21.52 dB and the selected LDPC code rate and unequally spaced symbol constellation pair includes an LDPC code rate below 7/10 and a one-thousand-twenty-four-point unequally spaced symbol constellations of the first type.

48. The method of claim 41, wherein: the plurality of predetermined LDPC code rate and unequally spaced symbol constellation pairs includes a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of a second type characterized in that no two constellation points within an unequally spaced symbol constellation of the second type have locations that are the same; and the method further comprises determining likelihoods using the receiver based upon an unequally spaced symbol constellation of the second type.

49. The method of claim 48, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of the second type comprises: a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixteen-point symbol constellations of the second type; a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixty-four-point symbol constellations of the second type; and a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different two-hundred-fifty-six-point symbol constellations of the second type.

50. The method of claim 49, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of a first type comprises: a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixty-four-point symbol constellations of the first type, a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different two-hundred-fifty-six-point symbol constellations of the first type, and a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different one-thousand-twenty-four-point symbol constellations of the first type.

51. The method of claim 49, wherein: the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixteen-point symbol constellations of the first type comprises multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 3/8 and less than or equal to 6/8 and a sixteen-point unequally spaced symbol constellation of the second type; the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixty-four-point symbol constellations of the first type comprises multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 3/6 and less than or equal to 5/6 and a sixty-four-point unequally spaced symbol constellation of the second type; and the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different two-hundred-and-fifty-six-point symbol constellations of the first type comprises multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 6/8 and less than or equal to 7/8 and a two-hundred-and-fifty-six-point unequally spaced symbol constellation of the second type.

52. The method of claim 49, wherein the SNR of the communication channel is: between 3.11 dB and 9.25 dB while using at least one of the multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 3/8 and less than or equal to 6/8 and a sixteen-point unequally spaced symbol constellation of the second type; between 9.00 dB and 15.93 dB while using at least one of the multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 3/6 and less than or equal to 5/6 and a sixty-four-point unequally spaced symbol constellation of the second type; and between 18.72 dB and 22.13 dB while using at least one of the multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 6/8 and less than or equal to 7/8 and a two-hundred-and-fifty-six-point unequally spaced symbol constellation of the second type.

53. The method of claim 48, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of the second type comprises a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different one-thousand-twenty-four-point symbol constellations.

54. The method of claim 53, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different one-thousand-twenty-four-point symbol constellations of the first type comprises multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 8/10 and less than or equal to 9/10 and a one-thousand-twenty-four-point unequally spaced symbol constellation of the second type.

55. The method of claim 53, wherein the receiver successfully decodes the signal received over the communication channel when the SNR of the communication channel is between 24.79 dB and 28.20 dB while using at least one of the multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 8/10 and less than or equal to 9/10 and a one-thousand-twenty-four-point unequally spaced symbol constellation of the second type.

56. The method of claim 33, wherein the symbol constellations is characterized by the assignment of labels and locations of the constellation points such that the symbol constellation is capable of providing greater parallel decoding capacity when operated at a symbol constellation operating SNR than an equally spaced symbol constellation when operated at the same SNR.

57. The method of claim 33, further comprising: selecting the symbol constellation from a plurality of unequally spaced symbol constellations using the receiver; wherein each of the plurality of unequally spaced symbol constellations is characterized by the assignment of labels and locations of the constellation points such that each of the plurality of unequally spaced symbol constellations is capable of providing greater parallel decoding capacity when operated at a symbol constellation operating SNR than the other plurality of unequally spaced symbol constellations when operated at the same SNR.

58. The method of claim 57, wherein the symbol constellation operating SNR is a channel SNR where the receiver is capable of using the respective unequally spaced symbol constellation to receive data at a frame error rate (FER) of 10.sup.−2 or lower.

59. The method of claim 33, wherein the symbol constellation is characterized by the assignment of labels and spacing of constellation points so as to maximize parallel decoding capacity at a symbol constellation operating SNR.

60. The method of claim 33, further comprising: selecting the symbol constellation from a plurality of unequally spaced symbol constellations using the receiver; wherein each of the plurality of unequally spaced symbol constellations is characterized by the assignment of labels and spacing of constellation points so as to maximize parallel decoding capacity at a symbol constellation operating SNR.

61. The method of claim 33, further comprising: selecting the symbol constellation from a plurality of unequally spaced symbol constellations using the receiver; wherein each of the plurality of unequally spaced symbol constellations is characterized by the assignment of labels and spacing of constellation points so as to maximize parallel decoding capacity at a symbol constellation operating SNR subject to at least one constraint.

62. The method of claim 33, wherein the symbol constellation is an orthogonalized unequally spaced Pulse Amplitude Modulated (PAM) constellation, wherein the PAM constellation comprises at least two constellation points having locations that are the same and different labels.

63. The method of claim 33, further comprising enabling the symbol constellation by an upgrade to at least one of software and firmware of the receiver.

64. The method of claim 33, further comprising: receiving data bits at a transmitter; encoding the data bits at the transmitter using the LDPC code; mapping the encoded bits to symbols at the transmitter based upon the symbol constellation; and transmitting a transmission signal via the communication channel based upon the mapped symbols using the transmitter.

65. A method of receiving data in a communication system, comprising: receiving signals at a receiver via a communication channel having a channel signal-to-noise ratio (SNR); transforming the received signals into received bits using the receiver; wherein the received signals are transformed into received bits based upon a symbol constellation and a low density parity check (LDPC) code having an associated LDPC code rate; wherein the symbol constellation includes constellation points at a plurality of unique point locations; wherein the plurality of unique point locations are unequally spaced; wherein the constellation points each have a location and a different label; and wherein the locations of at least two of the constellation points are the same.

66. The method of claim 65, where the location of at least one of the constellation points of the symbol constellation is different from the location of any other of the constellation points.

67. The method of claim 65, further comprising: selecting the symbol constellation from a plurality of symbol constellations each having a same number of constellation points using the receiver; wherein a first of the plurality of symbol constellations includes constellation points at a first set of unique point locations that are unequally spaced; wherein a second of the plurality of symbol constellations includes constellation points at a second set of unique point locations that are unequally spaced; and wherein the number of unique point locations in the first set of unique point locations is different from the number of unique point locations in the second set of unique point locations.

68. The method of claim 65, wherein the receiver successfully decodes the signal received over the communication channel when the SNR of the communication channel is equal to or less than 5.27 dB and the associated LDPC code rate is below 1/3 and the symbol constellation includes sixty-four constellation points.

69. The method of claim 65, wherein the receiver successfully decodes the signal received over the communication channel when the SNR of the communication channel is equal to or less than 15.42 dB and the associated LDPC code rate is below 5/8 and the symbol constellation includes two-hundred-fifty-six constellation points.

70. The method of claim 65, wherein the receiver successfully decodes the signal received over the communication channel when the SNR of the communication channel is equal to or less than 21.52 dB and the associated LDPC code rate is below 7/10 and the symbol constellation includes one-thousand-twenty-four constellation points.

71. The method of claim 65, further comprising: selecting an LDPC code rate and unequally spaced symbol constellation pair from a plurality of predetermined LDPC code rate and unequally spaced symbol constellation pairs, where each of the plurality of predetermined LDPC code rate and unequally spaced symbol constellation pairs comprises: an LDPC code rate from a plurality of LDPC code rates; and an unequally spaced symbol constellation from a plurality of unequally spaced symbol constellations; wherein the selected LDPC code rate and the unequally spaced symbol constellation pair includes the associated LDPC code rate of the LDPC code and the symbol constellation.

72. The method of claim 71, wherein each of the plurality of unequally spaced symbol constellations is only included in one of the plurality of predetermined LDPC code rate and unequally spaced symbol constellation pairs.

73. The method of claim 71, wherein the plurality of predetermined LDPC code rate and unequally spaced symbol constellation pairs includes a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of a first type characterized by at least two constellation points having locations that are the same and different labels.

74. The method of claim 73, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of a first type comprises: a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixty-four-point symbol constellations of the first type, and a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different two-hundred-fifty-six-point symbol constellations of the first type.

75. The method of claim 74, wherein: the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixty-four-point symbol constellations of the first type comprises multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate below 1/3 and a sixty-four-point unequally spaced symbol constellation of the first type; and the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different two-hundred-fifty-six-point symbol constellations of the first type and multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate below 5/8 and a two-hundred-and-fifty-six-point unequally spaced symbol constellation of the first type.

76. The method of claim 73, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of a first type comprises a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different one-thousand-twenty-four-point symbol constellations of the first type.

77. The method of claim 76, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different one-thousand-twenty-four-point symbol constellations of the first type and multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate below 7/10 and a one-thousand-twenty-four-point unequally spaced symbol constellation of the first type.

78. The method of claim 73, wherein the SNR of the communication channel is: equal to or less than 5.27 dB and the selected LDPC code rate and unequally spaced symbol constellation pair includes an LDPC code rate that is below 1/3 and a sixty-four-point unequally spaced symbol constellation of the first type; and equal to or less than 15.42 dB and the selected LDPC code rate and unequally spaced symbol constellation pair includes an LDPC code rate below 5/8 and a two-hundred-fifty-six-point unequally spaced symbol constellation of the first type.

79. The method of claim 78, wherein the SNR of the communication channel is equal to or less than 21.52 dB and the selected LDPC code rate and unequally spaced symbol constellation pair includes an LDPC code rate below 7/10 and a one-thousand-twenty-four-point unequally spaced symbol constellations of the first type.

80. The method of claim 73, wherein: the plurality of predetermined LDPC code rate and unequally spaced symbol constellation pairs includes a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of a second type characterized in that no two constellation points within an unequally spaced symbol constellation of the second type have locations that are the same; and the method further comprises determining likelihoods using the receiver based upon an unequally spaced symbol constellation of the second type.

81. The method of claim 80, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of the second type comprises: a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixteen-point symbol constellations of the second type; a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixty-four-point symbol constellations of the second type; and a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different two-hundred-fifty-six-point symbol constellations of the second type.

82. The method of claim 81, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of a first type comprises: a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixty-four-point symbol constellations of the first type, a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different two-hundred-fifty-six-point symbol constellations of the first type, and a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different one-thousand-twenty-four-point symbol constellations of the first type.

83. The method of claim 81, wherein: the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixteen-point symbol constellations of the first type comprises multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 3/8 and less than or equal to 6/8 and a sixteen-point unequally spaced symbol constellation of the second type; the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different sixty-four-point symbol constellations of the first type comprises multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 3/6 and less than or equal to 5/6 and a sixty-four-point unequally spaced symbol constellation of the second type; and the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different two-hundred-and-fifty-six-point symbol constellations of the first type comprises multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 6/8 and less than or equal to 7/8 and a two-hundred-and-fifty-six-point unequally spaced symbol constellation of the second type.

84. The method of claim 81, wherein the SNR of the communication channel is: between 3.11 dB and 9.25 dB while using at least one of the multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 3/8 and less than or equal to 6/8 and a sixteen-point unequally spaced symbol constellation of the second type; between 9.00 dB and 15.93 dB while using at least one of the multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 3/6 and less than or equal to 5/6 and a sixty-four-point unequally spaced symbol constellation of the second type; and between 18.72 dB and 22.13 dB while using at least one of the multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 6/8 and less than or equal to 7/8 and a two-hundred-and-fifty-six-point unequally spaced symbol constellation of the second type.

85. The method of claim 80, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include unequally spaced symbol constellations of the second type comprises a plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different one-thousand-twenty-four-point symbol constellations.

86. The method of claim 85, wherein the plurality of LDPC code rate and unequally spaced symbol constellation pairs that include multiple different one-thousand-twenty-four-point symbol constellations of the first type comprises multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 8/10 and less than or equal to 9/10 and a one-thousand-twenty-four-point unequally spaced symbol constellation of the second type.

87. The method of claim 85, wherein the receiver successfully decodes the signal received over the communication channel when the SNR of the communication channel is between 24.79 dB and 28.20 dB while using at least one of the multiple LDPC code rate and unequally spaced symbol constellation pairs that each include an LDPC code rate that is equal to or greater than 8/10 and less than or equal to 9/10 and a one-thousand-twenty-four-point unequally spaced symbol constellation of the second type.

88. The method of claim 65, wherein the symbol constellations is characterized by the assignment of labels and locations of the constellation points such that the symbol constellation is capable of providing greater parallel decoding capacity when operated at a symbol constellation operating SNR than an equally spaced symbol constellation when operated at the same SNR.

89. The method of claim 65, further comprising: selecting the symbol constellation from a plurality of unequally spaced symbol constellations using the receiver; wherein each of the plurality of unequally spaced symbol constellations is characterized by the assignment of labels and locations of the constellation points such that each of the plurality of unequally spaced symbol constellations is capable of providing greater parallel decoding capacity when operated at a symbol constellation operating SNR than the other plurality of unequally spaced symbol constellations when operated at the same SNR.

90. The method of claim 89, wherein the symbol constellation operating SNR is a channel SNR where the receiver is capable of using the respective unequally spaced symbol constellation to receive data at a frame error rate (FER) of 10.sup.−2 or lower.

91. The method of claim 65, wherein the symbol constellation is characterized by the assignment of labels and spacing of constellation points so as to maximize parallel decoding capacity at a symbol constellation operating SNR.

92. The method of claim 65, further comprising: selecting the symbol constellation from a plurality of unequally spaced symbol constellations using the receiver; wherein each of the plurality of unequally spaced symbol constellations is characterized by the assignment of labels and spacing of constellation points so as to maximize parallel decoding capacity at a symbol constellation operating SNR.

93. The method of claim 65, further comprising: selecting the symbol constellation from a plurality of unequally spaced symbol constellations using the receiver; wherein each of the plurality of unequally spaced symbol constellations is characterized by the assignment of labels and spacing of constellation points so as to maximize parallel decoding capacity at a symbol constellation operating SNR subject to at least one constraint.

94. The method of claim 65, wherein the symbol constellation is an orthogonalized unequally spaced Pulse Amplitude Modulated (PAM) constellation, wherein the PAM constellation comprises at least two constellation points having locations that are the same and different labels.

95. The method of claim 65, further comprising enabling the symbol constellation by an upgrade to at least one of software and firmware of the receiver.

96. The method of claim 1, further comprising: transforming encoded bits into symbols using the symbol constellation; and transmitting a transmission signal via the communication channel based upon the symbols.

Description

BRIEF DESCRIPTION OF DRAWINGS

[0016] FIG. 1 is a conceptual illustration of a communication system in accordance with an embodiment of the invention.

[0017] FIG. 2 is a conceptual illustration of a transmitter in accordance with an embodiment of the invention.

[0018] FIG. 3 is a conceptual illustration of a receiver in accordance with an embodiment of the invention.

[0019] FIG. 4a is a conceptual illustration of the joint capacity of a channel.

[0020] FIG. 4b is a conceptual illustration of the parallel decoding capacity of a channel.

[0021] FIG. 5 is a flow chart showing a process for obtaining a constellation optimized for capacity for use in a communication system having a fixed code rate and modulation scheme in accordance with an embodiment of the invention.

[0022] FIG. 6a is a chart showing a comparison of Gaussian capacity and PD capacity for traditional PAM-2,4,8,16,32.

[0023] FIG. 6b is a chart showing a comparison between Gaussian capacity and joint capacity for traditional PAM-2,4,8,16,32.

[0024] FIG. 7 is a chart showing the SNR gap to Gaussian capacity for the PD capacity and joint capacity of traditional PAM-2,4,8,16,32 constellations.

[0025] FIG. 8a is a chart comparing the SNR gap to Gaussian capacity of the PD capacity for traditional and optimized PAM-2,4,8,16,32 constellations.

[0026] FIG. 8b is a chart comparing the SNR gap to Gaussian capacity of the joint capacity for traditional and optimized PAM-2,4,8,16,32 constellations.

[0027] FIG. 9 is a chart showing Frame Error Rate performance of traditional and PD capacity optimized PAM-32 constellations in simulations involving several different length LDPC codes.

[0028] FIGS. 10a-10d are locus plots showing the location of constellation points of a PAM-4 constellation optimized for PD capacity and joint capacity versus user bit rate per dimension and versus SNR.

[0029] FIGS. 11a and 11b are design tables of PD capacity and joint capacity optimized PAM-4 constellations in accordance with embodiments of the invention.

[0030] FIGS. 12a-12d are locus plots showing the location of constellation points of a PAM-8 constellation optimized for PD capacity and joint capacity versus user bit rate per dimension and versus SNR.

[0031] FIGS. 13a and 13b are design tables of PD capacity and joint capacity optimized PAM-8 constellations in accordance with embodiments of the invention.

[0032] FIGS. 14a-14d are locus plots showing the location of constellation points of a PAM-16 constellation optimized for PD capacity and joint capacity versus user bit rate per dimension and versus SNR.

[0033] FIGS. 15a and 15b are design tables of PD capacity and joint capacity optimized PAM-16 constellations in accordance with embodiments of the invention.

[0034] FIGS. 16a-16d are locus plots showing the location of constellation points of a PAM-32 constellation optimized for PD capacity and joint capacity versus user bit rate per dimension and versus SNR.

[0035] FIGS. 17a and 17b are design tables of PD capacity and joint capacity optimized PAM-32 constellations in accordance with embodiments of the invention.

[0036] FIG. 18 is a chart showing the SNR gap to Gaussian capacity for traditional and capacity optimized PSK constellations.

[0037] FIG. 19 is a chart showing the location of constellation points of PD capacity optimized PSK-32 constellations.

[0038] FIG. 20 is a series of PSK-32 constellations optimized for PD capacity at different SNRs in accordance with embodiments of the invention.

[0039] FIG. 21 illustrates a QAM-64 constructed from orthogonal Cartesian product of two PD optimized PAM-8 constellations in accordance with an embodiment of the invention.

[0040] FIGS. 22a and 22b are locus plots showing the location of constellation points of a PAM-4 constellation optimized for PD capacity over a fading channel versus user bit rate per dimension and versus SNR.

[0041] FIGS. 23a and 23b are locus plots showing the location of constellation points of a PAM-8 constellation optimized for PD capacity over a fading channel versus user bit rate per dimension and versus SNR.

[0042] FIGS. 24a and 24b are locus plots showing the location of constellation points of a PAM-16 constellation optimized for PD capacity over a fading channel versus user bit rate per dimension and versus SNR.

DETAILED DESCRIPTION OF THE INVENTION

[0043] Turning now to the drawings, communication systems in accordance with embodiments of the invention are described that use signal constellations, which have unequally spaced (i.e. ‘geometrically’ shaped) points. In several embodiments, the locations of geometrically shaped points are designed to provide a given capacity measure at a reduced signal-to-noise ratio (SNR) compared to the SNR required by a constellation that maximizes drain. In many embodiments, the constellations are selected to provide increased capacity at a predetermined range of channel signal-to-noise ratios (SNR). Capacity measures that can be used in the selection of the location of constellation points include, but are not limited to, parallel decode (PD) capacity and joint capacity.

[0044] In many embodiments, the communication systems utilize capacity approaching codes including, but not limited to, LDPC and Turbo codes. As is discussed further below, direct optimization of the constellation points of a communication system utilizing a capacity approaching channel code, can yield different constellations depending on the SNR for which they are optimized. Therefore, the same constellation is unlikely to achieve the same coding gains applied across all code rates; that is, the same constellation will not enable the best possible performance across all rates. In many instances, a constellation at one code rate can achieve gains that cannot be achieved at another code rate. Processes for selecting capacity optimized constellations to achieve increased coding gains based upon a specific coding rate in accordance with embodiments of the invention are described below. In a number of embodiments, the communication systems can adapt location of points in a constellation in response to channel conditions, changes in code rate and/or to change the target user data rate.

Communication Systems

[0045] A communication system in accordance with an embodiment of the invention is shown in FIG. 1. The communication system 10 includes a source 12 that provides user bits to a transmitter 14. The transmitter transmits symbols over a channel to a receiver 16 using a predetermined modulation scheme. The receiver uses knowledge of the modulation scheme, to decode the signal received from the transmitter. The decoded bits are provided to a sink device that is connected to the receiver.

[0046] A transmitter in accordance with an embodiment of the invention is shown in FIG. 2. The transmitter 14 includes a coder 20 that receives user bits from a source and encodes the bits in accordance with a predetermined coding scheme. In a number of embodiments, a capacity approaching code such as a turbo code or a LDPC code is used. In other embodiments, other coding schemes can be used to providing a coding gain within the communication system. A mapper 22 is connected to the coder. The mapper maps the bits output by the coder to a symbol within a geometrically distributed signal constellation stored within the mapper. The mapper provides the symbols to a modulator 24, which modulates the symbols for transmission via the channel.

[0047] A receiver in accordance with an embodiment of the invention is illustrated in FIG. 3. The receiver 16 includes a demodulator 30 that demodulates a signal received via the channel to obtain symbol or bit likelihoods. The demapper uses knowledge of the geometrically shaped symbol constellation used by the transmitter to determine these likelihoods. The demapper 32 provides the likelihoods to a decoder 34 that decodes the encoded bit stream to provide a sequence of received bits to a sink.

Geometrically Shaped Constellations

[0048] Transmitters and receivers in accordance with embodiments of the invention utilize geometrically shaped symbol constellations. In several embodiments, a geometrically shaped symbol constellation is used that optimizes the capacity of the constellation. Various geometrically shaped symbol constellations that can be used in accordance with embodiments of the invention, techniques for deriving geometrically shaped symbol constellations are described below.

Selection of a Geometrically Shaped Constellation

[0049] Selection of a geometrically shaped constellation for use in a communication system in accordance with an embodiment of the invention can depend upon a variety of factors including whether the code rate is fixed. In many embodiments, a geometrically shaped constellation is used to replace a conventional constellation (i.e. a constellation maximized for d.sub.min) in the mapper of transmitters and the demapper of receivers within a communication system. Upgrading a communication system involves selection of a constellation and in many instances the upgrade can be achieved via a simple firmware upgrade. In other embodiments, a geometrically shaped constellation is selected in conjunction with a code rate to meet specific performance requirements, which can for example include such factors as a specified bit rate, a maximum transmit power. Processes for selecting a geometric constellation when upgrading existing communication systems and when designing new communication systems are discussed further below.

Upgrading Existing Communication Systems

[0050] A geometrically shaped constellation that provides a capacity, which is greater than the capacity of a constellation maximized for d.sub.min, can be used in place of a conventional constellation in a communication system in accordance with embodiments of the invention. In many instances, the substitution of the geometrically shaped constellation can be achieved by a firmware or software upgrade of the transmitters and receivers within the communication system. Not all geometrically shaped constellations have greater capacity than that of a constellation maximized for d.sub.min. One approach to selecting a geometrically shaped constellation having a greater capacity than that of a constellation maximized for d.sub.min is to optimize the shape of the constellation with respect to a measure of the capacity of the constellation for a given SNR. Capacity measures that can be used in the optimization process can include, but are not limited to, joint capacity or parallel decoding capacity.

Joint Capacity and Parallel Decoding Capacity

[0051] A constellation can be parameterized by the total number of constellation points, M, and the number of real dimensions, N.sub.dim. In systems where there are no belief propagation iterations between the decoder and the constellation demapper, the constellation demapper can be thought of as part of the channel. A diagram conceptually illustrating the portions of a communication system that can be considered part of the channel for the purpose of determining PD capacity is shown in FIG. 4a. The portions of the communication system that are considered part of the channel are indicated by the ghost line 40. The capacity of the channel defined as such is the parallel decoding (PD) capacity, given by:

[00001]$C_{\mathrm{PD}}=\underset{i=0}{\overset{l-1}{.Math.}}I\left(X_i;Y\right)$

where X.sub.i is the ith bit of the I-bits transmitted symbol, and Y is the received symbol, and I(A;B) denotes the mutual information between random variables A and B.

[0052] Expressed another way, the PD capacity of a channel can be viewed in terms of the mutual information between the output bits of the encoder (such as an LDPC encoder) at the transmitter and the likelihoods computed by the demapper at the receiver. The PD capacity is influenced by both the placement of points within the constellation and by the labeling assignments.

[0053] With belief propagation iterations between the demapper and the decoder, the demapper can no longer be viewed as part of the channel, and the joint capacity of the constellation becomes the tightest known bound on the system performance. A diagram conceptually illustrating the portions of a communication system that are considered part of the channel for the purpose of determining the joint capacity of a constellation is shown in FIG. 4b. The portions of the communication system that are considered part of the channel are indicated by the ghost line 42. The joint capacity of the channel is given by:

C.sub.JOINT=I(X;Y)

[0054] Joint capacity is a description of the achievable capacity between the input of the mapper on the transmit side of the link and the output of the channel (including for example AWGN and Fading channels). Practical systems must often ‘demap’ channel observations prior to decoding. In general, the step causes some loss of capacity. In fact it can be proven that C.sub.G≥C.sub.JIONTC.sub.PD. That is, C.sub.JOINT upper bounds the capacity achievable by C.sub.PD. The methods of the present invention are motivated by considering the fact that practical limits to a given communication system capacity are limited by C.sub.JOINT and C.sub.PD. In several embodiments of the invention, geometrically shaped constellations are selected that maximize these measures.

Selecting a Constellation Having an Optimal Capacity

[0055] Geometrically shaped constellations in accordance with embodiments of the invention can be designed to optimize capacity measures including, but not limited to PD capacity or joint capacity. A process for selecting the points, and potentially the labeling, of a geometrically shaped constellation for use in a communication system having a fixed code rate in accordance with an embodiment of the invention is shown in FIG. 5. The process 50 commences with the selection (52) of an appropriate constellation size M and a desired capacity per dimension η. In the illustrated embodiment, the process involves a check (52) to ensure that the constellation size can support the desired capacity. In the event that the constellation size could support the desired capacity, then the process iteratively optimizes the M-ary constellation for the specified capacity. Optimizing a constellation for a specified capacity often involves an iterative process, because the optimal constellation depends upon the SNR at which the communication system operates. The SNR for the optimal constellation to give a required capacity is not known a priori. Throughout the description of the present invention SNR is defined as the ratio of the average constellation energy per dimension to the average noise energy per dimension. In most cases the capacity can be set to equal the target user bit rate per symbol per dimension. In some cases adding some implementation margin on top of the target user bit rate could result in a practical system that can provide the required user rate at a lower rate. The margin is code dependent. The following procedure could be used to determine the target capacity that includes some margin on top of the user rate. First, the code (e.g. LDPC or Turbo) can be simulated in conjunction with a conventional equally spaced constellation. Second, from the simulation results the actual SNR of operation at the required error rate can be found. Third, the capacity of the conventional constellation at that SNR can be computed. Finally, a geometrically shaped constellation can be optimized for that capacity.

[0056] In the illustrated embodiment, the iterative optimization loop involves selecting an initial estimate of the SNR at which the system is likely to operate (i.e. SNR.sub.in). In several embodiments the initial estimate is the SNR required using a conventional constellation. In other embodiments, other techniques can be used for selecting the initial SNR. An M-ary constellation is then obtained by optimizing (56) the constellation to maximize a selected capacity measure at the initial SNR.sub.in estimate. Various techniques for obtaining an optimized constellation for a given SNR estimate are discussed below.

[0057] The SNR at which the optimized M-ary constellation provides the desired capacity per dimension η (SNR.sub.out) is determined (57). A determination (58) is made as to whether the SNR.sub.out and SNR.sub.in have converged. In the illustrated embodiment convergence is indicated by SNR.sub.out equaling SNR.sub.in. In a number of embodiments, convergence can be determined based upon the difference between SNR.sub.out and SNR.sub.in being less than a predetermined threshold. When SNR.sub.out and SNR.sub.in have not converged, the process performs another iteration selecting SNR.sub.out as the new SNR.sub.in (55). When SNR.sub.out and SNR.sub.in have converged, the capacity measure of the constellation has been optimized. As is explained in more detail below, capacity optimized constellation at low SNRs are geometrically shaped constellations that can achieve significantly higher performance gains (measured as reduction in minimum required SNR) than constellations that maximize d.sub.min.

[0058] The process illustrated in FIG. 5 can maximize PD capacity or joint capacity of an M-ary constellation for a given SNR. Although the process illustrated in FIG. 5 shows selecting an M-ary constellation optimized for capacity, a similar process could be used that terminates upon generation of an M-ary constellation where the SNR gap to Gaussian capacity at a given capacity is a predetermined margin lower than the SNR gap of a conventional constellation, for example 0.5 db. Alternatively, other processes that identify M-ary constellations having capacity greater than the capacity of a conventional constellation can be used in accordance with embodiments of the invention. A geometrically shaped constellation in accordance with embodiments of the invention can achieve greater capacity than the capacity of a constellation that maximizes d.sub.min without having the optimal capacity for the SNR range within which the communication system operates.

[0059] We note that constellations designed to maximize joint capacity may also be particularly well suited to codes with symbols over GF(q), or with multi-stage decoding. Conversely constellations optimized for PD capacity could be better suited to the more common case of codes with symbols over GF(2)

Optimizing the Capacity of an M-ARY Constellation at a Given SNR

[0060] Processes for obtaining a capacity optimized constellation often involve determining the optimum location for the points of an M-ary constellation at a given SNR. An optimization process, such as the optimization process 56 shown in FIG. 5, typically involves unconstrained or constrained non-linear optimization. Possible objective functions to be maximized are the Joint or PD capacity functions. These functions may be targeted to channels including but not limited to Additive White Gaussian Noise (AWGN) or Rayleigh fading channels. The optimization process gives the location of each constellation point identified by its symbol labeling. In the case where the objective is joint capacity, point bit labelings are irrelevant meaning that changing the bit labelings doesn't change the joint capacity as long as the set of point locations remains unchanged.

[0061] The optimization process typically finds the constellation that gives the largest PD capacity or joint capacity at a given SNR. The optimization process itself often involves an iterative numerical process that among other things considers several constellations and selects the constellation that gives the highest capacity at a given SNR. In other embodiments, the constellation that requires the least SNR to give a required PD capacity or joint capacity can also be found. This requires running the optimization process iteratively as shown in FIG. 5.

[0062] Optimization constraints on the constellation point locations may include, but are not limited to, lower and upper bounds on point location, peak to average power of the resulting constellation, and zero mean in the resulting constellation. It can be easily shown that a globally optimal constellation will have zero mean (no DC component). Explicit inclusion of a zero mean constraint helps the optimization routine to converge more rapidly. Except for cases where exhaustive search of all combinations of point locations and labelings is possible it will not necessarily always be the case that solutions are provably globally optimal. In cases where exhaustive search is possible, the solution provided by the non-linear optimizer is in fact globally optimal.

[0063] The processes described above provide examples of the manner in which a geometrically shaped constellation having an increased capacity relative to a conventional capacity can be obtained for use in a communication system having a fixed code rate and modulation scheme. The actual gains achievable using constellations that are optimized for capacity compared to conventional constellations that maximize d.sub.min are considered below.

Gains Achieved by Optimized Geometrically Spaced Constellations

[0064] The ultimate theoretical capacity achievable by any communication method is thought to be the Gaussian capacity, C.sub.G which is defined as:

C.sub.G=½log.sub.2(1+SNR)

[0065] Where signal-to-noise (SNR) is the ratio of expected signal power to expected noise power. The gap that remains between the capacity of a constellation and CG can be considered a measure of the quality of a given constellation design.

[0066] The gap in capacity between a conventional modulation scheme in combination with a theoretically optimal coder can be observed with reference to FIGS. 6a and 6b. FIG. 6a includes a chart 60 showing a comparison between Gaussian capacity and the PD capacity of conventional PAM-2, 4, 8, 16, and 32 constellations that maximize d.sub.min, Gaps 62 exist between the plot of Gaussian capacity and the PD capacity of the various PAM constellations. FIG. 6b includes a chart 64 showing a comparison between Gaussian capacity and the joint capacity of conventional PAM-2, 4, 8, 16, and 32 constellations that maximize d.sub.min, Gaps 66 exist between the plot of Gaussian capacity and the joint capacity of the various PAM constellations. These gaps in capacity represent the extent to which conventional PAM constellations fall short of obtaining the ultimate theoretical capacity i.e. the Gaussian capacity.

[0067] In order to gain a better view of the differences between the curves shown in FIGS. 6a and 6b at points close to the Gaussian capacity, the SNR gap to Gaussian capacity for different values of capacity for each constellation are plotted in FIG. 7. It is interesting to note from the chart 70 in FIG. 7 that (unlike the joint capacity) at the same SNR, the PD capacity does not necessarily increase with the number of constellation points. As is discussed further below, this is not the case with PAM constellations optimized for PD capacity.

[0068] FIGS. 8a and 8b summarize performance of constellations for PAM-4, 8, 16, and 32 optimized for PD capacity and joint capacity (it should be noted that BPSK is the optimal PAM-2 constellation at all code rates). The constellations are optimized for PD capacity and joint capacity for different target user bits per dimension (i.e. code rates). The optimized constellations are different depending on the target user bits per dimension, and also depending on whether they have been designed to maximize the PD capacity or the joint capacity. All the PD optimized PAM constellations are labeled using a gray labeling which is not always the binary reflective gray labeling. It should be noted that not all gray labels achieve the maximum possible PD capacity even given the freedom to place the constellation points anywhere on the real line. FIG. 8a shows the SNR gap for each constellation optimized for PD capacity. FIG. 8b shows the SNR gap to Gaussian capacity for each constellation optimized for joint capacity. Again, it should be emphasized that each ‘+’ on the plot represents a different constellation.

[0069] Referring to FIG. 8a, the coding gain achieved using a constellation optimized for PD capacity can be appreciated by comparing the SNR gap at a user bit rate per dimension of 2.5 bits for PAM-32. A user bit rate per dimension of 2.5 bits for a system transmitting 5 bits per symbol constitutes a code rate of 1/2. At that code rate the constellation optimized for PD capacity provides an additional coding gain of approximately 1.5 dB when compared to the conventional PAM-32 constellation.

[0070] The SNR gains that can be achieved using constellations that are optimized for PD capacity can be verified through simulation. The results of a simulation conducted using a rate ½ LDPC code in conjunction with a conventional PAM-32 constellation and in conjunction with a PAM-32 constellation optimized for PD capacity are illustrated in FIG. 9. A chart 90 includes plots of Frame Error Rate performance of the different constellations with respect to SNR and using different length codes (i.e. k=4,096 and k=16,384). Irrespective of the code that is used, the constellation optimized for PD capacity achieves a gain of approximately 1.3 dB, which closely approaches the gain predicted from FIG. 8a.

Capacity Optimized PAM Constellations

[0071] Using the processes outlined above, locus plots of PAM constellations optimized for capacity can be generated that show the location of points within PAM constellations versus SNR. Locus plots of PAM-4, 8, 16, and 32 constellations optimized for PD capacity and joint capacity and corresponding design tables at various typical user bit rates per dimension are illustrated in FIGS. 10a-17b. The locus plots and design tables show PAM-4,8,16,32 constellation point locations and labelings from low to high SNR corresponding to a range of low to high spectral efficiency.

[0072] In FIG. 10a, a locus plot 100 shows the location of the points of PAM-4 constellations optimized for Joint capacity plotted against achieved capacity. A similar locus plot 105 showing the location of the points of Joint capacity optimized PAM-4 constellations plotted against SNR is included in FIG. 10b. In FIG. 10c. the location of points for PAM-4 optimized for PD capacity is plotted against achievable capacity and in FIG. 10d the location of points for PAM-4 for PD capacity is plotted against SNR. At low SNRs, the PD capacity optimized PAM-4 constellations have only 2 unique points, while the Joint optimized constellations have 3. As SNR is increased, each optimization eventually provides 4 unique points. This phenomenon is explicitly described in FIG. 11a and FIG. 11b where vertical slices of FIGS. 10ab and 10cd are captured in tables describing some PAM-4 constellations designs of interest. The SNR slices selected represent designs that achieve capacities={0.5, 0.75, 1.0, 1.25, 1.5} bits per symbol (bps). Given that PAM-4 can provide at most log.sub.2(4)=2 bps, these design points represent systems with information code rates R={1/4, 3/8, 1/2, 5/8, 3/4} respectively.

[0073] FIGS. 12ab and 12cd present locus plots of PD capacity and joint capacity optimized PAM-8 constellation points versus achievable capacity and SNR. FIGS. 13a and 13b provide slices from these plots at SNRs corresponding to achievable capacities η={0.5, 1.0, 1.5, 2.0, 2.5} bps. Each of these slices correspond to systems with code rate R=η bps/log.sub.2(8), resulting in R={1/6, 1/3, 1/2, 2/3, 5/6}. As an example of the relative performance of the constellations in these tables, consider FIG. 13b which shows a PD capacity optimized PAM-8 constellation optimized for SNR=9.00 dB, or 1.5 bps. We next examine the plot provided in FIG. 8a and see that the gap of the optimized constellation to the ultimate, Gaussian, capacity (CG) is approximately 0.5 dB. At the same spectral efficiency, the gap of the traditional PAM-8 constellation is approximately 1.0 dB. The advantage of the optimized constellation is 0.5 dB for the same rate (in this case R=1/2). This gain can be obtained by only changing the mapper and demapper in the communication system and leaving all other blocks the same.

[0074] Similar information is presented in FIGS. 14abcd, and 15ab which provide loci plots and design tables for PAM-16 PD capacity and joint capacity optimized constellations. Likewise FIGS. 16abcd, 17ab provide loci plots and design tables for PAM-32 PD capacity and joint capacity optimized constellations.

Capacity Optimized PSK Constellations

[0075] Traditional phase shift keyed (PSK) constellations are already quite optimal. This can be seen in the chart 180 comparing the SNR gaps of tradition PSK with capacity optimized PSK constellations shown in FIG. 18 where the gap between PD capacity and Gaussian capacity is plotted for traditional PSK-4,8,16,32 and for PD capacity optimized PSK-4,8,16,32.

[0076] The locus plot of PD optimized PSK-32 points across SNR is shown in FIG. 19, which actually characterizes all PSKs with spectral efficiency η≤5. This can be seen in FIG. 20. Note that at low SNR (0.4 dB) the optimal PSK-32 design is the same as traditional PSK-4, at SNR=8.4 dB optimal PSK-32 is the same as traditional PSK-8, at SNR=14.8 dB optimal PSK-32 is the same as traditional PSK-16, and finally at SNRs greater than 20.4 dB optimized PSK-32 is the same as traditional PSK-32. There are SNRs between these discrete points (for instance SNR=2 and 15. dB) for which optimized PSK-32 provides superior PD capacity when compared to traditional PSK constellations.

[0077] We note now that the locus of points for PD optimized PSK-32 in FIG. 19 in conjunction with the gap to Gaussian capacity curve for optimized PSK-32 in FIG. 18 implies a potential design methodology. Specifically, the designer could achieve performance equivalent or better than that enabled by traditional PSK-4,8,16 by using only the optimized PSK-32 in conjunction with a single tuning parameter that controlled where the constellation points should be selected from on the locus of FIG. 19. Such an approach would couple a highly rate adaptive channel code that could vary its rate, for instance, rate 4/5 to achieve and overall (code plus optimized PSK-32 modulation) spectral efficiency of 4 bits per symbol, down to 1/5 to achieve an overall spectral efficiency of 1 bit per symbol. Such an adaptive modulation and coding system could essentially perform on the optimal continuum represented by the rightmost contour of FIG. 18.

Adaptive Rate Design

[0078] In the previous example spectrally adaptive use of PSK-32 was described. Techniques similar to this can be applied for other capacity optimized constellations across the link between a transmitter and receiver. For instance, in the case where a system implements quality of service it is possible to instruct a transmitter to increase or decrease spectral efficiency on demand. In the context of the current invention a capacity optimized constellation designed precisely for the target spectral efficiency can be loaded into the transmit mapper in conjunction with a code rate selection that meets the end user rate goal. When such a modulation/code rate change occurred a message could propagated to the receiver so that the receiver, in anticipation of the change, could select a demapper/decoder configuration in order to match the new transmit-side configuration.

[0079] Conversely, the receiver could implement a quality of performance based optimized constellation/code rate pair control mechanism. Such an approach would include some form of receiver quality measure. This could be the receiver's estimate of SNR or bit error rate. Take for example the case where bit error rate was above some acceptable threshold. In this case, via a backchannel, the receiver could request that the transmitter lower the spectral efficiency of the link by swapping to an alternate capacity optimized constellation/code rate pair in the coder and mapper modules and then signaling the receiver to swap in the complementary pairing in the demapper/decoder modules.

Geometrically Shaped QAM Constellations

[0080] Quadrature amplitude modulation (QAM) constellations can be constructed by orthogonalizing PAM constellations into QAM inphase and quadrature components. Constellations constructed in this way can be attractive in many applications because they have low-complexity demappers.

[0081] In FIG. 21 we provide an example of a Quadrature Amplitude Modulation constellation constructed from a Pulse Amplitude Modulation constellation. The illustrated embodiment was constructed using a PAM-8 constellation optimized for PD capacity at user bit rate per dimension of 1.5 bits (corresponds to an SNR of 9.0 dB) (see FIG. 13b). The label-point pairs in this PAM-8 constellation are {(000, −1.72), (001, −0.81), (010, 1.72), (011, −0.62), (100, 0.62), (101, 0.02), (110, 0.81), (111, −0.02)}. Examination of FIG. 21 shows that the QAM constellation construction is achieved by replicating a complete set of PAM-8 points in the quadrature dimension for each of the 8 PAM-8 points in the in-phase dimension. Labeling is achieved by assigning the PAM-8 labels to the LSB range on the in-phase dimension and to the MSB range on the quadrature dimension. The resulting 8×8 outer product forms a highly structured QAM-64 for which very low-complexity de-mappers can be constructed. Due to the orthogonality of the in-phase and quadrature components the capacity characteristics of the resulting QAM-64 constellation are identical to that of the PAM-8 constellation on a per-dimension basis.

N-Dimensional Constellation Optimization

[0082] Rather than designing constellations in 1-D (PAM for instance) and then extending to 2-D (QAM), it is possible to take direct advantage in the optimization step of the additional degree of freedom presented by an extra spatial dimension. In general it is possible to design N-dimensional constellations and associated labelings. The complexity of the optimization step grows exponentially in the number of dimensions as does the complexity of the resulting receiver de-mapper. Such constructions constitute embodiments of the invention and simply require more ‘run-time’ to produce.

Capacity Optimized Constellations for Fading Channels

[0083] Similar processes to those outlined above can be used to design capacity optimized constellations for fading channels in accordance with embodiments of the invention. The processes are essentially the same with the exception that the manner in which capacity is calculated is modified to account for the fading channel. A fading channel can be described using the following equation:

Y=a(t).Math.X+N

where X is the transmitted signal, N is an additive white Gaussian noise signal and a(t) is the fading distribution, which is a function of time.

[0084] In the case of a fading channel, the instantaneous SNR at the receiver changes according to a fading distribution. The fading distribution is Rayleigh and has the property that the average SNR of the system remains the same as in the case of the AWGN channel, E[X.sup.2]/E[N.sup.2]. Therefore, the capacity of the fading channel can be computed by taking the expectation of AWGN capacity, at a given average SNR, over the Rayleigh fading distribution of a that drives the distribution of the instantaneous SNR.

[0085] Many fading channels follow a Rayleigh distribution. FIGS. 22a-24b are locus plots of PAM-4, 8, and 16 constellations that have been optimized for PD capacity on a Rayleigh fading channel. Locus plots versus user bit rate per dimension and versus SNR are provided. Similar processes can be used to obtain capacity optimized constellations that are optimized using other capacity measures, such as joint capacity, and/or using different modulation schemes.