IP Library › Granted Patent US 12677168

Granted Patent B2

US 12677168 · App. 18/326,967 · Granted Jul 7, 2026

Methods and apparatuses for signaling with geometric constellations in a Rayleigh Fading Channel

Inventors: Maged F. Barsoum (San Jose, CA); Christopher R. Jones (Pacific Palisades, CA)

Assignee: Constellation Designs, LLC

H04W24/02 H04L1/0003 H04L1/0009 H04L5/006 H04L27/3405 H04W72/0453 H04W72/0473

View Patent ↗

Loading inventors, assignments & file history…

Monitor This Case

Get email alerts when status or documents change.

Order Certified Copies

Most orders are placed with the USPTO same day — all within 24 business hours.

Order via The Patent Place →

Pre-filled with this patent's details

Quick Facts

Patent No.: US 12677168
App. No.: 18/326,967
Filed: May 31, 2023
Granted: Jul 7, 2026
Kind: B2
Examiner: ODOM, CURTIS B
Art Unit: 2631
USPC: 375/268

External Links

USPTO Patent Center↗ Full Patent PDF↗

Abstract

Communication systems are described that use signal constellations, which have unequally spaced (i.e. ‘geometrically’ shaped) points. In many embodiments, the communication systems use specific geometric constellations that are capacity optimized at a specific SNR, over the Rayleigh fading channel. In addition, ranges within which the constellation points of a capacity optimized constellation can be perturbed and are still likely to achieve a given percentage of the optimal capacity increase compared to a constellation that maximizes d min , are also described. Capacity measures that are used in the selection of the location of constellation points include, but are not limited to, parallel decode (PD) capacity and joint capacity.

Claims (157)

1 . A communication system, comprising:

a transmitter capable of transmitting signals via a communication channel;

wherein the transmitter comprises:

a coder capable of receiving bits and outputting encoded bits using a Low Density Parity Check (LDPC) code;

a mapper, coupled to the coder, capable of mapping encoded bits to symbols in a non-uniform symbol constellation;

a modulator, coupled to the mapper, capable of producing a signal for transmission via the communication channel based upon symbols generated by the mapper;

a receiver capable of receiving signals via the communication channel at a channel signal-to-noise ratio (SNR), wherein the receiver comprises:

a demodulator capable of demodulating a received signal into a demodulated signal;

a demapper, coupled to the demodulator, capable of determining likelihoods using the non-uniform symbol constellation; and

a decoder, coupled to the demapper, capable of using likelihoods determined by the demapper to provide a sequence of received bits based upon the LDPC code;

wherein the non-uniform symbol constellation is constructed by orthogonalizing non-uniform Pulse Amplitude Modulation (PAM) constellations into Quadrature Amplitude Modulation (QAM) in-phase and quadrature components;

wherein points y(i) of the PAM constellations are defined such that y(i)=k(x(i)+r(i))+c, where k is a scaling factor, c=0, the value for −0.5320≤r(i)≤0.5320 and the value of x(i) is selected from the group consisting of:

−37.7602, −32.0093, −27.5785, −24.2219, −20.9075, −18.6078, −15.8834, −14.4076, −11.4808, −10.7002, −7.9842, −7.6972, −4.5506, −4.4707, −1.6645, −1.6412, 1.6435, 1.6666, 4.4731, 4.5532, 7.7004, 7.9858, 10.7014, 11.4808, 14.4074, 15.8829, 18.6066, 20.906, 24.2204, 27.577, 32.0068, 37.7537.

2 . The communication system of claim 1 , wherein the value of r(i) is less than or equal to 0.

3 . The communication system of claim 1 , wherein the value of r(i)=0.

4 . The communication system of claim 1 , wherein the transmitter is capable of selecting the non-uniform symbol constellation from a plurality of constellations including a plurality of non-uniform symbol constellations.

5 . The communication system of claim 1 , wherein the transmitter is capable of selecting an LDPC code rate and the non-uniform symbol constellation from a plurality of symbol constellations as a pair from a plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs.

6 . The communication system of claim 1 , wherein the symbols in the non-uniform symbol constellation are labelled using gray labels.

7 . The communication system of claim 1 , wherein the symbols in the non-uniform symbol constellation are labelled using binary reflective gray labels.

8 . The digital communication system of claim 1 , wherein the LDPC code has a code rate that is equal to or less than 0.6387.

9 . The digital communication system of claim 4 , wherein the non-uniform symbol constellation is characterized in that the non-uniform symbol constellation provides greater parallel decode capacity at a specific signal-to-noise ratio (SNR) compared to a QAM 1024 constellation that maximizes minimum distance (d min ) at the specific SNR.

10 . The digital communication system of claim 4 , wherein the non-uniform symbol constellation is characterized in that selection of the non-uniform symbol constellation from the plurality of symbol constellations in combination with an LDPC code rate that is equal to or less than 0.6387 enables a receiver to decode the transmitted signals when the communication channel has a signal-to-noise ratio (SNR) at the receiver that is between 22.4 dB and 23.2 dB.

11 . The digital communication system of claim 4 , wherein the transmitter is configured to select the non-uniform symbol constellation in combination with an LDPC code rate that is equal to or less than 0.6387.

12 . The communication system of claim 5 , wherein:

the transmitter is capable of selecting an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a second LDPC code rate and non-uniform symbol constellation pair having a code rate equal to or less than 0.27212 and a second non-uniform symbol constellation constructed by orthogonalizing second non-uniform PAM constellations into QAM in-phase and quadrature components;

wherein points y(i) of the second non-uniform PAM constellations are defined such that y(i)=k(x(i)+r(i))+c, where k is a scaling factor, c=0, the value for r(i) is −0.6870≤r(i)≤0.6870 and the value of x(i) is selected from the group consisting of:

−37.9896, −30.4644, −26.1314, −25.8467, −18.8919, −18.8919, −18.1165, −17.8829, −8.2966, −8.2966, −8.2966, −8.2966, −4.9384, −4.9384, −4.9004, −4.8867, 4.8863, 4.9008, 4.9384, 4.9384, 8.2966, 8.2966, 8.2966, 8.2966, 17.8840, 18.1156, 18.8918, 18.8918, 25.8467, 26.1314, 30.4644, and 37.9896.

13 . The communication system of claim 5 , wherein:

the transmitter is capable of selecting an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a third LDPC code rate and non-uniform symbol constellation pair having a code rate equal to or less than 0.34894 and a third non-uniform symbol constellation constructed by orthogonalizing third non-uniform PAM constellations into QAM in-phase and quadrature components;

wherein points y(i) of the third non-uniform PAM constellations are defined such that y(i)=k(x(i)+r(i))+c, where k is a scaling factor, c=0, the value for r(i) is −0.602≤r(i)≤0.602 and the value of x(i) is selected from the group consisting of:

−38.8088, −31.7370, −26.1892, −25.0630, −18.4668, −18.4668, −16.6894, −16.6675, −9.2115, −9.2115, −9.0522, −9.0480, −3.7549, −3.7549, −3.7549, −3.7549, 3.7539, 3.7539, 3.7539, 3.7539, 9.0476, 9.0509, 9.2107, 9.2107, 16.6677, 16.6875, 18.4662, 18.4662, 25.0617, 26.1893, 31.7383, and 38.8190.

14 . The communication system of claim 5 , wherein:

the transmitter is capable of selecting an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a fourth LDPC code rate and non-uniform symbol constellation pair having a code rate equal to or less than 0.40756 and a fourth non-uniform symbol constellation constructed by orthogonalizing fourth non-uniform PAM constellations into QAM in-phase and quadrature components;

wherein points y(i) of the fourth non-uniform PAM constellations are defined such that y(i)=k(x(i)+r(i))+c, where k is a scaling factor, c=0, the value for r(i) is −0.5190≤r(i)≤0.5190 and the value of x(i) is selected from the group consisting of:

−39.2317, −32.0637, −26.2140, −24.6366, −18.7399, −18.5633, −15.9554, −15.9554, −9.5560, −9.5560, −8.9852, −8.9797, −3.3963, −3.3963, −3.3645, −3.3634, 3.3646, 3.3653, 3.3966, 3.3966, 8.9803, 8.9851, 9.5562, 9.5562, 15.9543, 15.9543, 18.5618, 18.7381, 24.6352, 26.2126, 32.0635, and 39.2363.

15 . The communication system of claim 5 , wherein:

the transmitter is capable of selecting an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a fifth LDPC code rate and non-uniform symbol constellation pair having a code rate equal to or less than 0.54052 and a fifth non-uniform symbol constellation constructed by orthogonalizing fifth non-uniform PAM constellations into QAM in-phase and quadrature components;

wherein points y(i) of the fifth non-uniform PAM constellations are defined such that y(i)=k(x(i)+r(i))+c, where k is a scaling factor, c=0, the value for r(i) is −0.3410≤r(i)≤0.3410 and the value of x(i) is selected from the group consisting of:

−38.5955, −32.0616, −27.0615, −24.0453, −19.9129, −18.8201, −15.3494, −15.1509, −10.9280, −10.9088, −8.0512, −8.0512, −4.2608, −4.2608, −1.9778, −1.9769, 1.9789, 1.9801, 4.2617, 4.2617, 8.0509, 8.0509, 10.9084, 10.9264, 15.1515, 15.3492, 18.8213, 19.9127, 24.0456, 27.0586, 32.0594, 38.5954.

16 . The communication system of claim 5 , wherein:

the transmitter is capable of selecting an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a sixth LDPC code rate and non-uniform symbol constellation pair having a code rate equal to or less than 0.56526 and a sixth non-uniform symbol constellation constructed by orthogonalizing sixth non-uniform PAM constellations into QAM in-phase and quadrature components;

wherein points y(i) of the sixth non-uniform PAM constellations are defined such that y(i)=k(x(i)+r(i))+c, where k is a scaling factor, c=0, the value for r(i) is −0.31≤r(i)≤0.31 and the value of x(i) is selected from the group consisting of:

−38.3887, −32.0467, −27.2059, −24.0230, −20.1597, −18.7135, −15.4053, −15.0236, −11.0860, −11.0231, −8.0271, −8.0244, −4.4028, −4.4028, −1.8543, −1.8540, 1.8561, 1.8565, 4.4049, 4.4049, 8.0268, 8.0295, 11.0254, 11.0873, 15.0260, 15.4055, 18.7141, 20.1578, 24.0205, 27.2021, 32.0412, and 38.3824.

17 . The communication system of claim 5 , wherein:

the transmitter is capable of selecting an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a seventh LDPC code rate and non-uniform symbol constellation pair having a code rate equal to or less than 0.69882 and a seventh non-uniform symbol constellation constructed by orthogonalizing seventh non-uniform PAM constellations into QAM in-phase and quadrature components;

wherein points y(i) of the seventh non-uniform PAM constellations are defined such that y(i)=k(x(i)+r(i))+c, where k is a scaling factor, c=0, the value for r(i) is −0.441≤r(i)≤0.441 and the value of x(i) is selected from the group consisting of:

−36.8223, −31.6346, −27.5952, −24.3912, −21.3627, −18.9630, −16.4909, −14.6073, −12.2395, −10.7040, −8.5986, −7.3701, −5.1427, −4.1180, −2.0669, −1.1401, 1.1397, 2.0664, 4.1177, 5.1424, 7.3697, 8.5984, 10.7038, 12.2394, 14.6073, 16.4909, 18.9632, 21.3631, 24.3917, 27.5958, 31.6352, and 36.8223.

18 . The communication system of claim 5 , wherein:

the transmitter is capable of selecting an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a eighth LDPC code rate and non-uniform symbol constellation pair having a code rate equal to or less than 0.76978 and an eighth non-uniform symbol constellation constructed by orthogonalizing eighth non-uniform PAM constellations into QAM in-phase and quadrature components;

wherein points y(i) of the eighth non-uniform PAM constellations are defined such that y(i)=k(x(i)+r(i))+c, where k is a scaling factor, c=0, the value for r(i) is −0.2830≤r(i)≤0.2830 and the value of x(i) is selected from the group consisting of:

−35.7806, −31.1542, −27.5041, −24.5029, −21.7164, −19.3443, −17.0214, −15.0190, −12.8773, −11.1053, −9.1812, −7.5809, −5.6791, −4.1763, −2.3910, −0.9323, 0.9322, 2.3910, 4.1762, 5.6791, 7.5809, 9.1813, 11.1053, 12.8772, 15.0190, 17.0214, 19.3443, 21.7164, 24.5029, 27.5041, 31.1543, and 35.7805.

19 . The communication system of claim 5 , wherein:

the transmitter is capable of selecting an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a ninth LDPC code rate and non-uniform symbol constellation pair having a code rate equal to or less than 0.86236 and a ninth non-uniform symbol constellation constructed by orthogonalizing ninth non-uniform PAM constellations into QAM in-phase and quadrature components;

wherein points y(i) of the ninth non-uniform PAM constellations are defined such that y(i)=k(x(i)+r(i))+c, where k is a scaling factor, c=0, the value for r(i) is −0.185≤r(i)≤0.185 and the value of x(i) is selected from the group consisting of:

−34.7205, −30.6582, −27.3954, −24.6158, −22.0414, −19.7371, −17.5199, −15.4780, −13.4322, −11.5522, −9.6593, −7.8881, −6.0512, −4.3433, −2.5729, −0.8935, 0.8940, 2.5734, 4.3438, 6.0516, 7.8885, 9.6597, 11.5525, 13.4325, 15.4781, 17.5200, 19.7371, 22.0413, 24.6154, 27.3948, 30.6573, and 34.7190.

20 . A communication system, comprising:

a transmitter capable of transmitting signals via a communication channel;

wherein the transmitter comprises:

a coder capable of receiving bits and outputting encoded bits using a Low Density Parity Check (LDPC) code;

a mapper, coupled to the coder, capable of mapping encoded bits to symbols in a non-uniform symbol constellation; and

a modulator, coupled to the mapper, capable of producing a signal for transmission via the communication channel based upon symbols generated by the mapper;

wherein points y(i) of the PAM constellations are defined such that y(i)=k(x(i)+r(i)+c, where k is a scaling factor, c=0, the value for −0.5320≤r(i)≤0.5320 and the value of x(i) is selected from the group consisting of:

21 . The communication system of claim 20 , wherein the absolute value of r(i) is less than or equal to 0.2540.

22 . The communication system of claim 20 , wherein the value of r(i)=0.

23 . The communication system of claim 20 , wherein the transmitter is capable of selecting the non-uniform symbol constellation from a plurality of constellations including a plurality of non-uniform symbol constellations.

24 . The communication system of claim 20 , wherein the transmitter is capable of selecting an LDPC code rate and the non-uniform symbol constellation from a plurality of symbol constellations as a pair from a plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs.

25 . The communication system of claim 20 , wherein the symbols in the non-uniform symbol constellation are labelled using gray labels.

26 . The communication system of claim 20 , wherein the symbols in the non-uniform symbol constellation are labelled using binary reflective gray labels.

27 . The digital communication system of claim 20 , wherein the LDPC code has a code rate that is equal to or less than 0.6387.

28 . The digital communication system of claim 23 , wherein the non-uniform symbol constellation is characterized in that the non-uniform symbol constellation provides greater parallel decode capacity at a specific signal-to-noise ratio (SNR) compared to a QAM 1024 constellation that maximizes minimum distance (d min ) at the specific SNR.

29 . The digital communication system of claim 23 , wherein the non-uniform symbol constellation is characterized in that selection of the non-uniform symbol constellation from the plurality of symbol constellations in combination with an LDPC code rate that is equal to or less than 0.6387 enables a receiver to decode the transmitted signals when the communication channel has a signal-to-noise ratio (SNR) at the receiver that is between 22.4 dB and 23.2 dB.

30 . The digital communication system of claim 23 , wherein the transmitter is configured to select the non-uniform symbol constellation in combination with an LDPC code rate that is equal to or less than 0.6387.

31 . The communication system of claim 24 , wherein:

32 . The communication system of claim 24 , wherein:

the transmitter is capable of selecting an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a third LDPC code rate and non-uniform symbol constellation pair having a code rate equal to or less than 0.34894 and a third non-uniform symbol constellation constructed by orthogonalizing third non-uniform PAM constellations into QAM in-phase and quadrature components;

33 . The communication system of claim 24 , wherein:

the transmitter is capable of selecting an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a fourth LDPC code rate and non-uniform symbol constellation pair having a code rate equal to or less than 0.40756 and a fourth non-uniform symbol constellation constructed by orthogonalizing fourth non-uniform PAM constellations into QAM in-phase and quadrature components;

34 . The communication system of claim 24 , wherein:

the transmitter is capable of selecting an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a fifth LDPC code rate and non-uniform symbol constellation pair having a code rate equal to or less than 0.54052 and a fifth non-uniform symbol constellation constructed by orthogonalizing fifth non-uniform PAM constellations into QAM in-phase and quadrature components;

35 . The communication system of claim 24 , wherein:

the transmitter is capable of selecting an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a sixth LDPC code rate and non-uniform symbol constellation pair having a code rate equal to or less than 0.56526 and a sixth non-uniform symbol constellation constructed by orthogonalizing sixth non-uniform PAM constellations into QAM in-phase and quadrature components;

36 . The communication system of claim 24 , wherein:

the transmitter is capable of selecting an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a seventh LDPC code rate and non-uniform symbol constellation pair having a code rate equal to or less than 0.69882 and a seventh non-uniform symbol constellation constructed by orthogonalizing seventh non-uniform PAM constellations into QAM in-phase and quadrature components;

37 . The communication system of claim 24 , wherein:

the transmitter is capable of selecting an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is an eighth LDPC code rate and non-uniform symbol constellation pair having a code rate equal to or less than 0.76978 and an eighth non-uniform symbol constellation constructed by orthogonalizing eighth non-uniform PAM constellations into QAM in-phase and quadrature components;

38 . The communication system of claim 24 , wherein:

the transmitter is capable of selecting an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a ninth LDPC code rate and non-uniform symbol constellation pair having a code rate equal to or less than 0.86236 and a ninth non-uniform symbol constellation constructed by orthogonalizing ninth non-uniform PAM constellations into QAM in-phase and quadrature components;

39 . A communication system, comprising:

a receiver capable of receiving signals via a communication channel having a channel signal-to-noise ratio (SNR), wherein the receiver comprises:

a demodulator capable of demodulating a received signal into a demodulated signal;

a demapper, coupled to the demodulator, capable of determining likelihoods using a non-uniform symbol constellation; and

a decoder, coupled to the demapper, capable of using likelihoods determined by the demapper to provide a sequence of received bits based upon the LDPC code; and

40 . The communication system of claim 39 , wherein the absolute value of r(i) is less than or equal to 0.2540.

41 . The communication system of claim 39 , wherein the value of r(i)=0.

42 . The digital communication system of claim 39 , wherein the LDPC code rate has a code rate that is equal to or less than 0.6387.

43 . The digital communication system of claim 39 , wherein the receiver is configured to select the non-uniform symbol constellation from a plurality of symbol constellations.

44 . The digital communication system of claim 43 , wherein the non-uniform symbol constellation is characterized in that the non-uniform symbol constellation provides greater parallel decode capacity at a specific signal-to-noise ratio (SNR) compared to a QAM 1024 constellation that maximizes minimum distance (d min ) at the specific SNR.

45 . The digital communication system of claim 43 , wherein the receiver is configured to select the non-uniform symbol constellation from the plurality of symbol constellations in combination with an LDPC code rate that is equal to or less than 0.6387 and the receiver is capable of decoding the signals received via the communication channel using the LDPC code rate and the non-uniform symbol constellation when the communication channel SNR is between 22.4 dB and 23.2 dB.

46 . The digital communication system of claim 43 , wherein the receiver is configured to select the non-uniform symbol constellation in combination with an LDPC code rate that is equal to or less than 0.6387.

47 . The communication system of claim 43 , wherein the receiver is capable of selecting an LDPC code rate and the non-uniform symbol constellation from a plurality of symbol constellations as a pair from a plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs.

48 . The communication system of claim 47 , wherein each of the plurality of non-uniform symbol constellations is only included in one of the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs.

49 . The communication system of claim 47 , wherein:

the receiver is configured to select an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a second LDPC code rate and non-uniform symbol pair having a code rate equal to or less than 0.27212 and a second non-uniform symbol constellation constructed by orthogonalizing second non-uniform PAM constellations into QAM in-phase and quadrature components;

50 . The communication system of claim 47 , wherein:

the receiver is configured to select an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a third LDPC code rate and non-uniform symbol pair having a code rate equal to or less than 0.34894 and a third non-uniform symbol constellation constructed by orthogonalizing third non-uniform PAM constellations into QAM in-phase and quadrature components;

51 . The communication system of claim 47 , wherein:

the receiver is configured to select an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a fourth LDPC code rate and non-uniform symbol pair having a code rate equal to or less than 0.40756 and a fourth non-uniform symbol constellation constructed by orthogonalizing fourth non-uniform PAM constellations into QAM in-phase and quadrature components;

wherein points y(i) of the fourth non-uniform PAM constellations are defined such that y(i)=k(x(i)+r(i))+c, where k is a scaling factor, c=0, the value for r(i) is −0.5190≤(i)≤0.5190 and the value of x(i) is selected from the group consisting of:

52 . The communication system of claim 47 , wherein:

the receiver is configured to select an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a fifth LDPC code rate and non-uniform symbol pair having a code rate equal to or less than 0.54052 and a fifth non-uniform symbol constellation constructed by orthogonalizing fifth non-uniform PAM constellations into QAM in-phase and quadrature components;

53 . The communication system of claim 47 , wherein:

the receiver is configured to select an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a sixth LDPC code rate and non-uniform symbol pair having a code rate equal to or less than 0.56526 and a sixth non-uniform symbol constellation constructed by orthogonalizing sixth non-uniform PAM constellations into QAM in-phase and quadrature components;

54 . The communication system of claim 47 , wherein:

the receiver is configured to select an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a seventh LDPC code rate and non-uniform symbol pair having a code rate equal to or less than 0.69882 and a seventh non-uniform symbol constellation constructed by orthogonalizing seventh non-uniform PAM constellations into QAM in-phase and quadrature components;

55 . The communication system of claim 47 , wherein:

the receiver is configured to select an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a eighth LDPC code rate and non-uniform symbol pair having a code rate equal to or less than 0.76978 and a eighth non-uniform symbol constellation constructed by orthogonalizing eighth non-uniform PAM constellations into QAM in-phase and quadrature components;

wherein points y(i) of the eight non-uniform PAM constellations are defined such that y(i)=k(x(i)+r(i))+c, where k is a scaling factor, c=0, the value for r(i) is −0.2830≤r(i)≤0.2830 and the value of x(i) is selected from the group consisting of:

56 . The communication system of claim 47 , wherein:

the receiver is configured to select an alternative LDPC code rate and non-uniform symbol constellation pair from the plurality of predetermined LDPC code rate and non-uniform symbol constellation pairs, where the alternative LDPC code rate and non-uniform symbol constellation pair is a ninth LDPC code rate and non-uniform symbol pair having a code rate equal to or less than 0.86236 and a ninth non-uniform symbol constellation constructed by orthogonalizing eighth non-uniform PAM constellations into QAM in-phase and quadrature components;

57 . The digital communication system of claim 39 , wherein the symbols in the non-uniform symbol constellation are labelled using Gray labels.

58 . The digital communication system of claim 39 , wherein the symbols in the non-uniform symbol constellation are labelled using binary reflective Gray labels.