Methods and apparatuses for signaling with geometric constellations in a Rayleigh fading channel

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 quadrature amplitude modulation 1024-point symbol constellation (NU-QAM 1024);
• 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 the NU-QAM 1024 comprises an in-phase component and a quadrature component, where each component comprises 32 levels of amplitude such that the amplitudes scaled by a scaling factor are within 0.756 from the following set of amplitudes:
• −38.3887, −32.0467, −27.2059, −24.0230, −20.1597, −18.7135, −15.4053, −15.0236, −11.086, −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, 15.026, 15.4055, 18.7141, 20.1578, 24.0205, 27.2021, 32.0412, 38.3824.

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 quadrature amplitude modulation 1024-point symbol constellation (NU-QAM 1024);
• 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
• wherein the NU-QAM 1024 comprises an in-phase component and a quadrature component, where each component comprises 32 levels of amplitude such that the amplitudes scaled by a scaling factor are within 0.756 from the following set of amplitudes:
• −38.3887, −32.0467, −27.2059, −24.0230, −20.1597, −18.7135, −15.4053, −15.0236, −11.086, −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, 15.026, 15.4055, 18.7141, 20.1578, 24.0205, 27.2021, 32.0412, 38.3824.