Communication systems are described that use geometrically shaped constellations that have increased capacity compared to conventional constellations operating within a similar SNR band. In several embodiments, the geometrically shaped is optimized based upon a capacity measure such as parallel decoding capacity or joint capacity. In many embodiments, a capacity optimized geometrically shaped
constellation can be used to replace a conventional
constellation as part of a
firmware upgrade to transmitters and receivers within a communication
system. In a number of embodiments, the geometrically shaped
constellation is optimized for an
Additive White Gaussian Noise channel or a
fading channel. In numerous embodiments, the communication uses adaptive rate encoding and the location of points within the geometrically shaped constellation changes as the
code rate changes. One embodiment of the invention includes a
transmitter configured to transmit signals to a
receiver via a
communication channel, wherein the
transmitter, includes a coder configured to receive user bits and output encoded bits at an expanded output encoded
bit rate, a mapper configured to map encoded bits to symbols in a symbol constellation, a modulator configured to generate a
signal for transmission via the
communication channel using symbols generated by the mapper. In addition, the
receiver includes a demodulator configured to demodulate the received
signal via the
communication channel, a demapper configured to estimate likelihoods from the demodulated
signal, a decoder that is configured to estimate decoded bits from the likelihoods generated by the demapper. Furthermore, the symbol constellation is a capacity optimized geometrically spaced symbol constellation that provides a given capacity at a reduced signal-to-
noise ratio compared to a signal constellation that maximizes dmin.