Multi-dimensional continuous phase modulation in uplink of MIMO systems

Phase Modulation on the Hypersphere (PMH) is considered in which the instantaneous sum power is constant. It is shown that for an i.i.d. Gaussian channel, the capacity achieving input distribution is approximately uniform on a hypersphere when the number of receive antennas is much larger than the number of transmit antennas. Moreover, in the case that channel state information is not available at the transmitter, it is proven that the capacity achieving input distribution is exactly uniform on a hypersphere. Mutual information between input and output of PMH with discrete constellation for an i.i.d. Gaussian channel is evaluated numerically. Furthermore, a spherical spectral shaping method for PMH is proposed to have Continuous Phase Modulation on the Hypersphere (CPMH). In CPMH, the continuous time signal has a constant instantaneous sum power. It is shown that using a spherical low pass filter in spherical domain followed by a Cartesian filter results in very good spectral properties.