Joint Precoding of High Efficiency MPSK Transmitters in MISO Channels

This work addresses the multiple input single output (MISO) channel when the transmitted waveform at the antenna belongs to small discrete constellation. We do not allow precoding matrix between the modulators and the antennas that modifies the constellation at the antenna. In the proposed scheme, the constellation at each antenna is the original modulation by limiting the available precoding to relative phase shift and relative gain between the antennas. This way, simple modulation hardware can be used, as well as high efficiency power amplifiers working near saturation. The mutual information of the MISO Gaussian channel with channel state information (CSI) at the receiver and the transmitter is maximized by simple beamforming, if the input constellation is dense. We show that in case the constellation at the antenna is small, like MPSK, the beamforming solution does not lead to the highest rate. The optimal solution is obtained by precoding and joint signal shaping through nonuniform constellation transmission. A gain is achieved since the combination of few small constellations provides richer constellation in the receiver than the original constellation. We discuss in detail the case of a system with two transmitting antennas and one receiving antenna and QPSK modulation. We address two types of power constraints: the Tx power constraint and Rx power constraint, wherein in each case the capacity is maximized at the respective power constraint. We show that the optimal solution is highly dependent on the SNR and on the type of constraint. For example in the case of Tx power constraint, beamforming is the optimal solution in low SNR while in medium to high SNR nontrivial solutions are obtained. Additionally, the shaping distribution behaves inversely to the typical shaping: instead of Maxwell-Boltzmann distribution, we obtain what we call Inverse Maxwell-Boltzmann distribution.