In spatial modulation (SM), a single signal symbol is transmitted from a given physical antenna, where both the signal symbol and the antenna index carry information. SM with multiple active antennas (MA-SM) transmits several signal symbols from a combination of antennas at each channel use, thereby increasing the spectral efficiency. MA-SM is proposed in combination with a new 3-dimensional constellation, where signal symbols transmitted from a given antenna combination are rotated before transmission. In this paper, we derived an upper-bound on the error probability of the MA-SM as a function of the rotation angles. The search for the optimal rotation angles is modeled as a multi-objective optimization problem. We concluded based on both analytical and simulation results that the 3-dimensional constellation with the optimal angles achieved negligible improvement. Therefore, we do not recommend using the 3-dimensional constellation with the MA-SM system.

Spatial modulation (SM) is a multiple-input multiple-output (MIMO) technique that achieves a MIMO capacity by conveying information through antenna indices, while keeping the transmitter as simple as that of a single-input system. Quadrature SM (QSM) expands the spatial dimension of the SM into in-phase and quadrature dimensions, which are used to transmit the real and imaginary parts of a signal symbol, respectively. A parallel QSM (PQSM) was recently proposed to achieve more gain in the spectral efficiency. In PQSM, transmit antennas are split into parallel groups, where QSM is performed independently in each group using the same signal symbol. In this paper, we analytically model the asymptotic pairwise error probability of the PQSM. Accordingly, the constellation design for the PQSM is formulated as an optimization problem of the sum of multivariate functions. We provide the proposed constellations for several values of constellation size, number of transmit antennas, and number of receive antennas. The simulation results show that the proposed constellation outperforms the phase-shift keying (PSK) constellation by more than 10 dB and outperforms the quadrature-amplitude modulation (QAM) schemes by approximately 5 dB for large constellations and number of transmit antennas.