# Efficient Transmit Antenna Subset Selection for Multiuser Space-Time Line Code Systems
## 1 Introduction
It is shown in [23; 24] that to reduce the number of RF chains in linearly precoded multiuser MIMO systems and zero-forcing (ZF)-based precoded spatial modulation (PSM) MIMO systems, respectively, decreasing the number of active transmit antennas by TAS selection always degrades the bit error rate (BER) performance. Recently, two efficient TAS selection algorithms for PSM-based massive MIMO systems have been developed in [25]. However, the conventional various TAS selection algorithms presented in [25] and other studies on different MIMO systems are unsuitable for the MU-STLC systems owing to different transmission schemes. The reason for this unsuitability is mentioned in Section 3.1 after the system model of MU-STLC with TAS selection is described. It should be noted that one key issue in the antenna subset selection for the STLC transmission scheme is the optimal design of a proper selection criterion. In this paper, two efficient TAS selection schemes that have a better tradeoff between transmission performance and computational complexity are proposed for MU-STLC by exploiting incremental and decremental strategies, respectively, combined with the Woodbury formula. First, the Woodbury formula is utilized to reduce significantly the complexity of the conventional greedy TAS selection algorithm proposed in [14]. Second, we propose a decremental TAS selection scheme in which the Woodbury formula is also used for enormous complexity reduction. For decremental selection, the MU-STLC transmitter uses preprocessing matrices based on ZF and minimum mean square error (MMSE) senses. It is shown that the proposed decremental TAS selection algorithm based on MMSE can offer near-optimal bit error rate (BER) performance with low computational complexity. Furthermore, we show that reducing the number of activated transmit antennas through TAS selection always degrades the SINR and BER performance. The detection SINR loss experienced from TAS selection is analytically obtained. Simulation results demonstrate that the detection SINR penalty agrees with the analytical one. Without BER simulations for a large number of transmit antennas, such as massive MIMO, we can anticipate how much BER performance is degraded in terms of signal-to-noise ratio (SNR) owing to reducing the number of activated transmit antennas through TAS selection. The main contributions of this study are summarized as follows:* The effective TAS selection algorithms based on the incremental and decremental methods combined with the Woodbury formula have been designed for the MU-STLC systems;
* The computational complexity of the proposed TAS selection algorithms is analyzed and compared to the optimal one and the previous TAS selection scheme of [14]. The complexity comparison proves the efficiency of the proposed algorithms;
* The asymptotic received SINR loss is analytically provided and verified by simulation results. The rest of the paper is organized as follows. The system model of MU-STLC with TAS selection is described in Section 2. In Section 3, we propose incremental and decremental TAS selection algorithms to offer low complexity. In Section 4, the computational complexity of the proposed algorithms is analyzed and compared with that of the conventional greedy TAS selection scheme. In Section 5, a simulation-based comparison of the BER performance of the proposed algorithms and the previous method is provided. Concluding remarks are drawn in Section 6. Notations, we use lower-case and upper-case boldface letters for vectors and matrices, respectively. Superscripts \({}^{*}\), \({}^{T}\), and \({}^{H}\) denote the complex conjugate, transposition, and Hermitian transposition, respectively. The notations \(Tr(\cdot)\) and \((\cdot)^{-1}\) denote the trace and the inverse of a matrix, respectively. \(E[\cdot]\), \(|\cdot|\), and \(\|\cdot\|_{F}\) stand for the expectation, the absolute value, and the Frobenius norm, respectively.