## 2 ESM System Model and Conventional Detectors
### ESM System Framework  
In this paper, we consider the case of two transmit antennas (2-TX) in the ESM system, and the primary modulation is quadrature phase shift keying (QPSK). Based on the above, the transmitted vector set \(\mathbf{X}\) can be represented as  
\[\mathbf{X}\in\{\begin{bmatrix}\mathcal{C}_{4}\\ 0\end{bmatrix},\begin{bmatrix}\mathcal{D}_{2}\\ \mathcal{C}_{4}\end{bmatrix},\begin{bmatrix}\mathcal{B}_{2}^{0}\\ \mathcal{B}_{2}^{0}\end{bmatrix},\begin{bmatrix}\mathcal{B}_{2}^{1}\\ \mathcal{B}_{2}^{1}\end{bmatrix}\}, \tag{2}\]  
where \(\mathcal{C}_{4}\) denotes the QPSK signal constellation used as the primary constellation and \(\mathcal{B}_{2}^{0}\), \(\mathcal{B}_{2}^{1}\) represent two secondary constellations given by \(\mathcal{B}_{2}^{0}=\{+1,-1\}\), \(\mathcal{B}_{2}^{1}=\{+i,-i\}\). Clearly, the first two components of the transmission vector correspond to the scenario where a single antenna is activated, transmitting a symbol from the primary constellation. The last two components correspond to a scenario where two antennas are activated, and both transmission antennas can simultaneously transmit symbols from the secondary constellation. However, each of these symbols carries only 1 bit of information, which is half the amount carried by the symbols from the primary constellation. The goal of this approach is to match the bit quantity conveyed by the first and second components. Additionally, the derivation of the secondary constellation is primarily guided by the criterion of maximizing the minimum Euclidean distance between the primary and secondary constellations without increasing the transmitted signal power. We name \(\mathcal{B}_{2}^{0}\) as Binary Space Shift Keying 0 (BPSK0) and \(\mathcal{B}_{2}^{1}\) as Binary Space Shift Keying 1 (BPSK1). An example of the signal constellation implemented for this design can be observed in Figure 1, where four signal space elements are described in Table 1. In each time slot, the number of activated antennas is \(Np\) and this scheme transmits 4 bits per channel use (bpcu). The number of bits of information is divided into two parts, where one part is used for the selection of transmit antenna combinations (TACs), and the other part is used for carrying symbol bits. Based on this, the number of bits for TAC (\(B-TAC\)) and symbol (\(B-SYM\)) can be expressed as  
\[B-TAC=\log_{2}TAC, \tag{3}\]  
\[B-SYM=N_{p}\log_{2}M, \tag{4}\]  
where the conjugate transpose of matrix \(\mathbf{H}\) can be represented as \(\mathbf{H}^{H}\). This method is sensitive to channel noise, which limits its performance in the presence of strong noise effects. From this perspective, the MMSE method, which takes noise into account, yields better BER performance than ZF. The MMSE expression is as follows:  
\[\mathbf{\hat{x}}_{MMSE}=(\mathbf{H}^{H}\mathbf{H}+\sigma^{2}\mathbf{I})^{-1} \mathbf{H}^{H}\mathbf{y}. \tag{8}\]  
MMSE detection takes into account the impact of noise during the equalization process, but in doing so, the noise may be amplified, especially when the channel conditions are poor. Therefore, MMSE detection requires a higher degree of accuracy in channel state information (CSI), and from this perspective, the difficulty of implementing MMSE is greater.