# Hybrid Precoding Design for Wireless Fading Channels
## 2 System Model  
Since the emergence of orthogonal frequency division multiplexing (OFDM) technology, research on precoding techniques in transmitters with multiple antennas has assumed an OFDM system, even without specific mention. In an OFDM system, independent transmission, reception, and signal processing operations are performed for each subcarrier through fast Fourier transform (FFT) and inverse FFT (IFFT) operations. The time-domain channel between the transmitter and receiver is generally a multipath fading channel in OFDM systems; however, the appropriate selection of OFDM parameters results in a frequency flat-fading channel for each subcarrier. Therefore, the received signal considered in this paper refers to the received signal at each subcarrier, and the channel model applied to the received signal is assumed to be a frequency flat-fading channel. Figure 1 shows the system model considered in this paper. Assume that the base station is transmitting data to (K + J) users, the number of transmitting antennas at the base station is M, and the number of receiving antennas for each user is one. In cases where there are multiple receiving antennas, the proposed technique can be easily applied by modifying it, so, for simplicity, this paper only considers the case where there is one receiving antenna. Let the data symbol for the m-th user be \(s_{m}\), and its mean and variance are \(E[s_{m}]=0\) and \(E[|s_{m}|^{2}]=1\), respectively. If symbol-level-based precoding is applied to all user data at the transmitter, the SER performance of each user will be excellent, but the computational complexity at the transmitter will become too high. Therefore, average-based linear MMSE precoding is applied to the data \(\mathbf{s}_{1}=[s_{1},\ \ldots,s_{K}]^{T}\) for K users, and symbol-based precoding is applied to the data \(\mathbf{s}_{2}=[s_{K+1},\ \ldots,s_{K+J}]^{T}\) for the remaining J users to implement a trade-off between the SER performance and the computation complexity. Let us assume that the precoding matrix for linear precoding is \(\mathbf{W}_{1}\) of size M \(\times\) K and that the data vector obtained through symbol-level precoding is \(\mathbf{x}_{2}=[x_{1},\ \ldots,\ x_{M}]^{T}\in\mathbb{C}^{M\times 1}\). In this case, the transmission signal, \(\mathbf{c}\), can be expressed as follows:  
\[\mathbf{c}=\beta_{1}\mathbf{W}_{1}\mathbf{s}_{1}+\beta_{2}\mathbf{x}_{2}, \tag{1}\]  
where \(\beta_{1}\) represents a parameter for adjusting the transmission power for linear precoding to be \(P_{1}\), and \(\beta_{2}\) represents a parameter for adjusting the transmission power for symbol level precoding to be \(P_{2}\). Therefore, \(\beta_{1}\) and \(\beta_{2}\) are given as follows:  
\[E[||\beta_{1}\mathbf{W}_{1}\mathbf{s}_{1}||^{2}]=\beta_{1}^{2}\text{tr}( \mathbf{W}_{1}\mathbf{W}_{1}^{H})=P_{1}\Rightarrow\beta_{1}=\sqrt{\frac{P_{ 1}}{\text{tr}(\mathbf{W}_{1}\mathbf{W}_{1}^{H})}}, \tag{2}\]  
\[||\beta_{2}\mathbf{x}_{2}||^{2}=P_{2}\Rightarrow\beta_{2}=\sqrt{\frac{P_{2}}{ ||\mathbf{x}_{2}||^{2}}}, \tag{3}\]  
where the properties \(E[\mathbf{s}_{1}\mathbf{s}_{1}^{H}]=\mathbf{I}_{K}\) and \(\text{tr}[\mathbf{W}_{1}^{H}\mathbf{W}_{1}]=\text{tr}[\mathbf{W}_{1}\mathbf{ W}_{1}^{H}]\) are used. As can be seen in Equation (1), since the base station applies both average-based linear precoding and symbol-level precoding, this technique is called hybrid precoding in this paper. Assuming that the channel between the base station and the k-th user is \(\mathbf{h}_{k}\in\mathbb{C}^{M\times 1}\), the received signal, \(y_{k}\), by the k-th user is given as follows:  
\[y_{k}=\mathbf{h}_{k}^{H}(\beta_{1}\mathbf{W}_{1}\mathbf{s}_{1}+\beta_{2} \mathbf{x}_{2})+z_{k}, \tag{4}\]  
where \(z_{k}\) represents additive white Gaussian noise (AWGN) with a mean of 0 and a variance of \(\sigma_{z}^{2}\). Figure 1: System model considered in this paper, where the base station with multiple transmitting antennas is transmitting data to (K + J) users with one receiving antenna each.