# A Receiving Antenna Allocation Scheme for Downlink MU-MIMO-OFDM Transmission
## 2 System Model  
Figure 1 shows the system model of the downlink MU-MIMO-OFDM transmission wherein the AP communicates with \(N\) STAs. We assume that the AP has \(M\) antennas, and each STA has two antennas because most existing new smart phones, assumed as STA in this paper, are equipped with two antennas. \(\mathbf{H}_{n}(k)=[ \begin{array}{cc}\mathbf{h}_{1,n}(k)^{T}&\mathbf{h}_{2,n}(k)^{T}\end{array}  ]^{T}\in\mathbb{C}^{2\times M}\) denotes the channel response of the \(k\)th (\(k=1,\ldots,\ K\)) subcarrier between the AP and STA-\(n\). \(\mathbf{A}^{T}\) is the transpose operator of \(\mathbf{A}\). \(\mathbf{h}_{1,n}(k)\) and \(\mathbf{h}_{2,n}(k)\) are the channel responses between AP and the first and second antennas of STA-\(n\), respectively. Moreover, this paper assumes the downlink MU-MIMO-OFDM transmission with the receiving antenna allocation per subcarrier, so the AP simultaneously transmits individual data frames to the antenna combination, including the allocated receiving antennas at STAs to obtain the spatial diversity gain. On the other hand, each STA receives its own data via its allocated antenna. Although the STA can use the signal from the non-allocated antenna, this signal includes a large amount of ISI because the non-allocated antenna is not supported by the pre-coding for ISI calculation at the AP. Therefore, this paper focuses on the reception via the allocated antenna. The channel response between the AP and STA-\(n\) after applying the receiving antenna allocation yields, \(\mathbf{\hat{h}}_{n}(k)\). In the downlink MU-MIMO-OFDM transmission, the AP acquires channel responses to calculate the pre-coding weights before actual data transmission. Given the transmit signal, \(x_{n}(k)\), the received signal of the \(k\)th subcarrier at the STA-\(n\), \(y_{n}(k)\), is expressed as:  
\[y_{n}(k)=\sqrt{P_{n}}\mathbf{\hat{h}}_{n}(k)\mathbf{w}_{n}(k)x_{n}(k)+\sum_{l=1,l\
eq n }^{N}\sqrt{P_{l}}\mathbf{\hat{h}}_{n}(k)\mathbf{w}_{l}(k)x_{l}(k)+z_{n}(k), \tag{1}\]  
where \(P_{n}\) is the transmit power for STA-\(n\), \(\mathbf{w}_{n}(k)\in\mathbb{C}^{M\times 1}\), is the pre-coding weight of the \(k\)th subcarrier for STA-\(n\), and \(z_{n}(k)\) is the additive white Gaussian noise vector of the \(k\)th subcarrier with a variance of \(\sigma^{2}\). It is assumed that equal power is allocated to each STA, \(P=P_{n}\), to prevent unfairness in terms of the transmission quality between STAs. For practical use, the pre-coding weight assumed in this paper is calculated by channel inversion based on the ZF [9; 10], a low complexity weight calculation method given by:  
\[\mathbf{W}(k)=\left[ \begin{array}{cccc}\frac{\mathbf{w}_{n}(k)}{||\mathbf{w}_{1}(k)||_{F }}&\cdots&\frac{\mathbf{w}_{n}(k)}{||\mathbf{w}_{n}(k)||_{F}}&\cdots&\frac{\mathbf{w}_{N}( k)}{||\mathbf{w}_{N}(k)||_{F}}\end{array} \right], \tag{2}\]  
\[\left[ \begin{array}{cccc}\frac{\mathbf{w}_{1}(k)}{||\mathbf{w}_{1}(k)||_{F}}&\cdots &\frac{\mathbf{w}_{n}(k)}{||\mathbf{w}_{n}(k)||_{F}}&\cdots&\frac{\mathbf{w}_{N}(k)}{||\bm {w}_{N}(k)||_{F}}\end{array} \right]=\hat{\mathbf{H}}(k)^{H}\Big{(}\hat{\mathbf{H}}(k) \hat{\mathbf{H}}(k)^{H}\Big{)}^{-1}, \tag{3}\]  
where \(||\mathbf{A}||_{F}\) denotes the Frobenius norm of \(\mathbf{A}\), and \(\hat{\mathbf{H}}(k)\) is expressed as:  
\[\hat{\mathbf{H}}(k)=[ \begin{array}{cccc}\mathbf{\hat{h}}_{1}(k)^{T}&\cdots&\mathbf{\hat {h}}_{n}(k)^{T}&\cdots&\mathbf{\hat{h}}_{N}(k)^{T}\end{array} ]^{T}. \tag{4}\]  
Figure 1: System model in which the access point (AP), using \(M\) antennas, communicates with \(N\) stations (STAs) using two antennas on the downlink multiuser multiple input multiple output orthogonal frequency division multiplexing (MU-MIMO-OFDM) transmission. By using the pre-coding weight with Equation (2), the downlink MU-MIMO-OFDM transmission to the antenna combination is realized.