# A Receiving Antenna Allocation Scheme for Downlink MU-MIMO-OFDM Transmission
## 3 Proposed Scheme
### Receiving Antenna Decision Method  
Both the coordinated Tx-Rx BD techniques [9; 13] and DL-MU-MIMO transmissions with receiving antenna allocation/selection schemes [14; 15] must require information of the receiver weights or allocated receiving antennas before receiving actual data frames, so the transmission efficiency, which is the ratio of time spent transmitting actual data to total time, is degraded. To reduce this overhead, the proposed method determines the receiving antenna by comparing the received powers of each antenna at STA using a few OFDM signals. In detail, each STA determines the receiving antenna with minimum received power as calculated from OFDM signals among all antennas using:  
\[\left\{ \begin{array}{l}\hat{\mathbf{h}}_{n}(k)=\mathbf{h}_{1,n}(k)\ if\ |y_{1,n}(k)|\leq|y_{2,n}(k)|\\ \hat{\mathbf{h}}_{n}(k)=\mathbf{h}_{2,n}(k)\ if\ |y_{1,n}(k)|>|y_{2,n}(k)|\end{array}  \right., \tag{9}\]  
where \(y_{1,n}(k)\) and \(y_{2,n}(k)\) are the received signals of the first and second antennas, respectively. Moreover, for improving the accuracy of the receiving antenna decision, we assume that the pre-coding weight, \(\mathbf{w}(k)\), which defines the right singular vector corresponding to the minimum eigenvalue, is given by:  
\[\mathbf{w}t(k)=\hat{\mathbf{v}}_{N}(k), \tag{10}\]  
\[\hat{\mathbf{H}}(k)=\hat{\mathbf{U}}(k)\hat{\mathbf{\Sigma}}(k)\hat{\mathbf{V}}(k)^{H}, \tag{11}\]  
\[\hat{\mathbf{V}}(k)=[\ \ \hat{\mathbf{v}}_{1}(k)\ \ \ \cdots\ \ \ \hat{\mathbf{v}}_{N}(k)\ \ ], \tag{12}\]  
where \(\mathbf{A}^{H}\) denotes the Hermitian transpose operator of \(\mathbf{A}\), \(\hat{\mathbf{U}}(k)\in\mathbb{C}^{N\times N}\) is the matrix combined from the left singular vectors of the \(k\)th subcarrier, \(\hat{\mathbf{\Sigma}}(k)\in\mathbb{C}^{N\times N}\) is the diagonal matrix of the \(k\)th subcarrier in which the diagonal elements represent the square roots of the eigenvalues, and \(\hat{\mathbf{V}}(k)\in\mathbb{C}^{N\times N}\) is the matrix combined from the right singular vectors of the \(k\)th subcarrier. Actual propagation environments exhibit frequency selective fading, shadowing, transmit beamforming, etc., so the STAs may err in determining the receiving antenna. To further improve the accuracy of the antenna decision, this paper applies a receiving antenna allocation scheme that appliesthe receiving antenna decision method. In this proposal, the AP calculates the SINR to estimate the receiving antenna per subcarrier at each STA, given by:  
\[\left\{ \begin{array}{l}SINR_{1,n}(k,s(k))=\frac{\left|\sqrt{\mathcal{P}_{h,s(k) }}(k)\mathbf{w}_{n,s(k)}(k)\right|^{2}}{\sigma^{2}+\sum_{i=1,j\
eq n}^{N}\left| \sqrt{\mathcal{P}_{h,s(k)}(k)\mathbf{w}_{n,s(k)}(k)}\right|^{2}}\\ SINR_{n}(k,s(k))=\frac{\left|\sqrt{\mathcal{P}_{h,s(k)}(k)\mathbf{w}_{n,s(k)}(k)} \right|^{2}}{\sigma^{2}+\sum_{i=1,j\
eq n}^{N}\left|\sqrt{\mathcal{P}_{h,s(k)} (k)\mathbf{w}_{n,s(k)}(k)}\right|^{2}}\end{array} \right. \tag{13}\]  
where \(SINR_{1,n}(k,s(k))\) and \(SINR_{2,n}(k,s(k))\) are the SINR values of the first and second antennas, respectively. The AP estimates the receiving antenna decision error at STA by using Equation (13). In addition, this process is performed only for the STAs that fed both channel responses back to the AP. Finally, the antenna combination that maximizes the channel capacity is calculated by Equations (6) and (7) from candidates without the antennas that may cause errors in the antenna decision. We present an example frame format for the proposed method that suits IEEE 802.11 ac. This frame is used by the STAs that fed back both channel responses, while STAs that fed back the channel response of a single antenna uses a general receiving method with a selected antenna. Figure 3a,b show the example frame format with proposed/general pre-cording regions and the receiving process with the concept of the proposed antenna decision method, respectively.