## 3 Spatial Phase Coding
### Constructive Spatial Phase Coding  
This section describes the constructive SPC scheme. The transmitter uses the pre-coding vector to modify the phase of the channel coefficients. The pre-coding vector is determined by the phase relationship between the channel coefficients and depends on the feedback information. The receiver obtains feedback information through the channel estimation and transmits it to the transmitter. According to the number of feedback bits, the constructive SPC scheme is divided into constructive 1- and 2-bit feedback SPCs. The pre-coding vector \(C_{1}^{S}\) of the constructive 1-bit feedback SPC is as follows,  
\[C_{1}^{S}=\left\{ \begin{array}{rl}1,&0<|\alpha|\leq\pi/2\quad(\text{State 1})\\ \text{e}^{-j\pi},&\pi/2<|\alpha|\leq\pi\quad(\text{State 2})\end{array}  \right\}. \tag{3}\]  
The \(\alpha\) is the phase difference between the two channel coefficients. If the state of \(\alpha\) is the State 1, the phase of the channel coefficient is not modified. On the other hand, if the state of \(\alpha\) is the State 2, the phase of the channel coefficient is flipped by the \(C_{1}^{S}\) in this State 2 [17]. The pre-coding vector \(C_{2}^{S}\) of the constructive 2-bit feedback SPC is as follows,  
\[C_{2}^{S}=\left\{ \begin{array}{rl}1,&\left(\begin{array}{rl}0<|\alpha|\leq \pi/4\\ 7\pi/4<|\alpha|\leq 2\pi\end{array}\right)\quad(\text{State 1})\\ \text{e}^{-j\pi/2},&\pi/4<|\alpha|\leq 3\pi/4\quad(\text{State 2})\\ \text{e}^{-j\pi},&3\pi/4<|\alpha|\leq 5\pi/4\quad(\text{State 3})\\ \text{e}^{j\pi/2},&5\pi/4<|\alpha|\leq 7\pi/4\quad(\text{State 4})\end{array}  \right\}. \tag{4}\]  
If the state of \(\alpha\) is 1, the phase is not modified. On the other hand, if the state of \(\alpha\) is not 1, the phase is modified to each state [20]. The transmitter multiplies the transmission signal by the pre-coding vectors \(C_{1}^{S}\), \(C_{2}^{S}\), and transmits the signal. Therefore, the phase of the channel coefficients experienced by the OFDM sub-carriers is selectively flipped. When the signals are transmitted by multiplying the pre-coding vectors \(C_{1}^{S}\), \(C_{2}^{S}\), the SNR of the received signal can be improved because the amplitude of the superimposed channel coefficient can be increased. Figure 2 shows the constructive 1-bit and 2-bit feedback SPCs. The \(H_{1}\) and \(H_{2}\) refer to channel coefficients. The \(H_{c}\) refers to a superimposed channel coefficient that the constructive 1-bit or 2-bit feedback SPC is not applied. The \(H_{2}\)-\(C_{1}^{S}\) and \(H_{2}\)-\(C_{2}^{S}\) are the channel coefficients whose phase is modified by the pre-coding vectors. The \(H_{S}\) refers to the superimposed channel coefficient that the constructive 1-bit or 2-bit feedback SPC is applied. As shown in Figure 2, the \(H_{S}\) that the constructive SPC is applied is larger than the \(H_{c}\) that the constructive SPC is not applied [17]. So, the mobile can  
receive a signal having a high signal-to-noise ratio (SNR) when constructive SPC is applied. And, when the constructive 2-bit SPC scheme is applied, the magnitude of the superimposed channel coefficient can be effectively increased as compared with the constructive 1-bit SPC scheme. One of the pre-coding schemes that uses feedback information is the maximum ratio transmission (MRT) scheme [5]. The MRT scheme is an optimal pre-coding scheme which uses full feedback information. Although the scheme has a slightly better performance than the SPC, it has a very high implementation complexity. The pre-coding coefficient of MRT is as follows,  
\[\textbf{C}_{n}=\left(H_{n}^{(1)*},\,H_{n}^{(2)*},\,\ldots,\,H_{n}^{(m)*} \right)^{T}. \tag{5}\]  
The full channel information about \(H_{n}^{(m)*}\) must be estimated at the receiver. Thus, the MRT requires the most complex estimation with highest overhead of all pre-coding schemes. Moreover, the MRT requires full feedback of amplitude and phase of each channel. Therefore, the MRT requires a large amount of feedback information [17].