## 2 Alamouti Technique  
The Alamouti technique is one of the simplest techniques which is used for the transmit diversity, which does not need any feedback from the receiver; however, it needs a small computation complexity  
Figure 1: The link structures for different wireless communication systems. at the transmitter side. This technique improves the signal quality, capacity, data rate, and error performance at the receiver and requires no bandwidth expansion. The Alamouti scheme can be used to increase the coverage area in wireless systems. Besides, this scheme is effective in the systems in which its capacity is limited by the multipath phenomenon. [PERSON] studied two cases; in the first case, [PERSON] assumed a system with two transmitting antennas and one receive antenna, while the second case assumed there are two antennas in both transmitted and received sides. This system modulates m-data bits using \(M\)-ary modulation to generate the symbols, where \(m=\log_{2}M\), each symbol has a complex value. [PERSON] suggests a strategy in sending these symbols during two-time slots. At the first time slot, the first transmitter sends the first symbol \(s_{1}\), while the second transmitter sends the second symbol \(s_{2}\). After that, at the second time slot the first transmitter sends the minus complex conjugate of the second symbol \(-s_{2}^{*}\), while the second transmitter sends the complex conjugate of the first symbol as shown in the following equation [1; 2; 4]:  
\[S=\left( \begin{array}{cc}s_{1}&-s_{2}^{*}\\ s_{2}&s_{1}^{*}\end{array} \right) \tag{1}\]  
where the first and second rows represent the symbols that will be transmitted by the first and second transmitter, respectively. At the same time, the first column represents the first time slot, while the second row represents the second time slot. The key feature is that the transmitted symbols at first time slot are orthogonal to the second time slot (i.e., \(s_{1}s_{2}^{*}-s_{2}^{*}s_{1}=0\)), [PERSON] assumed that the channel between the transmitter and the receiver remains constant during the two periods. Concurrently, the channel is known by the receiver. The number of channels is equal to the number of the transmit antennas multiplied by the number of receiver antennas. Therefore, there will be two channels between two transmit antennas and one receiver antenna:  
\[H=\left[h_{1}\;h_{2}\right] \tag{2}\]  
where: \(h_{1}\) and \(h_{2}\) are the channels characteristics between the transmitted antennas of the first user and the receiver antenna of the second user and can be represented as a complex value:  
\[h_{1}=\alpha_{1}e^{i\theta_{1}} \tag{3}\]  
[PERSON] noticed that the equations above are equivalent to that obtained from MRC in equations. However, the only difference is the phase of the noise components that do not affect the SNR. Moreover, higher diversity is needed in many applications where more antennas are used to increase the diversity for both transmitting and receiving sides. In the case of having two transmitting and \(M\) receiving antenna, the diversity will be \(2M\). By using the same way of sending the symbols, the received signal for both antennas during two-time slots are:  
\[y_{1}=h_{1}s_{1}+h_{2}s_{2}+n_{1} \tag{11}\]  
\[y_{2}=-h_{1}s_{2}^{*}+h_{2}s_{1}^{*}+n_{2} \tag{12}\]  
\[y_{3}=h_{3}s_{1}+h_{4}s_{2}+n_{3} \tag{13}\]  
\[y_{4}=-h_{3}s_{2}^{*}+h_{4}s_{1}^{*}+n_{4} \tag{14}\]  
where: \(n_{1}\), \(n_{2}\), \(n_{3}\) and \(n_{4}\) are representing the interference and the noise at the receiver. \(h_{1}-h_{4}\) are the characteristics of channels between the two transmit antennas and the two receiver antennas.