# A New Generation of Fast and Low-Memory Smart Digital/Geometrical Beamforming MIMO Antenna
## 2 Digital/Geometrical Hybrid MIMO Beamforming Method
### Mathematical Modeling of the Smart MIMO System
Substituting the expressions for the distances in Equation (1) and rearranging the equation by assuming that all of the distances are equal to \(r\) for the amplitude component of the far electrical field, we obtain  
\[E=\frac{e^{-j\beta r}}{r}f_{tx}(\theta)\sum_{i=1}^{N}Z_{i}\mathrm{e}^{j\beta(x _{i}cos\varphi+y_{i}sin\varphi)}\left(f_{rx}(\theta)\sum_{j=1}^{M}W_{j}\, \mathrm{e}^{-j\beta(u_{j}cos\varphi+y_{j}sin\varphi)}\right) \tag{3}\]  
In order to maximize the directional gain at the receiver, we need to obtain  
\[E=\left|\frac{e^{-j\beta r}}{r}\right|\left|f_{tx}(\theta)\sum_{i=1}^{N}Z_{i} e^{j\beta(x_{i}cos\varphi+y_{i}sin\varphi)}\right|_{max}\left|f_{rx}( \theta)\sum_{j=1}^{M}W_{j}e^{-j\beta(u_{j}cos\varphi+y_{j}sin\varphi)}\right| _{max} \tag{4}\]  
Expressing the phase delay factor as \(W_{j}=e^{j\beta(u_{j}cos\varphi+y_{j}sin\varphi)}\), which is the conjugate value of each receiving antenna element weight, and multiplying each individual term with its conjugate, the maximum directional gain can be achieved. \[\left|E\right|_{max}=\left|\frac{e^{-j\beta r}}{r}\right|\left|f_{tx}(\theta) \sum_{i=1}^{N}Z_{i}e^{j\beta(x_{i}cos\varphi+y_{i}sin\varphi)}\right|_{max}f_ {rx}(\theta)M \tag{5}\]  
where \(M\) is the number of receiving antenna elements. At the transmitter, the directional beam is to be maximized towards the receiver. In order to obtain \(\left(\left|\sum_{i=1}^{N}Z_{i}e^{j\beta(x_{i}cos\varphi+y_{i}sin\varphi)} \right|_{max}\right)\), we enhanced the single beamforming and beamsteering towards receiver direction by optimizing the phase delay coefficients \(Z_{i}\). The directivity of an antenna is defined as the ratio of the power radiation intensity in a given direction from the antenna to the power radiation intensity averaged over all directions around the antenna. The directivity \(D\) of an antenna in a specified direction is defined from its definition, as below:  
\[D=\frac{\textit{Power Density at the specified Direction}}{\textit{Total Radiated Power}}\]  
The surface area of a sphere enclosed  
Based on the above definition, we derived Equation (6) for the directivity of the transmitter antenna. \[D=\frac{2}{\pi}\frac{\left[\left|f_{tx}(\theta)\right|^{2}\left|E_{\varphi} \right|^{2}\right]}{\frac{1}{PQ}\sum_{i=1}^{P}\sum_{j=1}^{Q}\left|f_{tx}( \theta)\right|^{2}\left|E_{\varphi}\right|^{2}sin\theta} \tag{6}\]  
where \(E_{\varphi}=\sum_{i=1}^{N}Z_{i}e^{j\beta(x_{i}cos\varphi+y_{i}sin\varphi)}\). The overall power gain \(G\) can be calculated for the system described in Equation (5) as per Equation (7) given below. \[G=D_{tx}\times M^{2}\times R \tag{7}\]  
where \(D_{tx}\) is the directivity of the transmitter antenna, \(M^{2}\) is the signal processing power gain at the receiver antenna, and \(R\) is the directivity obtained from the \(f_{rx}(\theta)\) function. The receiving antenna can also perform beamforming and direct the beam towards the transmitter direction. In that case, the overall gain of the system described in Equation (4) can be determined as given in Equation (8). \[G=D_{Tx}\times D_{Rx} \tag{8}\]  
where \(D_{Rx}\) is the directivity of receiver antenna. Having beamformed the transmitter and the receiver by steering the transmitter and receiver beams towards each other, the interaction noise received at the receiver is minimized.