# 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  
The proposed new digital/geometrical smart MIMO antenna scheme of this paper can be illustrated using Figure 3 in the XY coordinates system. The transmitter (Tx) has \(N\) number of multiple antenna elements, with each element positioned in the XY coordinates plane at \((x_{1},y_{1})\), \((x_{2},y_{2})\), \((x_{3},y_{3})\), \(\ldots\), \((x_{n},y_{n})\). The receiver (Rx) has \(M\) number of antenna elements, with each element positioned in the UV coordinates plane at \((u_{1},\,v_{1})\), \((u_{2},\,v_{2})\), \((u_{3},\,v_{3})\), \(\ldots\), \((u_{n},\,v_{n})\). \(r\) is the distance between the reference points of the transmitter antenna and the receiver antenna elements. Figure 1: Transmitter antenna of the proposed MIMO system. Figure 2: Receiver antenna of the proposed MIMO system. Referring to Figure 3, the distance between the first antenna element of the transmitter and the first antenna element of the receiver can be written as  
\[r_{1,1}=r-x_{1}\cos(\varphi)-y_{1}\sin(\varphi)+u_{1}\cos(\varphi)+v_{1}\sin(\varphi)\]  
The distance between the second antenna element of the transmitter and the first antenna element of the receiver can be written as  
\[r_{2,1}=r-x_{2}\cos(\varphi)-y_{2}\sin(\varphi)+u_{1}\cos(\varphi)+v_{1}\sin(\varphi)\]  
Similarly, the distance between the \(n^{\text{th}}\) antenna element of the transmitter and the \(m^{\text{th}}\) antenna element of the receiver can be written as  
\[r_{n,m}=r-x_{n}\cos(\varphi)-y_{n}\sin(\varphi)+u_{m}\cos(\varphi)+v_{m}\sin(\varphi)\]  
With the assumption that all of the transmitter antenna elements have the same length and the same current distribution function along the dipole elements, the vertical plane radiation function \(f_{tx}(\theta)\) for all of the transmitter elements are the same. Here, \(\theta\) is the angle that the observation point makes with the vertical axis of the dipole element. Similarly, with the same assumption, we can also say that the vertical plane radiation function \(f_{rx}(\theta)\) of the receiver antenna elements remains same for the receiver elements. Therefore, the entire electric field (\(E\)) at the receiver can be written as follows, by incorporating the transmitter and receiver models shown in Figures 1 and 2:  
\[ \begin{array}{l}E=\{f_{tx}(\theta)f_{rx}(\theta)&\left[\frac{e^{-j\mu_{1,1}} }{r_{1,1}}Z_{1}W_{1}+\frac{e^{-j\mu_{2,1}}}{r_{2,1}}Z_{2}W_{1}+\cdots\frac{e^{- j\mu_{n,1}}}{r_{n,1}}Z_{n}W_{1}\right]\\ &+\left[\frac{e^{-j\mu_{1,2}}}{r_{1,2}}Z_{1}W_{2}+\frac{e^{-j\mu_{2,2}}}{r_{2, 2}}Z_{2}W_{2}+\cdots\frac{e^{-j\mu_{n,2}}}{r_{n,2}}Z_{n}W_{2}\right]+\cdots\\ &+\left[\frac{e^{-j\mu_{1,m}}}{r_{1,m}}Z_{1}W_{m}+\frac{e^{-j\mu_{2,m}}}{r_{2, m}}Z_{2}W_{m}+\cdots+\frac{e^{-j\mu_{n,m}}}{r_{n,m}}Z_{n}W_{m}\right]\right\}\end{array}  \tag{1}\]  
where \(\beta\) is the phase constant of the radiating wave, \(W_{j}\) are the phase delay factors of the receiving antenna, and \(Z_{i}\) are the phase delay factors of the transmitting antenna. Here, the dipole current \(I_{i}\) is replaced with phase delay factors \(Z_{i}=AI_{i}\) and \(A\) is the constant related  
Figure 3: Schematic diagram of the MIMO antenna on XY and UV coordinate planes. to the respective current vectors \(I_{i}\) with the respective phase delay factors \(Z_{i}\). The vertical plane radiation function of a dipole antenna \(f(\theta)\) is given below [14]. \[f(\theta)=\frac{\cos\left(\frac{\beta l}{2}\cos(\theta)\right)-\cos\left(\frac{ \beta l}{2}\right)}{\sin(\theta)} \tag{2}\]  
where \(l\) is the length of the dipole element.