## 3 Collaborative Beamforming and Array Factor  
In long distance communications, antennas with high directivity are required. Such antennas are possible to construct by enlarging the dimensions of the radiating aperture. However, this approach may lead to the appearance of multiple side lobes. In addition, the antenna is usually large and difficult to fabricate [17]. Another way to increase the electrical size of an antenna is to construct it as an assembly of radiating elements in a proper electrical and geometrical configuration, also known as antenna array. Usually, the array elements are identical. The total field of an array is a vector superposition of the fields radiated by the individual elements. To provide a directive pattern, it is necessary that the partial fields (generated by the individual elements) interfere constructively in the desired direction and interfere destructively in the remaining space.  
Thus, to control the overall antenna pattern, we have different options to modify the partial fields, by geometrical configuration, relative placement of the elements, the excitation amplitude or phase of the individual elements and the individual pattern of each element.  
The electric field of a radiant element \(i\) is given by \(E_{i}=\frac{I_{i}}{r}e^{j\cdot(w\cdot t-k\cdot r)}\), where \(i\) is the element number, \(I_{i}\) is the current of this element (excitation or field amplitude), \(w=2\pi f_{c}\), where \(f_{c}\) is the carrier frequency, \(t\) is time, \(k=\frac{2\pi}{\lambda}\) and \(\lambda=\frac{c}{f_{c}}\), with \(c\) being the speed of light and \(r\) being the distance to the BS.  
If the elements are placed along one dimension, as a uniform linear array (1D), and the BS is with an angle \(\theta\) with the axis of the linear array as shown in Figure 2a, then the distances \(r\) from each element to the BS are different and can be calculated taking into account \(\theta\). These distance differences introduce different phase offsets that are given by \(l\cdot cos(\theta)\), where \(l\) is the element spacing. Then,assuming that the array elements are identical and their excitations are of the same amplitude with isotropic (ominidirectional) antennas, the superposition of the radiated fields is given by  
\[E=\sum_{i=0}^{N-1}E_{i}=E_{0}\cdot(\sum_{i=0}^{N-1}e^{j\cdot i(k\cdot l\cdot cos( \theta)-\beta)}), \tag{1}\]  
where \(N\) is the number of elements, \(E_{0}\) is the radiated field created by the first element located at the origin (\(i=0\)) and each element has a \(\beta\) progressive phase lead current excitation relative to the preceding one. This equation can be also expressed as the product of the field \(E_{0}\) and an array factor (AF) [17] of the set, as  
\[E=E_{0}\cdot AF_{1D}, \tag{2}\]  
where \(AF_{1D}\) is  
\[AF_{1D}=\sum_{i=0}^{N-1}e^{j\cdot i(k\cdot l\cdot cos(\theta)-\beta)}. \tag{3}\]  
We can generalize Equation (3) for a bidimensional uniform (2D) array of \(N_{x}\)x\(N_{y}\) elements as shown in Figure 2b, where \(N_{x}\) and \(N_{y}\) are the elements along the \(x\)- and \(y\)-axis, respectively. In this case, the projection of \(\overline{r}\) on plane \(XY\) is \(r\cdot sin(\theta)\). Then, if we arrange \(N_{y}\) 1D-arrays at equal \(l_{y}\) intervals along the \(y\) dimension, a rectangular array is obtained, and each array has also a progressive phase shift \(\beta_{y}\) in the \(y\)-axis. In case of equal excitation amplitudes, we have an uniform array, and the general expression for \(AF_{2D}\) is:  
\[AF_{2D}=(\sum_{i=0}^{N_{x}-1}e^{j\cdot i(k\cdot l_{x}\cdot sin(\theta)cos( \phi)-\beta_{x})})\cdot(\sum_{m=0}^{N_{y}-1}e^{j\cdot m(k\cdot l_{y}\cdot sin (\theta)sin(\phi)-\beta_{y})}). \tag{4}\]