# Analysis of Grating Lobe Effects on GEO DSC Distributed Antennas
## 3 Analysis of Grating Lobes in DSC Distributed Antennas  
### Grating Lobe Effect  
DSC distributed antennas, as a type of ultra-sparse array, generate numerous grating lobes after beamforming. Figure 4 illustrates the element distribution of an \(N\) elements linear array, with individual element excitation denoted by \(w_{0},w_{1},w_{2},\ldots,w_{N-1}\). Assume that each element is an isotropic point source and the array factor pattern represents the total field pattern. However, if the elements in the array are not point sources, the total field pattern needs to be obtained using the principle of pattern multiplication, where the array factor is multiplied by the element pattern. The array factor expression for this linear array is as the following [17]:  
\[Af=\sum_{n=0}^{N-1}I_{0}e^{jn(kd\sin\theta-\alpha_{0})} \tag{7}\]  
For Equation (6), let \(u=kd\cos\theta-\alpha\). The array factor function can be written as the following:  
\[Af=\sum_{n=0}^{N-1}I_{0}e^{jn} \tag{8}\]  
The summation is a geometric series, which can be further simplified. \[ \begin{split} Af&=I_{0}\sum\limits_{n=0}^{N-1}e^{ jn}\\ &=I_{0}\Big{(}1+e^{jn}+e^{j^{2n}}+\ldots+e^{j(N-1)n}\Big{)}\\ &=I_{0}\Big{(}\frac{1-e^{jNn}}{1-e^{jn}}\Big{)}\\ &=I_{0}\frac{e^{-\frac{jNn}{2}}}{e^{-\frac{jn}{2}}}\frac{\sin \Big{(}\frac{Nn}{2}\Big{)}}{\sin\Big{(}\frac{n}{2}\Big{)}}\end{split}  \tag{9}\]  
If the coordinate origin of the array is established at the center position, the result of Equation (9) is the following [15]:  
\[Af =I_{0}\frac{\sin\Big{(}\frac{Nu}{2}\Big{)}}{\sin\Big{(}\frac{u}{2} \Big{)}} \tag{10}\]  
As \(u\) approaches 0 and \(\pm n\pi\), the \(Af\) tends to \(I_{0}N\), and the value at this point is the maximum. Therefore, the normalized array factor for a uniform linear array is given by  
\[Af_{n} =\frac{Af}{\max(Af)}=\frac{\sin\Big{(}\frac{Nu}{2}\Big{)}}{N\sin \Big{(}\frac{u}{2}\Big{)}} \tag{11}\]  
Taking the modulus operation of Equation (11), we obtain the normalized array factor directional pattern function:  
\[|Af_{n}| =\left|\frac{\sin\Big{(}\frac{Nu}{2}\Big{)}}{N\sin\Big{(}\frac{u}{2} \Big{)}}\right| \tag{12}\]  
From Equation (10), it can be observed that the array factor is a function of \(u\), which is a periodic function with a period of \(2\pi\). The range of \(u\) extends throughout the entire real number line. However, in an actual array, the range of \(\theta\) is limited to the interval \([0,\pi]\), corresponding to a variation range of \([-kd-\alpha,kd-\alpha]\) for \(u\). This range is known as the visible region, while the regions outside this range are considered invisible. Therefore, the  
Figure 4: Distribution of the \(N\) units linear array element. phase \(\alpha\) can be used to control the initial position of the visible region and hence move the visible region, while the inter-element spacing \(d\) controls the size of the visible region. As shown in Figure 5, the schematic diagram illustrates the visible region of a 5-element uniform linear array. For the expression \(Af_{n}=\frac{\sin(Nu/2)}{N\sin(ku/2)}\), due to the limited range of \(u\), the theoretically possible regions do not exist in practice. In engineering applications, it is important to avoid the occurrence of grating lobes within the visible region. The main lobe of the array factor reaches its maximum value at \(u=0\). Since the array factor \(Af(u)\) is a periodic function with a period of \(2\pi\), the maximum value also occurs periodically, specifically, when \(u=2m\pi,m=0,\pm 1,\pm 2,\ldots\) When \(m=0,u=0\), it corresponds to the main lobe, while for other values of \(m\), it corresponds to the side lobes [18]. The presence of side lobes leads to a decrease in gain, dispersion of radiated energy, and can result in erroneous target localization and direction finding, making it necessary to suppress them. Based on the concept of the visible region mentioned earlier, the actual range of \(u\) is finite. Therefore, by limiting the range of the visible region to \(2\pi\), the occurrence of side lobes can be avoided.