## 3 Results
### Sampling of Continuous Apertures
#### 3.1.2 Example A.2  
We produce a LFoV array employing the radial wrapping technique [25]. The prescribed indices are \(SLL_{\text{L}}<-20\) dB, \(HPBW_{\text{L}}\)\(\sim\)0.6\({}^{\circ}\), and \(D_{\text{L}}>50\) dBi with \(\theta_{0}\)\(\sim\)8\({}^{\circ}\). Let us now follow the steps of the proposed design procedure:  
Step 1: The scaling factor is equal to 7.  
Step 2: Using Table 1, the respective FFoV indices are \(SLL_{\text{F}}<-20\) dB, \(HPBW_{\text{F}}\)\(\sim\)4.2\({}^{\circ}\), and \(D_{\text{F}}>33.1\) dBi.  
Step 3: Now the aperture distribution should be determined from the FFoV indices.  
We use a paraboloidal distribution for a circular-aperture array and from the \(SLL\) we determine the exponent \(n\). From Table 1.2 of [4], we copy that for \(SLL=[-17.6,-24.6,-30.6]\) and the respective \(n\) takes the values \(n=[0,\,1,\,2]\). Using interpolation for \(SLL_{\text{F}}=-20\) dB, we find \(n\)\(\sim\)0.34. Additionally, from \(HPBW_{\text{F}}\), we determine the radius of the FFoV circular-aperture array. Indeed, again from Table 1.2 of [4], we copy that for \(n=[0,\,1,\,2]\), and it is \(HPBW=[58.9,\,72.7,\,84.3]/(\text{circular-aperture diameter in wavelengths})\). Thus, using interpolation, we obtain \(HPBW_{\text{F}}\)\(\sim\)63.63/(circular-aperture diameter in wavelengths). Now, solving for the circular-aperture diameter of the FFoV array, we find that it is approximately equal to 15.15\(\lambda\), and thus the respective diameter of the LFoV array is \(\mu\) times bigger. This parameter is used in the next step by the radial wrapping technique to determine the final result.  
Step 4: In order to apply the sampling procedure, we need to determine the number of elements and choose a proper array element placement method. We start from the determination of the number and size of the elements.  
Following the same path as in Step 3 of the previous example, we can find the number of elements \(N_{\text{e}}=625\) and an initial side-length equal to 3.59\(\lambda\).  
Before we can employ the radial wrapping, we should provide an initial uniform array. This is done as follows: From a regular square grid array having elements with inter-distance equal to \(\lambda/2\), we use the elements that are inside a circular disc of radius \(R_{0}=5.47\lambda\). This value of \(R_{0}\) is determined so that the circular-aperture array has a number of elements approximately equal to the value determined previously. In this way, the number of elements is 640.  
Next, we employ the procedure described in [25].  
Figure 4: Power pattern: (**a**) red: without element pattern, green: with element pattern, and (**b**) layout of the final array.  
Step 5: Last, we employ the scaling transformation. The final layout is given in Figure 5 along with the achieved power pattern. The final side-length, \(s=3.22\lambda\), is determined as described in Step 5 of the previous example while \(D=49.33\) dBi. Note that, although the achieved directivity is slightly below the desired value, we present this example in order to show that at the end of the design procedure, a fine tuning, with respect to the number of the elements, may be appropriate.  
Before we proceed to the next examples, a note is in order. Here, using paraboloidal functions to represent the aperture taper, we were able to contribute a closed form formula that provides the element placement. Indeed, an element whose center is initially placed at the point (\(r_{1}\), \(\varphi\)) is finally placed at a point (\(r_{2}\), \(\varphi\)), where for \(r_{1}\) and \(r_{2}\) the following holds:  
\[r_{2}=\sqrt{1-\sqrt[n+1]{1-r_{1}^{2}}}. \tag{23}\]  
The relative proof can be found in the Appendix A.