# On the Trade-Off between the Main Parameters of Planar Antenna Arrays
## 4 Improving the Scanning Performance of Sparse Arrays  
If we look at Figure 6, we can see that the threshold on the directivity has lowered the side-lobes in the region \(w\leq 1\) compared to the sidelobes of the layout of Figure 5. This happens because the compact formula \(D=1/\mathbf{a^{\dagger}Sa}\) is the result of the integration of the array factor in the visible range. Imagine now to perform a \"scaling\" of the positions of the radiating elements of a factor \(\varsigma\), so that the scaled coordinates of the radiating elements are \((x^{\prime}_{n},y^{\prime}_{n})=(\varsigma x_{n},\varsigma y_{n})\). If we chose \(\varsigma=1+u_{s}\), without changing the excitation of the radiators, we will obtain an array that radiates a \"scaled\" array factor, that is shrunk of the same factor \(\varsigma\): the pattern radiated by the starting array in the angular region \(w\leq 1+u_{s}\) will be radiated by the scaled array in the angular region \(w\leq 1\). Let us now introduce a \"dummy directivity\"  
\[D_{\varsigma}=\frac{1}{\mathbf{a^{\dagger}S_{\varsigma}a}} \tag{22}\]  
with \(\mathbf{S}_{\varsigma}\) is a square matrix, the entries of which are:  
\[s_{\varsigma,m,n}=\frac{\sin(\beta\varsigma\rho_{m,n})}{\beta\varsigma\rho_{m,n}} \tag{23}\]  
and obviously, \(D_{\varsigma=1}=D\). It turns out that the equation of dummy directivity has a form as simple as the directivity in Equation (10), but allows a significant improvement of the scanning performances. To show this improvement, we will consider the synthesis of a new sparse array, in which we add two further convex thresholds to the original IDEA. The first is \(D_{\varsigma=1}\geq 29\), the second is \(D_{\varsigma=1.766}\geq 28.4\). The obtained layout is shown in Figure 8; the achieved layout employs 346 elements and shows an excitation dynamic \(\rho=6.1\) dB, the minimum inter-element distance is 0.48\(\lambda\), and the computation time was about six hours; again, the beam-width and side-lobe level constraints are perfectly verified. Figure 8: The 346 elements sparse layout. Left subplot: position and normalized excitation amplitude (dB) of the synthesized layout; center subplot: normalized Array Factor in the \(uv\) plane; right subplot: some relevant cuts. Negative values of \(w\) are related to the cuts \(\phi^{\prime}=\phi+\pi\). It is also possible to introduce further convex constraints using Equation (22). We can consider the synthesis with the thresholds \(D_{\xi=1}\geq 29\), \(D_{\xi=1.766}\geq 28.4\) and \(D_{\xi=1.383}\geq 28.5\). The obtained layout is shown in Figure 9; it employs 370 elements and shows an excitation dynamic \(\rho=5.9\) dB, the minimum inter-element distance is 0.49\(\lambda\), and the computation time was about four hours; even in this case, the beam-width and side-lobe level constraints are perfectly verified. In Figure 10, we can see the comparison of the scanning performances of all the sparse layouts synthesized up to this point. The use of the constraints on the dummy directivity allows to create arrays that have a much lower number of radiating elements than the fully populated array, but the directivity performance is very good. The previous layouts were aimed to reduce the number of radiating elements as much as possible. In other applications, the number of radiators may not be the key parameter in the design, but we could be interested in maximizing the directivity. In this case, we could perform the synthesis maximizing the dummy directivity of the array \(D_{\xi=1.766}\), instead of performing the minimizations of (12) and (17). Applying this approach to the same pattern specifications of the previous cases, it is possible to obtain an array of 622 elements (Figure 11), which shows an excitation dynamic \(\rho=11.5\) dB and a minimum inter-element distance of 0.39\(\lambda\); the computation time was about sixteen hours. Even in this case, the beam-width and side-lobe level constraints are perfectly verified.