# Antenna Arrays for Line-of-Sight Massive MIMO: Half Wavelength Is Not Enough
## 5 Exploiting Sparse Arrays
Not all of the combinations of \([\alpha_{1};\cdots;\alpha_{P}]\) would lead to a monotonically-increasing vector of positions \(\mathbf{x}_{s}\) in the range \([-L,L]\); for instance, it is easy to demonstrate that if \(P=1\), any value of \(\alpha_{1}\) in the range:  
\[-\frac{1}{8}<\alpha_{1}<\frac{1}{4} \tag{9}\]  
would lead to a set of positions in the range \([-L,L]\) satisfying the monotonicity constraint, but for higher values of \(P\), it would be non-trivial to achieve a formula like Equation (9) in closed form. In the remaining part of the paper, we will focus on the relatively easy case of \(P=1\); this choice is justified by the results of some numerical tests, which have shown only a marginal improvement when using \(P>1\). In particular, values of \(\alpha_{1}<0\) would lead to antenna arrays with a higher density of elements at the extrema, while values of \(\alpha_{1}>0\) would lead to antenna arrays with a higher density of elements at the center. Following the guidelines discussed before, we will consider now a parametric analysis of the performances, in terms of average condition number, on a set of 1000 configurations of \(K=50\) user terminals, as a function of \(\alpha_{1}\). The result is depicted in Figure 10: apart from values very close to \(\alpha_{1}=0\), corresponding to the equispaced array, we achieve a significant advantage in the mean condition number for both positive and negative values of \(\alpha_{1}\) (with a slight advantage for the latter). To get better insight into this surprising result and to better understand the effect of the non-linear spacing on the array performances, we will consider the same analysis developed in the previous two paragraphs on a test array obtained with \(\alpha=-0.03\) with an average inter-element distance  
Figure 10: Average condition number as a function of \(\alpha_{1}\) for a number of users \(K=50\) and \(d_{0}=2\lambda\). \(d_{0}=2\lambda\). The choice of this particular value of \(\alpha_{1}\) is driven by the desire of reducing the spreading of the inter-element distance, in particular in this case, we have a minimum inter-element distance (at the edges) of \(1.53\lambda\) and a maximum inter-element distance (at the center) of \(2.24\lambda\). In Figure 11, we can see that the capability of the array to distinguish terminals sharing the same angular position with respect to the BS has been marginally improved with respect to the case of Figure 3. The difference with respect to the corresponding equispaced array is instead much clearer from the analysis of Figure 12: now, the effect of the grating lobes is replaced by some sectors of the plane that show a lower (but non-null) correlation coefficient. In Figure 13, we have instead depicted the effect of the non-linear spacing for a variable number of the average inter-element distance \(d\) and the number of users \(K\): the oscillatory behavior of Figure 6 has now disappeared, and the condition number is now a monotonic function of the number of terminals and the average inter-element distance. The improvement with respect to the equispaced array is even more significant when the number of user terminals is higher. It is worth underlining that the use of a non-equispaced array is beneficial even when using an inter-element distance only slightly higher than half-wavelength, so to achieve a performance improvement, we are not required to employ large array sizes. Figure 11: Two terminal correlation analysis for the \(2\lambda\) non-equispaced array with \(\alpha_{1}=-0.03\). (**Left**) Image map with the correlation level for variable \(r_{1}\) and \(r_{2}\); (**Right**) plot of the correlation coefficient for some fixed values of \(r_{2}\). Figure 12: Image map of the correlation coefficient when the first terminals are fixed in the position marked with the circle and the second user is moved on the plane for the \(2\lambda\) non-equispaced array with \(\alpha_{1}=-0.03\).