# Antenna Arrays for Line-of-Sight Massive MIMO: Half Wavelength Is Not Enough
## 5 Exploiting Sparse Arrays  
In the previous section, we have shown that increasing the size of the array is beneficial to the overall MU-MIMO channel matrix condition number: even if we have to deal with the effect of grating lobes, the possibility to discriminate sources that share a similar angular direction, as well as the reduced width of the main beam lead to a better conditioned matrix. It is trivial to recognize that, if we could get rid of the grating lobes, we should expect a further improvement of the performances. To this aim, we would have two different possibilities: in the first case, we could employ a sub-array architecture [35], but we should also deal with the reduced scanning capability of our array (and hence, the dimension of the sector we could cover); the second possibility would be to employ a non-equispaced, sparse array. The synthesis of sparse arrays is one of the most studied antenna topics in the last 50 years, and a very large number of methods and algorithms has been developed for linear and planar arrays [36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47]. Unfortunately, none of these algorithms can be directly applied to an MU-MIMO system: our aim is to obtain the antenna positions, within a given range, that allow the best performances of a multi-user scenario, and this problem cannot be directly translated into a classical antenna pattern synthesis problem. To solve this issue, we propose a novel paradigm for antenna array synthesis, inspired by the results presented in the previous paragraph: we will be seeking for the antenna positions providing the best performances on a relatively large set of user terminal scenarios. We could, for example, optimize the antenna array to provide the lowest average condition number of the channel matrix, or we could look for the antenna configuration that guarantees the lowest 0.99-quantile of the condition number. Once we have defined our optimization objective, we could use an evolutionary search method to achieve the sought antenna positions. Actually, the direct application of this approach could be very ineffective, since the number of unknowns we have to deal with is of the order of the number of antennas, and evolutionary algorithms are known to require a computational time that increases exponentially with the number of unknowns [48]. Figure 9: Singular values analysis for \(d=2\lambda\) and \(K=50\). (**Left**) Image map of the distribution of each one of the 50 singular values. (**Right**) solid line, average singular values; dashed lines, average singular values \(\pm\) the standard deviation of the distribution. A much more convenient, yet sub-optimal, choice is to define a proper parametrization of the antenna positions, thus reducing the search space of the evolutionary algorithm to a small number of parameters and leading to a reasonable computational time. A simple and effective parametrization of the antenna positions can be found using Tchebyshev polynomials. If we are looking for a symmetric array, the position of the \(n\)-th antenna \(x_{s,n}\) could be found has:  
\[x_{s,n}=L\left(\left(1-\sum_{p=1}^{P}\alpha_{p}\right)u_{n}+\sum_{p=1}^{P} \alpha_{p}T_{2p+1}(u_{n})\right) \tag{7}\]  
with:  
\[L=\frac{d_{0}}{2(N-1)} \tag{8}\]  
where \(T_{p}(\cdot)\) is the Tchebyshev polynomial of order \(p\), \([\alpha_{1},\cdots,\alpha_{P}]\) is the set of \(P\) parameters defining the position of the antenna elements, \(d_{0}\) is the desired average inter-element distance, \(N\) is the overall number of antennas and \([u_{1};\cdots;u_{N}]\) is an equispaced ordered sampling of the interval \([-1,1]\).