# Antenna Arrays for Line-of-Sight Massive MIMO: Half Wavelength Is Not Enough
## 3 User Correlation Analysis  
From this section on, we will choose \(f=60\) GHz as the working frequency and a base station array of 200 elements. The choice of considering a fixed number of antennas at the BS has been made just to compare different antenna geometries with the same system complexity. In order to get better insight into the effect of the antenna shape on the system performance, we will start considering the simple case of only two terminals, and we will calculate the correlation coefficient between the two columns of \(\mathbf{H}\), as:  
\[\chi_{1,2}=\frac{|\mathbf{h}_{1}\cdot\mathbf{h}_{2}^{*}|}{||\mathbf{h}_{1}|| \ ||\ |\mathbf{h}_{2}||}. \tag{4}\]  
The use of this performance metric is due to the fact that it provides a simple way to check the effectiveness of a maximal ratio combining approach at the BS; furthermore, as will be clear in the following, it allows us to provide a simple and effective graphical description of the correlation between user terminal positions. Let us first consider the case of an equispaced array, with the inter-element distance \(d=\lambda/2\); in Figure 2, it is possible to see the case in which the two terminals are aligned along the \(y\) axis, and have only different distances (\(r_{1}\) and \(r_{2}\)) from the BS. In the left subplot, we can see an image map representing the correlation coefficient of Equation (4) for variable \(r_{1}\) and \(r_{2}\), while in the right subplot, we can see the plot of the correlation coefficient for some fixed values of \(r_{2}\). This figure confirms that it is possible to distinguish between two users sharing the same angular position with respect to the center of the BS, but this feature can be exploited only when one terminal is very close to the BS with respect to the other. It has to be underlined that this behavior has been obtained when the two terminals lie on the \(y\) axis; for other directions, it is possible to obtain different plots, showing a slightly increased correlation, not reported here for the lack of space. Let us now consider an equispaced array with a larger inter-element distance, say \(d=2\lambda\). From an antenna point of view, increasing the inter element distance, without changing the number of elements, increases the far-field distance of the array, which is conventionally calculated as:  
\[r_{FAR}=\frac{2[(N-1)d]^{2}}{\lambda}. \tag{5}\]  
Figure 1: Massive MIMO system scenario. Circles: base station antennas; squares: terminals. This means that an increase of the inter-element distance of a four factor increases the far field distance of a 16 factor. In Figure 3, we can see the image map and the plot of the correlation for the larger array. It is clear that we are now allowed to easily distinguish two users that share the same angular position with respect to the BS, much more than in the previous case. The previous result has shown that the increase of the inter-element distance seems to have a beneficial effect on the capability of decorrelating users. To confirm this feature, let us see what happens if we consider the first terminal in a fixed position with respect to the array, and we move the second terminal on the \(xy\) plane. In Figure 4, we can see the case of a \(\lambda/2\) equispaced array: apart from an angular sector around the broadside direction of the array, the correlation is very low, thus confirming the very good results, in terms of favorable propagation, found by other researchers. If we instead employ the \(2\lambda\) equispaced array, we obtain the result depicted in Figure 5: apart from the broadside direction, which shows a slightly lower correlation level with respect to the \(\lambda/2\) case, there are other directions in space that present a non-negligible correlation. These regions are related to the unavoidable grating lobes, appearing when we consider equispaced elements with an inter-element distance greater than half-wavelength. Clearly, the presence of grating lobes causes ambiguities and must be avoided.