# Antenna Arrays for Line-of-Sight Massive MIMO: Half Wavelength Is Not Enough
## 6 Circular Arrays  
In the previous sections, we have considered linear BS arrays for a sectoral coverage; if the terminals are displaced in a circular area, it could be convenient to use a circular array. The increase of the antenna dimension, without changing the number of active elements, can be very beneficial also in this case.  
A circular array does not show grating lobes, so it will not be necessary to consider non-equispaced positioning of the array elements on the circle, but the increase in size would allow a better discrimination of the users, because of the increased far field distance of the array.  
To better compare the results of the present section to the results of the previous ones, we will always consider \(N=200\) radiating elements, displaced on a circle of radius \(R=Nd/(2\pi)\) where \(d\) is the (approximate) inter-element distance.  
For the sake of compactness of the paper, we will skip the correlation analysis, and we will only focus on the condition number analysis of the MU-MIMO channel matrix. In particular, we will consider a set of 1000 different propagation scenarios, where a variable number \(K\) of randomly-positioned user terminals is displaced between a distance \(r_{min}=5\) m and \(r_{max}=200\) m from the center of the BS antenna.  
In Figure 16, we can see the average condition number that we can obtain for a variable inter-element distance \(d\) and a variable number of users \(K\). This result confirms the importance of increasing the size of the antenna array as much as possible if we are interested in improving the conditioning of the channel matrix. It is also worth noting that this result is surprisingly very similar to the result obtained with the non-equispaced antenna array of Figure 13.  
In Figure 17, we instead depict the distribution of the condition number for a variable antenna distance \(d\) for the specific case \(K=50\); even in this case, it is surprising to see the behavior of the condition number that is very similar to the result depicted in Figure 14.  
Finally, in Figures 18 and 19, we depict the distribution of the singular values for the two specific cases of \(d=\lambda/2\) and \(d=2\lambda\), confirming the advantage of an increased antenna array dimension.  
Figure 16: Mean condition number of the channel matrix as a function of the average inter-element distance \(d\) and the number of users \(K\) for a circular array.  
Figure 19: Singular values analysis for the circular array with \(d=2\lambda\) 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.  
Figure 17: Distribution of the condition number as a function of the average distance \(d\) for the circular array with \(K=50\). Each column of the matrix in the image map is a histogram of the condition number.  
Figure 18: Singular values analysis for the circular array with \(d=\lambda/2\) 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.