# Antenna Arrays for Line-of-Sight Massive MIMO: Half Wavelength Is Not Enough
## 4 Multi User Channel Matrix  
Let us now consider a number \(K\) of terminals, uniformly randomly displaced in an angular sector \(\varphi\in[-\varphi_{s},\varphi_{s}]\), with a minimum distance \(r_{min}\) and a maximum distance \(r_{max}\) from the center of the BS array. For all of the simulations that will be discussed in this section, we will use \(\varphi_{s}=\pi/3\), \(r_{min}=5\) m and \(r_{max}=200\) m; the choice of these parameters has been done in order to simulate a sectorized coverage of a large planar area, like a plaza or a park. It has to be underlined that differently from [28], we will consider a fixed number of antennas at the BS, in order to compare systems with the same complexity (and cost). Figure 4: Image map of the correlation coefficient when the first terminal is fixed in the position marked with the circle and the second user is moved on the plane for the \(\lambda/2\) equispaced array. Figure 5: Image map of the correlation coefficient when the first terminal is fixed in the position marked with the circle and the second user is moved on the plane for the \(2\lambda\) equispaced array. In order to better emphasize the possibility to discriminate different users according to their distance, we will employ \"Normalization 1\" from [20]. In this way, all of the columns of the channel matrix \(\mathbf{H}\) will have the same norm, thus removing the unbalance of channel attenuation between users due to their different relative distance from the BS. \(M=1000\) different random scenarios have been considered in the numerical simulations. As a first test, in Figure 6, we show as a function of the inter-element distance \(d\) and the number of users \(K\) the average condition number of the channel matrix, defined as:  
\[\rho_{dB}=20*log_{10}\left(\frac{\sigma_{1}}{\sigma_{K}}\right) \tag{6}\]  
where \(\sigma_{1}\) and \(\sigma_{K}\) are the greatest and smallest singular values of the channel matrix. The choice of the condition number is due to the fact that it provides a direct indication of the orthogonality of the columns of the channel matrix, and it is related to the stability of its inversion [29], needed in many multi-user MIMO transmission schemes [30]. The main advantage in using the condition number is that the presented results are independent of the actual SNR at the receiver; it is indeed true that the condition number provides only a rough indication of the channel quality, since it gives only the spread of the singular values and not their actual distribution, but its value is strongly related with the performances of MIMO precoders and detectors [31; 32; 33]. We see that the average condition number is an increasing function of the number of users, but it is interesting to note that (besides some oscillations), it is also a decreasing function of the inter-element distance \(d\) at the base station, confirming the advantages of a larger array at the BS foreseen in the previous paragraph. A brief discussion of the oscillations is now needed. Taking as a reference the \(K=100\) curve of Figure 6, we can see that the condition number increases up to about \(d=0.7\lambda\), then it decreases up to \(d=1.1\lambda\), then it increase again up to \(d=1.3\lambda\), and so on. As we have seen  
It is important to recall that a very low condition number is necessary to apply the Maximal Ratio Combining (MRC) algorithm at the BS, and a low condition number is also necessary to correctly calculate the matrix inversion needed to efficiently use Zero Forcing (ZF) at the BS [11; 34]. To get better insight into this result, let us focus on the case of \(K=50\); in Figure 7, we show the distribution of the condition number as a function of the distance \(d\) (each column of the matrix in the image map is a histogram of the condition number). The increase of \(d\) from \(\lambda/2\) to \(2\lambda\) reduces the average value of the condition number of about 15 dB, and also, its spreading decreases.