# Antenna Arrays for Line-of-Sight Massive MIMO: Half Wavelength Is Not Enough
## 7 Evaluating the Spectral Efficiency  
Up to this point, we have focused on the singular values of the channel matrix and its condition number. As stated before, this choice provides results independent of the SNR at the receiver.  
A further figure of merit that is often used in MIMO system analysis is spectral efficiency. In this section, we show how the better conditioning of the channel matrix influences the spectral efficiency of massive MIMO systems. In particular, we will evaluate the average maximum achievable rate using a Zero Forcing (ZF) receiver [49], considering a set of 1000 different scenarios for the user terminals:  
\[C_{ZF}=\sum_{k=1}^{K}log\left(1+\frac{\rho}{N[(\mathbf{H}^{H}\mathbf{H})^{-1} ]_{kk}}\right) \tag{10}\]  
where \(\rho\) is the average SNR at the receivers and \([\mathbf{A}]_{ij}\) takes the \((i,j)\)-entry of the matrix \(\mathbf{A}\).  
In Figure 20, we compare the performances of the equispaced array, with inter-element distances (\(\lambda/2\), \(\lambda\), \(2\lambda\)), with the performances of the non-equispaced array, calculated with a \(\alpha_{1}=-0.03\) and average inter-element distance of (\(\lambda\), \(2\lambda\)); the SNR at the receiver is 5 dB. It is worth noting that the maximum spectral efficiency is increased from \(C=146.1\) bit/s/Hz/cell for \(K=45\) of the \(\lambda/2\) equispaced array to \(C=162.6\) bit/s/Hz/cell for \(K=55\) for the \(2\lambda\) non equispaced array.  
We can obtain an average spectral efficiency gain of about 11% just positioning the radiating element of the BS in an optimal fashion.  
In Figure 21, we present the same results, but comparing only the \(\lambda/2\) equispaced array and the \(2\lambda\) non-equispaced array. In this plot, we also represent, by means of two colored areas, the spreading of the spectral efficiency: the areas represent a range of a standard deviation with respect to the mean. It is interesting to compare the \"top\" border of these two areas, which gives us an indication of the performances achievable by a smart user scheduling scheme: the advantage of the non-equispaced architecture is again clear.  
As a final remark, in our numerical tests, we have also considered the Dirty Paper Coding (DPC) capacity [30], which is calculated supposing the use of the optimum receiver. In this case, the results are stable with respect to the inter-element distance and marginally depend on the use of non-equispaced architectures.  
The reason for this fact is that only the greatest singular values of the channel matrix affect the DPC, and from the plots in Figures 8, 9 and 15, we can see that the distribution of the elements of the BS array influences mostly the distribution of the smallest singular values.  
Figure 20: Average maximum achievable Zero Forcing (ZF) rate as a function of the number of user terminals \(K\).  
Actually, communication schemes that achieve the DPC capacity are very hard to implement in the massive MIMO framework, where linear processing (like ZF) is preferred for computing reasons, and for this reason, we decided to show only the ZF performances.