# A Survey on Massive MIMO Systems in Presence of Channel and Hardware Impairments
## 2 Basic Concepts
### Uplink Transmission  
Figure 1 shows uplink transmission in massive MIMO systems. In the uplink, the users transmit their data and after passing through the channel, the received signals at the BS antennas are passed through a detection block to detect the data of each user. The \(M\times 1\) vector of received signal at the BS is  
\[\mathbf{y}^{(ul)}=\sqrt{p^{(ul)}}\mathbf{H}^{(ul)}\mathbf{x}^{(ul)}+\mathbf{j }^{(ul)}, \tag{1}\]  
where \(\mathbf{y}^{(ul)}=[y_{1}^{(ul)},y_{2}^{(ul)},...,y_{M}^{(ul)}]\) and \(\mathbf{H}^{(ul)}\) is the \(M\times K\) channel matrix between the BS antennas and the \(K\) users in the uplink. The \(m\)th row and \(k\)th column element of this matrix is the channel coefficient between the \(m\)th BS antenna and the \(k\)th user, and is defined as \(h_{m,k}^{(ul)}=\sqrt{d_{k}^{(ul)}}g_{m,k}^{(ul)}\), where \(d_{k}^{(ul)}\) and \(g_{m,k}^{(ul)}\) indicate the large scale and small scale fadings, respectively. The parameters \(\mathbf{j}^{(ul)}\), \(\mathbf{x}^{(ul)}=[x_{1}^{(ul)},x_{2}^{(ul)},...,x_{K}^{(ul)}]\), and \(p^{(ul)}\) are the \(M\times 1\) noise vector, \(K\times 1\) vector of the data sent by the \(K\) users, and the uplink transmission power of the users, respectively. In [22] it is shown that when \(M>>K\), then the columns of the matrix \(\mathbf{H}^{(ul)}\) become orthogonal and therefore with simple linear detectors the multi-user interference can be removed. In other words if we rewrite \(\mathbf{H}^{(ul)}\) as \(\mathbf{H}^{(ul)}=\mathbf{G}^{(ul)}(\mathbf{D}^{(ul)})^{\frac{1}{2}}\), where \(\mathbf{G}^{(ul)}\) is a \(M\times K\) matrix with elements \(\left[\mathbf{G}^{(ul)}\right]_{m,k}=g_{m,k}^{(ul)}\) and \(\mathbf{D}^{(ul)}\) is a \(K\times K\) diagonal matrix with diagonal elements \(d_{k}^{(ul)}\), then with MF detector whose matrix is \(\left(\mathbf{H}^{(ul)}\right)^{H}\), we have  
\[\frac{\left(\mathbf{H}^{(ul)}\right)^{H}\mathbf{H}^{(ul)}}{M}=\left(\mathbf{D }^{(ul)}\right)^{\frac{1}{2}}\frac{\left(\mathbf{G}^{(ul)}\right)^{H}\mathbf{ G}^{(ul)}}{M}(\mathbf{D}^{(ul)})^{\frac{1}{2}}\approx\mathbf{D}^{(ul)}, \text{M}\gg\text{K}.\]  
Since \(\mathbf{D}^{(ul)}\) is a diagonal matrix, it can be concluded that the interference between the users is completely removed. Therefore, it has been shown that the capacity of the \(k\)th user is [5] \begin{table}
\begin{tabular}{l l} \hline \hline
**Challenge** & **Paper** \\ \hline Waveform & [16] \\ Detection and Precoding & [18; 19] \\ Hardware impairment & This survey \\ Channel measurement and modeling & [17] \\ Channel aging and time variation & This survey \\ Pilot contamination & [20] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Surveys on massive multiple input multiple output (MIMO) challenges. \[C_{k}^{(ul)}=\log_{2}(1+Mp^{(ul)}d_{k}^{(ul)}). \tag{2}\]  
For a single-input single-output (SISO) system with similar conditions the capacity is  
\[C_{1}^{(ul)}=\log_{2}(1+p^{(ul)}d_{1}^{(ul)}). \tag{3}\]  
Comparing (2) and (3), we can conclude that in massive MIMO systems the users can reduce their transmit power by \(M\) to have a performance similar to the SISO system. However, this is for the case where we have perfect channel knowledge and with imperfect channel knowledge this factor reduces [22]. We should note that other linear detectors like zero forcing (ZF) and minimum mean square error (MMSE) can also be used in massive MIMO systems. These detectors have similar behaviour with large number of BS antennas, being able to reduce substantially the multi-user interference. The matrix of ZF and MMSE detector, with respect to the uplink channel matrix, are defined as (4) and (5), respectively.