# A Survey on Massive MIMO Systems in Presence of Channel and Hardware Impairments
## 2 Basic Concepts
### Downlink Transmission  
Figure 2 shows downlink transmission in massive MIMO systems. In the downlink, the data which have to be sent to each user are passed through a precoding block at the BS. Then the signal of each BS antenna is transmitted through the channel and received by the users. The precoder is employed to direct the signal of each user to its corresponding user. All the discussions for the uplink can be extended to the downlink with the only difference being the fact that in the downlink we have a precoding block instead of the detection block. As an example we assume to use MF precoder and therefore the \(M\times 1\) vector of the precoded signal at the BS is  
\[\mathbf{y}^{(dl)}=(\mathbf{H}^{(dl)})^{H}\mathbf{x}^{(dl)}, \tag{6}\]  
where \(\mathbf{y}^{(dl)}=[y_{1}^{(dl)},y_{2}^{(dl)},...,y_{M}^{(dl)}]\) and \(\mathbf{(H}^{(dl)})^{H}\) and \(\mathbf{x}^{(dl)}=[x_{1}^{(dl)},x_{2}^{(dl)},...,x_{K}^{(dl)}]\) are the \(M\times K\) MF precoder matrix and \(K\times 1\) vector of users' data, respectively. \(\mathbf{H}^{(dl)}\) is the \(K\times M\) downlink channel matrix between the BS antennas and the \(K\) users. The \(k\)th row and \(m\)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  
Figure 1: Block diagram of massive multiple input multiple output (MIMO) system in uplink.  
\(h_{k,m}^{(dl)}=\sqrt{d_{k}^{(dl)}}g_{k,m}^{(dl)}\), where \(d_{k}^{(dl)}\) and \(g_{k,m}^{(dl)}\) are the large scale and small scale fadings, respectively. In this case when \(M>>K\), the rows of the matrix \(\mathbf{H}^{(dl)}\) become approximately orthogonal and, therefore, a simple linear precoder can mitigate multi-user interference. In other words, if we rewrite \(\mathbf{H}^{(dl)}\) as \(\mathbf{H}^{(dl)}=(\mathbf{D}^{(dl)})^{\frac{1}{2}}\mathbf{G}^{(dl)}\), where \(\mathbf{G}^{(dl)}\) is a \(K\times M\) matrix with elements \(\left[\mathbf{G}^{(dl)}\right]_{k,m}=g_{k,m}^{(dl)}\) and \(\mathbf{D}^{(dl)}\) is a \(K\times K\) diagonal matrix with diagonal elements \(d_{k}^{(dl)}\), then after passing the precoded signal through the channel we have [5]  
\[\frac{\mathbf{H}^{(dl)}(\mathbf{H}^{(dl)})^{H}}{M}=(\mathbf{D}^{(dl)})^{\frac{ 1}{2}}\frac{\mathbf{G}^{(dl)}(\mathbf{G}^{(dl)})^{H}}{M}(\mathbf{D}^{(dl)})^{ \frac{1}{2}}\approx\mathbf{D}^{(dl)}\text{, }\text{M}\gg\text{K}.\]  
Since \(\mathbf{D}^{(dl)}\) is a diagonal matrix, the interference between the users is completely removed and each user can receive its desired signal.  
We should note that other linear precoders such as ZF and MMSE are used in massive MIMO systems. These precoders have similar behaviour at large number of BS antennas and remove the interference between the users very well. The matrix of ZF and MMSE precoder, with respect to the downlink channel matrix, are defined as (7) and (8), respectively.  
\[\mathbf{W}_{P}^{ZF}=(\mathbf{H}^{(dl)})^{H}\left(\mathbf{H}^{(dl)}(\mathbf{H}^{(dl)})^{H}\right) ^{-1}. \tag{7}\]  
\[\mathbf{W}_{P}^{MMSE}=(\mathbf{H}^{(dl)})^{H}\left(\mathbf{H}^{(dl)}(\mathbf{H}^{(dl)})^{H}+ \frac{\sigma_{\eta}^{2}}{\sigma_{x}^{2}}\mathbf{I}_{k}\right)^{-1}. \tag{8}\]