# A Low Complexity Channel Estimation and Detection for Massive MIMO Using SC-FDE
## 2 System and Signal Characterization
Using the EGC receiver, as: \[\mathbf{B}_{k}=\exp\Big{\{}j\arg\!\left(\mathbf{H}_{k}^{H}\right)\Big{\}},\] (9)  
A disadvantage of the ZF relies on the need to compute the pseudo-inverse of the channel matrix, for each frequency component, which corresponds to a high processing power capability. This article mitigates this limitation by using the MRC and EGC algorithms. For m-MIMO, with \(R\)\(\gg\)1, with small correlation between the channels between different transmitting and receiving antennas, the elements outside the main diagonal of  
\[\widetilde{X}_{k}=\left[Y_{k}H_{k}^{*}\right]\beta_{k}^{(2)}, \tag{2}\]0  
are much lower than the ones at its diagonal, where \((i,i)^{th}\) elements of the matrix \(\mathbf{A}\) are defined as [15]:  
1. For MRC: \(\left[\mathbf{A}\right]_{i,i}=\left[\mathbf{H}\right]_{i,i}^{H}\). For EGC: \([\mathbf{A}]_{i,i}=\exp\left(j\arg\left([\mathbf{H}]_{i,i}\right)\right)\), that is, they have absolute value 1 and phase identical to the corresponding element of the matrix \(\mathbf{H}\). Based on \(\mathbf{A}_{k}^{H}\mathbf{H}_{k^{\prime}}\), we can implement the MRC or EGC, in the frequency domain. Nevertheless, for moderate values of \(T/R\), the residual interference can still present a certain level. In order to mitigate this, we implement the iterative receiver (interference canceller), depicted in Figure 2, as:  
\[\widetilde{X}_{k}=\left[Y_{k}H_{k}^{*}\right]\beta_{k}^{(2)}, \tag{2}\]1  
where the frequency domain estimated data symbols are \(\mathbf{\widetilde{X}}_{k}=\left[\mathbf{\widetilde{X}}_{k}^{(1)},\ldots, \mathbf{\widetilde{X}}_{k}^{(R)}\right]^{T}\). The interference cancellation matrix \(\mathbf{C}_{k}\) can be computed by  
\[\widetilde{X}_{k}=\left[Y_{k}H_{k}^{*}\right]\beta_{k}^{(2)}, \tag{2}\]2  
where \(\mathbf{I}\) is an \(R\times R\) identity matrix. This interference canceller is implemented using \(\mathbf{\widetilde{X}}_{k}=\left[\mathbf{\widetilde{X}}_{0},\ldots,\mathbf{ \widetilde{X}}_{N-1}\right]\), with \(\mathbf{\widetilde{X}}_{k}\) denoting the frequency-domain average values conditioned to the FDE output for the previous iteration [11], with \(\mathbf{\widetilde{X}}_{k}=DFT[\mathbf{\widetilde{x}}_{n}]\). Note that \(\mathbf{\widetilde{x}}_{n}\) can be obtained as defined in [15]. For the first iteration, there is no information about the transmitted symbols and \(\mathbf{\widetilde{X}}_{k}=0\). It is worth noting that an increase in the number of transmitting antennas results in an increase of the symbols rate. Moreover, increasing the number of receiving antennas results in an increase of diversity and, as a result, in performance improvement. Apart from the EGC and MRC, the Matched Filter Bound (MFB) curve is a way to measure the channel modeled by the sum of delayed and independently Rayleigh-fading rays [15]. For a massive MIMO scheme with a large number of antennas at the receiver side, this technique does not require special pilot design. Nevertheless, a problem arises for a large number of transmit antennas with uncorrelated channels, since we should have orthogonal pilots for the different transmit antennas. This means switching off all antennas, but one in a successive way. The proposed technique is generic, and valid for all frequencies, which are suitable for mm-Wave associated with massive MIMO, due to the smaller wavelength. This allows smaller antennas and packing more antennas in a given device. The main difference when mm-Wave is employed relies on the type of channel, which, in general, has a lower number of multipath components.