# A Low Complexity Channel Estimation and Detection for Massive MIMO Using SC-FDE
## 2 System and Signal Characterization  
It is assumed a MIMO scenario (Figure 1), with \(T\) transmit antennas and \(R\) receive antennas, using SC-FDE signals with the Quadrature Phase Shift Keying (QPSK) modulation. We consider an \(N\)-length time-domain block signal to be transmitted of the form \(\{x_{n};n=0,1,\ldots,N-1\}\). The corresponding frequency-domain block is obtained as \(\left\{X_{k};k=0,1,\ldots,N-1\right\}=DFT\{x_{n};n=0,1,\ldots,N-1\}\), i.e., by performing the Discrete Fourier Transform (DFT) of the time-domain block. After removing the cyclic prefix, and assuming a cyclic prefix longer than the overall channel impulse response of each channel, the received frequency-domain signal comes,  
\[Y_{k}=X_{k}H_{k}+N_{k}, \tag{1}\]  
where \(H_{k}\) denotes the channel frequency response for the \(k\)-th subcarrier (which is assumed invariant during the transmission of a given block), i.e., \(\left\{H_{k};k=0,1,\ldots,N-1\right\}=DFT\{h_{n};n=0,1,\ldots,N-1\}\). Moreover, \(N_{k}\) is the frequency-domain block channel noise for that subcarrier. The received time-domain signal can be obtained from (1) as \(\left\{y_{n};n=0,1,\ldots,N-1\right\}=IDFT\{Y_{x^{i}};k=0,1,\ldots,N-1\}\)[15]. Assuming the conventional linear FDE for SC schemes, the post-processing comes,  
\[\widetilde{X}_{k}=\left[Y_{k}H_{k}^{*}\right]\beta_{k}^{(2)}, \tag{2}\]  
Figure 1: Block diagram of massive Multiple Input–Multiple Output (m-MIMO) with Single-Carrier with Frequency-Domain Equalization (SC-FDE) signals. where \(\beta_{k}^{(2)}=\left(\alpha+\left(\left|H_{k}\right|^{2}\right)\right)^{-1}\). As expected,  
\[\widetilde{X}_{k}=X_{k}\big{|}H_{k}\big{|}^{2}\beta_{k}^{(2)}+N_{k}^{eq}, \tag{3}\]  
This means that the received symbol corresponds to the transmitted one, multiplied by \(\big{|}H_{k}\big{|}^{2}\) that can be viewed as a gradient, with a noise factor \(\beta_{k}^{(2)}\), and added by the noise \(N_{k}^{eq}\). In addition, we define \(\alpha\) as  
\[\alpha=E\Big{[}\big{|}N_{k}\big{|}^{2}\Big{]}/E\Big{[}\big{|}X_{k}\big{|}^{2} \Big{]}, \tag{4}\]  
\(N_{k}^{eq}\) denotes the equivalent noise for detection purposes, with \(E\Big{[}\big{|}N_{k}^{eq}\big{|}^{2}\Big{]}=\left[2\sigma_{N}^{2}\big{|}H_{k} \big{|}^{2}\right]\beta_{k}^{(2)}\), and with \(\sigma_{N}^{2}=E\Big{[}\big{|}N_{k}\big{|}^{2}\Big{]}_{\overline{2}}\). This article focus on the scenario with \(T\) data streams, where \(R\)\(\gg\)\(T\). The \(t\)th antenna has a block of \(N\) data symbols \(\left\{x_{n}^{(t)};n=0,1,\ldots,N-1\right\}\) to send. At the BS, the received block associated to the \(r\)th user is represented by \(\left\{y_{n}^{(r)};k=0,1,\ldots,N-1\right\}\). As with other SC-FDE schemes, a cyclic prefix longer than the maximum overall channel impulse response is appended to each transmitted block and removed at the receiver. In this case, the corresponding frequency-domain block \(\left\{Y_{k}^{(r)};k=0,1,\ldots,N-1\right\}\) satisfies  
\[\mathbf{Y}_{k}=\left[Y_{k}^{(1)},\ldots,Y_{k}^{(R)}\right]^{T}=\mathbf{H}_{k} \mathbf{X}_{k}+\mathbf{N}_{k}, \tag{5}\]  
where \(\mathbf{H}_{k}\) denotes the \(T\times R\) channel matrix for the \(k\)th frequency, with \((r,t)\)_th_ element \(\mathbf{H}_{k}^{(t,r)}\). The transmitted symbols comes \(\mathbf{X}_{k}=\left[X_{k}^{(1)},\ldots,X_{k}^{(T)}\right]^{T}\). Let us consider the frequency domain estimated data symbols \(\widetilde{\mathbf{X}}_{k}=\left[\widetilde{X}_{k}^{(1)},\ldots,\widetilde{ X}_{k}^{(R)}\right]^{T}\). Assuming a non-iterative receiver, we have:  
\[\widetilde{\mathbf{X}}_{k}=\mathbf{B}_{k}\mathbf{Y}_{k}, \tag{6}\]  
where \(\mathbf{B}_{k}\) is defined as follows:  
1. For the ZF receiver, as: \[\mathbf{B}_{k}=\left(\mathbf{H}_{k}^{H}\mathbf{H}_{k}\right)^{-1}\mathbf{H}_ {k}^{H},\] (7)
2. Using the MRC receiver, as: \[\mathbf{B}_{k}=\mathbf{H}_{k}^{H},\] (8)
3.