# Reduced Complexity Detection in MIMO Systems with SC-FDE Modulations and Iterative DFE Receivers
## 2 System Characterization  
The system is characterized by a MIMO communication scheme, where \(P\) independent data streams are sent to \(R\) receiving antennas, as illustrated in Figure 1.  
With the purpose of performing the multi-streaming detection and/or separation, this scheme can be framed into BS cooperation [5] or other MIMO systems, as shown in Figures 2 and 3, respectively. In BS cooperation systems, \(P\) MTs share the same physical channel and transmit to \(R\) separate BSs. Instead of having one MT assigned exclusively to a given BS and the signals from other MTs being considered interference, in BS cooperation systems, the transmission is performed in the same frequency, and the combined signals are detected by a Central Processing Unit (CPU). Figure 1: MIMO communication system with multi-stream detection. Figure 2: BS cooperation system. On the other hand, in conventional MIMO or massive MIMO systems, one can have \(P\) multiple streams transmitted to a multi-reception array with up to \(R\) antennas. In general, we have \(R\geq P\), unless we have an overloaded system, and both \(R\) and \(P\) can be much greater than one, namely in massive MIMO schemes. In the adopted transmission scheme, the communication between the \(P\) transmitting and \(R\) receiving antennas is subject to frequency-selective multipath channels. Each transmission link employs an \(N\)-symbol-sized SC-FDE block modulation scheme with an appropriate length Cyclic Prefix (CP) appended to each block. After the removal of the samples associated with the CP, the useful time-domain received data stream at the \(r\)-th antenna can be expressed by:  
\[y_{n}^{(r)}=\sum_{p=1}^{P}\xi_{p,r}s_{n,p}\vartriangle h_{n,p}^{(r)}+v_{n}^{(r)}, \tag{1}\]  
with \(\vartriangle\) denoting the cyclic convolution concerning the index \(n\). \(h_{n,p}^{(r)}\) indicates the Channel Impulse Response (CIR) between \(p\) and \(r\) for the \(n\)-th time-domain element. The data symbol \(s_{n,p}\) is selected from a constellation scheme according to a given mapping rule such as Quadrature Phase Shift Keying (QPSK) with Gray mapping. The Additive White Gaussian Noise (AWGN) component is denoted by \(v_{n}^{(r)}\), and \(\xi_{p,r}\) corresponds to a weighting parameter that accounts for the combined effects of power control and propagation loss, with the average received power associated with the \(p\)-th MT at the \(r\)-th BS corresponding to \(\left|\xi_{p,r}\right|^{2}\). Moreover, thanks to the CP, the frequency-domain version of (1) can be expressed as:  
\[Y_{k}^{(r)}=\sum_{p=1}^{P}S_{k,p}H_{k,p}^{eq(r)}+N_{k}^{(r)}, \tag{2}\]  
where \(\{Y_{k}^{(r)};k=0,1,...,N-1\}\) is the Discrete Fourier Transform (DFT) of the useful time-domain received block \(\{y_{n}^{(r)}\}\) (\(r=1,2,...,R\)) and \(S_{k,p}\) corresponds to the DFT of the time-domain data block \(\{s_{n,p};n=0,1,...,N-1\}\) that is associated with the \(p\)-th transmitting antenna \((p=1,...,P)\). \(N_{k}^{(r)}\) corresponds to the frequency-domain noise component associated with the \(r\)-th antenna and the \(k\)-th frequency and:  
\[H_{k,p}^{eq(r)}=\xi_{p,r}H_{k,p}^{(r)}, \tag{3}\]  
Figure 3: Massive MIMO scheme. where \(H_{k,p}^{(r)}\) denotes the channel frequency response between the \(p\)-th MT and the \(r\)-th BS, for the \(k\)-th frequency, where we have a normalized channel frequency response, so as \(\mathbb{E}\left[\left|H_{k,p}^{(r)}\right|^{2}\right]=1\). The detection blocks associated with each receiving antenna can have nonlinear devices, namely those associated with the quantization effects in low resolution ADCs [16; 17]. Let us admit a scenario where the received data streams at each reception branch \(r\) passes through an ADC, being sampled and quantized before the multi-stream detection is performed. Let us admit a scenario where the \(r\)-th received time-domain data stream passes through an ADC.