# Reduced Complexity Detection in MIMO Systems with SC-FDE Modulations and Iterative DFE Receivers
## 2 System Characterization
At the ADC output, the \(n\)-th time-domain sample associated with the \(r\)-th stream can be expressed as:  
\[y_{n}^{Q(r)}=f_{\mathcal{Q}}\left(\frac{\text{Re}\{y_{n}^{(r)}\}}{\sigma_{y}^ {(r)}}\right)\sigma_{y}^{(r)}+jf_{\mathcal{Q}}\left(\frac{\text{Im}\{y_{n}^{( r)}\}}{\sigma_{y}^{(r)}}\right)\sigma_{y}^{(r)}, \tag{4}\]  
where \(\sigma_{y}^{(r)}\) indicates the variance of the input signal. Furthermore, \(f_{\mathcal{Q}}(\cdot)\) denotes the quantization characteristics (or other nonlinear characteristic). In this paper, we consider a uniform \"mid-rise\" quantizer with a normalized saturation level of \(\frac{A_{M}}{\sigma_{y}^{(r)}}\) and \(m\) bits of resolution for the real and imaginary parts of a given quantized sample. Since we are considering severely time-dispersive channels with rich multipath propagation environment (the typical channel conditions in SC-FDE schemes), the received time-domain samples \(y_{n}^{(r)}\) can be accounted for as samples of a zero-mean complex Gaussian process, i.e., \(y_{n}^{(r)}\sim\mathcal{CN}\left(0,2\sigma_{y}^{(r)^{2}}\right)\), with:  
\[ \begin{split} 2\sigma_{y}^{(r)^{2}}=\mathbb{E}\left[\left|y_{n}^{(r)} \right|^{2}\right]&=\sum_{p=1}^{p}\left|\xi_{p,r}\right|^{2} \mathbb{E}\left[\left|s_{n,p}\right|^{2}\right]+\mathbb{E}\left[\left|v_{n}^{ (r)}\right|^{2}\right]\\ &=\sum_{p=1}^{p}\left|\xi_{p,r}\right|^{2}2\sigma_{s}^{2}+2\sigma _{v}^{2},\end{split}  \tag{5}\]  
where:  
\[\sigma_{s}^{2}=\mathbb{E}\left[\left|\text{Re}\{s_{n}\}\right|^{2}\right]= \mathbb{E}\left[\left|\text{Im}\{s_{n}\}\right|^{2}\right] \tag{6}\]  
and:  
\[\sigma_{v}^{2}=\mathbb{E}\left[\left|\text{Re}\{v_{n}\}\right|^{2}\right]= \mathbb{E}\left[\left|\text{Im}\{v_{n}\}\right|^{2}\right], \tag{7}\]  
correspond to the symbol's and noise variance, respectively. According to [PERSON]'s theorem [18], the Gaussian nature of \(y_{n}^{(r)}\) allows the quantized signals to be decomposed as the sum of uncorrelated useful and distortion terms, which leads to:  
\[y_{n}^{Q(r)}\approx\omega y_{n}^{(r)}+d_{n}^{(r)}, \tag{8}\]  
with \(d_{n}^{(r)}\) denoting the quantization noise term. The \(\alpha\) parameter is a constant that depends only on the nonlinear characteristic and is given by:  
\[\alpha=\int_{-\infty}^{\infty}\omega f_{\mathcal{Q}}(\omega)\frac{1}{\sqrt{2 \pi}}e^{-\frac{\omega^{2}}{2}}d\omega. \tag{9}\]  
In the frequency-domain, the sample associated with the quantized version of the signal \(Y_{k}^{(r)}\) corresponds to:  
\[Y_{k}^{Q(r)}\approx\alpha Y_{k}^{(r)}+D_{k}^{(r)}\approx\alpha\sum_{p=1}^{p}S _{k,p}H_{k,p}^{eq(r)}+N_{k}^{Tot(r)}, \tag{10}\]which is the DFT of the time-domain signal \(y_{n}^{Q(r)}\). \(N_{k}^{Tot(r)}=\alpha N_{k}^{(r)}+D_{k}^{(r)}\) accounts for the global noise from the transmitted and quantized signals. The statistical characterization of \(D_{k}^{(r)}\) can be done as described in [12]. Essentially, \(D_{k}^{(r)}\) is approximately complex Gaussian with zero mean and variance \(\sigma_{D}^{(r)^{2}}(k)\), which is a function of \(r\) and \(k\). The exact computation of \(\sigma_{D}^{(r)^{2}}(k)\) can be obtained using the method of [12]. When the combined signals from all \(P\) and \(R\) antennas are taken into account, (10) can be expressed in a matrix format, given by:  
\[\mathbf{Y}_{k}^{Q}=\mathbf{H}_{k}^{T}\mathbf{S}_{k}+\alpha\mathbf{N}_{k}+ \mathbf{D}_{k}, \tag{11}\]  
where \(\mathbf{Y}_{k}^{Q}=\left[Y_{k}^{Q(1)}\,\...\,Y_{k}^{Q(R)}\right]^{T}\), \(\mathbf{N}_{k}=\left[N_{k}^{(1)}\,\...\,N_{k}^{(R)}\right]^{T}\) and \(\mathbf{D}_{k}=\left[D_{k}^{(1)}\,\...\,D_{k}^{(R)}\right]^{T}\) are column vectors of size \(R\times 1\) and \(\mathbf{S}_{k}=\left[S_{k,1},\...\,S_{k,P}\right]^{T}\) is a column vector of size \(P\times 1\). Furthermore, the channel associated with the \(k\)-th frequency component is represented by the \(R\times P\) matrix:  
\[\mathbf{H}_{k}^{T}= \begin{bmatrix}H_{k,1}^{eq(1)}&\dots&H_{k,P}^{eq(1)}\\ \vdots&\ddots&\vdots\\ H_{k,1}^{eq(R)}&\dots&H_{k,P}^{eq(R)}\end{bmatrix} , \tag{12}\]  
with \(H_{k,p}^{eq(r)}=\alpha\zeta_{p,r}H_{k,p}^{(r)}\).