## 2 System Model  
We consider a coded OFDM system with \(N_{s}\) subcarriers and iterative processing as illustrated in Figure 1. For a conventional SIC receiver, the number of feedback candidates is reduced to one in the multi-feedback generation block. The information bits are encoded as \(b_{m}\) by the channel encoder and then interleaved as \(u_{m}\) through the random interleaver, where the subscript \(m\) denotes the \(m\)-th bit in the sequence. Each group of \(c\) bits is modulated by the symbol mapper onto one symbol \(s_{k}\) on the \(k\)-th subcarrier at the \(i\)-th OFDM symbol, and then, the inverse fast Fourier transform (IFFT) is performed to obtain the serial data stream. Hence, the signals can be written as:  
\[a_{n}=\frac{1}{\sqrt{N_{s}}}\sum_{k=0}^{N_{s}-1}s_{k}e^{j\frac{2\pi}{N_{s}}kn} \tag{1}\]  
where the quantity \(a_{n}\) is transmitted over a time-varying multi-path channel. The cyclic-prefix (CP) is inserted after Equation (1). Once the distorted transmitted signals reach the receiver, the CP is removed. Then, the received signals during the \(i\)-th OFDM symbol are represented as:  
\[r_{n}=\sum_{l=0}^{L-1}h_{tl}(n,l)a_{n-l}+z_{n}\ \ 0\leq n<N_{s}-1 \tag{2}\]  
where the quantity \(z_{n}\) denotes one sample of additive white Gaussian noise (AWGN). The received signals are split into several segments, which go through the PFFT for channel estimation. The summation of the outputs of PFFT blocks is the same as the conventional FFT. Thus, the output is used for equalization as used in conventional OFDM systems. We assume that the \(k\)-th subcarrier is the desired one and omit the noise for simplicity. Substituting Equation (1) into Equation (2), Equation (2) after FFT becomes [4]:  
\[ \begin{split} y_{k}&=\frac{1}{N_{s}}\sum_{n=0}^{N_{ s}-1}\sum_{l=0}^{L-1}\sum_{q=0}^{N_{s}-1}h_{tl}(n,l)s(q)e^{j2\pi(qn+ql)/N_{s}}e^{-j2 \pi kn/N_{s}}\\ &=\underbrace{h_{\textit{df}}(0,k)s(k)}_{\text{desired signal}}+ \underbrace{\sum_{q=0,q\
eq k}^{N_{s}-1}h_{\textit{df}}(k-q,q)s(q)}_{\text{ICI components}},\end{split}  \tag{3}\]where \(h_{\mathit{df}}(k-q,q)=\frac{1}{N_{s}}\sum_{n=0}^{N_{s}-1}\sum_{l=0}^{L-1}h_{ \mathit{tl}}(n,l)e^{-j2\pi((k-q)n+ql)/N_{s}}\), and the quantity \(h_{\mathit{tl}}(n,l)\) denotes the channel impulse response for the \(n\)-th time index in one OFDM symbol and the \(l\)-th channel path. If we let \(d=k-q\) and \(k=q\), \(h_{\mathit{df}}(d,k)=\frac{1}{N_{s}}\sum_{n=0}^{N_{s}-1}\sum_{l=0}^{L-1}h_{ \mathit{tl}}(n,l)e^{-j2\pi(dn+kl)}\). As described in [4], the quantities \(d\) and \(k\) in \(h_{\mathit{df}}(d,k)\) can be interpreted as the \"Doppler\" index and the subcarrier index, respectively. We can also rewrite Equation (3) in a matrix form:  
\[\mathbf{y}=\underbrace{\mathbf{FH}_{\mathit{tl}}\mathbf{F}^{H}}_{\mathbf{H_{ df}}}\mathbf{s}+\mathbf{v}, \tag{4}\]  
where the matrix \(\mathbf{H}_{\mathit{tl}}\in\mathbb{C}_{N_{s}\times N_{s}}\) is illustrated as a circulant matrix in the time domain, and \(\mathbf{s}\in\mathbb{C}_{N_{s}\times 1}\), \(\mathbf{y}\in\mathbb{C}_{N_{s}\times 1}\), \(\mathbf{v}\in\mathbb{C}_{N_{s}\times 1}\), \(\mathbf{F}\in\mathbb{C}_{N_{s}\times N_{s}}\). The symbol \(\mathbb{C}_{n\times n}\) denotes the matrix with the size of \(n\) by \(n\). The matrix \(\mathbf{H}_{\mathit{df}}\) denotes the equivalent frequency channel matrix with the size of \(N_{s}\times N_{s}\). The output of MBMF block is the output of the equalizer, which yields the estimated symbols \(\hat{s}_{k}\). The bit LLRs corresponding to these symbols can be obtained and deinterleaved for the channel decoder using the MAP algorithm [25]. The output of the channel decoder, after the interleaver, is also used for the interference cancellation. The interference is regenerated by the multi-feedback generation block. For the conventional SIC receiver, the number of feedback candidates is reduced to one in the output of the multi-feedback generation block. In the following, the processing is based on the frequency domain, unless otherwise specified.