## 4 Channel Estimations  
This section briefly introduces two channel estimation methods. Using training symbols, the channel estimations for PPC processing can be obtained using both the LS method [22] and the MP algorithm [16].  
By combining M observed symbols, Equation (1) is rewritten as:  
\[\begin{bmatrix}V_{k}^{n-M+1}\\ V_{k}^{n-M}\\ \vdots\\ V_{k}^{n}\end{bmatrix}=\begin{bmatrix}I_{n-M+1}&\cdots&I_{n-M-L+2}\\ \vdots&&\vdots\\ I_{n}&\cdots&I_{n-L+1}&\begin{bmatrix}H_{k}^{0}\\ H_{k}^{1}\\ \vdots\\ H_{k}^{L-1}\end{bmatrix}+\begin{bmatrix}W_{k}^{n-M+1}\\ W_{k}^{n-M}\\ \vdots\\ W_{k}^{n}\end{bmatrix}\end{bmatrix} \tag{2}\]  
which is simplified to:  
\[V_{k}={H}_{k}+W_{k} \tag{3}\]  
In the channel estimation problem, the information matrix \(I\) is known as training symbols. An estimation of \(\hat{H}_{k}\) can be obtained by solving LS problem:  
\[\hat{H}_{k}=\arg\min_{\hat{H}_{k}}\left\{\left\|\left\|V_{k}-{I}\tilde{H}_{k} \right\|^{2}\right\}\right. \tag{4}\]  
which gives the solution:  
\[\hat{H}_{k}=\begin{bmatrix}I^{n}I\end{bmatrix}^{-1}I^{n}V_{k} \tag{5}\]  
In practice, the LS method is sensitive to noise. When a channel is sparse, the CIR consists of a large number of zeros among several dominant taps, and the LS method will suffer from the noise between dominant taps. Besides, the LS method involves matrix inversion, and it sometimes suffers ill-conditioned problem of a matrix of large eigenvalue spread.  
Figure 7: Block diagram of PPC-McDFE.  
To exploit the sparse property of channels, the channel estimation problem can be reconsidered as an approximation problem. It is assumed that the received signal vector is approximated by:  
\[\left(\hat{V}_{k}\right)_{M}=\sum_{i=0}^{M-1}\hat{H}_{k}^{p_{i}}\left(I\right)_{ p_{i}} \tag{6}\]  
where \(\left(I\right)_{p_{i}}\) is the \(p_{i}\) th column of information matrix \(I\). Finding the approximation of \(\left(\hat{V}_{k}\right)_{M}\) that minimizes \(\left\|\left(\hat{V}_{k}\right)_{M}-V_{k}\right\|\) is an non-deterministic polynomial-time hard problem [23, 24], which means there is unknown polynomial time algorithm that can solve this problem. MP [16] is a greedy algorithm that achieves non-optimal yet computational efficient approximation of \(V_{k}\).  
The MP algorithm selects one column in matrix \(I\) which is best aligned with residual signal \(r_{p-1}\), where \(r_{0}=V_{k}\) at initial step. In practice at the \(p\)th step, the selected \(l_{p}\)th column of \(I\) is determined by:  
\[l_{p}=\arg\max_{l}\left\{\left|\left(I\right)_{l}^{H}r_{p-1}\right|\left/ \left|\left(I\right)_{l}\right|\right\|\right\} \tag{7}\]  
Correspondingly, the tap value \(\hat{H}_{k}^{l_{p}}\) is estimated by:  
\[\hat{H}_{k}^{l_{p}}=\frac{\left(I\right)_{l_{p}}^{H}r_{p-1}}{\left|\left|\left( I\right)_{l_{p}}\right|^{2}} \tag{8}\]  
and \(r_{p}\) is updated by:  
\[r_{p}=r_{p-1}-\frac{\left(I\right)_{l_{p}}^{H}r_{p-1}}{\left|\left(I\right)_{ l_{p}}\right|^{2}}\left(I\right)_{l_{p}} \tag{9}\]  
This iteration is terminated until the preset \(P\) taps have been estimated. In practice, one column in \(I\) is probably selected more than once. To deal with this problem, we can either exclude previously selected columns in the search press shown in Equation (7), or the tap value calculated in Equation (8) can be added to the value found in previous steps [25]. In this paper, we use the former method.