The channel taps are classified into postcursor (causal) and the precursor (anti-causal) parts.
\begin{equation}y_{k}=\sum_{l=-L_{2}}^{l=L_{1}} h_{l} x_{k-l}+n_{k}\end{equation}
here $h_{l}$ is the $l^{t h}$ tap of the channel and $L_{1}$ and $L_{2}$ are the lengths of the causal and anti-causal part of the channel respectively and $L=L_{1}+L_{2}+1$ is the total length of the channel. In addition, $n_{k}$ represents an additive Gaussian white noise with zero mean and variance $\sigma_{n}^{2} .$ Arranging the equation above into a matrix form we have
\begin{equation}\begin{aligned} \mathbf{y}_{k} &=\mathbf{H} \mathbf{x}_{k}+\mathbf{n}_{k} \\ \mathbf{y}_{k} &=\left[y_{k-K_{1}} y_{k-K_{1}+1} \cdots y_{k+K_{2}}\right]^{T} \\ \mathbf{x}_{k} &=\left[x_{k-K_{1}-L_{1}} x_{k-K_{1}-L_{1}+1} \cdots x_{k+K_{2}+L 2}\right]^{T} \\ \mathbf{n}_{k} &=\left[n_{k-K_{1}} n_{k-K_{1}+1} \cdots n_{k+K_{2}}\right]^{T} \\ \mathbf{H} &=\left[\begin{array}{cccccc} h_{L_{1}} & \cdots & h_{-L_{2}} & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & h_{L_{1}} & \cdots & h_{-L_{2}} \end{array}\right] \end{aligned}\end{equation} The anti causal part of the channel span means that at a time $k$, the output of the channel depends on previous symbols and future symbols ? how can this be ?