Performance of Viterbi detector over non-minimum phase channels For sequences that are transmitted over channels with memory $\mu=n$ and response H=$[h_0 h_1 \ldots h_n]$,Viterbi algorithm implements Maximum Likelihood (ML) detection and BCJR implements Maximum A-Posteriori (MAP) Detector. Trellis over which these detectors operate have total states = $2^n$.  

While applying any trellis based detector either ML or MAP detectors, channels are always considered causal i.e. $h_i=0 \quad \forall i&lt;0$. More over equalizers/detectors are tested for channels that have $h_0=\max(h_i)\quad \forall i=0, \ldots,N$. 

My question is related BER vs SNR performance of these trellis based detectors (that jointly equalize and detect) when applied to channels that are not causal and have a peak at $t=k,k\neq 0$. Since a delay to shift response to right has no effect on the structure of trellis, one can easily achieve this. But now the peak is no more at $t=0$. Following is an example to make it more clear.

Let $h_{ij}$ represent the $j^{th}$ coefficient for channel $i$. Consider two channels,  

\begin{align}H_1 &amp;=\left[h_{10} \space h_{11} \space h_{12}\right]=[4 \space 2 \space 1]\\H_2&amp;=\left[h_{20} \space h_{21} \space h_{22}\right]=[2 \space 4 \space 1]\end{align}

The performance of ML and MAP detector on $H_1$ is better than that of $H_2$ which is obviously because of high ISI in $H_2$ as compared to $H_1$. 

What transformation on $H_2$ can be done so that the resulting channel has a BER vs SNR that is close to $H_1$?
The Viterbi algorithm for optimum detection in the presence of intersymbol interference (ISI) is always preceded by a whitened matched filter (cf. this article by G.D. Forney). So if you use - as mentioned in a comment - the equivalent discrete-time channel, which then also includes the whitened matched filter (WMF), then, due to the properties of the WMF, this model must always be causal and minimum-phase. So, regardless of the actual impulse response of the channel, the complete equivalent discrete-time model (including the WMF) can never be non-causal or non-minimum-phase if implemented correctly.