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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. $h_i=0 \quad \forall i<0$. 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 &=\left[h_{10} \space h_{11} \space h_{12}\right]=[4 \space 2 \space 1]\\H_2&=\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 $H_1$?

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  • $\begingroup$ Minimum phase does not imply $h_0=\max\{h_i\}$. A counter-example: h=[1,1.5,0.6] is minimum-phase. $\endgroup$ – Matt L. Jun 6 '15 at 19:56
  • $\begingroup$ Is the channel known? Practical approaches would be to use an equalizer before the Viterbi decoder, or combine the two, incorporating the known channel response in the trellis model. This is sometimes known as Viterbi equalization. $\endgroup$ – Jason R Jun 6 '15 at 19:58
  • $\begingroup$ @ Matt…rigorous definition of minimum phase (MP) includes position of poles/zeros wrt unit circle. For my application, this does not matter to me. Also they effect if we are trying to use linear filters to equalize but since I am interested in trellis based equalizers/detectors so I am not very much worried about this aspect. But thanks for pointing it out. I should had been more careful in using MP term $\endgroup$ – NAASI Jun 6 '15 at 20:09
  • $\begingroup$ @ Jason…. Yes channel response is know and you are right in pointing out the viterbi equalization terminology. I will include that in my original post. Would you like to comment on the last para of my original post that talks about the performance comarion between $H1$ & $H2$ $\endgroup$ – NAASI Jun 6 '15 at 20:12
  • $\begingroup$ Do you use a whitened matched filter before the Viterbi decoder? $\endgroup$ – Matt L. Jun 6 '15 at 21:29
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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.

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