# State definition in Viterbi Algorithm?

An ISI channel of length 'L+1' $\mathbf{h}=[h_0 \space h_1 \space … \space h_L]$can always be modeled as a finite state machine(FSM) of memory 'L'. For a binary input $x\in\{+1,-1\}$,a state can be defined as $\mathbf{s}=[x_{1}\space x_{2} \space … \space x_{L}]$.

Myint, L.M.M.; Supnithi, P.; Tantaswadi, P., "An Inter-Track Interference Mitigation Technique Using Partial ITI Estimation in Patterned Media Storage," Magnetics, IEEE Transactions on , vol.45, no.10, pp.3691,3694, Oct. 2009

in which $\mathbf{h}=[h_{k-1} \space h_k \space h_{k+1}]=[\beta \space b \space \beta]$, where the authors define gain=$b$ and ISI-coefficient=$\beta$ (ref:eq 3,4,5) . the noiseless channel output is defined as

$$y_{k} = bx_k+\beta(x_{k-1}+x_{k+1})\\ r_{k} = y_k + n_k$$

and create a 4-state trellis using $[x_{k-1} \space x_{k+1}]$.

The branch metric is computed as $\lambda_k=(r_k-y_k)^2$.

This is approach is different from the conventional state definition $[x_{k-1} \space x_{k}]$. Since this paper is published in IEEE transactions on magnetics, I am hesitant to say that there is a mistake here but on the other hand this definition doesn't sound right to me.

I will be very grateful if someone can comment on this.

$\color{red}{New \space Edit:}$ While searching for the answer I came across another paper (cf. this article)

Rubsamen, M.; Gene, J.M.; Winzer, P.J.; Essiambre, R., "ISI Mitigation Capability of MLSE Direct-Detection Receivers," Photonics Technology Letters, IEEE , vol.20, no.8, pp.656,658, April15, 2008 doi: 10.1109/LPT.2008.919597

That takes a similar approach.

• What doesn't sound right to you? An ISI channel of length 3 would be modeled as a finite state machine with memory length 2, which would have four states. – Jason R Jun 10 '15 at 14:01
• they are using $h_{k-1},h_{k+1}$ coefficients for defining states. wrt $h_k$, $h_{k-1}$ is past and $h_{k+1}$ is future. This is the point of confusion for me. The gain coefficient $b$ is in the middle. normally we would consider $h_k,h_{k-1}$ for state definition – NAASI Jun 10 '15 at 14:08
• I have updated my question and have included important definitions for making question more clear – NAASI Jun 10 '15 at 14:25
• It is often the case that one wants the largest entry in $\mathbf h$ to be called $h_0$ so that the peak response to input $x_k$ is at time $k$ rather than at time $k+m$. This simplifies the ISI analysis, but for ML sequence estimation using the Viterbi algorithm, it is necessary to re-adapt. – Dilip Sarwate Jun 10 '15 at 14:31
• @ Dilip, I get your point but using this new definition we would be computing the likelihood of making transitions because of bit at t=k+1 but assigning the value to bit at t=k. I can't find the justification for these mismatched indices. – NAASI Jun 10 '15 at 14:36