Maximum Likelihood Sequence Estimation

A binary sequence of five symbols $\underline{I}$ (elements are drawn from $I_n \in \{-1,1\}$) is transmitted over a channel which is characterized by tapped delay-line with coefficients $a=5$ and $b=-4$. The observation is further corrupted by additive white Gaussian noise. The following sequence $\underline{r}$ is observed at the output of the tapped delay line

$$\underline{r} = \{7,-1,-4,-5,7,-2\}$$

1. Given the observed sequence $\underline{r}$, determine the most likely input sequence $\underline{I}$. Clearly, show how you arrived at your solution.
2. Draw and clearly label a trellis diagram and indicate the path through the trellis which corresponds to the most likely sequence.
3. What is the Euclidean distance associated with the two sequences $\underline{I}_1 = \{1,1,-1,-1,1\}$ and $\underline{I}_2 = \{1,1,1,-1,1\}$, respectively? Explain.
4. For the remainder of the problem the coefficients of the tapped delay-line are time-varying. In the first, third, and fifth symbol period the channel coefficients are $a=5$ and $b=-4$, in the second, fourth, and sixth symbol period the channel coefficients are $a=4$ and $b=-5$.

   Explain how the Viterbi Algorithm must be modified to account for the time-varying coefficients.

5. The observed sequence is $\underline{r} = \{6,-8,8,0,-8,4\}$. Find the most likely input sequence $\underline{I}$.