Maximum Likelihood Sequence Estimation

A binary sequence of five symbols $I_n, n=1,\ldots,5$ (elements are drawn from $I_n \in \{-1,1\}$) is transmitted over a channel which is characterized by tapped delay-line with coefficients $\underline{f} = \{\frac{3}{5},\frac{4}{5}\}$. The observation is further corrupted by additive white Gaussian noise. The following sequence $R_n$ is observed at the output of the tapped delay line

$$R_n = \{1,\frac{3}{2},\frac{1}{4},- \frac{5}{4},- \frac{1}{2}, \frac{3}{2}\}$$

1. Given the observed sequence $R_n$, determine the most likely input sequence $I_n$. 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. For the remainder of the problem the coefficients $\underline{f}$ of the tapped delay-line are unknown. However it is known that the first five input symbols are
   
   $$I_n = \{1,-1,1,-1,-1\}.$$
   
   
   In total, the input sequence is eight symbols long and the observed output sequence is
   
   $$R_n = \{1, 0.1, 0.3, 0, -1.4, 0, 1.8, -0.4, -0.4\}$$
   
   

   Find the best estimate for the channel coefficients based on the knowledge of the first five symbols in the sequence.

4. Use your estimate for the channel coefficients to determine the most likely sequence of the remaining three symbols.