Maximum-Likelihood Sequence Estimation for Convolutional Codes

Consider a rate $\frac{1}{2}$ convolutional coder with constraint length $K=3$ and generator polynomials in octal representation $g_1=5$ and $g_2=7$.

1. Draw a block diagram of this encoder and label it accurately.
2. Determine the output from the encoder if the input sequence is $I=\{1,0,0,1,1\}$. Assume that the register is initially loaded with $0$-bits and that $0$-bits are transmitted after the sequence as stop bits .
3. Assume now that the sequence $V=\{1,1,0,1,1,0,0,1,1,1\}$ is received. Again $0$-start and stop bits have been added to the sequence to force the encoder into the all $0$-state at the beginning and end of the transmission. Thus, there are three information bits between the start and stop bits.

   Draw a trellis diagram, showing clearly the possible states of the encoder, the possible transitions and the encoder outputs associated with the transitions.

4. Determine the most likely input sequence.