Continuous Phase Modulation

Equally likely random symbols $I_n \in \{-1,1\}$ are to be transmitted at a rate of one symbol per $T$ seconds over an additive white Gaussian noise channel. The symbols are modulated using partial response ($L=2$) continuous phase modulation with modulation index $h=\frac{1}{2}$ and rectangular pulses.

1. Sketch the phase of the transmitted signal resulting from the sequence
   
   $$\underline{I} = \{1,-1,1,1 -1,-1,-1,1\}.$$
   
   

2. Show that for an arbitrary input sequence $\underline{I}$ the phase can be expressed as
   
   $$\phi(t,\underline{I}) = \theta_n + 2 \pi h \sum_{k=0}^{L-1} I_{n-k} q(t-(n-k)T) \quad \mbox{for $nT \leq t < (n+1)T$}.$$
   
   
   Provide an explicit expression for $\theta_n$.
3. Which values can $\theta_n$ take on?
4. Phase transitions between times $nT$ and $(n+1)T$ depend only on $\theta_n$, $I_n$, and $I_{n-1}$. Hence, the evolution of the phase $\phi(t,\underline{I})$ can be described as a path through a suitably chosen trellis. How would you define the states of the trellis?
5. Draw a trellis diagram showing all possible state transitions between times $nT$ and $(n+1)T$.
6. To employ the Viterbi algorithm for demodulating CPM signals the metrics (distances) associated with the state transitions are required. Define these distances precisely and explain how they can be obtained.