$(10$ points$)$ Ring Counter

$(3$ points$)$ Figure $2$ shows the $4-$bit switch$-$tail ring counter $($known otherwise as Johnson Counter$)$ where $"Q_D$ is fed to $D_A.$ What is the state diagram for $Q_A{Q}_B{Q}_c{Q}_D?$ In general, a $k-$bit switch$-$tail ring counter goes through a sequence of $2k$ states.

$(2$ points$)$ Identify the logic functions $($with minimal number of gate inputs$)$ that can decode each of the Johnson counter outputs $Q_A{Q}_B{Q}_c{Q}_D?$

$(2$ points$)$ What are the unused states in the counter circuit of Figure $2?$

$(3$ points$)$ What are the counter states $Q_A{Q}_B{Q}_c{Q}_D$ should the counter start from one of the unused states? Show transition paths that begin from any three unused states and their respective next states.

Figure $2$