2.5 Step 5: Derive Boolean Equations

Once the truth tables for the state transitions and the output logic have been created, they can be used to derive Boolean equations that define the logical behavior of the system. One expression needs to be defined for each of the bits of the encoded state to define how each state will transition to the next. Sum-of-Products (SOP) expressions for X₀ and X₁ can be derived by examining the state transition table:

X_{0} = {\bar{E}}_{} {\bar{X}}_{2} {\bar{X}}_{1} {\bar{X}}_{0} + {\bar{E}}_{} {\bar{X}}_{2} X_{1} {\bar{X}}_{0}

X_{1} = {\bar{E}}_{} {\bar{X}}_{2} {\bar{X}}_{1} X_{0} + {\bar{E}}_{} {\bar{X}}_{2} X_{1} {\bar{X}}_{0}

Because $X_{2}$ has five $1$ terms, a Karnaugh map can be used to get the simplest possible Boolean equation:

X_{2} = E_{} {\bar{X}}_{2} + E_{} {\bar{X}}_{1} {\bar{X}}_{0}

Once the next-state logic has been defined, the logic for controlling the output can be determined using the output logic truth table:

Equation 2-1. Output Logic Expressions

N S G = {\bar{X}}_{2} {\bar{X}}_{1} {\bar{X}}_{0}

N S Y = {\bar{X}}_{2} {\bar{X}}_{1} X_{0}

E W G = {\bar{X}}_{2} X_{1} {\bar{X}}_{0}

E W Y = {\bar{X}}_{2} X_{1} X_{0}

Since the error state is the only state where $X_{2}$ is a $1$ , $X_{2}$ can be used directly as the ERR output:

E R R = X_{2}