How to Generate a State and Output Equations from a State Transition Table?
- Solve each column using the Karnaugh Map (K-map) method to generate the corresponding equation in the simplest form.
- The right half of the state table, represents all the next state variables and combinational outputs.
- Each column represents one combinational output or one flip-flop.
- For flip-flops, the equation represents the next state of that flip-flop.
- For combinational outputs, the equation represents the output of that combinational circuit.
Example: Generate State and Output Equations from the State Table
Continuing our design example, we have the following state table:
Analyzing the state table, we realize that we are working with two flip-flops and one combinational output. Consequently, we must derive two equations for the next-state variables and one for the output. The state table organizes these parameters across specific columns. Specifically, the first three columns track current state variables \(\mathsf{A(t)}\) and \(\mathsf{B(t)}\) and the input \(\mathsf{x}\). Following this, the next two columns record upcoming state variables \(\mathsf{A(t+1)}\) and \(\mathsf{B(t+1)}\). Finally, the last column displays the combinational output \(\mathsf{y}\). Because there are three total input and current state variables, a three-input Karnaugh map (K-map) is required. By evaluating these columns through the K-map, we can systematically solve for the simplest next-state and output equations.
The interactive animation below highlights columns four, five, and six. Additionally, it demonstrates how their data populates a three-input Karnaugh map. Furthermore, the K-map visualizes the optimal cell groupings. Consequently, the resulting logic equations are displayed directly alongside for clear analysis.