How to Generate a Logic Diagram from State and Output Equations?
- Identify the number of inputs, outputs and flip-flops in the state/output equations.
- For D Flip-flops:
- Draw as many flip-flops as the number of state equations.
- Draw the logic diagram of the circuit represented by the state equation and attach its output to the input of the corresponding D flip-flop.
- Draw the logic diagram of the circuit represented by each output equation.
Example: Extract the Logic Diagram from the State and Output Equations
We begin with the following equations:
\begin{align*}
\mathsf{A(t+1) }& \mathsf{ =Ax+Bx=(A+B)x}\\
\mathsf{B(t+1) }& \mathsf{ =\overline{A}x}\\
\mathsf{y }& \mathsf{ =A\overline{x}+B\overline{x}=(A+B)\overline{x}}
\end{align*}
Step 1: Identify the number of inputs, outputs and flip-flops in the logic diagram
From the equations we can surmise that:
- Total Inputs: 1 (External input \(\mathsf{x}\))
- Total Flip-Flops: 2 flip-flops (Outputs \(\mathsf{A}\) and \(\mathsf{B}\))
- Total Combinational Outputs: 1 (Output \(\mathsf{y}\))
Step 2 & 3:
To begin, draw two D-type flip-flops. Subsequently, construct the logic circuit represented by the next-state equations. Afterward, connect its output directly to the input of each D flip-flop. Furthermore, name the resulting output signals \(\mathsf{A}\) and \(\mathsf{B}\) to align with the equations. Next, take the final output equation and draw its corresponding circuit layout. Finally, designate the output of that circuit as \(\mathsf{y}\), establishing it as the primary output of the entire system.
The circuit is drawn here with each state and output equation written along side it:
