Question 9.5: Develop a synchronous 3-bit up/down counter with a Gray code...

Step 1: The state diagram is shown in Figure 9–33. The 1 or 0 beside each arrow indicates the state of the UP/\overline{\text{DOWN}} control input, Y.

Step 2: The next-state table is derived from the state diagram and is shown in Table 9–12. Notice that for each present state there are two possible next states, depending on the UP/\overline{\text{DOWN}} control variable, Y.

Step 3: The transition table for the J-K flip-flops is repeated in Table 9–13.

Step 4: The Karnaugh maps for the J and K inputs of the flip-flops are shown in Figure 9–34. The UP/\overline{\text{DOWN}} control input, Y, is considered one of the state variables along with Q_{0},  Q_{1},  and  Q_{2}. Using the next-state table, the information in the “Flip-Flop Inputs” column of Table 9–13 is transferred onto the maps as indicated for each present state of the counter.

Step 5: The 1s are combined in the largest possible groupings, with “don’t cares” (Xs) used where possible. The groups are factored, and the expressions for the J and K inputs are as follows:
J_{0}  =  Q_{2} Q_{1} Y  +  Q_{2} \overline{Q}_{1}  \overline{Y}  +  \overline{Q}_{2}  \overline{Q}_{1} Y  +  \overline{Q}_{2} Q_{1} \overline{Y}         K_{0}  =  \overline{Q}_{2}  \overline{Q}_{1}  \overline{Y}  +  \overline{Q}_{2} Q_{1} Y  +  Q_{2} \overline{Q}_{1} Y  +  Q_{2} Q_{1} \overline{Y}

J_{2}  =  Q_{1} \overline{Q}_{0} Y  +  \overline{Q}_{1}  \overline{Q}_{0}  \overline{Y}                             K_{2}  =  Q_{1} \overline{Q}_{0}  \overline{Y}  +  \overline{Q}_{1}  \overline{Q}_{0} Y
Step 6: The J and K equations are implemented with combinational logic. This step is the Related Problem.