BCD Counter: Next-State Table Derivation
Digital clocks, seven-segment displays, and test equipment readouts rely on Binary-Coded Decimal (BCD) counters to track human-readable digits. A hardware-optimized BCD counter minimizes logic by treating invalid states as don't-cares rather than explicitly routing them to zero.
Construct the combinational next-state logic for a BCD counter. The circuit evaluates the current state (Q3, Q2, Q1, Q0) and computes the next state (D3, D2, D1, D0). The sequence must count from 0 to 9 and wrap back to 0. Treat states 10 through 15 as don't-cares to minimize your logic gates.
| Q3 | Q2 | Q1 | Q0 | D3 | D2 | D1 | D0 | |------|------|------|------|------|------|------|------| | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | | 1 | 0 | 1 | 0 | X | X | X | X | | 1 | 0 | 1 | 1 | X | X | X | X | | 1 | 1 | 0 | 0 | X | X | X | X | | 1 | 1 | 0 | 1 | X | X | X | X | | 1 | 1 | 1 | 0 | X | X | X | X | | 1 | 1 | 1 | 1 | X | X | X | X |
Constraints
- Implement only the combinational next-state logic.
- Do not use sequential elements like flip-flops or latches.
- Optimize the logic by exploiting states 10 through 15 as don't-cares.
Topics
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