BCD Counter Gate Count Analysis: Simplified vs. Binary
Hardware counters often implement decimal arithmetic directly to avoid binary-to-decimal conversion overhead in display systems. The logic for a 4-bit synchronous binary counter and a BCD (modulo-10) counter both rely on four flip-flops: Q3, Q2, Q1, and Q0. The binary counter sequences from 0000 to 1111. The BCD counter sequences from 0000 to 1001, treating states 1010 through 1111 as don't-cares for logic minimization.
Derive the fully optimized sum-of-products (SOP) Boolean expression for the D3 input of both counters. Compare their combinational logic complexity and identify the source of any structural simplification.
Constraints
- Assume standard D-type flip-flops are used for all state bits.
- Do not count inputs to NOT gates when calculating total gate inputs.
- Treat states 10 through 15 strictly as don't-cares for the BCD counter K-maps.
Topics
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