BCD to Decimal Converter: Design a Combinational Circuit

(a) Here is the truth table for the combinational circuit that takes a BCD digit (A, B, C, D) and outputs a decimal number represented by two BCD digits (S, T, U, V and W, X, Y, Z) that is 1 more than four times the input:

| A | B | C | D | S | T | U | V | W | X | Y | Z ||---|---|---|---|---|---|---|---|---|---|---|---|| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 || 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 || 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 || 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 || 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 || 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 || 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 || 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 || 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 || 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 || 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 || 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 || 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 || 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 || 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 || 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

(b) From the truth table, we can derive the minimum sum-of-products expressions for each output:

S = A'BC'D' + A'BCD' + AB'C'D' + AB'CD' + ABC'D + ABCD T = A'BC'D' + A'BCD' + AB'C'D + AB'CD + ABCD U = A'BC'D' + AB'C'D' + AB'CD' + ABC'D + ABCD V = A'BC'D + A'BCD' + AB'C'D' + AB'CD' + ABC'D + ABCD W = A'BC'D' + A'BCD' + AB'CD' + ABC'D + ABCD X = A'B'C'D' + A'B'CD + A'BCD' + AB'C'D + AB'CD + ABCD Y = A'B'C'D' + A'B'CD + A'BCD' + AB'C'D' + AB'CD' + ABC'D' + ABCD Z = A'B'C'D' + A'B'CD' + A'BC'D + A'BCD' + AB'CD' + ABC'D' + ABCD