Q15Digital Electronics

A multiplexer is not merely a data router; it is a highly powerful, universal combinational logic generator. While a 16:1 MUX (which has 4 select lines) could effortlessly implement any 4-variable Boolean function simply by connecting its 16 inputs directly to VDD (1) or Ground (0) based on the minterms, this is highly inefficient. We are challenged to implement a complex 4-variable function, $F(A,B,C,D)=\Sigma m(0,1,2,5,7,8,9,14,15)$, utilizing only a much smaller 8:1 MUX. This requires aggressively applying Shannon's Expansion Theorem to fold the fourth variable ($D$) directly into the data input lines.

Step 1: MUX Configuration and Variable Assignment

An 8:1 MUX possesses exactly 3 select lines ($S_2,S_1,S_0$) and 8 data input lines ($I_0$ through $I_7$). We must strictly assign the three most significant variables of our function ($A,B,C$) to the select lines: - $S_2=A$ - $S_1=B$ - $S_0=C$

This assignment implies that the 8 inputs ($I_0\dots I_7$) directly correspond to the 8 possible binary combinations of $A,B,C$. The final output $F$ must now be mathematically evaluated solely in terms of the remaining variable, $D$.

Step 2: The Implementation Table Methodology

We construct a strict Implementation Table. We list the 8 MUX inputs ($I_0\dots I_7$) as columns. We list the states of the fourth variable ($D^{\prime }$ and $D$) as two horizontal rows. We then sequentially populate the grid with the decimal minterm numbers (0-15).

Inputs: | $I_0$ | $I_1$ | $I_2$ | $I_3$ | $I_4$ | $I_5$ | $I_6$ | $I_7$ ($A,B,C$) | (000)| (001)| (010)| (011)| (100)| (101)| (110)| (111) --------------------------------------------------------- Row $D^{\prime }$ (0):| 0 | 2 | 4 | 6 | 8 | 10 | 12 | 14 Row $D$ (1): | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15

Step 3: Minterm Circling and Input Derivation

We meticulously circle the specific numbers in the grid that strictly correspond to the provided minterms of our function: 0, 1, 2, 5, 7, 8, 9, 14, 15.

• Column $I_0$: Both 0 and 1 are circled. Result: $I_0$ must be tied to logic 1. ($D^{\prime }+D=1$)
• Column $I_1$: Only 2 is circled (top row). Result: $I_1$ must be tied to $D^{\prime }$.
• Column $I_2$: Only 5 is circled (bottom row). Result: $I_2$ must be tied to $D$.
• Column $I_3$: Only 7 is circled (bottom row). Result: $I_3$ must be tied to $D$.
• Column $I_4$: Both 8 and 9 are circled. Result: $I_4$ must be tied to logic 1.
• Column $I_5$: Neither 10 nor 11 are circled. Result: $I_5$ must be tied to logic 0.
• Column $I_6$: Neither 12 nor 13 are circled. Result: $I_6$ must be tied to logic 0.
• Column $I_7$: Both 14 and 15 are circled. Result: $I_7$ must be tied to logic 1.

By physically wiring the 8 inputs of the MUX exactly according to this derived sequence (1, $D^{\prime }$, $D$, $D$, 1, 0, 0, 1), the smaller 8:1 MUX will flawlessly execute the massive 4-variable Boolean logic function.