Function Realization with an 8-to-1 Multiplexer: 25-Comp-A2 Digital Systems Design, December 2014, Q2

Question 2: Function Realization with an 8-to-1 Multiplexer (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A 3-variable function $F(A,B,C)$ specified by the K-map-style table above (Gray-coded columns $AB=00,01,11,10$; rows $C=0,1$).

Find. (a) $F$ as a Boolean expression in $A,B,C$; (b) a realization of $F$ using one 8-to-1 multiplexer.

Approach. Read the eight cells off the table as minterms of $(A,B,C)$, write the canonical sum, then — because a 3-variable function maps onto an 8-to-1 MUX one-for-one — drive the select lines directly from $A,B,C$ and tie each data input to the truth-table value of its own minterm; no algebraic reduction is needed for this direct implementation.

Part (a) — read the minterms off the table. Taking $A,B$ from the column header and $C$ from the row, the eight cells read as: $(A,B,C){=}(0,0,0){\to}0$, $(0,0,1){\to}1$, $(0,1,0){\to}1$, $(0,1,1){\to}0$, $(1,0,0){\to}0$, $(1,0,1){\to}1$, $(1,1,0){\to}1$, $(1,1,1){\to}0$. The four 1-cells give the canonical sum-of-products directly: $$F(A,B,C)=\boxed{\bar A\bar BC+\bar AB\bar C+A\bar BC+AB\bar C}=\Sigma m(1,2,5,6)$$
Part (b) — map $A,B,C$ onto the MUX select lines. An 8-to-1 MUX has three select lines $S_2,S_1,S_0$ and eight data inputs $D_0\ldots D_7$, with output $=D_i$ where $i=4S_2+2S_1+S_0$. Since $F$ has exactly three variables and the MUX has exactly $2^3=8$ data inputs, connect $S_2{=}A,\ S_1{=}B,\ S_0{=}C$ directly — every possible $(A,B,C)$ combination then selects exactly one data input, so no reduction/Shannon-expansion trick is required.
Tie each data input to its own truth-table value. Reading the minterm index $i=4A+2B+C$ against the table gives the constant to hard-wire onto each $D_i$ (0 to ground, 1 to $V_{CC}$): $$D_0=0,\ D_1=\boxed{1},\ D_2=\boxed{1},\ D_3=0,\ D_4=0,\ D_5=\boxed{1},\ D_6=\boxed{1},\ D_7=0$$ This is exactly the four minterms found in Step 1, confirming the two parts are consistent.

Fig. Q2-b — 8:1 MUX realizing $F(A,B,C)=\Sigma m(1,2,5,6)$: select lines $S_2S_1S_0=ABC$ chosen directly; data inputs tied to the truth-table value of their own minterm ($D_1{=}D_2{=}D_5{=}D_6{=}1$, the rest $0$).

Final Results — Question 2
Quantity	Value
$F(A,B,C)$ (canonical SOP)	$\bar A\bar BC+\bar AB\bar C+A\bar BC+AB\bar C=\Sigma m(1,2,5,6)$
MUX select mapping	$S_2{=}A,\ S_1{=}B,\ S_0{=}C$
Data inputs tied HIGH	$D_1,D_2,D_5,D_6$
Data inputs tied LOW	$D_0,D_3,D_4,D_7$

Topic: Combinational function realization with a multiplexer.

Key relations: $n$-variable function $\to$ $2^n{:}1$ MUX: connect all $n$ variables to the select lines, wire $D_i=F(\text{minterm }i)$; MUX output $=D_{4S_2+2S_1+S_0}$ (3-select case).

Why this works: A MUX is a physical realization of a truth table — each select combination is a unique address that steers exactly one data line to the output, so tying $D_i$ to the desired output value for minterm $i$ reproduces the table exactly, with no minimization needed.

Common pitfall: Using only $n{-}1$ variables on the select lines and the last variable (or its complement) on the data inputs — a valid, more economical technique for an $2^{n-1}{:}1$ MUX, but unnecessary and easy to mis-apply here since the MUX size already matches the variable count exactly.

Source: Mano & Ciletti, Digital Design, Multiplexers / combinational-circuit implementation with MSI.

Question 2 of 6: Function Realization with an 8-to-1 Multiplexer

Question 2: Function Realization with an 8-to-1 Multiplexer (20 marks)