Question $3(30\%)$

Design a system that has as its inputs a number from $1$ to $10$ and provides as its outputs $($A$,$ B$,$

$C,$dots,$H$ the signals to drive the display described below.

The inputs are labeled $W,x,Y,$ and $Z$ and are normal binary. The input combinations $0000,1011,$

$1100,1101,1110,$ and $1111$ will never occur; they are to be treated as don't cares.

The display allows for the representation of Roman numerals $($except that IIX is used to represent

$8,$ whereas it is normally written as VIII$).$ There are a total of eight segments in the display, labeled

A through $H,$ as shown in figure above.

There are two ways to represent a numeral on this display $-$ left and right. For example, to represent

a $5$ on this display, for left representation, light segments A and $C($or $E$ and $G)$ and right

representation, light segments $B$ and $D($or $F$ and $H).$

The following illustration shows all digits as they should be coded for each of these, with a lit

segment represented by a bold line and an unlit segment represented by a dashed line.

Lelt

$6$

$7$

$8$ Version High: To light a segment, a $1$ is placed on the appropriate display input $(A,B,$dots,$H).$

Version Low: To light a segment, a $0$ is placed on the appropriate display input $(A,B,$dots,$H).$

Note that version low is just a complement to version high.

Answer the following questions:

$($a$)$ Establish a truth table of $8$ functions with four input variables for High$-$Left display.

$($b$)$ Minimize each function independently with KM$.$ Identify possible shared terms$*$ and fill in

the shared term table$**$ as follows. Realize the eight$-$output circuit with five $3-$input NAND

gates and a maximum of eleven $2-$input NAND gates. Show maps, equations, and circuit

diagrams.

A shared term is not a common term we learned in the lecture; it is just identical product terms

that exist in multiple independent functions from $KM.$ No loop splitting is required.

$**$ The shared term table below is useful to illustrate the shared terms

Example:

$x=ABC{C}^{\text{'}}+AC+BD$

$Y=B{C}^{\text{'}}+AC+AC$

$Z=AB+BD+B{C}^{\text{'}}$

Remark: $AC$ and $AC\text{'}$ use different gates. $($c$)$ Realize the circuit for the Low$-$Left display with six $3-$input NAND gates and a maximum of

twelve $2-$input NAND gates. Identify possible shared terms. Show KM maps, equations, and

the shared terms table in $($b$)$ and the circuit diagram.