Page 72 - DCAP108_DIGITAL_CIRCUITS_AND_LOGIC_DESIGNS

Unit 4: Minimization of Boolean Algebra

To summarize: Notes
Looping a pair of adjacent 1s in a K-map eliminates the variable that appears in complemented
and uncomplemented form.
Looping Groups of Four (Quads)
A K-map may contain a group four 1s that are adjacent to each other. This group is called a quad.
Figure 4.19 shows several examples of quads. In part (a) the four 1s are vertically adjacent and in
part (b) they are horizontally adjacent. The K-map in Figure 4.19(c) contains four 1s in a square,
and they are considered adjacent to each other. The four 1s in Figure 4.19(d) are also adjacent, as
are those in Figure 4.19 (e) because, as pointed out earlier, the top and bottom rows are considered
to be adjacent to each other as are the leftmost and rightmost columns. Pair two horizontally or
vertically adjacent 1s on a Karnaugh map.

Resistive divider must be constructed in such a way that a 1 in the 21 bit
position will cause a change of +7 *1/7=+ 2 V in the analog output voltage.

When a quad is looped, the resultant term will contain only the variables–do not change form for
all the squares in the quad. For example, in Figure 4.19 ( the four squares which contain a 1 are
ABCABC, ABC, and ABC. Examination of these terms reveals that only the variable C remains
,
unchanged (both A and B) appear in complemented and uncomplemented form).
Figure 4.18: Examples of Looping Groups of Four 1s (Quads)

C C CD CD CD CD
(a) (b)
0 1
AB AB 0 0 0 0
AB 0 1 AB 0 0 0 0
0 1
AB AB 1 1 1 1
0 1
AB AB 0 0 0 0
X=C X= AB

(c) CD CD CD CD (d) CD CD CD CD

AB 0 0 0 0 AB 0 0 0 0

AB 0 1 1 0 AB 0 0 0 0
AB 0 1 1 0 AB 1 0 0 1

AB 0 0 0 0 AB 1 0 0 1

X= AD X= AD
CD CD CD CD
(e)
AB 1 0 0 1

AB 0 0 0 0
AB 0 0 0 0

AB 1 0 0 1

X=BD

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