Page 71 - DCAP108_DIGITAL_CIRCUITS_AND_LOGIC_DESIGNS

Digital Circuits and Logic Design

Notes 4.3.2 Looping
The expression for output X can be simplified by properly combining those squares in the K-map
which contain 1s. The process for combining these 1s is called looping.
Looping Groups of Two (Pairs)
Figure 4.18(a) is the K map for a particular three-variable truth table. This map contains a pair of
1s that are vertically adjacent to each other; the first represents ABC and the second represents
ABC. Note that in these two terms only the A variable appears in both normal and complemented
form (B and C remain unchanged). These two terms can be looped (combined) to give a resultant
that eliminates the A variable since it appears in both uncomplemented and complemented forms.
This is easily proved as follows:
X = ABC + ABC
= BC A( + A)
= BC ()1 = BC
This same principle holds true for any pair of vertically or horizontally adjacent 1s.
Figure 4.18 (b) shows an example of two horizontally adjacent 1s. These two can be looped and
the C variable eliminated since it appears in both its uncomplemented and complemented forms
to give a resultant of X = AB .
Another example is shown in Figure 4.18 (c). In a K-map the top row and bottom row of squares
are considered to be adjacent. Thus, the two 1s in this map can be looped to provide a resultant
of ABC + ABC = BC.
Figure 4.17: Examples of Looping Pairs of Adjacent 1s

Figure 4.18 (d) shows a K-map that has two pairs of 1s which can be looped. The two 1s in the top
row are horizontally adjacent. The two 1s in the bottom row are also adjacent, since in a K-map
the leftmost column and rightmost column of sequence are considered to be adjacent. When the
top pair of ls is looped, the D able is eliminated (since it appears as both D and D ) to give the
term ABC. ping the bottom pair eliminates the C variable to give the term ABD. These terms
are ORed to give the final result for X.

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