Page 122 - Bebop to The Boolean Boogie An Unconventional Guide to Electronics Fundamentals, Components, and Processes

Karnaugh Maps 103

associated with the relevant input combinations as
question marks in the map (Figure 10-8). C K 00 01 11 10
The ? characters indicate don’t cure states, which
can be considered to represent either 0 or 1 values at 011 I I I I
the designer’s discretion. In the example shown in
Figure 10-8, we have no interest in the ? character at 11
a = 0, b = 0, = 1, d = 0 or the ? character at a = 0,
c
b = 1, c = 1, d = 1, because neither of these can be 10
used to form a larger group. However, if we decide
that the other three ? characters are going to repre-
sent 1 values, then they can be used to form larger
groups, which allows us to minimize the function to
a greater degree than would otherwise be possible.
It should be noted that many electronics refer- Figure 10-8. Karnaugh Map
ences use X characters to represent don’t care states. for an incompletely
specified function
Unfortunately, this may lead to confusion as design
tools such as logic simulators use X characters to
represent don’t know states. Unless otherwise indicated, this book will use ? and
X to represent don’t cure and don’t know states, respectively.

Populating Maps Using Os Versus 1s
When we were extracting Boolean equations from truth tables in the
previous chapter, we noted that in the case of a function whose output is logic I
fewer times than it is logic 0, is generally easier to extract a sum-of-products
it
equation. Similarly, if the output is logic 0 fewer times than it is logic I, it is
generally easier to extract a product-of-sums equation.
The same thing applies to a Karnaugh Map. If the output is logic 1 fewer
times than it is logic 0, then it’s probably going to be a lot easier to populate
the map using logic 1‘s. Alternatively, if the output is logic 0 fewer times than
it is logic I, then populating the map using logic Os may not be a bad idea.
When a Karnaugh Map is populated using the 1s assigned to the truth
table’s output, the resulting Boolean expression is extracted from the map in
sum-of-products form. By comparison, if the Karnaugh Map is populated using
the Os assigned to the truth table’s output, then the groupings of Os are used to
generate expressions in product-of-sums form (Figure 10-9).