Page 249 - Engineering Digital Design

220 CHAPTER 5 / FUNCTION MINIMIZATION

which is a three-level minimum with a gate/input tally of 5/ 1 1 excluding possible inverters.
No attempt is made to examine bond sets other than [a, b}. Consequently, Eq. (5.48) cannot
necessarily be regarded as an exact minimum.
The EQPOS CRMT form for bond set {a, b] is obtained from Eqs. (5.20) and (5.21) and is

fab = So O (£ + £i) O (a + £2) O (« + & + g 3), (5.49)

for which the g coefficients are

g Q = c®e = cQe gi = c © e Q c ® e = 0

After introducing these coefficients into Eq. (5.49) there results the mixed polarity CRMT
result

fab = c O e O (b + c O <?) O a O (a + b + e)
= dOeOcO(b+c)O(b+e)Q(a+b+e)
= dQeO(b + c)O(a + b + e). (5.50)

This is a three-level EQPOS minimum with a gate/input tally of 5/1 1.
Now it is desirable to compare the CRMT minimum forms of Eqs. (5.48) and (5.50)
with the EV K-map and two-level results. Reading the loops of Fig. 5.6b in maxterm code
(or the submaps in Fig. 5.6a) gives

f K-map = (b + a®c® e)(a + b + e), (5.51)

which is a four-level function having a gate/input tally of 5/ 1 1 excluding possible inverters.
By comparison, the computer-minimized two-level POS minimum result is

f pos = ( a+b + c + e)(a +b + c + e)(a + c + e}(a + b + c + e) (5.52)

and has a gate/input tally of 5/19. The SOP minimum result (not shown) has a gate/input
tally of 7/22. No attempt is made to minimize function / by the EXSOP minimization
approach, which is best accomplished by computer algorithmic means.
Figure 5.6 illustrates how the CRMT method can be applied to a five- variable function
having a two- variable bond set, {a, b} in this case. Shown in Fig. 5.7a is the conventional
K-map array suitable for an eight variable function F having the bond set {w, x, y, z}, and in
Fig. 5.7b its fourth-order compression, also for bond set {w, x, y, z}. These K-map formats
also suggest a means by which functions with more than eight variables can be minimized
by the CRMT method, providing one can deal with EV K-maps within EV K-maps. Large
numbers of don't cares would greatly reduce the complexity of the K-map cell entries (and
hence / coefficients) in Fig. 5.7b.