Page 245 - Engineering Digital Design

216 CHAPTER 5 / FUNCTION MINIMIZATION

From the conventional K-map in Fig. 5.5a and counting O's within a given domain, the g
coefficients are found to be go = g2 = g4 = gv = gio = 0 with all the rest being logic 1.
Introducing these values in the R-M expansion gives the minimum result

G 4 = 0 O Y O X Q (W + Z) O (W + Y)
= 0 O X O (W + Z) O (W + Y)
= XQ(W + Z)Q(W + Y), (5.40)

as before in Eq. (5.39).
The result previously obtained for G YZ can also be obtained by using the CRMT approach
in a somewhat different way. The plan is to obtain the result for the SOP CRMT expansion
("for the O's") and then complement that result to produce the EQPOS CRMT expansion
form. Complementing each of the four EV cell entries in Fig. 5.5b gives

Gy Z(EPOS) = (YZ)X + (YZ)(W © X) + (YZ)(W 0 X} + (YZ)X
= go © Zgi © Yg 2 © YZg 3, (5.41)

with g values

go = /o = X g 2 = 0/(2, 0)=W®X®X = W
) = w®x@x = w #3 = e/(3 -O) = X®W®X®W = Q,

where use is made of g\ = ©/(I, 0) = W in the last term for g 3. Introducing these values
into Eq. (5.41) gives

G YZ(EPOS) = X 0 ZW® YW,

resulting in the EQPOS expression

= XQ(W + Z)Q(W + Y), (5.42)

where an odd number of complementations (operators and operands) have been performed
to complement the function. Notice that the / coefficients are also the complements of
those required for the EQPOS expansion, as they must be, since the cells of the EV K-map
in Fig. 5.5b were complemented.
It is interesting to compare the results just obtained for G with those read from the EV
K-maps in Figs. 5.5b and c, and with two-level POS minimization. Following the procedure
given by [3, 4], the results for GWX and Gyz are read directly in maxterm code from the
K-maps (see K-map loopings) as

G K. map wx = [W + (X® Y)][(W + (X © Z)] (5.43)