Page 262 - Engineering Digital Design

PROBLEMS 233

(a) W(A, £, C, D) = £m(3, 6, 9, 12)
M 2
(b) X(A, B, C, D) = H ( > 3, 4, 5, 7, 8, 9, 11, 14, 15)
(c) F(A, 5,C, D) = £>(1,2,4,7, 11, 13, 14)
(d) Z(A, 5, C, D) = n Af(0, 3, 4, 6, 9, 10, 13)
5.7 The following incompletely specified function contains XOR-type patterns:

G(A, B, C, D) = ]~| M(0, 1, 2, 3, 8, 11, 12, 13) • 0(4, 5, 6,7).

(a) Compress the following function into a second-order K-map of axes A, B and loop
out a gate-minimum expression by using XOR-type patterns where appropriate.
(Hint: Consider both minterm and maxterm codes and the best use of the don't
cares when looping out XOR-type patterns for a gate-minimum result.)
(b) Use the same K-map to extract minimum SOP and POS expressions for this
function. Compare the gate/input tallies (exclusive of possible inverters) for the
XOR result with those for the SOP and POS results. What do you conclude from
these comparisons?
(c) Construct the logic circuit for both the XOR result and the SOP result, assuming
that the inputs and output are all active high.
5.8 Use XOR-type patterns to extract a gate-minimum expression for each of the three
functions represented in Fig. P5.3. Use the gate/input tally (exclusive of inverters)
to compare the multilevel result with that for the two-level SOP and POS minimum
result. Note that compound XOR-type patterns may exist. [Hint: For / 2 and /?, it will
be necessary to make use of Eqs. (3.27).]

5.9 A computer program has been written that will yield a minimum solution to a combi-
national logic function, but only in SOP form. It accepts the data in either conventional
(1's and O's) form or in two-level EV SOP form—it does not recognize the XOR or
EQV operators.
(1) Given the functions FI and F? represented by the EV K-maps in Fig. P5.4, extract
a gate-minimum expression from each in maxterm code by using the pencil-and-
paper method and XOR-type patterns.
(2) By following Example 2 in Section 4.8, outline the procedure required to "trick"
the computer program into yielding a two-level minimum expression from the
K-maps in Fig. P5.4, that can be easily converted to minimum POS form. (Hint:
It will be necessary to complement the subfunction in each cell of the K-map and
represent it in SOP form.)

\BC \ BC \ BC
AX °° 01 11 10 A\ 00 01 11 10 A\ °° 01 11 10
0 X X®Y 0 0 0 X 0 X®Y 0 z®x 0 X 0
*
1 X x+y Y tf 1 Y X 0 X 1 X 0 X®Y Y©Z
/ f
/ ti /
(a) ' (b) - (c)
FIGURE P5.3