Page 447 - Engineering Digital Design

PROBLEMS 417

9.7 Find the minimum POS expression for the function in Fig. P9.2. How many static
0-hazards does this function have? What do you conclude as to the relative complexity
of the SOP and POS expressions for this function when account is taken of hazard
cover?

9.8 The following six- variable function has several static 1 -hazards. Construct a table
listing the coupled variable, coupled terms, initial and final states, and the hazard
cover for each of the hazards.

F = ABCDF + ABCE + ABCF + ABCE + DEF + CDE

9.9 The following multilevel functions have one or more static hazards. For each expres-
sion (without using a K-map), determine the coupled variable, coupled terms, the
initial and final states of the hazardous transition (read in alphabetical order), and
the hazard cover to be added to the expression. To do this, follow the examples in
Eq. (9.12), (9.13), and (9.14), whichever is relevant to the particular function.
(a) G = WXY eXTZe WYZ 0 WY
(b) T = (A + B + C) 0 (A + B + C) O (A + D} Q (B + D)

9.10 The following three-level XOR-type function has two static 1 -hazards:

/ = (Y e W) © (XZ) + WY.

(a) Construct the lumped path delay diagram (LPDD) for this function. From the
LPDD determine the hazard cover and initial and final states for each of the static
hazards. Follow the example in Fig. 9. 10. (Hint: Keep in mind the bidirectionality
of the static hazards in XOR-type functions and read the states for function / in
the order of WXYZ).
(b) By using two binary decision diagrams (BDDs), show the binary decisions re-
quired for each static 1 -hazard formation.
9.11 The following three-level function has both a static 1 -hazard and a static 0-hazard:

F = [x e WY] e

(a) Construct the LPDD for this function (exactly as written). Then, determine the
hazard cover and initial and final states for each of the static hazards. Read the
states in alphabetical order. Follow the example in Fig. 9.13 and Eq. (9.19).
(b) Use a timing diagram to show the development of the two hazards, similar to the
example in Fig. 9. 14. Then by adding the hazard cover, show that the hazards are
eliminated. Assume that all inputs and the output are active high.
(c) By using a binary decision diagram (BDD), show the binary decisions required
for the static 1 -hazard formation.
9.12 The following four-level function has three static 1 -hazards, one static 0-hazard, and
one dynamic hazard:

Y = [B 0 (AD)] © \AB 0 ACD 0 BCD]