Table of Contents
Chapter 3 Simplification of Switching Functions
Simplification Goals
Example 3.1
Minimization Methods
Minimum SOP and POS Representations
Karnaugh Maps
Figure 3.1 Venn diagram and equivalent K-map for two variables
Figure 3.2 Venn diagram and equivalent K-map for three variables
Figure 3.3 (a) -- (d) K-maps for four and five variables
Figure 3.3 (e) -- (f) K-maps for six variables
Plotting (Mapping) Functions in Canonical Form on a K-map
Figure 3.4 Plotting functions on K-maps
Figure 3.5 K-maps for f(a,b,Q,G) in Example 3.4 (a) Minterm form. (b) Maxterm form.
Figure 3.6 K-map of Figure 3.5(a) with variables reordered: f(Q,G,b,a).
Plotting Functions in Algebraic Form
Figure 3.7 -- Example 3.6. (a) Venn diagram form. (b) Sum of minterms. (c) Maxterms.
Figure 3.8 -- Example 3.7. (a) Maxterms, (b) Minterms, (c) Minterms of f ?.
Figure 3.9 -- Example 3.8. (a) K-map of f?, (b) K-map of f.
Simplification of Switching Functions Using K-maps
Figure 3.10 K-map for Example 3.9
Simplification Guidelines for K-maps
Prime Implicants and Covers
Figure 3.11 K-map illustrating implicants
Algorithm 3.1 -- Generating and Selecting Prime Implicants
Figure 3.12 -- Example 3.10 (Illustrating Algorithm 3.1)
Algorithm 3.2 -- Generating and Selecting Prime Implicants (Revisited)
Figure 3.13 -- Example 3.11 (Illustrates Algorithm 3.2)
Figure 3.14 -- Example 3.12 f(A,B,C,D) = ?m(0,5,7,8,10,12,14,15)
Figure 3.15 -- Example 3.13 f(A,B,C,D) = ?m(1,2,3,6) = A?C + BC?
Figure 3.16 -- Example 3.14 f(A,B,C,D) = B?D? + B?C? + BCD
Figure 3.17 -- Example 3.15 Function with no essential prime implicants.
Figure 3.18 -- Example 3.16 Minimizing a five-variable function. f(A,B,C,D,E) = ?m(0,2,4,7,10,12,13,18,23,26,28,29)
Prime Implicates and Covers
Algorithm 3.3 -- Generating and Selecting Prime Implicates
Algorithm 3.4 -- Generating and Selecting Prime Implicates (Revisited)
Example 3.17 -- Find the minimum POS form of the function f(A,B,C,D) = ?M(0,1,2,3,6,9,14)
Algorithm 3.5 -- Finding MPOS of f from f?
Example 3.18 -- Find the MPOS of the following function using Algorithm 3.5 f(A,B,C,D) = ?M(0,1,2,3,6,9,14)
Example 3.19 -- Minimum covers of f(A,B,C,D) = ? M (3,4,6,8,9,11,12,14) and its complement.
Figure 3.22 Finding a minimal POS expression for a 5-variable function.
Figure 3.23 Deriving POS and SOP forms of a function.
Example 3.22 -- Minimizing a Function with Don’t Cares. f(A,B,C,D) = ?m(1,3,4,7,11) + d(5,12,13,14,15) = ?M(0,2,6,8,9,10) ? D(5,12,13,14,15)
Example 3.23 -- Design a circuit to distinguish BCD digits ? 5 from those ? 5.
Example 3.23 (concluded)
Timing Hazards in Combinational Logic Circuits
Figure 3.27 (a)--(b) Illustration of a static hazard.
Figure 3.27 (c) Illustration of a static hazard (con’t)
Figure 3.27 (d) Illustration of a static hazard (con’t).
Figure 3.28 Identifying hazards on a K-map.
Figure 3.29 Hazard-free network.
Figure 3.30 (a)--(b) Example of a static-0 hazard.
Figure 3.30 (c)--(d) Example of a static-0 hazard (con’t).
Figure 3.31 Dynamic hazards.
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Author: Dr Bill Carroll
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