电子科技大学 姜书艳 数字逻辑4-3

Chapter 4 Combinational logic design principles
Karnaugh map and minimal logic design
How to make a minimal circuit ?
F XYZ 1,5,7 x' y' z xy' z xyz
figure 4-11 12 13
How to make a minimal one ?
Karnaugh map
2D Figure for truth table
Properties of Karnaugh map
The coordinates are ordered in Gray codes;
Minimal product
F ' A'B B D' A'C 'D
A B' B' D A C D'
F A'B 'B D'' A'C 'D '
Prime implicant: a maxim imply; Complete sum: sum of all the prime implicant; Distinguished 1-cell: only in one prime implicant; Essential prime implicant :with distinguished 1-cell in it.
Each cell differs from its neighbors in only one variable!
Standard logic equation and their Karnaugh map
Minterm list Maxterm list
F A, B,C 1,2,3,6
F x' y' z xy' z xyz y' z xyz F y' xyz y' x z
How to make a minimal circuit ?
One logic function may have many equivalent equations and circuit
Each set must be maximal !
(Any 1s can be reused in the process !)
Minimal sum
Step ቤተ መጻሕፍቲ ባይዱ :Make rectangular sets of 1s
Minimal sum
Step 2 :write minimal sum from sets of 1s Write each set as a product; for 2i-set,the product have (ni )variables! Sum all these products.
F A, B,C 0,4,5,6
Logic equations and their Karnaugh map
Sum of product Product of sum
F A, B, C A'C' B C
F A, B, C A C B C
Minimal sum
Get a minimal product-sum equation
Step 1 :Make rectangular sets of 1s Cover all the 1s in the Kanaugh maps; The cell-numbers in any sets must be 2i ; The number of sets must be minimal !
Same logic with different forms
Properties of neighbor cells
F m 2 m6 A'B C ' A B C B C'
F m2 m6 m3 m7 B C ' B C B
Can be minimized by karnaugh map
Minimal product Get a minimal sum-product equation
Step 1
get minimal sum for inverse function
Step 2
use DeMorgen’s theorem to get minimal
product of the function
Minimal sum
F B C ' A'C
F A C ' B'C
Minimal sum
F B 'D ' A D B 'D
Minimal sum
Minimal sum
Some concept
Imply :a rectangular set of 1s;