Lecture 11: Digital Logic Design
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1 Lecture 11: Digital Logic Design
2 Today s Focus: Truth Table (Simplified) Boolean Expression
3 From Truth Table Boolean Expression Sum of the Product F= ABC + BCD + DEF Product of the Sum F= (A+B+C) (B+C+D) (D+E+F)
4 Sum of the Product Each row of the truth table represents a product term Product term -- each row in which the output column is a 1 contributes a single ANDed term of input variables to the Boolean expressions If the column associated with variable X has a 0 in it, the expression X is part of the ANDed term., otherwise, X is part of the ANDed term Sum of the product Product terms are ORed together
5 Examples literal
6 Another Example 1 1 Carry = A B + AB Sum = A B + A B+ AB Carry = A B+AB Sum = AB
7 One More 1 1
8 What About Product of the Sum 1 1 Carry = A B + AB Sum = A B + A B+ AB Carry = (A+B)(A +B ) = C Sum = (A+B)(A+B )(A +B) =S C S C = A B+AB S = AB
9 Product of the Sum Each row of the truth table represents a sum term Sum term -- each row in which the output column is a 0 contributes a single ORed term of input variables to the Boolean expressions If the column associated with variable X has a 0 in it, the expression X is part of the ORed term.; If X has a 1 in it, then X is part of the ORed term Product of the Sum Sum terms are ANDed together
10 From Truth Table Boolean Expression Sum of the Product Find rows with output of 1 each product term, input X, x=0 use X, x=1, use X OR all the product terms together Product of the Sum Find rows with output of 0 each sum term, input X, x=0 use X, x=1, use X AND all the sum terms together
11 How is this related to Logic Design?
12 Logic Design Process Function definition Truth table Boolean expression Logic block adder?
13 Truth Table Boolean Expression
14 From Truth Table to Minimized Boolean Expression OR
15 K-Map: A systematic way to simplify Boolean expressions Directly from Truth Table
16 CS 30 Graphing Boolean Expressions
17 Mapping Truth Tables to Tubes Adjacent plane A off set on set B
18 Mapping Truth Tables to Tubes Adjacent plane A off set on set B
19 CS 30 3-variable example
20 Adjacencies of higher dimensions Eliminate 2 variables, reduce the expression into a single variable What about higher dimensions. The problem for humans is the difficulty of visualizing adjacencies in more than three dimensions. CS 30
21 CS 30 Graphing Boolean becomes much more complex as the input increases
22 From Multi-dimention to Two-dimention
23 K-MAP
24 K-Map (general idea)
25 Properties of K-Map Any two adjacent (horizontal or vertical, but not diagonal) elements are distance 1 apart in the equivalent cube representation
26 Rules of Simplification Grouping together adjacent cells containing 1 Groups may not include any cell containing 0. Groups may be horizontal or vertical, but not diagonal.
27 Groups must contain 1, 2, 4, 8, or in general 2^n cells.
28 Each group should be as large as possible.
29 Each cell containing a one must be in at least one group. Groups may overlap.
30 Groups may wrap around the table. The leftmost cell in a row may be grouped with the rightmost cell and the top cell in a column may be grouped with the bottom cell.
31 There should be as few groups as possible, as long as this does not contradict any of the previous rules.
32 CS 30 Mapping from truth table to k-map
33 Boolean Minimization via K-MAP A B B A
34 CS 30 Mapping from truth table to k-map
35 Additional examples F = A C + B C
36
37 Examples Avoid redundant coverage!
38 Exercise F = C + A B D + B D
39 Using K-map to perform complement complement
40 Don t cares X: don t care. Do not confuse this with an undefined value or a don t know. Any actual implementation of the circuit will generate some output for the don t-care cases. In a truth table, an X simply means that we have a choice of assigning a 0 or 1 to the truth table entry. We should choose the value that will lead to the simplest implementation.
41 CS 30 Choosing don t cares
42 Summary Review: transistors and gates Combinational logic Vs. sequential logic Boolean Algebra Laws of boolean algebra Realizing Boolean Expressions using Gates NAND, NOR, AND, OR, NOT K-map 1,2,3,4 variables.
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