Lecture 11: Digital Logic Design
Today’s Focus: Truth Table (Simplified) Boolean Expression
CS 30
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)
CS 30
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
CS 30
Examples
literal CS 30
Another Example
1
0 1 1 0 CS 30
1
0 0 0 1
Carry = A’B’+ AB Sum = A’B’ + A’B+ AB’
Carry = A’B+AB’ Sum = AB
One More
1
1
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
0 1 1 0 CS 30
S
0 0 0 1
C = A’B+AB’ S = AB
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
CS 30
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
CS 30
How is this related to Logic Design?
CS 30
Logic Design Process
Function definition
adder
CS 30
Boolean expression
Truth table
?
Logic block
Truth Table
CS 30
Boolean Expression
From Truth Table to Minimized Boolean Expression
OR
CS 30
K-Map: A systematic way to simplify Boolean expressions Directly from Truth Table
CS 30
Graphing Boolean Expressions
CS 30
Mapping Truth Tables to Tubes
Adjacent plane
A
on set off set
B CS 30
Mapping Truth Tables to Tubes
Adjacent plane
A
on set off set
B CS 30
3-variable example
CS 30
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
Graphing Boolean becomes much more complex as the input increases
CS 30
From Multi-dimention to Two-dimention
CS 30
K-MAP
CS 30
K-Map (general idea)
CS 30
Properties of K-Map
Any two adjacent (horizontal or vertical, but not diagonal) elements are distance 1 apart in the equivalent cube representation
CS 30
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.
CS 30
Groups must contain 1, 2, 4, 8, or in general 2^n cells.
CS 30
Each group should be as large as possible.
CS 30
Each cell containing a one must be in at least one group.
Groups may overlap.
CS 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.
CS 30
There should be as few groups as possible, as long as this does not contradict any of the previous rules.
CS 30
Mapping from truth table to k-map
CS 30
Boolean Minimization via K-MAP
AA CS 30
B
B’
Mapping from truth table to k-map
CS 30
Additional examples
F = A•C + B •C CS 30
CS 30
Examples
Avoid redundant coverage!
CS 30
Exercise
F =C + A•B•D+ B •D
CS 30
Using K-map to perform complement
complement
CS 30
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.
CS 30
Choosing don’t cares
CS 30
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….
CS 30
