lecture 11: digital logic designhtzheng/teach/cs64s11/pdf/... · 2011-05-05 · sum of the...
TRANSCRIPT
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Lecture 11: ���Digital Logic Design
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CS 30
���Today’s Focus:���
���Truth Table ���
(Simplified) Boolean Expression
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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
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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
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Examples
literal
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Another Example
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1 1 Carry = A’B’+ AB Sum = A’B’ + A’B+ AB’
0 0 1 0 1 0 0 1
Carry = A’B+AB’ Sum = AB
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One More
1 1
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What About Product of the Sum
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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
0 0 1 0 1 0 0 1
C = A’B+AB’ S = AB
C S
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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
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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
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How is this related to ���
Logic Design?
CS 30
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Logic Design Process
Function definition
Truth table
Boolean expression
Logic block
CS 30
adder ?
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Truth Table
CS 30
Boolean Expression
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From Truth Table to Minimized Boolean Expression
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OR
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K-Map: ���A systematic way to simplify ���
Boolean expressions ���Directly from Truth Table
CS 30
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Graphing Boolean Expressions
CS 30
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B
Mapping Truth Tables to Tubes
A
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Adjacent plane
on set
off set
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Mapping Truth Tables to Tubes
A
B
CS 30
Adjacent plane
on set
off set
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3-variable example
CS 30
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Adjacencies of higher dimensions
CS 30
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.
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Graphing Boolean becomes much more complex as the input increases
CS 30
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From Multi-dimention to Two-dimention
CS 30
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K-MAP
CS 30
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K-Map (general idea)
CS 30
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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
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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.
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Groups must contain 1, 2, 4, 8, or in general 2^n cells.
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Each group should be as large as possible.
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Each cell containing a one must be in at least one group.
Groups may overlap.
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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.
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There should be as few groups as possible, as long as this does not contradict any of the previous rules.
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Mapping from truth table to k-map
CS 30
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Boolean Minimization via K-MAP
CS 30
A BA B’
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Mapping from truth table to k-map
CS 30
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Additional examples
CBCAF •+•=
CS 30
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CS 30
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Examples
Avoid redundant coverage!
CS 30
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Exercise
DBDBACF •+••+=CS 30
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Using K-map to perform complement
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complement
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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
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Choosing don’t cares
CS 30
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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