Lecture 11: Digital Logic Design

Lecture 11: Digital Logic Design  Today’s Focus:  Truth Table   (Simplified) Boolean Expression CS 30 From Truth Table  Boolean Ex...
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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)

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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

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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

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How is this related to  Logic Design?

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Logic Design Process

Function definition

adder

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Boolean expression

Truth table

?

Logic block

Truth Table

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Boolean Expression

From Truth Table to Minimized Boolean Expression

OR

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K-Map:  A systematic way to simplify  Boolean expressions  Directly from Truth Table

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Graphing Boolean Expressions

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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

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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

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From Multi-dimention to Two-dimention

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K-MAP

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K-Map (general idea)

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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

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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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 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

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Boolean Minimization via K-MAP

AA CS 30

B

B’

Mapping from truth table to k-map

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Additional examples

F = A•C + B •C CS 30

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Examples

 Avoid redundant coverage!

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Exercise

F =C + A•B•D+ B •D

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Using K-map to perform complement

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.

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Choosing don’t cares

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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….

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