combinational logic 1

CombinationalLogic 1

2

Topics• Basics of digital logic • Basic functions♦ Boolean algebra♦ Gates to implement Boolean functions

• Identities and Simplification

3

Binary Logic• Binary variables♦ Can be 0 or 1 (T or F, low or high)♦ Variables named with single letters in

examples♦ Use words when designing circuits

• Basic Functions♦ AND♦ OR♦ NOT

4

AND Operator• Symbol is dot♦ Z = X · Y

• Or no symbol♦ Z = XY

• Truth table ->• Z is 1 only if♦ Both X and Y are 1

5

OR Operator• Symbol is +♦ Not addition♦ Z = X + Y

• Truth table ->• Z is 1 if either 1♦ Or both!

6

NOT Operator• Unary• Symbol is bar (or ’)♦ Z = X’

• Truth table ->• Inversion

7

Gates• Circuit diagrams are

traditionally used to document circuits

• Remember that 0 and 1 are represented by voltages

8

AND Gate

Timing Diagrams

9

OR Gate

10

Inverter

11

More Inputs• Work same way• What’s output?

12

Digital Circuit Representation: Schematic

13

Digital Circuit Representation: Boolean Algebra

• For now equations with operators AND, OR, and NOT

• Can evaluate terms, then final OR

• Alternate representations next

ZY X F

14

Digital Circuit Representation: Truth Table

• 2n rowswhere n # ofvariables

15

Functions• Can get same truth table with

different functions

• Usually want simplest function♦ Fewest gates or using particular types

of gates♦ More on this later

ZY X F ))(( ZXY X F

16

Identities• Use identities to manipulate

functions• On previous slide, I used

distributive law

to transform fromZY X F ))(( ZXY X F

))(( ZX YX YZ X

to

17

Table of Identities

18

Duals• Left and right columns are

duals• Replace AND with OR, 0s with

1s

19

Single Variable Identities

20

Commutative• Order independent

21

Associative• Independent of order in which

we group

• So can also be written asand

ZYX XYZ

22

Distributive

• Can substitute arbitrarily large algebraic expressions for the variables

23

DeMorgan’s Theorem• Used a lot• NOR equals invert AND

• NAND equals invert OR

24

Truth Tables for DeMorgan’s

25

Algebraic Manipulation• Consider function

XZZYX YZ X F

26

Simplify Function

XZZ Z YX F )(

XZYX F 1

XZYX F

Apply

Apply

Apply

XZZYX YZ X F

27

Fewer GatesXZYX F

28

Consensus Theorem

• The third term is redundant♦ Can just drop

• Proof in book, but in summary♦ For third term to be true, Y & Z both 1♦ Then one of the first two terms must be

1!

ZXXYYZZXXY

29

Complement of a Function

• Definition:

1s & 0s swapped in truth table

30

Truth Table of the Complement of a Function

X Y Z F = X + Y’Z F’

0 0 0 0 10 0 1 1 00 1 0 0 10 1 1 0 11 0 0 1 01 0 1 1 01 1 0 1 01 1 1 1 0

31

Algebraic Form for Complement• Mechanical way to derive

algebraic form for the complement of a function

1. Take the dual• Recall: Interchange AND & OR, and 1s & 0s

2. Complement each literal (a literal is a variable complemented or not; e.g. x , x’ , y, y’ each is a literal)

32

Example: Algebraic form for the complement of a function

F = X + Y’Z• To get the complement F’

1. Take dual of right hand side X . (Y’ + Z)2. Complement each literal: X’ . (Y

+ Z’) F’ = X’ . (Y + Z’)

Mechanically Go From Truth Table to Function

34

From Truth Table to Function

• Consider a truth table• Can implement F

by taking OR of all terms that correspond to rows for which F is 1 “Standard Form” of

the function

XYZZXYZYXZY XZ YX F

35

Standard Forms• Not necessarily simplest F• But it’s mechanical way to go

from truth table to function

• Definitions:♦ Product terms – AND ĀBZ♦ Sum terms – OR X + Ā♦ This is logical product and sum, not

arithmetic

36

Definition: Minterm

• Product term in which all variables appear once (complemented or not)

• For the variables X, Y and Z example minterms : X’Y’Z’, X’Y’Z, X’YZ’, …., XYZ

37

Definition: Minterm (continued)

MinTerm

Each minterm represents exactly one combination of the binary variables in a truth table.

38

Truth Tables of Minterms

39

Number of Minterms

• For n variables, there will be 2n minterms

• Minterms are labeled from minterm 0, to minterm 2n-1♦m0 , m1 , m2 , … , m2n-2 , m2n-1

• For n = 3, we have♦m0 , m1 , m2 , m3 , m4 , m5 , m6 ,

m7

40

Definition: Maxterm

• Sum term in which all variables appear once (complemented or not)

• For the variables X, Y and Z the maxterms are:

X+Y+Z , X+Y+Z’ …. , X’+Y’+Z’

41

Definition: Maxterms (continued)

mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm,mxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx,mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm

Maxterm

42

Truth Tables of Maxterms

43

Minterm related to Maxterm

• Minterms and maxterms with same subscripts are complements

• Example33 MZYXYZXm

Mjm j

44

Standard Form of F:Sum of Minterms

• OR all of the minterms of truth table for which the functionvalue is 1

• F = m0 + m2 + m5 + m7

45

Complement of F• Not surprisingly, just sum of

the other minterms• In this caseF’ = m1 + m3 + m4 + m6

46

Product of Maxterms• Recall that maxterm is true

except for its own row• So M1 is only false for 001

47

Product of Maxterms• F = m0 + m2 + m5 + m7

• Remember:♦ M1 is only false for 001♦ M3 is only false for 011♦ M4 is only false for 100♦ M6 is only false for 110

• Can express F as AND of M1, M3, M4, M66431 MMMMF

))(( ZYXZYXF ))(( ZYXZYX

or

48

Recap• Working (so far) with AND, OR,

and NOT• Algebraic identities• Algebraic simplification• Minterms and maxterms• Can now synthesize function

from truth table