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Combinational Logic 1. Today. Basics of digital logic Basic functions Boolean algebra Gates to implement Boolean functions Identities and Simplification. Digital circuits are a hardware that manipulate binary info. Each basic unit in a circuit is called: Gate.

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today
Today
  • Basics of digital logic
  • Basic functions
    • Boolean algebra
    • Gates to implement Boolean functions
  • Identities and Simplification

Digital Logic, Fall 2005Digital Logic

slide3
Digital circuits are a hardware that manipulate binary info.
  • Each basic unit in a circuit is called: Gate.
  • Boolean Algebra: a mathematical system that describe the binary logic system.

Digital Logic, Fall 2005Digital Logic

binary logic
Binary Logic
  • Deals with binary variables and mathematical logic.
  • Binary variables
    • Can be 0 or 1 (T or F, low or high)
    • Variables named with single letters in examples
    • Really use words when designing circuits

Basic Functions

    • AND
    • OR
    • NOT

Digital Logic, Fall 2005Digital Logic

slide5
AND
  • Symbol is dot
    • Z = A · B
  • Or no symbol
    • Z = AB
  • Truth table ->
  • Z is 1 only if
    • Both A and B are 1
  • Truth table: a table of combinations of the binary variables showing the relationship between the values of variables and the result.

Digital Logic, Fall 2005Digital Logic

slide6
OR
  • Symbol is +
    • Not addition
    • Z = A + B
  • Truth table ->
  • Z is 1 if either 1
    • Or both!

Digital Logic, Fall 2005Digital Logic

slide7
NOT
  • Unary
  • Symbol is bar
    • Z = Ā
  • Truth table ->
  • Inversion

Digital Logic, Fall 2005Digital Logic

gates
Gates
  • Electronic circuits that operate on one or more inputs to produce an output.
  • Remember that 0 and 1 are represented by voltages

Digital Logic, Fall 2005Digital Logic

and gate
AND Gate

Timing Diagrams

Digital Logic, Fall 2005Digital Logic

or gate
OR Gate

Digital Logic, Fall 2005Digital Logic

inverter
Inverter

Digital Logic, Fall 2005Digital Logic

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

Digital Logic, Fall 2005Digital Logic

representation schematic
Representation: Schematic

Digital Logic, Fall 2005Digital Logic

representation boolean algebra
Representation: Boolean Algebra
  • Deals with binary variables and logic operations.
  • For now equations with operators AND, OR, and NOT
  • Boolean function: described by Boolean equation.
  • Boolean equation: express logical relationship between binary variables

Term

Boolean function

Digital Logic, Fall 2005Digital Logic

representation truth table
Representation: Truth Table
  • 2n rows

where n # of

variables

Digital Logic, Fall 2005Digital Logic

functions
Functions
  • Can get same truth table with different functions
  • Usually want simplest
    • Fewest gates or using particular types of gates
    • More on this later

Digital Logic, Fall 2005Digital Logic

slide17
Combinational Logic Circuits: Circuit gates interconnected by wires that carry logic signals.

Digital Logic, Fall 2005Digital Logic

identities
Identities
  • Use identities to manipulate functions
  • I used distributive law

to transform from

to

Digital Logic, Fall 2005Digital Logic

table of identities
Table of Identities

Digital Logic, Fall 2005Digital Logic

duals
Duals
  • Left and right columns are duals
  • Replace AND with OR, 0s with 1s

Digital Logic, Fall 2005Digital Logic

single variable identities
Single Variable Identities

Digital Logic, Fall 2005Digital Logic

commutative
Commutative
  • Order independent

Digital Logic, Fall 2005Digital Logic

associative
Associative
  • Independent of order in which we group
  • So can also be written as

and

Digital Logic, Fall 2005Digital Logic

distributive
Distributive
  • Can substitute arbitrarily large algebraic expressions for the variables

Digital Logic, Fall 2005Digital Logic

demorgan s theorem
DeMorgan’s Theorem
  • Used a lot
  • NOR equals invert AND
  • NAND equals invert OR

Digital Logic, Fall 2005Digital Logic

truth tables for demorgan s
Truth Tables for DeMorgan’s

Digital Logic, Fall 2005Digital Logic

algebraic manipulation
Algebraic Manipulation
  • Consider function

Digital Logic, Fall 2005Digital Logic

simplify function
Simplify Function

Apply

Apply

Apply

Digital Logic, Fall 2005Digital Logic

fewer gates
Fewer Gates

Digital Logic, Fall 2005Digital Logic

consensus theorem
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!

Digital Logic, Fall 2005Digital Logic

complement of a function
Complement of a Function
  • Definition: 1s & 0s swapped in truth table
  • Mechanical way to derive algebraic form
    • Take the dual
      • Recall: Interchange AND & OR, and 1s & 0s
    • Complement each literal

Digital Logic, Fall 2005Digital Logic

complement of a function32
Complement of a Function
  • Definition: 1s & 0s swapped in truth table
  • Mechanical way to derive algebraic form for the complement of a function
    • Take the dual
      • Recall: Interchange AND & OR, and 1s & 0s
    • Complement each literal (a literal is a variable complemented or not; e.g. x , x’ , y, y’ each is a literal)

Digital Logic, Fall 2005Digital Logic

truth table of the complement of a function
Truth Table of the Complement of a Function

Digital Logic, Fall 2005Digital Logic

algebraic form for the complement of a function
Algebraic form for the Complement of a Function
  • F = X + Y’Z
  • Take dual of right hand side to get the complement F’
    • F’ = X’ . (Y + Z’)

Digital Logic, Fall 2005Digital Logic

from truth table to function
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

Digital Logic, Fall 2005Digital Logic

standard forms
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

Digital Logic, Fall 2005Digital Logic

definition minterm
Definition: Minterm
  • Product term in which all variables appear once (complemented or not)
  • Represents exactly one combination of the binary variables in a truth table. Its value is 1 only for that combination

Digital Logic, Fall 2005Digital Logic

number of minterms
Number of Minterms
  • For n variables, there will be 2n minterms
  • Minterms are labeled from minterm 0, m0 to to minterm 2n-1, m2n-1

Digital Logic, Fall 2005Digital Logic

definition maxterms
Definition: Maxterms
  • Sum term in which all variables appear once (complemented or not)

Digital Logic, Fall 2005Digital Logic

minterm related to maxterm
Minterm related to Maxterm
  • Minterm and maxterm with same subscripts are complements
  • Example

Digital Logic, Fall 2005Digital Logic

standard form of f sum of minterms
Standard Form of F:Sum of Minterms
  • OR all of the minterms of truth table for which the function

value is 1

  • F = m0 + m2 + m5 + m7
  • A function that includes all the

minterms is equal to logic 1

Ex: G(X,Y)=Σm(0,1,2,3)=1

Digital Logic, Fall 2005Digital Logic

complement of f
Complement of F
  • Not surprisingly, just sum of the other minterms
  • In this case

F’ = m1 + m3 + m4 + m6

Digital Logic, Fall 2005Digital Logic

product of maxterms
Product of Maxterms
  • Recall that maxterm is true except for its own row
  • So M1 is only false for 001

Digital Logic, Fall 2005Digital Logic

product of maxterms45
Product of Maxterms
  • Can express F as AND of all Maxterms of rows that should evaluate to 0

or

Digital Logic, Fall 2005Digital Logic

recap
Recap
  • Working (so far) with AND, OR, and NOT
  • Algebraic identities
  • Algebraic simplification
  • Minterms and maxterms
  • Can now synthesize function (and gates) from truth table

Digital Logic, Fall 2005Digital Logic