Combinational Logic 1

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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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CombinationalLogic 1

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

Digital Logic, Fall 2005Digital Logic

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
• 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

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.

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OR
• Symbol is +
• Z = A + B
• Truth table ->
• Z is 1 if either 1
• Or both!

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

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

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AND Gate

Timing Diagrams

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OR Gate

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Inverter

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

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Representation: Schematic

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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

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Representation: Truth Table
• 2n rows

where n # of

variables

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

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

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

to transform from

to

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Table of Identities

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

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Single Variable Identities

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Commutative
• Order independent

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

and

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Distributive
• Can substitute arbitrarily large algebraic expressions for the variables

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

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Truth Tables for DeMorgan’s

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Algebraic Manipulation
• Consider function

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Simplify Function

Apply

Apply

Apply

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Fewer Gates

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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!

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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 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)

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Truth Table of the Complement of a Function

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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’)

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Mechanically Go 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

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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

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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

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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

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

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

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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

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

F’ = m1 + m3 + m4 + m6

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

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

or

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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

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