Combinational Logic 1

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

Binary Logic Binary variables Basic Functions 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

AND Operator Symbol is dot Or no symbol Truth table -> Z = X · Y Or no symbol Z = XY Truth table -> Z is 1 only if Both X and Y are 1

OR Operator Symbol is + Truth table -> Z is 1 if either 1 Not addition Z = X + Y Truth table -> Z is 1 if either 1 Or both!

NOT Operator Unary Symbol is bar (or ’) Truth table -> Inversion Z = X’ Truth table -> Inversion

Gates Circuit diagrams are traditionally used to document circuits Remember that 0 and 1 are represented by voltages

AND Gate Timing Diagrams

OR Gate

Inverter

More Inputs Work same way What’s output?

Digital Circuit Representation: Schematic

Digital Circuit Representation: Boolean Algebra For now equations with operators AND, OR, and NOT Can evaluate terms, then final OR Alternate representations next

Digital Circuit Representation: Truth Table 2n rows where n # of variables

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

Identities Use identities to manipulate functions On previous slide, I used distributive law to transform from to

Table of Identities

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

Single Variable Identities

Commutative Order independent

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

Distributive Can substitute arbitrarily large algebraic expressions for the variables

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

Truth Tables for DeMorgan’s

Algebraic Manipulation Consider function

Simplify Function Apply Apply Apply

Fall 2005 Fewer Gates

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!

Complement of a Function Definition: 1s & 0s swapped in truth table

Truth Table of the Complement of a Function X Y Z F = X + Y’Z F’ 1

Algebraic Form for Complement 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)

Example: Algebraic form for the complement of a function F = X + Y’Z To get the complement F’ Take dual of right hand side X . (Y’ + Z) Complement each literal: X’ . (Y + Z’) F’ = X’ . (Y + Z’)

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

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

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

Definition: Minterm (continued) Each minterm represents exactly one combination of the binary variables in a truth table.

Truth Tables of Minterms

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

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’

Definition: Maxterms (continued) mmmmmmmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmm,m xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx ,mmmmmmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmmmmmm Maxterm

Truth Tables of Maxterms

Minterm related to Maxterm Minterms and maxterms with same subscripts are complements Example

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

Complement of F Not surprisingly, just sum of the other minterms In this case F’ = m1 + m3 + m4 + m6

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

Product of Maxterms or 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, M6 or

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