COMP541 Combinational Logic - II

1 / 34

# COMP541 Combinational Logic - II - PowerPoint PPT Presentation

COMP541 Combinational Logic - II. Montek Singh Jan 19, 2010. Today. Basics of Boolean Algebra (review) Identities and Simplification Basics of Logic Implementation Minterms and maxterms Going from truth table to logic implementation. Identities. Use identities to manipulate functions

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'COMP541 Combinational Logic - II' - melaney

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### COMP541Combinational Logic - II

Montek Singh

Jan 19, 2010

Today
• Basics of Boolean Algebra (review)
• Identities and Simplification
• Basics of Logic Implementation
• Minterms and maxterms
• Going from truth table to logic implementation
Identities
• Use identities to manipulate functions
• You can use distributive law …

… to transform from

to

Duals
• Left and right columns are duals
• Replace AND and OR, 0s and 1s
Commutativity
• Operation is independent of order of variables
Associativity
• Independent of order in which we group
• So can also be written as

and

Distributivity
• Can substitute arbitrarily large algebraic expressions for the variables
• Distribute an operation over the entire expression
DeMorgan’s Theorem
• Used a lot
• NOR  invert, then AND
• NAND  invert, then OR
Algebraic Manipulation
• Consider function
Simplify Function

Apply

Apply

Apply

Consensus Theorem
• The third term is redundant
• Can just drop
• Proof summary:
• For third term to be true, Y & Z both must be 1
• Then one of the first two terms is already 1!
Complement of a Function
• Definition: 1s & 0s swapped in truth table
• Mechanical way to derive algebraic form
• Take the dual
• Recall: Interchange AND and OR, and 1s & 0s
• Complement each literal

### Mechanically Go From Truth Table to Function

From Truth Table to Func
• Consider a truth table
• Can implement F by taking OR of all terms that are 1
Standard Forms
• Not necessarily simplest F
• But it’s a 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)
Number of Minterms
• For n variables, there will be 2n minterms
• Like binary numbers from 0 to 2n-1
• Often numbered same way (with decimal conversion)
Maxterms
• Sum term in which all variables appear once (complemented or not)
Minterm related to Maxterm
• Minterm and maxterm with same subscripts are complements
• Example
Sum of Minterms
• Like Slide 18
• OR all of the minterms of truth table row with a 1
• “ON-set minterms”
Sum of Products
• Simplifying sum-of-minterms can yield a sum of products
• Difference is each term need not be a minterm
• i.e., terms do not need to have all variables
• A bunch of ANDs and one OR
Two-Level Implementation
• Sum of products has 2 levels of gates
More Levels of Gates?
• What’s best?
• More gate delays (more on this later)
• But maybe we only have 2-input gates
• So multi-input ANDs and ORs have to be decomposed
Complement of a Function
• Definition: 1s & 0s swapped in truth table
• Mechanical way to derive algebraic form
• Take the dual
• Recall: Interchange AND and OR, and 1s & 0s
• Complement each literal
Complement of F
• Not surprisingly, just sum of the other minterms
• “OFF-set minterms”
• In this case

m1 + m3 + m4 + m6

Product of Maxterms
• Recall that maxterm is true except for its own case
• So M1 is only false for 001
Product of Maxterms
• Can express F as AND of all rows that should evaluate to 0

or

Product of Sums
• Result: another standard form
• ORs followed by AND
• Terms do not have to be maxterms
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
Next Time
• Lab Prep
• Demo lab software