1 / 19

Lecture No. 9

Lecture No. 9. Computer Logic Design Boolean Algebra and Logic Simplification. Veracity of DeMorgan's Theorems. First Theorem Second Theorem Alternative Method – use Truth Tables. Application of DeMorgan's Theorems. Apply to any number of variables Apply to combination of variables.

marsden-cooley
Download Presentation

Lecture No. 9

An Image/Link below is provided (as is) to download presentation 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. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lecture No. 9 Computer Logic Design Boolean Algebra and Logic Simplification

  2. Veracity of DeMorgan's Theorems • First Theorem • Second Theorem • Alternative Method – use Truth Tables

  3. Application of DeMorgan's Theorems • Apply to any number of variables • Apply to combination of variables

  4. Application of DeMorgan's Theorems • Finding Complement of a Function • Example:

  5. Application of DeMorgan's Theorems Shortcut for finding Complement of a Function • Take dual of the function • swap 1’s and 0’s • Swap AND and OR gates • Helpful to add parenthesis • Complement each literal

  6. Types of Boolean Expressions • Define Domain of an expression • set of all variables (complemented or otherwise) • Boolean expressions may be expressed as: • Sum-of-Products (SOP) Form • Product-of-Sums (POS) Form • Each form may contain single variable terms • May contain complemented and un-complemented terms • A SOP and POS expression can’t have a term of more than one variable having an over bar extending over the entire term

  7. Sum-of-Products (SOP) Form • Two or more product terms summed by Boolean addition • Any expression -> SOP using Boolean algebra • Examples: * A + BC

  8. Sum-of-Products (SOP) Form • Conversion to SOP Form:

  9. Implementation of SOP Expression B+AC+AD

  10. Standard SOP Form & Minterms • SOP expressions containing all Variables in the Domain in each term are in Standard Form. • Standard product terms are also called Minterms. • Any non-standard SOP expression may be converted to Standard form by applying Boolean Algebra Rule 6 to it. • Example:

  11. Standard SOP Form • Example: Determine Standard SOP expression SHORTCUT: Introduce all possible combinations of the missing variables AND’ed with the original term

  12. Characteristics of a Minterm • Minterm is a standard product term in which all variables appear exactly once (complemented or uncomplemented) • Represents exactly one combination of the binary variables in a truth table for which the function produces a “1” output. That is the binary representation or value. • Has value of 1 for that combination and 0 for all others • For n variables, there are 2n distinct minterms • Example:

  13. Product-of-Sums (POS) Form • Two or more sum terms multiplied by Boolean multiplication • Any expression -> POS using Boolean algebra • Examples: (A+B)(B+C)(A+B+C)

  14. Product-of-Sums (POS) Form • Conversion to POS Form:

  15. Implementation of POS expression (A+B)(B+C+D)(A+C)

  16. Standard POS Form & Maxterms • POS expressions containing all Variables in the Domain in each term are in Standard Form. • Standard sum terms are also called Maxterms. A Maxterm is a NOT Minterm. • Any non-standard POS expression may be converted to Standard form by applying Boolean Algebra Rule 8 and Rule 12A+BC=(A+B)(A+C) to it.

  17. Standard POS Form • Example: {Rule 8} {Rule 12} SHORTCUT: Introduce all possible combinations of the missing variables OR’ed with the original term

  18. Characteristics of a Maxterm • Maxterm is a standard sum term in which all variables appear exactly once (complemented or uncomplemented) • Represents exactly one combination of the binary variables in a truth table for which the function produces a “0” output. That is the binary representation or value. • Has value of 0 for that combination and 1 for all others • For n variables, there are 2n distinct maxterms • Example:

  19. Why Standard SOP and POS Forms? • Direct mapping of Standard Form expressions and Truth Table entries. • Alternate Mapping methods for simplification of expressions • Minimal Circuit implementation by switching between Standard SOP or POS • PLD based function implementation

More Related