1 / 29

CS 325: CS Hardware and Software Organization and Architecture

CS 325: CS Hardware and Software Organization and Architecture. Gates and Boolean Algebra Part 2. Outline. Sum of Products (SOP) Fan-in, Fan-out Cascading to Reduce Inputs Boolean Algebra Laws Gate Reduction using Boolean Algebra. Circuits from SOP Functions. Why simplify circuits?

dean-king
Télécharger la présentation

CS 325: CS Hardware and Software Organization and Architecture

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. CS 325: CS Hardware and SoftwareOrganization and Architecture Gates and Boolean Algebra Part 2

  2. Outline • Sum of Products (SOP) • Fan-in, Fan-out • Cascading to Reduce Inputs • Boolean Algebra Laws • Gate Reduction using Boolean Algebra

  3. Circuits from SOP Functions • Why simplify circuits? • NAND and NOR gates are simpler (faster, smaller) than NOT AND and NOT OR. • Reduction in complexity when using a small number of gate types. • Goal: To implement circuit using a small complete set of operators. • NAND and NOR are both complete since any Boolean function can be implemented with either. • Faster to use small number of inputs to a gate (fan-in), and small number of gate inputs from a gate output (fan-out) • Typically, fan-in and fan-out < 10.

  4. Logic Gate Fan-n and Fan-out

  5. Gate Cascading to Reduce Inputs Implementing 3-input AND and OR functions with 2-input gates ABC = (AB)C A+B+C = (A+B)+C Implementing a 3-input NAND function with 2-input gates. NO! Correct

  6. Basic Laws of Boolean Algebra • Boolean Algebra follows many algebra rules which can be used to make simpler circuits.

  7. Basic Laws of Boolean Algebra • Boolean Algebra follows many algebra rules which can be used to make simpler circuits.

  8. Basic Laws of Boolean Algebra • Boolean Algebra follows many algebra rules which can be used to make simpler circuits.

  9. Basic Laws of Boolean Algebra • Boolean Algebra follows many algebra rules which can be used to make simpler circuits.

  10. Basic Laws of Boolean Algebra • Boolean Algebra follows many algebra rules which can be used to make simpler circuits.

  11. Basic Laws of Boolean Algebra • Boolean Algebra follows many algebra rules which can be used to make simpler circuits.

  12. Basic Laws of Boolean Algebra • Boolean Algebra follows many algebra rules which can be used to make simpler circuits.

  13. Basic Laws of Boolean Algebra • Boolean Algebra follows many algebra rules which can be used to make simpler circuits.

  14. Basic Laws of Boolean Algebra • Boolean Algebra follows many algebra rules which can be used to make simpler circuits.

  15. Basic Laws of Boolean Algebra • Boolean Algebra follows many algebra rules which can be used to make simpler circuits. • Example: AB + AC Three gates • = A(B + C), Distributive Law Two gates

  16. Gate Reduction • AB + AC Three gates • = A(B + C), Distributive Law Two gates

  17. Equivalent Gates/Symbols • Using Boolean Laws (identities), alternative symbols for some gates can be derived:

  18. Functionally Complete Sets of Gates • Not all gate types are typically implemented in circuit design. • Simpler if only 1 or 2 types of gates are used. • A functionally complete set of gates means that any Boolean function can be implemented using only the gates in that set. • Examples of functionally complete sets: • AND, OR, NOT • AND, NOT • OR, NOT • NAND • NOR

  19. NAND and NOR Completeness

  20. Implement XOR with NANDs • Exclusive-OR (XOR) example: • Step 1: build truth table • Step 2: find SOP and build circuit using AND and OR.

  21. Implement XOR with NANDs • Apply Boolean Algebra rules: so, De Morgan’s Law: so, • The last formula is 3 NAND gates.

  22. Implement XOR with NANDs • Logic circuits implementing XOR:

  23. Simplification • Boolean functions, and therefore circuits, can usually be manipulated using Boolean laws into simpler functions. Distributive Law Inverse Law Distributive Law Inverse Law Idempotent Law Identity Law • How to check for correctness?

  24. Checking Logic for Correctness • We can check our solution using a truth table • Checking

  25. Checking Logic for Correctness • We can check our solution using a truth table • Checking Correct

  26. Another Example • Draw the logic gate diagram for the following Boolean function: • Circuit Reduction: Distributive Law Idempotent Law Idempotent Law Distributive Law • Check with Truth table for correctness.

  27. Checking Logic for Correctness • We can check our solution using a truth table • Checking

  28. Checking Logic for Correctness • We can check our solution using a truth table • Checking Correct

  29. Another Example • Draw the logic gate diagram for the following Boolean function: • Circuit Reduction: Distributive Law Idempotent Law Absorption Law Distributive Law Idempotent Law Absorption Law Absorption Law • Check with Truth table for correctness.

More Related