1 / 21

Digital Circuit Design Jeffrey N. Denenberg Lecture #3

Understand Boolean algebra for designing basic circuits. Learn logic symbols, theorems, DeMorgan's rules, and circuit analysis techniques.

pfelton
Download Presentation

Digital Circuit Design Jeffrey N. Denenberg Lecture #3

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. Digital Circuit Design Jeffrey N. Denenberg Lecture #3 Boolean algebraCombinational-circuit analysis

  2. Boolean algebra • a.k.a. “switching algebra” • deals with Boolean values -- 0, 1 • Positive-logic convention • analog voltages LOW, HIGH --> 0, 1 • Negative logic -- seldom used • Signal values denoted by variables(X, Y, FRED, etc.) Boolean - Combinational Logic

  3. Boolean operators • Complement: X¢ (opposite of X) • AND: X × Y • OR: X + Y • Huntington Postulates: 1-6 Ch. 2.2Postulates and Theorems: Table 2-1 binary operators, describedfunctionally by truth table. Boolean - Combinational Logic

  4. More definitions • Literal: a variable or its complement • X, X¢, FRED¢, CS_L • Expression: literals combined by AND, OR, parentheses, complementation • X+Y • P × Q × R • A + B × C • ((FRED × Z¢) + CS_L × A × B¢× C + Q5) × RESET¢ • Equation: Variable = expression • P = ((FRED × Z¢) + CS_L × A × B¢× C + Q5) × RESET¢ Boolean - Combinational Logic

  5. Logic symbols Boolean - Combinational Logic

  6. Theorems • See Table 2-1 • Proofs by perfect induction Boolean - Combinational Logic

  7. More Theorems • Again Table 2-1 Boolean - Combinational Logic

  8. Duality • Swap 0 & 1, AND & OR • Result: Theorems still true • Why? • Each axiom (A1-A5) has a dual (A1¢-A5¢) • Counterexample:X + X × Y = X (T9)X × X + Y = X (dual)X + Y = X (T3¢)???????????? X + (X×Y) = X (T9)X× (X + Y) = X (dual)(X× X) + (X× Y) = X (T8)X+ (X× Y) = X (T3¢) parentheses,operator precedence! Boolean - Combinational Logic

  9. N-variable Theorems • Prove using finite induction • Most important: DeMorgan theorems Boolean - Combinational Logic

  10. DeMorgan Symbol Equivalence Boolean - Combinational Logic

  11. Likewise for OR • (be sure to check errata!) Boolean - Combinational Logic

  12. DeMorgan Symbols Boolean - Combinational Logic

  13. Even more definitions (Sec. 4.1.6) • Product term • Sum-of-products expression • Sum term • Product-of-sums expression • Normal term • Minterm (n variables) • Maxterm (n variables) Boolean - Combinational Logic

  14. Truth table vs. minterms & maxterms • Table 2-3 Boolean - Combinational Logic

  15. Combinational analysis Boolean - Combinational Logic

  16. Signal expressions • Multiply out:F = ((X + Y¢) × Z) + (X¢× Y × Z¢) = (X × Z) + (Y¢× Z) + (X¢× Y × Z¢) Boolean - Combinational Logic

  17. New circuit, same function Boolean - Combinational Logic

  18. “Add out” logic function • Circuit: Boolean - Combinational Logic

  19. Shortcut: Symbol substitution Boolean - Combinational Logic

  20. Different circuit, same function Boolean - Combinational Logic

  21. Another example Boolean - Combinational Logic

More Related