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Digital Circuit Design Jeffrey N. Denenberg Lecture #3

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

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Digital Circuit Design Jeffrey N. Denenberg Lecture #3

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  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

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