CPE100: Digital Logic Design I

Transcription

1 Chapter 2 Professor Brendan Morris, SEB 326, brendan.morris@unlv.edu CPE: Digital Logic Design I Section 4: Dr. Morris Combinational Logic Design Chapter 2 <>

2 Chapter 2 :: Topics Introduction Boolean Equations Boolean Algebra From Logic to Gates Multilevel Combinational Logic X s and Z s, Oh My Karnaugh Maps Combinational Building Blocks Timing Chapter 2 <2>

3 Introduction A logic circuit is composed of: Inputs Outputs Functional specification Timing specification inputs functional spec timing spec outputs Chapter 2 <3>

4 Circuits Nodes Inputs: A, B, C Outputs: Y, Z A E n Internal: n B E3 Y Circuit elements C E2 Z E, E2, E3 Each a circuit Chapter 2 <4>

5 Types of Logic Circuits Combinational Logic (Ch 2) Memoryless Outputs determined by current values of inputs Sequential Logic (Ch 3) Has memory Outputs determined by previous and current values of inputs inputs functional spec timing spec outputs Chapter 2 <5>

6 Rules of Combinational Composition Every element is combinational Every node is either an input or connects to exactly one output The circuit contains no cyclic paths E.g. no connection from output to internal node Example: Chapter 2 <6>

7 Boolean Equations Functional specification of outputs in terms of inputs Example: S = F(A, B, C in ) C out = F(A, B, C in ) A S B C L C C out in A B C in S C out S = A B C in C out = AB + AC in + BC in Chapter 2 <7>

8 Functional specification Goals: Systematically express logical functions using Boolean equations To simplify Boolean equations Chapter 2 <8>

9 Administrative Notes Note: New homework instructions starting with HW3 Homework is due at the beginning of class Homework must be organized, legible (messy is not), and stapled to be graded Chapter 2 <9>

10 Some Definitions Complement: variable with a bar over it A, B, C Literal: variable or its complement A, A, B, B, C, C Implicant: product of literals ABC, AC, BC Minterm: product that includes all input variables ABC, ABC, ABC Maxterm: sum that includes all input variables (A+B+C), (A+B+C), (A+B+C) Chapter 2 <>

11 Canonical Sum-of-Products (SOP) Form All equations can be written in SOP form Each row has a minterm A minterm is a product (AND) of literals Each minterm is TRUE for that row (and only that row) A B Y minterm A B A B A B A B minterm name m m m 2 m 3 Chapter 2 <>

12 Canonical Sum-of-Products (SOP) Form All equations can be written in SOP form Each row has a minterm A minterm is a product (AND) of literals Each minterm is TRUE for that row (and only that row) Form function by ORing minterms where the output is TRUE A B Y Y = F(A, B) = minterm A B A B A B A B minterm name m m m 2 m 3 Chapter 2 <2>

13 Canonical Sum-of-Products (SOP) Form All equations can be written in SOP form Each row has a minterm A minterm is a product (AND) of literals Each minterm is TRUE for that row (and only that row) Form function by ORing minterms where the output is TRUE Thus, a sum (OR) of products (AND terms) A B Y minterm A B A B A B A B minterm name m m m 2 m 3 Y = F(A, B) = AB + AB = Σ(m, m 3 ) Chapter 2 <3>

14 SOP Example Steps: Find minterms that result in Y= Sum TRUE minterms A B Y Y = F(A, B) = Chapter 2 <4>

15 Aside: Precedence AND has precedence over OR In other words: AND is performed before OR Example: Y = ҧ A B + A B Equivalent to: Y = AB ҧ + (AB) Chapter 2 <5>

16 Canonical Product-of-Sums (POS) Form All Boolean equations can be written in POS form Each row has a maxterm A maxterm is a sum (OR) of literals Each maxterm is FALSE for that row (and only that row) A B Y maxterm A + B A + B A + B A + B maxterm name M M M 2 M 3 Chapter 2 <6>

17 Canonical Product-of-Sums (POS) Form All Boolean equations can be written in POS form Each row has a maxterm A maxterm is a sum (OR) of literals Each maxterm is FALSE for that row (and only that row) Form function by ANDing the maxterms for which the output is FALSE Thus, a product (AND) of sums (OR terms) A B Y maxterm A + B A + B A + B A + B maxterm name M M M 2 M 3 Y = M M 2 = A + B ( A ҧ + B) Chapter 2 <7>

18 SOP and POS Comparison Sum of Products (SOP) Implement the ones of the output Sum all one terms OR results in one Product of Sums (POS) Implement the zeros of the output Multiply zero terms AND results in zero Chapter 2 <8>

19 Boolean Equations Example You are going to the cafeteria for lunch You will eat lunch (E=) If it s open (O=) and If they re not serving corndogs (C=) Write a truth table for determining if you will eat lunch (E). O C E Chapter 2 <9>

20 Boolean Equations Example You are going to the cafeteria for lunch You will eat lunch (E=) If it s open (O=) and If they re not serving corndogs (C=) Write a truth table for determining if you will eat lunch (E). O C E Chapter 2 <2>

21 SOP & POS Form SOP sum-of-products O C E POS product-of-sums O C E minterm O C O C O C O C maxterm O + C O + C O + C O + C Chapter 2 <2>

22 SOP & POS Form SOP sum-of-products O C E minterm O C O C O C O C POS product-of-sums O C E maxterm O + C O + C O + C O + C Chapter 2 <22>

23 SOP & POS Form SOP sum-of-products O C E minterm O C O C O C O C E = OC = Σ(m 2 ) POS product-of-sums O C E maxterm O + C O + C O + C O + C Chapter 2 <23>

24 SOP & POS Form SOP sum-of-products O C E minterm O C O C O C O C E = OC = Σ(m 2 ) POS product-of-sums O C E maxterm O + C O + C O + C O + C E = (O + C)(O + C)(O + C) = Π(M, M, M3) Chapter 2 <24>

25 Boolean Algebra Axioms and theorems to simplify Boolean equations Like regular algebra, but simpler: variables have only two values ( or ) Duality in axioms and theorems: ANDs and ORs, s and s interchanged Chapter 2 <25>

26 Boolean Axioms Chapter 2 <26>

27 Duality Duality in Boolean axioms and theorems: ANDs and ORs, s and s interchanged Chapter 2 <27>

28 Boolean Axioms Chapter 2 <28>

29 Boolean Axioms Dual: Exchange: and + and Chapter 2 <29>

30 Boolean Axioms Dual: Exchange: and + and Chapter 2 <3>

31 Basic Boolean Theorems B = B Chapter 2 <3>

32 Basic Boolean Theorems: Duals Dual: Exchange: and + and Chapter 2 <32>

33 T: Identity Theorem B = B B + = B Chapter 2 <33>

34 T: Identity Theorem B = B B + = B B = B B = B Chapter 2 <34>

35 Switching Algebra Simplification of digital logic connecting wires with a on/off switch X = (switch open) X = (switch closed) Chapter 2 <35>

36 Series Switching Network: AND Switching circuit in series performs AND is connected to 2 iff A AND B are Chapter 2 <36>

37 Parallel Switching Network: OR Switching circuit in parallel performs OR is connected to 2 if A OR B is Chapter 2 <37>

38 T: Identity Theorem B = B B + = B B = B B = B Chapter 2 <38>

39 T2: Null Element Theorem B = B + = Chapter 2 <39>

40 T2: Null Element Theorem B = B + = B = B = Chapter 2 <4>

41 T3: Idempotency Theorem B B = B B + B = B Chapter 2 <4>

42 T3: Idempotency Theorem B B = B B + B = B B = B B B B = B Chapter 2 <42>

43 T4: Involution Theorem B = B Chapter 2 <43>

44 T4: Involution Theorem B = B B = B Chapter 2 <44>

45 T5: Complements Theorem B B = B + B = Chapter 2 <45>

46 T5: Complements Theorem B B = B + B = B = B B B = Chapter 2 <46>

47 Recap: Basic Boolean Theorems Chapter 2 <47>

48 Boolean Theorems of Several Vars Number Theorem Name T6 B C = C B Commutativity T7 (B C) D = B (C D) Associativity T8 B (C + D) = (B C) + (B D) Distributivity T9 B (B+C) = B Covering T (B C) + (B C) = B Combining T B C + (B D) + (C D) = B C + B D Consensus Chapter 2 <48>

49 Boolean Theorems of Several Vars Number Theorem Name T6 B C = C B Commutativity T7 (B C) D = B (C D) Associativity T8 B (C + D) = (B C) + (B D) Distributivity T9 B (B+C) = B Covering T (B C) + (B C) = B Combining T B C + (B D) + (C D) = B C + B D Consensus How do we prove these are true? Chapter 2 <49>

50 How to Prove Boolean Relation Method : Perfect induction Method 2: Use other theorems and axioms to simplify the equation Make one side of the equation look like the other Chapter 2 <5>

51 Proof by Perfect Induction Also called: proof by exhaustion Check every possible input value If two expressions produce the same value for every possible input combination, the expressions are equal Chapter 2 <5>

52 Example: Proof by Perfect Induction Number Theorem Name T6 B C = C B Commutativity B C BC CB Chapter 2 <52>

53 Example: Proof by Perfect Induction Number Theorem Name T6 B C = C B Commutativity B C BC CB Chapter 2 <53>

54 Boolean Theorems of Several Vars Number Theorem Name T6 B C = C B Commutativity T7 (B C) D = B (C D) Associativity T8 B (C + D) = (B C) + (B D) Distributivity T9 B (B+C) = B Covering T (B C) + (B C) = B Combining T B C + (B D) + (C D) = B C + B D Consensus Chapter 2 <54>

55 T7: Associativity Number Theorem Name T7 (B C) D = B (C D) Associativity Chapter 2 <55>

56 T8: Distributivity Number Theorem Name T8 B (C + D) = (B C) + (B D) Distributivity Chapter 2 <56>

57 T9: Covering Number Theorem Name T9 B (B+C) = B Covering Chapter 2 <57>

58 T9: Covering Number Theorem Name T9 B (B+C) = B Covering Prove true by: Method : Perfect induction Method 2: Using other theorems and axioms Chapter 2 <58>

59 T9: Covering Number Theorem Name T9 B (B+C) = B Covering Method : Perfect Induction B C (B+C) B(B+C) Chapter 2 <59>

60 T9: Covering Number Theorem Name T9 B (B+C) = B Covering Method : Perfect Induction B C (B+C) B(B+C) Chapter 2 <6>

61 T9: Covering Number Theorem Name T9 B (B+C) = B Covering Method 2: Prove true using other axioms and theorems. Chapter 2 <6>

62 T9: Covering Number Theorem Name T9 B (B+C) = B Covering Method 2: Prove true using other axioms and theorems. B (B+C) = B B + B C T8: Distributivity = B + B C T3: Idempotency = B ( + C) T8: Distributivity = B () T2: Null element = B T: Identity Chapter 2 <62>

63 T: Combining Number Theorem Name T (B C) + (B C) = B Combining Prove true using other axioms and theorems: Chapter 2 <63>

64 T: Combining Number Theorem Name T (B C) + (B C) = B Combining Prove true using other axioms and theorems: B C + B C = B (C+C) T8: Distributivity = B () T5 : Complements = B T: Identity Chapter 2 <64>

65 T: Consensus Number Theorem T (B C) + (B D) + (C D) = (B C) + B D Name Consensus Prove true using () perfect induction or (2) other axioms and theorems. Chapter 2 <65>

66 Recap: Boolean Thms of Several Vars Number Theorem Name T6 B C = C B Commutativity T7 (B C) D = B (C D) Associativity T8 B (C + D) = (B C) + (B D) Distributivity T9 B (B+C) = B Covering T (B C) + (B C) = B Combining T B C + (B D) + (C D) = B C + B D Consensus Chapter 2 <66>

67 Boolean Thms of Several Vars: Duals # Theorem Dual Name T6 B C = C B B+C = C+B Commutativity T7 (B C) D = B (C D) (B + C) + D = B + (C + D) Associativity T8 B (C + D) = (B C) + (B D) B + (C D) = (B+C) (B+D) Distributivity T9 B (B+C) = B B + (B C) = B Covering T (B C) + (B C) = B (B+C) (B+C) = B Combining T (B C) + (B D) + (C D) = (B C) + (B D) (B+C) (B+D) (C+D) = (B+C) (B+D) Dual: Replace: with + with Consensus Chapter 2 <67>

68 Boolean Thms of Several Vars: Duals # Theorem Dual Name T6 B C = C B B+C = C+B Commutativity T7 (B C) D = B (C D) (B + C) + D = B + (C + D) Associativity T8 B (C + D) = (B C) + (B D) B + (C D) = (B+C) (B+D) Distributivity T9 B (B+C) = B B + (B C) = B Covering T (B C) + (B C) = B (B+C) (B+C) = B Combining T (B C) + (B D) + (C D) = (B C) + (B D) (B+C) (B+D) (C+D) = (B+C) (B+D) Consensus Dual: Replace: with + with Warning: T8 differs from traditional algebra: OR (+) distributes over AND ( ) Chapter 2 <68>

69 Boolean Thms of Several Vars: Duals # Theorem Dual Name T6 B C = C B B+C = C+B Commutativity T7 (B C) D = B (C D) (B + C) + D = B + (C + D) Associativity T8 B (C + D) = (B C) + (B D) B + (C D) = (B+C) (B+D) Distributivity T9 B (B+C) = B B + (B C) = B Covering T (B C) + (B C) = B (B+C) (B+C) = B Combining T (B C) + (B D) + (C D) = (B C) + (B D) (B+C) (B+D) (C+D) = (B+C) (B+D) Consensus Axioms and theorems are useful for simplifying equations. Chapter 2 <69>

70 Simplifying an Equation Reducing an equation to the fewest number of implicants, where each implicant has the fewest literals Chapter 2 <7>

71 Simplifying an Equation Reducing an equation to the fewest number of implicants, where each implicant has the fewest literals Recall: Implicant: product of literals ABC, AC, BC Literal: variable or its complement A, A, B, B, C, C Chapter 2 <7>

72 Simplifying an Equation Reducing an equation to the fewest number of implicants, where each implicant has the fewest literals Recall: Implicant: product of literals ABC, AC, BC Literal: variable or its complement A, A, B, B, C, C Also called: minimizing the equation Chapter 2 <72>

73 Simplification methods Distributivity (T8, T8 ) B (C+D) = BC + BD B + CD = (B+ C)(B+D) Covering (T9 ) A + AP = A Combining (T) PA + PA = P Chapter 2 <73>

74 Simplification methods Distributivity (T8, T8 ) B (C+D) = BC + BD B + CD = (B+ C)(B+D) Covering (T9 ) A + AP = A Combining (T) PA + PA = P Expansion P = PA + PA A = A + AP Duplication A = A + A Chapter 2 <74>

75 Simplification methods Distributivity (T8, T8 ) B (C+D) = BC + BD B + CD = (B+ C)(B+D) Covering (T9 ) A + AP = A Combining (T) PA + PA = P Expansion P = PA + PA A = A + AP Duplication A = A + A A combination of Combining/Covering PA + A = P + A Chapter 2 <75>

76 Simplification methods A combination of Combining/Covering PA + A = P + A Proof: PA + A = PA + (A + AP) T9 Covering = PA + PA + A T6 Commutativity = P(A + A) + A T8 Distributivity = P() + A T5 Complements = P + A T Identity Chapter 2 <76>

77 T: Consensus Number Theorem T (B C) + (B D) + (C D) = (B C) + B D Name Consensus Prove using other theorems and axioms: Chapter 2 <77>

78 T: Consensus Number Theorem T (B C) + (B D) + (C D) = (B C) + B D Name Consensus Prove using other theorems and axioms: B C + B D + C D = BC + BD + (CDB+CDB) T: Combining = BC + BD + BCD+BCD T6: Commutativity = BC + BCD + BD + BCD T6: Commutativity = (BC + BCD) + (BD + BCD) T7: Associativity = BC + BD T9 : Covering Chapter 2 <78>

79 Recap: Boolean Thms of Several Vars # Theorem Dual Name T6 B C = C B B+C = C+B Commutativity T7 (B C) D = B (C D) (B + C) + D = B + (C + D) Associativity T8 B (C + D) = (B C) + (B D) B + (C D) = (B+C) (B+D) Distributivity T9 B (B+C) = B B + (B C) = B Covering T (B C) + (B C) = B (B+C) (B+C) = B Combining T (B C) + (B D) + (C D) = (B C) + (B D) (B+C) (B+D) (C+D) = (B+C) (B+D) Consensus Chapter 2 <79>

80 Simplification methods Distributivity (T8, T8 ) B (C+D) = BC + BD B + CD = (B+ C)(B+D) Covering (T9 ) A + AP = A Combining (T) PA + PA = P Expansion P = PA + PA A = A + AP Duplication A = A + A A combination of Combining/Covering PA + A = P + A Chapter 2 <8>

81 Simplifying Boolean Equations Example : Y = AB + AB Chapter 2 <8>

82 Simplifying Boolean Equations Example : Y = AB + AB Y = A T: Combining or = A(B + B) T8: Distributivity = A() T5 : Complements = A T: Identity Chapter 2 <82>

83 Simplification methods Distributivity (T8, T8 ) B (C+D) = BC + BD B + CD = (B+ C)(B+D) Covering (T9 ) A + AP = A Combining (T) PA + PA = P Expansion P = PA + PA A = A + AP Duplication A = A + A A combination of Combining/Covering PA + A = P + A Chapter 2 <83>

84 Simplifying Boolean Equations Example 2: Y = A(AB + ABC) Chapter 2 <84>

85 Simplifying Boolean Equations Example 2: Y = A(AB + ABC) = A(AB( + C)) T8: Distributivity = A(AB()) T2 : Null Element = A(AB) T: Identity = (AA)B T7: Associativity = AB T3: Idempotency Chapter 2 <85>

86 Simplification methods Distributivity (T8, T8 ) B (C+D) = BC + BD B + CD = (B+ C)(B+D) Covering (T9 ) A + AP = A Combining (T) PA + PA = P Expansion P = PA + PA A = A + AP Duplication A = A + A A combination of Combining/Covering PA + A = P + A Chapter 2 <86>

87 Simplifying Boolean Equations Example 3: Y = A BC + A Recall: A = A Chapter 2 <87>

88 Simplifying Boolean Equations Example 3: Y = A BC + A Recall: A = A = A T9 Covering: X + XY = X or = A (BC + ) T8: Distributivity = A () T2 : Null Element = A T: Identity Chapter 2 <88>

89 Simplification methods Distributivity (T8, T8 ) B (C+D) = BC + BD B + CD = (B+ C)(B+D) Covering (T9 ) A + AP = A Combining (T) PA + PA = P Expansion P = PA + PA A = A + AP Duplication A = A + A A combination of Combining/Covering PA + A = P + A Chapter 2 <89>

90 Simplifying Boolean Equations Example 4: Y = AB C + ABC + A BC Chapter 2 <9>

91 Simplifying Boolean Equations Example 4: Y = AB C + ABC + A BC = AB C + ABC + ABC + A BC T3 : Idempotency = (AB C+ABC) + (ABC+A BC) T7 : Associativity = AC + BC T: Combining Chapter 2 <9>

92 Simplification methods Distributivity (T8, T8 ) B (C+D) = BC + BD B + CD = (B+ C)(B+D) Covering (T9 ) A + AP = A Combining (T) PA + PA = P Expansion P = PA + PA A = A + AP Duplication A = A + A A combination of Combining/Covering PA + A = P + A Chapter 2 <92>

93 Simplifying Boolean Equations Example 5: Y = AB + BC +B D + AC D Chapter 2 <93>

94 Simplifying Boolean Equations Example 5: Y = AB + BC +B D + AC D Method : Y = AB + BC + B D + (ABC D + AB C D ) T: Combining = (AB + ABC D ) + BC + (B D + AB C D ) T6: Commutativity T7: Associativity = AB + BC + B D T9: Covering Method 2: Y = AB + BC + B D + AC D + AD T: Consensus = AB + BC + B D + AD T9: Covering = AB + BC + B D T: Consensus Chapter 2 <94>

95 Simplification methods Distributivity (T8, T8 ) B (C+D) = BC + BD B + CD = (B+ C)(B+D) Covering (T9 ) A + AP = A Combining (T) PA + PA = P Expansion P = PA + PA A = A + AP Duplication A = A + A A combination of Combining/Covering PA + A = P + A Chapter 2 <95>

96 Simplifying Boolean Equations Example 6: Y = (A + BC)(A + DE) Apply T8 first when possible: W+XZ = (W+X)(W+Z) Chapter 2 <96>

97 Simplifying Boolean Equations Example 6: Y = (A + BC)(A + DE) Apply T8 first when possible: W+XZ = (W+X)(W+Z) Make: X = BC, Z = DE and rewrite equation Y = (A+X)(A+Z) substitution (X=BC, Z=DE) = A + XZ T8 : Distributivity = A + BCDE substitution or Y = AA + ADE + ABC + BCDE T8: Distributivity = A + ADE + ABC + BCDE T3: Idempotency = A + ADE + ABC + BCDE = A + ABC + BCDE T9 : Covering = A + BCDE T9 : Covering Chapter 2 <97>

98 Y = (A + BC)(A + DE) or Simplifying Boolean Equations Example 6: Apply T8 first when possible: W+XZ = (W+X)(W+Z) Make: X = BC, Z = DE and rewrite equation Y = (A+X)(A+Z) substitution (X=BC, Z=DE) Y = A + XZ T8 : Distributivity = A + BCDE substitution = AA + ADE + ABC + BCDE T8: Distributivity = A + ADE + ABC + BCDE T3: Idempotency = A + ADE + ABC + BCDE = A + ABC + BCDE T9 : Covering = A + BCDE T9 : Covering This is called multiplying out an expression to get sum-of-products (SOP) form. Chapter 2 <98>

99 Reminder Midterm : Thursday, Oct. 5 th In class: hour and 5 minutes Chap 2.6 Closed book, closed notes No calculator Boolean Theorems & Axioms document will be attached as last page of the exam for your convenience Chapter 2 <99>

100 Multiplying Out: SOP Form An expression is in simplified sum-ofproducts (SOP) form when all products contain literals only. SOP form: Y = AB + BC + DE NOT SOP form: Y = DF + E(A +B) SOP form: Z = A + BC + DE F Chapter 2 <>

101 Multiplying Out: SOP Form Example: Y = (A + C + D + E)(A + B) Apply T8 first when possible: W+XZ = (W+X)(W+Z) Chapter 2 <>

102 Multiplying Out: SOP Form Example: Y = (A + C + D + E)(A + B) Apply T8 first when possible: W+XZ = (W+X)(W+Z) Make: X = (C+D+E), Z = B and rewrite equation Y = (A+X)(A+Z) substitution (X=(C+D+E), Z=B) = A + XZ T8 : Distributivity = A + (C+D+E)B substitution = A + BC + BD + BE T8: Distributivity or Y = AA+AB+AC+BC+AD+BD+AE+BE T8: Distributivity = A+AB+AC+AD+AE+BC+BD+BE T3: Idempotency = A + BC + BD + BE T9 : Covering Chapter 2 <2>

103 Canonical SOP & POS Form SOP sum-of-products O C E minterm O C O C O C O C E = OC = Σ(m 2 ) POS product-of-sums O C E maxterm O + C O + C O + C O + C E = (O + C)(O + C)(O + C) = Π(M, M, M3) Chapter 2 <3>

104 Factoring: POS Form An expression is in simplified productof-sums (POS) form when all sums contain literals only. POS form: Y = (A+B)(C+D)(E +F) NOT POS form: Y = (D+E)(F +GH) POS form: Z = A(B+C)(D+E ) Chapter 2 <4>

105 Factoring: POS Form Example : Y = (A + B CDE) Apply T8 first when possible: W+XZ = (W+X)(W+Z) Chapter 2 <5>

106 Factoring: POS Form Example : Y = (A + B CDE) Apply T8 first when possible: W+XZ = (W+X)(W+Z) Make: X = B C, Z = DE and rewrite equation Y = (A+XZ) substitution (X=B C, Z=DE) = (A+B C)(A+DE) T8 : Distributivity = (A+B )(A+C)(A+D)(A+E) T8 : Distributivity Chapter 2 <6>

107 Factoring: POS Form Example 2: Y = AB + C DE + F Apply T8 first when possible: W+XZ = (W+X)(W+Z) Chapter 2 <7>

108 Factoring: POS Form Example 2: Y = AB + C DE + F Apply T8 first when possible: W+XZ = (W+X)(W+Z) Make: W = AB, X = C, Z = DE and rewrite equation Y = (W+XZ) + F substitution W = AB, X = C, Z = DE = (W+X)(W+Z) + F T8 : Distributivity = (AB+C )(AB+DE)+F substitution = (A+C )(B+C )(AB+D)(AB+E)+F T8 : Distributivity = (A+C )(B+C )(A+D)(B+D)(A+E)(B+E)+F T8 : Distributivity = (A+C +F)(B+C +F)(A+D+F)(B+D+F)(A+E+F)(B+E+F) T8 : Distributivity Chapter 2 <8>

109 Boolean Thms of Several Vars: Duals # Theorem Dual Name T6 B C = C B B+C = C+B Commutativity T7 (B C) D = B (C D) (B + C) + D = B + (C + D) Associativity T8 B (C + D) = (B C) + (B D) B + (C D) = (B+C) (B+D) Distributivity T9 B (B+C) = B B + (B C) = B Covering T (B C) + (B C) = B (B+C) (B+C) = B Combining T (B C) + (B D) + (C D) = (B C) + (B D) (B+C) (B+D) (C+D) = (B+C) (B+D) Consensus Axioms and theorems are useful for simplifying equations. Chapter 2 <9>

110 Simplification methods Distributivity (T8, T8 ) B (C+D) = BC + BD B + CD = (B+ C)(B+D) Covering (T9 ) A + AP = A Combining (T) PA + PA = P Expansion P = PA + PA A = A + AP Duplication A = A + A A combination of Combining/Covering PA + A = P + A Chapter 2 <>

111 DeMorgan s Theorem Number Theorem Name T2 B B B 2 = B +B +B 2 DeMorgan s Theorem Chapter 2 <>

112 DeMorgan s Theorem Number Theorem Name T2 B B B 2 = B +B +B 2 DeMorgan s Theorem The complement of the product is the sum of the complements Chapter 2 <2>

113 DeMorgan s Theorem: Dual # Theorem Dual Name T2 B B B 2 = B +B +B 2 B +B +B 2 = B B B 2 DeMorgan s Theorem Chapter 2 <3>

114 DeMorgan s Theorem: Dual # Theorem Dual Name T2 B B B 2 = B +B +B 2 B +B +B 2 = B B B 2 The complement of the product is the sum of the complements DeMorgan s Theorem Chapter 2 <4>

115 DeMorgan s Theorem: Dual # Theorem Dual Name T2 B B B 2 = B +B +B 2 B +B +B 2 = B B B 2 The complement of the product is the sum of the complements. Dual: The complement of the sum is the product of the complements. DeMorgan s Theorem Chapter 2 <5>

116 DeMorgan s Theorem Y = AB = A + B A B A B Y Y Y = A + B = A B A B A B Y Y Chapter 2 <6>

117 DeMorgan s Theorem Example Y = (A+BD)C Chapter 2 <7>

118 DeMorgan s Theorem Example Y = (A+BD)C = (A+BD) + C = (A (BD)) + C = (A (BD)) + C = ABD + C Chapter 2 <8>

119 DeMorgan s Theorem Example 2 Y = (ACE+D) + B Chapter 2 <9>

120 DeMorgan s Theorem Example 2 Y = (ACE+D) + B = (ACE+D) B = (ACE D) B = ((AC+E) D) B = ((AC+E) D) B = (ACD + DE) B = ABCD + BDE Chapter 2 <2>

121 Bubble Pushing Backward: Body changes Adds bubbles to inputs A B Y A B Y Forward: Body changes Adds bubble to output A B Y A B Y Chapter 2 <2>

122 Bubble Pushing What is the Boolean expression for this circuit? A B C D Y Chapter 2 <22>

123 Bubble Pushing What is the Boolean expression for this circuit? A B C D Y Y = AB + CD Chapter 2 <23>

124 Bubble Pushing Rules Begin at output, then work toward inputs Push bubbles on final output back Draw gates in a form so bubbles cancel A B C Y D Chapter 2 <24>

125 Bubble Pushing Example A B C D Y Chapter 2 <25>

126 Bubble Pushing Example A B no output bubble C D Y Chapter 2 <26>

127 Bubble Pushing Example A B no output bubble C D Y A B bubble on input and output C D Y Chapter 2 <27>

128 Bubble Pushing Example A B no output bubble C D Y A B C D A B C D no bubble on input and output Y = ABC + D bubble on input and output Y Y Chapter 2 <28>

129 Canonical SOP & POS Form Revisited SOP sum-of-products O C E minterm O C O C O C O C How do we implement this logic function with gates? E = OC = Σ(m 2 ) POS product-of-sums O C E maxterm O + C O + C O + C O + C E = (O + C)(O + C)(O + C) = Π(M, M, M3) Chapter 2 <29>

130 From Logic to Gates Two-level logic: ANDs followed by ORs Example: Y = ABC + ABC + ABC A B C A B C minterm: ABC minterm: ABC minterm: ABC Y Chapter 2 <3>

131 Circuit Schematics Rules Inputs on the left (or top) Outputs on right (or bottom) Gates flow from left to right Straight wires are best Y = തB ҧ C + A തB Chapter 2 <3>

132 Circuit Schematic Rules (cont.) Wires always connect at a T junction A dot where wires cross indicates a connection between the wires Wires crossing without a dot make no connection wires connect at a T junction wires connect at a dot wires crossing without a dot do not connect Chapter 2 <32>

133 Multiple-Output Circuits Example: Priority Circuit Output asserted corresponding to most significant TRUE input A 3 A 2 A A PRIORITY CiIRCUIT Y 3 Y 2 Y Y A 3 A 2 A A Y 3 Y 2 Y Y Chapter 2 <33>

134 Multiple-Output Circuits Example: Priority Circuit Output asserted corresponding to most significant TRUE input A 3 A 2 A A PRIORITY CiIRCUIT Y 3 Y 2 Y Y A 3 A 2 A A Y 3 Y 2 Y Y Chapter 2 <34>

135 Priority Circuit Hardware A 3 A 2 A A Y 3 Y 2 Y Y A 3 A 2 A A Y 3 Y 2 Y Y Chapter 2 <35>

136 Don t Cares Simplify truth table by ignoring entries A 3 A 2 A A Y 3 Y 2 Y Y A 3 A 2 X A A X X X Y 3 Y 2 Y Y X X Much easier to read off Boolean equations A 3 A 2 A A Y 3 Y 2 Y Y = A 3 = A 3 A 2 = A 3 A 2 A = A 3 A 2 A A Chapter 2 <36>

137 Contention: X Contention: circuit tries to drive output to and Actual value somewhere in between Could be,, or in forbidden zone Might change with voltage, temperature, time, noise Often causes excessive power dissipation A = Y = X B = Warnings: Contention usually indicates a bug. X is used for don t care and contention - look at the context to tell them apart Chapter 2 <37>

138 Floating: Z Floating, high impedance, open, high Z Floating output might be,, or somewhere in between A voltmeter won t indicate whether a node is floating Tristate Buffer E A E A Y Z Z Y Note: tristate buffer has an enable bit (E) to turn on the gate Chapter 2 <38>

139 shared bus Tristate Busses Floating nodes are used in tristate busses en Many different drivers Exactly one is active at once processor to bus from bus video to bus from bus en2 Ethernet en3 to bus from bus memory en4 to bus from bus Chapter 2 <39>

140 Karnaugh Maps (K-Maps) Boolean expressions can be minimized by combining terms PA + PA = P K-maps minimize equations graphically Put terms to combine close to one another A B C Y Y C AB Y = Aҧ തB C ҧ + Y AB C ABC ABC ҧ A തBC = ABC ABC ҧ A തB(C + ABC ABC ҧ C) ABC ABC Chapter 2 <4>

141 K-Map Circle s in adjacent squares In Boolean expression, include only literals whose true and complement form are not in the circle A B C Y Y C Y = AB ҧ A തB C not included because both C and Cҧ included in circle Chapter 2 <4>

142 3-Input K-Map Y AB C ABC ABC ABC ABC ABC ABC ABC ABC A Truth Table B C Y K-Map Y AB C Chapter 2 <42>

143 K-Map Definitions Complement: variable with a bar over it A, B, C Literal: variable or its complement A, A, B, B, C, C Implicant: product of literals ABC, AC, BC Prime implicant: implicant corresponding to the largest circle in a K-map Chapter 2 <43>

144 K-Map Rules Every must be circled at least once Each circle must span a power of 2 (i.e., 2, 4) squares in each direction Each circle must be as large as possible A circle may wrap around the edges A don't care (X) is circled only if it helps minimize the equation Chapter 2 <44>

145 4-Input K-Map A B C D Y Y AB CD Chapter 2 <45>

146 4-Input K-Map A B C D Y Y AB CD Chapter 2 <46>

147 4-Input K-Map A B C D Y Y CD AB Y = AC + ABD + ABC + BD Chapter 2 <47>

148 K-Maps with Don t Cares A B C D Y X X X X X X X Y AB CD Chapter 2 <48>

149 K-Maps with Don t Cares A B C D Y X X X X X X X Y AB CD X X X X X X X Chapter 2 <49>

150 K-Maps with Don t Cares A B C D Y X X X X X X X Y AB CD X X X X X Y = A + BD + C X X Chapter 2 <5>

151 4-Input K-Map: POS & SOP Form A B C D Y Y AB CD Chapter 2 <5>

152 4-Input K-Map: POS Form A B C D Y Y CD AB Y = AC + ABD + ABC + BD Chapter 2 <52>

153 4-Input K-Map: POS Form A B C D Y Y AB CD Chapter 2 <53>

154 Canonical POS Expansion Add literal/complement terms to reverse simplification ( expand literal) Example Y = C Y = C + AAҧ Y = C + A (C + A) ҧ Y = [ C + A + B തB](C + A) ҧ Y = C + A + B C + A + തB C + Aҧ Chapter 2 <54>

155 Combinational Building Blocks Multiplexers Decoders Chapter 2 <55>

156 Multiplexer (Mux) Selects between one of N inputs to connect to output log 2 N-bit required to select input control input S Example: 2: Mux (2 inputs to output) N = 2 log 2 2 = control bit required S D D D D S Y Y S Y D D Chapter 2 <56>

157 Multiplexer Implementations Logic gates Sum-of-products form Y D D S Tristates For an N-input mux, use N tristates Turn on exactly one to select the appropriate input S Y = D S + D S D Y D D S D Y Chapter 2 <57> 2-<57>

158 Logic using Multiplexers Using the mux as a lookup table Zero outputs tied to GND One output tied to VDD A B Y Y = AB A B Y Chapter 2 <58>

159 Logic using Multiplexers Reducing the size of the mux Y = AB A B Y A Y A B B Y Chapter 2 <59>

160 Decoders N inputs, 2 N outputs One-hot outputs: only one output HIGH at once Example 2:4 Decoder (2 inputs to 4 outputs) A i decimal value selects the corresponding output A A 2:4 Decoder Y 3 Y 2 Y Y A A Y 3 Y 2 Y Y Chapter 2 <6>

161 Decoder Implementation A A Y 3 Y 2 Y Y Chapter 2 <6>

162 Logic Using Decoders OR minterms A B 2:4 Decoder Minterm AB AB AB AB Y = AB + AB = A B Y XNOR function Chapter 2 <62>

163 Timing Delay between input change and output changing A Y delay A Y Time How to build fast circuits? Chapter 2 <63>

164 Propagation & Contamination Delay Propagation delay: t pd = max delay from input to final output Contamination delay: t cd = min delay from input to initial output change A Y A Y t pd Note: Timing diagram shows a signal with a high and low and transition time as an X. Cross hatch indicates unknown/changing values t cd Time Chapter 2 <64>

165 Propagation & Contamination Delay Delay is caused by Capacitance and resistance in a circuit Speed of light limitation Reasons why t pd and t cd may be different: Different rising and falling delays Multiple inputs and outputs, some of which are faster than others Circuits slow down when hot and speed up when cold Chapter 2 <65>

166 Critical (Long) & Short Paths Critical Path A B C D n Short Path n2 Y Critical (Long) Path: t pd = 2t pd_and + t pd_or Short Path: t cd = t cd_and Chapter 2 <66>

167 Glitches When a single input change causes an output to change multiple times Chapter 2 <67>

168 Glitch Example What happens when A =, C =, B falls? A B Y C Y C AB Y = AB + BC Chapter 2 <68>

169 Glitch Example (cont.) A = B = C = Critical Path n n2 Y = Short Path B n2 n Note: n is slower than n2 because of the extra inverter for B to go through Y glitch Time Chapter 2 <69>

170 Fixing the Glitch Y C AB Consensus term AC Y = AB + BC + AC A = B = Y = C = Chapter 2 <7>

171 Why Understand Glitches? Glitches shouldn t cause problems because of synchronous design conventions (see Chapter 3) It s important to recognize a glitch: in simulations or on oscilloscope Can t get rid of all glitches simultaneous transitions on multiple inputs can also cause glitches Chapter 2 <7>