# Combinational Logic Circuits

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1 Chapter 2 Combinational Logic Circuits J.J. Shann

2 Chapter Overview 2-1 Binary Logic and Gates 2-2 Boolean Algebra 2-3 Standard Forms 2-4 Two-Level Circuit Optimization 2-5 Map Manipulation 補 充 資 料 :Quine-McCluskey Method 2-6 Multiple-Level Circuit Optimization 2-7 Other Gate Types 2-8 Exclusive-OR Operator and Gates 2-9 High-Impedance Outputs 2-10 Chapter Summary J.J. Shann 2-2

3 2-1 Binary Logic and Gates Digital circuits: hardware components that manipulate binary information are implemented using transistors and interconnections in integrated circuits. Logic gate: basic ckt that performs a specific logical op J.J. Shann 2-3

4 A. Binary Logic Binary logic: deals w/ binary variables and the ops of mathematical logic applied to these variables. binary variable: variable that take on two discrete values resembles binary arithmetic, but should no be confused w/ each other. J.J. Shann 2-4

5 Basic Logic Operations Basic logical ops: p.31, Table 2-1 AND:, identical to binary multiplication OR:, resemble binary addition In binary logic, In binary arithmetic: NOT: complement;, J.J. Shann 2-5

6 Truth Table Truth table: a table of combinations of the binary variables showing the relationship b/t the values that the variables take on and the values of the result of the op. 2 n rows, n: # variables E.g.: truth table for AND op J.J. Shann 2-6

7 B. Logic Gates Logic gate: is an electronic ckt that operate on one or more input signals to produce an output signal. The input terminals of logic gates accept binary signals within the allowable range and respond at the output terminals w/ binary signals that fall within a specified range. E.g.: transition region J.J. Shann 2-7

8 Graphic symbols of 3 basic logic gates: Timing diagram: J.J. Shann 2-8

9 Multiple-input logic gates: AND and OR gates may have 2 inputs. E.g.: J.J. Shann 2-9

10 2-2 Boolean Algebra Boolean algebra: is an algebra dealing w/ binary variables and logic ops binary variables: are designated by letters of the alphabet logic ops: AND, OR, NOT Boolean expression: an algebraic expression formed by using binary variables, the constants 0 and 1, the logic op symbols, and parentheses. J.J. Shann 2-10

11 Boolean Function Boolean function: can be described by a Boolean equation, a truth table, or a logic ckt diagram Single-output Boolean function Multiple-output Boolean function J.J. Shann 2-11

12 Boolean Equation Boolean equation: consists of a binary variable identifying the function followed by an equal sign and a Boolean expression. expresses the logical relationship b/t binary variables can be expressed in a variety of ways * Obtain a simpler expression for the same function. E.g.: F (, Y, Z) YZ terms J.J. Shann 2-12

13 Truth table for a function: is unique a list of all combinations of 1 s and 0 s that can be assigned to the binary variables and a list that shows the value of the function for each binary combination. E.g.: F (, Y, Z) YZ J.J. Shann 2-13

14 Logic circuit diagram: An algebraic expression for a Boolean function A ckt diagram composed of logic gates Circuit gates are interconnected by wires that carry logic signals. E.g.: F (, Y, Z) YZ Combinational logic circuits J.J. Shann 2-14

15 Example Present a set of requirements under which an insurance policy will be issued: The applicant must be 1. a married female 25 years old or over, or 2. a female under 25, or 3. a married male under 25 who has not been involved in a car accident, or 4. a married male who has been involved in a car accident, or 5. a married male 25 years or over who has not been involved in a car accident. J.J. Shann 2-15

16 <Ans.> Define variables: 4; w, x, y, z w 1 if applicant has been involved in a car accident x 1 if applicant is married y 1 if applicant is a male z 1 if applicant is under 25 Find a Boolean expression which assumes the value 1 whenever the policy should be issued: x y z 1. a married female 25 years old or over y z 2. a female under 25 w x y z 3. a married male under 25 who has not been involved in a car accident w x y 4. a married male who has been involved in a car accident w x y z 5. a married male 25 years or over who has not been involved in a car accident F x y z y z w x y z w x y w x y z J.J. Shann 2-16

17 Simplify the expression and suggest a simpler set of requirements: F x y z y z w x y z w x y w x y z x y z w 1 if applicant has been involved in a car accident x 1 if applicant is married y 1 if applicant is a male z 1 if applicant is under 25 The applicant must be 1. married or 2. a female under 25. J.J. Shann 2-17

18 A. Basic Identities of Boolean Algebra Basic identities of Boolean Algebra:, Y, Z: binary variable dual J.J. Shann 2-18

19 Duality Principle Dual: The dual of an algebraic expression is obtained by interchanging OR and AND ops and replacing 1 s by 0 s and 0 s by 1 s (OR AND, 0 1) E.g.: dual Duality principle: 0 1 A Boolean equation remains valid if we take the dual of the expressions on both sides of the equals sign. E.g.: J.J. Shann 2-19

20 Verification of Identities Verification of an identity: by truth table or verified identities E.g.: 0 0, , E.g.: DeMorgan s theorem ( Y) Y J.J. Shann 2-20

21 General DeMorgan s Theorem DeMorgan s theorem: Y Y Y Y General DeMorgan s theorem: n 1 n n 1 n J.J. Shann 2-21

22 B. Algebraic Manipulation Simplification of Boolean functions: Boolean algebra is a useful tool for simplifying digital ckts. E.g.: F YZ YZ Z F YZ YZ Z Y ( Z Z ) Z 14 : x( y z) xy xz Y 1 Z 7 : x x 1 Y Z 2 : x 1 x J.J. Shann 2-22

23 J.J. Shann 2-23

24 Implementing Boolean Equations When a Boolean equation is implemented w/ logic gates: each term requires a gate each variable within the term designates an input to the gate literal: a single variable within a term that may or may not be complemented E.g.: F YZ YZ Z 3 terms & 8 literals 2 terms & 4 literals F Y Z * Reduce # of terms, # of literals, or both in a Boolean expression is often possible to obtain a simpler ckt J.J. Shann 2-24

25 Reducing Expressions by Boolean Algebra Boolean algebra may be applied to reduce an expression for obtaining a simpler ckt: Computer tools for synthesizing logic ckt Manual method: cut-and-try E.g.s: Y ( ( ( Y Y Y Y ) Y )( (1 Y ) ( Y ) ( Y )( Y ) Y ) Y Y Y ) dual YY Y Y J.J. Shann 2-25

26 Consensus Theorem xy x z yz xy x z (x y) (x z) (y z) (x y) (x z) xy x z yz xy x z yz(x x ) xy x z xyz x yz xy xyz x z x yz xy (1 z) x z (1 y) xy x z can be used to eliminate redundant terms from Boolean expressions. * Venn diagram J.J. Shann 2-26

27 E.g.: Simplify the expression (p.2-15) F x y z y z w x y z w x y w x y z x y z J.J. Shann 2-27

28 C. Complement of a Function Complement of a function F: interchange of 1 s to 0 s and 0 s to 1 s for the values of F in the truth table can be derived algebraically by applying DeMorgan s theorem interchange AND and OR ops and complement each variable and constant take the dual of the function equation and complement each literal J.J. Shann 2-28

29 Examples Complementing functions by applying DeMorgan s theorem: E.g.s: F YZ 1 YZ F 1 ( YZ YZ) ( YZ ) ( YZ) ( Y Z) ( Y Z ) F 2 ( YZ YZ) F 2 ( ( YZ YZ)) ( YZ YZ) ( YZ ) ( YZ) ( Y Z) ( Y Z ) J.J. Shann 2-29

30 J.J. Shann 2-30 Complementing functions by using duals: E.g.s: ) ( ) ( each literal complement ) ( ) ( of dual F Z Y Z Y Z Y Z Y F YZ YZ F ) ( ) ( each literal complement ) ( ) ( of dual ) ( F Z Y Z Y Z Y Z Y F YZ YZ F

31 2-3 Standard Forms Standard forms: Contain product terms & sum terms Two types: sum-of-products form product-of-sums form Product term: a logical product consisting of an AND op among literals E.g.: Y Z Sum terms a logical sum consisting of an OR op among literals E.g.: YZ J.J. Shann 2-31

32 A. Minterms and Maxterms Minterm: m j a product term in which all the variables appear exactly once, either complemented or uncomplemented represents exactly one combination of the binary variables in a truth table has the value 1 for that combination and 0 for all others There are 2 n distinct minterms for n variables. Symbol: m j, j: denotes the decimal equivalent of the binary combination for which the minterm has the value 1 E.g.: Minterms for 3 variables J.J. Shann 2-32

33 E.g.: Minterms for 3 variables J.J. Shann 2-33

34 Maxterms Maxterm: M j a sum term that contains all the variables in complemented or uncomplemented form represents exactly one combination of the binary variables in a truth table has the value 0 for that combination and 1 for all others There are 2 n distinct maxterms for n variables. Symbol: M j, j: denotes the decimal equivalent of the binary combination for which the maxterm has the value 0 E.g.: Maxterms for 3 variables J.J. Shann 2-34

35 E.g.: Maxterms for 3 variables J.J. Shann 2-35

36 Minterms vs. Maxterms A minterm a function, not equal to 0, having the min # of 1 s in its truth table A maxterm a function, not equal to 1, having the max # of 1 s in its truth table M j (m j ) E.g.: For 3 variables & j 3 m 3 YZ m YZ Y Z 3 M 3 J.J. Shann 2-36

37 Representing Boolean Function by m j s or M j s Given the truth table of a function a Boolean function in sum of minterms or in product of maxterms Sum of minterms: a logical sum of all the minterms that produce a 1 in the function. Product of maxterms: a logical product of all the maxterms that produce a 0 in the function. J.J. Shann 2-37

38 E.g.: Represent F and F in sum-of-minterms form F Y Z Y Z YZ YZ m 0 m 2 m 5 m 7 m(0,2,5,7) logical sum (Boolean OR) of the minterms F YZ YZ Y Z Y Z m 1 m3 m4 m6 m(1,3,4,6) J.J. Shann 2-38

39 J.J. Shann 2-39 E.g.: Represent F in product-of-maxterms form ),,, M( Z Y Z Y Z Y Z Y M m M M M M m m m m m m m m F j j ) )( )( )( ( ) ( Q ) (1,3,4,6 ),, ( m Z Y F logical product (Boolean AND) of the maxterms

40 Conversion to Sum-of-minterms Form Conversion: by means of truth table or Boolean algebra E Y Z Method 1: by truth table E.g.:Convert to sum-of-minterms form E Y Z truth table m(0,1,2,4,5 ) J.J. Shann 2-40

41 E.g.:Convert E Y Z to sum-of-minterms form Method 2: by Boolean algebra E Y Z Y ( )( Z Z ) Z( Y Y ) YZ YZ YZ YZ YZ YZ YZ YZ YZ YZ YZ m(0,1,2,4,5) J.J. Shann 2-41

42 Summary of the Properties of Minterms There are 2 n minterms for n Boolean variables. These minterms can be evaluated from the binary numbers from 0 to 2 n 1. Any Boolean function can be expressed as a logical sum of minterms. The complement of a function contains those minterms not included in the original function. A function that includes all the 2 n minterms is equal to logic 1. E.g.: G(, Y) Σm(0,1,2,3) 1 J.J. Shann 2-42

43 B. Sum of Products (SoP) Sum-of-products form: a standard form that contains product terms w/ any number of literals. These AND terms are ORing together. E.g.: F (, Y, Z) Y Y Z Y Sum-of-minterms form: is a special case of SoP form is obtained directly from a truth table contains the max # of literals in each term and usually has more product terms than necessary simplify the expression to reduce the # of product terms and the # of literals in the terms a simplified expression in sum-of-products form J.J. Shann 2-43

44 Logic Diagram for an SoP form Logic diagram for an SoP form: consists of a group of AND gates followed by a single OR gate. 2-level implementation or 2-level ckt Assumption: The input variables are directly available in their complemented and uncomplemented forms. ( Inverters are not included in the diagram) E.g.: F Y Y Z Y J.J. Shann 2-44

45 C. Product of Sums (PoS) Product-of-sums form: a logical product of sum terms & each logical sum term may have any # of distinct literals. 2-level gating structure Special case: Product-of-maxterms form E.g.: F ( Y Z) ( Y Z ) J.J. Shann 2-45

46 D. Nonstandard Form Nonstandard form: E.g.: F AB C(D E) 3-level Nonstandard form Standard form: By using the distributive laws E.g.: F AB C(D E) AB CD DE 2-level J.J. Shann 2-46

47 2-4 Two-Level Circuit Optimization Representation of a Boolean function: Truth table: unique Algebraic expression: many different forms digital logic circuit Minimization of Boolean function: Algebraic manipulation: literal minimization ( 2-2) use the rules and laws of Boolean algebra Disadv.: It lacks specific rules to predict each succeeding step in the manipulation process. Map method: gate-level minimization ( 2-4~2-5) a simple straightforward procedure using Karnaugh map (K-map) Disadv.: Maps for more than 4 variables are not simple to use. Tabular method: Quine-McCluskey method ( 補 充 資 料 ) * The simplest algebraic expression: not unique J.J. Shann 2-47

48 A. Cost Criteria Two cost criteria: i. Literal cost the # of literal appearances in a Boolean expression ii. Gate input cost ( ) the # of inputs to the gates in the implementation J.J. Shann 2-48

49 Literal Cost Literal cost: the # of literal appearances in a Boolean expression E.g.: F F Adv.: is very simple to evaluate by counting literal appearances Disadv.: does not represent ckt complexity accurately in all cases E.g.: G G ( A AB C( D AB CD ABCD B)( B E) CE ABCD C)( C 5 literals 6 literals 8 literals D)( D A) 8 literals J.J. Shann 2-49

50 Gate Input Cost Gate input cost: the # of inputs to the gates in the implementation For SoP or PoS eqs, the gate input cost can be found by the sum of all literal appearances the # of terms excluding terms that consist only of a single literal the # of distinct complemented single literals (optional) E.g.: p.2-46 G ABCD ABCD G ( A B)( B C)( C 8 2 ( D)( D 8 4 ( is a good measure for contemporary logic implementation A) is proportional to the # of transistors and wires used in implementing a logic ckt. (especially for ckt 2 levels) 4) gate input counts 4) J.J. Shann 2-50

51 B. The Map Method Map method: Karnaugh map simplification a simple straightforward procedure K-map: a pictorial form of a truth table a diagram made up of squares Each square represents one minterm of the function. The simplified expressions produced by the map are always in one of the two standard forms: SOP (sum of products) or POS (product of sums) J.J. Shann 2-51

52 (a) Two-Variable Map Two-variable map: 4 squares, one for each minterm. A function of 2 variables can be represented in the map by marking the squares that correspond to the minterms of the function. J.J. Shann 2-52

53 Example E.g.: <Ans> F 1 m 3 Y F 2 m 1 m 2 m Y Y Y 3 by combining squares in the map Y ( Y Y ) by applying Boolean algebra Y ( )( Y Y ) J.J. Shann 2-53

54 (b) Three-Variable Map Three-variable map: 8 minterms 8 squares Only one bit changes in value from one adjacent column to the next Any two adjacent squares in the map differ by only one variable, which is primed in one square and unprimed in the other. E.g.: m 5 & m 7 Note: Each square has 3 adjacent squares. The right & left edges touch each other to form adjacent squares. E.g.: m 4 m 0, m 5, m 6 J.J. Shann 2-54

55 Map Minimization of SOP Expression Basic property of adjacent squares: Any two adjacent squares in the map differ by only one variable: primed in one square and unprimed in the other E.g.: m 5 YZ, m7 YZ Any two minterms in adjacent squares that are ORed together can be simplified to a single AND term w/ a removal of the different variable. E.g.: m 5 m7 YZ YZ Z( Y Y ) Z J.J. Shann 2-55

56 Procedure of map minimization of SOP expression: i. A 1 is marked in each minterm that represents the function. Two ways: (1) Convert each minterm to a binary number and then mark a 1 in the corresponding square. (2) Obtain the coincidence of the variables in each term. ii. Find possible adjacent 2 k squares: 2 adjacent squares (i.e., minterms) remove 1 literal 4 adjacent squares (i.e., minterms) remove 2 literal 2 k adjacent squares (i.e., minterms) remove k literal The larger the # of squares combined, the less the # of literals in the product (AND) term. * It is possible to use the same square more than once. J.J. Shann 2-56

57 Example 2-3 Example 2-3: Simplify the Boolean function F(, Y, Z) Σm(2,3,4,5) <Ans.> F Y Y J.J. Shann 2-57

58 4-minterm Product terms Product terms using 4 minterms: E.g.: F(, Y, Z) Σm(0,2,4,6) m 0 m 2 m 4 m 6 Y Z Y Z Y Z Y Z Z( Y Y ) Z( Y Y ) Z Z Z( ) Z J.J. Shann 2-58

59 E.g.: F(, Y, Z) Σm(0,1,2,3,6,7) J.J. Shann 2-59

60 Example 2-4 Example 2-4: Simplify the Boolean function F F 1 <Ans.> (, Y, Z) 2 (, Y, Z) m(3,4,6,7) m(0,2,4,5,6) J.J. Shann 2-60

61 Non-unique Optimized Expressions There may be alternative ways of combining squares to product equally optimized expressions: E.g.: F(, Y, Z) m(1,3,4,5,6 ) F(, Y, Z) m(1,3,4,5,6 ) Z Z Y Z Z YZ J.J. Shann 2-61

62 Simplifying Functions not Expressed as Sumof-minterms Form If a function is not expressed as a sum of minterms: use the map to obtain the minterms of the function & then simplify the function E.g.: Given the Boolean function <Ans.> F Z Y YZ YZ F Z Y YZ YZ 1, 3 2, 3 5 3, 7 m(1,2,3,5,7 ) Z Y J.J. Shann 2-62

63 (c) Four-Variable Map Four-variable map: 16 minterms 16 squares Note: Each square has 4 adjacent squares. The map is considered to lie on a surface w/ the top and bottom edges, as well as the right and left edges, touching each other to form adjacent squares. E.g.: m 8 m 0, m 9, m 10, m 12 J.J. Shann 2-63

64 Example 2-5 Simplify the Boolean function F(W,, Y, Z) Σ(0,1,2,4,5,6,8,9,12,13,14) <Ans.> F Y W Z Z J.J. Shann 2-64

65 Example 2-6 Simplify the Boolean function <Ans.> F ABC BCD ABC ABCD F BD BC ACD J.J. Shann 2-65

66 (d) Five-Variable Map ( 補 充 資 料 ) Five-variable map: 32 minterms 32 squares Each square has 5 adjacent squares. J.J. Shann 2-66

67 Alternatives: Five-variable map Each square has 5 adjacent squares. Consider the two half maps as being one on top of the other. Any 2 squares that fall one over the other are considered adjacent. E.g.: m 8 m 0, m 9, m 10, m 12, m 24 Six-variable map: 64 minterms 64 squares J.J. Shann 2-67

68 Example Simplify the Boolean function F(A,B,C,D,E) Σ(0,2,4,6,9,13,21,23,25,29,31) <Ans.> F A B E BD E ACE J.J. Shann 2-68

69 Relationship b/t the # of adjacent squares and the # of literals in the term: Any 2 k adjacent squares, for k o, 1,, n, in an n-variable map, will represent an area that gives a term of n k literals. J.J. Shann 2-69

70 Maps for more than 4 variables are not as simple to use: Employ computer programs specifically written to facilitate the simplification of Boolean functions w/ a large # of variables. 補 充 Quine-McCluskey Method (p.7-87~7-99) J.J. Shann 2-70

71 2-5 Map Manipulation When choosing adjacent squares in a map: Ensure that all the minterms of the function are covered when combining the squares. Minimize the # of terms in the expression. avoid any redundant terms whose minterms are already covered by other terms J.J. Shann 2-71

72 A. Essential Prime Implicants Implicant: A product term is an implicant of a function if the function has the value 1 for all minterms of the product term. Prime implicant: PI a product term obtained by combining the max. possible # of adjacent squares in the map Essential prime implicant: EPI, must be included If a minterm in a square is covered by only one PI, that PI is said to be essential. Look at each square marked w/ a 1 and check the # of PIs that cover it. J.J. Shann 2-72

73 Example: p.2-57 Find the PIs and EPIs of the Boolean function F(, Y, Z) Σm(1,3,4,5,6) <Ans.> F(, Y, Z) m(1,3,4,5,6 ) Z Z Y Z Z YZ 4 PIs : Z, Z, Y, YZ 2 EPIs : Z ( m3), Z ( m6 ) * * J.J. Shann 2-73

74 Example Find the PIs and EPIs of the Boolean function F(A,B,C,D) Σ(0,2,3,5,7,8,9,10,11,13,15) <Ans.> 6 PIs: BD, B'D', CD, B'C, AD, AB' 2 EPIs: BD (m5), B'D' (m0) J.J. Shann 2-74

75 Example (Cont ) Find the PIs and EPIs of the Boolean function F(A,B,C,D) Σ(0,2,3,5,7,8,9,10,11,13,15) <Ans.> 6 PIs: BD, B'D', CD, B'C, AD, AB' 2 EPIs: BD (m 5 ), B'D' (m 0 ) J.J. Shann 2-75

76 B. Finding the Simplified Expression Procedure for finding the simplified expression from the map: (SoP form) i. Determine all PIs. ii. The simplified expression is obtained from the logical sum of all the EPIs plus other PIs that may be needed to cover any remaining minterms not covered by the EPIs. There may be more than one expression that satisfied the simplification criteria. J.J. Shann 2-76

77 Example 2-7 Simplify the Boolean function F(A,B,C,D) Σ(1,3,4,5,6,7,12,14) <Ans.> PIs: A'D, BD', A'B EPIs: A'D (m 1, m 3 ), BD' (m 12, m 14 ) m 1, m 3, m 5, m 7, m 4, m 6, m 12, m 14 m 1, m 3, m 4, m 5, m 6, m 7, m 12, m 14 F AD BD J.J. Shann 2-77

78 Example 2-8 Simplify the Boolean function F(A,B,C,D) Σ(0,5,10,11,12,13,15) <Ans.> EPIs : ABCD BCD ABC ABC Combine PIs that contains m 15 ACD F ABCD BCD ABC ABC or ABD J.J. Shann 2-78

79 Selection Rule of Nonessential PIs Selection rule of Nonessential PIs: Minimize the overlap among PIs as much as possible. Make sure that each PI selected includes at least 1 minterm not included in any other PI selected. It results in a simplified, although not necessarily minimum cost, SoP expression. J.J. Shann 2-79

80 Example 2-9 Simplify the Boolean function F(A,B,C,D) Σm(0,1,2,4,5,10,11,13,15) <Ans.> EPI : AC (m 1, m 4 ) F( A, B, C, D) AC ABD ABC ABD J.J. Shann 2-80

81 Example: p.2-74 Simplify the Boolean function F(A,B,C,D) Σ(0,2,3,5,7,8,9,10,11,13,15) <Ans.> 6 PIs: BD, B'D', CD, B'C, AD, AB' 2 EPIs: BD, B'D' EPIs: BD, B'D' m 0,m 2, m 5, m 7, m 8, m 10, m 13, m 15 Combine PIs that contains m 3,m 9, m 11 (CD, B'C, AD, AB') F BD B'D' CD AD BD B'D' CD AB' BD B'D' B'C AD BD B'D' B'C AB' J.J. Shann 2-81

82 C. Product-of-Sums (PoS) Optimization Approach 1: Simplified F' in the form of sum of products Apply DeMorgan's theorem F (F')' F': sum of products F: product of sums Approach 2: combinations of the maxterms of F E.g.: i. A 0 is marked in each maxterm that represents the function. ii. Find possible adjacent 2 k squares and realize each set as a sum (OR) term, w/ variables being complemented. CD M 0 M 1 (ABCD)(ABCD') (ABC)(DD') ABC AB M 0 M 1 M 3 M 2 01 M 4 M 5 M 7 M 6 11 M 12 M 13 M 15 M M 8 M 9 M 11 M 10 J.J. Shann 2-82

83 Example 3-8 Simplify the Boolean function in (a) SoP and (b) PoS: F(A,B,C,D) Σm(0,1,2,5,8,9,10) <Ans.> (a) F B D B C A C D (b) Approach 1: F AB CD BD F ( F ) ( A B)( C D)( B D) Approach 2: Think in terms of maxterms J.J. Shann 2-83

84 Gate implementation: (a) F B D B C A C D (b) F ( A B)( C D)( B D) The implementation of a function in a standard form is said to be a two-level implementation. Assumption: The input variables are directly available in their complement Inverters are not needed. Determine which form will be best for a function. J.J. Shann 2-84

85 D. Don t-care Conditions Don t care condition: the unspecified minterms of a function is represented by an E.g.: A 4-bit decimal code has 6 combinations which are not used. Incompletely specified function: has unspecified outputs for some input combinations Simplification of an incompletely specified function: When choosing adjacent squares to simplify the function in the map, the s may be assumed to be either 0 or 1, whichever gives the simplest expression. An need not be used at all if it does not contribute to covering a larger area. J.J. Shann 2-85

86 Example 2-11 Simplify the Boolean function F (w,x,y,z) Σm(1,3,7,11,15) which has the don t-care conditions d (w,x,y,z) Σm(0,2,5). <Ans.> * The outputs in a particular implementation of the function are only 0 s and 1 s. (a) SOP: F (w,x,y,z) yz w x Σm(0,1,2,3,7,11,15) F (w,x,y,z) yz w z Σm(1,3,5,7,11,15) (b) POS: F (w,x,y,z) z (w y) Σm(1,3,5,7,11,15) J.J. Shann 2-86

87 補 充 資 料 : Quine-McCluskey Method & CAD Tools for Simplification J.J. Shann

88 A. Quine-McCluskey Method (for SOP) Tabular method to systematically find all PIs Stage 1: Find all prime implicants Step 1: Fill Column 1 with minterm and don t-care condition indices. Group by number of 1's. Step 2: Apply Uniting Theorem (xy xy x) Compare elements of group w/ N 1's against those w/ N1 1's. Differ by one bit implies adjacent. Eliminate variable and place in next column. When used in a combination, mark with a check. If cannot be combined, mark with a star. These are the prime implicants (PI). Stage 2: Find the minimum cover (SOP) J.J. Shann 2-88

89 Stage 1: Finding All Prime Implicants (1/4) F(A,B,C,D) Σm(4,5,6,8,9,10,13) Σd(0,7,15) Stage 1: Find all prime implicants Step 1: Fill Column 1 with minterm and don t-care condition indices. Group by number of 1's. Column Implication Table J.J. Shann 2-89

90 Stage 1: Finding All Prime Implicants (2/4) Step 1: Fill Column 1 with minterm and don t-care condition indices. Group by number of 1's. Step 2: Apply Uniting Theorem Compare elements of group w/ N 1's against those w/ N1 1's. Differ by one bit implies adjacent. Eliminate variable and place in next column. E.g., 0000 vs yields vs yields -000 When used in a combination, mark with a check. If cannot be combined, mark w/ a star (i.e., PI). Implication Table Column 1 Column Repeat until no further combinations can be made. J.J. Shann 2-90

91 Stage 1: Finding All Prime Implicants (3/4) Step 1: Fill Column 1 with minterm and don t-care condition indices. Group by number of 1's. Step 2: Apply Uniting Theorem Compare elements of group w/ N 1's against those w/ N1 1's. Differ by one bit implies adjacent. Eliminate variable and place in next column. E.g., 0000 vs yields vs yields -000 When used in a combination, mark with a check. If cannot be combined, mark w/ a star (i.e., PI). Implication Table Column 1 Column 2 Column * * -000 * * * * * Repeat until no further combinations can be made. J.J. Shann 2-91

92 Stage 1: Finding All Prime Implicants (4/4) Prime Implicants: 0-00 A' C' D' -000 B' C' D' A' B -1-1 B D 100- A B' C' 10-0 A B' D' 1-01 A C' D J.J. Shann 2-92

93 Find all PIs by using K-map: AB A CD C B AB A CD C B D D F(A,B,C,D) Σm(4,5,6,8,9,10,13) Σd(0,7,15) Prime Implicants: 0-00 A' C' D' -000 B' C' D' 100- A B' C' 10-0 A B' D' 1-01 A C' D A' B -1-1 B D J.J. Shann 2-93

94 Stage 2: Finding the Minimum Cover Stage 2: Find smallest set of prime implicants that cover the minterms. Recall that essential prime implicants must be in all covers. Another tabular method the prime implicant chart Prime Implicants: 0-00 A' C' D' -000 B' C' D' A' B -1-1 B D 100- A B' C' 10-0 A B' D' 1-01 A C' D J.J. Shann 2-94

95 Prime Implicant Chart rows prime implicants columns minterms only (don t-care conditions are not included) Place an if minterm is covered by the PI. F(A,B,C,D) Σm(4,5,6,8,9,10,13) Σd(0,7,15) Prime Implicants: 0-00 A' C' D' -000 B' C' D' 100- A B' C' 10-0 A B' D' 1-01 A C' D A' B -1-1 B D J.J. Shann 2-95

96 Prime Implicant Chart If column has a single, then the PI associated w/ the row is essential. (EPI) It must appear in minimum cover. J.J. Shann 2-96

97 Prime Implicant Chart Eliminate all columns covered by EPI. J.J. Shann 2-97

98 Prime Implicant Chart Find minimum set of rows that cover the remaining columns F A B' B' D' D' A C' C' D A' A' B J.J. Shann 2-98

99 B. CAD Tools for Simplification ESPRESSO Method Problem with Quine-McCluskey: the # of prime implicants grows rapidly with the # of inputs Upper bound: 3 n /n, where n is the number of inputs Finding a minimum cover is NP-complete, i.e., a computational expensive process not likely to yield to any efficient algorithm. Espresso: trades solution speed for minimality of answer don't generate all prime implicants (Quine-McCluskey Stage 1) judiciously select a subset of primes that still covers the ON-set operates in a fashion not unlike a human finding primes in a K-map J.J. Shann 2-99

100 Espresso Inputs and Outputs F(A,B,C,D) Σm(4,5,6,8,9,10,13) d(0,7,15) Espresso Input.i 4.o 1.ilb a b c d.ob f.p e -- # inputs -- # outputs -- input names -- output name -- number of product terms -- A'BC'D' -- A'BC'D -- A'BCD' -- AB'C'D' -- AB'C'D -- AB'CD' -- ABC'D -- A'B'C'D' don't care -- A'BCD don't care -- ABCD don't care -- end of list Espresso Output.i 4.o 1.ilb a b c d.ob f.p e F A C' D A B' D' A' B J.J. Shann 2-100

101 2-6 Multiple-Level Circuit Optimization 2-level ckt optimization: can reduce the cost of combinational logic ckts Multi-level ckts: ckts w/ more than 2 levels There are often additional cost saving available J.J. Shann 2-101

102 Example E.g.: G ABC ABD E ACF ADF <Ans.> 2-level implementation: gate-input cost 17 Multi-level implementation: distributive law G ABC ABD E ACF ADF 17 AB C D) E AF( C D) ( 13 ( AB AF)( C D) E 11 A( B F)( C D) E 9 Gate input count (GIC) J.J. Shann 2-102

103 (17) (13) G ABC ABD E ACF ADF G AB( C D) E A( C D) F (11) (9) G ( AB AF)( C D) E G A( B F)( C D) E J.J. Shann 2-103

104 Multiple-level Optimization Multiple-level ckt optimization (simplification): is based on the use of a set of transformations that are applied in conjunction w/ cost evaluation to find a good, but not necessarily optimum solution. J.J. Shann 2-104

105 for GIC for delay Multilevel Optimization Transformations Transformations: Factoring: is finding a factored form from either a SoP or PoS expression for a function Decomposition: is the expression of a function as a set of new functions Extraction: is the expression of multiple functions as a set of new functions Substitution of a function G into a function F: is expressing F as a function of G and some or all of the original variables of F Elimination: flattening or collapsing is the inverse of substitution function G in an expression for function F is replaced by the expression for G J.J. Shann 2-105

106 J.J. Shann Example 2-12 E.g. 2-12: Multilevel optimization transformations <Ans.> BCF BCE ABF ABE ABCD H BCDEF ADF ADE ACF ACE G ) ( ) )( ( )) ( ) ( ( ) ( B A EF B D C A BCDEF F E D C A BCDEF F E D F E C A BCDEF DF DE CF CE A BCDEF ADF ADE ACF ACE G F E CD 2 1 factoring factoring factoring GIC decomposition & substitution

107 J.J. Shann (Cont d) C A F E CD )) )( ( ) ( ( )) ( ) ( ( ) ( F E C A CD A B F E C F E A ACD B CF CE AF AE ACD B BCF BCE ABF ABE ABCD H ( 2) A B H B A G F E CD B A G factoring factoring factoring extraction of G & H: substitution:

108 G H ACE ACF ADE ADF BCDEF ABCD ABE ABF BCE BCF G H A 1 2 B B( A 1 3 2) 1 CD 1 2 GIC 48 2 E F 3 A C GIC 25 (w/ shared terms) ( 31 w/o shared terms) J.J. Shann 2-108

109 Key to successful transformations: is the determination of the factors to be used in decomposition or extraction and choice of the transformation sequence to apply J.J. Shann 2-109

110 Transformation for Delay Reduction Path delay: the length of time it takes for a change in a signal to propagate down a path through the gates Critical path: the longest path(s) through a ckt In a large proportion of designs, the length of the longest path(s) through the ckt is often constrained. The # of gates in series may need to be reduced. Elimination transformation J.J. Shann 2-110

111 Elimination transform: replaces intermediate variables, i, w/ the expressions on their right hand sides or removes other factoring of some variables Reduces the # of gates in series Determination of factor or combination of factors should be eliminated: look at the effect of gate input count The increase in gate input count for the combinations of eliminations that reduce the problem path lengths by at lease one gate are of interest. J.J. Shann 2-111

112 Example 2-13 E.g.: Transformation for delay reduction Assumption: Ignore the delay of NOT gate. G A 1 2 B 1 2 H B( A 1 3 2) CD E F A C GIC 25 J.J. Shann 2-112

113 G A 1 H B( A 1 3 2) 1 CD 2 B 2 E F GIC 25 3 A C <Ans.> 3 combinations of eliminations that reduce 1 gate delay: i. removal of the factor B in H GIC increases 0 ii. removal of the intermediate variables 1, 2, 3 GIC increases 12 (?) iii. removal of B, 1, 2, 3 GIC increases 12 (?) 1 2 G A 1 H B A B B 3 G ACE ACF ADE ADF BCDEF H B( ACD AE AF CE CF) CD E F A C G H ACE ACF ADE ADF BCDEF ABCD ABE ABF BCE BCF J.J. Shann 2-113

114 G A 1 2 B 1 2 H B( A 1 3 2) CD E F A C GIC 25 Eliminate the factor B G A 1 2 B 1 2 H BA 1 B 3 2 GIC 25 J.J. Shann 2-114

115 2-7 Other Gate Types Basic logic operations: AND, OR, NOT All possible functions of n binary variables: n binary variables 2 n distinct minterms 2 2n possible functions E.g.: n 2 4 minterms 16 possible functions J.J. Shann 2-115

116 J.J. Shann 2-116

117 The 16 functions can be subdivided into 3 categories: 1. 2 functions that produce a constant 0 or functions w/ unary operations complement (NOT) and transfer functions w/ binary operators that define eight different operations AND, OR, NAND, NOR, exclusive-or, equivalence, inhibition, and implication. J.J. Shann 2-117

118 Digital Logic Gates Boolean expression: AND, OR and NOT operations It is easier to implement a Boolean function in these types of gates. Consider the construction of other types of logic gates: the feasibility and economy of implementing the gate w/ electronic components the basic properties of the binary operations E.g.: commutativity, associativity, the ability of the gate to implement Boolean functions alone or in conjunction w/ other gates the convenience of representing gate functions that are frequently used J.J. Shann 2-118

119 Primitive Digital Logic Gates AND OR NOT (inverter) Buffer amplify an electrical signal to permit more gates to be attached to the output or to decrease the time it takes for signals to propagate through the ckt 3-state Buffer ( 2-9) NAND NOR * NAND or NOR gates alone can implement any Boolean function are much more widely used than AND and OR gates. negation indicator J.J. Shann 2-119

120 Universal gate: a gate type that alone can be used to implement all Boolean functions E.g.: NAND gate, NOR gate Proof of a universal gate: Show that the logical ops of AND, OR, and NOT can be obtained w/ the universal gate only. E.g.: Show that the NAND gate is a universal gate J.J. Shann 2-120

121 E.g.: Show that the NAND gate is a universal gate J.J. Shann 2-121

122 E.g.: Simplify the Boolean function and implement it by NAND gates F(, Y, Z) m(0,2,4,5,6) <Ans.> NAND gates SoP form Y Z NAND gates J.J. Shann 2-122

123 Method 1: Y Z AND OR Y Z Y Z J.J. Shann 2-123

124 Method 2: Y Z Y Z Y Z J.J. Shann 2-124

125 Complex Digital Logic Gates Exclusive-OR ( 2-8) NOR ( 2-8) AND-OR-Invert (AOI) the complement of SoP E.g.: 2-1 AOI F Y Z E.g.: AOI F TUV W YZ OR-AND-Invert (OAI) the dual of AOI the complement of PoS AND-OR (AO) OR-AND (OA) J.J. Shann 2-125

126 Adv. of using complex gates: reduce the ckt complexity needed for implementing specific Boolean functions in order to reduce integrated ckt (IC) cost reduce the time required for signals to propagate through a ckt J.J. Shann 2-126

127 2-8 Exclusive-OR Operator and Gates Exclusive-OR: OR, Y Y is equal to 1 if exactly one input variable is equal to 1 Exclusive-NOR: NOR, equivalence Y Y Y Y is equal to 1 if both and Y are equal to 1 or if both are equal to 0. OR & NOR are the complement to each other. Y Y Y ( Y )( Y ) Y Y They are particularly useful in arithmetic operations and error-detection and correction ckts. J.J. Shann 2-127

128 Properties of OR Identities: Y Y Y Y can be verified by using a truth table or by replacing the op by its equivalent Boolean expression Commutativity and associativity: A B B A (A B) C A (B C) A B C OR gates w/ three or more inputs J.J. Shann 2-128

129 Implementations of OR function OR function is usually constructed w/ other types of gates: J.J. Shann 2-129

130 Odd Function Multiple-variable OR operation: odd function equal to 1 if the input variables have an odd # of 1 s E.g.: 3-variable OR Y Z ( Y Y ) Z ( Y Y ) Z ( Y Y ) Z Y Z Y Z YZ YZ m(1,2,4,7) is equal to 1 if only one variable is equal to 1 or if all three variables are equal to 1. J.J. Shann 2-130

131 E.g.: 4-variable OR A B C D (AB A B) (CD C D) (AB A B)(CD C D ) (AB A B )(CD C D) Σ(1, 2, 4, 7, 8, 11, 13, 14) n-variable OR function: the logical sum of the 2 n /2 minterms whose binary numerical values have an odd # of 1 s. J.J. Shann 2-131

132 Even Function Multiple-variable NOR op: an even function E.g.: 4-variable NOR n-variable NOR function: the logical sum of the 2 n /2 minterms whose binary numerical values have an even # of 1 s. J.J. Shann 2-132

133 Logic Diagram of Odd & Even Functions Logic diagram of odd & even functions: Multiple-input Odd function: J.J. Shann 2-133

134 Example Simplify the Boolean function and implement it by OR gates and other gates F(W,, Y, Z) Σ(1, 4, 7, 8, 13) <Ans.> YZ W F Y (W Z) WZ ( Y) or Y (W Z) W YZ J.J. Shann 2-134

135 2-9 High-Impedance Outputs Output values of a gate: 0, 1 Hi-Z: high-impedance state behaves as an open ckt, i.e., looking back into the ckt, the output appears to be disconnected * Gates w/ only logic 0 and 1 outputs cannot have their outputs connected together. * Gates w/ Hi-Z output values can have their outputs connected together, provided that no 2 gates drive the line at the same time to opposite 0 and 1 values. Structures that provide Hi-Z: 3-state buffers transmission gates J.J. Shann 2-135

136 Three-State Buffers 3-state buffer: J.J. Shann 2-136

137 Multiplexer constructed by 3- state buffers: E.g.: 2-to-1 MU Input 0 Input 1 MU Output Selection input J.J. Shann 2-137

138 Transmission Gates Transmission gate (TG): J.J. Shann 2-138

139 OR gate constructed from transmission gates: A A C 0, F A C 1, F A F AC A C J.J. Shann 2-139

140 2-10 Chapter Summary Primitive logic ops: AND, OR, NOT Boolean algebra Minterm & maxterm standard forms SoP and PoS standard forms 2-level gate ckts 2 cost measures for ckt optimization: # of input literals to a ckt total # of inputs to the gates in a ckt J.J. Shann 2-140

141 Circuit optimization: 2-level ckt optimization Boolean algebra manipulation K-map Quine-McCluskey method Multi-level ckt optimization Other logic gates: NAND, NOR OR, NOR Hi-Z outputs: 3-state buffer transmission gates J.J. Shann 2-141

142 Problems Sections Exercises 2-1 ~ ~ ~ ~ ~ ~ 2-35 J.J. Shann 2-142

143 Homework (1/2) Part I: (a) 與 (c) 用 algebraic manipulation 來 證 明 ( 證 明 兩 個 Boolean equations 相 等 ) (a) and (d) ( 化 簡 Boolean equation) (minterms and maxterms) (sum-of-products and product-of-sums) 6. 找 出 下 列 式 子 的 complement ( 證 明 DeMorgan's law) (a) x'y' xy x'y (b) ab(c'dcd') a'b'(c'd)(cd') 7. 用 真 值 表 驗 證 底 下 的 式 子 xy x'y yz xy x'y J.J. Shann 2-143

144 Homework (2/2) Part II: 1. 使 用 K-Map 化 簡 F(a,b,c,d) b cd abc a bc abc d --> 給 定 布 林 方 程 式 來 做 化 簡 (c) --> 給 定 布 林 方 程 式 來 做 化 簡 (a) --> 給 定 布 林 方 程 式 ( 輸 入 是 sum of minterm) 來 做 化 簡,PoS form (c) --> 給 定 布 林 方 程 式 來 做 化 簡, 並 轉 成 SoP and PoS forms (a) --> 給 定 布 林 方 程 式 與 don't-card conditions 來 做 化 簡, 並 轉 成 SoP and PoS forms (a) -->decomposition (a) -->elimination > 使 用 excusinve-or 與 AND, 並 在 最 少 # of gate inputs 的 前 提 下, 實 做 一 個 布 林 方 程 式 J.J. Shann 2-144

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