# Gate-Level Minimization

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1 Gate-Level Minimization ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Fall,

2 Outlines The Map Method Four-Variable Map Five-Variable Map Product-of-Sums Simplification Don t-care Conditions NAND and NOR Implementation Other Two-Level Implementations Exclusive-OR Function DCD-03-2

3 The Map Method Boolean function sum of minterms sum of products (or product of sum) in the simplest form a minimum number of terms a minimum number of literals The simplified expression may not be unique Gate-level minimization refers to the design task of finding an optimal gate-level implementation of Boolean functions describing a digital circuit. Logic minimization algebraic approach: lack specific rules Karnaugh map approach a simple straight forward procedure a pictorial form of a truth table applicable if the # of variables < 7 A diagram made up of squares each square represents one minterm DCD-03-3

4 Two-Variable Map A two-variable map four minterms x' = row 0; x = row 1 y' = column 0; y = column 1 a truth table in square diagram DCD-03-4

5 Eight minterms Gray code sequence Three-Variable Map Any two adjacent squares in the map differ by only one variable e.g., m 5 and m 7 can be simplified m 5 + m 7 = xy'z + xyz = xz (y'+y) = xz yz DCD-03-5

6 Example 3.1 F(x,y,z) = S(2,3,4,5) = x'y + xy' DCD-03-6

7 Three-Variable Map m 0 and m 2 (m 4 and m 6 ) are adjacent m 0 + m 2 = x'y'z' + x'yz' = x'z' (y'+y) = x'z' m 4 + m 6 = xy'z' + xyz' = xz' (y'+y) = xz' yz DCD-03-7

8 Example 3.2 F(x,y,z) = S(3,4,6,7) = yz+ xz' DCD-03-8

9 Four adjacent squares 2, 4, 8 and 16 squares Three-Variable Map m 0 +m 2 +m 4 +m 6 = x'y'z'+x'yz'+xy'z'+xyz' = x'z'(y'+y) +xz'(y'+y) = x'z' + xz = z' m 1 +m 3 +m 5 +m 7 = x'y'z+x'yz+xy'z+xyz =x'z(y'+y) + xz(y'+y) =x'z + xz = z yz DCD-03-9

10 Example 3.3 F(x,y,z) = S(0,2,4,5,6)= z'+ xy' DCD-03-10

11 Example 3.4 Given F = A'C + A'B + AB'C + BC Express it in sum of minterms => S(1,2,3,5,7) Find the minimal sum of products expression => F=C+A B DCD-03-11

12 The map Four-Variable Map 16 minterms combinations of 2, 4, 8, and 16 adjacent squares DCD-03-12

13 Example 3.5 Simplify the Boolean Function: F(w,x,y,z) = S(0,1,2,4,5,6,8,9,12,13,14)=y'+w'z'+xz' DCD-03-13

14 Example 3.6 Simplify the Boolean Function: Given F = A B C + B CD + A BCD + AB C => F = B D + B C + A CD DCD-03-14

15 Systematic Simplification Implicant a product term of which if the function has the value 1 for all minterms. A Prime Implicant is a product term obtained by combining the maximum possible number of adjacent squares in the map into a rectangle with the number of squares a power of 2 remove any literal not a implicant. A prime implicant is called an Essential Prime Implicant if it is the only prime implicant that covers (includes) one or more minterms. Prime Implicants and Essential Prime Implicants can be determined by inspection of a K-Map. A set of prime implicants "covers all minterms" if, for each minterm of the function, at least one prime implicant in the set of prime implicants includes the minterm. Source: M. Morris Mano and Charles R. Kime, Logic and Computer Design Fundamentals. DCD-03-15

16 Example of Prime Implicants Find ALL Prime Implicants BD CD C ESSENTIAL Prime Implicants C B D BD A AB 1 1 B D AD BC BD A Minterms covered by single prime implicant 1 1 B D Source: M. Morris Mano and Charles R. Kime, Logic and Computer Design Fundamentals. DCD-03-16

17 Example of Prime Implicants Find all prime implicants for: F(A, B,C, D) = Sm (0,2,3,8,9,10,11,12,13,14,15) BD 1 C 1 BC 1 A 1 1 A D Source: M. Morris Mano and Charles R. Kime, Logic and Computer Design Fundamentals. B DCD-03-17

18 Example of Prime Implicants Find all prime implicants for: G(A, B,C, D) = Sm Hint: There are seven prime implicants! (0,2,3,4,7,12,13,14,15) C A B D Source: M. Morris Mano and Charles R. Kime, Logic and Computer Design Fundamentals. DCD-03-18

19 Prime Implicants Selection Rule Find all prime implicants. Include all essential prime implicants in the solution Select a minimum cost set of non-essential prime implicants to cover all minterms not yet covered: Minimize the overlap among prime implicants as much as possible. Make sure that each prime implicant selected includes at least one minterm not included in any other prime implicant selected avoid redundancy. Source: M. Morris Mano and Charles R. Kime, Logic and Computer Design Fundamentals. DCD-03-19

20 Selection Rule Example Simplify F(A, B, C, D) given on the K-map. C Selected Essential C A B A B D Minterms covered by essential prime implicants Source: M. Morris Mano and Charles R. Kime, Logic and Computer Design Fundamentals. D DCD-03-20

21 Simplification Using Prime Implicants Consider F( A, B, C, D ) = (0, 2,3,5,7,8,9,10,11,13,15) The simplified expression may not be unique F = BD+B'D'+CD+AD = BD+B'D'+CD+AB = BD+B'D'+B'C+AD = BD+B'D'+B'C+AB' Digital Circuit Design DCD-03-21

22 Five-Variable Map Map for more than four variables becomes complicated five-variable map: two four-variable map (one on the top of the other) DCD-03-22

23 Example 3.7 F = S(0,2,4,6,9,13,21,23,25,29,31)=A'B'E'+BD'E+ACE DCD-03-23

24 Another Map for Example 3.7 DCD-03-24

25 Relationship between # of Adjacent Squares and # of the Literals Digital Circuit Design DCD-03-25

26 Product of Sums Simplification Approach 1: Complement from minterms Step 1: Simplify F' in the form of sum of products Step 2: Apply DeMorgan's theorem F = (F')' => F': sum of products => F: product of sums Approach 2: Duality from maxterms Combinations of maxterms M 0 M 1 = (A+B+C+D)(A+B+C+D') = (A+B+C)+(DD') = A+B+C CD 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 DCD-03-26

27 Given F = S(0,1,2,5,8,9,10) Approach 1: Step 1: F' = AB+CD+BD' Example 3.8 Step 2: Apply DeMorgan's theorem; F=(A'+B')(C'+D')(B'+D) Approach 2: Think in terms of maxterms DCD-03-27

28 Implementation of Example 3.8 DCD-03-28

29 Truth Table of Function F Consider the function defined in Table 3.2. In sum-of-minterm: F( x, y, z ) = (1,3,4,6) In product-of-maxterm: F ( x, y, z) = (0,2,5,7) Taking the complement of F F( x, y, z) = ( x z )( x z) DCD-03-29

30 Truth Table of Function F Consider the function defined in Table 3.2. Combine the 1 s: F( x, y, z) = x' z xz' Combine the 0 s : F' ( x, y, z) = xz x' z' DCD-03-30

31 Don't-Care Conditions The value of a function is not specified for certain combinations of variables BCD; : don't care The don't care conditions can be utilized in logic minimization can be implemented as 0 or 1 DCD-03-31

32 Example 3.9 Given F(w,x,y,z) = S(1,3,7,11,15) and d(w,x,y,z) = S(0,2,5) F = yz + w'x'; F = yz + w'z F = S(0,1,2,3,7,11,15) ; F = S(1,3,5,7,11,15) Either expression is acceptable DCD-03-32

33 NAND Implementation NAND gate is a universal gate can implement any digital system Two graphic symbols for a NAND gate DCD-03-33

34 Two-level Implementation NAND-NAND = sum of products Example: F = AB+CD F = ((AB)' (CD)' )' =AB+CD DCD-03-34

35 Example 3.10 F( x, y, z ) = (1,2,3,4,5,7) F( x, y, z) = xy x y z DCD-03-35

36 General Design Procedure Procedure 1. Convert all AND gates to NAND gates with AND-inverter graphic symbols. 2. Convert all OR gates to NAND gates with inverter-or graphic symbols. 3. Check all the bubbles in the diagram. For every bubble that is not compensated by another small circle along the same line, insert an inverter or complement the input literal. DCD-03-36

37 Multilevel NAND Circuits Implementing F = A(CD + B) + BC using NAND gate only DCD-03-37

38 Multilevel NAND Circuits Implementing F = (AB +A B)(C+ D ) using NAND gate only DCD-03-38

39 NOR Implementation NOR function is the dual of NAND function. The NOR gate is also universal. Two graphic symbols for a NOR gate DCD-03-39

40 Two-level Implementation Implementing F = (A + B)(C + D)E using NOR gate only. DCD-03-40

41 Multilevel NOR Circuits Implementing F = (AB +A B)(C + D ) using NOR gate only. DCD-03-41

42 Other Two-level Implementations Digital Circuit Design Wired logic A wire connection between the outputs of two gates Open-collector TTL NAND gates: wired-and logic The NOR output of ECL gates: wired-or logic F = ( AB) ( CD) = ( AB CD) = ( A B )( C D ) F = ( A B) ( C D) = [( A B)( C D)] AND-OR-INVERT function OR-AND-INVERT function DCD-03-42

43 Nondegenerate Forms Considering four types of goats: AND, OR, NAND, NOR, 16 possible combinations of two-level forms are obtained. Eight of them: degenerate forms = a single operation The eight nondegenerate forms AND-OR, OR-AND, NAND-NAND, NOR-NOR, NOR-OR, NAND- AND, OR-NAND, AND-NOR AND-OR and NAND-NAND = sum of products OR-AND and NOR-NOR = product of sums NOR-OR, NAND-AND, OR-NAND, AND-NOR =? DCD-03-43

44 AND-OR-Invert Implementation AND-OR-INVERT (AOI) Implementation NAND-AND = AND-NOR = AOI F = (AB+CD+E)' F' = AB+CD+E (sum of products) DCD-03-44

45 OR-AND-Inverter Implementation OR-AND-INVERT (OAI) Implementation OR-NAND = NOR-OR = OAI F = ((A+B)(C+D)E)' F' = (A+B)(C+D)E (product of sums) simplified F' in products of sum DCD-03-45

46 Tabular Summary DCD-03-46

47 Example 3-11 Example 3.11 F' = x'y+xy'+z (F': sum of products) F = (x'y+xy'+z)' (F: AOI implementation) F = x'y'z' + xyz' (F: sum of products) F' = (x+y+z)(x'+y'+z) (F': product of sums) F = ((x+y+z)(x'+y'+z))' (F: OAI) DCD-03-47

48 Example 3.11 DCD-03-48

49 Exclusive-OR Function Exclusive-OR (XOR) x y = xy'+x'y Exclusive-NOR (XNOR) (x y)' = xy + x'y' Some identities x 0 = x x 1 = x' x x = 0 x x' = 1 x y' = (x y)' x' y = (x y)' Commutative and associative A B = B A (A B) C = A (B C) = A B C DCD-03-49

50 Exclusive-OR Function Implement (x'+y')x + (x'+y')y = xy'+x'y = x y DCD-03-50

51 Odd and Even Functions A B C = (AB'+A'B)C' +(AB+A'B')C = AB'C'+A'BC'+ABC+A'B'C = S(1,2,4,7) An odd number of 1's DCD-03-51

52 Logic Diagram of Odd and Even Functions Logic diagram of odd and even functions DCD-03-52

53 Four-Variable Exclusive-OR Function Four-variable Exclusive-OR function A B C D = (AB +A B) (CD +C D) = (AB +A B)(CD+C D )+(AB+A B )(CD +C D) Digital Circuit Design DCD-03-53

54 Parity Generation and Checking Parity Generation and Checking a parity bit: P = x y z parity check: C = x y z P C=1: an odd number of data bit error C=0: correct or an even # of data bit error DCD-03-54

55 Parity Generation and Checking DCD-03-55

56 Conclusions From this lecture, you have learned the follows: Four-Variable Map Five-Variable Map Product-of-Sums Simplification Don t-care Conditions NAND and NOR Implementation Other Two-Level Implementations Exclusive-OR Function DCD-03-56

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