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1 Computer Engineering 1 (ECE290) Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization HOANG Trang 2008 Pearson Education, Inc.

2 Overview Part 1 Gate Circuits and Boolean Equations Binary Logic and Gates Boolean Algebra Standard Forms Part 2 Circuit Optimization Two-Level Optimization K-map Map Manipulation Part 3 Additional Gates and Circuits Other Gate Types Exclusive-OR Operator and Gates High-Impedance Outputs Chapter 2 - Part 2 2

3 Circuit Optimization Goal: To obtain the simplest implementation for a given function Optimization requires a cost criterion to measure the simplicity of a circuit Distinct cost criteria we will use: Literal cost (L) Gate input cost (G) Gate input cost with NOTs (GN) Chapter 2 - Part 2 3

4 Literal Cost Literal a variable or it complement Literal cost the number of literal appearances in a Boolean expression corresponding to the logic circuit diagram Examples: F = BD + A BC + A C D L = 8 F = BD + A BC + A B D+ AB C L = F = (A + B)(A + D)(B + C + D)( B + C + D) L = Which solution is best? Chapter 2 - Part 2 4

5 Gate Input Cost Gate input costs - the number of inputs to the gates in the implementation corresponding exactly to the given equation or equations. (G - inverters not counted, GN - inverters counted) For SOP and POS equations, it can be found from the equation(s) by finding the sum of: all literal appearances the number of terms excluding single literal terms,(g) and optionally, the number of distinct complemented single literals (GN). Example: G=L+term, GN=L+term + complement! F=BD+A B C+A C D G=8 8, GN=11 F = BD + A BC + A B D+ AB C G =, GN = F = (A + B )(A + D)(B + C + D )( B + C+ D) G =,GN= Which solution is best? Chapter 2 - Part 2 5

6 Cost Criteria (continued) Example 1: GN BC+ L = 5 = G + 2 = 9 F = A + B C G = L + 2 = 7 B C A F L (literal count) counts the AND inputs and the single literal OR input. G (gate input count) adds the remaining OR gate inputs GN(gate input count with NOTs) adds the inverter inputs Chapter 2 - Part 2 6

7 Cost Criteria (continued) Example 2: A B F=ABC+ A BC C L = 6 G = 8 GN = 11 F F = (A + C)( B + C)( A + B) L = 6 G = 9 GN = 12 Same function and same literal cost But first circuit has better gate input count and better gate input count with NOTs G, GN: is proportional to the number of transistors and wires used to implement a logic circuit Sl Select tit! => G, GN: good measure to choose! A B C F Chapter 2 - Part 2 7

8 Boolean Function Optimization Minimizing the gate input (or literal) cost of a (a set of) Boolean equation(s) reduces circuit cost. We choose gate input cost. Boolean Algebra and graphical techniques are tools to minimize cost criteria values. Some important questions: When do we stop trying to reduce the cost? Do we know when we have a minimum cost? Treat optimum or near-optimum cost functions for two-level (SOP and POS) circuits first. Introduce a graphical technique using Karnaugh maps (K-maps, for short) Chapter 2 - Part 2 8

9 Karnaugh Maps (K-map) A K-map is a collection of squares Each square represents a minterm The collection of squares is a graphical representation of a Boolean function Adjacent squares differ in the value of only one variable (Gray Code) Alternative algebraic expressions for the same function are derived by recognizing patterns of squares The K-map can be viewed as A reorganized version of the truth table A topologically-warped Venn diagram as used to visualize sets in algebra of sets Chapter 2 - Part 2 9

10 Some Uses of K-Maps Provide a means for: Finding optimum or near optimum SOP and POS standard forms, and two-level AND/OR and OR/AND circuit implementations for functions with small numbers of variables Visualizing concepts related to manipulating Boolean expressions Demonstrating concepts used by computer- aided design (CAD) programs to simplify large circuits Chapter 2 - Part 2 10

11 Two Variable Maps A 2-variable Karnaugh Map: Note that minterm m0 and minterm m1 are adjacent x = 0 m 0 = m 1 = and differ in the value of the x y x y variable y x = 1 m 2 = m 3 = x x y Similarly, minterm m0 and y y = 0 y = 1 minterm m2 differ in the x variable. Also, m1 and m3 differ in the x variable as well. Finally, m2 and m3 differ in the value of the variable y Chapter 2 - Part 2 11

12 K-Map and Truth Tables The K-Map is just a different form of the truth table. Example Two variable function: We choose a,b,c and d from the set {0,1} to implement e a particular a function, F(x,y). Function Table K-Map Input Function y = 0 y = 1 Values (x,y) Value F(x,y) x = 0 a b 00 0 a 0 1 b x = 1 c d 1 0 c The K-map: 1 1 d a reorganized version of the truth table Chapter 2 - Part 2 12

13 K-Map Function Representation Example: G(x,y) = x + y G = x+y y = 0 y = 1 x = x = For G(x,y), two pairs of adjacent cells containing 1 s can be combined using the Minimization Theorem x x Theorem: x y y y x y y y G(x, y) x y x y xy x y x y Duplicate xy Chapter 2 - Part 2 13

14 Three Variable Maps A three-variable K-map: yz=00 yz=01 yz=11 yz=10 x=0 m 0 m 1 m 3 m 2 x=1 m 4 m 5 m 7 m 6 Where each minterm corresponds to the product terms: yz=00 yz=01 yz=11 yz=10 x=0 x y z x=1 x y z x y z x y z x y Note that if the binary value for an index differs in only one bit position, the minterms are adjacent on the K- Map x y z x y z x y z z Chapter 2 - Part 2 14

15 Alternative Map Labeling useful x y y x z z z x y z x 0 1 y z Chapter 2 - Part 2 15

16 Example Functions By convention, we represent the minterms of F by a "1" in the map and leave the minterms of F blank Example: F(x, y, z) Example: G(a, b,c) m (2,3,4,5) (3,4,6,7) Learn the locations of the 8 indices based on the variable order shown (x, most significant and z, least significant) on the map boundaries m y x z y x z Chapter 2 - Part 2 16

17 Simplify? => Combining Squares By combining squares reduce number of literals in a product term, reducing the literal cost, thereby reducing the other two cost criteria Chapter 2 - Part 2 17

18 Example: Combining Squares Example: Let F m(2,3,6,7) x y Applying the Minimization Theorem three times: F(x, y, z) x y z x y z yz yz y z x y z x y z Thus the four terms that form a 2 2 square correspond to the term "y". Chapter 2 - Part 2 18

19 Three-Variable Maps Example Shapes of 2-cell Rectangles: y x z Chapter 2 - Part 2 19

20 Three-Variable Maps Example Shapes of 4-cell Rectangles: y x z Chapter 2 - Part 2 20

21 Three Variable Maps K-Maps can be used to simplify Boolean functions by systematic methods. Terms are selected to cover the 1s in the map. Example: Simplify F(x, (, y, z) m(1,2,3,5,7 (,,,, ) x y z y 1 1 x z F(x, y, z) z x y Chapter 2 - Part 2 21

22 Three-Variable Map Simplification Use a K-map to find an optimum SOP equation for F(X, Y, Z) m(01246 (0,1,2,4,6,7) Chapter 2 - Part 2 22

23 Four Variable Maps Map and location of minterms: Y X W Z Chapter 2 - Part 2 23

24 Four-Variable Maps K-map: useful WX YZ Chapter 2 - Part 2 24

25 Four-Variable Maps Example of combining squares Y W X Z Chapter 2 - Part 2 25

26 Four-Variable Maps Example of combining squares Y X W Z Chapter 2 - Part 2 26

27 Four-Variable Map Simplification F(W, X, Y, Z) m (0, 2,4,5,6,7,8,10,13,15) Chapter 2 - Part 2 27

28 Four-Variable Map Simplification F(W, X, Y, Z) m (3,4,5,7,9,13,14,15) Chapter 2 - Part 2 28

29 Systematic Simplification: prime implicant, essential A Prime Implicant (PI) => a product term obtained by combining i the maximum possible number of adjacent squares (the number of squares a power of 2) 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. Chapter 2 - Part 2 29

30 Example of Prime Implicants BD Find ALL Prime Implicants CD C ESSENTIAL Prime Implicants C BD BD A AB D AD BC B BD A D Minterms covered by single prime implicant Chapter 2 - Part 2 30 B

31 Prime Implicant Practice Find all prime implicants for: F(A, B,C, D) m (0,2,3,8,9,10,11,12,13,14,15) Chapter 2 - Part 2 31

32 Another Example Find all prime implicants for: G(A, B,C, D) (02347 (0,2,3,4,7,12,13,14,15) ) Hint: There are seven prime implicants! m Chapter 2 - Part 2 32

33 Five Variable or More K-Maps 5 variables => use two adjacent K-maps. It becomes harder to visualize adjacent minterms for selecting PIs. 6 variables => use 4 K-Maps. V = 0 V = 1 Y Y X X W W Z Z Chapter 2 - Part 2 33

34 Don't Cares (X) in K-Maps Sometimes a function table or map contains entries for which it is known: the input values for the minterm will never occur, or The output value for the minterm is not used => the output value need not be defined Instead, the output value is defined as a don't care ( X ): value 0 or 1. Placing X in the function table or map => the cost (L, G, GN) of the logic circuit may be lowered. Chapter 2 - Part 2 34

35 X in K-Maps: example Example 1: A logic function has: + input: the binary codes for the BCD digits. Only the codes for 0 through 9 are used. The six codes, 1010 through 1111 never occur, => output values for these codes are X to represent don t cares. Chapter 2 - Part 2 35

36 Example: X BCD 5 or More w K-Map: Function F1(w,x,y,z) is defined as "5 or more" over BCD inputs. 1. X used for the 6 non-bcd combinations as 1 F1(wxyz)=w+xz+xy (w,x,y,z) w x z + x y G=7 y X X X X X X z x 2. X are treated as "0s." x,y, z) w x z w x y wx y G = 12 F 2 2( (w, => F1 is lower in cost than F2 For this particular function, cost G for the POS solution for F 1 (w,x,y,z) is not changed by using the don't cares. Placing X in the function table or map => the cost (L, G, GN) of the logic circuit may be lowered Chapter 2 - Part 2 36

37 Optimization Algorithm Find all prime implicants. Include all essential prime implicants in the solution Sl Select a minimum ii cost set of non-essential prime implicants to cover all minterms not yet covered: Obtaining an optimum solution: See Reading Supplement - More on Optimization Obtaining a good simplified solution: Use the Selection Rule Chapter 2 - Part 2 37

38 Prime Implicant Selection Rule Minimize the overlap among prime implicants as much as possible. In particular, in the final solution, make sure that each prime implicant selected includes at least one minterm not included in any other prime implicant selected. Chapter 2 - Part 2 38

39 Selection Rule Example Simplify F(A, B, C, D) given on the K- map. Selected Essential C C A B 1 1 A B D D Minterms covered by essential prime implicants Chapter 2 - Part 2 39

40 Selection Rule Example with Don't Cares Simplify F(A, B, C, D) given on the K-map. C Selected Essential C 1 x 1 x A 1 x x 1 1 x x 1 B x x A 1 1 x 1 1 x B D D Minterms covered by essential prime implicants Chapter 2 - Part 2 40

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