1 / 26

Chapter 2 Minimization of Switching Functions

Chapter 2 Minimization of Switching Functions. 2.1 Boolean Algebra. Table 1 Boolean Operators for Variables x 1 and x 2. Table 2 Truth Table for AND, OR, NOT, Exclusive-OR, and Exclusive-NOR Operations. Axiom 1: Boolean set definition

piper
Download Presentation

Chapter 2 Minimization of Switching Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 2 Minimization of Switching Functions

  2. 2.1 Boolean Algebra Table 1 Boolean Operators for Variables x1 and x2 Table 2 Truth Table for AND, OR, NOT, Exclusive-OR, and Exclusive-NOR Operations

  3. Axiom 1: Boolean set definition • The set B contains at least two elements x1 and x2, where x1≠x2. • Axiom 2: Closure laws • For every x1,x2 ∈ B • (a) x1 + x2∈ B • (b) x1 • x2∈ B • Axiom 3: Identity laws • For every x1 ∈ B • (a) x1 + 0 = 0 + x1 = x1 • (b) x1 • 1 = 1 • x1 = x1

  4. Axiom 4: Commutative laws • For every x1,x2∈ B • (a) x1 + x2 = x2 + x1 • (b) x1 • x2 = x2 • x1 • Axiom 5: Associative laws • For every x1, x2, x3∈ B • (a) (x1 + x2) + x3 = x1 + (x2 + x3) • (b) (x1 • x2) • x3 = x1 • (x2 • x3) • Axiom 6: Distributive laws • For every x1, x2, x3∈ B • (a) x1 + (x2 • x3) = (x1 + x2) • (x1 + x3) • (b) x1 • (x2 + x3) = (x1 • x2) + (x1 • x3)

  5. Axiom 7: Complementation laws • For every x1 , x1' ∈ B • (a) x1 + x1' =1 • (b) x1 • x1' = 0 • Theorem 1: 0 and 1 associated with a variable • (a) x1 + 1 =1 • (b) x1 • 0 = 0 • Theorem 2: 0 and 1 complement • (a) 0' = 1 • (b) 1' = 0 • Theorem 3: Idempotent laws • (a) x1 + x1 = x1 • (b) x1 • x1 = x1

  6. Theorem 4: Involution law • x1''= x1 • Theorem 5: Absorption law 1 • For every x1, x2∈ B • (a) x1 + (x1 • x2) = x1 (Does not imply x1 • x2 = 0 ) • (b) x1 • (x1 + x2) = x1 • Theorem 6: Absorption law 2 • (a) x1 + (x1' • x2) = x1 + x2 • (b) x1 • (x1' + x2) = x1 • x2 • Theorem 7: DeMorgan’s laws • (a) (x1 + x2) ' = x1' • x2' • (b) (x1 • x2) ' = x1' + x2'

  7. Minterm • A minterm is the Boolean product of n variables and contains all n variables of the function exactly once, either true or complemented. • Example: x1x2'x3 • Maxterm • A maxterm is the Boolean sum of n variables and contains all n variables of the function exactly once, either true or complemented. • Example: x1 + x2' + x3 • Product term • A product term is the Boolean product of variables containing a subset of the possible variables or their complements. • Example: x1'x3(does not contain all the variables)

  8. Sum term • A sum term is the Boolean sum of variables containing a subset of the possible variables or their complements. • Example: x1'+x3 (does not contain all the variables) • Sum of minterms • A sum of minterms is an expression in which each term contains all the variables, either true or complemented. • Example: z1(x1,x2,x3) = x1'x2x3+ x1x2'x3' + x1x2x3 • Sum of products • A sum of products is an expression in which at least one term does not contain all the variables; that is, at least one term is a proper subset of the possible variables or their complements. • Example: z1(x1,x2,x3) = x1'x2x3+ x2'x3' + x1x2x3

  9. Product of maxterms • A product of maxterms is an expression in which each term contains all the variables, either true or complemented. • Example: z1(x1,x2,x3) = (x1+x2+x3) (x1+x2'+x3') (x1+x2+x3) • Product of sums • A product of sums is an expression in which at least one term does not contain all the variables; that is, at least one term is a proper subset of the possible variables or their complements. • Example: z1(x1,x2,x3) = (x1'+x2+x3) (x2'+x3') (x1+x2+x3)

  10. Table 3 Summary of Boolean Algebra Axioms and Theorems

  11. 2.2 Algebraic Minimization • The number of terms and variables that are necessary to generate a Boolean function can be minimized by algebraic manipulation. • Example x1x2 + x1'x3 + x2x3 = x1x2 + x1'x3 + x2x3 = x1x2 + x1'x3 + x2x3(x1 + x1')Complementation law = x1x2 + x1'x3 + x1x2x3 + x1'x2x3Distributive law = x1x2 + x1x2x3 + x1'x3 + x1'x2x3Commutative law = x1x2(1 + x3) + x1'x3(1 + x2) Distributive law = (x1x2 • 1) + (x1'x3 • 1) Theorem 1 = x1x2 + x1'x3Identity law

  12. 2.3 Karnaugh Maps • The Karnaugh map provides a simplified method for minimizing Boolean functions. • A Karnaugh map provides a geometrical representation of a Boolean function. • The Karnaugh map is arranged as an array of squares (or cells) in which each square represents a binary value of the input variables.

  13. (a) (b) (c) Figure 1 Two-variable Karnaugh map: (a) truth table; (b) minterm placement; and (c) minterms

  14. (a) (b) (c) Figure 2 Karnaugh maps showing minterm locations: (a) two variables; (b) three variables; (c) four variables.

  15. (d) Figure 2 Karnaugh maps showing minterm locations: (d) five variables.

  16. (e) Figure 2 Karnaugh maps showing minterm locations: (e) alternative map for five variables.

  17. Example • The following function will be minimized using a 4-variable Karnaugh map: • z1(x1,x2,x3, x4) = x2x3' + x2x3x4' + x1x2'x3 + x1x3x4' • Karnaugh map representation of the function • z1(x1,x2,x3, x4) = x2x3' + x2x3x4' + x1x2'x3 + x1x3x4' • z1(x1,x2,x3, x4) = x2x3' + x2x4' + x1x2'x3 z1

  18. 2.3.1 Map-Entered Variables • Variables may also be entered in a Karnaugh map as map-entered variables, together with 1s and 0s. • A map of this type is more compact than a standard Karnaugh map, but contains the same information. • A map containing map-entered variables is particularly useful in analyzing and synthesizing synchronous sequential machines. • When variables are entered in a Karnaugh map, two or more squares can be combined only if the squares are adjacent and contain the same variable(s).

  19. Example • The following Boolean equation will be minimized using a 3-variable Karnaugh map with x4 as a map-entered variable: z1(x1,x2,x3, x4) = x1x2'x3 x4' + x1x2 + x1'x2'x3'x4' + x1'x2'x3'x4 z1(x1,x2,x3, x4) = x1x2'x3(x4') + x1x2 + x1'x2'x3'(x4') + x1'x2'x3'(x4) z1

  20. Location 0 : x4' + x4 = 1 • Location 7 : 1 = 1 + x4' • Minterm 0 equates to x1'x2'x3'. • Minterm locations 5 and 7 to be combined as x1x3x4'. • Minterms 6 and 7 combine to yield the term x1x2. • The minimized equation for z1 is : z1(x1,x2,x3,x4) = x1x3x4' + x1x2 + x1'x2'x3' z1

  21. 2.4 Quine-McCluskey Algorithm • The Quine-McCluskey algorithm is a tabular method of obtaining a minimal set of prime implicants that represents the Boolean function. • The method consists of two steps: • first obtain a set of prime implicants for the function; • then obtain a minimal set of prime implicants that represents the function. • The rationale for the Quine-McCluskey method relies on the repeated application of the distributive and complementation laws.

  22. The following function will be minimized using the Quine-McCluskey method: Table 4 Minterms Listed in Groups

  23. Figure 3 essential and nonessential prime implicants Figure 4 secondary essential prime implicants with minimal cover for remaining prime implicants

  24. f(x1,x2,x3,x4) = x2x3x4' + x2'x3' + x1'x3x4 z1 Figure 5 Karnaugh map for Example

  25. 2.4.1 Petrick Algorithm • The function may not always contain an essential prime implicant, or the secondary essential prime implicants may not be intuitively obvious. • The technique for obtaining a minimal cover of secondary prime implicants is called the Petrick algorithm

  26. Example Function is covered = (pi1 + pi2)(pi1 + pi3)(pi2 + pi4)(pi2 + pi3)(pi1 + pi4) All minterms are covered = pi1pi2 + pi2pi3pi4 + pi1pi3pi4

More Related