1 / 47

ELEC 2200-002 Digital Logic Circuits Fall 2008 Logic Minimization (Chapter 3)

ELEC 2200-002 Digital Logic Circuits Fall 2008 Logic Minimization (Chapter 3). Vishwani D. Agrawal James J. Danaher Professor Department of Electrical and Computer Engineering Auburn University, Auburn, AL 36849 http://www.eng.auburn.edu/~vagrawal vagrawal@eng.auburn.edu.

lacy
Download Presentation

ELEC 2200-002 Digital Logic Circuits Fall 2008 Logic Minimization (Chapter 3)

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. ELEC 2200-002Digital Logic CircuitsFall 2008Logic Minimization (Chapter 3) Vishwani D. Agrawal James J. Danaher Professor Department of Electrical and Computer Engineering Auburn University, Auburn, AL 36849 http://www.eng.auburn.edu/~vagrawal vagrawal@eng.auburn.edu ELEC2200-002 Lecture 5

  2. Understanding Minimization • Logic function: a AND b NOT AND OR F AND c AND d ELEC2200-002 Lecture 5

  3. Understanding Minimization • Reducing products: Distributivity Complementation Identity Complementation Distribitivity Consensus theorem Distributivity Complement, identity ELEC2200-002 Lecture 5

  4. Understanding Minimization • Reduced SOP: b NOT OR F AND d ELEC2200-002 Lecture 5

  5. Understanding Minimization b d • Reducing literals: bd Absorption theorem b NOT OR F d ELEC2200-002 Lecture 5

  6. Logic Minimization • Generally means • In SOP form: • Minimize number of products (reduce gates) and • Minimize literals (reduce gate inputs) • In POS form: • Minimize number of sums (reduce gates) and • Minimize literals (reduce gate inputs) ELEC2200-002 Lecture 5

  7. Product or Implicant or Cube • Any set of literals ANDed together. • Minterm is a special case where all variables are present. It is the largest product. • A minterm is also called a 0-implicant of 0-cube. • A 1-implicant or 1-cube is a product with one variable eliminated: • Obtained by combining two adjacent 0-cubes • ABCD + ABCD = ABC(D+D) = ABC ELEC2200-002 Lecture 5

  8. Cubes (Implicants) of 4 Variables A Minterm or 0-implicant or 0-cube AB CD 1 1 What is this? 1 C 1 D 1 1-implicant or 1-cube ABC B ELEC2200-002 Lecture 5

  9. Growing Cubes, Reducing Products 1-implicant or 1-cube AB C A 2-implicant or 2-cube BC 1 1 C 1 1 D 1-implicant or 1-cube ABC B What is this? ELEC2200-002 Lecture 5

  10. Largest Cubes or Smallest Products A 1 1 1 1 What is this? C 1 1 D 1 1 3-implicant or 3-cube B B ELEC2200-002 Lecture 5

  11. Implication and Covering • A larger cube covers a smaller cube if all minterms of the smaller cube are included in the larger cube. • A smaller cube implies (or subsumes) a larger cube if all minterms of the smaller cube are included in the larger cube. ELEC2200-002 Lecture 5

  12. Implicants of a Function • Minterms, products, cubes that imply the function. A 1 1 1 1 C 1 1 1 D 1 B ELEC2200-002 Lecture 5

  13. Prime Implicant (PI) • A cube or implicant of a function that cannot grow larger by expanding into other cubes. A A PI 1 1 1 1 1 1 1 1 1 1 1 1 C C 1 1 1 1 1 1 1 1 D D 1 1 1 1 B B ELEC2200-002 Lecture 5

  14. Essential Prime Implicant (EPI) • If among the minterms subsuming a prime implicant (PI), there is at least one minterm that is covered by this and only this PI, then the PI is called an essential prime implicant (EPI). • Also called essential prime cube (EPC). A EPI C Why not this? B ELEC2200-002 Lecture 5

  15. Redundant Prime Implicant (RPI) • If each minterm subsuming a prime implicant (PI) is also covered by other essential prime implicants, then that PI is called a redundant prime implicant (RPI). • Also called redundant prime cube (RPC). A EPI C RPI B ELEC2200-002 Lecture 5

  16. Selective Prime Implicant (SPI) • A prime implicant (PI) that is neither EPI nor RPI is called a selective prime implicant (SPI). • Also called selective prime cube (SPC). • SPIs occur in pairs. A EPI C B SPI ELEC2200-002 Lecture 5

  17. Minimum Sum of Products (MSOP) • Identify all prime implicants (PI) by letting minterms and implicants to grow. • Construct MSOP with PI only : • Cover all minterms • Use only essential prime implicants (EPI) • Use no redundant prime implicant (RfroPI) • Use cheaper selective prime implicants (SPI) • A good heuristic – Choose EPI in ascending order, starting from 0-implicant, then 1-implicant, 2-implicant, . . . ELEC2200-002 Lecture 5

  18. Example: F=m(1,3,4,5,8,9,13,15) A MSOP: F = AB D +A BC + A B D + ABC D C B RPI ELEC2200-002 Lecture 5

  19. Example: F=m(1,3,5,7,8,10,12,13,14) SPI A MSOP: F = A D + A D + A BC D C B ELEC2200-002 Lecture 5

  20. Functions with Don’t Care Minterms • F(A,B,C) = m(0,3,7) + d(4,5) • Include don’t care minterms when beneficial. A A C C B B F = B C +ABC F = B C +BC ELEC2200-002 Lecture 5

  21. Five-Variable Functions • F(A,B,C,D,E) = m(0,1,4,5,6,13,14,15,22,24,25,28,29,30,31) A = 0 B A = 1 B E E D D C C F = ABD + A BD+ B C E + C DE ELEC2200-002 Lecture 5

  22. Multiple-Output Minimization ELEC2200-002 Lecture 5

  23. Individual Output Minimization Need five products. F1 A F2 A D D C C B B ELEC2200-002 Lecture 5

  24. Global Minimization Need four products. F1 A F2 A D D C C B B ELEC2200-002 Lecture 5

  25. Minimized SOP and POS • F(A,B,C,D) =  m(1,3,4,7,11) + d(5,12,13,14,15) =  M(0,2,6,8,9,10) D(5,12,13,14,15) A A D D C C B B F = BD + CD + AC F = (B + D)(C + D)(A + C) F = BC +AD + C D ELEC2200-002 Lecture 5

  26. SOP and POS Circuits • F(A,B,C,D) =  m(1,3,4,7,11)+d(5,12,13,14,15) =  M(0,2,6,8,9,10) D(5,12,13,14,15) F = BC +AD + C D F = (B + D)(C + D)(A + C) A C B A C F F D D B Are two circuits functionally Identical? ELEC2200-002 Lecture 5

  27. Absorption Theorem • For two Boolean variables: A, B • A + A B = A • Proof: A B AB ELEC2200-002 Lecture 5

  28. Consensus Theorem • For three Boolean variables: A, B, C • A B +A C + B C = A B +A C • Proof: BC A B AB AC C ELEC2200-002 Lecture 5

  29. Growing Implicants to PI • F = AB +CD +ABCD initial implicants = AB +ABCD + BCD+CD consensus th. = AB + BCD +CD absorption th. = AB + BCD +CD + BD consensus th. = AB +CD + BD absorption th. A A A D D D C C C B B B ELEC2200-002 Lecture 5

  30. Identifying EPI • Find all prime implicants. • From primary implicant SOP, remove a PI. • Apply consensus theorem to the remaining SOP. • If the removed PI is generated, then it is either an RPI or an SPI. • If the removed PI is not generated, then it is an EPI ELEC2200-002 Lecture 5

  31. Example • PI SOP: F = A D +AC +CD • Is AD an EPI? F – {AD} = AC +CD, no new PI can be generated Hence, AD is an EPI. A D C AD B ELEC2200-002 Lecture 5

  32. Example (Cont.) • PI SOP: F = A D +AC +CD • Is CD an EPI? F – {CD } = A D +AC = A D +AC +C D (Consensus th.) HenceC D is not an EPI (it is an RPI) A D C B CD ELEC2200-002 Lecture 5

  33. Finding MSOP • Start with minterm or cube SOP representation of Boolean function. • Find all prime implicants (PI). • Include all EPI’s in MSOP. • Find the set of uncovered minterms, {UC}. • MSOP is minimum if {UC} is empty. DONE. • For a minterm in {UC}, include the largest PI from remaining PI’s (non-EPI’s) in MSOP. • Go to step 4. ELEC2200-002 Lecture 5

  34. Finding Uncovered Minterms, {UC} • {UC} = ({PI} # {PMSOP}) # {DC} • Where: • {PI} is set of all prime implicants of the function. • {PMSOP} is any partial SOP. • {DC} is set of don’t care minterms. • Sharp (#) operation between Boolean expressions X and Y, X # Y, is the set of minterms covered by X that are not covered by Y. ELEC2200-002 Lecture 5

  35. Example: # Operation • AD #CD = {A B C D, AB C D} • CD # AD = {A BC D,A BC D} A D C AD B CD ELEC2200-002 Lecture 5

  36. Minterms Covered by a Product • A product from which k variables have been eliminated, covers 2kminterms. • Example: For four variables, A, B, C, D Product AC covers 22 = 4 minterms: • AB CD • AB C D • A B CD • A B C D Obtained by inserting the eliminated variables in all possible ways. A D C B ELEC2200-002 Lecture 5

  37. Quine-McCluskey Willard V. O. Quine 1908 – 2000 Edward J. McCluskey b. 1929 Agrawal: ELEC2200-002 Lecture 5

  38. Quine-McCluskey Tabular Minimization Method • W. V. Quine, “The Problem of Simplifying Truth Functions,” American Mathematical Monthly, vol. 59, no. 10, pp. 521-531, October 1952. • E. J. McCluskey, “Minimization of Boolean Functions,” Bell System Technical Journal, vol. 35, no. 11, pp. 1417-1444, November 1956. • Textbook, Section 3.9, pp. 211-225. ELEC2200-002 Lecture 5

  39. Q-M Tabular Minimization • Minimizes functions with many variables. • Begin with minterms: • Step 1: Tabulate minterms in groups of increasing number of true variables. • Step 2: Conduct linear searches to identify all prime implicants (PI). • Step 3: Tabulate PI’s vs. minterms to identify EPI’s. • Step 4: Tabulate non-essential PI’s vs. minterms not covered by EPI’s. Select minimum number of PI’s to cover all minterms. ELEC2200-002 Lecture 5

  40. F(A,B,C,D) =  m(2,4,6,8,9,10,12,13,15) • Q-M Step 1: Group minterms with 1 true variable, 2 true variables, etc. ELEC2200-002 Lecture 5

  41. Q-M Step 2 • Find all implicants by combining minterms, and then products that differ in one variable: For example, • 2 and 6, orAB CD and A B CD → A CD, written as 0 – 1 0. • Try combining a minterm (or product) with all minterms (or products) listed below. • Include resulting products in the next list. • If minterm (or product) does not combine with any other, mark it as PI. • Check the minterm (or product) and repeat for all other minterms (or products). ELEC2200-002 Lecture 5

  42. Step 2 Executed on Example ELEC2200-002 Lecture 5

  43. Step 3 Identify EPI ELEC2200-002 Lecture 5

  44. Step 4 Covering Remaining Minterms Integer linear program (ILP), available from Matlab and other sources: Define integer {0,1} variables, xk = 1, select PIk; xk = 0, do not select PIk. Minimize k xk, subject to following constraints: x2 + x3 ≥ 1 x4 + x5 ≥ 1 x2 + x4 ≥ 1 x3 + x6 ≥ 1 A solution is x3 = x4 = 1, x2 = x5 = x6 = 0 ELEC2200-002 Lecture 5

  45. Linear Programming (LP) • A mathematical optimization method for problems where some “cost” depends on a large number of input variables. • An easy to understand introduction is: • S. I. Gass, An Illustrated Guide to Linear Programming, New York: Dover Publications, 1970. • Very useful tool for a variety of engineering design problems. • Available in software packages like Matlab. • Courses available in Math, Business and Engineering departments. ELEC2200-002 Lecture 5

  46. Q-M Solution and Verification • F(A,B,C,D) = PI1 + PI3 + PI4 + PI7 = 1-0- + -010 + 01-0 + 11-1 = AC +B CD +A BD + A B D • See Karnaugh map. A D C B ELEC2200-002 Lecture 5

  47. Further Reading • Incompletely specified functions: See Example 3.25, pages 218-220. • Multiple outputs: See Example 3.26, pages 220-222. ELEC2200-002 Lecture 5

More Related