1 / 21

Key Observation

1. 10. 01. 1. 1. 1. 1. Key Observation. Adjacencies in the K-Map. 111. 011. B. BC. 00. 01. 11. 10. A. 010. 110. 0. 000. 010. 100. 001. B. 001. 101. 1. 101. C. 000. C. 100. A. Any two adjacant cells in the K-map differ in exactly one variable.

gaerwn
Download Presentation

Key Observation

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. 1 10 01 1 1 1 1 Key Observation Adjacencies in the K-Map 111 011 B BC 00 01 11 10 A 010 110 0 000 010 100 001 B 001 101 1 101 C 000 C 100 A Any two adjacant cells in the K-map differ in exactly one variable. The uniting theorem can be applied when adjacent cells contain (function =1) to eliminat the changing variable.

  2. Design Examples Two Bit Comparator Truth Table Block Diagram Design Steps: 1- Simplify to reduce cost (three 4-variable K-maps, one for each output) 2- Implement using a suitable design style (e.g. 2-level AND-OR, NAND-NAND, or multilevel techniques)

  3. Design Examples Two Bit Comparator (continued) F1 = A' B' C' D' + A' B C' D + A B C D + A B' C D' F2 = A' B' D + A' C F3 = B C' D' + A C' + A B D' f1 = AC(BD+bd) + ac(bd+BD) = (AC+ac) (bd+BD) = (a c) (b d)

  4. Design Examples Two-Bit Adder Truth Table Block Diagram

  5. A A A AB AB AB 00 01 11 10 00 01 11 10 00 01 11 10 CD CD CD 00 0 0 0 0 00 0 0 1 1 00 0 1 1 0 01 0 0 1 0 01 0 1 0 1 01 1 0 0 1 D D D 11 0 1 1 1 11 1 0 1 0 11 1 0 0 1 C C C 10 0 0 1 1 10 1 1 0 0 10 0 1 1 0 B B B K-map for X K-map for Y K-map for Z Design Example Two-Bit Adder (Continued) X = A C + B C D + A B D Z = B D' + B' D = B xor D Y = A' B' C + A B' C' + A' B C' D + A' B C D' + A B C' D' + A B C D

  6. B B ABC 000 001 011 010 100 101 111 110 DE 00 1 1 01 E 11 1 1 1 1 D 1 1 1 1 10 C C B B ABC 000 001 011 010 100 101 111 110 DE 1 1 1 1 00 01 1 1 1 1 1 1 11 D 1 1 1 1 10 C C 5-Variable K-Maps Constructed from two 4-variable K-Maps. A = 0 A = 1 f = bCDE + Bd E g = ABe + Bce + bE

  7. 5-Variable K-Maps Another View ƒ(A,B,C,D,E) = Sm(2,5,7,8,10,13,15,17,19,21,23,24,29 31) ƒ(A,B,C,D,E) = C E + A B' E + B C' D' E' + A' C' D E'

  8. 6-Variable K-Maps ƒ(A,B,C,D,E,F) = Sm(2,8,10,18,24, 26,34,37,42,45,50, 53,58,61)

  9. 6-Variable K-Maps ƒ(A,B,C,D,E,F) = Sm(2,8,10,18,24, 26,34,37,42,45,50, 53,58,61) = D' E F' + A D E' F + A' C D' F'

  10. CD Definitions • Implicant: Single 1 entry or any group of 1’s that can be combined together in a K-map to form a product term. • (Dual): Single 0 entry or any group of 0’s that can be combined together in a K-map to form a sum term. • Prime Implicant: an implicant that cannot be combined with another • implicant to eliminate a term • Essential Prime Implicant: a prime implicant that contains a 1 entry • not covered by any other implicant. Example C 6 Prime Implicants: 00 01 11 10 AB A' B C’ , C D, A‘ D , B C' D’ , A C , AB D' 00 0 1 1 0 01 1 1 1 0 B essential 11 1 0 1 1 A Minimum cover = A’ D + A C + B C ‘ D ’ 10 0 0 1 1 D

  11. CD Definitions • Implicant: Single 1 entry or any group of 1’s that can be combined together in a K-map to form a product term. • (Dual): Single 0 entry or any group of 0’s that can be combined together in a K-map to form a sum term. • Prime Implicant: an implicant that cannot be combined with another • implicant to eliminate a term • Essential Prime Implicant: a prime implicant that contains a 1 entry • not covered by any other implicant. Example C 00 01 11 10 AB 00 0 1 1 0 01 1 1 1 0 B 11 1 0 1 1 A 10 0 0 1 1 D

  12. Illustrating the Definitions Prime Implicants: B D, C D, A C, B' C essential Essential primes form the minimum cover 5 Prime Implicants: B D, A B C', A C D, A' B C, A' C' D essential Essential implicants form minimum cover

  13. CD Illustrating the Definitions C Prime Implicants: 00 01 11 10 AB A D’, AB, A C, BD 00 0 0 0 0 essential 01 0 1 1 0 B 11 1 1 1 1 Essential primes form the minimum cover A 10 1 0 1 1 D Try the other axis 5 Prime Implicants: B D, A B C', A C D, A' B C, A' C' D essential Essential implicants form minimum cover

  14. Quine-McCluskey (Tabular) Method systematically finds all prime implicants Example: Simplify ƒ(A,B,C,D) = Sm(4,5,6,8,9,10,13) + Sd(0,7,15) Implication Table Column I 0000 0100 1000 0101 0110 1001 1010 0111 1101 1111 Step 1: List minterms and don’t cares using their binary representation, and group according to number of 1’s Principle: Combine terms in adjacent groups which differ in a single variable, and eliminate the changing variable. Remarks: Only terms in adjacent groups have to be compared with one another. Terms in non-adjacent groups differ in more than one variable. Thus can not be combined

  15. Q & M Method ƒ(A,B,C,D) = Sm(4,5,6,8,9,10,13) + Sd(0,7,15) • Step 2: • Compare elements of a group with • k 1's against those with k+1 1's. • If they differ by one bit, eliminate • changing variable and place reduced • term in next column. • E.g., 0000 vs. 0100 yields 0-00 • 0000 vs. 1000 yields -000 Implication Table Column I Column II 0000 ¦ 0-00 -000 0100 ¦ 1000 ¦ 010- 01-0 0101 ¦ 100- 0110 ¦ 10-0 1001 ¦ 1010 ¦ 01-1 -101 0111 ¦ 011- 1101 ¦ 1-01 1111 ¦ -111 11-1 When a term is used in a combination, mark that term with a check. If cannot be combined, mark term with a star. These are the prime implicants.

  16. Q & M Method • Step 2 Continues: • Compare elements of a group in • Col. II with • k 1's against those with k+1 1's. • If they differ by one bit, eliminate • changing variable and place reduced • term in next column. • E.g., 010- vs. 011- yields 01- - • -101 vs. -111 yields -1-1 Implication Table Column I Column II Column III 0000 ¦ 0-00 * 01-- * -000 * 0100 ¦ -1-1 * 1000 ¦ 010- ¦ 01-0 ¦ 0101 ¦ 100- * 0110 ¦ 10-0 * 1001 ¦ 1010 ¦ 01-1 ¦ -101 ¦ 0111 ¦ 011- ¦ 1101 ¦ 1-01 * 1111 ¦ -111 ¦ 11-1 ¦

  17. Q & M - Prime Implicant Chart x x x x x x 5,7,13,15 (-1-1) 4,5,6,7(01--) 9,13(1-01) 8,10(10-0) 8,9(10-0) 0,8(-000) 0,4(0-00) 5,7,13,15 (-1-1) 4,5,6,7(01--) 9,13(1-01) 8,10(10-0) 8,9(10-0) 0,8(-000) 0,4(0-00) x x x x x x x x x x x x x x x x x x x x 4 5 6 8 9 10 13 4 5 6 8 9 10 13 rows = prime implicants columns = minterms place an "X" if minterm is covered by the prime implicant If column has a single X, than the implicant associated with that row is essential. It must appear in minimum cover

  18. Description T ruth T able If X = 0 then X ' = 1 X X T rue If X = 1 then X ' = 0 0 1 1 0 Description T ruth T able Z = 1 if X or Y X Y Z (or both) are 1 0 0 0 X + Y 0 1 1 T rue 1 0 1 1 1 1 X Y Gate Logic • Review Switches NOT X False X Description Truth Table Switches Z = 1 if X and Y X Y Z false are both 1 0 0 0 AND X • Y 0 1 0 1 0 0 true 1 1 1 X Y Switches False OR

  19. c s e d w a d e’ s 1 a c’ w Example : Logic circuit & its Switch realization s = a (d+w)(c’+ e’) Gate realization Switch realization

  20. Description T ruth T able Z = 1 if X is 0 T rue X Y Z or Y is 0 0 0 1 X • Y 0 1 1 1 0 1 1 1 0 X Y Logic Functions: NAND Switches NAND False Description Truth Table Switches Z = 1 if X and Y X Y Z false AND are both 1 0 0 0 X • Y 0 1 0 1 0 0 true 1 1 1 X Y Note that switch circuit of the NAND gate is the complement of the switch circuit for the AND gate.

  21. Description T ruth T able Z = 1 if both X X Y Z T rue and Y are 0 0 0 1 0 1 0 X + Y 1 0 0 1 1 0 X Y Description T ruth T able Z = 1 if X or Y X Y Z (or both) are 1 0 0 0 X + Y 0 1 1 T rue 1 0 1 1 1 1 X Y Logic Functions: NAND, NOR, XOR, XNOR Switches NOR False Switches False OR Note that switch circuit of the NOR gate is the complement of the switch circuit for the OR gate.

More Related