Multivariable K Maps, CSE 140: Components and Design Techniques for - PowerPoint PPT Presentation

Multivariable K Maps, CSE 140: Components and Design Techniques for Digital Systems Diba Mirza Dept. of Computer Science and Engineering University of California, San Diego 1

Another 3-Input Corresponding K-map truth table Id a b c f (a,b,c) 0 0 0 0 0 (0,0) (0,1) (1,1) (1,0) 1 0 0 1 0 0 2 6 4 c = 0 2 0 1 0 1 0 1 X 1 3 0 1 1 0 4 1 0 0 1 1 3 7 5 c = 1 0 0 1 1 5 1 0 1 1 6 1 1 0 X 7 1 1 1 1 2

Another 3-Input Corresponding K-map truth table b = 1 Id a b c f (a,b,c) 0 0 0 0 0 (0,0) (0,1) (1,1) (1,0) 1 0 0 1 0 0 2 6 4 c = 0 2 0 1 0 1 0 1 X 1 3 0 1 1 0 4 1 0 0 1 1 3 7 5 c = 1 0 0 1 1 5 1 0 1 1 6 1 1 0 X 7 1 1 1 1 a = 1 f(a,b,c) = a + bc ’ 3

Proof of Consensus Theorem using K Maps Consensus Theorem: A ’ B+AC+BC=A ’ B+AC A B C Y Y Y AB AB 0 0 0 0 00 01 11 10 00 01 11 10 C 0 0 1 0 C 0 1 0 1 0 0 1 0 0 0 ABC ABC ABC ABC 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 ABC ABC ABC ABC 1 1 0 0 1 1 1 1 4

Procedure for finding the minimal function via K-maps (formal terms) Y AB CD 00 01 11 10 1. Convert truth table to K-map 00 1 0 0 1 2. Include all essential primes 01 0 1 0 1 3. Include non essential primes as needed to completely cover the on-set 11 1 1 0 0 (all cells of value one) 10 1 1 0 1 5

Definition: Implicant • Implicant: A product term obtained by combining min terms for which the value of the function is either 1 or don’t care. In other words a group of adjacent ‘1’ or ‘X’ cells Y AB CD 00 01 11 10 Q: Is this an implicant? 00 1 0 0 1 A. Yes 01 0 1 0 1 B. No 11 1 1 0 0 10 1 1 0 1 6

Definition: Implicant • Implicant: A product term obtained by combining min terms for which the value of the function is either 1 or don’t care. In other words a group of adjacent ‘1’ or ‘X’ cells Y AB CD 00 01 11 10 Q: Is this an implicant? 00 1 0 0 1 A. Yes 01 0 1 0 1 B. No 11 1 1 0 0 10 1 1 0 1 7

Definition: Prime Implicant • Implicant: A product term obtained by combining min terms for which the value of the function is either 1 or don’t care. In other words a group of adjacent ‘1’ or ‘X’ cells • Prime Implicant: An implicant that cannot be fully covered by a larger implicant. Y Q: Is this a prime implicant? AB CD 00 01 11 10 00 1 0 0 1 A. Yes 01 0 1 0 1 B. No 11 1 1 0 0 10 1 1 0 1 8

Definition: Prime Implicant • Implicant: A product term obtained by combining min terms for which the value of the function is either 1 or don’t care. In other words a group of adjacent ‘1’ or ‘X’ cells • Prime Implicant: An implicant that cannot be fully covered by a larger implicant. Q: Is this a prime implicant? Y AB CD 00 01 11 10 00 1 0 0 1 A. Yes 01 0 1 0 1 B. No 11 1 1 0 0 10 1 1 0 1 9

Definition: Essential Prime 1. Prime Implicant: A group of adjacent ones that cannot be fully covered by any other large group of ones 2. Essential Prime Implicant or Essential Prime: Prime implicants that contain at least one element (a ‘1’ cell) that cannot be covered by any other prime implicant Y AB CD 00 01 11 10 00 1 0 0 1 01 0 1 0 1 11 1 1 0 0 10 1 1 0 1 10

Definition: Essential Prime 1. Prime Implicant: A group of adjacent ones that cannot be fully covered by any other large group of ones 2. Essential Prime Implicant or Essential Prime: Prime implicants that contain at least one element (a ‘1’ cell) that cannot be covered by any other prime implicant Q: Are the corner four essential Y AB CD 00 01 11 10 prime implicants? 00 1 0 0 1 A. Yes 01 0 1 0 1 B. No 11 1 1 0 0 10 1 1 0 1 11

Include all essential prime implicants! Y AB 00 01 11 10 CD 00 1 0 0 1 01 0 1 0 1 11 1 1 0 0 10 1 1 0 1 Y = AC + ABD + ABC + BD 12

Reading the reduced K-map Y AB 00 01 11 10 CD 00 1 0 0 1 01 0 1 0 1 11 1 1 0 0 10 1 1 0 1 Y = AC + ABD + ABC + BD 13

Definition: Non-Essential Prime Non Essential Prime Implicant : Prime implicant that has no element that cannot be covered by other prime implicant Q: Is the blue group a non-essential prime implicant? ab 00 01 11 10 cd A. Yes 1 1 1 00 B. No 1 1 01 11 1 1 1 1 10 1 14

Definition: Non-Essential Prime Non Essential Prime Implicant : Prime implicant that has no element that cannot be covered by other prime implicant Q: Is the blue group a non-essential prime implicant? ab 00 01 11 10 cd A. Yes 1 1 1 00 B. No 1 1 01 11 1 1 1 1 10 1 15

Procedure for finding the minimal function via K-maps (formal terms) Y AB CD 00 01 11 10 1. Convert truth table to K-map 00 1 0 0 1 2. Include all essential primes 01 0 1 0 1 3. Include non essential primes as needed to completely cover the on-set 11 1 1 0 0 (all cells of value one) 10 1 1 0 1 16

K-maps with Don ’ t Cares A B C D Y Y 0 0 0 0 1 0 0 0 1 0 AB 00 01 11 10 CD 0 0 1 0 1 0 0 1 1 1 00 1 0 X 1 0 1 0 0 0 0 1 0 1 X 0 1 1 0 1 01 0 X X 1 0 1 1 1 1 1 0 0 0 1 1 0 0 1 1 11 1 1 X X 1 0 1 0 X 1 0 1 1 X 1 1 0 0 X 10 1 1 X X 1 1 0 1 X 1 1 1 0 X 1 1 1 1 X 17

K-maps with Don ’ t Cares Y AB A B C D Y 00 01 11 10 CD 0 0 0 0 1 0 0 0 1 0 00 1 0 X 1 0 0 1 0 1 0 0 1 1 1 0 1 0 0 0 01 0 X X 1 0 1 0 1 X 0 1 1 0 1 0 1 1 1 1 11 1 1 X X 1 0 0 0 1 1 0 0 1 1 1 0 1 0 X 10 1 1 X X 1 0 1 1 X 1 1 0 0 X 1 1 0 1 X Y = A + BD + C 1 1 1 0 X 1 1 1 1 X 18

Reducing Canonical expressions Given f(a,b,c,d) = Σm (0, 1, 2, 8, 14)+Σd (9, 10) 1. Draw K-map ab 11 00 01 10 cd 00 01 11 10 19

Reducing Canonical Expressions Given f(a,b,c,d) = Σm (0, 1, 2, 8, 14)+ Σd (9, 10) 1. Draw K-map ab 11 00 01 10 cd 0 4 12 8 00 1 5 13 9 01 3 7 15 11 11 2 6 14 10 10 20

Reducing Canonical Expressions Given f(a,b,c,d) = Σm (0, 1, 2, 8, 14)+Σd (9, 10) 1. Draw K-map ab 11 00 01 10 cd 0 4 12 8 1 0 0 1 00 1 5 13 9 01 1 0 0 X 3 7 15 11 11 0 0 0 0 2 6 14 10 10 1 0 1 X 21

Reducing Canonical Expressions 1. Draw K-map Are all the Prime implicants 2. Identify Prime implicants also essential? A. Yes B. No ab 11 00 01 10 cd 0 4 12 8 1 0 0 1 00 PI Q: Do the E-primes cover the entire on set? 1 5 13 9 01 1 0 0 X A. Yes B. No 3 7 15 11 11 0 0 0 0 2 6 14 10 10 1 0 1 X 22

Reducing Canonical Expressions Prime implicants: Σm (0, 1, 8, 9), Σm (0, 2, 8, 10), Σm (10, 14) 1. Essential Primes: Σm (0, 1, 8, 9), Σm (0, 2, 8, 10), Σm (10, 14) 2. ab 11 00 01 10 cd 0 4 12 8 1 0 0 1 00 1 5 13 9 01 1 0 0 X 3 7 15 11 11 0 0 0 0 2 6 14 10 10 1 0 1 X 23