hapter 5 Karnaugh Map and Minimization Procedures
Lesson 1 KARNAUGH MAP h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 2
Outline Three variable Karnaugh map Four variable Karnaugh map Five/Six Variable Karnaugh Map h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 3
Map from 3 variables Truth table or SOP form Boolean Expression A two-dimensional map built from a truth table or 3 variables SOP form Boolean Expression Since number of rows in three variable (three inputs) truth table are 8, the map has 8 cells Two cells horizontal and four cells vertical. [It can also be vice versa.] h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 4
Map for F = A.B. + A.B. AB AB 00 0 1 AB 01 1 AB 11 AB 10 1 Σ m(3, 4) h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 5
Map for F = A.B. + A.B. +A.B. + A.B. + A.B. AB 0 1 AB 00 1 AB 01 1 AB 11 1 Σ m(0, 3, 4, 5, 7 ) AB 10 1 1 h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 6
Filling the cell with 1s 1 When output is 1 for a given combination of A, B and, we place 1 at the corresponding cell. 2 omplete the step 1 for all the rows of truth table with outputs = 1. h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 7
orresponding miniterms of the cells AB AB 00 0 m0 1 m1 AB 01 m2 m3 AB 11 m6 m7 AB 10 m4 m5 h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 8
Map for F = (A+ B + ). (A+ B+ ) AB 0 1 A+B 00 0 A+B 01 A+B 11 0 A+B 10 Π M(1, 6) h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 9
Filling the cell with 0s from POS form 1 When output is 0 for a given combination of A, B and, we place 0 at the corresponding cell. 2 omplete the step 1 for all 8 rows of truth table with outputs = 0. h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 10
orresponding Maxterms of the cells AB A+B 00 0 M0 1 M1 A+B 01 M2 M3 A+B 11 M6 M7 A+B 10 M4 M5 h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 11
Outline Three variable Karnaugh map Four variable Karnaugh map Five/Six Variable Karnaugh Map h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 12
Map from 4 variables Truth table or SOP form Boolean Expression A two-dimensional map built from a truth table or 4 variables SOP form Boolean Expression Since number of rows in a four variable (three inputs) truth table are 16, the map has 16 cells Four cells horizontal and four cells vertical. h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 13
Map for F = A.B..D + A.B..D + A.B..D Σ m(3, 6, 8) AB D D D D 00 01 11 10 AB 00 1 AB 01 1 AB 11 AB 10 1 h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 14
Map for F = A.B..D + A.B..D + A.B..D Σ m(4, 8, 10) AB AB 00 AB 01 1 D D D D 00 01 11 10 AB 11 AB 10 1 1 h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 15
Filling the cell with 1s 1. When output is 1 for a given combination of A, B, and D, we place 1 at the corresponding cell. 2. omplete the step 1 for all 16 rows of truth table with outputs = 1. h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 16
orresponding Miniterms of the cells D D D D AB 00 01 11 10 AB 00 m0 m1 m3 m2 AB 01 m4 m5 m7 m6 AB 11 m12 m13 AB 10 m8 m9 m15 m11 m14 m10 h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 17
Filling the cell with 0s from POS form 1 When output is 0 for a given combination of A, B and, we place 0 at the corresponding cell. 2 omplete the step 1 for all the rows of truth table with outputs = 0. h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 18
Map for F = (A+B++D). (A+B++D) AB +D 00 A+B 00 0 A+B 01 A+B 11 +D 01 Π M(0, 10) +D 11 +D 10 A+B 10 0 h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 19
orresponding Maxterms of the cells +D +D +D +D AB 00 01 11 10 A+B 00 M0 M1 M3 M2 A+B 01 M4 M5 M7 M6 A+B 11 M12 M13 M15 M14 A+B 10 M8 M9 M11 M10 h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 20
Outline Three variable Karnaugh map Four variable Karnaugh map Five/Six Variable Karnaugh Map h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 21
Five Variable Map - Left part A = 0 DE DE DE DE B 00 01 11 10 B 00 m0 m1 m3 m2 B 01 m4 m5 m7 m6 B 11 m12 m13 B 10 m8 m9 m15 m11 m14 m10 h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 22
Five Variable Map - Left part A = 1 DE DE DE DE B 00 01 11 10 B 00 m16 m17 m19 m18 B 01 m20 m21 m23 m22 B 11 m28 m29 B 10 m24 m25 m31 m27 m30 m26 h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 23
Six Variable Map - Left Top part A, B = 0, 0 D EF EF EF EF 00 01 11 10 D 00 m0 m1 m3 m2 D 01 m4 m5 m7 m6 D 11 m12 m13 D 10 m8 m9 m15 m11 m14 m10 h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 24
Six Variable Map - Right Top part A, B = 0, 1 D EF EF EF EF 00 01 11 10 D 00 m16 m17 m19 m18 D 01 m20 m21 m23 m22 D 11 m28 m29 D 10 m24 m25 m31 m27 m30 m26 h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 25
Six Variable Map - Left Bottom part A, B = 1, 1 D D 00 D 01 EF EF EF EF 00 01 11 10 D 11 m63 m62 D 10 m56 m57 m59 m58 h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 26
Six Variable Map - Right Bottom part A, B = 1, 0 D EF EF EF EF 00 01 11 10 D 00 D 01 D 11 D 10 m32 m40 m34 m42 h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 27
Summary h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 28
Karnaugh Map has cells. On moving from one cell to nearby cell, a variable complements First column and last column adjacent First row and last row adjacent Each cell represent one miniterm or one Maxterm Map reflect the truth table Map reflects the Boolean expression for output in SOP or POS form h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 29
End of Lesson 1 THREE VARIABLE KARNAUGH MAP h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 30
THANK YOU h05l1-"digital Principles and Design", Raj Kamal, Pearson Education, 2006 31