1. Minimization Karnaugh Maps

2. Minimization
Logic Function minimization : Simplifying the logic function to reduce the number and size of gates.
Minimization methods: 1- Using theorems T9,T9’, T10,T10’ 2- Karnaugh map

3. Algebraic simplification
Starting from a truth table, or a minterm (maxterm) list
Reduce the cost of a 2-level circuit:
Minimize # of first-level gates
Minimize # of inputs of gates
Do not consider the cost of input inverters: assuming that both true and complemented versions of all input variables are available.

4. Algebraic simplification
T10, X.Y+X.Y’=X T10’: (X+Y).(X+Y’)=X
Generalization of the theorems:
(Given product term).Y+(Given product term).Y’= Given product term
(Given product term+Y).(Given product term+Y’)= Given product term
Reduce number of gates and gate inputs

5. Prime detector

6. Algebraic simplification
Theorem T10,
Reduce number of gates and gate inputs

7. Resulting circuit

8. Karnaugh Maps
Karnaugh Map : a representation of the truth table by a matrix of squares (cells) , where each square corresponds to a minterm ( or a maxterm) of the logic function.
For n-variable function, we need 2^n rows truth table and 2^n squares (cells).
The square number is equivalent to the row number in the truth table
To represent a logic function, the truth table values are copied into their corresponding cells .
The arrangements of the squares help to identify the input variable redundancy ( X.Y.Z+X.Y.Z’=X.Y )

9. Two-variable Karnaugh map
Example : F = X.Y’+X.Y
Simplification : F = X(Y+Y’) = X.1 = X
Also proven by T(10) X X
ROW X Y F 1 Y 2 1 1 1 1 3 2 1 1 1 Y 1 3 1 1 1

10. Two-variable Karnaugh map
Example : F = X’.Y’+X’.Y
Simplification : F = X’(Y+Y’) = X’.1 = X’
Exercise : Can we simplify FX,Y= SX,Y (1,2) ? X X
ROW X Y F 1 Y 1 2 1 1 1 1 1 3 2 1 1 Y 1 3 1 1

11. 3-variable Karnaugh map
Can we simplify FX,Y,Z = Sx,y,z (1,3,5,7) ?

12. Three-variable Karnaugh map
Example : F=X’.Y’.Z’+X’.Y’.Z+X.Y’.Z’+X.Y’.Z
Row X Y Z F
F = X’.Y’.(Z’+Z)+X.Y’.(Z’+Z)=X’.Y’+X.Y’=(X’+X).Y’ = Y’ X XY 00 01 11 10 Z 2 6 4 1 1 1 3 7 5 1 Z 1 1 Y

13. 4-variable Karnaugh map

14. Five-variable Karnaugh map
Five variable K-map is formed using two connected 4-variable maps: V
VWX W W
000
001
011
010
100
101
111
110 YZ 4 12 8 16 20 28 24 00 1 5 13 9 17 21 29 25 01 Z 3 7 15 11 19 23 31 27 11 Y 2 6 14 10 18 22 30 26 10 X X

15. Next:
Simplifying SOP