ECE 3110: Introduction to Digital Systems

1. ECE 3110: Introduction to Digital Systems Karnaugh Maps

2. Previous… • Combinational Circuit Synthesis • Algebra Minimization

3. 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 )

4. Two-variable Karnaugh map • Example : F = XY’+XY • Simplification : F = X(Y+Y’) = X.1 = X X X ROW X Y F 0 1 Y 0 0 0 0 0 2 0 0 1 1 0 1 0 1 3 1 Y 2 1 0 1 0 1 3 1 1 1 ECE311 Ch4

5. Two-variable Karnaugh map • Example : F = X’Y’+X’Y • Simplification : F = X’(Y+Y’) = X’.1 = X’ • Excercise : Simplify FX,Y= S (1,2) ! X X ROW X Y F 0 1 Y 0 0 0 1 0 2 0 1 0 1 0 1 1 1 3 1 Y 2 1 0 0 1 0 3 1 1 0 ECE311 Ch4

6. 3-variable Karnaugh map

7. Example : F=X’.Y’.Z’+X’.Y’.Z+X.Y’.Z’+X.Y’.Z Row X Y Z F 0 0 0 0 1 1 0 0 1 1 2 0 1 0 0 3 0 1 1 0 4 1 0 0 1 5 1 0 1 1 6 1 1 0 0 7 1 1 1 0 F = X’.Y’.(Z’+Z)+X.Y’.(Z’+Z)=X’.Y’+X.Y’=(X’+X).Y’ = Y’ Three-variable Karnaugh map X XY 00 01 11 10 Z 0 2 6 4 0 1 0 0 1 1 3 7 5 1 Z 1 0 0 1 Y

8. 4-variable Karnaugh map

9. Five-variable Karnaugh map V • Five variable K-map is formed using two connected 4-variable maps: VWX W W 000 001 011 010 100 101 111 110 YZ 0 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

10. Karnaugh-map usage • Plot 1s corresponding to minterms of function. • Circle largest possible rectangular sets of 1s. • # of 1s in set must be power of 2 • OK to cross edges • Read off product terms, one per circled set. • Variable is 1 ==> include variable • Variable is 0 ==> include complement of variable • Variable is both 0 and 1 ==> variable not included • Circled sets and corresponding product terms are called “prime implicants” • Minimum number of gates and gate inputs

11. Prime-number detector (again)

12. When we solved algebraically, we missed one simplification -- the circuit below has three less gate inputs.

13. Simplifying the Sum of Products • Two main steps :1) Combining/Grouping the 1-cells.2) Writing the product term for each group. • Rules : ( for n-variable function ) • 1) The group size must be a power of 2.2) A set of 2^i cells can be combined if there are ( i ) variables that take all possible combinations within the set and the remaining ( n-i ) variables have the same value within that set. • 3) The corresponding product term for each group contains (n-i) literals: - The variable is complemented if it is 0 in the combined cells - Tha variable is uncomplemented if it’s 1 in the combined cells - The variable is not included in the product term if it takes the values 0 and 1 within the combined cells

14. X XY 00 01 11 10 Z 0 2 6 4 0 1 1 1 1 1 3 7 5 1 Z 0 0 0 1 Y Example 1 • The canonical sum is : F=X’Y’Z’+X’YZ’+XY’Z’+XY’Z+XYZ’ • Combine cells (0,2,6,4)X=0,1 , Y= 0,1 , Z=0Product Term : Z’ • Combine cells ( 4,5 )Z=0,1 , Y=0 , X=1Product Term : XY’ • F= XY’+Z’

15. Next: • Simplifying SOP • Read Ch. 4.2--4.5 • HW #6