1. ECE- 1551 Digital logic Lecture 10: Karnaugh MAps
Assistant Prof. Fareena Saqib
Florida Institute of Technology
Fall 2015, 09/22/2015

2. Recap
Karnaugh Maps

3. Agenda Don’t Care Conditons Karnaugh map of Product of Maxterms
Exclusive OR Function
Code Convertors

4. Karnaugh Maps bc 00 01 11 10 a 1 1 1 Pair 1 Quad bc 00 01 11 10 a 1 11 1
1 Pair
1 Quad bc 00 01 11 10 a 1 1
1 Quad

5. Karnaugh Maps cd 00 01 11 10 ab 00 1 2 Quads Not 4 pairs 01 11 10 cd
1 Octal 01 11 10

6. Karnaugh Maps cd 00 01 11 10 ab 00 1
2 Quads rather than 1 quad and 1 pair 01 11 10

7. Configurations with 3 variables1 1 1 1 1 1

8. Configurations with 4 variablescd cd 00 01 11 10 00 01 11 10 ab ab 00 1 00 1 01 01 11 11 10 10 cd cd 00 01 11 10 00 01 11 10 ab ab 00 1 00 1 01 01 11 11 10 10

9. Don’t Care Conditions
Logical sum of minterms associated with a boolean function specifies the condition under which the function is 1. The function is considered 0 for the rest of conditions.
In practice function is not specified for some of the combinations, such functions are called incompletely specified functions.
Those unspecified minterms of a functions are called don’t-care conditions.
Example Binary digits representation in binary.

10. Boolean Function with Don’t-Care Minterms
F(a,b,c,d) = m0+m3+m6+m9+m11+m13+m14 Don’t-care = m5+m7+m10+12 ab 00 01 11 10 cd 00 1 X 01 11 10
F = bc’d + b’cd +ac’d + a’b’c’

11. Boolean Function with Don’t-Care Minterms
F(w,x,y,z) = m5+m7+9+m11+m13+m14 Don’t-care = m2+m6+m10+12+m15 wx yz 00 01 11 10 00 X 1 01 11 10
F = xz+wz+wy

12. Karnaugh Map with product of Maxterms
F(a,b,c,d) = M0.M3.M6.M9 Don’t-care = M3.M13 ab cd 00 01 11 10 00 1 X 01 11 10
F = (a+b+d’) (a’+b’+c) (a’+c’+d’)

13. Exclusive OR Function Plot following expression on a K-map
Z = (A.B) XOR(C+D)
= (AB)(C+D)’ + (AB)’(C+D)
= ABC’D’ + (A’+B’)(C+D)
= ABC’D’ + A’C + A’D+B’C+B’D
Express in sum of minterms
Simplify with expression using K-maps.
…. Discussed in class.

14. Design Code Converters
Binary codes and how to develop code converters.
An n-bit binary code is a group of n bits that assumes up to 2^n distinct combinations of 1’s and 0’s. With each combination representing one element o the set that is being coded
Combinational
Logic IN
OUT

15. Binary Codes BCD, Binary representation of decimal numbers. Excess 3
Gray Code
ASCII

16. Excess 3 - Specification
Specification: Excess‐3 is an unweighted code in which each coded combination is obtained from the corresponding binary value plus 3.
Application:its selfcomplementing property. Example 9’s complement of 3 is 6 and 6 9’s complement is 3.
Excess 3 representation of 3 is 0110 and of 6 is 1001
9’s complement of 3 only requires flipping the bits from 1 to 0 and 0 to 1 (as we did in 1’s complement).
Its not true in BCD where 3 is 0011 and 6 is We cannot directly calculate the 9’s complement.

17. Excess 3 Specification to truth table1 X

18. Excess 3 – Minimization using K-maps
b1b0
b1b0
b3b2
b3b2 00 01 11 10 00 01 11 10 cd 00 1 X 00 1 X 01 01 11 11 10 10 e2
e0 = b0’
e1 = (b1 xor b0)’ e3
b1b0
b3b2
b1b0 00 01 11 10 00 01 11 10
b3b2 00 1 X 00 1 X 01 01 11 11 10 10
e2 = b2’b1+b2b1’b0
e3 = b3+b2b0+b2b1

19. Excess 3
Design circuit.
Code Converters
Discussed in class