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

Recap Karnaugh Maps http://district.bluegrass.kctcs.edu/kevin.dunn/files/3_Basic_Logic_Gates/3_Basic_Logic_Gates_print.html

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

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

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

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

Configurations with 3 variables 1 1 1 1 1 1

Configurations with 4 variables cd 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

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.

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’

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

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’)

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.

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

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

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 0110. We cannot directly calculate the 9’s complement.

Excess 3 Specification to truth table 1 X

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

Excess 3 Design circuit. Code Converters Discussed in class