ECE 171: Introduction to Digital Circuits

Fall 1999

Rev: 10.28.99

Lecture Notes 9

Last Time

• Truth Table Sum of Products NAND Circuits
• Truth Table Product of Sums NOR Circuits
• Boolean Algebra Simplification
  • 23 Rules
  • Several Examples
• Simplifying Sum of Products Expressions

This Time

• Project 1 Assigned
• 4 More BA Rules
• TT POS Simplify w/BA NOR LD
• Karnaugh Maps
• Review of Exam 1
•  

Boolean Algebra

Write out the 23 rules discussed last time.

Show how XOR and XNOR gates can be thought of as programmable inverters where one of the inputs determines whether the gate acts like an inverter or a buffer.

Add the following rules:

25. 1 XOR A = A'
26. 0 XOR A = A
27. 1 XNOR A = A
28. 0 XNOR A = A'

Derive rule 24 by the following steps:

24. (A + B)(A + B' + C) = (A + B)(A + C) + (A + B) B'
= (A + B)(A + C) + AB' + BB'
= AA + AB + AC + BC + AB' + 0
= A + AB + AC + BC + AB'
= A + AB + AC + BC
= AA + AB + AC + BC
= (A + B)(A + C)

POS Minimization

• Have students pick entries for a three-input truth table
• Write out the POS expression (this also serves as a review for earlier)
• Draw logic diagram and do a gate count
• Simplify the expression using Boolean Algebra and show how to use rules 23 & 24.
• Draw a logic diagram for the simplified expression
• Do a gate count to show the savings

Karnaugh Maps

• An alternative to Boolean algebra
• Used for simplification
• More methodical than BA
• Easier than BA
• Faster than BA
• Main drawback is that it is limited to 6 variables, and most people wouldn't do more than 5

2-Variable K-Maps

Show what a K-map looks like for an XOR. Show how to translate entries in a truth table to entries in the K-map.

Write a truth table for an OR and show how you would simplify with BA.

Write a K-map for an OR.

• Circle largest square or rectangular group of ones that you can.
• Each group must contain 1, 2, 4, 8, 16, 32,... ones. Thus, you can not circle a group of 3 ones or 6 ones.
• For each group, write a product using only terms that define the group.

Show how to write the minimal sum for an OR circuit directly.

3-Variable K-Maps

Have students pick the entries for a tall 3-variable K-map.
Show how the numbering is tricky along the side.
Show how to number the cells so that the K-map can be filled in directly from a function specified using notation.
Show how to write the minimal sum.

Repeat for a wide 3-variable K-map.

Explain why the numbering is strange: to gaurantee that only one digit changes as you move between cells.

4-Variable K-Maps

Work through one or two examples with 4-variable K-maps.
Have students pick entries.
Show how to number the cells.
If examples do not illustrate the following points, make up one additional example that does.

• Groups of ones can wrap around left-right or top-bottom.
• The minimal sum is often not unique and there may be more than one.