ECE 171: Introduction to Digital Circuits

Fall 1999

Rev: 10.27.99

Lecture Notes 8

Last Time

• Translations
  • Logic Diagrams
  • Truth Tables
  • Boolean Expressions
  • Sum of Products (SOP)
• Review for Midterm

This Time

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

Sum of Products (SOP)

• Boolean functions can be written using the greek letter
• For example,
  F = (0,1,2,7) = f(A,B,C)
  says that rows 0, 1, 2, and 7 in a truth table have ones and all other rows contain zeros.
  The notation f(A,B,C) indicates that the input variables are A, B, and C where A is the most significant bit and C is the least significant bit.
• Sometimes the book adds a lowercase letter m, which stands for minterm. For example, the book would have written the SOP above as
  F = m(0,1,2,7)
• From a truth table, you should be able to immediately write down the sum of products. For each row that the function has a one, write down a product with all of the input variables. For each variable that has a zero in row of interest, complement the variable. Thus, if the row corresponding to 2, that is A=0, B=1, C=0, contained a one then the corresponding product would be A' B C'.

Have students pick four numbers between 0 and 7 and fill in the truth table. Show the four functions, or minterms, that make up the function and show that the OR of these functions is equivalent to the original function.

• From a sum of products expression you can directly draw a two layer logic diagram where the first layer has a separate AND gate for each minterm and the second layer has a single OR gate with an input for each AND gate. A number of inverters may also be necessary.
• Show that the AND gates and OR gate can be converted to NAND gates without changing the expression.

Product of Sums (POS)

• Boolean functions can be written using the greek letter
• For example,
  F = (0,1,2,7) = f(A,B,C)
  says that rows 0, 1, 2, and 7 in a truth table have zeros and all other rows contain ones. Note that this is just the opposite of the notation. Thus,
  F = (0,1,2,7) can also be written as F = (3,4,5,6)
• Sometimes the book adds an uppercase letter M, which stands for maxterm. For example, the book would have written the POS above as
  F = M(0,1,2,7)
• From a truth table, you should be able to immediately write down the product of sums. For each row that the function has a zero, write down a sum with all of the input variables. For each variable that has a one in row of interest, complement the variable. Thus, if the row corresponding to 2, that is A=0, B=1, C=0, contained a one then the corresponding product would be A + B' + C. Note that this is the opposite of what was done for sum of products.

Have students pick four numbers between 0 and 7 and fill in the truth table. Show the four functions, or maxterms, that make up the function and show that the AND of these functions is equivalent to the original function.

• From a product of sums expression you can directly draw a two layer logic diagram where the first layer has a separate OR gate for each minterm and the second layer has a single AND gate with an input for each OR gate. A number of inverters may also be necessary.
• Show that the OR gates and AND gate can be converted to NOR gates without changing the expression.

Boolean Algebra

• George Boole invented in 1854 to give expression to the logic of reasoning. Was interested in philosophical implications.
• Claude Shannon, a very influential researcher that you will hear more about in other classes, realized it could be applied to switched circuits in 1938

Write out the 22 rules in the book. Derive the non-obvious rules by either previous rules or use a truth table proof. Add the following rule

23. (A+B) (A+B') = A

Have students pick the relationships for 2-3 examples with 2 or 3 different variables. Use the variables X,Y, and Z to avoid confusion with the rules.

SOP Minimization

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